Add files using upload-large-folder tool

-dFST4oBgHgl3EQfcTgr/content/tmp_files/2301.13802v1.pdf.txt

@@ -0,0 +1,651 @@
+1
+Armouring of a frictional interface by mechanical noise
+Elisa El Sergany, Matthieu Wyart, Tom W.J. de Geus
+Physics Institute, ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL) Switzerland
+Abstract
+A dry frictional interface loaded in shear often displays stick-slip. The amplitude of this cycle depends
+on the probability that a slip event nucleates into a rupture, and on the rate at which slip events are
+triggered. This rate is determined by the distribution P(x) of soft spots which yields if the shear stress is
+increased by some amount x. In minimal models of a frictional interface that include disorder, inertia and
+long-range elasticity, we discovered an ‘armouring’ mechanism, by which the interface is greatly stabilised
+after a large slip event: P(x) then vanishes at small arguments, as P(x) ∼ xθ [1]. The exponent θ > 0,
+which exists only in the presence of inertia (otherwise θ = 0), was found to depend on the statistics of
+the disorder in the model, a phenomenon that was not explained. Here, we show that a single-particle
+toy model with inertia and disorder captures the existence of a non-trivial exponent θ > 0, which we can
+analytically relate to the statistics of the disorder.
+1
+Introduction
+We study systems in which disorder and elasticity
+compete, leading to intermittent, avalanche-type re-
+sponse under loading.
+Examples include an elastic
+line being pulled over a disordered pinning potential,
+or frictional interfaces
+[2–4].
+When subject to an
+external load f, such systems are pinned by disor-
+der when the load is below a critical value fc.
+At
+f > fc, the system moves forward at a finite rate. At
+f = fc the system displays a crackling-type response
+described by avalanches whose sizes and durations are
+distributed according to powerlaws.
+A key aspect of such systems is the distribution
+of soft spots [5]. If we define x as the force increase
+needed to trigger an instability locally, then increas-
+ing the remotely applied force by ∆f will trigger
+na ∝
+� ∆f
+0
+P(x)dx avalanches, with P(x) the probabil-
+ity density of x. The relevant behaviour of P(x) there-
+fore is that at small x. Let us assume that P(x) ∼ xθ
+at small x, such that na ∝ (∆f)θ+1.
+Classical models used to study the depinning tran-
+sition consider an over-damped dynamics [2]. In that
+case, it can be shown that θ = 0 [2]. This result is
+not true for certain phenomena, including the plastic-
+ity of amorphous solids or mean-field spin glasses. In
+these cases, due to the fact that elastic interactions
+are long-range and can vary in sign (which is not the
+case for the depinning transition, where a region that
+is plastically rearranged can only destabilise other re-
+gions), one can prove that θ > 0, as reviewed in [5, 6].
+Recently, we studied simple models of dry fric-
+tional interface [1, 7]. We considered disorder, long-
+range elastic interactions along the interface. These
+interactions are strictly positive as in the usual class
+of the depinning transition. However, we studied the
+role of inertia, that turns out to have dramatic ef-
+fects.
+Inertia causes transient overshoots and un-
+dershoots of the stress resulting from a local plas-
+tic event. It thus generates a mechanical noise, that
+lasts until damping ultimately takes place. Remark-
+ably, we found that right after system-spanning slip
+events, θ > 0 [1] in the presence of inertia.
+Intu-
+itively, such an ‘armouring’ mechanism results from
+the mechanical noise stemming inertial effects, that
+destabilises spots close to an instability (i.e. small
+x), thus depleting P(x) at small argument.
+This
+property is consequential: the number of avalanches
+of plastic events triggered after a system-spanning
+rupture is very small. As a consequence, the inter-
+face can increase its load when driven quasistatically
+in a finite system, without much danger of trigger-
+ing large slip events. The interface therefore present
+larger stick-slip cycles due to this effect, as sketched in
+Fig. 1. Thus, one of the central quantities governing
+the stick-slip amplitude is θ [1].
+Our previous model [1] divided the interface in
+blocks whose mechanical response was given by a po-
+tential energy landscape that, as a function of slip,
+comprised a sequence of parabolic wells with equal
+curvature. We drew the widths w of each well ran-
+domly from a Weibull distribution, such that its dis-
+arXiv:2301.13802v1  [cond-mat.soft]  31 Jan 2023
+
+2
+tribution Pw(w) ∼ wk at small k.
+We empirically
+found θ ≃ 2.5 for k = 1 and θ ≃ 1.4 for k = 0.2.
+Here we present a toy model for a region of space
+that stops moving at the end of a large slip event. In
+the most idealised view, we describe this region as a
+single particle that moves over a disordered potential
+energy landscape, and that slows down due to dissipa-
+tion. We model this potential energy landscape by a
+sequence of parabolic potentials that have equal cur-
+vature κ but different widths taken from Pw(w), with
+w the width of a parabola. In this model, x = κw/2
+and is thus proportional to the width of the well in
+which the particle stops.
+Below we prove that for
+such a model, P(x) ∼ xk+2 if Pw(w) ∼ wk.
+This
+result explains both why θ > 0 and why this expo-
+nent in non-universal, as it depends on k that char-
+acterises the disorder. Although this prediction does
+not match quantitatively our previous observations,
+the agreement is already noticeable for such a sim-
+ple model. We support our argument with analytical
+proofs, and verify our conclusion numerically.
+The
+generality of our argument suggests that the presence
+of a non-trivial exponent θ may hold in other depin-
+ning systems, as long a inertia is present.
+force
+slip
+(a)
+(b)
+avalanche
+slip
+Figure 1. (a) Sketch of stick-slip response: “slip” events
+punctuate periods in which the interface is macroscopi-
+cally stuck, but microscopic events (“avalanches”) do oc-
+cur. The number of avalanches na ∝ (∆f)θ+1, which can
+be linked to (b) the distribution of soft spots. x is thereby
+the amount of force needed to trigger an instability locally.
+Right after a large slip event, its distribution empirically
+scales like P(x) ∼ xθ at small x as indicated (log-scale
+implied).
+2
+Model
+During a big slip event, all regions in space are moving
+but eventually slow down and stop. We model this by
+considering a single region in space in which a particle
+of finite mass is thrown into the potential energy land-
+scape at a finite velocity. In the simplest case, this
+particle is “free”, such that it experiences no external
+driving and stops due to dissipation, see Fig. 2. This
+corresponds to the Prandtl-Tomlinson [8–10] model
+that describes the dynamics of one (driven) particle
+in a potential energy landscape. The equation of mo-
+tion of the “free” particle reads
+m¨r = fe(r) − η ˙r.
+(1)
+with r the particle’s position, m its mass, and η
+a damping coefficient.
+fe(r) is the restoring force
+due to the potential energy landscape.
+We con-
+sider a potential energy landscape that consists of a
+sequence of finite-sized, symmetric, quadratic wells,
+such that the potential energy inside a well i is given
+by U(r) = (κ/2)(r − ri
+min)2 + U i
+0 for ri
+y < r ≤ ri+1
+y
+,
+with wi ≡ ri+1
+y
+− ri
+y the width of the well, κ the
+elastic constant, ri
+min ≡ (ri
+y + ri+1
+y
+)/2 the position of
+the center of the well, and U i
+0 = κ(wi)2/8 an unim-
+portant offset.
+The elastic force deriving from this
+potential energy is fe(r) ≡ −∂xU(r) = κ(ri
+min − r).
+With κ is constant, the landscape is parameterised by
+the distance between two subsequent cusps wi, which
+we assume identically distributed (iid) according to
+a distribution Pw(w). We consider underdamped dy-
+namics corresponding to η2 < 4mκ. Within a well,
+the dynamics is simply that of a underdamped oscil-
+lator, as recalled in Appendix A.
+0.0
+0.5
+1.0
+1.5
+2.0
+λt
+0
+1
+2
+3
+E/⟨U⟩
+0
+4
+8
+r/⟨w⟩
+−1
+0
+U/⟨U⟩
+Figure 2. Evolution of the kinetic energy E as a function
+of position r (in red) of the “free” particle ‘thrown’ into
+a potential energy landscape (shown in the inset). Every
+entry into a new well is indicated using a marker. A thin
+green line shows the evolution of the total energy (with
+the definition of the inset, it has the local minimum of the
+last well as arbitrary offset).
+
+3
+3
+Stopping well
+Distribution.
+We are interested in the width of the
+well in which the particle eventually stops. Suppose
+that a particle enters a well of width w with a kinetic
+energy E. The particle stops in that well if E < Ec(w),
+with Ec the minimum kinetic energy with which the
+particle needs to enter a well of width w to be able to
+exit. The distribution of wells in which particles stop
+in that case is
+Ps(w) ∼ Pw(w)P(E < Ec(w)),
+(2)
+with Pw(w) the probability density of well widths,
+and Ps(w) the probability of well widths in which the
+particle stops. Within one well, the particle is sim-
+ply a damped harmonic oscillator as has been studied
+abundantly. In the limit of a weakly damped system,
+the amount of kinetic energy lost during one cycle is
+∆E = κw2(1 − exp(−2π/Q))/8 with the quality fac-
+tor Q =
+�
+4mκ/η2 − 1. The minimal kinetic energy
+with which the particle needs to enter the well in or-
+der to be able to exist is thus Ec = ∆E ∝ w2 (see
+Appendix B for the exact calculation of Ec). Further-
+more, if P(E) is a constant at small argument (as we
+will argue below), then
+P(E < Ec(w)) =
+� Ec
+0
+P(E)dE ∼ Ec(w).
+(3)
+Therefore, the particle stops in a well whose width is
+distributed as
+Ps(w) ∼ w2Pw(w).
+(4)
+Central result.
+Once stopped, the force, x, by
+which we need to tilt the well in which the particle
+stopped, in order for it to exit again is x = κw/2 1,
+such that our central result is that
+P(x) ∼ x2Pw(x).
+(5)
+For example, if Pw(w) ∼ wk at small w, we predict
+that
+P(x) ∼ x2+k.
+(6)
+Energy at entry.
+We will now argue that the den-
+sity of kinetic energy with which the particle enters
+the final well, P(E), is finite at small E.
+For one
+realisation, E results from passing many wells with
+random widths. If its kinetic energy is much larger
+than the potential energy of the typical wells, it will
+1Without external forces, the particle ends in the local min-
+imum – the center of the well.
+not stop. We thus consider that the particle energy
+has decreased up to some typical kinetic energy E0 of
+the order of the typical potential energy κ⟨w2⟩/8. If
+the particle exits the next well, at exit it will have a
+kinetic energy K = E0 − ∆E(E0, w). For a given E0
+and distributed w, we have:
+P(E) =
+�
+dw Pw(w) δ(K(E0, w) − E).
+(7)
+It thus implies that:
+P(E = 0) = Pw(w∗)/
+��∂wK
+��
+w=w∗
+(8)
+w∗ is the well width for which the particle reaches the
+end of the well with zero velocity, i.e. E0 = Ec(w∗).
+By assumption, Pw(w∗) > 0. Furthermore we prove
+in Appendix C that ∂wK|w=w∗ = κw∗/2 > 0. Over-
+all, it implies that P(E = 0) > 0, i.e. P(E) does not
+vanish as E → 0, from which our conclusions follow.
+Here we give a simple argument for ∂wK|w=w∗ =
+κw∗/2 > 0. Given E0, but an infinitesimally smaller
+well of width w∗ −δw, the particle will enter the next
+well. Because the velocity is negligible in the vicinity
+of w∗, the damping is negligible. Therefore, δK is of
+the order of the difference in potential energy on a
+scale δw, δU = U(w∗) − U(w∗ − δw) ≈ κw∗δw/2, as
+we illustrate in Fig. 3. We thus find that ∂wK|w=w∗ =
+limδw→0 δK/δw = κw∗/2.
+−w∗/2
+0
+w∗/2
+position
+0
+energy
+1
+2κw∗δw
+δw
+1
+2κw∗δw
+δw
+potential energy U
+kinetic energy E
+total energy E + U
+Figure 3. Evolution of the kinetic energy E (red), po-
+tential energy U (black), and total energy E + V (green)
+for a particle that has entered a well of width w∗ with a
+kinetic energy E0 = Ec(w∗) such that it stops just. Con-
+sequently, ∂r(E + V )|w∗/2 = 0, which can be decomposed
+in ∂rV |w∗/2 = κw∗/2 such that ∂rE|w∗/2 = −κw∗/2, as
+indicated using thin lines.
+
+4
+4
+Numerical support
+Objective.
+We now numerically verify our predic-
+tion that P(x) ∼ xk+2 (Eq. (6)). We simulate a large
+number of realisations of a potential energy landscape
+constructed from randomly drawn widths (consider-
+ing different distributions Pw(w)) and constant cur-
+vature. We study the distribution of stopping wells
+if a “free” particle is ‘thrown’ into the landscape at
+a high initial velocity (much larger than vc(⟨w⟩) such
+that particle transverses many wells before stopping).
+Map.
+We find an analytical solution for Eq. (1) in
+the form of a map. In particular, we derive the evo-
+lution of the position in a well based on an initial po-
+sition −w/2 and velocity in Appendix A. This maps
+the velocity with which the particle enters a well at
+position w/2, to an exit velocity which corresponds
+to the entry velocity of the next well, etc.
+Stopping well.
+We record the width of the stop-
+ping well, x, and the velocity V with which the parti-
+cle enters the final well. We find clear evidence for the
+scaling P(x) ∼ xk+2 in Fig. 4. Perturbing the evolu-
+tion with random force kicks2 changes nothing to our
+observations, as included in Fig. 4 (see caption). We,
+furthermore, show that the probability density of the
+kinetic energy with which the particle enters the final
+well, P(E), is constant as small argument in Fig. 5.
+5
+Concluding remarks
+Our central result is that P(x) ∼ x2Pw(x) in our
+toy model. For a disorder Pw(w) ∼ wk we thus find
+P(x) ∼ xk+2. We expect this result to qualitatively
+apply to generic depinning systems in the presence of
+inertia. In particular they are qualitatively (but not
+quantitatively) consistent with our previous empiri-
+cal observations θ ≃ 2.5 for k = 1 [1] and θ ≃ 1.4
+for k = 0.2. A plausible limitation of our approach
+is underlined by the following additional observation:
+in Ref. [1], it was found that for x to be small, the
+stopping well was typically small (by definition), but
+also that the next well had to be small. Such corre-
+lations can exist only if the degree of freedom con-
+sidered had visited the next well, before coming back
+and stopping. This scenario cannot occur in our sim-
+ple description where the particle only moves forward,
+except when it oscillates in its final well.
+2Such the for each well is tilted with a random force that we
+take independent and identically distributed (iid) according to
+a normal distribution with zero mean.
+10−2
+100
+x
+10−5
+100
+P(x)/cx
+1
+2
+1
+3
+1
+4
+k = 0 1 2
+weibull
+power
+uniform
+Figure 4.
+Width of the stopping well, x, for different
+Pw(w): a uniform, Weibull, and powerlaw distribution,
+that scale as Pw(w) ∼ wk at small w, as indicated in the
+legend (the bottom row for each distribution corresponds
+to perturbing the dynamcs with random force kicks, tilt-
+ing individual wells by a force F = N(0, 0.1), with N the
+normal distribution; the top row corresponds to F = 0).
+To emphasise the scaling, the distributions have been
+rescaled by a fit of the prefactors: P(x) = cxxk+2. Fur-
+thermore, we use m = κ = 1, η = 0.1, v0 = N(100, 10),
+and ⟨w⟩ ≈ 1.
+10−5
+10−1
+E
+10−2
+100
+P(E)/ce
+Figure 5.
+The kinetic energy with which the particle
+enters the well in which it stops for different realisations,
+P(E), normalised by its prefactor ce (that is here simply
+the density of the first bin). See Fig. 4 for legend.
+
+5
+References
+[1] T.W.J. de Geus, M. Popovi´c, W. Ji, A. Rosso, and
+M. Wyart. How collective asperity detachments nucleate slip
+at frictional interfaces. Proc. Natl. Acad. Sci., 116(48):
+23977–23983, 2019. doi: 10.1073/pnas.1906551116.
+arXiv: 1904.07635.
+[2] D.S. Fisher. Collective transport in random media: From
+superconductors to earthquakes. Phys. Rep., 301(1–3):
+113–150, 1998. doi: 10.1016/S0370-1573(98)00008-8.
+arXiv: cond-mat/9711179.
+[3] O. Narayan and D.S. Fisher. Threshold critical dynamics of
+driven interfaces in random media. Phys. Rev. B, 48(10):
+7030–7042, 1993. doi: 10.1103/PhysRevB.48.7030.
+[4] M. Kardar. Nonequilibrium dynamics of interfaces and lines.
+Phys. Rep., 301(1–3):85–112, 1998.
+doi: 10.1016/S0370-1573(98)00007-6.
+arXiv: cond-mat/9704172.
+[5] M. M¨uller and M. Wyart. Marginal Stability in Structural,
+Spin, and Electron Glasses. Annu. Rev. Condens. Matter
+Phys., 6(1):177–200, 2015.
+doi: 10.1146/annurev-conmatphys-031214-014614.
+[6] Alberto Rosso, James P Sethna, and Matthieu Wyart.
+Avalanches and deformation in glasses and disordered
+systems. arXiv preprint: 2208.04090, 2022.
+doi: 10.48550/arXiv.2208.04090.
+[7] T.W.J. de Geus and M. Wyart. Scaling theory for the
+statistics of slip at frictional interfaces. Phys. Rev. E, 106
+(6):065001, 2022. doi: 10.1103/PhysRevE.106.065001.
+arXiv: 2204.02795.
+[8] L. Prandtl. Ein Gedankenmodell zur kinetischen Theorie der
+festen K¨orper. Z. angew. Math. Mech., 8(2):85–106, 1928.
+doi: 10.1002/zamm.19280080202.
+[9] G.A. Tomlinson. CVI. A molecular theory of friction. The
+London, Edinburgh, and Dublin Philosophical Magazine
+and Journal of Science, 7(46):905–939, 1929.
+doi: 10.1080/14786440608564819.
+[10] V.L. Popov and J.A.T. Gray. Prandtl-Tomlinson Model: A
+Simple Model Which Made History. In E. Stein, editor, The
+History of Theoretical, Material and Computational
+Mechanics - Mathematics Meets Mechanics and
+Engineering, volume 1, pages 153–168. Springer Berlin
+Heidelberg, 2014. ISBN 978-3-642-39904-6
+978-3-642-39905-3. doi: 10.1007/978-3-642-39905-3 10.
+A
+Analytical solution
+Anasatz.
+We look for the solution of a general lin-
+ear equation of motion in one well
+m¨r + η ˙r + κr − F = 0.
+(9)
+where the position r is expressed relative to the local
+minimum in potential energy. The external force F
+tilts the potential energy landscape and will be used
+as a perturbation to check the robustness of our ar-
+gument. An ansatz to this differential equation is
+r(τ) = αe−λ−τ + βe−λ+τ + ∆r,
+(10)
+where ∆r = F/κ, and τ is the time that the parti-
+cle has spent since the entry in the current well at
+r(τ = 0) ≡ −w/2. We denote the particle’s veloc-
+ity v(τ) ≡ ˙r(τ), whereby we take v(τ = 0) ≡ v0.
+Substituting this ansatz in Eq. (9) leads to λ± =
+(η ±
+�
+η2 − 4mκ)/(2m). The prefactors α and β are
+set by the initial conditions such that
+α, β = ±r0λ± + v0
+λ+ − λ−
+,
+(11)
+with r0 ≡ −w/2 − ∆r.
+Underdamped.
+We recognise that if λ± are real
+(η2 > 4mκ), the dynamics are overdamped and the
+velocity decays exponentially.
+Conversely, the un-
+derdamped dynamics that we consider correspond to
+η2 < 4mκ 3.
+Oscillator.
+In the underdamped case, λ± and α, β
+are complex conjugates. This allows us to simplify
+the solution by expressing those coefficients as λ± =
+λ ± iω and α, β = (L/2)e±iφ as follows 4
+r(τ) = Le−λτ cos(ωτ + φ) + ∆r.
+(12)
+We remark that the velocity v(τ) can be expressed as
+a phase shift with respect to r(τ) 5
+v(τ) = −Ae−λτ cos
+�
+ωτ + φ − arctan
+�ω
+λ
+��
+(13)
+with A = λL
+�
+1 + (ω/λ)2. We summarise the ampli-
+tudes, frequency, and phase. From λ± we find
+λ =
+η
+2m,
+ω2 = κ
+m −
+� η
+2m
+�2
+.
+(14)
+Furthermore, Eq. (11) gives
+α, β = 1
+2
+�
+r0 ∓ i(λr0 + v0)/ω
+�
+,
+(15)
+such that
+L2 = 4
+�
+Re(α)2 + Im(α)2�
+(16)
+=
+�
+(ωr0)2 + (λr0 + v0)2�
+/ω2,
+(17)
+and
+φ = χπ + arctan (Im(α)/Re(α))
+(18)
+= χπ − arctan (λ/ω + v0/(ωr0)) ,
+(19)
+where χ depends on α 6.
+3Note that our solution warrants some caution for critical
+damping η2 = 4mκ.
+4cos(z) = (eiz + e−iz)/2
+5a cos(z) + b sin(z) = sgn(a)
+√
+a2 + b2 cos (z − arctan(b/a))
+6Re(α) ≥ 0 → χ = 0. (Re(α) < 0, Im(α) ≥ 0) → χ = 1.
+(Re(α) < 0, Im(α) < 0) → χ = −1.
+
+6
+B
+Exiting well
+We will show that the minimum kinetic energy with
+which the particle needs to enter a well to be able
+to exit Ec ∝ w2, whereby we consider F = 0. The
+particle exits the well if v0 > vc. vc thus corresponds
+to the case that r(τe) = w/2 = −r0 for which v(τe) =
+0. Let us make the ansatz that v0 = vc = w/τc and
+look for the solution of τc.
+We note that on the interval τ ∈ [0, τe) the posi-
+tion is strictly monotonically increasing. The solution
+of v(τn) = 0 corresponds to
+ωτn + φ − arctan(ω/λ) = (n + 1/2)π,
+n ∈ Z. (20)
+The for us relevant solution is τe = min(τn > 0) 7 .
+r(τe) = w/2 corresponds to
+Le−λτe = w/(2c0),
+(21)
+with
+c0 = cos((n + 1/2)π + arctan(ω/λ)).
+(22)
+Using the definition of τe in Eq. (20) leads to
+Leλφ/ω = w/(2c0c1),
+(23)
+with
+c1 = e−λ(n+1/2)π/ω−(λ/ω) arctan(ω/λ).
+(24)
+Furthermore,
+L2 = 1
+4
+�
+ω2 + (2/τc − λ)2�
+(w/ω)2,
+(25)
+φ = sign (2/τc − λ) π + arctan
+��
+2/τc − λ
+�
+/ω
+�
+, (26)
+such that
+L′eλφ/ω = 1/(2c0c1),
+(27)
+with L′ = L/w.
+Eq. (27) is w independent and can
+be solved for τc = τc(λ, ω, n), proving that vc = w/τc.
+This results thus corresponds to
+Ec = 1
+2mv2
+c = m
+2τ 2c
+w2.
+(28)
+as we used to go from Eq. (2) to obtain Eq. (4) using
+Eq. (3).
+7i.e. n = ±1 depending on (r0, v0)
+C
+Entry kinetic energy
+With K the kinetic energy at exiting the well, we show
+that ∂wK
+��
+w=w∗ > 0. We again consider F = 0. In
+particular, we show that
+∂wK = m ve ∂wve > 0.
+(29)
+The derivative of the velocity as a function of the well
+size is
+∂rv = ∂τv/∂τr = a(τ)/v(τ),
+(30)
+where the acceleration a(τ) ≡ ¨r(τ) = αλ2
+−e−λ−τ +
+βλ2
++e−λ+τ. By evaluating this expression with initial
+conditions r(τ = 0) = −w∗/2 and v(τ = 0) = 0, we
+find
+∂wK
+��
+w=w∗ = m (αλ2
+− + βλ2
++) = −m
+2 (λ2 + ω2) w∗.
+(31)
+From the definitions of λ and ω in Eq. (14), we thus
+find
+∂wK
+��
+w=w∗ = −κw∗/2,
+(32)
+as we argued above to show that P(E = 0) > 0 using
+Eq. (8).
+

-dFST4oBgHgl3EQfcTgr/content/tmp_files/load_file.txt

@@ -0,0 +1,328 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf,len=327
+page_content='1  Armouring of a frictional interface by mechanical noise  Elisa El Sergany, Matthieu Wyart, Tom W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' de Geus  Physics Institute, ´Ecole Polytechnique F´ed´erale de Lausanne (EPFL) Switzerland  Abstract  A dry frictional interface loaded in shear often displays stick-slip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' The amplitude of this cycle depends  on the probability that a slip event nucleates into a rupture, and on the rate at which slip events are  triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' This rate is determined by the distribution P(x) of soft spots which yields if the shear stress is  increased by some amount x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' In minimal models of a frictional interface that include disorder, inertia and  long-range elasticity, we discovered an ‘armouring’ mechanism, by which the interface is greatly stabilised  after a large slip event: P(x) then vanishes at small arguments, as P(x) ∼ xθ [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' The exponent θ > 0,  which exists only in the presence of inertia (otherwise θ = 0), was found to depend on the statistics of  the disorder in the model, a phenomenon that was not explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Here, we show that a single-particle  toy model with inertia and disorder captures the existence of a non-trivial exponent θ > 0, which we can  analytically relate to the statistics of the disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  1  Introduction  We study systems in which disorder and elasticity  compete, leading to intermittent, avalanche-type re-  sponse under loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Examples include an elastic  line being pulled over a disordered pinning potential,  or frictional interfaces  [2–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  When subject to an  external load f, such systems are pinned by disor-  der when the load is below a critical value fc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  At  f > fc, the system moves forward at a finite rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' At  f = fc the system displays a crackling-type response  described by avalanches whose sizes and durations are  distributed according to powerlaws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  A key aspect of such systems is the distribution  of soft spots [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' If we define x as the force increase  needed to trigger an instability locally, then increas-  ing the remotely applied force by ∆f will trigger  na ∝  � ∆f  0  P(x)dx avalanches, with P(x) the probabil-  ity density of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' The relevant behaviour of P(x) there-  fore is that at small x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Let us assume that P(x) ∼ xθ  at small x, such that na ∝ (∆f)θ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Classical models used to study the depinning tran-  sition consider an over-damped dynamics [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' In that  case, it can be shown that θ = 0 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' This result is  not true for certain phenomena, including the plastic-  ity of amorphous solids or mean-field spin glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' In  these cases, due to the fact that elastic interactions  are long-range and can vary in sign (which is not the  case for the depinning transition, where a region that  is plastically rearranged can only destabilise other re-  gions), one can prove that θ > 0, as reviewed in [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Recently, we studied simple models of dry fric-  tional interface [1, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' We considered disorder, long-  range elastic interactions along the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' These  interactions are strictly positive as in the usual class  of the depinning transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' However, we studied the  role of inertia, that turns out to have dramatic ef-  fects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Inertia causes transient overshoots and un-  dershoots of the stress resulting from a local plas-  tic event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' It thus generates a mechanical noise, that  lasts until damping ultimately takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Remark-  ably, we found that right after system-spanning slip  events, θ > 0 [1] in the presence of inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Intu-  itively, such an ‘armouring’ mechanism results from  the mechanical noise stemming inertial effects, that  destabilises spots close to an instability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' small  x), thus depleting P(x) at small argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  This  property is consequential: the number of avalanches  of plastic events triggered after a system-spanning  rupture is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' As a consequence, the inter-  face can increase its load when driven quasistatically  in a finite system, without much danger of trigger-  ing large slip events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' The interface therefore present  larger stick-slip cycles due to this effect, as sketched in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Thus, one of the central quantities governing  the stick-slip amplitude is θ [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Our previous model [1] divided the interface in  blocks whose mechanical response was given by a po-  tential energy landscape that, as a function of slip,  comprised a sequence of parabolic wells with equal  curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' We drew the widths w of each well ran-  domly from a Weibull distribution, such that its dis-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='13802v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='soft]  31 Jan 2023  2  tribution Pw(w) ∼ wk at small k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  We empirically  found θ ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='5 for k = 1 and θ ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='4 for k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Here we present a toy model for a region of space  that stops moving at the end of a large slip event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' In  the most idealised view, we describe this region as a  single particle that moves over a disordered potential  energy landscape, and that slows down due to dissipa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' We model this potential energy landscape by a  sequence of parabolic potentials that have equal cur-  vature κ but different widths taken from Pw(w), with  w the width of a parabola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' In this model, x = κw/2  and is thus proportional to the width of the well in  which the particle stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Below we prove that for  such a model, P(x) ∼ xk+2 if Pw(w) ∼ wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  This  result explains both why θ > 0 and why this expo-  nent in non-universal, as it depends on k that char-  acterises the disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Although this prediction does  not match quantitatively our previous observations,  the agreement is already noticeable for such a sim-  ple model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' We support our argument with analytical  proofs, and verify our conclusion numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  The  generality of our argument suggests that the presence  of a non-trivial exponent θ may hold in other depin-  ning systems, as long a inertia is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  force  slip  (a)  (b)  avalanche  slip  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' (a) Sketch of stick-slip response: “slip” events  punctuate periods in which the interface is macroscopi-  cally stuck, but microscopic events (“avalanches”) do oc-  cur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' The number of avalanches na ∝ (∆f)θ+1, which can  be linked to (b) the distribution of soft spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' x is thereby  the amount of force needed to trigger an instability locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Right after a large slip event, its distribution empirically  scales like P(x) ∼ xθ at small x as indicated (log-scale  implied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  2  Model  During a big slip event, all regions in space are moving  but eventually slow down and stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' We model this by  considering a single region in space in which a particle  of finite mass is thrown into the potential energy land-  scape at a finite velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' In the simplest case, this  particle is “free”, such that it experiences no external  driving and stops due to dissipation, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' This  corresponds to the Prandtl-Tomlinson [8–10] model  that describes the dynamics of one (driven) particle  in a potential energy landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' The equation of mo-  tion of the “free” particle reads  m¨r = fe(r) − η ˙r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  (1)  with r the particle’s position, m its mass, and η  a damping coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  fe(r) is the restoring force  due to the potential energy landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  We con-  sider a potential energy landscape that consists of a  sequence of finite-sized, symmetric, quadratic wells,  such that the potential energy inside a well i is given  by U(r) = (κ/2)(r − ri  min)2 + U i  0 for ri  y < r ≤ ri+1  y  ,  with wi ≡ ri+1  y  − ri  y the width of the well, κ the  elastic constant, ri  min ≡ (ri  y + ri+1  y  )/2 the position of  the center of the well, and U i  0 = κ(wi)2/8 an unim-  portant offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  The elastic force deriving from this  potential energy is fe(r) ≡ −∂xU(r) = κ(ri  min − r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  With κ is constant, the landscape is parameterised by  the distance between two subsequent cusps wi, which  we assume identically distributed (iid) according to  a distribution Pw(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' We consider underdamped dy-  namics corresponding to η2 < 4mκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Within a well,  the dynamics is simply that of a underdamped oscil-  lator, as recalled in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='0  λt  0  1  2  3  E/⟨U⟩  0  4  8  r/⟨w⟩  −1  0  U/⟨U⟩  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Evolution of the kinetic energy E as a function  of position r (in red) of the “free” particle ‘thrown’ into  a potential energy landscape (shown in the inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Every  entry into a new well is indicated using a marker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' A thin  green line shows the evolution of the total energy (with  the definition of the inset, it has the local minimum of the  last well as arbitrary offset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  3  3  Stopping well  Distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  We are interested in the width of the  well in which the particle eventually stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Suppose  that a particle enters a well of width w with a kinetic  energy E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' The particle stops in that well if E < Ec(w),  with Ec the minimum kinetic energy with which the  particle needs to enter a well of width w to be able to  exit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' The distribution of wells in which particles stop  in that case is  Ps(w) ∼ Pw(w)P(E < Ec(w)),  (2)  with Pw(w) the probability density of well widths,  and Ps(w) the probability of well widths in which the  particle stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Within one well, the particle is sim-  ply a damped harmonic oscillator as has been studied  abundantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' In the limit of a weakly damped system,  the amount of kinetic energy lost during one cycle is  ∆E = κw2(1 − exp(−2π/Q))/8 with the quality fac-  tor Q =  �  4mκ/η2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' The minimal kinetic energy  with which the particle needs to enter the well in or-  der to be able to exist is thus Ec = ∆E ∝ w2 (see  Appendix B for the exact calculation of Ec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Further-  more, if P(E) is a constant at small argument (as we  will argue below), then  P(E < Ec(w)) =  � Ec  0  P(E)dE ∼ Ec(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  (3)  Therefore, the particle stops in a well whose width is  distributed as  Ps(w) ∼ w2Pw(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  (4)  Central result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Once stopped, the force, x, by  which we need to tilt the well in which the particle  stopped, in order for it to exit again is x = κw/2 1,  such that our central result is that  P(x) ∼ x2Pw(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  (5)  For example, if Pw(w) ∼ wk at small w, we predict  that  P(x) ∼ x2+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  (6)  Energy at entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  We will now argue that the den-  sity of kinetic energy with which the particle enters  the final well, P(E), is finite at small E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  For one  realisation, E results from passing many wells with  random widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' If its kinetic energy is much larger  than the potential energy of the typical wells, it will  1Without external forces, the particle ends in the local min-  imum – the center of the well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  not stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' We thus consider that the particle energy  has decreased up to some typical kinetic energy E0 of  the order of the typical potential energy κ⟨w2⟩/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' If  the particle exits the next well, at exit it will have a  kinetic energy K = E0 − ∆E(E0, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' For a given E0  and distributed w, we have:  P(E) =  �  dw Pw(w) δ(K(E0, w) − E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  (7)  It thus implies that:  P(E = 0) = Pw(w∗)/  ��∂wK  ��  w=w∗  (8)  w∗ is the well width for which the particle reaches the  end of the well with zero velocity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' E0 = Ec(w∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  By assumption, Pw(w∗) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Furthermore we prove  in Appendix C that ∂wK|w=w∗ = κw∗/2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Over-  all, it implies that P(E = 0) > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' P(E) does not  vanish as E → 0, from which our conclusions follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Here we give a simple argument for ∂wK|w=w∗ =  κw∗/2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Given E0, but an infinitesimally smaller  well of width w∗ −δw, the particle will enter the next  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Because the velocity is negligible in the vicinity  of w∗, the damping is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Therefore, δK is of  the order of the difference in potential energy on a  scale δw, δU = U(w∗) − U(w∗ − δw) ≈ κw∗δw/2, as  we illustrate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' We thus find that ∂wK|w=w∗ =  limδw→0 δK/δw = κw∗/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  −w∗/2  0  w∗/2  position  0  energy  1  2κw∗δw  δw  1  2κw∗δw  δw  potential energy U  kinetic energy E  total energy E + U  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Evolution of the kinetic energy E (red), po-  tential energy U (black), and total energy E + V (green)  for a particle that has entered a well of width w∗ with a  kinetic energy E0 = Ec(w∗) such that it stops just.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Con-  sequently, ∂r(E + V )|w∗/2 = 0, which can be decomposed  in ∂rV |w∗/2 = κw∗/2 such that ∂rE|w∗/2 = −κw∗/2, as  indicated using thin lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  4  4  Numerical support  Objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  We now numerically verify our predic-  tion that P(x) ∼ xk+2 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' (6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' We simulate a large  number of realisations of a potential energy landscape  constructed from randomly drawn widths (consider-  ing different distributions Pw(w)) and constant cur-  vature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' We study the distribution of stopping wells  if a “free” particle is ‘thrown’ into the landscape at  a high initial velocity (much larger than vc(⟨w⟩) such  that particle transverses many wells before stopping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  We find an analytical solution for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' (1) in  the form of a map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' In particular, we derive the evo-  lution of the position in a well based on an initial po-  sition −w/2 and velocity in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' This maps  the velocity with which the particle enters a well at  position w/2, to an exit velocity which corresponds  to the entry velocity of the next well, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Stopping well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  We record the width of the stop-  ping well, x, and the velocity V with which the parti-  cle enters the final well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' We find clear evidence for the  scaling P(x) ∼ xk+2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Perturbing the evolu-  tion with random force kicks2 changes nothing to our  observations, as included in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' 4 (see caption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' We,  furthermore, show that the probability density of the  kinetic energy with which the particle enters the final  well, P(E), is constant as small argument in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  5  Concluding remarks  Our central result is that P(x) ∼ x2Pw(x) in our  toy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' For a disorder Pw(w) ∼ wk we thus find  P(x) ∼ xk+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' We expect this result to qualitatively  apply to generic depinning systems in the presence of  inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' In particular they are qualitatively (but not  quantitatively) consistent with our previous empiri-  cal observations θ ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='5 for k = 1 [1] and θ ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='4  for k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' A plausible limitation of our approach  is underlined by the following additional observation:  in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' [1], it was found that for x to be small, the  stopping well was typically small (by definition), but  also that the next well had to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Such corre-  lations can exist only if the degree of freedom con-  sidered had visited the next well, before coming back  and stopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' This scenario cannot occur in our sim-  ple description where the particle only moves forward,  except when it oscillates in its final well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  2Such the for each well is tilted with a random force that we  take independent and identically distributed (iid) according to  a normal distribution with zero mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  10−2  100  x  10−5  100  P(x)/cx  1  2  1  3  1  4  k = 0 1 2  weibull  power  uniform  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Width of the stopping well, x, for different  Pw(w): a uniform, Weibull, and powerlaw distribution,  that scale as Pw(w) ∼ wk at small w, as indicated in the  legend (the bottom row for each distribution corresponds  to perturbing the dynamcs with random force kicks, tilt-  ing individual wells by a force F = N(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='1), with N the  normal distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' the top row corresponds to F = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  To emphasise the scaling, the distributions have been  rescaled by a fit of the prefactors: P(x) = cxxk+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Fur-  thermore, we use m = κ = 1, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='1, v0 = N(100, 10),  and ⟨w⟩ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  10−5  10−1  E  10−2  100  P(E)/ce  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  The kinetic energy with which the particle  enters the well in which it stops for different realisations,  P(E), normalised by its prefactor ce (that is here simply  the density of the first bin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' 4 for legend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  5  References  [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' de Geus, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Popovi´c, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Ji, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Rosso, and  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Wyart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' How collective asperity detachments nucleate slip  at frictional interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=', 116(48):  23977–23983, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='1906551116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  arXiv: 1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='07635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Fisher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Collective transport in random media: From  superconductors to earthquakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=', 301(1–3):  113–150, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='1016/S0370-1573(98)00008-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  arXiv: cond-mat/9711179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  [3] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Narayan and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Fisher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Threshold critical dynamics of  driven interfaces in random media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' B, 48(10):  7030–7042, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='1103/PhysRevB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='7030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Kardar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Nonequilibrium dynamics of interfaces and lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=', 301(1–3):85–112, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='1016/S0370-1573(98)00007-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  arXiv: cond-mat/9704172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' M¨uller and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Wyart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Marginal Stability in Structural,  Spin, and Electron Glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Matter  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=', 6(1):177–200, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='1146/annurev-conmatphys-031214-014614.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  [6] Alberto Rosso, James P Sethna, and Matthieu Wyart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Avalanches and deformation in glasses and disordered  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' arXiv preprint: 2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='04090, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='04090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  [7] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' de Geus and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Wyart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Scaling theory for the  statistics of slip at frictional interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' E, 106  (6):065001, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='1103/PhysRevE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='065001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  arXiv: 2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='02795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  [8] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Prandtl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Ein Gedankenmodell zur kinetischen Theorie der  festen K¨orper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=', 8(2):85–106, 1928.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='1002/zamm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='19280080202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  [9] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Tomlinson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' CVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' A molecular theory of friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' The  London, Edinburgh, and Dublin Philosophical Magazine  and Journal of Science, 7(46):905–939, 1929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='1080/14786440608564819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  [10] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Popov and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Prandtl-Tomlinson Model: A  Simple Model Which Made History.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' In E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Stein, editor, The  History of Theoretical, Material and Computational  Mechanics - Mathematics Meets Mechanics and  Engineering, volume 1, pages 153–168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Springer Berlin  Heidelberg, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' ISBN 978-3-642-39904-6  978-3-642-39905-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='1007/978-3-642-39905-3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  A  Analytical solution  Anasatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  We look for the solution of a general lin-  ear equation of motion in one well  m¨r + η ˙r + κr − F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  (9)  where the position r is expressed relative to the local  minimum in potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' The external force F  tilts the potential energy landscape and will be used  as a perturbation to check the robustness of our ar-  gument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' An ansatz to this differential equation is  r(τ) = αe−λ−τ + βe−λ+τ + ∆r,  (10)  where ∆r = F/κ, and τ is the time that the parti-  cle has spent since the entry in the current well at  r(τ = 0) ≡ −w/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' We denote the particle’s veloc-  ity v(τ) ≡ ˙r(τ), whereby we take v(τ = 0) ��� v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Substituting this ansatz in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' (9) leads to λ± =  (η ±  �  η2 − 4mκ)/(2m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' The prefactors α and β are  set by the initial conditions such that  α, β = ±r0λ± + v0  λ+ − λ−  ,  (11)  with r0 ≡ −w/2 − ∆r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Underdamped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  We recognise that if λ± are real  (η2 > 4mκ), the dynamics are overdamped and the  velocity decays exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Conversely, the un-  derdamped dynamics that we consider correspond to  η2 < 4mκ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  In the underdamped case, λ± and α, β  are complex conjugates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' This allows us to simplify  the solution by expressing those coefficients as λ± =  λ ± iω and α, β = (L/2)e±iφ as follows 4  r(τ) = Le−λτ cos(ωτ + φ) + ∆r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  (12)  We remark that the velocity v(τ) can be expressed as  a phase shift with respect to r(τ) 5  v(τ) = −Ae−λτ cos  �  ωτ + φ − arctan  �ω  λ  ��  (13)  with A = λL  �  1 + (ω/λ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' We summarise the ampli-  tudes, frequency, and phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' From λ± we find  λ =  η  2m,  ω2 = κ  m −  � η  2m  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  (14)  Furthermore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' (11) gives  α, β = 1  2  �  r0 ∓ i(λr0 + v0)/ω  �  ,  (15)  such that  L2 = 4  �  Re(α)2 + Im(α)2�  (16)  =  �  (ωr0)2 + (λr0 + v0)2�  /ω2,  (17)  and  φ = χπ + arctan (Im(α)/Re(α))  (18)  = χπ − arctan (λ/ω + v0/(ωr0)) ,  (19)  where χ depends on α 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  3Note that our solution warrants some caution for critical  damping η2 = 4mκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  4cos(z) = (eiz + e−iz)/2  5a cos(z) + b sin(z) = sgn(a)  √  a2 + b2 cos (z − arctan(b/a))  6Re(α) ≥ 0 → χ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' (Re(α) < 0, Im(α) ≥ 0) → χ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  (Re(α) < 0, Im(α) < 0) → χ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  6  B  Exiting well  We will show that the minimum kinetic energy with  which the particle needs to enter a well to be able  to exit Ec ∝ w2, whereby we consider F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' The  particle exits the well if v0 > vc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' vc thus corresponds  to the case that r(τe) = w/2 = −r0 for which v(τe) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' Let us make the ansatz that v0 = vc = w/τc and  look for the solution of τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  We note that on the interval τ ∈ [0, τe) the posi-  tion is strictly monotonically increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' The solution  of v(τn) = 0 corresponds to  ωτn + φ − arctan(ω/λ) = (n + 1/2)π,  n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' (20)  The for us relevant solution is τe = min(τn > 0) 7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  r(τe) = w/2 corresponds to  Le−λτe = w/(2c0),  (21)  with  c0 = cos((n + 1/2)π + arctan(ω/λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  (22)  Using the definition of τe in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' (20) leads to  Leλφ/ω = w/(2c0c1),  (23)  with  c1 = e−λ(n+1/2)π/ω−(λ/ω) arctan(ω/λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  (24)  Furthermore,  L2 = 1  4  �  ω2 + (2/τc − λ)2�  (w/ω)2,  (25)  φ = sign (2/τc − λ) π + arctan  ��  2/τc − λ  �  /ω  �  , (26)  such that  L′eλφ/ω = 1/(2c0c1),  (27)  with L′ = L/w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' (27) is w independent and can  be solved for τc = τc(λ, ω, n), proving that vc = w/τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  This results thus corresponds to  Ec = 1  2mv2  c = m  2τ 2c  w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  (28)  as we used to go from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' (2) to obtain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' (4) using  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  7i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' n = ±1 depending on (r0, v0)  C  Entry kinetic energy  With K the kinetic energy at exiting the well, we show  that ∂wK  ��  w=w∗ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' We again consider F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' In  particular, we show that  ∂wK = m ve ∂wve > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  (29)  The derivative of the velocity as a function of the well  size is  ∂rv = ∂τv/∂τr = a(τ)/v(τ),  (30)  where the acceleration a(τ) ≡ ¨r(τ) = αλ2  −e−λ−τ +  βλ2  +e−λ+τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' By evaluating this expression with initial  conditions r(τ = 0) = −w∗/2 and v(τ = 0) = 0, we  find  ∂wK  ��  w=w∗ = m (αλ2  − + βλ2  +) = −m  2 (λ2 + ω2) w∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content='  (31)  From the definitions of λ and ω in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' (14), we thus  find  ∂wK  ��  w=w∗ = −κw∗/2,  (32)  as we argued above to show that P(E = 0) > 0 using  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}
+page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dFST4oBgHgl3EQfcTgr/content/2301.13802v1.pdf'}

-tFST4oBgHgl3EQfcjgx/content/2301.13803v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:9ec4a491cbb15125a132c600492dde2bebbbf9da238559cfff708730b0a82390
+size 2428479

.gitattributes

@@ -5449,3 +5449,37 @@ j9E3T4oBgHgl3EQf5Quy/content/2301.04780v1.pdf filter=lfs diff=lfs merge=lfs -tex
 SdA0T4oBgHgl3EQfD_9b/content/2301.02011v1.pdf filter=lfs diff=lfs merge=lfs -text
 19AzT4oBgHgl3EQfDfqo/content/2301.00978v1.pdf filter=lfs diff=lfs merge=lfs -text
 gNE5T4oBgHgl3EQfhg_G/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+j9E3T4oBgHgl3EQf5Quy/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+0tE1T4oBgHgl3EQflAQk/content/2301.03279v1.pdf filter=lfs diff=lfs merge=lfs -text
+eNFAT4oBgHgl3EQfZR37/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+l9FIT4oBgHgl3EQfsisx/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+e9FST4oBgHgl3EQfGTh-/content/2301.13721v1.pdf filter=lfs diff=lfs merge=lfs -text
+7dAyT4oBgHgl3EQfQvYd/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+eNFAT4oBgHgl3EQfZR37/content/2301.08545v1.pdf filter=lfs diff=lfs merge=lfs -text
+0tE1T4oBgHgl3EQflAQk/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+vtFQT4oBgHgl3EQfvDZj/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+wdAzT4oBgHgl3EQfQPvW/content/2301.01197v1.pdf filter=lfs diff=lfs merge=lfs -text
+j9AzT4oBgHgl3EQfbfze/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+tNAyT4oBgHgl3EQfmvgr/content/2301.00475v1.pdf filter=lfs diff=lfs merge=lfs -text
+VdAzT4oBgHgl3EQfJ_tt/content/2301.01089v1.pdf filter=lfs diff=lfs merge=lfs -text
+U9AyT4oBgHgl3EQfV_cW/content/2301.00152v1.pdf filter=lfs diff=lfs merge=lfs -text
+q9FRT4oBgHgl3EQfezfQ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+JNAzT4oBgHgl3EQfVPyV/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+vtE_T4oBgHgl3EQf-xzf/content/2301.08389v1.pdf filter=lfs diff=lfs merge=lfs -text
+INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf filter=lfs diff=lfs merge=lfs -text
+1tFLT4oBgHgl3EQfqC-n/content/2301.12138v1.pdf filter=lfs diff=lfs merge=lfs -text
+VNAzT4oBgHgl3EQfX_zt/content/2301.01329v1.pdf filter=lfs diff=lfs merge=lfs -text
+e9FST4oBgHgl3EQfGTh-/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+-tFST4oBgHgl3EQfcjgx/content/2301.13803v1.pdf filter=lfs diff=lfs merge=lfs -text
+19AzT4oBgHgl3EQfDfqo/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+1tFLT4oBgHgl3EQfqC-n/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+vtE_T4oBgHgl3EQf-xzf/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+btE2T4oBgHgl3EQfwQhi/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+x9E2T4oBgHgl3EQfhQeR/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf filter=lfs diff=lfs merge=lfs -text
+y9E5T4oBgHgl3EQfNw4V/content/2301.05491v1.pdf filter=lfs diff=lfs merge=lfs -text
+fNFST4oBgHgl3EQfGjh7/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+7dAyT4oBgHgl3EQfQvYd/content/2301.00050v1.pdf filter=lfs diff=lfs merge=lfs -text
+vdFJT4oBgHgl3EQffCyN/content/2301.11555v1.pdf filter=lfs diff=lfs merge=lfs -text
+69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf filter=lfs diff=lfs merge=lfs -text
+PNFPT4oBgHgl3EQfnjVw/content/2301.13130v1.pdf filter=lfs diff=lfs merge=lfs -text

0tE1T4oBgHgl3EQflAQk/content/2301.03279v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:4b35fe7cdf77abb096c8a1c1f82b57e9a5007a02bd901b10514d1d302f6be07f
+size 243087

0tE1T4oBgHgl3EQflAQk/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:e081cdbe200fa67261ee3bf58c34b90a7e1dbef9733117036648bfa55d019705
+size 3866669

19AzT4oBgHgl3EQfDfqo/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:368a380856cc20211a7e2963ed9669bd16c326d9c7f17a261e5567fc7095deff
+size 1507373

1tFLT4oBgHgl3EQfqC-n/content/2301.12138v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b3388d4e19dca1f5b895bd808c806d688b7f445aeef0a565fd5d2639b97553f0
+size 4008479

1tFLT4oBgHgl3EQfqC-n/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:8cca964cdac233dfb088458b921bc29b937ea62d3f04f2ba2e50f501c6cd33b0
+size 8978477

1tFLT4oBgHgl3EQfqC-n/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:95342751e3089557556c92b7310483caf0c05aae676d1a9e962d091cafdbf2f9
+size 333061

2NE2T4oBgHgl3EQfNgYt/content/tmp_files/2301.03737v1.pdf.txt

@@ -0,0 +1,4539 @@
+arXiv:2301.03737v1  [hep-ph]  10 Jan 2023
+EPHOU-23-001
+Quark hierarchical structures in modular symmetric
+flavor models at level 6
+Shota Kikuchi, Tatsuo Kobayashi, Kaito Nasu,
+Shohei Takada, and Hikaru Uchida
+Department of Physics, Hokkaido University, Sapporo 060-0810, Japan
+Abstract
+We study modular symmetric quark flavor models without fine-tuning. Mass matrices are
+written in terms of modular forms, and modular forms in the vicinity of the modular fixed
+points become hierarchical depending on their residual charges. Thus modular symmetric
+flavor models in the vicinity of the modular fixed points have a possibility to describe
+mass hierarchies without fine-tuning. Since describing quark hierarchies without fine-tuning
+requires Zn residual symmetry with n ≥ 6, we focus on Γ6 modular symmetry in the
+vicinity of the cusp τ = i∞ where Z6 residual symmetry remains. We use only modular
+forms belonging to singlet representations of Γ6 to make our analysis simple. Consequently,
+viable quark flavor models are obtained without fine-tuning.
+
+1
+Introduction
+The origin of quark and lepton flavor structures such as hierarchical masses and mixing angles
+is one of challenging issues in particle physics.
+Indeed, many works were done in order to
+solve the problem. Among such works, modular symmetric flavor models are interesting. In
+these flavor models, the quark and lepton mass matrices are written in terms of modular forms,
+which are holomorphic functions of the modulus τ [1] 1. It is well known that the finite modular
+groups ΓN for N = 2, 3, 4, 5 are isomorphic to the non-Abelian finite groups S3, A4, S4 and
+A5, respectively [17]. This is interesting since the non-Abelian finite groups are long familiar
+in flavor models for quarks and leptons [18–28]. Inspired by this point, the modular symmetric
+lepton flavor models have been proposed in Γ2 ≃ S3 [29], Γ3 ≃ A4 [1], Γ4 ≃ S4 [30] and
+Γ5 ≃ A5 [31, 32]. In addition, modular symmetries at levels 6 [33] and 7 [34] were studied.
+Furthermore, modular forms of other groups were also studied [7,35–38].
+Using these various modular forms, the mass ratios and flavor mixing of quarks and leptons
+have been discussed successfully in these years. Phenomenological studies have been developed
+in many works and interesting results have been obtained [39–78]. However, one needs to fine-
+tune coefficients of modular forms in Yukawa couplings in order to describe the hierarchical
+structure of fermion masses, in particular quark mass hierarchies.
+Describing the lepton flavors without fine-tuning on modular invariant models was proposed
+in Ref. [79].
+Authors focused on the vicinity of three modular fixed points, τ = i, ω (=
+e2πi/3) and i∞ where residual symmetries remain [42]. The values of modular forms become
+hierarchical as close to these modular fixed points due to approximate residual symmetries.
+Then, the hierarchy among values of the modular forms is determined by charges of residual
+symmetries at the modular fixed points. For instance the modular forms of Γ4 ≃ S4 with Z4
+(T-transformation) charges 0, 1, 2 and 3 can take the sizes 1, ε, ε2 and ε3, respectively in the
+vicinity of τ = i∞, where ε expresses the deviation from the modular fixed points. Indeed viable
+lepton models on the double covering groups of ΓN, Γ′
+3 ≃ A′
+4, Γ′
+4 ≃ S′
+4 and Γ5 ≃ A′
+5, were studied
+in Ref. [79]. This is one successful way to generate hierarchical structures without fine-tuning.
+Nevertheless the realization of the quark flavor structure is not straightforward. Experiments
+show mass hierarchies of up sector quarks as mu/mt ∼ 10−5 and mc/mt ∼ 10−2-10−3 and ones
+of down sector quarks as md/mb ∼ 10−3 and ms/mb ∼ 10−2 [80]. Suppose that ε = O(0.1).
+Then, we could explain these mass ratios except mu/mt ∼ 10−5. However, ε5 does not appear
+in the framework of the finite modular group of level N less than 6 since the present residual
+symmetries Z2, Z3 and ZN at τ = i, ω and i∞ do not possess the charge larger than 4.
+Thus describing the quark flavor structure without fine-tuning requires the way generating
+hierarchical mass ratios including ε5 = O(10−5).
+One way is to relax the quark mass eigenvalues by tuning the values of coupling constants in
+Yukawa couplings. In Ref. [81], the quark flavor model with A4 modular symmetry was studied
+and succeeded to generate both up and down sector quark mass hierarchies by adjusting one
+1The modular flavor symmetry was also studied from the top-down approach such as string theory [2–16].
+1
+
+coupling constant ratio denoted by gu/gd to O(10). As a result, quark mass hierarchies originate
+from following two origins,
+(i) The vacuum expectation value (VEV) of the modulus τ (the deviation from the modular
+fixed points),
+(ii) The coupling constants in Yukawa couplings.
+Another way is to introduce the finite modular symmetry including Zn residual symmetry
+with n ≥ 6. In such models, the modular forms in the vicinity of the symmetric points can
+take the sizes 1, ε, · · ·, εn−1 depending on their residual charges. Hence, it may be possible
+to generate mass hierarchies in both up and down sector quarks without fine-tuning using the
+hierarchical values of the modular forms up to ε5. Note that in this way quark mass hierarchies
+simply originate from (i) above. In this paper, we study the modular symmetric quark flavor
+model with the finite modular group of level 6, Γ6 ≃ S3 × A4. The finite modular symmetry
+Γ6 breaks into Z6 (T-transformation) residual symmetry at τ = i∞ and can generate the
+hierarchical values of the modular forms up to ε5 in the vicinity of τ = i∞.
+This paper is organized as follows. In section 2, we study generic aspects for the modular
+symmetric quark flavor models being able to realize both up and down sector quark mass
+hierarchies without fine-tuning along the lines proposed in Ref. [79]. In section 3, we study quark
+flavor models with the finite modular group Γ6. Section 4 is our conclusion. We summarize
+group theoretical aspects of Γ6 in appendix A and the modular forms of level 6 in appendix B.
+2
+Hierarchical quark mass matrices without fine-tuning
+In this section, we present modular symmetric quark flavor models without fine-tuning. We
+start from the following assignment of modular weights to supermultiplets:
+• quark doublets Q = (Q1, Q2, Q3) are assigned into three-dimensional (reducible or irre-
+ducible ) representation of a finite modular group with weight −kQ,
+• up sector quark singlets uR = (u1
+R, u2
+R, u3
+R) are assigned into three-dimensional (reducible
+or irreducible ) representation of a finite modular group with weight −ku,
+• down sector quark singlets dR = (d1
+R, d2
+R, d3
+R) are assigned into three-dimensional (re-
+ducible or irreducible ) representation of a finite modular group with weight −kd,
+• each of up and down sector Higgs fields Hu,d is assigned into one-dimensional representa-
+tions of a finite modular group with weight −kHu,d.
+Note that three-dimensional representations are constructed by combining singlets, doublets
+and triplets of any finite modular groups. The most general form of the superpotential relevant
+2
+
+to up sector quark masses is written as
+Wu =
+�
+ri
+
+Y (kYu)
+ri
+�
+Q1
+Q2
+Q3�
+
+
+α11
+ri
+α12
+ri
+α13
+ri
+α21
+ri
+α22
+ri
+α23
+ri
+α31
+ri
+α32
+ri
+α33
+ri
+
+
+
+
+u1
+R
+u2
+R
+u3
+R
+
+ Hu
+
+
+1
+,
+(1)
+where Y (kYu)
+ri
+denotes the modular forms of irreducible representation ri for weight kYu = kQ +
+ku + kHu. Some of coupling constants αij may be related each other when quark doublets Q
+and/or up sector quark singlets uR belong to multiplets. Similarly the superpotential relevant
+to down sector quark masses is written as
+Wd =
+�
+ri
+
+Y
+(kYd)
+ri
+�
+Q1
+Q2
+Q3�
+
+
+β11
+ri
+β12
+ri
+β13
+ri
+β21
+ri
+β22
+ri
+β23
+ri
+β31
+ri
+β32
+ri
+β33
+ri
+
+
+
+
+d1
+R
+d2
+R
+d3
+R
+
+ Hd
+
+
+1
+,
+(2)
+with kYd = kQ + kd + kHd. They lead to the up and down sector quark mass matrices, Mu and
+Md,
+�
+Q1
+Q2
+Q3�
+Mu
+
+
+u1
+R
+u2
+R
+u3
+R
+
+ =
+�
+ri
+
+Y (kYu)
+ri
+�
+Q1
+Q2
+Q3�
+
+
+α11
+ri
+α12
+ri
+α13
+ri
+α21
+ri
+α22
+ri
+α23
+ri
+α31
+ri
+α32
+ri
+α33
+ri
+
+
+
+
+u1
+R
+u2
+R
+u3
+R
+
+ ⟨Hu⟩
+
+
+1
+, (3)
+�
+Q1
+Q2
+Q3�
+Md
+
+
+d1
+R
+d2
+R
+d3
+R
+
+ =
+�
+ri
+
+Y
+(kYd)
+ri
+�
+Q1
+Q2
+Q3�
+
+
+β11
+ri
+β12
+ri
+β13
+ri
+β21
+ri
+β22
+ri
+β23
+ri
+β31
+ri
+β32
+ri
+β33
+ri
+
+
+
+
+d1
+R
+d2
+R
+d3
+R
+
+ ⟨Hd⟩
+
+
+1
+.
+(4)
+We expect that the coefficients αij and βij are of O(1), because we do not explain quark
+mass hierarchies by using hierarchies of these coefficients. In particular, we restrict all coupling
+constants αij and βij to ±1 and study the realization of the orders of mass ratios and the
+Cabibbo, Kobayashi, Maskawa (CKM) matrix elements. Then free parameter is only the value
+of the modulus τ (and the choices of +1 or −1 in coupling constants αij and βij).
+(mu, mc, mt)/mt
+(1.26 × 10−5, 7.38 × 10−3, 1)
+(md, ns, mb)/mb
+(1.12 × 10−3, 2.22 × 10−2, 1)
+Table 1: Observed values of quark masses [80].
+In order to realize hierarchical quark masses as shown in Table 1 without fine-tuning, it is
+necessary to generate hierarchies by values of the modular forms. Actually such hierarchical
+values of the modular forms can be realized in a vicinity of three modular fixed points, τ = i,
+ω and i∞. This can be understood as follows. As an example let us consider Zn symmetric
+point and quark doublets Q, up sector quark singlets uR and up-type Higgs field Hu with the
+following Zn residual charges,
+Q : (1, n − 1, 0),
+uR : (1, 0, 0),
+Hu : 0.
+(5)
+3
+
+Then the entities of the up sector quark mass matrix, Mij
+u , must have the following Zn residual
+charges to make Lagrangian modular invariant,
+Mij
+u :
+
+
+n − 2
+n − 1
+n − 1
+0
+1
+1
+n − 1
+0
+0
+
+ .
+(6)
+In the vicinity of Zn symmetric point, the modular forms with Zn residual charge q, f(τ), can
+be expanded by the deviation from the symmetric point to the power of q [79]:
+1. τ ∼ i: f(τ) ∼ εq, ε ≡ τ−i
+τ+i,
+2. τ ∼ ω: f(τ) ∼ εq, ε ≡
+τ−ω
+τ−ω2,
+3. τ ∼ i∞: f(τ) ∼ εq, ε ≡ e−2πImτ/N (N is a level of the finite modular group).
+Thus the above up sector quark mass matrix can be evaluated as
+Mij
+u ∼
+
+
+εn−2
+εn−1
+εn−1
+1
+ε
+ε
+εn−1
+1
+1
+
+ ,
+(7)
+in the vicinity of Zn symmetric point. Similarly, for the down sector quark mass matrix as
+well as lepton mass matrices, the modular forms take hierarchical values depending on their
+residual charges and lead to hierarchical mass matrices as close to the modular fixed points.
+In Ref. [79], lepton flavor models without fine-tuning around the vicinity of the modular fixed
+points was studied.
+On the other hand, it is difficult to realize quark mass hierarchies by the values of the
+modular forms in the vicinity of τ = i and ω. To realize both up and down sector quark mass
+hierarchies in Table 1 simultaneously, we may need ε to the fifth power, when ε = O(0.1). Hence
+we need five different residual charges. This requirement excludes the vicinity of τ = i and ω
+since they correspond to Z2 and Z3 symmetries, respectively. In other words, such hierarchical
+masses can be realized in the vicinity of the cusp τ = i∞ with ZN charge for N ≥ 6.
+Let us discuss the candidates of the modular symmetry.
+As mentioned above the level
+of the modular symmetry must be lager than 5. Here we focus on the levels 6 and 7, that
+is, Γ6 ≃ S3 × A4 and Γ7 ≃ PSL(2, Z7) as the candidates of the modular symmetry. As the
+irreducible representations less than four dimension, Γ7 ≃ PSL(2, Z7) has only one singlet 1
+and two triplets 3 and ¯3 [34]; this variety of irreducible representations may not be enough
+to find the models being able to realize both up and down sector quark masses. In contrast,
+Γ6 ≃ S3 × A4 has six singlets, 10
+0, 10
+1, 10
+2, 11
+0, 11
+1 and 11
+2, three doublets, 20, 21 and 22, and two
+triplets, 30 and 31 [33]. They would be sufficient to find realistic models. In the following, we
+consider the models with Γ6 modular symmetry and realize quark flavors without fine-tuning
+in the vicinity of τ = i∞.
+4
+
+3
+The models with Γ6 modular symmetry
+Here we study the models with Γ6 modular symmetry and realize the quark flavor structure
+without fine-tuning. As we have mentioned in the previous section, we restrict all couplings αij
+and βij to ±1 in quark mass matrices to avoid fine-tuning by them. We study the realization
+of the orders of mass ratios and mixing angles. We use only the modulus τ (and the choices of
++1 or -1 in αij and βij) as a free parameter.
+In Γ6 modular symmetry, ε to the power up to 5 can appear in mass matrices. Indeed six
+Γ6 singlets with six different T-charges correspond to different powers of ε in the vicinity of
+τ = i∞ as shown in Table 2.
+singlet
+10
+0
+11
+2
+10
+1
+11
+0
+10
+2
+11
+1
+T-charge
+0
+1
+2
+3
+4
+5
+order
+1
+ε
+ε2
+ε3
+ε4
+ε5
+Table 2: T-charges of six Γ6 singlets and their orders in the vicinity of τ = i∞.
+To realize the quark flavor structure, let us consider the following four types of mass matrices,
+Type I:
+Mu ∝
+
+
+ε5
+ε3−a+b
+εb
+±ε5+a−b
+±ε3
+±εa
+±ε5−b
+±ε3−a
+±1
+
+ ,
+Md ∝
+
+
+ε3
+ε2−a+b
+εb
+±ε3+a−b
+±ε2
+±εa
+±ε3−b
+±ε2−a
+±1
+
+ ,
+(8)
+Type II:
+Mu ∝
+
+
+ε5
+ε3−a+b
+εb
+±ε5+a−b
+±ε3
+±εa
+±ε5−b
+±ε3−a
+±1
+
+ ,
+Md ∝
+
+
+ε4
+ε2−a+b
+εb
+±ε4+a−b
+±ε2
+±εa
+±ε4−b
+±ε2−a
+±1
+
+ ,
+(9)
+Type III:
+Mu ∝
+
+
+ε5
+ε2−a+b
+εb
+±ε5+a−b
+±ε2
+±εa
+±ε5−b
+±ε2−a
+±1
+
+ ,
+Md ∝
+
+
+ε3
+ε2−a+b
+εb
+±ε3+a−b
+±ε2
+±εa
+±ε3−b
+±ε2−a
+±1
+
+ ,
+(10)
+Type IV:
+Mu ∝
+
+
+ε5
+ε2−a+b
+εb
+±ε5+a−b
+±ε2
+±εa
+±ε5−b
+±ε2−a
+±1
+
+ ,
+Md ∝
+
+
+ε4
+ε2−a+b
+εb
+±ε4+a−b
+±ε2
+±εa
+±ε4−b
+±ε2−a
+±1
+
+ ,
+(11)
+where ± corresponds any possible combinations of signs and a, b ∈ {0, 1, 2, 3, 4, 5}. Note that
+it is always possible to fix the signs of (1,1), (1,2) and (1,3) components to +1 by the basis
+transformation for right-handed quarks. We set powers of ε on diagonal components in up and
+down sector quark mass matrices to (5,3,0) and (3,2,0) for type I, (5,3,0) and (4,2,0) for type
+II, (5,2,0) and (3,2,0) for type III and (5,2,0) and (4,2,0) for type IV in order to realize their
+hierarchical masses. Here, we use only six Γ6 singlets, 10
+0, 10
+1, 10
+2, 11
+0, 11
+1 and 11
+2, as irreducible
+representations to make our analysis simple. Note that again powers of ε in mass matrices are
+determined by Z6 charges of entities of mass matrices. Thus mass matrices of each type can be
+5
+
+led by the following assignments,
+Type I :
+Q = (1b mod 2
+b mod 3, 1a mod 2
+a mod 3, 10
+0), uR = (15−b mod 2
+5−b mod 3, 13−a mod 2
+3−a mod 3, 10
+0), dR = (13−b mod 2
+3−b mod 3, 12−a mod 2
+2−a mod 3, 10
+0),
+(12)
+Type II :
+Q = (1b mod 2
+b mod 3, 1a mod 2
+a mod 3, 10
+0), uR = (15−b mod 2
+5−b mod 3, 13−a mod 2
+3−a mod 3, 10
+0), dR = (14−b mod 2
+4−b mod 3, 12−a mod 2
+2−a mod 3, 10
+0),
+(13)
+Type III :
+Q = (1b mod 2
+b mod 3, 1a mod 2
+a mod 3, 10
+0), uR = (15−b mod 2
+5−b mod 3, 12−a mod 2
+2−a mod 3, 10
+0), dR = (13−b mod 2
+3−b mod 3, 12−a mod 2
+2−a mod 3, 10
+0),
+(14)
+Type IV :
+Q = (1b mod 2
+b mod 3, 1a mod 2
+a mod 3, 10
+0), uR = (15−b mod 2
+5−b mod 3, 12−a mod 2
+2−a mod 3, 10
+0), dR = (14−b mod 2
+4−b mod 3, 12−a mod 2
+2−a mod 3, 10
+0).
+(15)
+On the other hand, it is not always true that mass matrices in four types are definitely realized
+by the above assignments. It depends on weights of the Yukawa couplings. All of the singlet
+modular forms of Γ6 with certain Z6 charges do not exist for weights less than 14 as shown in
+appendix B. For instance, the modular forms of weight 12 belong to 11
+1 do not exist. Yukawa
+couplings of the weights less than 14 can lead to mass matrices with some zeros due to this
+shortage of the modular forms on low weights. We study the case of Yukawa couplings of the
+weight 14 in subsection 3.1 and one of the weights less than 14 in subsection 3.2.
+3.1
+Weight 14
+First of all, we study the models with Yukawa couplings of weight 14 to avoid zero textures in
+mass matrices of four types. We choose τ = 3.2i as a benchmark point of the modulus. At
+weight 14, seven singlet modular forms, Y (14)
+10
+0 , Y (14)
+11
+2i , Y (14)
+10
+1 , Y (14)
+11
+0 , Y (14)
+10
+2 , Y (14)
+11
+1
+and Y (14)
+11
+2ii, exist
+and they are approximated by ε as
+Y (14)
+10
+0 /Y (14)
+10
+0
+= 1 → 1,
+Y (14)
+11
+2i /Y (14)
+10
+0
+= 0.172 → ε,
+(16)
+Y (14)
+10
+1 /Y (14)
+10
+0
+= 0.0208 → ε2,
+Y (14)
+11
+0 /Y (14)
+10
+0
+= 0.00358 → ε3,
+(17)
+Y (14)
+10
+2 /Y (14)
+10
+0
+= 0.000435 → ε4,
+Y (14)
+11
+1 /Y (14)
+10
+0
+= 0.0000746 → ε5,
+(18)
+Y (14)
+11
+2ii/Y (14)
+10
+0
+= 0.00000156 → ε7,
+(19)
+at τ = 3.2i. Note that Y (14)
+11
+2ii ∼ ε7 originates from Y (6)
+11
+0 Y (8)
+10
+2 ∼ ε3 · ε4 while Y (14)
+11
+2i ∼ ε originates
+from Y (6)
+11
+2 Y (8)
+10
+0 ∼ ε · 1. εn for n > 5 can appear when the different modular forms of the same
+irreducible representations exist. In what follows, we ignore Y (14)
+11
+2ii because it belongs to the
+same representation as Y (14)
+11
+2i and Y (14)
+11
+2i >> Y (14)
+11
+2ii.
+6
+
+3.1.1
+Type I: (5,3,0) and (3,2,0)
+The mass matrices of type I are given by
+Mu =
+
+
+
+
+
+α11Y (14)
+11
+1
+α12Y (14)
+13+a−b mod 2
+3+a−b mod 3
+α13Y (14)
+16−b mod 2
+6−b mod 3
+α21Y (14)
+11−a+b mod 2
+1−a+b mod 3
+α22Y (14)
+11
+0
+α23Y (14)
+16−a mod 2
+6−a mod 3
+α31Y (14)
+11+b mod 2
+1+b mod 3
+α32Y (14)
+13+a mod 2
+3+a mod 3
+α33Y (14)
+10
+0
+
+
+
+
+ ,
+(20)
+Md =
+
+
+
+
+
+β11Y (14)
+11
+0
+β12Y (14)
+14+a−b mod 2
+4+a−b mod 3
+β13Y (14)
+16−b mod 2
+6−b mod 3
+β21Y (14)
+13−a+b mod 2
+3−a+b mod 3
+β22Y (14)
+10
+1
+β23Y (14)
+16−a mod 2
+6−a mod 3
+β31Y (14)
+13+b mod 2
+3+b mod 3
+β32Y (14)
+14+a mod 2
+4+a mod 3
+β33Y (14)
+10
+0
+
+
+
+
+ ,
+(21)
+where αij and βij are coupling constants which we restrict to ±1. The hierarchical mass matrices
+in Eq. (8) can be obtained by choosing +1 or −1 appropriately in αij and βij. As a result, we
+find best-fit mass matrices at τ = 3.2i,
+Mu/Y (14)
+10
+0
+=
+
+
+
+
+Y (14)
+11
+1
+Y (14)
+10
+2
+Y (14)
+11
+0
+Y (14)
+10
+2
+Y (14)
+11
+0
+Y (14)
+10
+1
+−Y (14)
+10
+1
+−Y (14)
+11
+2i
+Y (14)
+10
+0
+
+
+
+ /Y (14)
+10
+0
+=
+
+
+0.0000746
+0.000435
+0.00358
+0.000435
+0.00358
+0.0208
+−0.0208
+−0.172
+1
+
+
+∼
+
+
+ε5
+ε4
+ε3
+ε4
+ε3
+ε2
+−ε2
+−ε
+1
+
+ ,
+(22)
+Md/Y (14)
+10
+0
+=
+
+
+
+
+Y (14)
+11
+0
+Y (14)
+11
+0
+Y (14)
+11
+0
+Y (14)
+10
+1
+−Y (14)
+10
+1
+−Y (14)
+10
+1
+−Y (14)
+10
+0
+Y (14)
+10
+0
+−Y (14)
+10
+0
+
+
+
+ /Y (14)
+10
+0
+=
+
+
+0.00358
+0.00358
+0.00358
+0.0208
+−0.0208
+−0.0208
+−1
+1
+−1
+
+
+∼
+
+
+ε3
+ε3
+ε3
+ε2
+−ε2
+−ε2
+−1
+1
+−1
+
+ .
+(23)
+These mass matrices correspond to a = 2, b = 3 and can be realized by
+Q = (11
+0, 10
+2, 10
+0),
+uR = (10
+2, 11
+1, 10
+0),
+dR = (10
+0, 10
+0, 10
+0),
+(24)
+and their mass matrices are written by,
+Mu =
+
+
+
+
+α11Y (14)
+11
+1
+α12Y (14)
+10
+2
+α13Y (14)
+11
+0
+α21Y (14)
+10
+2
+α22Y (14)
+11
+0
+α23Y (14)
+10
+1
+α31Y (14)
+10
+1
+α32Y (14)
+11
+2i
+α33Y (14)
+10
+0
+
+
+
+ ,
+Md =
+
+
+
+
+β11Y (14)
+11
+0
+β12Y (14)
+11
+0
+β13Y (14)
+11
+0
+β21Y (14)
+10
+1
+β22Y (14)
+10
+1
+β23Y (14)
+10
+1
+β31Y (14)
+10
+0
+β32Y (14)
+10
+0
+β33Y (14)
+10
+0
+
+
+
+ ,
+(25)
+7
+
+with the following choises of +1 or −1 in coupling constants,
+
+
+α11
+α12
+α13
+α21
+α22
+α23
+α31
+α32
+α33
+
+ =
+
+
+1
+1
+1
+1
+1
+1
+−1
+−1
+1
+
+ ,
+
+
+β11
+β12
+β13
+β21
+β22
+β23
+β31
+β32
+β33
+
+ =
+
+
+1
+1
+1
+1
+−1
+−1
+−1
+1
+−1
+
+ .
+(26)
+They lead to the following quark mass ratios,
+(mu, mc, mt)/mt = (2.11 × 10−5, 7.07 × 10−3, 1),
+(27)
+(md, ms, mb)/mb = (2.91 × 10−3, 1.97 × 10−2, 1),
+(28)
+and the absolute values of the CKM matrix elements,
+|VCKM| =
+
+
+0.973
+0.231
+0.000681
+0.231
+0.973
+0.0270
+0.00690
+0.0261
+1.00
+
+ .
+(29)
+Results are shown in Table 3. Our purpose is to derive quark masses and mixing angles
+without fine-tuning. Thus, we have fixed αij, βij = ±1 to make our point clear. If we vary
+αij, βij = O(1) without fixing αij, βij = ±1, we can obtain more realistic values. Of course, we
+have ambiguity in normalization of modular forms, although we expect naturally that normal-
+ization factors would not lead a large hierarchy. Our values appear at a high energy scale such
+as the GUT scale. Renormalization group effects change values by some factors, although such
+radiative corrections may be realized by varying αij, βij = O(1).
+Obtained values
+Observed values
+(mu, mc, mt)/mt
+(2.11 × 10−5, 7.07 × 10−3, 1)
+(1.26 × 10−5, 7.38 × 10−3, 1)
+(md, ms, mb)/mb
+(2.91 × 10−3, 1.97 × 10−2, 1)
+(1.12 × 10−3, 2.22 × 10−2, 1)
+|VCKM|
+
+
+
+
+0.973
+0.231
+0.000681
+0.231
+0.973
+0.0270
+0.00690
+0.0261
+1.00
+
+
+
+
+
+
+
+
+0.974
+0.227
+0.00361
+0.226
+0.973
+0.0405
+0.00854
+0.0398
+0.999
+
+
+
+
+Table 3: The mass ratios of the quarks and the absolute values of the CKM matrix elements
+at the benchmark point τ = 3.2i in the best-fit model by Eqs. (24) and (26) of type I with
+Yukawa couplings of weight 14. Observed values are shown in Ref. [80].
+8
+
+3.1.2
+Type II: (5,3,0) and (4,2,0)
+The mass matrices of type II are given by Eq. (20) and
+Md =
+
+
+
+
+
+β11Y (14)
+10
+2
+β12Y (14)
+14+a−b mod 2
+4+a−b mod 3
+β13Y (14)
+16−b mod 2
+6−b mod 3
+β21Y (14)
+12−a+b mod 2
+2−a+b mod 3
+β22Y (14)
+10
+1
+β23Y (14)
+16−a mod 2
+6−a mod 3
+β31Y (14)
+12+b mod 2
+2+b mod 3
+β32Y (14)
+14+a mod 2
+4+a mod 3
+β33Y (14)
+10
+0
+
+
+
+
+ .
+(30)
+The hierarchical mass matrices in Eq. (9) can be obtained by choosing +1 or −1 appropriately
+in αij and βij. As a result, we find best-fit mass matrices at τ = 3.2i,
+Mu/Y (14)
+10
+0
+=
+
+
+
+
+Y (14)
+11
+1
+Y (14)
+10
+2
+Y (14)
+11
+0
+Y (14)
+10
+2
+Y (14)
+11
+0
+−Y (14)
+10
+1
+−Y (14)
+10
+1
+−Y (14)
+11
+2i
+−Y (14)
+10
+0
+
+
+
+ /Y (14)
+10
+0
+=
+
+
+0.0000746
+0.000435
+0.00358
+0.000435
+0.00358
+−0.0208
+−0.0208
+−0.172
+−1
+
+
+∼
+
+
+ε5
+ε4
+ε3
+ε4
+ε3
+−ε2
+−ε2
+−ε
+−1
+
+ ,
+(31)
+Md/Y (14)
+10
+0
+=
+
+
+
+
+Y (14)
+10
+2
+Y (14)
+11
+0
+Y (14)
+11
+0
+−Y (14)
+11
+0
+Y (14)
+10
+1
+Y (14)
+10
+1
+Y (14)
+11
+2i
+Y (14)
+10
+0
+−Y (14)
+10
+0
+
+
+
+ /Y (14)
+10
+0
+=
+
+
+0.000435
+0.00358
+0.00358
+−0.00358
+0.0208
+0.0208
+0.172
+1
+−1
+
+
+∼
+
+
+ε4
+ε3
+ε3
+−ε3
+ε2
+ε2
+ε
+1
+−1
+
+ .
+(32)
+These mass matrices correspond to a = 2, b = 3 and can be realized by
+Q = (11
+0, 10
+2, 10
+0),
+uR = (10
+2, 11
+1, 10
+0),
+dR = (11
+1, 10
+0, 10
+0),
+(33)
+and their mass matrices are written by,
+Mu =
+
+
+
+
+α11Y (14)
+11
+1
+α12Y (14)
+10
+2
+α13Y (14)
+11
+0
+α21Y (14)
+10
+2
+α22Y (14)
+11
+0
+α23Y (14)
+10
+1
+α31Y (14)
+10
+1
+α32Y (14)
+11
+2i
+α33Y (14)
+10
+0
+
+
+
+ ,
+Md =
+
+
+
+
+β11Y (14)
+10
+2
+β12Y (14)
+11
+0
+β13Y (14)
+11
+0
+β21Y (14)
+11
+0
+β22Y (14)
+10
+1
+β23Y (14)
+10
+1
+β31Y (14)
+11
+2i
+β32Y (14)
+10
+0
+β33Y (14)
+10
+0
+
+
+
+ ,
+(34)
+with the following choises of +1 or −1 in coupling constants,
+
+
+α11
+α12
+α13
+α21
+α22
+α23
+α31
+α32
+α33
+
+ =
+
+
+1
+1
+1
+1
+1
+−1
+−1
+−1
+−1
+
+ ,
+
+
+β11
+β12
+β13
+β21
+β22
+β23
+β31
+β32
+β33
+
+ =
+
+
+1
+1
+1
+−1
+1
+1
+1
+1
+−1
+
+ .
+(35)
+9
+
+They lead to the following quark mass ratios,
+(mu, mc, mt)/mt = (2.14 × 10−5, 7.00 × 10−3, 1),
+(36)
+(md, ms, mb)/mb = (7.16 × 10−4, 2.11 × 10−2, 1),
+(37)
+and the absolute values of the CKM matrix elements,
+|VCKM| =
+
+
+0.982
+0.190
+0.00309
+0.190
+0.982
+0.0200
+0.00683
+0.0191
+1.00
+
+ .
+(38)
+Results are shown in Table 4.
+Obtained values
+Observed values
+(mu, mc, mt)/mt
+(2.14 × 10−5, 7.00 �� 10−3, 1)
+(1.26 × 10−5, 7.38 × 10−3, 1)
+(md, ms, mb)/mb
+(7.16 × 10−4, 2.11 × 10−2, 1)
+(1.12 × 10−3, 2.22 × 10−2, 1)
+|VCKM|
+
+
+
+
+0.982
+0.190
+0.00309
+0.190
+0.982
+0.0200
+0.00683
+0.0191
+1.00
+
+
+
+
+
+
+
+
+0.974
+0.227
+0.00361
+0.226
+0.973
+0.0405
+0.00854
+0.0398
+0.999
+
+
+
+
+Table 4: The mass ratios of the quarks and the absolute values of the CKM matrix elements
+at the benchmark point τ = 3.2i in the best-fit model by Eqs. (33) and (35) of type II with
+Yukawa couplings of weight 14. Observed values are shown in Ref. [80].
+3.1.3
+Type III: (5,2,0) and (3,2,0)
+The mass matrices of type III are given by Eq. (21) and
+Mu =
+
+
+
+
+
+α11Y (14)
+11
+1
+α12Y (14)
+14+a−b mod 2
+4+a−b mod 3
+α13Y (14)
+16−b mod 2
+6−b mod 3
+α21Y (14)
+11−a+b mod 2
+1−a+b mod 3
+α22Y (14)
+10
+1
+α23Y (14)
+16−a mod 2
+6−a mod 3
+α31Y (14)
+11+b mod 2
+1+b mod 3
+α32Y (14)
+14+a mod 2
+4+a mod 3
+α33Y (14)
+10
+0
+
+
+
+
+ .
+(39)
+10
+
+The hierarchical mass matrices in Eq. (10) can be obtained by choosing +1 or −1 appropriately
+in αij and βij. As a result, we find best-fit mass matrices at τ = 3.2i,
+Mu/Y (14)
+10
+0
+=
+
+
+
+
+Y (14)
+11
+1
+Y (14)
+11
+0
+Y (14)
+11
+0
+Y (14)
+10
+2
+−Y (14)
+10
+1
+−Y (14)
+10
+1
+Y (14)
+10
+1
+Y (14)
+10
+0
+−Y (14)
+10
+0
+
+
+
+ /Y (14)
+10
+0
+=
+
+
+0.0000746
+0.00358
+0.00358
+0.000435
+−0.0208
+−0.0208
+0.0208
+1
+−1
+
+
+∼
+
+
+ε5
+ε3
+ε3
+ε4
+−ε2
+−ε2
+ε2
+1
+−1
+
+ ,
+(40)
+Md/Y (14)
+10
+0
+=
+
+
+
+
+Y (14)
+11
+0
+Y (14)
+11
+0
+Y (14)
+11
+0
+Y (14)
+10
+1
+Y (14)
+10
+1
+−Y (14)
+10
+1
+Y (14)
+10
+0
+−Y (14)
+10
+0
+Y (14)
+10
+0
+
+
+
+ /Y (14)
+10
+0
+=
+
+
+0.00358
+0.00358
+0.00358
+0.0208
+0.0208
+−0.0208
+1
+−1
+1
+
+
+∼
+
+
+ε3
+ε3
+ε3
+ε2
+ε2
+−ε2
+1
+−1
+1
+
+ .
+(41)
+These mass matrices correspond to a = 2, b = 3 and can be realized by
+Q = (11
+0, 10
+2, 10
+0),
+uR = (10
+2, 10
+0, 10
+0),
+dR = (10
+0, 10
+0, 10
+0),
+(42)
+and their mass matrices are written by,
+Mu =
+
+
+
+
+α11Y (14)
+11
+1
+α12Y (14)
+11
+0
+α13Y (14)
+11
+0
+α21Y (14)
+10
+2
+α22Y (14)
+10
+1
+α23Y (14)
+10
+1
+α31Y (14)
+10
+1
+α32Y (14)
+10
+0
+α33Y (14)
+10
+0
+
+
+
+ ,
+Md =
+
+
+
+
+β11Y (14)
+11
+0
+β12Y (14)
+11
+0
+β13Y (14)
+11
+0
+β21Y (14)
+10
+1
+β22Y (14)
+10
+1
+β23Y (14)
+10
+1
+β31Y (14)
+10
+0
+β32Y (14)
+10
+0
+β33Y (14)
+10
+0
+
+
+
+ ,
+(43)
+with the following choises of +1 or −1 in coupling constants,
+
+
+α11
+α12
+α13
+α21
+α22
+α23
+α31
+α32
+α33
+
+ =
+
+
+1
+1
+1
+1
+−1
+−1
+1
+1
+−1
+
+ ,
+
+
+β11
+β12
+β13
+β21
+β22
+β23
+β31
+β32
+β33
+
+ =
+
+
+1
+1
+1
+1
+1
+−1
+1
+−1
+1
+
+ .
+(44)
+They lead to the following quark mass ratios,
+(mu, mc, mt)/mt = (1.04 × 10−4, 2.12 × 10−2, 1),
+(45)
+(md, ms, mb)/mb = (2.91 × 10−3, 1.97 × 10−2, 1),
+(46)
+and the absolute values of the CKM matrix elements,
+|VCKM| =
+
+
+0.967
+0.255
+0.00000171
+0.255
+0.967
+0.00706
+0.00180
+0.00682
+1.00
+
+ .
+(47)
+11
+
+Results are shown in Table 5.
+Obtained values
+Observed values
+(mu, mc, mt)/mt
+(1.04 × 10−4, 2.12 × 10−2, 1)
+(1.26 × 10−5, 7.38 × 10−3, 1)
+(md, ms, mb)/mb
+(2.91 × 10−3, 1.97 × 10−2, 1)
+(1.12 × 10−3, 2.22 × 10−2, 1)
+|VCKM|
+
+
+
+
+0.967
+0.255
+0.00000171
+0.255
+0.967
+0.00706
+0.00180
+0.00682
+1.00
+
+
+
+
+
+
+
+
+0.974
+0.227
+0.00361
+0.226
+0.973
+0.0405
+0.00854
+0.0398
+0.999
+
+
+
+
+Table 5: The mass ratios of the quarks and the absolute values of the CKM matrix elements
+at the benchmark point τ = 3.2i in the best-fit model by Eqs. (42) and (44) of type III with
+Yukawa couplings of weight 14. Observed values are shown in Ref. [80].
+3.1.4
+Type IV: (5,2,0) and (4,2,0)
+The mass matrices of type IV are given by Eqs. (39) and (30). The hierarchical mass matrices
+in Eq. (11) can be obtained by choosing +1 or −1 appropriately in αij and βij. As a result, we
+find best-fit mass matrices at τ = 3.2i,
+Mu/Y (14)
+10
+0
+=
+
+
+
+
+Y (14)
+11
+1
+Y (14)
+11
+0
+Y (14)
+10
+0
+Y (14)
+10
+2
+Y (14)
+10
+1
+Y (14)
+11
+1
+Y (14)
+11
+1
+−Y (14)
+11
+0
+−Y (14)
+10
+0
+
+
+
+ /Y (14)
+10
+0
+=
+
+
+0.0000746
+0.00358
+1
+0.000435
+0.0208
+0.0000746
+0.0000746
+−0.00358
+−1
+
+
+∼
+
+
+ε5
+ε3
+1
+ε4
+ε2
+ε5
+ε5
+−ε3
+−1
+
+ ,
+(48)
+Md/Y (14)
+10
+0
+=
+
+
+
+
+Y (14)
+10
+2
+Y (14)
+11
+0
+Y (14)
+10
+0
+Y (14)
+11
+0
+−Y (14)
+10
+1
+−Y (14)
+11
+1
+Y (14)
+10
+2
+Y (14)
+11
+0
+−Y (14)
+10
+0
+
+
+
+ /Y (14)
+10
+0
+=
+
+
+0.000435
+0.00358
+1
+0.00358
+−0.0208
+−0.0000746
+0.000435
+0.00358
+−1
+
+
+∼
+
+
+ε4
+ε3
+1
+ε3
+−ε2
+−ε5
+ε4
+ε3
+−1
+
+ .
+(49)
+These mass matrices correspond to a = 5, b = 0 and can be realized by
+Q = (10
+0, 11
+2, 10
+0),
+uR = (11
+2, 11
+0, 10
+0),
+dR = (10
+1, 11
+0, 10
+0),
+(50)
+12
+
+and their mass matrices are written by,
+Mu =
+
+
+
+
+α11Y (14)
+11
+1
+α12Y (14)
+11
+0
+α13Y (14)
+10
+0
+α21Y (14)
+10
+2
+α22Y (14)
+10
+1
+α23Y (14)
+11
+1
+α31Y (14)
+11
+1
+α32Y (14)
+11
+0
+α33Y (14)
+10
+0
+
+
+
+ ,
+Md =
+
+
+
+
+β11Y (14)
+10
+2
+β12Y (14)
+11
+0
+β13Y (14)
+10
+0
+β21Y (14)
+11
+0
+β22Y (14)
+10
+1
+β23Y (14)
+11
+1
+β31Y (14)
+10
+2
+β32Y (14)
+11
+0
+β33Y (14)
+10
+0
+
+
+
+ ,
+(51)
+with the following choises of +1 or −1 in coupling constants,
+
+
+α11
+α12
+α13
+α21
+α22
+α23
+α31
+α32
+α33
+
+ =
+
+
+1
+1
+1
+1
+1
+1
+1
+−1
+−1
+
+ ,
+
+
+β11
+β12
+β13
+β21
+β22
+β23
+β31
+β32
+β33
+
+ =
+
+
+1
+1
+1
+1
+−1
+−1
+1
+1
+−1
+
+ .
+(52)
+They lead to the following quark mass ratios,
+(mu, mc, mt)/mt = (7.46 × 10−5, 1.47 × 10−2, 1),
+(53)
+(md, ms, mb)/mb = (1.01 × 10−3, 1.54 × 10−2, 1),
+(54)
+and the absolute values of the CKM matrix elements,
+|VCKM| =
+
+
+0.974
+0.226
+0.0000000194
+0.226
+0.974
+0.000158
+0.0000358
+0.000154
+1.00
+
+ .
+(55)
+Results are shown in Table 6.
+Obtained values
+Observed values
+(mu, mc, mt)/mt
+(7.46 × 10−5, 1.47 × 10−2, 1)
+(1.26 × 10−5, 7.38 × 10−3, 1)
+(md, ms, mb)/mb
+(1.01 × 10−3, 1.54 × 10−2, 1)
+(1.12 × 10−3, 2.22 × 10−2, 1)
+|VCKM|
+
+
+
+
+0.974
+0.226
+0.0000000194
+0.226
+0.974
+0.000158
+0.0000358
+0.000154
+1.00
+
+
+
+
+
+
+
+
+0.974
+0.227
+0.00361
+0.226
+0.973
+0.0405
+0.00854
+0.0398
+0.999
+
+
+
+
+Table 6: The mass ratios of the quarks and the absolute values of the CKM matrix elements
+at the benchmark point τ = 3.2i in the best-fit model by Eqs. (50) and (52) of type IV with
+Yukawa couplings of weight 14. Observed values are shown in Ref. [80].
+3.2
+Weights less than 14
+Next we study the models with Yukawa couplings of weights less than 14. In this case, some
+of entities in mass matrices vanish because there do not exist modular forms of proper weights
+13
+
+and representations. As an example, let us consider the case that Yukawa couplings for the up
+sector have the weight 8 and ones for the down sector have the weight 10. We choose τ = 3.7i
+as a benchmark point of the modulus. At weight 8, four singlet modular forms, Y (8)
+10
+0 , Y (8)
+11
+2 , Y (8)
+10
+1
+and Y (8)
+10
+2 , exist and they are approximated by ε as
+Y (8)
+10
+0 /Y (8)
+10
+0 = 1 → 1,
+Y (8)
+11
+2 /Y (8)
+10
+0 = −0.0719 → ε,
+(56)
+Y (8)
+10
+1 /Y (8)
+10
+0 = 0.00732 → ε2,
+Y (8)
+10
+2 /Y (8)
+10
+0 = 0.0000535 → ε4,
+(57)
+at τ = 3.7i. At weight 10, five singlet modular forms, Y (10)
+10
+0 , Y (10)
+11
+2 , Y (10)
+10
+1 , Y (10)
+11
+0
+and Y (10)
+11
+1 , exist
+and they are approximated by ε as
+Y (10)
+10
+0 /Y (10)
+10
+0
+= 1 → 1,
+Y (10)
+11
+2 /Y (10)
+10
+0
+= 0.102 → ε,
+(58)
+Y (10)
+10
+1 /Y (10)
+10
+0
+= 0.00732 → ε2,
+Y (10)
+11
+0 /Y (10)
+10
+0
+= 0.000744 → ε3,
+(59)
+Y (10)
+11
+1 /Y (10)
+10
+0
+= 0.00000544 → ε5,
+(60)
+at τ = 3.7i. As a result, we find the following best-fit mass matrices of type III,
+Mu/Y (8)
+10
+0 =
+
+
+
+
+0
+0
+Y (8)
+10
+1
+Y (8)
+10
+2
+Y (8)
+10
+1
+Y (8)
+11
+2
+0
+−Y (8)
+11
+2
+−Y (8)
+10
+0
+
+
+
+ /Y (8)
+10
+0 =
+
+
+0
+0
+0.00732
+0.0000535
+0.00732
+−0.0719
+0
+0.0719
+−1
+
+
+∼
+
+
+0
+0
+ε2
+ε4
+ε2
+−ε
+0
+ε
+−1
+
+ ,
+(61)
+Md/Y (10)
+10
+0
+=
+
+
+
+
+Y (10)
+11
+0
+Y (10)
+11
+0
+Y (10)
+10
+1
+−Y (10)
+10
+1
+−Y (10)
+10
+1
+Y (10)
+11
+2
+Y (10)
+11
+2
+−Y (10)
+11
+2
+Y (10)
+10
+0
+
+
+
+ /Y (10)
+10
+0
+=
+
+
+0.000744
+0.000744
+0.00732
+−0.00732
+−0.00732
+0.102
+0.102
+−0.102
+1
+
+
+∼
+
+
+ε3
+ε3
+ε2
+−ε2
+−ε2
+ε
+ε
+−ε
+1
+
+ .
+(62)
+These mass matrices correspond to a = 1, b = 2 and can be realized by
+Q = (10
+2, 11
+1, 10
+0),
+uR = (11
+0, 11
+1, 10
+0),
+dR = (11
+1, 11
+1, 10
+0),
+(63)
+and their mass matrices,
+Mu =
+
+
+
+
+0
+0
+α13Y (8)
+10
+1
+α21Y (8)
+10
+2
+α22Y (8)
+10
+1
+α23Y (8)
+11
+2
+0
+α32Y (8)
+11
+2
+α33Y (8)
+10
+0
+
+
+
+ ,
+Md =
+
+
+
+
+β11Y (10)
+11
+0
+β12Y (10)
+11
+0
+β13Y (10)
+10
+1
+β21Y (10)
+10
+1
+β22Y (10)
+10
+1
+β23Y (10)
+11
+2
+β31Y (10)
+11
+2
+β32Y (10)
+11
+2
+β33Y (10)
+10
+0
+
+
+
+ ,
+(64)
+14
+
+with the following choises of +1 or −1 in coupling constants,
+
+
+-
+-
+α13
+α21
+α22
+α23
+-
+α32
+α33
+
+ =
+
+
+-
+-
+1
+1
+1
+1
+-
+−1
+−1
+
+ ,
+
+
+β11
+β12
+β13
+β21
+β22
+β23
+β31
+β32
+β33
+
+ =
+
+
+1
+1
+1
+−1
+−1
+1
+1
+−1
+1
+
+ .
+(65)
+They lead to the following up quark and down quark mass ratios,
+(mu, mc, mt)/mt = (1.27 × 10−5, 2.18 × 10−3, 1),
+(66)
+(md, ms, mb)/mb = (1.44 × 10−3, 1.74 × 10−2, 1),
+(67)
+and the absolute values of the CKM matrix elements,
+|VCKM| =
+
+
+0.974
+0.227
+0.00741
+0.227
+0.973
+0.0300
+0.0140
+0.0276
+1.00
+
+ .
+(68)
+Results are shown in Table 7. Thus it is also possible to realize a realistic quark flavor structure
+in the models with Yukawa couplings of weights less than 14 despite some zeros in mass matrices.
+Here we studied the case that Yukawa couplings for the up sector have weight 8 and ones for
+the down sector have weight 10 but other cases may be available for realization of the quark
+flavor structure.
+Obtained values
+Observed values
+(mu, mc, mt)/mt
+(1.27 × 10−5, 2.18 × 10−3, 1)
+(1.26 × 10−5, 7.38 × 10−3, 1)
+(md, ms, mb)/mb
+(1.44 × 10−3, 1.74 × 10−2, 1)
+(1.12 × 10−3, 2.22 × 10−2, 1)
+|VCKM|
+
+
+
+
+0.974
+0.227
+0.00741
+0.227
+0.973
+0.0300
+0.0140
+0.0276
+1.00
+
+
+
+
+
+
+
+
+0.974
+0.227
+0.00361
+0.226
+0.973
+0.0405
+0.00854
+0.0398
+0.999
+
+
+
+
+Table 7: The mass ratios of the quarks and the absolute values of the CKM matrix elements at
+the benchmark point τ = 3.7i in the best-fit model by Eqs. (63) and (65) of type III with up
+sector Yukawa couplings of weight 8 and down sector Yukawa couplings of weight 10. Observed
+values are shown in Ref. [80].
+3.3
+Comment on the origin of Γ6 modular symmetry
+Here we comment on a plausible origin of Γ6 modular symmetry of the theories. For example,
+some modular forms are derived from the torus compactification T 2
+1 ×T 2
+2 ×T 2
+3 of the low-energy
+effective theory of the superstring theory with magnetic flux background [6–11]. The group Γ6
+15
+
+may originate from one of T 2
+i , while the others T 2
+j lead to a trivial symmetry. Alternatively,
+since Γ6 ≃ S3 × A4 ≃ Γ2 × Γ3, it may be expected that Γ2 ≃ S3 originates from one torus T 2
+1
+and Γ3 ≃ A4 originates from another torus T 2
+2 with the moduli stabilization τ1 = τ2 ≡ τ. Then
+T 2
+3 contributes to the group symmetry trivially.
+4
+Conclusion
+We have discussed the possibility to describe mass hierarchies of both up and down sector quarks
+as well as mixing angles without fine-tuning. Describing the quark flavor structure requires Zn
+residual symmetry with n ≥ 6. We have studied the modular symmetric quark flavor models
+of Γ6 ≃ S3 × A4 in the vicinity of the cusp τ = i∞ where Z6 residual symmetry remains. Then
+the values of the modular forms become hierarchical as close to the cusp depending on their Z6
+residual charges.
+In order to obtain viable models, we consider four types of quark mass matrices; the diagonal
+components in up and down sector quark mass matrices are written by ε with the powers of
+(5, 3, 0) and (3, 2, 0) respectively for type I, (5, 3, 0) and (4, 2, 0) for type II, (5, 2, 0) and (3, 2, 0)
+for type III, and (5, 2, 0) and (4, 2, 0) for type IV. The powers of non-diagonal components in up
+and down sector quark mass matrices have been treated as model depending values. When we
+assign the irreducible representations into quarks and Higgs fields, powers of ε in mass matrix
+components are determined by residual charges of mass matrix components. For simplicity, we
+have used only six singlets 10
+0, 10
+1, 10
+2, 11
+0, 11
+1 and 11
+2 as the irreducible representations of Γ6.
+In addition, we have restricted the values of the coupling constants to ±1 to avoid fine-tuning
+by them.
+Firstly, we have investigated the case that up and down sector Yukawa couplings have weight
+14. In such cases, mass matrices have no zeros, that is, all of their components are written in
+terms of the modular forms for Γ6 of weight 14. Consequently, we have obtained viable models
+at τ = 3.2i for each type without fine-tuning.
+Second, we have shown the viable model in the case that Yukawa couplings of the up sector
+have weight 8 and ones of the down sector have weight 10. In this case some components of
+mass matrices can become zero because there do not exist modular forms of proper weights
+and representations. As a result, we have obtained the viable model at τ = 3.7i despite three
+zeros in the up quark mass matrix, Eq. (61).
+Thus, the modular symmetric quark flavor models based on Γ6 in the vicinity of the cusp τ =
+i∞ lead to successful quark mass matrices without fine-tuning. As we have commented in the
+end of previous section, Γ6 ≃ S3 × A4 ≃ Γ2 × Γ3 may originate from the torus compactification
+T 2
+1 ×T 2
+2 ×T 2
+3 of the low-energy effective theory of superstring theory. Motivated this point, the
+modular flavor models based on the direct product of finite modular groups, ΓN1 × ΓN2 × ΓN3,
+may be interesting. Also we can extend our analysis to the lepton sector. We will study them
+in the near future.
+16
+
+In our models, the important parameter is the modulus τ. It must be stabilized such that
+the proper mass hierarchies are realized. We would study such modulus stabilization elsewhere2.
+Acknowledgement
+This work was supported by JSPS KAKENHI Grant Numbers JP22J10172 (SK) and JP20J20388
+(HU), and JST SPRING Grant Number JPMJSP2119(KN).
+Appendix
+A
+Tensor product of Γ6 group
+Here, we give a review on group theoretical aspects of Γ6. The generators of Γ6 are denoted by
+S and T, and they satisfy the following algebraic relations:
+S2 = (ST)3 = T 6 = ST 2ST 3ST 4ST 3 = 1,
+S2T = TS2.
+(69)
+In Γ6 group, there are 12 irreducible representations, six singlets 10
+0, 10
+1, 10
+2, 11
+0, 11
+1 and 11
+2,
+three doublets 20, 21 and 22, two triplets 30 and 31 and one six-dimensional representation 6.
+Each irreducible representation is given by
+1r
+k : S = (−1)r,
+T = (−1)rωk,
+(70)
+2k : S = 1
+2
+�−1
+√
+3
+√
+3
+1
+�
+,
+T = ωk
+�1
+0
+0
+−1
+�
+,
+(71)
+3r : (−1)ra3,
+(−1)rb3,
+(72)
+6 : 1
+2
+� −a3
+√
+3a3
+√
+3a3
+a3
+�
+,
+T =
+�b3
+0
+0
+−b3
+�
+,
+(73)
+where r = 0, 1, k = 0, 1, 2 and
+a3 = 1
+3
+
+
+−1
+2
+2
+2
+−1
+2
+2
+2
+−1
+
+ ,
+b3 =
+
+
+1
+0
+0
+0
+ω
+0
+0
+0
+ω2
+
+ .
+(74)
+In this basis, the Kronecker products between irreducible representations are:
+1r
+i ⊗ 1s
+j = 1t
+m,
+1r
+i ⊗ 2j = 2m,
+1r
+i ⊗ 3s = 3t,
+1r
+i ⊗ 6 = 6,
+(75)
+2i ⊗ 2j = 10
+m ⊕ 11
+m ⊕ 2m,
+2i ⊗ 3r = 6,
+2i ⊗ 6 = 30 ⊕ 31 ⊕ 6,
+(76)
+3r ⊗ 3s = 1t
+0 ⊕ 1t
+1 ⊕ 1t
+2 ⊕ 3t
+1 ⊕ 3t
+2,
+3r ⊗ 6 = 20 ⊕ 21 ⊕ 22 ⊕ 6 ⊕ 6,
+(77)
+6 ⊗ 6 = 10
+0 ⊕ 10
+1 ⊕ 10
+2 ⊕ 11
+0 ⊕ 11
+1 ⊕ 11
+2 ⊕ 20 ⊕ 21 ⊕ 22 ⊕ 30 ⊕ 30 ⊕ 31 ⊕ 31 ⊕ 6 ⊕ 6,
+(78)
+2See for modulus stabilization in modular flavor symmetric models Refs. [82–86] .
+17
+
+where i, j = 0, 1, 2, r, s = 0, 1, m = i + j (mod 3) and t = r + s (mod 2). In the following, we
+show the Clebsch-Gordon (CG) coefficients of these products.
+(α1)1r
+i ⊗
+�β1
+β2
+�
+2j
+= α1P r
+2
+�β1
+β2
+�
+2m
+,
+(α1)1r
+i ⊗
+
+
+β1
+β2
+β3
+
+
+3s
+= α1P i
+3
+
+
+β1
+β2
+β3
+
+
+3t
+,
+(α1)1r
+i ⊗
+
+
+
+
+
+
+
+
+
+β1
+β2
+β3
+β4
+β5
+β6
+
+
+
+
+
+
+
+
+
+6
+= α1P6(r, i)
+
+
+
+
+
+
+
+
+
+β1
+β2
+β3
+β4
+β5
+β6
+
+
+
+
+
+
+
+
+
+6
+,
+�α1
+α2
+�
+2i
+⊗
+�β1
+β2
+�
+2j
+= 1
+√
+2
+(α1β1 + α2β2)10m ⊕ 1
+√
+2
+(α1β2 − α2β1)11m ⊕ 1
+√
+2
+� α1β1 − α2β2
+−α1β2 − α2β1
+�
+2m
+,
+�α1
+α2
+�
+2i
+⊗
+
+
+β1
+β2
+β3
+
+
+3r
+= P6(r, i)
+
+
+
+
+
+
+
+
+
+α1β1
+α1β2
+α1β3
+α2β1
+α2β2
+α2β3
+
+
+
+
+
+
+
+
+
+6
+,
+�α1
+α2
+�
+2i
+⊗
+
+
+
+
+
+
+
+
+
+β1
+β2
+β3
+β4
+β5
+β6
+
+
+
+
+
+
+
+
+
+6
+= P i
+3
+√
+2
+
+
+α1β1 + α2β4
+α1β2 + α2β5
+α1β3 + α2β6
+
+
+30
+⊕ P i
+3
+√
+2
+
+
+α1β4 − α2β1
+α1β5 − α2β2
+α1β6 − α2β3
+
+
+31
+⊕ P6(0, i)
+√
+2
+
+
+
+
+
+
+
+
+
+α1β1 − α2β4
+α1β2 − α2β5
+α1β3 − α2β6
+−α1β4 − α2β1
+−α1β5 − α2β2
+−α1β6 − α2β3
+
+
+
+
+
+
+
+
+
+6
+,
+18
+
+
+
+α1
+α2
+α3
+
+
+3r
+⊗
+
+
+β1
+β2
+β3
+
+
+3s
+= 1
+√
+3(α1β1 + α2β3 + α3β2)1t
+0 ⊕ 1
+√
+3(α1β2 + α2β1 + α3β3)1t
+1
+⊕ 1
+√
+3
+(α1β3 + α2β2 + α3β1)1t
+2 ⊕ 1
+√
+3
+
+
+2α1β1 − α2β3 − α3β2
+−α1β2 − α2β1 + 2α3β3
+−α1β3 + 2α2β2 − α3β1
+
+
+3t
+1
+⊕ 1
+√
+2
+
+
+−α2β3 + α3β2
+−α1β2 + α2β1
+α1β3 − α3β1
+
+
+3t
+2
+,
+
+
+α1
+α2
+α3
+
+
+3r
+⊗
+
+
+
+
+
+
+
+
+
+β1
+β2
+β3
+β4
+β5
+β6
+
+
+
+
+
+
+
+
+
+6
+= P r
+2
+√
+3
+�α1β1 + α2β3 + α3β2
+α1β4 + α2β6 + α3β5
+�
+20
+⊕ P r
+2
+√
+3
+�α1β2 + α2β1 + α3β3
+α1β5 + α2β4 + α3β6
+�
+21
+⊕ P r
+2
+√
+3
+�α1β3 + α2β2 + α3β1
+α1β6 + α2β5 + α3β4
+�
+22
+⊕ P6(r, 0)
+√
+2
+
+
+
+
+
+
+
+
+
+α1β1 − α3β2
+−α2β1 + α3β3
+−α1β3 + α2β2
+α1β4 − α3β5
+−α2β4 + α3β6
+−α1β6 + α2β5
+
+
+
+
+
+
+
+
+
+6
+⊕ P6(r, 0)
+√
+2
+
+
+
+
+
+
+
+
+
+α2β3 − α3β2
+α1β2 − α2β1
+−α1β3 + α3β1
+α2β6 − α3β5
+α1β5 − α2β4
+−α1β6 + α3β4
+
+
+
+
+
+
+
+
+
+6
+,
+19
+
+
+
+
+
+
+
+
+
+
+α1
+α2
+α3
+α4
+α5
+α6
+
+
+
+
+
+
+
+
+
+6
+⊗
+
+
+
+
+
+
+
+
+
+β1
+β2
+β3
+β4
+β5
+β6
+
+
+
+
+
+
+
+
+
+6
+=
+1
+√
+6(α1β1 + α2β3 + α3β2 + α4β4 + α5β6 + α6β5)10
+0
+⊕ 1
+√
+6(α1β2 + α2β1 + α3β3 + α4β5 + α5β4 + α6β6)10
+1
+⊕ 1
+√
+6(α1β3 + α2β2 + α3β1 + α4β6 + α5β5 + α6β4)10
+2
+⊕ 1
+√
+6(α1β4 + α2β6 + α3β5 − α4β1 − α5β3 − α6β2)11
+0
+⊕ 1
+√
+6(α1β5 + α2β4 + α3β6 − α4β2 − α5β1 − α6β3)11
+1
+⊕ 1
+√
+6(α1β6 + α2β5 + α3β4 − α4β3 − α5β2 − α6β1)11
+2
+⊕
+1
+√
+6
+� α1β1 + α2β3 + α3β2 − α4β4 − α5β6 − α6β5
+−(α1β4 + α2β6 + α3β5 + α4β1 + α5β3 + α6β2)
+�
+20
+⊕
+1
+√
+6
+� α1β2 + α2β1 + α3β3 − α4β5 − α5β4 − α6β6
+−(α1β5 + α2β4 + α3β6 + α4β2 + α5β1 + α6β3)
+�
+21
+⊕
+1
+√
+6
+� α1β3 + α2β2 + α3β1 − α4β6 − α5β5 − α6β4
+−(α1β6 + α2β5 + α3β4 + α4β3 + α5β2 + α6β1)
+�
+22
+⊕
+1
+2
+√
+3
+
+
+2α1β1 − α2β3 − α3β2 + 2α4β4 − α5β6 − α6β5
+2α3β3 − α1β2 − α2β1 + 2α6β6 − α4β5 − α5β4
+2α2β2 − α1β3 − α3β1 + 2α5β5 − α4β6 − α6β4
+
+
+30
+⊕ 1
+2
+
+
+α2β3 − α3β2 + α5β6 − α6β5
+α1β2 − α2β1 + α4β5 − α5β4
+−α1β3 + α3β1 − α4β6 + α6β4
+
+
+30
+⊕ 1
+2
+
+
+α2β6 − α3β5 − α5β3 + α6β2
+α1β5 − α2β4 − α4β2 + α5β1
+−α1β6 + α3β4 + α4β3 − α6β1
+
+
+31
+⊕
+1
+2
+√
+3
+
+
+2α1β4 − α2β6 − α3β5 − 2α4β1 + α5β3 + α6β2
+−α1β5 − α2β4 + 2α3β6 + α4β2 + α5β1 − 2α6β3
+−α1β6 + 2α2β5 − α3β4 + α4β3 − 2α5β2 + α6β1
+
+
+31
+⊕
+1
+2
+√
+3
+
+
+
+
+
+
+
+
+
+2α1β1 − α2β3 − α3β2 − 2α4β4 + α5β6 + α6β5
+−α1β2 − α2β1 + 2α3β3 + α4β5 + α5β4 − 2α6β6
+−α1β3 + 2α2β2 − α3β1 + α4β6 − 2α5β5 + α6β4
+−2α1β4 + α2β6 + α3β5 − 2α4β1 + α5β3 + α6β2
+α1β5 + α2β4 − 2α3β6 + α4β2 + α5β1 − 2α6β3
+α1β6 − 2α2β5 + α3β4 + α4β3 − 2α5β2 + α6β1
+
+
+
+
+
+
+
+
+
+6
+⊕ 1
+2
+
+
+
+
+
+
+
+
+
+α2β3 − α3β2 − α5β6 + α6β5
+α1β2 − α2β1 − α4β5 + α5β4
+−α1β3 + α3β1 + α4β6 − α6β4
+−α2β6 + α3β5 − α5β3 + α6β2
+−α1β5 + α2β4 − α4β2 + α5β1
+α1β6 − α3β4 + α4β3 − α6β1
+
+
+
+
+
+
+
+
+
+6
+.
+20
+
+Here we have used the notations,
+P2 =
+� 0
+1
+−1
+0
+�
+,
+P3 =
+
+
+0
+0
+1
+1
+0
+0
+0
+1
+0
+
+ ,
+P6(r, i) =
+� 03
+13
+−13
+03
+�r �P3
+03
+03
+P3
+�i
+.
+(79)
+Further details can be found in Ref. [33].
+B
+Modular forms of Γ6
+Here we give a review on the modular forms of Γ6. The modular forms of level 6 of even weights
+can be constructed from the products of the Dedekind eta function [33],
+η(τ) = q1/24
+∞
+�
+n=1
+(1 − qn),
+q = e2πiτ.
+(80)
+Using η, four linearly independent modular forms of weight 2 can be written down as
+Y (2)
+30 (τ) =
+
+
+−Y 2
+1
+√
+2Y1Y2
+Y 2
+2
+
+ ,
+Y (2)
+11
+2 (τ) = Y3Y6 − Y4Y5,
+Y (2)
+20 (τ) = 1
+√
+2
+�Y1Y4 − Y2Y3
+Y1Y6 − Y2Y5
+�
+,
+(81)
+Y (2)
+6 (τ) = 1
+√
+2
+
+
+
+
+
+
+
+
+
+Y1Y4 + Y2Y3
+√
+2Y2Y4
+−
+√
+2Y1Y3
+Y1Y6 + Y2Y5
+√
+2Y2Y6
+−
+√
+2Y1Y5
+
+
+
+
+
+
+
+
+
+,
+(82)
+where
+Y1(τ) = 3η3(3τ)
+η(τ) + η3(τ/3)
+η(τ) ,
+(83)
+Y2(τ) = 3
+√
+2η3(3τ)
+η(τ) ,
+(84)
+Y3(τ) = 3
+√
+2η3(6τ)
+η(2τ) ,
+(85)
+Y4(τ) = −3η3(6τ)
+η(2τ) − η3(2τ/3)
+η(2τ) ,
+(86)
+Y5(τ) =
+√
+6η3(6τ)
+η(2τ) −
+√
+6η3(3τ/2)
+η(τ/2) ,
+(87)
+Y6(τ) = −
+√
+3η3(6τ)
+η(2τ) + 1
+√
+3
+η3(τ/6)
+η(τ/2) − 1
+√
+3
+η3(2τ/3)
+η(2τ)
++
+√
+3η3(3τ/2)
+η(τ/2) .
+(88)
+21
+
+Then we can construct the modular forms of weight 4 by the CG coefficients shown in appendix
+A as
+Y (4)
+10
+0 (τ) =
+�
+Y (2)
+20 Y (2)
+20
+�
+10
+0
+,
+Y (4)
+10
+1 (τ) =
+�
+Y (2)
+11
+2 Y (2)
+11
+2
+�
+10
+1
+,
+Y (4)
+20 (τ) =
+�
+Y (2)
+20 Y (2)
+20
+�
+20 ,
+(89)
+Y (4)
+22 (τ) =
+�
+Y (2)
+11
+2 Y (2)
+20
+�
+22 ,
+Y (4)
+30 (τ) =
+�
+Y (2)
+20 Y (2)
+6
+�
+30 ,
+Y (4)
+31 (τ) =
+�
+Y (2)
+11
+2 Y (2)
+30
+�
+31 ,
+(90)
+Y (4)
+6i (τ) =
+�
+Y (2)
+11
+2 Y (2)
+6
+�
+6 ,
+Y (4)
+6ii (τ) =
+�
+Y (2)
+20 Y (2)
+30
+�
+6 .
+(91)
+Note that Y (4)
+6i
+and Y (4)
+6ii stand for two linearly independent six-dimensional modular forms of
+weight 4. We use the same convention for other modular forms. Similarly, we construct the
+modular forms of weight 6 as
+Y (6)
+10
+0 (τ) =
+�
+Y (2)
+20 Y (4)
+20
+�
+10
+0
+,
+Y (6)
+11
+0 (τ) =
+�
+Y (2)
+11
+2 Y (4)
+10
+1
+�
+11
+0
+,
+Y (6)
+11
+2 (τ) =
+�
+Y (2)
+11
+2 Y (4)
+10
+0
+�
+11
+2
+,
+(92)
+Y (6)
+20 (τ) =
+�
+Y (2)
+20 Y (4)
+10
+0
+�
+20 ,
+Y (6)
+21 (τ) =
+�
+Y (2)
+20 Y (4)
+10
+1
+�
+21 ,
+Y (6)
+22 (τ) =
+�
+Y (2)
+11
+2 Y (4)
+20
+�
+22 ,
+(93)
+Y (6)
+30i(τ) =
+�
+Y (2)
+30 Y (4)
+10
+1
+�
+30 ,
+Y (6)
+30ii(τ) =
+�
+Y (2)
+30 Y (4)
+10
+0
+�
+30 ,
+Y (6)
+31 (τ) =
+�
+Y (2)
+11
+2 Y (4)
+30
+�
+31 ,
+(94)
+Y (6)
+6i (τ) =
+�
+Y (2)
+20 Y (4)
+31
+�
+6 ,
+Y (6)
+6ii (τ) =
+�
+Y (2)
+6 Y (4)
+10
+0
+�
+6 ,
+Y (6)
+6iii(τ) =
+�
+Y (2)
+30 Y (4)
+20
+�
+6 .
+(95)
+In Table 8 we summarize the modular forms of level 6 of even weights up to 6.
+Modular form Y (kY )
+r
+kY = 2
+Y (2)
+11
+2 , Y (2)
+20 , Y (2)
+30 , Y (2)
+6
+kY = 4
+Y (4)
+10
+0 , Y (4)
+10
+1 , Y (4)
+20 , Y (4)
+22 , Y (4)
+30 , Y (4)
+31 , Y (4)
+6i , Y (4)
+6ii
+kY = 6
+Y (6)
+10
+0 , Y (6)
+11
+0 , Y (6)
+11
+2 , Y (6)
+20 , Y (6)
+21 , Y (6)
+22 , Y (6)
+30i, Y (6)
+30ii, Y (6)
+31 , Y (6)
+6i , Y (6)
+6ii , Y (6)
+6iii
+Table 8: The modular forms of level 6 of even weights up to 6.
+Also we construct the singlet modular forms of weights 8, 10, 12 and 14 which we have used
+in our analysis. First, the singlet modular forms of weight 8 are given by
+Y (8)
+10
+0 =
+�
+Y (4)
+10
+0 Y (4)
+10
+0
+�
+10
+0
+,
+Y (8)
+10
+1 =
+�
+Y (4)
+10
+0 Y (4)
+10
+1
+�
+10
+1
+,
+Y (8)
+10
+2 =
+�
+Y (4)
+10
+1 Y (4)
+10
+1
+�
+10
+2
+,
+Y (8)
+11
+2 =
+�
+Y (4)
+20 Y (4)
+22
+�
+11
+2
+.
+(96)
+The singlet modular forms of weight 10 are given by
+Y (10)
+10
+0
+=
+�
+Y (4)
+10
+0 Y (6)
+10
+0
+�
+10
+0
+,
+Y (10)
+10
+1
+=
+�
+Y (4)
+10
+1 Y (6)
+10
+0
+�
+10
+1
+,
+Y (10)
+11
+0
+=
+�
+Y (4)
+10
+0 Y (6)
+11
+0
+�
+11
+0
+,
+(97)
+Y (10)
+11
+1
+=
+�
+Y (4)
+10
+1 Y (6)
+11
+0
+�
+11
+1
+,
+Y (10)
+11
+2
+=
+�
+Y (4)
+10
+0 Y (6)
+11
+2
+�
+11
+2
+.
+(98)
+22
+
+The singlet modular forms of weight 12 are given by
+Y (12)
+10
+0i =
+�
+Y (6)
+10
+0 Y (6)
+10
+0
+�
+10
+0
+,
+Y (12)
+10
+0ii =
+�
+Y (6)
+11
+0 Y (6)
+11
+0
+�
+10
+0
+,
+Y (12)
+10
+1
+=
+�
+Y (6)
+11
+2 Y (6)
+11
+2
+�
+10
+1
+,
+(99)
+Y (12)
+10
+2
+=
+�
+Y (6)
+11
+0 Y (6)
+11
+2
+�
+10
+2
+,
+Y (12)
+11
+0
+=
+�
+Y (6)
+10
+0 Y (6)
+11
+0
+�
+11
+0
+,
+Y (12)
+11
+2
+=
+�
+Y (6)
+10
+0 Y (6)
+11
+2
+�
+11
+2
+.
+(100)
+The singlet modular forms of weight 14 are given by
+Y (14)
+10
+0
+=
+�
+Y (6)
+10
+0 Y (8)
+10
+0
+�
+10
+0
+,
+Y (14)
+10
+1
+=
+�
+Y (6)
+10
+0 Y (8)
+10
+1
+�
+10
+1
+,
+Y (14)
+10
+2
+=
+�
+Y (6)
+10
+0 Y (8)
+10
+2
+�
+10
+2
+,
+(101)
+Y (14)
+11
+0
+=
+�
+Y (6)
+11
+0 Y (8)
+10
+0
+�
+11
+0
+,
+Y (14)
+11
+1
+=
+�
+Y (6)
+11
+0 Y (8)
+10
+1
+�
+11
+1
+,
+Y (14)
+11
+2i =
+�
+Y (6)
+11
+2 Y (8)
+10
+0
+�
+11
+2
+,
+(102)
+Y (14)
+11
+2ii =
+�
+Y (6)
+11
+0 Y (8)
+10
+2
+�
+11
+2
+.
+(103)
+In Table 9 we summarize the singlet modular forms of level 6 of weights 8, 10, 12 and 14.
+Modular form Y (kY )
+r
+kY = 8
+Y (8)
+10
+0 , Y (8)
+10
+1 , Y (8)
+10
+2 , Y (8)
+11
+2
+kY = 10
+Y (10)
+10
+0 , Y (10)
+10
+1 , Y (10)
+11
+0 , Y (10)
+11
+1 , Y (10)
+11
+2
+kY = 12
+Y (12)
+10
+0i , Y (12)
+10
+0ii, Y (12)
+10
+1 , Y (12)
+10
+2 , Y (12)
+11
+0 , Y (12)
+11
+2
+kY = 14
+Y (14)
+10
+0 , Y (14)
+10
+1 , Y (14)
+10
+2 , Y (14)
+11
+0 , Y (14)
+11
+1 , Y (14)
+11
+2i , Y (14)
+11
+2ii
+Table 9: The singlet modular forms of level 6 of weights 8, 10, 12 and 14.
+23
+
+References
+[1] F. Feruglio, in From My Vast Repertoire ...: Guido Altarelli’s Legacy, A. Levy, S. Forte,
+Stefano, and G. Ridolfi, eds., pp.227–266, 2019, arXiv:1706.08749 [hep-ph].
+[2] S. Ferrara, D. Lust, A. D. Shapere and S. Theisen, Phys. Lett. B 225, 363 (1989).
+[3] S. Ferrara, D. Lust and S. Theisen, Phys. Lett. B 233 (1989), 147-152.
+[4] W. Lerche, D. Lust and N. P. Warner, Phys. Lett. B 231 (1989), 417-424.
+[5] J. Lauer, J. Mas and H. P. Nilles, Nucl. Phys. B 351, 353 (1991).
+[6] T. Kobayashi, S. Nagamoto, S. Takada, S. Tamba and T. H. Tatsuishi, Phys. Rev. D 97,
+no. 11, 116002 (2018) [arXiv:1804.06644 [hep-th]].
+[7] T. Kobayashi and S. Tamba, Phys. Rev. D 99 (2019) no.4, 046001 [arXiv:1811.11384
+[hep-th]].
+[8] H. Ohki, S. Uemura and R. Watanabe, Phys. Rev. D 102, no.8, 085008 (2020)
+[arXiv:2003.04174 [hep-th]].
+[9] S. Kikuchi, T. Kobayashi, S. Takada, T. H. Tatsuishi and H. Uchida, Phys. Rev. D 102,
+no.10, 105010 (2020) [arXiv:2005.12642 [hep-th]].
+[10] S. Kikuchi, T. Kobayashi, H. Otsuka, S. Takada and H. Uchida, JHEP 11, 101 (2020)
+[arXiv:2007.06188 [hep-th]].
+[11] S. Kikuchi, T. Kobayashi and H. Uchida, Phys. Rev. D 104, no.6, 065008 (2021)
+[arXiv:2101.00826 [hep-th]].
+[12] Y. Almumin, M. C. Chen, V. Knapp-Perez, S. Ramos-Sanchez, M. Ratz and S. Shukla,
+JHEP 05 (2021), 078 [arXiv:2102.11286 [hep-th]].
+[13] A. Baur, H. P. Nilles, A. Trautner and P. K. S. Vaudrevange, Nucl. Phys. B 947 (2019),
+114737 [arXiv:1908.00805 [hep-th]].
+[14] H. P. Nilles, S. Ramos-S´anchez and P. K. S. Vaudrevange, Nucl. Phys. B 957 (2020),
+115098 [arXiv:2004.05200 [hep-ph]].
+[15] A. Baur, M. Kade, H. P. Nilles, S. Ramos-Sanchez and P. K. S. Vaudrevange, JHEP 02
+(2021), 018 [arXiv:2008.07534 [hep-th]].
+[16] H. P. Nilles, S. Ramos–S´anchez and P. K. S. Vaudrevange, Nucl. Phys. B 966 (2021),
+115367 [arXiv:2010.13798 [hep-th]].
+24
+
+[17] R. de Adelhart Toorop, F. Feruglio and C. Hagedorn, Nucl. Phys. B 858, 437 (2012)
+[arXiv:1112.1340 [hep-ph]].
+[18] G. Altarelli and F. Feruglio, Rev. Mod. Phys. 82 (2010) 2701 [arXiv:1002.0211 [hep-ph]].
+[19] H. Ishimori, T. Kobayashi, H. Ohki, Y. Shimizu, H. Okada and M. Tanimoto, Prog. Theor.
+Phys. Suppl. 183 (2010) 1 [arXiv:1003.3552 [hep-th]].
+[20] H. Ishimori, T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu and M. Tanimoto, Lect. Notes
+Phys. 858 (2012) 1, Springer.
+[21] T. Kobayashi, H. Ohki, H. Okada, Y. Shimizu and M. Tanimoto, Lect. Notes Phys. 995
+(2022) 1, Springer doi:10.1007/978-3-662-64679-3.
+[22] D. Hernandez and A. Y. Smirnov, Phys. Rev. D 86 (2012) 053014 [arXiv:1204.0445 [hep-
+ph]].
+[23] S. F. King and C. Luhn, Rept. Prog. Phys. 76 (2013) 056201 [arXiv:1301.1340 [hep-ph]].
+[24] S. F. King, A. Merle, S. Morisi, Y. Shimizu and M. Tanimoto, New J. Phys. 16, 045018
+(2014) [arXiv:1402.4271 [hep-ph]].
+[25] M. Tanimoto, AIP Conf. Proc. 1666 (2015) 120002.
+[26] S. F. King, Prog. Part. Nucl. Phys. 94 (2017) 217 [arXiv:1701.04413 [hep-ph]].
+[27] S. T. Petcov, Eur. Phys. J. C 78 (2018) no.9, 709 [arXiv:1711.10806 [hep-ph]].
+[28] F. Feruglio and A. Romanino, arXiv:1912.06028 [hep-ph].
+[29] T. Kobayashi, K. Tanaka and T. H. Tatsuishi, Phys. Rev. D 98, no. 1, 016004 (2018)
+[arXiv:1803.10391 [hep-ph]].
+[30] J. T. Penedo and S. T. Petcov, Nucl. Phys. B 939, 292 (2019) [arXiv:1806.11040 [hep-ph]].
+[31] P. P. Novichkov, J. T. Penedo, S. T. Petcov and A. V. Titov, JHEP 1904, 174 (2019)
+[arXiv:1812.02158 [hep-ph]].
+[32] G. J. Ding, S. F. King and X. G. Liu, Phys. Rev. D 100 (2019) no.11, 115005
+[arXiv:1903.12588 [hep-ph]].
+[33] C. C. Li, X. G. Liu and G. J. Ding, JHEP 10 (2021), 238 doi:10.1007/JHEP10(2021)238
+[arXiv:2108.02181 [hep-ph]].
+[34] G. J.
+Ding,
+S.
+F. King,
+C.
+C.
+Li and Y. L.
+Zhou,
+JHEP 08
+(2020),
+164
+doi:10.1007/JHEP08(2020)164 [arXiv:2004.12662 [hep-ph]].
+25
+
+[35] X. G. Liu and G. J. Ding, JHEP 08, 134 (2019) [arXiv:1907.01488 [hep-ph]].
+[36] P. P. Novichkov, J. T. Penedo and S. T. Petcov, Nucl. Phys. B 963 (2021), 115301
+[arXiv:2006.03058 [hep-ph]].
+[37] X. G. Liu, C. Y. Yao and G. J. Ding, Phys. Rev. D 103, no.5, 056013 (2021)
+[arXiv:2006.10722 [hep-ph]].
+[38] X. G. Liu, C. Y. Yao, B. Y. Qu and G. J. Ding, Phys. Rev. D 102, no.11, 115035 (2020)
+[arXiv:2007.13706 [hep-ph]].
+[39] J. C. Criado and F. Feruglio, SciPost Phys. 5 (2018) no.5, 042 [arXiv:1807.01125 [hep-ph]].
+[40] T. Kobayashi, N. Omoto, Y. Shimizu, K. Takagi, M. Tanimoto and T. H. Tatsuishi, JHEP
+11 (2018), 196 [arXiv:1808.03012 [hep-ph]].
+[41] G. J. Ding, S. F. King and X. G. Liu, JHEP 1909 (2019) 074 [arXiv:1907.11714 [hep-ph]].
+[42] P. P. Novichkov, J. T. Penedo, S. T. Petcov and A. V. Titov, JHEP 1904 (2019) 005
+[arXiv:1811.04933 [hep-ph]].
+[43] T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto and T. H. Tatsuishi, JHEP 02 (2020),
+097 [arXiv:1907.09141 [hep-ph]].
+[44] X. Wang and S. Zhou, JHEP 05 (2020), 017 [arXiv:1910.09473 [hep-ph]].
+[45] P. Chen, G. J. Ding, J. N. Lu and J. W. F. Valle, Phys. Rev. D 102 (2020) no.9, 095014
+[arXiv:2003.02734 [hep-ph]].
+[46] I. de Medeiros Varzielas, S. F. King and Y. L. Zhou, Phys. Rev. D 101 (2020) no.5, 055033
+[arXiv:1906.02208 [hep-ph]].
+[47] T. Asaka,
+Y. Heo,
+T. H. Tatsuishi and T. Yoshida,
+JHEP 2001 (2020) 144
+[arXiv:1909.06520 [hep-ph]].
+[48] T. Asaka, Y. Heo and T. Yoshida, Phys. Lett. B 811 (2020), 135956 [arXiv:2009.12120
+[hep-ph]].
+[49] F. J. de Anda, S. F. King and E. Perdomo, Phys. Rev. D 101 (2020) no.1, 015028
+[arXiv:1812.05620 [hep-ph]].
+[50] T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto and T. H. Tatsuishi, PTEP 2020,
+no.5, 053B05 (2020) [arXiv:1906.10341 [hep-ph]].
+[51] P. P. Novichkov, S. T. Petcov and M. Tanimoto, Phys. Lett. B 793 (2019) 247
+[arXiv:1812.11289 [hep-ph]].
+26
+
+[52] T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto, T. H. Tatsuishi and H. Uchida, Phys.
+Lett. B 794 (2019) 114 [arXiv:1812.11072 [hep-ph]].
+[53] H. Okada and M. Tanimoto, Phys. Lett. B 791 (2019) 54 [arXiv:1812.09677 [hep-ph]].
+[54] H. Okada and M. Tanimoto, Eur. Phys. J. C 81 (2021) no.1, 52 [arXiv:1905.13421 [hep-
+ph]].
+[55] T. Nomura and H. Okada, Phys. Lett. B 797, 134799 (2019) [arXiv:1904.03937 [hep-ph]].
+[56] H. Okada and Y. Orikasa, Phys. Rev. D 100, no.11, 115037 (2019) [arXiv:1907.04716
+[hep-ph]].
+[57] T. Nomura and H. Okada, Nucl. Phys. B 966 (2021), 115372 [arXiv:1906.03927 [hep-ph]].
+[58] T. Nomura, H. Okada and O. Popov, Phys. Lett. B 803 (2020) 135294 [arXiv:1908.07457
+[hep-ph]].
+[59] J. C. Criado, F. Feruglio and S. J. D. King, JHEP 2002 (2020) 001 [arXiv:1908.11867
+[hep-ph]].
+[60] S. F. King and Y. L. Zhou, Phys. Rev. D 101 (2020) no.1, 015001 [arXiv:1908.02770
+[hep-ph]].
+[61] G. J. Ding, S. F. King, X. G. Liu and J. N. Lu, JHEP 1912 (2019) 030 [arXiv:1910.03460
+[hep-ph]].
+[62] I. de Medeiros Varzielas, M. Levy and Y. L. Zhou, JHEP 11 (2020), 085 [arXiv:2008.05329
+[hep-ph]].
+[63] D. Zhang, Nucl. Phys. B 952 (2020) 114935 [arXiv:1910.07869 [hep-ph]].
+[64] T. Nomura, H. Okada and S. Patra, Nucl. Phys. B 967 (2021), 115395 [arXiv:1912.00379
+[hep-ph]].
+[65] T. Kobayashi, T. Nomura and T. Shimomura, Phys. Rev. D 102 (2020) no.3, 035019
+[arXiv:1912.00637 [hep-ph]].
+[66] J. N. Lu, X. G. Liu and G. J. Ding, Phys. Rev. D 101 (2020) no.11, 115020
+[arXiv:1912.07573 [hep-ph]].
+[67] X. Wang, Nucl. Phys. B 957 (2020), 115105 [arXiv:1912.13284 [hep-ph]].
+[68] S. J. D. King and S. F. King, JHEP 09 (2020), 043 [arXiv:2002.00969 [hep-ph]].
+[69] M. Abbas, Phys. Rev. D 103 (2021) no.5, 056016 [arXiv:2002.01929 [hep-ph]].
+27
+
+[70] H. Okada and Y. Shoji, Phys. Dark Univ. 31 (2021), 100742 [arXiv:2003.11396 [hep-ph]].
+[71] H. Okada and Y. Shoji, Nucl. Phys. B 961 (2020), 115216 [arXiv:2003.13219 [hep-ph]].
+[72] G. J. Ding and F. Feruglio, JHEP 06 (2020), 134 [arXiv:2003.13448 [hep-ph]].
+[73] H. Okada and M. Tanimoto, [arXiv:2005.00775 [hep-ph]].
+[74] H. Okada and M. Tanimoto, Phys. Rev. D 103 (2021) no.1, 015005 [arXiv:2009.14242
+[hep-ph]].
+[75] K. I. Nagao and H. Okada, JCAP 05 (2021), 063 [arXiv:2008.13686 [hep-ph]].
+[76] X. Wang, B. Yu and S. Zhou, Phys. Rev. D 103 (2021) no.7, 076005 [arXiv:2010.10159
+[hep-ph]].
+[77] H. Okada and M. Tanimoto, JHEP 03 (2021), 010 [arXiv:2012.01688 [hep-ph]].
+[78] C. Y. Yao, J. N. Lu and G. J. Ding, JHEP 05 (2021), 102 [arXiv:2012.13390 [hep-ph]].
+[79] P.
+P.
+Novichkov,
+J.
+T.
+Penedo
+and
+S.
+T.
+Petcov,
+JHEP
+04
+(2021),
+206
+doi:10.1007/JHEP04(2021)206 [arXiv:2102.07488 [hep-ph]].
+[80] P. A. Zyla et al. [Particle Data Group], PTEP 2020 (2020) no.8, 083C01
+[81] S. T. Petcov and M. Tanimoto, [arXiv:2212.13336 [hep-ph]].
+[82] T. Kobayashi, Y. Shimizu, K. Takagi, M. Tanimoto and T. H. Tatsuishi, Phys. Rev. D 100,
+no.11, 115045 (2019) [erratum: Phys. Rev. D 101, no.3, 039904 (2020)] [arXiv:1909.05139
+[hep-ph]].
+[83] K. Ishiguro, T. Kobayashi and H. Otsuka, JHEP 03, 161 (2021) [arXiv:2011.09154 [hep-
+ph]].
+[84] H. Abe, T. Kobayashi, S. Uemura and J. Yamamoto, Phys. Rev. D 102, no.4, 045005
+(2020) [arXiv:2003.03512 [hep-th]].
+[85] P. P. Novichkov, J. T. Penedo and S. T. Petcov, JHEP 03, 149 (2022) [arXiv:2201.02020
+[hep-ph]].
+[86] K. Ishiguro, H. Okada and H. Otsuka, JHEP 09, 072 (2022) [arXiv:2206.04313 [hep-ph]].
+28
+

4NFKT4oBgHgl3EQf9C5Z/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:a04ee84131e21b3d7e08811799e7a868b09995dc39fa9f5484244fcbc822bab7
+size 67112

69E1T4oBgHgl3EQfnASE/content/2301.03304v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:92410a6bb34295da4282c12a96d17744e0398b9cedfa2655f3c8b98ae978e3f4
+size 19524360

6NE1T4oBgHgl3EQfBQJe/content/tmp_files/2301.02849v1.pdf.txt

@@ -0,0 +1,1136 @@
+Temperature and density effects on the two-nucleon momentum correlation function
+from excited single nuclei
+Ting-Ting Wang(王婷婷),1 Yu-Gang Ma(马余刚),1, 2 De-Qing Fang(方德清),1, 2 and Huan-Ling Liu(刘焕玲)3
+1Key Laboratory of Nuclear Physics and Ion-Beam Application (MOE),
+Institute of Modern Physics, Fudan University, Shanghai 200433, China
+2Shanghai Research Center for Theoretical Nuclear Physics，NSFC and Fudan University, Shanghai 200438, China
+3Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China
+(Dated: January 10, 2023)
+Two-nucleon momentum correlation functions are investigated for different single thermal sources
+at given initial temperature (T) and density (ρ). To this end, the space-time evolutions of various
+single excited nuclei at T = 1 − 20 MeV and ρ = 0.2 - 1.2 ρ0 are simulated by using the ther-
+mal isospin-dependent quantum molecular dynamics (ThIQMD) model. Momentum correlation
+functions of identical proton-pairs (Cpp(q)) or neutron-pairs (Cnn(q)) at small relative momenta
+are calculated by Lednick´y and Lyuboshitz analytical method. The results illustrate that Cpp(q)
+and Cnn(q) are sensitive to the source size (A) at lower T or higher ρ, but almost not at higher
+T or lower ρ. And the sensitivities become stronger for smaller source. Moreover, the T, ρ and A
+dependencies of the Gaussian source radii are also extracted by fitting the two-proton momentum
+correlation functions, and the results are consistent with the above conclusions.
+I.
+INTRODUCTION
+Properties of nuclear matter is one of the most in-
+teresting topics in heavy-ion physics
+[1–4] and lots of
+works have been done around zero temperature, includ-
+ing the nuclear equation of state (EOS). However, the
+studies on properties of nuclear matter at finite tem-
+peratures are relatively limited.
+Many previous works
+mainly focus on the temperature dependence of hot nu-
+clear matter and the nuclear liquid-gas phase transition
+(LGPT) [5–14], the ratio between shear viscosity over
+entropy density (η/s) [15–19], as well as the nuclear gi-
+ant dipole resonance [20–22] etc. Among above works,
+the relationship between the phase transition tempera-
+ture and the source size has been investigated [5].
+In
+Ref. [5], the finite-size scaling effects on nuclear liquid-
+gas phase transition probes are investigated by study-
+ing de-excitation processes of the thermal sources by the
+isospin-dependent quantum molecular dynamics model
+(IQMD). Several probes, including the total multiplicity
+derivative, second moment parameter, intermediate mass
+fragment multiplicity, Fisher,s power-law exponent as
+well as nuclear Zipf ,s law exponent of Ma [9] were ex-
+plored, and the phase transition temperatures were then
+obtained.
+Recently, the deep neural network has also
+been used to determine the nuclear liquid gas phase tran-
+sition [23] and to estimate the temperature of excited
+nuclei by the charge multiplicity distribution of emitted
+fragments [24]. The latter work proposed that the charge
+multiplicity distribution can be used as a thermometer of
+heavy-ion collisions.
+Considering that the intermediate-state at high tem-
+perature and density in the evolution process of nuclear
+reactions cannot be directly measured, one always ex-
+plore properties of nuclear matter and the dynamical
+description of heavy-ion collisions through the analysis
+of the final-state products.
+As well known, the two-
+particle momentum correlation function in the final-state
+has been extensively used as a probe of the space-time
+properties and characteristics of the emission source [25–
+27]. The two-proton momentum correlation function has
+been explored systematically by a lot of experiments as
+well as different models, several reviews can be found
+in Refs. [28–31]. In various studies on the momentum
+correlation function, impacts of the impact parameter,
+the total momentum of nucleon pairs, the isospin of the
+emission source, the nuclear symmetry energy, the nu-
+clear equation of state (EOS) as well as the in-medium
+nucleon-nucleon cross section have been discussed in lit-
+erature [32–38].
+Even more, nuclear structure effects
+were also carefully investigated, such as the effects from
+binding energy and separation energy of the nucleus [39],
+density distribution of valence neutrons in neutron-rich
+nuclei [40], as well as high momentum tail of the nucleon-
+momentum distribution [41] etc. Two-proton momentum
+correlation function was also constructed in few-body re-
+actions as well as α-clustered nucleus induced collisions
+[42–46]. In addition, momentum correlation function be-
+tween two light charged particles also offers a unique
+tool to investigate dynamical expansion of the reaction
+zone [38].
+Here we extend the momentum correlation method
+of final-state interaction to study the time-spatial infor-
+mation of the finite-temperature nuclear systems which
+have different initial density. The purpose of the present
+paper is to systematically investigate the relationship
+between two-particle momentum correlation functions
+and system parameters, such as the source-temperature,
+density as well as system-size in a framework of the
+thermal isospin-dependent quantum molecular dynamics
+(ThIQMD) model
+[5, 14, 17]. In addition, the Gaus-
+sian source radii are quantitatively extracted by assump-
+tion of Gaussian source fits to the momentum correla-
+tion function distributions. In this article, the evolution
+process of excited nuclear sources at given initial tem-
+arXiv:2301.02849v1  [nucl-th]  7 Jan 2023
+
+2
+peratures varying from 1 MeV to 20 MeV are studied.
+The present work selects six different nuclear systems
+with similar ratio of neutron to proton numbers, i.e.,
+N/Z ∼ 1.3, which include (A, Z) = (36, 15), (52, 24),
+(80, 33), (100, 45), (112, 50), and (129, 54) nuclei. Then,
+Lednick´y-Lyuboshitz theoretical approach [47] is applied
+for calculating two-particle momentum correlation func-
+tions which are constructed based on phase-space infor-
+mation from the evolution process of single excited nu-
+clear sources by the ThIQMD model.
+The rest of this article is organized as follows. In Sec-
+tion II, we firstly describe the thermal isospin-dependent
+quantum molecular dynamics model [14, 17], then briefly
+introduce the momentum correlation technique using
+Lednick´y and Lyuboshitz analytical formalism. In Sec-
+tion III, we show the results of the ThIQMD plus
+the LL method for the source-temperature dependence
+of two-particle momentum correlation function.
+The
+two-particle momentum correlation functions of differ-
+ent system-sizes at different initial densities are sys-
+tematically discussed. The detailed analysis of the ex-
+tracted Gaussian source radii are presented under differ-
+ent source-temperature and density.
+Furthermore, the
+momentum correlation function of two-neutron is also
+analyzed. Finally, Section IV gives a summary of the
+paper.
+II.
+MODELS AND FORMALISM
+A.
+THE ThIQMD MODEL
+In this paper, the thermal isospin-dependent Quan-
+tum Molecular Dynamics transport model is used as
+the event generator, which has been applied success-
+fully to study the LGPT [5, 24].
+In the following
+discussion, we introduce this model briefly.
+As well
+known, isospin-dependent Quantum Molecular Dynam-
+ics (IQMD) model was used to describe the collision
+process between two nuclei.
+The Quantum Molecular
+Dynamics transport model is a n-body transport theory,
+which describes heavy-ion reaction dynamics from inter-
+mediate to relativistic energies [48–51]. In the present
+work, we use a single excited source in the ThIQMD
+which is different from the traditional IQMD. Usually,
+the ground state of the initial nucleus is considered to be
+T = 0 MeV in the traditional IQMD model. However,
+the ThIQMD model developed by Fang, Ma, and Zhou
+in Ref. [17] is used to simulate single thermal source at
+different temperatures and densities.
+The main parts of QMD transport model include the
+following issues: the initialization of the projectile and
+the target, nucleon propagation under the effective po-
+tential, the collisions between the nucleons in the nuclear
+medium and the Pauli blocking effect. In the ThIQMD,
+instead of using the Fermi-Dirac distribution for T = 0
+MeV with the nucleon’s maximum momentum limited
+by P i
+F (⃗r) = ℏ
+�
+3π2ρi(⃗r)
+�1/3, the initial momentum of nu-
+cleons is sampled by the Fermi-Dirac distribution at finite
+temperature:
+n (ek) =
+g (ek)
+e
+ek−µi
+T
++ 1
+,
+(1)
+where the kinetic energy ek =
+p2
+2m, p and m is the mo-
+mentum and mass of the nucleon, respectively. g (ek) =
+V
+2π2
+� 2m
+ℏ2
+� 3
+2 √ek represents the state density with the vol-
+ume of the source V = 4
+3πr3 where r = rV A
+1
+3 (rV is a
+parameter to adjust the initial density).
+In addition, the chemical potential µi is determined by
+the following equation:
+1
+2π2
+�2m
+ℏ2
+� 3
+2 � ∞
+0
+√ek
+e
+ek−µi
+T
++ 1
+dek = ρi.
+(2)
+where i = n or p refer to the neutron or proton.
+In the ThIQMD model, the interaction potential is
+also represented by the form as follows:
+U = USky + UCoul + UY uk + USym + UMDI,
+(3)
+where USky, UCoul, UY uk, USym, and UMDI are the
+density-dependent Skyrme potential, the Coulomb po-
+tential, the surface Yukawa potential, the isospin asym-
+metry potential, and the momentum-dependent interac-
+tion, respectively. Among these potentials, the Skyrme
+potential, the Coulomb potential and the momentum-
+dependent interaction can be written as follows:
+USky = α( ρ
+ρ0
+) + β( ρ
+ρ0
+)γ,
+(4)
+where ρ and ρ0 are total nucleon density and its normal
+value at the ground state, i.e., 0.16 fm−3, respectively.
+The above parameters α, β, and γ with an incompress-
+ibility parameter K are related to the nuclear equation
+of state [52–58].
+USym = Csym
+(ρn − ρp)
+ρ0
+τz,
+(5)
+UCoul = 1
+2 (1 − τz) Vc,
+(6)
+where ρn and ρp are neutron and proton densities, re-
+spectively, τz is the z-th component of the isospin degree
+of freedom for the nucleon, which equals 1 or −1 for a
+neutron or proton, respectively, and Csym is the symme-
+try energy coefficient. UCoul is the Coulomb potential
+where Vc is its parameter for protons.
+UMDI = δ · ln2 �
+ϵ · (∆p)2 + 1
+�
+· ρ
+ρ0
+,
+(7)
+where ∆p is the relative momentum, δ and ϵ can be found
+in Refs. [48, 49].
+Their values of the above potential
+parameters are all listed in Table I:
+
+3
+TABLE I. The value of the interaction potential parameters.
+α
+β
+γ
+K
+δ
+ϵ
+(MeV ) (MeV )
+(MeV ) (MeV ) ((GeV/c)−2)
+−390.1
+320.3
+1.14
+200
+1.57
+500
+B.
+LEDNICK ´Y AND LYUBOSHITZ
+ANALYTICAL FORMALISM
+Next, we briefly review the method for the two-particle
+momentum correlation function proposed by Lednick´y
+and Lyuboshitz [47, 59, 60]. The momentum correlation
+technique in nuclear collisions is based on the principle
+as follows: when they are emitted at small relative mo-
+mentum, the two-particle momentum correlation is deter-
+mined by the space-time characteristics of the production
+processes owing to the effects of quantum statistics (QS)
+and final-state interactions (FSI) [61, 62].
+Therefore,
+the two-particle momentum correlation function can be
+expressed through a square of the symmetrizied Bethe-
+Salpeter amplitude averaging over the four coordinates
+of the emitted particles and the total spin of the two-
+particle system, which represents the continuous spec-
+trum of the two-particle state.
+In this theoretical approach, the final-state interactions
+of the particle pairs is assumed independent in the pro-
+duction process. According to the conditions in Ref. [63],
+the correlation function of two particles can be written
+as the expression:
+C (k∗) =
+�
+S (r∗, k∗) |Ψk∗ (r∗)|2 d4r∗
+�
+S (r∗, k∗) d4r∗
+,
+(8)
+where r∗ = x1 − x2 is the relative distance of the two
+particles in the pair rest frame (PRF) at their kinetic
+freeze-out, k∗ is half of the relative momentum between
+two particles in the PRF, S (r∗, k∗) is the probability to
+emit a particle pair with given r∗ and k∗, i.e., the source
+emission function, and Ψk∗ (r∗) is the equal-time (t∗ = 0)
+reduced Bethe-Salpeter amplitude which can be approx-
+imated by the outer solution of the scattering problem in
+the PRF [64, 65]. This approximation is valid on condi-
+tion |t∗| ≪ m (r∗)2, which is well fulfilled for sufficiently
+heavy particles like protons or kaons and reasonably ful-
+filled even for pions [59]. In the above limit, the asymp-
+totic solution of the wave function of the two charged
+particles approximately takes the expression:
+Ψk∗ (r∗) = eiδc�
+Ac (λ)×
+�
+e−ik∗r∗F (−iλ, 1, iξ) + fc (k∗)
+˜G (ρ, λ)
+r∗
+�
+.
+(9)
+In the above equation, δc = arg Γ (1 + iλ) is the Coulomb
+s-wave phase shift with λ = (k∗ac)−1 where ac is the two-
+particle Bohr radius, Ac (λ) = 2πλ [exp (2πλ) − 1]−1 is
+the Coulomb penetration factor, and its positive (neg-
+ative) value corresponds to the repulsion (attraction).
+˜G (ρ, λ) =
+�
+Ac (λ) [G0 (ρ, λ) + iF0 (ρ, λ)] is a combina-
+tion of regular (F0) and singular (G0) s-wave Coulomb
+functions [59, 60]. F (−iλ, 1, iξ) = 1 + (−iλ) (iξ) /1!2 +
+(−iλ) (−iλ + 1) (iξ)2 /2!2 + · · · is the confluent hyperge-
+ometric function with ξ = k∗r∗ + ρ, ρ = k∗r∗.
+fc (k∗) =
+�
+Kc (k∗) − 2
+ac
+h (λ) − ik∗Ac (λ)
+�−1
+(10)
+is the s-wave scattering amplitude renormalizied by
+the long-range Coulomb interaction,
+with h (λ)
+=
+λ2 �∞
+n=1
+�
+n
+�
+n2 + λ2��−1−C −ln [λ] where C = 0.5772 is
+the Euler constant. Kc (k∗) =
+1
+f0 + 1
+2d0k∗2 +Pk∗4 +· · · is
+the effective range function, where d0 is the effective ra-
+dius of the strong interaction, f0 is the scattering length
+and P is the shape parameter. The parameters of the
+effective range function are important parameters char-
+acterizing the essential properties of the FSI, and can
+be extracted from the correlation function measured ex-
+perimentally [38, 65–67].
+For n-n momentum correlation functions which include
+uncharged particle, only the short-range particle inter-
+action works. For p-p momentum correlation functions,
+both the Coulomb interaction and the short-range parti-
+cle interaction dominated by the s-wave interaction are
+taken into account.
+III.
+ANALYSIS AND DISCUSSION
+Within
+the
+framework
+of
+the
+thermal
+isospin-
+dependent
+quantum
+molecular
+dynamics
+model
+[5,
+14, 17], the two-particle momentum correlation func-
+tions are calculated by using the phase-space infor-
+mation from the freeze-out stage of the excited nu-
+clear source at an initial temperature varying from 1
+MeV
+to 20 MeV
+and/or density varying from ρ =
+0.2ρ0 to 1.2ρ0.
+This work performs calculations for
+thermal source systems with different mass including
+(A, Z)
+=
+(36, 15), (52, 24), (80, 33), (100, 45), (112, 50),
+and (129, 54).
+We firstly calculated the proton-proton momentum
+correlation function Cpp(q) for finite-size systems at tem-
+peratures ranging from 1 to 20 MeV .
+In Fig. 1, the
+results of Cpp(q) for temperature of 2, 4, 6, 8, 10 and
+12 MeV at different values of density (0.2ρ0 - 1.2ρ0)
+are presented.
+The proton-proton momentum correla-
+tion function exhibits a peak at relative momentum q =
+20 MeV/c, which is due to the strong final-state s-wave
+attraction together with the suppression at lower rela-
+tive momentum as a result of Coulomb repulsion and the
+antisymmetrization wave function between two protons.
+The shape of the two-proton momentum correlation func-
+tions is consistent with many previous experimental data
+in heavy-ion collisions, eg. Ref. [68]. For protons which
+are emitted from the lower temperature (T < 8 MeV )
+source in Fig. 1 (a)-(c), the general trend is very similar.
+The figure shows that Cpp(q) increases as ρ increases for
+
+4
+0.5
+1.0
+1.5
+2.0
+2.5
+0.5
+1.0
+1.5
+2.0
+2.5
+20
+40
+60
+80
+0.5
+1.0
+1.5
+2.0
+2.5
+20
+40
+60
+80
+(a) T = 2.0 MeV
+ ρ = 0.2 ρ0 
+ ρ = 0.4 ρ0 
+ ρ = 0.6 ρ0 
+ ρ = 0.8 ρ0 
+ ρ = 1.0 ρ0 
+ ρ = 1.2 ρ0
+Cpp(q)
+(d) T = 8.0 MeV
+(b) T = 4.0 MeV
+(e) T = 10.0 MeV
+(c) T = 6.0 MeV
+(f) T = 12.0 MeV
+q (MeV/c)
+FIG. 1.
+The proton-proton momentum correlation function
+(Cpp(q)) at different densities (i.e., 0.2ρ0, 0.4ρ0, 0.6ρ0, 0.8ρ0,
+1.0ρ0, and 1.2ρ0) for the smaller nucleus (A=36, Z=15) with
+fixed source-temperatures T = 2 MeV (a), 4 MeV (b), 6
+MeV (c), 8 MeV (d), 10 MeV (e) and 12 MeV (f), respec-
+tively. The freeze-out time is taken to be 200 fm/c.
+fixed T (T < 8 MeV ). The increase of the density in-
+dicates that the geometrical size becomes smaller for a
+source with fixed neutrons and protons, which makes the
+strength of the momentum correlation function stronger.
+Finally, the p-p momentum correlation function becomes
+almost one at q > 60 MeV/c.
+For larger T (T > 8
+MeV ) in Fig. 1 (d)-(f), the difference of Cpp(q) between
+different densities becomes smaller.
+From Fig. 1, it is
+found that the Cpp(q) almost keep the same above T = 8
+MeV for different densities and the p-p momentum cor-
+relation function becomes almost unique above approx-
+imately q = 30 MeV/c.
+It indicates that the emitted
+proton is not affected by the change of density when the
+source temperature beyond certain value (T ≈ 8 MeV in
+present work). In order to understand which one of the
+two factors (i.e., temperature and density) has larger in-
+fluence, the two-particle momentum correlation in fig. 2
+is plotted by exchanging of the two input parameters.
+From fig. 2, we can intuitively observe dependence of the
+two-particle momentum correlation on the source tem-
+perature. The dependence of Cpp(q) on the source tem-
+perature is stronger than on density.
+In other words,
+the Cpp(q) is more sensitive to T than to density ρ. In
+0.5
+1.0
+1.5
+2.0
+2.5
+0.5
+1.0
+1.5
+2.0
+2.5
+20
+40
+60
+80
+0.5
+1.0
+1.5
+2.0
+2.5
+20
+40
+60
+80
+(d) ρ = 0.8 ρ0
+(a) ρ = 0.2 ρ0
+ Τ =  2 MeV 
+ Τ =  8 MeV
+ Τ =  4 MeV 
+ Τ =  10 MeV
+ Τ =  6 MeV 
+ Τ =  12 MeV 
+Cpp(q)
+(c) ρ = 0.6 ρ0
+(b) ρ = 0.4 ρ0
+(f) ρ = 1.2 ρ0
+(e) ρ = 1.0 ρ0
+q (MeV/c)
+ 
+FIG. 2.
+Similar to Fig. 1, but at different source-
+temperatures (T = 2, 4, 6, 8, 10 and 12 MeV ) with different
+fixed densities, namely ρ = 0.2ρ0 (a), 0.4 ρ0 (b), 0.6 ρ0 (c),
+0.8 ρ0 (d), 1.0 ρ0 (e), and 1.2ρ0 (f).
+addition, for larger ρ from fig. 2 (a) to (f), the differ-
+ence of Cpp(q) between different densities becomes bigger.
+Next, we explore whether the phenomenon exists in mo-
+mentum correlation functions for the uncharged-particle
+pairs. Fig. 3 presents the neutron-neutron momentum
+correlation functions (Cnn(q)) for temperature of 2, 4,
+6, 8, 10 and 12 MeV at different values of density, re-
+spectively. For neutron-neutron momentum correlation
+function, it peaks at q ≈ 0 MeV/c caused by the s-wave
+attraction. Although the Cnn(q) has different shape com-
+pared with the p-p momentum correlation function, it has
+the similar dependence on the source temperature and
+density. The similar trend in Cpp(q) and Cnn(q) shows
+the close emission mechanism in the evolution process.
+Fig. 4 shows the results of a larger system at differ-
+ent source-temperature and density, and a similar be-
+havior of Cpp(q) is demonstrated. We also observe that
+the proton-proton momentum correlation in larger-size
+system ((A, Z) = (129, 54)) in Fig. 4 becomes weaker
+in comparison with the smaller-size source ((A, Z) =
+(36, 15)) in Fig. 1. In view of the above phenomenon,
+Fig. 5 describes the relationship between system-size and
+momentum correlation function in more details.
+The
+decreasing of Cpp(q) as the system-size increasing for a
+fixed value of T or ρ can be clearly seen in Fig. 5 (g),
+
+5
+2
+4
+6
+8
+10
+12
+14
+2
+4
+6
+8
+10
+12
+14
+20
+40
+60
+80
+2
+4
+6
+8
+10
+12
+14
+20
+40
+60
+80
+(b) T = 4.0 MeV
+(a) T = 2.0 MeV
+ ρ = 0.2 ρ0 
+ ρ = 0.4 ρ0 
+ ρ = 0.6 ρ0 
+ ρ = 0.8 ρ0 
+ ρ = 1.0 ρ0 
+ ρ = 1.2 ρ0
+(d) T = 8.0 MeV
+(e) T = 10.0 MeV
+(c) T = 6.0 MeV
+q (MeV/c)
+Cnn(q)
+(f) T = 12.0 MeV
+FIG. 3.
+The neutron-neutron (n-n) momentum correlation
+functions (Cnn(q)) in the same conditions as Fig. 1.
+which is consistent with the previous results of Gaussian
+source [37, 38, 69]. In Fig. 5 (a)-(i), with larger temper-
+ature or lower density, the difference of Cpp(q) between
+different T or ρ becomes smaller, respectively. The Gaus-
+sian source radii are extracted for further discussion later
+in this article.
+From the above plots, we can extract Cmax(q), i.e.,
+the maximum value of Cpp(q) as well as the full width at
+half maximum (FWHM) of Cpp(q) distribution, i.e. at
+Cpp(q) = [Cmax(q)−1]/2. The source-temperature T de-
+pendence of Cmax(q) and FWHM for the proton-proton
+momentum correlation function with different density are
+given in Fig. 6. As shown in Fig. 6 (a) and (b), both
+Cmax(q) and FWHM decrease gradually with the in-
+creasing of T. In addition, both of them increase grad-
+ually with density.
+At high temperature, the change
+of Cmax(q) and FWHM is very small and not plotted
+in the figure.
+Of course, the behavior of the Cmax(q)
+and FWHM with T and ρ can also be clearly seen in
+Fig. 2, and the increasing of Cmax(q) and FWHM are
+generally inversely proportional to Gaussian radius r0 as
+shown later.
+Similarly, the system-size A dependence
+of Cmax(q) and FWHM for the proton-proton momen-
+tum correlation function at T = 2MeV and ρ = 0.6ρ0
+is shown in Fig. 7.
+The dependence of Cmax(q) and
+FWHM on system-size A is quite similar to the temper-
+ature dependence in Fig. 6. The Cmax(q) and FWHM
+0.5
+1.0
+1.5
+2.0
+2.5
+0.5
+1.0
+1.5
+2.0
+2.5
+20
+40
+60
+80
+0.5
+1.0
+1.5
+2.0
+2.5
+20
+40
+60
+80
+(a) T = 2.0 MeV
+ ρ = 0.2 ρ0 
+ ρ = 0.4 ρ0 
+ ρ = 0.6 ρ0 
+ ρ = 0.8 ρ0 
+ ρ = 1.0 ρ0 
+ ρ = 1.2 ρ0
+(d) T = 8.0 MeV
+(b) T = 4.0 MeV
+(e) T = 10.0 MeV
+(c) T = 6.0 MeV
+(f) T = 12.0 MeV
+Cpp(q)
+q (MeV/c)
+FIG. 4.
+Same to Fig. 1, but for a larger system (A=129,
+Z=54).
+values become smaller for larger systems.
+Fig. 8 shows the source-temperature, density, and
+system-size dependence of Gaussian radii extracted from
+two-particle momentum correlation functions,
+where
+panels (a) and (b) are results with the smaller source
+size and the larger source size, respectively. The radii
+are extracted by a Gaussian source assumption, i.e.,
+S(r) ≈ exp[−r2/(4r2
+0)], where r0 is the Gaussian source
+radius from the proton-proton momentum correlation
+functions.
+The theoretical calculations for Cpp(q) was
+performed by using the Lednick´y and Lyuboshitz an-
+alytical method.
+The best fitting radius is judged by
+finding the minimum of the reduced chi-square between
+the ThIQMD calculations and the Gaussian source as-
+sumption. Since the effect of the strong FSI scales as
+fc (k∗) /r∗ in Eq. (9), one may read the sensitivity of
+the correlation function to the temperature T, density
+ρ and atomic number A from their effects on the Gaus-
+sian radius r0. One may observe a linear dependence on
+these parameters up to T ≈ 8 MeV and then a lost of
+sensitivity in a plateau region at higher temperatures in
+Fig. 8. As the density decreases, the decreasing speed of
+the Gaussian radius of the small system is larger than
+that of the larger system. Fig. 9 shows the Gaussian ra-
+dius of the different system-size varies with the temper-
+ature in panels (a)-(c) or density in panels (d)-(f). The
+Gaussian source radius is consistent with the system-size,
+
+6
+0.5
+1.0
+1.5
+2.0
+0.5
+1.0
+1.5
+2.0
+40
+80
+0.5
+1.0
+1.5
+2.0
+40
+80
+40
+80
+(a)
+ρ = 1.0ρ0
+ρ = 0.6ρ0
+ρ = 0.2ρ0
+T = 6 MeV
+T = 4 MeV
+ A = 36     
+ A = 52 
+ A = 80     
+ A = 96   
+ A = 100   
+ A = 112   
+ A = 129
+ 
+ 
+ 
+ 
+T = 2 MeV
+(b)
+(c)
+(d)
+(g)
+(e)
+(f)
+(h)
+(i)
+Cpp(q)
+q (MeV/c)
+FIG. 5.
+Cpp(q) of different source size systems at fixed temperatures (i.e., from left column to right column, they correspond
+to T = 2, 4 and 6 MeV , respectively) or fixed densities (i.e., from top row to bottom row, they correspond to ρ = 0.2ρ0, 0.6ρ0,
+1.0ρ0, respectively).
+i.e., at higher temperature or larger density, the differ-
+ences of Gaussian source between different system sizes
+are bigger in the low density and low temperature region,
+but the difference in opposite conditions almost disap-
+pear. In other words, the sensitivity of the source radii
+to the system size seem to be different in the different
+regions of temperatures and densities. For example, the
+sensitivity is better in the region of lower T and higher ρ
+(Fig. 9(b) and (c)), or it is better in the higher T region
+for the lower ρ (Fig. 9(a)), or it is better in the higher ρ
+region for the lower T (Fig. 9(d)).
+From the above discussion, it is demonstrated that
+the strength of the two-particle momentum correlation
+function is affected by the source temperature, density
+and system size. The two-particle momentum correlation
+function strength is larger for a single source with lower
+temperature, higher density or smaller mass number as
+shown in Fig. 1-5.
+Otherwise, the strength becomes
+smaller. To some extents, the strong correlation between
+two particles is mainly caused by the closed position of
+each other in phase space in both coordinate and momen-
+tum. Varying only one in the three condition parameters
+(temperature, density and system size), lower temper-
+ature means smaller momentum space, higher density
+means smaller coordinate space and small system size
+also mean smaller coordinate space to keep fixed den-
+sity compared with large system size.
+The dependen-
+cies of the two-particle momentum correlation function
+strength on the source temperature, density and system
+size could be explained by the change of the phase space
+sizes. Two particles emitted from small phase space will
+have strong correlation and those from large phase space
+will have weak correlation. For example, the increase of
+the Cpp(q) strength with the increase of the density for
+
+7
+1.0
+1.5
+2.0
+2.5
+2
+3
+4
+5
+6
+4
+6
+8
+10
+12
+(a)
+ ρ = 0.2 ρ0 
+ ρ = 0.4 ρ0 
+ ρ = 0.6 ρ0 
+ ρ = 0.8 ρ0 
+ ρ = 1.0 ρ0 
+ ρ = 1.2 ρ0
+Cmax(q)
+(b)
+FWHM (Mev/c)
+T (MeV)
+FIG. 6.
+Source-temperature T dependencies of Cmax(q)
+(a) and of FWHM (b) of Cpp(q) distributions at different
+densities (0.2ρ0 - 1.2ρ0) for the (A = 35, Z = 16) system.
+1.4
+1.6
+1.8
+40
+60
+80
+100
+120
+6.0
+6.5
+7.0
+7.5
+8.0
+8.5
+(a)
+y = b+a*x    value
+    a
+       -0.0055
+    b
+        2.0358
+Cmax(q)
+ Cmax(q)
+ Fitting curve
+(b)
+y = b+a*x    value
+    a
+       -0.022
+    b
+        8.913
+A
+FWHM (Mev/c)
+ FWHM
+ Fitting curve
+FIG. 7.
+Cmax(q) (a) and FWHM (b) for different source-
+size systems at given T = 2 MeV and ρ = 0.6ρ0.
+2
+3
+4
+5
+6
+7
+4
+8
+12
+16
+20
+2
+3
+4
+5
+6
+7
+(a) A=36 Z=15
+r0 (fm)
+(b) A=129 Z=54
+ ρ = 0.2 ρ0 
+ ρ = 0.4 ρ0
+ ρ = 0.6 ρ0 
+ ρ = 0.8 ρ0
+ ρ = 1.0 ρ0 
+ ρ = 1.2 ρ0
+T (MeV)
+FIG. 8.
+Gaussian source radius as a function of temperature
+at different densities (ρ = 0.2ρ0, 0.4ρ0, 0.6ρ0, 0.8ρ0, 1.0ρ0,
+1.2ρ0) for a fixed source size. Panel (a) and (b) correspond to
+the smaller source size with (A = 36, Z = 15) and the larger
+source size with (A = 129, Z = 54), respectively.
+a fixed system size could be explained by the decreasing
+of the coordinate space as shown in Fig. 1 (a). And the
+small Cpp(q) strength at temperature higher than 8 MeV
+could be caused by the large momentum space compared
+with lower temperatures as shown in Fig. 1 (d-f). The
+decrease of the Cpp(q) strength with the increase of the
+system size for a fixed density could also be explained by
+the increasing of the coordinate space as shown in Fig. 5
+(g). Thus it is concluded that the phase space size for
+the emitted nucleons have strong effect on strength of the
+two-particle momentum correlation function, which can
+also be seen in the extracted Gaussian radii as shown in
+Fig. 8.
+IV.
+SUMMARY
+In summary, the two-particle momentum correlation
+functions for single excited sources are investigated using
+the Lednick´y and Lyuboshitz analytical formalism with
+the phase-space information at the freeze-out stage for
+different initial temperatures and densities in a frame-
+work of the ThIQMD transport approach. We mainly
+performed a series of studies focusing on the varied ef-
+fects of source temperature, density and system-size on
+the two-particle momentum correlation functions. The
+results reflect that the shape of the two-proton momen-
+
+8
+2
+3
+4
+5
+6
+7
+2
+3
+4
+5
+6
+7
+4
+8
+12
+16
+20
+2
+3
+4
+5
+6
+7
+0.2 0.4 0.6 0.8 1.0 1.2
+(a) ρ = 0.2 ρ0
+ A = 36    
+ A = 52 
+ A = 80    
+ A = 96 
+ A = 100 
+ A = 112
+ A = 129
+(d) T = 2 MeV
+(c) ρ = 1.0 ρ0
+(b) ρ = 0.6 ρ0
+r0 (fm)
+(e) T = 6 MeV
+(f) T = 10 MeV 
+T (MeV)
+ρ / ρ0
+FIG. 9.
+Gaussian source radius as a function of temperature
+or density at different source-size systems.
+Left and right
+columns correspond to r0 at different densities, i.e ρ = 0.2 ρ0
+(a), 0.6 ρ0 (b), and 1.0 ρ0 (c) as well as different temperatures,
+i.e. T = 2 (d), 6 (e), and 10 (f) MeV , respectively.
+tum correlation function is in accordance with the previ-
+ous experimental data in heavy-ion collisions [68].
+At
+the same time, the trend of the relationship between
+the two-proton momentum correlation and system-size
+is consistent with previous simulations [37, 38, 69]. At
+low source-temperature, the larger density makes the
+two-particle momentum correlation stronger. However,
+at higher source temperature, the effect becomes almost
+disappear. Both proton-proton correlations and neutron-
+neutron correlations have the similar responses to tem-
+perature and density.
+This work also shows that the
+emission source is not much influenced by density above
+a certain temperature for a single excited source. In the
+same way, the emission source are softly influenced by
+temperature below a given density for a single excited
+source. In one word, the dependence of the two-particle
+momentum correlation function on the source tempera-
+ture, density and system size could be explained by the
+change of the coordinate and/or momentum phase space
+sizes. In the end, the Gaussian radii are extracted to ex-
+plore the emission source sizes in single excited systems.
+Gaussian radii become larger in the larger systems. The
+dependence of the extracted Gaussian radius on source-
+temperature and density is consistent with behavior of
+the two-proton momentum correlation function as dis-
+cussed in the texts.
+ACKNOWLEDGMENTS
+This work was supported in part by the National Nat-
+ural Science Foundation of China under contract Nos.
+11890710, 11890714, 11875066, 11925502, 11961141003,
+11935001, 12147101 and 12047514, the Strategic Pri-
+ority Research Program of CAS under Grant No.
+XDB34000000, National Key R&D Program of China un-
+der Grant No. 2016YFE0100900 and 2018YFE0104600,
+Guangdong Major Project of Basic and Applied Basic
+Research No. 2020B0301030008, and the China PostDoc-
+toral Science Foundation under Grant No. 2020M681140.
+[1] G. Giuliani, H. Zheng, and A. Bonasera, Prog. Part.
+Nucl. Phys. 76, 116 (2014).
+[2] B. A. Li, L. W. Chen, and C. M. Ko, Phys. Rep. 464,
+113 (2008).
+[3] B. Borderie and M. F. Rivet, Prog. Part. Nucl. Phys. 61,
+551 (2008).
+[4] C. W. Ma and Y.-G. Ma, Prog. Part. Nucl. Phys. 99,
+120(2018).
+[5] H. L. Liu, Y. G. Ma, and D. Q. Fang, Phys. Rev. C 99,
+054614 (2019).
+[6] J. E. Finn, S. Agarwal, A. Bujak et al., Phys. Rev. Lett.
+49, 1321 (1982).
+[7] P. J . Siemens, Nature 305, 410 (1983).
+[8] Y. G. Ma, A. Siwek, J. P´eter et al., Phys. Lett. B 390,
+41 (1997).
+[9] Y. G. Ma, Phys. Rev. Lett. 83, 3617 (1999).
+[10] P. Chomaz, V. Duflot, and F. Gulminelli, Phys. Rev.
+Lett. 85, 3587 (2000).
+[11] J. B. Natowitz, K. Hagel, Y. G. Ma, M. Murray, L. Qin,
+R. Wada, J. Wang, Phys. Rev. Lett. 89, 212701 (2002).
+[12] Y. G. Ma, J.B. Natowitz, R. Wada et al., Phys. Rev. C
+71, 054606 (2005).
+[13] B. Borderie, J. Frankland, Prog. Part. Nucl. Phys. 105,
+82 (2019).
+[14] Zhen-Fang Zhang, De-Qing Fang, and Yu-Gang Ma,
+Nucl. Sci. Tech. 29, 78 (2018).
+[15] N. Auerbach, S. Shlomo, Phys. Rev. Lett. 103, 172501
+(2009).
+[16] N. D. Dang, Phys. Rev. C 84, 034309 (2011).
+[17] D. Q. Fang, Y. G. Ma, C. L. Zhou, Phys. Rev. C 89,
+047601 (2014).
+[18] X. G. Deng, Y. G. Ma, M. Veselsk´y, Phys. Rev. C 94,
+044622 (2016).
+[19] D. Mondal, D. Pandit, S. Mukhopadhyay et al., Phys.
+Rev. Lett. 118, 192501 (2017).
+
+9
+[20] A. Bracco, J. J. Gaardhje, A. M. Bruce et al., Phys. Rev.
+Lett. 62, 2080 (1989).
+[21] P. F. Bortignon, A. Bracco, D. Brink, R. A. Broglia,
+Phys. Rev. Lett. 67, 3360 (1991).
+[22] O. Wieland, A. Bracco, F. Camera et al., Phys. Rev.
+Lett. 97, 012501 (2006).
+[23] Rui Wang, Yu-Gang Ma, R. Wada et al., Phys. Rev. Res.
+2, 043202 (2020).
+[24] Y. D. Song, R. Wang, Y. G. Ma, X. G. Deng, H. L. Liu,
+Phys. Lett. B 814, 136084 (2021).
+[25] R. Kotte et al., Eur. Phys. J. A 23, 271 (2005).
+[26] R. Ghetti et al., Phys. Rev. Lett. 91, 092701 (2003).
+[27] D. Gourio et al., Eur. Phys. J. A 7, 245 (2000).
+[28] D. H. Boal, C. K. Gelbke, B. K. Jennings, Rev. Mod.
+Phys. 62, 553 (1990).
+[29] U. A. Wiedemann, U. Heinz, Phys. Rep. 319, 145 (1999).
+[30] M. A. Lisa, S. Pratt, R. Soltz, U. Wiedemann, Ann. Rev.
+Nucl. Part. Sci. 55, 357 (2005).
+[31] G. Verde, A. Chbihi, R. Ghetti, J. Helgesson, Eur. Phys.
+J. A 30, 81 (2006).
+[32] W. G. Gong, W. Bauer, C. K. Gelbke, and S. Pratt, Phys.
+Rev. C 43, 781 (1991).
+[33] G. Verde, D. A. Brown, P. Danielewicz, C. K. Gelbke,
+W. G. Lynch, and M. B. Tsang, Phys. Rev. C 65, 054609
+(2002).
+[34] Y. G. Ma et al., Phys. Rev. C 73, 014604 (2006).
+[35] R. Ghetti et al., Phys. Rev. C. 69, 031605(R) (2004).
+[36] L. W. Chen, V. Greco, C. M. Ko, and B. A. Li, Phys.
+Rev. Lett. 90, 162701 (2003).
+[37] Ting-Ting Wang, Yu-Gang Ma, Chun-Jian Zhang and
+Zheng-Qiao Zhang, Phys. Rev. C 97, 034617 (2018).
+[38] Ting-Ting Wang, Yu-Gang Ma, and Zheng-Qiao Zhang,
+Phys. Rev. C 99, 054626 (2019).
+[39] Y. B. Wei, Y. G. Ma, W. Q. Shen et al., Phys. Lett. B
+586, 225 (2004).
+[40] X. G. Cao, X. Z. Cai, Y. G. Ma, D. Q. Fang, G. Q. Zhang,
+W. Guo, J. G. Chen, and J. S. Wang, Phys. Rev. C 86,
+044620 (2012).
+[41] Gao-Feng Wei, Xi-Guang Cao, Qi-Jun Zhi, Xin-Wei Cao,
+and Zheng-Wen Long, Phys. Rev. C 101, 014613 (2020).
+[42] B. S. Huang, Y. G. Ma, Phys. Rev. C 101, 034615 (2020).
+[43] D. Q. Fang, Y. G. Ma, X. Y. Sun et al., Phys. Rev. C
+94, 044621 (2016).
+[44] Bo-Song Huang, Yu-Gang Ma, Chin. Phys. C 44, 094105
+(2020).
+[45] L. Shen, B. S. Huang, Y. G. Ma, Phys. Rev. C 105,
+014603 (2022).
+[46] J. J. He, S. Zhang, Y. G. Ma, J. H. Chen, C. Zhong, Eur.
+Phys. J. A 56, 52 (2020).
+[47] R. Lednick´y, Nucl. Phys. A 774, 189 (2006).
+[48] J. Aichelin, A. Rosenhauer, G. Peilert, H. Stoecker, and
+W. Greiner, Phys. Rev. Lett. 58, 1926 (1987).
+[49] J. Aichelin, Phys. Rep. 202, 233 (1991).
+[50] G. Peilert, H. St¨ocker, W. Greiner, A. Rosenhauer, A.
+Bohnet, and J. Aichelin, Phys. Rev. C 39, 1402 (1989).
+[51] Zhao-Qing Feng, Nucl. Sci. Tech. 29, 40 (2018).
+[52] Bao-An Li, Lie-Wen Chen, Che Ming Ko, Phys. Rep.
+464, 113 (2008).
+[53] J. Liu, C. Gao, N. Wan and C. Xu, Nucl. Sci. Tech. 32,
+117 (2021).
+[54] Fan Zhang, Jun Su, Nucl. Sci. Tech. 31, 77 (2020) .
+[55] Hao Yu, De-Qing Fang, Yu-Gang Ma, Nucl. Sci. Tech.
+31, 61 (2020).
+[56] Gao-Feng Wei, Qi-Jun Zhi, Xin-Wei Cao, Zheng-Wen
+Long, Nucl. Sci. Tech. 31, 71 (2020) .
+[57] Chong-Ji Jiang, Yu Qiang, Da-Wei Guan et al., Chin.
+Phys. Lett. 38, 052101 (2021) .
+[58] Jun Xu, Chin. Phys. Lett. 38, 042101 (2021) .
+[59] R. Lednick´y, Phys. Part. Nucl. 40, 307 (2009).
+[60] R. Lednick´y, Phys. Ato. Nucl. 71, 1572 (2008).
+[61] S. E. Koonin, Phys. Lett. B 70, 43 (1977).
+[62] R. Lednick´y, Braz. J. Phys. 37, 939 (2007).
+[63] R. Lednick´y, V.L. Lyuboshitz, B. Erazmus, and D.
+Nouais, Phys. Lett. B 373, 30 (1996).
+[64] R. Lednick´y, V.L. Lyuboshitz, Sov. J. Nucl. Phys. 35,
+770 (1982).
+[65] L. Adamczyk et al. (STAR Collaboration), Nature 527,
+345 (2015).
+[66] B. Erazmus, L. Martin, R. Lednicky, N. Carjan, Phys.
+Rev. C 49, 349 (1994).
+[67] J. Arvieux, Nucl. Phys. A 221, 253 (1974).
+[68] R. Ghetti et al., Nucl. Phys. A 674, 277 (2000).
+[69] Long Zhou, De-Qing Fang, Nucl. Sci. Tech. 21, 52 (2020).
+

7dAyT4oBgHgl3EQfQvYd/content/2301.00050v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:4e41c97a3da87f0414c27724fd1389402c049091f24a13bd9415addc9df8be98
+size 10181532

7dAyT4oBgHgl3EQfQvYd/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b089c632e6e647d666510f9fbcde81cbc4b72d865917fbc9a5d5da347b50b7cf
+size 3735597

8NFLT4oBgHgl3EQfsy_c/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:10fb9b2e9b175aae52d32c6fe1ea726ead27e25f3f8c53495e6f4110e19b55c0
+size 152610

8dE3T4oBgHgl3EQfRwni/content/tmp_files/2301.04426v1.pdf.txt

@@ -0,0 +1,2081 @@
+Flagged observation analyses as a tool for scoping and communication in 
+Integrated Ecosystem Assessments   
+ 
+Solvang, Hiroko Kato 
+Institute of Marine Research, P.O. Box 1870 Nordnes, N-5817 Bergen, Norway 
+[email protected] 
+Arneberg, Per 
+Institute of Marine Research, Fram Centre, P.O. Box 6606, 9296 Langnes, Norway 
+[email protected] 
+ 
+Abstract 
+Working groups for integrated ecosystem assessments are often challenged with understanding and 
+assessing recent change in ecosystems. As a basis for this, the groups typically have at their disposal 
+many time series and will often need to prioritize which ones to follow up for closer analyses and 
+assessment. Here we provide a procedure termed Flagged observation analyses that can be applied to 
+all the available time series to help identifying time series that should be prioritized. The statistical 
+procedure first applies a structural time series model including a stochastic trend model to the data to 
+estimate the long-term trend. The model adopts a state space representation, and the trend component 
+is estimated by a Kalman filter algorithm. The algorithm obtains one- or more-years-ahead prediction 
+values using all past information from the data. Thus, depending on the number of years the investigator 
+wants to consider as “the most recent”, the expected trend for these years is estimated through the 
+statistical procedure by using only information from the years prior to them. Forecast bands are 
+estimated around the predicted trends for the recent years, and in the final step, an assessment is made 
+on the extent to which observations from the most recent years fall outside these forecast bands. Those 
+that do, may be identified as flagged observations. A procedure is also presented for assessing whether 
+the combined information from all of the most recent observations form a pattern that deviates from the 
+predicted trend and thus represents an unexpected tendency that may be flagged. In addition to form the 
+basis for identifying time series that should be prioritized in an integrated ecosystem assessment, 
+flagged observations can provide the basis for communicating with managers and stakeholders about 
+recent ecosystem change. Applications of the framework are illustrated with two worked examples. 
+ 
+Keywords: Trend estimation, Kalman filter, Prediction, Forecast band, Stakeholder, Climate change
+ 
+ 
+ 
+Introduction  
+Against a background of increasing impact from climate change and other anthropogenic drivers, 
+causing elevated rates of change in marine ecosystems [1-6] leading to patterns of variability beyond 
+the range of the Holocene [7-12], ecosystem-based management (EBM) is increasingly identified as a 
+needed framework for management of marine socio-ecological systems [13]. Integrated ecosystem 
+
+assessments (IEA) have been developed to provide the scientific basis for EBM [14], and numerous 
+groups of scientists working with IEA have been established, such as the regional IEA groups within 
+the International Council for the Exploration of the Sea (ICES, Walther and Möllmann (15)).  
+Among the core activities of IEA groups are analyses of time series to summarize changes that have 
+occurred in recent decades in ecosystems, highlight possible connections between physical, biological, 
+and human ecosystem components [14, 16]. Emphasis is put on keeping an open communication 
+management and stakeholders [13, 17, 18]. As the groups typically have at their disposal a large number 
+of time series [16, 19], it will often be necessary to prioritize a subset of them for more extensive 
+analyses and communication purposes [20, 21]. Prioritization should preferably be done using a 
+standardized framework applied to all time series. Here we present an approach which is based analyses 
+of patterns of recent change, where the aim is to identify time series in which the most recent values 
+deviate significantly from an expected trend, possibly indicating unexpected change. This should be of 
+high relevance for IEA groups, as they are often challenged with understanding and interpreting recent 
+change [14].  
+Our approach is based on first estimating trends of time series before assessing whether the most recent 
+observations deviate significantly from these trends. Since temporal changes in ecosystems can take the 
+form of long-term movements as well as short- or mid-term cyclic periods and noise components, 
+different definitions of trends have been used in marine IEAs [16]. In the field of statistical time series 
+analysis, the long-term movements are commonly classified as ‘trends’, while short- or mid-term cyclic 
+periods are not, due to the different assumptions about the statistical properties. When investigating a 
+trend in time series data, it can therefore be useful to separately identify non-stationary trends and 
+stationary cyclic components. This decomposition is performed by a framework called ‘structural time 
+series modelling’, which is using a state space representation where the state of each component is 
+estimated by the Kalman filter algorithm [22]. The concept is basically different from applying an 
+Autoregressive integrated moving average model to adjust with the aim of studying stationary processes 
+from nonstationary time series data [23]. 
+The Kalman filter algorithm can make one- or multistep-ahead predictions in the numerical procedure. 
+The numerical procedure introduced in this paper uses prediction values and forecast uncertainty bands 
+
+to assess the status of a recent observation, which determines whether the most recent observation 
+follows the prediction or deviates from it [24], thus giving an indication whether change that is 
+unexpected from the predicted trend, is occurring in the time series. We call the significant deviated 
+observations a “Flagged Observation (FO)” and the approach “Flagged Observation analysis”. The 
+interpretation of a FO is not equivalent to the types of early warning signals that have been proposed 
+with the aim of predicting critical transitions in marine populations or ecosystems [25, 26], nonlinear 
+ecological change [27] or early warning signs based on theoretical framework in social-ecological 
+networks [28], but is, as described above, a practical tool for IAE groups for prioritizing time series for 
+in depth analyses, communication and other purposes.  
+In this article, we first introduce the numerical procedure and then demonstrate two examples using 
+time series data for, respectively, the Atlantic Multi-decadal Oscillation (AMO) and the Norwegian Sea 
+ecosystem.  
+ 
+Statistical method  
+The statistical method includes first a procedure for trend estimation and second a procedure for flagged 
+observations analyses (FO analyses) based on multistep ahead prediction values. The output from these 
+analyses can be used to identify observations (for example years) that deviate significantly from the 
+expected trend. In addition, it can also be interesting to explore whether observations from the predicted 
+years together form a pattern where all are consecutively either above or below the predicted trend in a 
+way that is not expected. This is equivalent to asking whether there is an unexpected tendency for the 
+most recent years. The probability for observing this is smaller than the probability of detecting an FO 
+for a single observation. The details are given as below: 
+Trend estimation procedure 
+The observation model of a time series is given by 
+( )
+( )
+( ),
+1,
+,
+y n
+t n
+u n
+n
+N
+=
++
+=
+⋅⋅⋅
+   , 
+                                                                        (1) 
+
+where ( )
+t n is the trend component and ( )
+u n is the residual component at time step n, assuming Gaussian 
+white noise. In this article, we introduce a stochastic trend model given by a dth-order difference 
+equation model and a method for estimating the trend [29, 30]. 
+The stochastic differential trend model is defined by the dth-order difference equation, which was posed 
+as a smoothing problem by [31]. This model allows for more flexible trends than does the polynomial 
+regression model. The stochastic trend model for k variables is expressed in the following way: 
+( )
+( ),
+1,
+, ,
+i
+d
+i
+t
+t n
+v n
+i
+k
+∇
+=
+=
+⋅⋅⋅
+                                                                                                   (2) 
+where ∇ is a difference operator 
+( )
+( )
+(
+1)
+t n
+t n
+t n
+∇
+=
+−
+−
+ and 
+( )
+itv n  is assumed to be a white noise 
+sequence. If 
+1
+d =
+, ( )
+(
+1)
+t n
+t n
+≈
+−
+ and the trend is known as a random walk model. If 
+2
+d =
+, 
+( )
+2 (
+1)
+(
+2)
+0
+t n
+t n
+t n
+−
+−
++
+−
+≈
+ [29]. Provided that the variance of 
+( )
+itv n  is sufficiently small, ( )
+it n  
+yields a smooth trend. We choose the second order difference stochastic model to estimate trend in this 
+study. 
+The model can be represented in linear state space form [22, 29, 30], as 
+( )
+(
+1)
+( ),
+( )
+( )
+( ),
+z n
+Fz n
+Gv n
+y n
+Hz n
+w n
+=
+−
++
+=
++
+ 
+                                                                                                     (3) 
+where ( )
+z n is the state vector corresponding to ( )
+t n , ( )
+v n  is the system noise vector with mean 0 and 
+unknown variance 
+2
+v
+σ , F ,G , and H indicate integers or matrices, and 
+( )
+w n  is observation error 
+with mean 0 and unknown variance 
+2
+w
+σ . The state ( )
+z n corresponds to the trend component, which we 
+cannot directly observe from the data. The component is modelled by the dth-order differential equation 
+model. Corresponding to the above dth-order difference equation, when 
+1
+d =
+ in equation (2),  
+( )
+( )
+t n
+z n
+
+
+
+
+=
+, 
+1
+F
+G
+H
+=
+=
+=  . 
+ 
+When 
+2
+d =
+, the state vector and matrices are as follows: 
+
+( )
+( )
+(
+1)
+t n
+z n
+t n
+
+
+= 
+
+−
+
+ ,
+2
+1
+1
+0
+F
+−
+
+
+= 
+
+
+,
+1
+0
+G
+ 
+=  
+   , and 
+1
+0
+H
+ 
+=  
+���   . 
+ 
+The observation error corresponds to ( )
+u n in equation (1) in this case. The trend component is estimated 
+with a Kalman filter, which is a powerful numerical algorithm that recursively operates the state 
+estimation, prediction and filtering:  
+prediction: 
+( |
+1)
+(
+1|
+1)
+( |
+1)
+(
+1|
+1)
+'
+z n n
+Fz n
+n
+V n n
+FV n
+n
+GQG
+−
+=
+−
+−
+−
+=
+−
+−
++
+                                                                                        (4) 
+filtering: 
+1
+( )
+( |
+1)
+'(
+( |
+1)
+'
+)
+( | )
+( |
+1)
+( ( )
+( |
+1))
+( | )
+(
+) ( |
+1)
+K n
+V n n
+H
+H V n n
+H
+R
+z n n
+z n n
+K y n
+H z n n
+V n n
+I
+K H V n n
+−
+=
+−
+−
++
+=
+−
++
+−
+−
+=
+−
+−
+                                                                          (5) 
+Here, ( |
+1)
+z n n −
+ and 
+( |
+1)
+V n n −
+ correspond to the conditional mean and conditional variance of the 
+state, R  is set as the observation error, and K  is called the Kalman gain. Setting the initial state 
+(
+)
+1| 0
+z
+ and variance 
+(
+)
+1| 0
+V
+as zero and the pre-determined system noise, the Kalman gain is 
+calculated by the initial variance and observation noise, and the filtering value (
+)
+1|1
+z
+ is obtained by 
+using observation 
+( )
+1
+y
+and the calculated gain from the filtering procedure (5). Then the next 
+prediction values (
+)
+2 |1
+z
+and 
+(
+)
+2 |1
+V
+are calcualted using (
+)
+1|1
+z
+ and 
+(
+)
+1|1
+V
+in the prediction 
+procedure (4). The iterative calculation procedure for the state is continued until n
+N
+=
+. Recursive 
+methods based on state space representations are known to be very efficient for calculating the 
+likelihood functions of discrete-time Gaussian proceeses. The state space model and the Kalman filter 
+provide an efficient method for the computation of the likelihood of the time series models [29]. In this 
+case, the trend model includes the parameter vector 
+2
+2
+( ,
+,
+)
+v
+w
+d
+θ
+σ
+σ
+=
+. The log-likelihood function ( )
+l θ  
+of the model is given by 
+
+1
+1
+1
+2
+( )
+log
+( ( ) |
+(
+1), ),
+1
+1
+log
+exp
+( )' ( )
+( )
+,
+2
+(2 ) det ( )
+N
+n
+N
+n
+l
+f y n
+Y n
+y n
+n
+y n
+n
+θ
+θ
+π
+=
+−
+=
+=
+−
+
+
+
+
+
+
+=
+−
+∆
+Σ
+∆
+
+
+
+
+
+
+Σ
+
+
+
+
+∑
+∑
+  
+where 
+(
+1)
+( (1), (2),
+, (
+1))
+Y n
+y
+y
+y n
+−
+=
+⋅⋅⋅
+−
+,
+( )
+( )
+( |
+1)
+y n
+y n
+Hz n n
+∆
+=
+−
+−
+, 
+and 
+2
+( )
+( )
+( |
+1)
+'( )
+w
+n
+H n V n n
+H n
+σ
+Σ
+=
+−
++
+. The flexibility of the estimated trend depends on 
+2
+v
+σ , which can be 
+determined by maximum likelihood within an arbitry variance range. The variance 
+2
+w
+σ  can be directly 
+set to the variance of the observation. If it is necessary to compare differernt order differential stochastic 
+model, the optimum differential order d is identified by the AIC [32] , which was formulated by the 
+maximum log-likelihood and number of parameters for d and the variances of system noise and 
+observation noise, given by AIC( )
+2 ( )
+2
+number of parameters
+c
+l θ
+= −
++
+×
+ . 
+After identifying the optimum trend model, the smoothed trend is estimated by a fixed-interval 
+smoother algorithm [33]: 
+1
+( )
+( | )
+' (
+1| )
+( |
+)
+( | )
+( )( (
+1|
+)
+(
+1| ))
+( |
+)
+( | )
+( )( (
+1|
+)
+(
+1| )) ( )'
+A n
+V n n F V n
+n
+z n N
+z n n
+A n
+z n
+N
+z n
+n
+V n N
+V n n
+A n V n
+N
+V n
+n A n
+−
+=
++
+=
++
++
+−
++
+=
++
++
+−
++
+ 
+In this study, we set the differential order as 
+2
+d =
+to obtain smooth trend for all time series data as 
+introduced in Kitagawa and Gersch (33) and take a procedure finding an optimum Q controlling 
+variability for the trend estimation within a range. 
+ 
+FO analysis by multistep-ahead prediction values 
+Using the Kalman filter algorithm directly, it can give only one-ahead prediction. However, it can be 
+expanded to multistep-ahead (j-ahead) prediction (
+1
+j >
+). Let us consider a relevant situation. With the 
+Kalman filter, one-ahead prediction for (
+)
+1
+z n +
+is obtained by (
+)
+1|
+z n
+n
++
+ and variance (
+)
+1|
+V n
+n
++
+. 
+If the data
+(
+)
+1
+y n +
+are not observed, the calculation is formally conducted by assuming 
+
+(
+)
+( )
+1
+Y n
+Y n
++
+=
+, 
+where 
+(
+)
+( )
+(1), (2),
+, ( )
+Y n
+y
+y
+y n
+=
+⋅⋅⋅
+. 
+Accordingly, 
+it 
+is 
+clear 
+that 
+(
+)
+(
+)
+1|
+1
+1|
+z n
+n
+z n
+n
++
++
+=
++
+and
+(
+)
+(
+)
+1|
+1
+1|
+V n
+n
+V n
+n
++
++
+=
++
+. Then, two-ahead prediction 
+(
+)
+2 |
+z n
+n
++
+and variance 
+(
+)
+2 |
+V n
+n
++
+ are obtained by (
+)
+1|
+z n
+n
++
+ and (
+)
+1|
+V n
+n
++
+. In general, we 
+assume ( )
+(
+1)
+(
+)
+Y n
+Y n
+Y n
+j
+=
++
+= ⋅⋅⋅ =
++
+ to obtain the j-ahead prediction, and the prediction step is 
+iteratively conducted j times. The algorithm used to predict states (
+)
+(
+)
+(
+)
+1 ,
+2 ,
+,
+z n
+z n
+z n
+j
++
++
+⋅⋅⋅
++
+, 
+based on the data ( )
+Y n  observed until time point n , is expressed as follows: 
+(
+)
+(
+)
+(
+)
+(
+)
+|
+1|
+,
+|
+1|
+'
+',
+1,
+, .
+z n
+i n
+Fz n
+i
+n
+V n
+i n
+FV n
+i
+n F
+GQG
+i
+j
++
+=
++ −
++
+=
++ −
++
+=
+⋅⋅⋅
+                         
+Now, let the mean and variance for the prediction (
+)
+y n
+j
++
+ of the data denote  
+(
+)
+( )
+(
+)
+E
+|
+y n
+j
+Y n
++
+and 
+(
+)
+( )
+(
+)
+Cov
+|
+y n
+j
+Y n
++
+, where 
+( )
+E ⋅  and 
+( )
+Cov ⋅  are notations for expectation and variance-
+covariance matrix (but in this study only variance because the data are univariate). Using the 
+observation equation (3), the mean of (
+)
+y n
+j
++
+ is expressed by 
+(
+)
+(
+)
+(
+)
+( )
+(
+)
+(
+)
+|
+=E
+|
+|
+.
+y n
+j n
+Hz n
+j
+w n
+j
+Y n
+Hz n
+j n
++
++
++
++
+=
++
+                                            (6) 
+The variance of (
+)
+y n
+j
++
+is given by 
+(
+)
+(
+)
+(
+)
+( )
+(
+)
+(
+)
+( )
+(
+)
+(
+)
+(
+)
+( )
+(
+)
+(
+)
+(
+)
+( )
+(
+)
+(
+)
+( )
+(
+)
+(
+)
+(
+)
+|
+=Cov
+|
+,
+Cov
+|
+'
+Cov
+,
+|
+Cov
+,
+|
+' Cov
+|
+,
+|
+'
+.
+d n
+j n
+Hz n
+j
+w n
+j
+Y n
+H
+z n
+j
+Y n
+H
+H
+z n
+j
+w n
+j
+Y n
+w n
+j
+z n
+j
+Y n
+H
+w n
+j
+Y n
+H V n
+j n H
+R n
+j
++
++
++
++
+=
++
++
++
++
++
++
++
++
++
+=
++
++
++
+       (7) 
+Therefore, the prediction distribution for (
+)
+y n
+j
++
+ based on data ( )
+Y n  is a normal distribution with 
+mean (
+)
+|
+y n
+j n
++
+ and variance 
+(
+)
+|
+d n
+j n
++
+or a standard deviation 
+(
+)
+|
+d n
+j n
++
+. The forecast 
+bands (FBs), e.g. mean ± 1.96 (~2) × standard deviation that corresponds to around 95-96% forecast 
+interval [34], are easily calculated by equations (6) and (7). Note that we define the values given by 
+
+multistep-ahead prediction procedure as ‘forecast value’ because it is calculated by using the previous 
+data in the algorithm.  
+Assessing unexpected tendencies - joint probability for detected FO 
+As the time series data is sampled as one sample at one time point, the joint probability that all of the 
+most recent years fall either above or below the predicted trend (i.e., whether there is an unexpected 
+tendency for the most recent years) should be calculated by random variables, which is generated from 
+a normal distribution. We provide the following procedure to calculate the joint probabilities in this 
+way: 
+1. Generate 10000 random number set 
+jr  by a normal distribution with mean (
+| )
+y n
+j n
++
+ and 
+variance (
+| )
+d n
+j n
++
+of the prediction values at n
+j
++
+; 
+2. The probability value 
+jp is calculated by the formula 
+        
+#{
+(
+| )}     if  (
+| ) is upper over FB
+10000
+#{
+(
+| )}     if  (
+| ) is lower under FB
+10000
+j
+j
+j
+r
+y n
+j n
+y n
+j n
+p
+r
+y n
+j n
+y n
+j n
+
+≤
++
+
++
+
+= 
+≥
++
+
++
+
+ ; 
+3. The joint probability values 
+1
+(
+,
+,
+)
+J
+P p
+p
+L
+are calculated for 
+1,
+,
+j
+J
+= L
+by 
+        
+1
+1
+(
+,
+,
+)
+J
+J
+P p
+p
+p
+p
+=
+×
+×
+L
+L
+; and     
+4. The procedure from 1 to 3 is iterated for 1000 and the averaged joint probability values are 
+calculated by 
+         
+1000
+1
+1
+1
+1
+(
+,
+,
+)
+(
+,
+,
+)
+1000
+J
+b
+J
+b
+P p
+p
+P p
+p
+=
+=
+∑
+L
+L
+. 
+ 
+A conceptual outline of this study’s analysis procedure for FO analysis is given in Fig 1. The numerical 
+procedure is implemented using MATLAB code [35], which is summarized in Supplementary folder. 
+Illustrative examples 
+The dataset for the Atlantic Multi-decadal Oscillation (AMO) is based on index monthly raw data [36], 
+while for the Norwegian Sea ecosystem, we use the yearly data assembled by the ICES integrated 
+ecosystem assessment working group for the Norwegian Sea (WGINOR, ICES (37)). Abbreviations for 
+the Norwegian Sea data used in this article is summarized in Supplementary Table1. In the examples, 
+
+we do not attempt to fully interpret any FOs revealed, but comment on the possible background for 
+some of them to illustrate the context for further work in IEA groups. 
+The Atlantic Multi-Decadal Oscillation (AMO) 
+AMO is a pronounced signal of climate variability in the North Atlantic Sea surface temperature (SST) 
+[38]. The monthly data recorded from December in 1869 to March in 2021 has been published under 
+the NCAR CLIMATE data guide [39]. We extracted monthly raw data for the period 1980-2020, from 
+which we calculated annual means, giving a time series with 31-time points. 
+To illustrate how inferences may differ for different time periods, both seven-years and three-years 
+predictions are shown for this example.  Thus, we first set the specific time point j as 2014 and 2017, 
+respectively. The stochastic difference trend model was applied to the data until j-1 time points (that is, 
+2013 and 2016). To optimize Q  for the model, we set 0.05
+0.5
+Q
+≤
+≤
+ as a search range. The 
+calculated maximum log-likelihood and optimum Q  for each dataset are summarized in Table 1a. The 
+details for the log-likelihood in the range of Q are summarized in Table 2.  
+Using the parameters of the model, the Kalman filter algorithm was run to calculate seven-years- and 
+three-years-ahead predictions. Fig 2 presents the outputs of this.  
+For the case of seven-years-ahead prediction, none of the observations for the most recent years fall 
+outside the 95% or 80% FBs, but the observation for 2015 fall outside the 70% FB. Thus, only a single 
+possible FO is identified when looking at each of the seven most recent years individually, and this is 
+due to a marked decrease in SST in 2015 that contrasts the slightly increasing trend predicted for 2014-
+2020. However, all observations for the seven most recent years fall below the predicted trend (Fig 2). 
+The joint probability for this pattern is small (Table 2), suggesting that there is a tendency in the data 
+that differs from the predicted trend and thus represents a FO. For the case of three-years-ahead 
+predictions, none of the observations from the most recent years fall outside any of the FBs, and they 
+are spread evenly around the predicted trend (Fig 2). Thus, while there are ample indications that the 
+seven most recent observations deviate from the trend predicted for the last seven years, no such pattern 
+is seen for the trend predicted for the last three years, suggesting that an assessment group may need to 
+look differently at change over these two time periods.  
+
+Norwegian Sea ecosystem 
+The Norwegian Sea is located West and Northwest of Norway, bordered by the North Sea and the 
+Atlantic Ocean to the south, the Greenland Sea to the west and the Arctic Ocean and Barents Sea to the 
+north and east. It is a deep-sea area with three species of mainly planktivorous pelagic fish making up 
+the economically most important fish stocks: mackerel (Scomber scombrus), Norwegian spring-
+spawning herring (Clupea harengus) and blue whiting (Micromesistius poutassou). Ocean currents are 
+dominated by relatively warm and saline Atlantic water masses flowing in from the south and colder 
+and fresher Arctic water masses flowing in from the northwest [40]. There is considerable negative 
+density dependence acting on biomass within the three pelagic fish stocks, presumably through 
+intraspecific competition over food [41]. While there are also indications of competition among the 
+stocks, most strongly between mackerel and herring [41], other work has suggested that interspecific 
+competition is less significant [42]. The combined biomass of the three species has increased over the 
+last decades while zooplankton biomass has declined, and it has been hypothesized that the pelagic fish 
+biomass may have exceeded the carrying capacity of the system [21]. The climate has historically varied 
+between cold and warm phases, with plankton and fish productivity tending to increase in the warmer 
+phases [43]. Here, we analyse time series on spawning stock biomass, recruitment and growth (age and 
+weight at age 6) for the three pelagic fish stocks, zooplankton biomass, and three key variables for the 
+physical environment: heat content, freshwater content and the North Atlantic Oscillation index. The 
+observations have been recorded annually, although the starting/ending years of observation vary 
+among the time series.  
+For the current work, we used time series with different start years and 2019 as the last year [37]. As 
+one of the main aims of IEAs in the Norwegian Sea has been to provide background information for 
+advisory work for operational fisheries management [44, 45], change over a short period of the most 
+recent years is typically of interest. The conditions for making the prediction were therefore set to three-
+years-ahead predictions for 2017-2019 using the data observed up to 2016. The calculated maximum 
+likelihood, and optimum Q  for each data are summarized in Table 1. The
+2
+w
+σ  was calculated using the 
+observations until 2016.  
+
+Using the parameters of the model, the Kalman filter algorithm was run to calculate three-years-ahead 
+predictions. Fig 3 presents the outputs of this. Looking at variables for the physical environment, FOs 
+for individual years were observed for relative freshwater content (RFW), where the observation for 
+2018 fall outside the 80% FB and for 2019 outside the 95% FB. In addition, observations for all of the 
+last three years fall well above the predicted trend, and the joint probability for this pattern is low (Table 
+2), indicating an unexpected tendency in the data when compared with the expected trend. While 
+freshening of the Norwegian Sea had been going on for nearly a decade before 2019 [46], these 
+unexpected increases in freshwater content point to a recent intensification of this that might require the 
+attention in IEAs of the Norwegian Sea. 
+For zooplankton biomass (ZooB), two of the most recent years fall above the 80% FB and above the 
+70% FB, with a low overall probability of the pattern (Table 2), indicating, again, an unexpected upward 
+tendency in the data. This suggests that an IEA should pay attention to a possible recovery of the 
+zooplankton biomass, which declined sharply in the early 2000s and remained at low levels in the 
+following years [18].  
+Looking at variables for pelagic fish stocks, little evidence for unexpected changes is seen for herring 
+and mackerel. For the four variables related to mackerel, no years fall outside any of the FBs and 
+observations generally lie relatively close to or on both sides of the predicted trends (Fig 3). A similar 
+pattern is seen for herring, except one estimate of recruitment falling above the 70% FB (Fig 3). As 
+pelagic fish recruitment is highly variable, a single observation falling outside the expected trend may 
+not be reason for flagging this variable for prioritization in an IEA.  
+A different picture emerges for blue whiting. For spawning stock biomass (BWB), two of the three most 
+recent years fall above the 70% FB and the third year also well above the predicted trend (Fig 3). The 
+joint probability of this pattern is low (Table 2), suggesting that there is a tendency for an increase in 
+biomass beyond the expected. At the same time, there are indications of unexpected declines in blue 
+whiting individual growth, shown for weight at age 6 (BWW), where all observations fall below the 
+70% FB (Fig 3) and the joint probability for the pattern is low (Table 2). These changes may be linked, 
+as increases in biomass tend to be associated with decreases in growth, possibly though intraspecific 
+competition over food [41]. In addition, for blue withing recruitment (BWR), two observations fall 
+
+below the 70% FB and one below the 80% FB (Fig 3) with a low joint probability for the overall pattern 
+(Table 2), indicating that the decline in recruitment for the most recent years represent an unexpected 
+tendency. Although pelagic fish recruitment remains hard to forecast (e.g. [47]), it is interesting to note 
+that considerable progress has been made in predicting variation in the geographical location of blue 
+whiting spawning habitat, which may be linked to recruitment success [48-50], thus offering a possible 
+avenue for more detailed assessments and studies following up the changes in blue whiting recruitment.  
+ 
+Discussion 
+We have introduced a time series analysis procedure for making predictions for a specific time period 
+using a structural time series model including a trend model. Based on this, we have outlined a 
+framework for investigating whether the most recent observations deviate from the predicted trend for 
+this time period and thus represent possible flagged observations (FOs). This includes assessing both 
+whether single years represent FOs or whether all of the observations from the recent years together 
+represent an unexpected tendency that is classified as a FO. The trend was estimated using a stochastic 
+trend model and observed time series data, and the specific-years-ahead predictions were systematically 
+calculated according to the iterative procedure of the Kalman filter algorithm. The statistical analysis is 
+followed by a qualitative evaluation of each FO, where it may be decided to follow some of them up by 
+more detailed analyses within an integrated ecosystem assessment (IEA). We note that applications may 
+also extend beyond IEAs to other areas of science and advisory processes where an overview of recent 
+change is required across multiple time series. 
+The time series available from marine ecosystems that are relevant for analyses described here are 
+typically short (i.e., < 50 time points) [19, 51]. This puts constraints on the types of time series analyses 
+that can be performed. For example, null hypothesis testing using a frequentist statistical approach can 
+produce misleading results, including false positive and negative results [52]. The procedure described 
+here does not include null hypothesis statistical testing, and the type of structural time series model used 
+by us is not based on a frequentist framework but corresponds to a Bayesian approach [53], which may 
+properly assess trends in short time series that cannot be analysed using a frequentist approach [52]. An 
+
+alternative method to the one used here could have been the Box Jenkins model, which transforms a 
+non-stationary mean time series to a stationary process [54]. However, effective fitting using this 
+method, again, requires longer time series than what is normally available in marine ecosystems [19, 
+51]. Thus, for short time series, the Bayesian framework used here appears to be more robust than 
+alternative approaches. 
+To study and assess recent change, IEA groups often rely on examinations of anomaly plots. In such 
+plots, recent change appears as deviations from a long-term mean, often estimated for the whole length 
+of the time series (see e.g. Bulgin, Merchant (55) for an application to global sea surface temperatures, 
+SST). By focusing on deviations from the expected trend for the most recent years (where the expected 
+trend is estimated by using information from the whole time series), FO analysis can provide a different 
+perspective of recent change. For example, using a seven-years prediction, the FO analysis indicates 
+that there is a tendency in North Atlantic Sea SST towards a more negative trend than what should be 
+expected for the last 7 years of the time series. We argue that the same interpretation is less evident 
+from an anomaly plot of the same time series (see Supplementary Fig 1). Thus, while positive and 
+negative values indicate how observations deviate from a constant mean value in an anomaly plot, the 
+trend changes through time, making it harder to assess how the most recent observations deviate from 
+the trend that should be expected for these recent years. Recognizing that anomaly plots are important 
+for a large range of purposes within IEAs, we emphasize that FO analysis can provide useful additional 
+information for the practical work in IEA groups, in particular in the light of the challenge they are 
+often faced with of understanding and assessing the most recent development of an ecosystem [14]. 
+Since the cumulative output of FO analysis also aim at giving a sweeping overview of the recent 
+dynamics of all the measured elements in an ecosystem by highlighting the variables that exhibit 
+unexpected change while at the same time showing trends and data for those that do not, the approach 
+should be useful for facilitating the necessary dialogue between scientists and stakeholders about recent 
+ecosystem change within the process of an IEA. Such overviews can also contribute to the scientific 
+output used to educate and inform the public and the political system during parts of policymaking 
+processes related to for example ecosystem-based management [56]. 
+ 
+
+Acknowledgements 
+We would like to thank Benjamin Planque, who motivated us to consider this approach to analysing the 
+time series data compiled by the ICES integrated ecosystem assessment working groups and Mette 
+Mauritzen and Daniel Howell for comments on an earlier draft of the paper. This study was carried out 
+as part of the project “Sustainable multi-species harvest from the Norwegian Sea and adjacent 
+ecosystems”, funded by The Research Council of Norway (pr. nr. 299554). 
+ 
+References 
+1. 
+Bindoff NL, Cheung WW, Kairo JG, Arístegui J, Guinder VA, Hallberg R, et al. Changing 
+ocean, marine ecosystems, and dependent communities. In: Pörtner H-O, Roberts DC, Masson-
+Delmotte V, Zhai P, Tignor M, Poloczanska E, et al., editors. IPCC special report on the ocean and 
+cryosphere in a changing climate2019. p. 477-587. 
+2. 
+Hoegh-Guldberg O, Cai R, Poloczanska ES, Brewer PG, Sundby S, Hilmi K, et al. The ocean. 
+In: Barros VR, Field CB, Dokken DJ, Mastrandrea MD, Mach KJ, Bilir TE, et al., editors. Climate 
+Change 2014: Impacts, Adaptation, and Vulnerability Part B: Regional Aspects Contribution of 
+Working Group II to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change. 
+Cambridge, United Kingdom and New York, NY, USA: Cambridge University Press; 2014. p. 1655-
+731. 
+3. 
+Meredith M, Sommerkorn M, Cassota S, Derksen C, Ekaykin A, Hollowed A, et al. Polar 
+regions. In: Pörtner H-O, Roberts DC, Masson-Delmotte V, Zhai P, Tignor M, Poloczanska E, et al., 
+editors. IPCC special report on the ocean and cryosphere in a changing climate2019. 
+4. 
+O’Hara CC, Frazier M, Halpern BS. At-risk marine biodiversity faces extensive, expanding, 
+and intensifying human impacts. Science. 2021;372(6537):84-7. doi: 10.1126/science.abe6731. 
+5. 
+Pörtner H-O, Karl DM, Boyd P, Cheung W, Lluch-Cota S, Nojiri Y, et al. Ocean systems. In: 
+Barros VR, Field CB, Dokken DJ, Mastrandrea MD, Mach KJ, Bilir TE, et al., editors. Climate 
+Change 2014: Impacts, Adaptation, and Vulnerability Part B: Regional Aspects Contribution of 
+Working Group II to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change. 
+Cambridge, United Kingdom and New York, NY, USA: Cambridge University Press; 2014. p. 411-
+84. 
+6. 
+Koul V, Sguotti C, Årthun M, Brune S, Düsterhus A, Bogstad B, et al. Skilful prediction of 
+cod stocks in the North and Barents Sea a decade in advance. Communications Earth & Environment. 
+2021;2(1):140. doi: 10.1038/s43247-021-00207-6. 
+7. 
+McLeod KL, Lubchenco J, Palumbi SR, Rosenberg AA. Scientific Consensus Statement on 
+Marine Ecosystem-Based Management. Signed by 217 academic scientists and policy experts with 
+relevant expertise and published by the Communication Partnership for Science and the Sea at 
+http://compassonline.org/?q=EBM. 2005. 
+8. 
+O’Boyle R, Jamieson G. Observations on the implementation of ecosystem-based 
+management: Experiences on Canada's east and west coasts. Fisheries Research. 2006;79(1):1-12. doi: 
+https://doi.org/10.1016/j.fishres.2005.11.027. 
+9. 
+Tallis H, Levin PS, Ruckelshaus M, Lester SE, McLeod KL, Fluharty DL, et al. The many 
+faces of ecosystem-based management: Making the process work today in real places. Marine Policy. 
+2010;34(2):340-8. doi: https://doi.org/10.1016/j.marpol.2009.08.003. 
+10. 
+Winther J-G, Dai M, Rist T, Hoel AH, Li Y, Trice A, et al. Integrated ocean management for 
+a sustainable ocean economy. Nature Ecology & Evolution. 2020;4(11):1451-8. doi: 10.1038/s41559-
+020-1259-6. 
+
+11. 
+Fulton EA, Punt AE, Dichmont CM, Harvey CJ, Gorton R. Ecosystems say good 
+management pays off. Fish and Fisheries. 2019;20(1):66-96. doi: https://doi.org/10.1111/faf.12324. 
+12. 
+Holsman KK, Haynie AC, Hollowed AB, Reum JCP, Aydin K, Hermann AJ, et al. 
+Ecosystem-based fisheries management forestalls climate-driven collapse. Nat Commun. 
+2020;11(1):4579. doi: 10.1038/s41467-020-18300-3. 
+13. 
+Dickey-Collas M, Link JS, Snelgrove P, Roberts JM, Anderson MR, Kenchington E, et al. 
+Exploring ecosystem-based management in the North Atlantic. Journal of Fish Biology. 
+2022;101(2):342-50. doi: https://doi.org/10.1111/jfb.15168. 
+14. 
+Levin PS, Fogarty MJ, Murawski SA, Fluharty D. Integrated Ecosystem Assessments: 
+Developing the Scientific Basis for Ecosystem-Based Management of the Ocean. Plos Biology. 
+2009;7(1):23-8. doi: e1000014 
+10.1371/journal.pbio.1000014. PubMed PMID: WOS:000262811000004. 
+15. 
+Walther YM, Möllmann C. Bringing integrated ecosystem assessments to real life: a scientific 
+framework for ICES. Ices Journal of Marine Science. 2014;71(5):1183-6. doi: 10.1093/icesjms/fst161. 
+PubMed PMID: WOS:000338630800019. 
+16. 
+ICES. Report of the Workshop on integrated trend analyses in support to integrated ecosystem 
+assessment (WKINTRA). ICES CM 2018/IEASG: 15: 23pp. 2018. doi: 
+https://doi.org/10.17895/ices.pub.8278. 
+17. 
+Dickey-Collas M. Why the complex nature of integrated ecosystem assessments requires a 
+flexible and adaptive approach. ICES Journal of Marine Science. 2014;71(5):1174-82. doi: 
+10.1093/icesjms/fsu027. 
+18. 
+ICES. Working Group on the Integrated Assessments of the Norwegian Sea (WGINOR; 
+outputs from 2020 meeting). ICES Scientific Reports. 3:35. 114 pp. . 2021. 
+19. 
+ICES. The second workshop on integrated trend analyses in support to integrated ecosystem 
+as-sessment (WKINTRA2). ICES Scientific Reports. 1:86. 2019. 
+20. 
+Holsman K, Samhouri J, Cook G, Hazen E, Olsen E, Dillard M, et al. An ecosystem-based 
+approach to marine risk assessment. Ecosystem Health and Sustainability. 2017;3(1):e01256-n/a. doi: 
+10.1002/ehs2.1256. 
+21. 
+ICES. Report of the Working Group on Integrated Ecosystem Assessments for the Norwegian 
+Sea (WGINOR). ICES WGINOR REPORT 2018 26-30 November 2018. Reykjavik, Iceland. ICES 
+CM 2018/IEASG:10. 123 pp. 2019. 
+22. 
+Harvey AC. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge: 
+Cambridge University Press; 1990. 
+23. 
+Hamilton JD. Time Series Analysis. New Jersey, United States: Princeton University Press; 
+1994. 
+24. 
+Mitchell PL, Sheehy JE. Comparison of predictions and observations to assess model 
+performance: a method of empirical validation. In: Kropff MJ, Teng P, Aggarwal PK, Bouma J, 
+Bouman BAM, Jones JW, et al., editors. Applications of Systems Approaches at the Filed Level: 
+Kluwer Academic Publishers; 1997. p. 437-51. 
+25. 
+Clements CF, McCarthy MA, Blanchard JL. Early warning signals of recovery in complex 
+systems. Nat Commun. 2019;10(1):1681. doi: 10.1038/s41467-019-09684-y. 
+26. 
+Clements CF, Ozgul A. Indicators of transitions in biological systems. Ecol Lett. 
+2018;21(6):905-19. doi: 10.1111/ele.12948. 
+27. 
+Litzow MA, Hunsicker ME. Early warning signals, nonlinearity, and signs of hysteresis in 
+real ecosystems. Ecosphere. 2016;7(12). doi: 10.1002/ecs2.1614. PubMed PMID: 
+WOS:000390136700014. 
+28. 
+Suweis S, D'Odorico P. Early Warning Signs in Social-Ecological Networks. PLoS One. 
+2014;9(7). doi: 10.1371/journal.pone.0101851. PubMed PMID: WOS:000339378400043. 
+29. 
+Kitagawa G, Gersch W. Smoothness Priors Analysis of Time Series. Lecture Notes in 
+Statistics 116. New York: Springer-Verlag 1996. 
+30. 
+Kato H, Naniwa S, Ishiguro M. A bayesian multivariate nonstationary time series model for 
+estimating mutual relationships among variables. Journal of Econometrics. 1996;75(1):147-61. doi: 
+https://doi.org/10.1016/0304-4076(95)01774-7. 
+31. 
+Whittaker E. On a new method of graduation. Proceedings of the Edinburgh Mathematical 
+Society. 1923;41:63-75. doi: 10.1017/S0013091500077853. 
+
+32. 
+Akaike H. A new look at the statistical model identification. IEEE Transactions on Automatic 
+Control. 1974;19(6):716-23. doi: 10.1109/TAC.1974.1100705. 
+33. 
+Kitagawa G, Gersch W. A smoothness priors state-space modelling of time-series with trend 
+and seasonality. Journal of the American Statistical Association. 1984;79(386):378-89. doi: 
+10.2307/2288279. PubMed PMID: WOS:A1984SW65000024. 
+34. 
+Hyndman RJ, Athanasopoulos G. Forecasting. Principles and practice. 2nd Edition (May 6, 
+2018). Melbourne: OTexts; 2018. 
+35. 
+MATLAB. Natick, Massachusetts: The MathWorks Inc. ver. R2018b. 
+36. 
+Trenberth KE, Shea DJ. Atlantic hurricanes and natural variability in 2005. Geophys Res Lett. 
+2006;33(12). doi: https://doi.org/10.1029/2006GL026894. 
+37. 
+ICES. Working Group on the Integrated Assessments of the Norwegian Sea (WGINOR; 
+outputs from 2019 meeting). ICES Scientific Reports. 2:29. 46 pp. . 2020. 
+38. 
+Dijkstra HA, te Raa L, Schmeits M, Gerrits J. On the physics of the Atlantic Multidecadal 
+Oscillation. Ocean Dynamics. 2006;56(1):36-50. doi: 10.1007/s10236-005-0043-0. 
+39. 
+Schneider DP, Deser C, Fasullo J, Trenberth KE. Climate Data Guide Spurs Discovery and 
+Understanding. Eos, Transactions American Geophysical Union. 2013;94(13):121-2. doi: 
+https://doi.org/10.1002/2013EO130001. 
+40. 
+Skjoldal HRe. The Norwegian Sea Ecosystem. Trondheim: Tapir Academic Press; 2004. 
+41. 
+Huse G, Holst JC, Utne K, Nottestad L, Melle W, Slotte A, et al. Effects of interactions 
+between fish populations on ecosystem dynamics in the Norwegian Sea - results of the INFERNO 
+project Preface. Marine Biology Research. 2012;8(5-6):415-9. doi: 10.1080/17451000.2011.653372. 
+PubMed PMID: WOS:000303560300001. 
+42. 
+Planque B, A. F, B. H, Mousing E, C. H, C. B, et al. Quantification of trophic interactions in 
+the Norwegian Sea pelagic food-web over multiple decades. ICES J Mar Sci. 2022. doi: 
+https://doi.org/10.1093/icesjms/fsac111. 
+43. 
+Drinkwater KF, Miles M, Medhaug I, Otterå OH, Kristiansen T, Sundby S, et al. The Atlantic 
+Multidecadal Oscillation: Its manifestations and impacts with special emphasis on the Atlantic region 
+north of 60°N. Journal of Marine Systems. 2014;133:117-30. doi: 
+https://doi.org/10.1016/j.jmarsys.2013.11.001. 
+44. 
+ICES. Working Group on the Integrated Assessments of the Norwegian Sea (WGINOR; 
+outputs from 2021 meeting). ICES Scientific Reports. 4:35. 48pp. 
+https://doi.org/10.17895/ices.pub.19643271. 2022. 
+45. 
+ICES. Working Group on Widely Distributed Stocks (WGWIDE). ICES Scientific Reports. 
+3:95. 874 pp. http://doi.org/10.17895/ices.pub.8298. 2021. 
+46. 
+Mork KA, Skagseth Ø, Søiland H. Recent Warming and Freshening of the Norwegian Sea 
+Observed by Argo Data. Journal of Climate. 2019;32(12):3695-705. doi: 10.1175/jcli-d-18-0591.1. 
+47. 
+Garcia T, Planque B, Arneberg P, Bogstad B, Skagseth Ø, Tiedemann M. An appraisal of the 
+drivers of Norwegian spring-spawning herring (Clupea harengus) recruitment. Fisheries 
+Oceanography. 2020;30(2):159-73. doi: 10.1111/fog.12510. 
+48. 
+Miesner AK, Payne MR. Oceanographic variability shapes the spawning distribution of blue 
+whiting (Micromesistius poutassou). Fisheries Oceanography. 2018;27(6):623-38. doi: 
+doi:10.1111/fog.12382. 
+49. 
+Hatun H, Payne MR, Jacobsen JA. The North Atlantic subpolar gyre regulates the spawning 
+distribution of blue whiting (Micromesistius poutassou). Canadian Journal of Fisheries and Aquatic 
+Sciences. 2009;66(5):759-70. doi: 10.1139/f09-037. PubMed PMID: WOS:000267811500005. 
+50. 
+ICES. Interim Report of the Working Group on Seasonal to Decadal Prediction of Marine 
+Ecosystems (WGS2D), 27–31 August 2018, ICES Headquarters, Copenhagen, Denmark. ICES CM 
+2018/EPDSG:22. 42 pp. 2018. 
+51. 
+Samhouri JF, Andrews KS, Fay G, Harvey CJ, Hazen EL, Hennessey SM, et al. Defining 
+ecosystem thresholds for human activities and environmental pressures in the California Current. 
+Ecosphere. 2017;8(6):e01860. doi: 10.1002/ecs2.1860. 
+52. 
+Hardison S, Perretti CT, DePiper GS, Beet A. A simulation study of trend detection methods 
+for integrated ecosystem assessment. ICES Journal of Marine Science. 2019. doi: 
+10.1093/icesjms/fsz097. 
+53. 
+West M, Harrison J. Bayesian Forecasting and Dynamic Models. New York: Springer; 1997. 
+54. 
+Box G, Jenkins G. Time Series Analysis: Forecasting and Control. San Francisco: Holden-
+Day; 1970. 
+
+55. 
+Bulgin CE, Merchant CJ, Ferreira D. Tendencies, variability and persistence of sea surface 
+temperature anomalies. Scientific Reports. 2020;10(1):7986. doi: 10.1038/s41598-020-64785-9. 
+56. 
+Cormier R, Kelble CR, Anderson MR, Allen JI, Grehan A, Gregersen O. Moving from 
+ecosystem-based policy objectives to operational implementation of ecosystem-based management 
+measures. Ices Journal of Marine Science. 2017;74(1):406-13. doi: 10.1093/icesjms/fsw181. PubMed 
+PMID: WOS:000397136400039. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Table 1. Calculated log-likelihood (LL) and the optimum Q  by applying stochastic trend model for 
+2
+d =
+for the AMO and Norwegian Sea ecosystem datasets. 
+ 
+Dataset 
+Variable 
+Q 
+Maximum log-
+likelihood 
+AMO 
+AMOS 
+0.50 
+-51.2 
+Norwegian Sea 
+ecosystem 
+(WGINOR) 
+RHC 
+0.05 
+-82.8 
+RFW 
+0.08 
+-82.6 
+NAO 
+0.05 
+-161.3 
+ZooB 
+0.09 
+-29.1 
+MacB 
+0.12 
+-43.3 
+MacR 
+0.06 
+-42.1 
+MacW 
+0.07 
+-44.1 
+MacL 
+0.07 
+-71.8 
+HerB 
+0.06 
+-125.7 
+HerR 
+0.05 
+-42.6 
+HerW 
+0.11 
+-87.1 
+HerL 
+0.08 
+-93.7 
+BWB 
+0.13 
+-42.4 
+BWR 
+0.16 
+-48.7 
+BWW 
+0.09 
+-43.8 
+BWL 
+0.08 
+-57.3 
+ 
+ 
+ 
+ 
+Table 2. Averaged p-values and joint probability for the prediction and observation for 
+AMOS, RFW, ZooB, BWB, BWR, and BWW. 
+ 
+Year 
+AMOS 
+RFW 
+ZooB 
+BWB 
+BWR 
+BWW 
+2014 
+0.19 
+ 
+2015 
+0.14 
+2016 
+0.29 
+2017 
+0.42 
+0.22 
+0.12 
+0.13 
+0.1003 
+0.12 
+2018 
+0.22 
+0.061 
+0.28 
+0.16 
+0.1237 
+0.11 
+2019 
+0.28 
+0.023 
+0.13 
+0.29 
+0.1415 
+0.13 
+2020 
+0.42 
+ 
+Joint 
+probability 
+8.24e-05 
+3.06e-04 
+0.0045 
+0.0063 
+0.0018 
+0.0016 
+ 
+ 
+ 
+
+ 
+ 
+Fig1. Outline of proposed Flagged observation (FO) analysis 
+
+Time series data
+2.5
+Estimationof thetrend
+1.5
+Ecosystem-based
+in the period 1950 - 2016
+fisheries management
+and
+0.
+prediction of the data
+contribution
+in the period 2017-2019
+1960
+1970
+1980
+1990
+2000
+2010
+12020
+Yea
+Dataintheperiod1950-2016
+RealdataoutsideofFBs
+deviatedfromrecent3-year
+Estimated trend =(n)
+trend - Flagged observation
+17
+RealdatawithinFBsfollowed
+recent 3-year trend
+Forecast bands (FBs)
+y(n+jln)+2×/d(n+j|n)
+1619
+Prediction of the data
+y(n+ jln), j=1, 2, 3 
+ 
+         
+ 
+    
+Fig 2. The estimated three-years (upper) and seven-years (lower) prediction values of sea 
+surface temperature and the most recent observations for the dataset on the Atlantic 
+Multi-decadal Oscillation. The solid grey lines indicate the observations used for making 
+the forecast values and the black points indicate the observations that were plotted for 
+comparison with the prediction values (dotted blue line). The dotted grey line presents 
+the prediction value 
+and the solid blue lines present the smoothed trend 
+estimates obtained by a fixed-interval smoother algorithm.  The band coloured by light-
+blue presents the 95% FB, 80% and approximately 70% FBs, which upper and lower limits 
+were calculated by mean 
+± 1.96 (1.28, 1) × standard deviation 
+, 
+where 
+or 
+. The observations in the years applying the prediction are 
+shown with black points.   
+ 
+ 
+
+0.6
+0.4
+0.2
+0.2
+-0.4
+-0.6
+-0.8
+-1
+1980
+17
+20
+lyearelsius]
+.0.5
+-0.5
+1980
+06
+0913
+1619
+[year]Climate: 
+ 
+Plankton: 
+ 
+ 
+Fig 3 Continued. 
+   
+          
+ 
+    
+ 
+   
+ 
+
+RHC
+225
+-wr
+[108
+20
+15
+10
+5
+0
+-5
+-10
+-15
+1951
+1619
+[year]RFW
+E
+1.5
+1
+0.5
+0.5
+-1.5
+2
+1951
+1619
+[year]NAO
+[djfm]
+5
+-2
+3
+1907
+1619
+[year]ZooB
+235
+[gm-2
+30
+25
+20
+15
+10
+1995
+16
+19
+[year]Pelagic fish: 
+Fig 3 Continued. 
+    
+      
+ 
+         
+ 
+   
+      
+ 
+  
+           
+ 
+ 
+
+MacB
+6
+5
+1
+3
+2
+1
+0
+1980
+1619
+[year]X106
+MacR
+4.597
+o!l
+4.596
+4.595
+4.594
+4.593
+4.592
+4.591
+4.59
+4.589
+4.588
+1980
+1619
+[vear]MacW
+0.1
+0.05
+0
+-0.05
+0.1
+-0.15
+1980
+16
+19
+[year]MacL
+48
+46
+44
+42
+40
+38
+1963
+16 19
+[year]HerB
+tonnes
+45
+40
+[million t
+35
+30
+25
+20
+15
+10
+5
+1907
+1619
+[year]X108
+HerR
+2.7868
+O
+2.7866
+2.7864
+2.7862
+2.786
+2.7858
+2.7856
+2.7854
+1988
+16
+19
+[year]Herw
+0.2
+0.15
+0.1
+0.05
+0
+-0.05
+0.1
+-0.15
+0.2
+1950
+1619
+[year]HerL
+44
+42
+40
+38
+36
+34
+1944
+1619
+[year]Pelagic fish: 
+Fig 3. The estimated three-years prediction values and the three most recent observations for 
+data on variables related to climate, plankton and pelagic fish in the Norwegian Sea 
+ecosystem. The solid grey lines indicate the observations used for making the forecast values 
+and the black points indicate the observations that were plotted for comparison with the 
+prediction values (dotted blue line). The dotted grey line presents the prediction value 
+and the solid blue lines present the smoothed trend estimates obtained by a 
+fixed-interval smoother algorithm.  The band coloured by light-blue presents the 95% FB, 80% 
+and approximately 70% FBs, which upper and lower limits were calculated by mean 
+± 1.96 (1.28, 1) × standard deviation 
+, where 
+. The 
+observations in the years applying the prediction are shown with black points.   
+ 
+ 
+ 
+ 
+ 
+    
+     
+ 
+  
+        
+ 
+ 
+
+BWB
+[milliont
+10
+8
+6
+4
+2
+0
+2
+1981
+1619
+[year]X108
+BWR
+3.1656
+3.1652
+3.165
+3.1648
+3.1646
+3.1644
+3.1642
+1981
+1619
+lyearBWW
+0.06
+0.04
+0.02
+0
+0.02
+-0.04
+-0.06
+-0.08
+1981
+16
+19
+[year]BWL
+48
+46
+44
+42
+40
+38
+1972
+1619
+[year]Supplementary Fig 1 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Supplementary Fig 1. Bar plots for anomalies of AMOS’s yearly data from 1980 to 2020. The y-
+axis indicates the value subtracting mean value of yearly data from 1980 to 2020. The x-axis 
+indicates year. The negative/positive values correspond lower/higher temperature to the mean 
+value. 
+
+0.4
+0.3
+0.2
+0.1
+0
+-0.1
+0.2
+-0.3
+-0.4
+1980
+1985
+1990
+1995
+20002
+2005
+2010
+2015
+2020Supplementary Table. Abbreviations used in figures and tables 
+Type 
+Abbreviation 
+in figures 
+and tables 
+Explanation of relevant data 
+ 
+Climate 
+RHC 
+Relative heat content in 108 Jm-2 
+RFW 
+Relative freshwater content in m 
+NAO 
+North Atlantic Oscillation expressed as djfm 
+Zooplankton 
+ZooB 
+Total zooplankton biomass the Norwegian Sea in in May, g m-2  
+ 
+ 
+ 
+ 
+ 
+ 
+Pelagic fish 
+MacB 
+Spawning stock biomass of Mackerel in million tonnes 
+MacR 
+Recruitment of Mackerel per year class at age 0 in millions 
+MacW 
+Weight of Mackerel at age 6 in the stock (in kg) 
+MacL 
+Length of Mackerel at age 6 in cm 
+HerB 
+Spawning stock biomass of Herring in million tonnes 
+HerR 
+Recruitment of Herring per year class at age 2 in millions 
+HerW 
+Weight of Herring at age 6 in the stock (in kg) 
+HerL 
+Length of Herring at age 6 in cm 
+BWB 
+Spawning stock biomass of Blue whiting in million tonnes 
+BWR 
+Recruitment of Blue whiting per year class at age 1 in millions 
+BWW 
+Weight of Blue whiting age 6 in the catch (in kg) 
+BWL 
+Length of Blue whiting at age 6 in cm 
+ 
+ 
+ 
+ 
+

ANFQT4oBgHgl3EQf8jdP/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:015d95bcc843dcce8b460633ddac3e462342a33ddf18b8dd39a336e1e6a7dd6a
+size 98288

AtE0T4oBgHgl3EQfPgDv/content/tmp_files/2301.02181v1.pdf.txt

@@ -0,0 +1,107 @@
+A Critical Appraisal of Data Augmentation Methods
+for Imaging-Based Medical Diagnosis Applications
+Tara M. Pattilachan∗¶, Ugur Demir†¶, Elif Keles†, Debesh Jha†, Derk Klatte‡, Megan Engels‡,
+Sanne Hoogenboom‡, Candice Bolan‡, Michael Wallace§, Ulas Bagci†
+∗University of Central Florida, FL, USA,
+†Machine and Hybrid Intelligence Lab, Northwestern University, IL, USA,
+‡ Mayo Clinic, Jacksonville, FL, USA,
+§ Mayo Clinic, Sheikh Shakhbout Medical City, UAE
+Abstract—Current data augmentation techniques and trans-
+formations are well suited for improving the size and quality
+of natural image datasets but are not yet optimized for medical
+imaging. We hypothesize that sub-optimal data augmentations
+can easily distort or occlude medical images, leading to false
+positives or negatives during patient diagnosis, prediction, or
+therapy/surgery evaluation. In our experimental results, we found
+that utilizing commonly used intensity-based data augmentation
+distorts the MRI scans and leads to texture information loss,
+thus negatively affecting the overall performance of classification.
+Additionally, we observed that commonly used data augmenta-
+tion methods cannot be used with a plug-and-play approach in
+medical imaging, and requires manual tuning and adjustment.
+Index Terms—Augmentation, diagnosis, deep learning, IPMN
+I. INTRODUCTION
+In this study, we investigate how commonly used data
+augmentation methods affect medical imaging based diagnosis
+tasks, the following methods are used: RandAugment [1],
+AutoAugment [2], Fast AutoAugment [3], Trivial Augment [4]
+and AugMix [5].
+Fig. 1. Overall architecture for ResNeST [6] training with data augmentation.
+II. METHOD
+Figure 1 shows the block diagram of the proposed method.
+In this study, we tested commonly used data augmentation
+methods RandAugment [1], AutoAugment [2], Fast AutoAug-
+ment [3], Trivial Augment [4] and AugMix [5], and their
+impact on the MRI based IPMN classification problem. The
+classification problem was modelled via a deep learning archi-
+tecture, called ResNeST [6], where we classified the images
+into normal, low grade and high grade categories.
+A. Results
+Table I exhibits the performance of each augmentation
+method and the baseline. It can be observed that the baseline
+TABLE I
+MRI BASED IPMN CLASSIFICATION RESULTS WITH RESPECT TO DATA
+AUGMENTATION METHODS.
+.
+Method
+Accuracy
+Baseline
+61.70% ± 9.42
+RandAug
+55.30% ± 9.29
+RandAug - geometric
+61.67% ± 8.37
+Auto Augment
+51.03% ± 10.31
+Fast Auto Augment
+55.34% ± 12.78
+Trivial Augment
+51.75% ± 9.30
+AugMix
+54.56% ± 8.19
+achieves an accuracy of 61.70% ± 9.42. However, other
+augmentation methods such as RandAug, Auto Augment, Fast
+Auto Augment, Trivial Augment and AugMix are causing
+significant performance drop (refer Table I). Table shows
+that when the intensity based augmentation is removed per-
+formance stays close to the baseline in the “RandAug -
+geometric” experiments where it achieves 61.67% ± 8.37.
+III. CONCLUSION
+In this study, we raised critical appraisals for the role of
+data augmentation for medical imaging tasks. We analyzed five
+commonly used data augmentation approaches and their effect
+on the performance of the MRI based IPMN classification
+problem. Our study in the controlled experiments showed that
+the commonly used data augmentation methods are designed
+specifically for natural images and they can have adverse
+effects in medical diagnosis tasks if used without modification.
+Acknowledgement: This project is supported by the NIH
+funding: R01-CA246704 and R01-CA240639.
+REFERENCES
+[1] E. D. Cubuk, B. Zoph, J. Shlens, and Q. Le, “Randaugment: Practical au-
+tomated data augmentation with a reduced search space,” in Proceedings
+of the Advances in NIPS, vol. 33, 2020, pp. 18 613–18 624.
+[2] E. D. Cubuk, B. Zoph, D. Mane, V. Vasudevan, and Q. V. Le, “Autoaug-
+ment: Learning augmentation policies from data,” in Proceedings of the
+CVPR, 2019.
+[3] S. Lim, I. Kim, T. Kim, C. Kim, and S. Kim, “Fast autoaugment,” in
+Proceedings of the NIPS, vol. 32, 2019.
+[4] S. G. M¨uller and F. Hutter, “Trivialaugment: Tuning-free yet state-of-the-
+art data augmentation,” in Proceedings of the ICCV, 2021, pp. 774–782.
+[5] D. Hendrycks*, N. Mu*, E. D. Cubuk, B. Zoph, J. Gilmer, and B. Lak-
+shminarayanan, “Augmix: A simple method to improve robustness and
+uncertainty under data shift,” in Proceedings of the ICLR, 2020.
+[6] H. Zhang, C. Wu, Z. Zhang, Y. Zhu, Z. Zhang, H. Lin, Y. Sun, T. He,
+J. Muller, R. Manmatha, M. Li, and A. Smola, “Resnest: Split-attention
+networks,” arXiv preprint arXiv:2004.08955, 2020.
+arXiv:2301.02181v1  [eess.IV]  14 Dec 2022
+
+Input
+Augmented Input
+Normal
+LowGrade
+Data
+ResNeST
+Augmentation
+High Grade
+..

AtE0T4oBgHgl3EQfPgDv/content/tmp_files/load_file.txt

@@ -0,0 +1,121 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf,len=120
+page_content='A Critical Appraisal of Data Augmentation Methods  for Imaging-Based Medical Diagnosis Applications  Tara M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Pattilachan∗¶,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Ugur Demir†¶,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Elif Keles†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Debesh Jha†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Derk Klatte‡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Megan Engels‡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  Sanne Hoogenboom‡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Candice Bolan‡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Michael Wallace§,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Ulas Bagci†  ∗University of Central Florida,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' FL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' USA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  †Machine and Hybrid Intelligence Lab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Northwestern University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' IL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' USA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  ‡ Mayo Clinic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Jacksonville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' FL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' USA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  § Mayo Clinic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Sheikh Shakhbout Medical City,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' UAE  Abstract—Current data augmentation techniques and trans-  formations are well suited for improving the size and quality  of natural image datasets but are not yet optimized for medical  imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' We hypothesize that sub-optimal data augmentations  can easily distort or occlude medical images, leading to false  positives or negatives during patient diagnosis, prediction, or  therapy/surgery evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' In our experimental results, we found  that utilizing commonly used intensity-based data augmentation  distorts the MRI scans and leads to texture information loss,  thus negatively affecting the overall performance of classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  Additionally, we observed that commonly used data augmenta-  tion methods cannot be used with a plug-and-play approach in  medical imaging, and requires manual tuning and adjustment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  Index Terms—Augmentation, diagnosis, deep learning, IPMN  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' INTRODUCTION  In this study, we investigate how commonly used data  augmentation methods affect medical imaging based diagnosis  tasks, the following methods are used: RandAugment [1],  AutoAugment [2], Fast AutoAugment [3], Trivial Augment [4]  and AugMix [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Overall architecture for ResNeST [6] training with data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' METHOD  Figure 1 shows the block diagram of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  In this study, we tested commonly used data augmentation  methods RandAugment [1], AutoAugment [2], Fast AutoAug-  ment [3], Trivial Augment [4] and AugMix [5], and their  impact on the MRI based IPMN classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' The  classification problem was modelled via a deep learning archi-  tecture, called ResNeST [6], where we classified the images  into normal, low grade and high grade categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Results  Table I exhibits the performance of each augmentation  method and the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' It can be observed that the baseline  TABLE I  MRI BASED IPMN CLASSIFICATION RESULTS WITH RESPECT TO DATA  AUGMENTATION METHODS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  Method  Accuracy  Baseline  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='70% ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='42  RandAug  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='30% ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='29  RandAug - geometric  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='67% ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='37  Auto Augment  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='03% ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='31  Fast Auto Augment  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='34% ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='78  Trivial Augment  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='75% ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='30  AugMix  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='56% ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='19  achieves an accuracy of 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='70% ± 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' However, other  augmentation methods such as RandAug, Auto Augment, Fast  Auto Augment, Trivial Augment and AugMix are causing  significant performance drop (refer Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Table shows  that when the intensity based augmentation is removed per-  formance stays close to the baseline in the “RandAug -  geometric” experiments where it achieves 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='67% ± 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' CONCLUSION  In this study, we raised critical appraisals for the role of  data augmentation for medical imaging tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' We analyzed five  commonly used data augmentation approaches and their effect  on the performance of the MRI based IPMN classification  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Our study in the controlled experiments showed that  the commonly used data augmentation methods are designed  specifically for natural images and they can have adverse  effects in medical diagnosis tasks if used without modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  Acknowledgement: This project is supported by the NIH  funding: R01-CA246704 and R01-CA240639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  REFERENCES  [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Cubuk, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Zoph, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Shlens, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Le, “Randaugment: Practical au-  tomated data augmentation with a reduced search space,” in Proceedings  of the Advances in NIPS, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' 33, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' 18 613–18 624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  [2] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Cubuk, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Zoph, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Mane, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Vasudevan, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Le, “Autoaug-  ment: Learning augmentation policies from data,” in Proceedings of the  CVPR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Lim, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Kim, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Kim, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Kim, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Kim, “Fast autoaugment,” in  Proceedings of the NIPS, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' M¨uller and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Hutter, “Trivialaugment: Tuning-free yet state-of-the-  art data augmentation,” in Proceedings of the ICCV, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' 774–782.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  [5] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Hendrycks*, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Mu*, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Cubuk, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Zoph, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Gilmer, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Lak-  shminarayanan, “Augmix: A simple method to improve robustness and  uncertainty under data shift,” in Proceedings of the ICLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Wu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Zhu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Sun, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' He,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Muller, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Manmatha, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Li, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content=' Smola, “Resnest: Split-attention  networks,” arXiv preprint arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='08955, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='02181v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='IV]  14 Dec 2022  Input  Augmented Input  Normal  LowGrade  Data  ResNeST  Augmentation  High Grade  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE0T4oBgHgl3EQfPgDv/content/2301.02181v1.pdf'}

BdE1T4oBgHgl3EQfVgTd/content/tmp_files/2301.03104v1.pdf.txt

@@ -0,0 +1,2028 @@
+arXiv:2301.03104v1  [math.AG]  8 Jan 2023
+ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE
+ANGELO FELICE LOPEZ* AND DEBADITYA RAYCHAUDHURY**
+Abstract. We study varieties X ⊆ PN of dimension n such that TX(k) is an Ulrich vector bundle for
+some k ∈ Z. First we give a sharp bound for k in the case of curves. Then we show that k ≤ n + 1 if
+2 ≤ n ≤ 12. We classify the pairs (X, OX(1)) for k = 1 and we show that, for n ≥ 4, the case k = 2
+does not occur.
+1. Introduction
+Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1. As is well known, the study of
+vector bundles on X can give important geometrical information about X itself. Regarding this, one
+of the most interesting family of vector bundles associated to X and its embedding, that received a lot
+of attention lately, is that of Ulrich vector bundles, that is bundles E such that Hi(E(−p)) = 0 for all
+i ≥ 0 and 1 ≤ p ≤ n. The study of such bundles is closely related with several areas of commutative
+algebra and algebraic geometry, and often gives interesting consequences on the geometry of X and on
+the cohomology of sheaves on X (see for example in [ES, Be1, CMRPL] and references therein).
+Perhaps the most challenging question in these matters is whether every X ⊆ PN carries an Ulrich
+vector bundle (see for example [ES, page 543]). It comes therefore very natural to ask if usual vector
+bundles associated to X can be Ulrich. Also, since Ulrich vector bundles are globally generated, it is
+better to consider twisted versions of the usual bundles associated to X. The cases of the (twisted)
+normal, cotangent, restricted tangent and cotangent bundles have been dealt with in [Lop], with an
+essentially complete classification.
+In this paper we study the more delicate question: for which integers k one has that TX(k) is an
+Ulrich vector bundle?
+Ulrich vector bundles have special cohomological features, but also numerical ones. This makes the
+above question rather tricky. It is easy to show that k ≥ 0 unless (X, OX(1), k) = (P1, OP1(1), −2).
+In the case k = 0, a recent result [BMPT, Prop. 4.1, Thm. 4.5] gives a classification: (X, OX(1)) =
+(P1, OP1(3)), (P2, OP2(2)) (we will give a new and simple proof in section 8; another proof is given in
+[C2]). On the other hand, for k ≥ 1, the question is more subtle as we will see below.
+In the case of curves, one sees that k = 1 is not possible (see Lemma 4.3(i)), while the cases k = 2, 3
+can be dealt with on any curve (see Lemma 5.1 and Example 5.2). On the other hand, the following
+sharp bound holds, showing that for curves k can be as large as wanted.
+Theorem 1.
+Let X ⊆ PN be a smooth irreducible curve of genus g. If TX(k) is an Ulrich line bundle, then
+(1.1)
+k ≤
+√8g + 1 − 1
+2
+and equality holds if and only if k is even and either X is one of the curves (5.1) lying on a smooth
+cubic or X is a curve of type (k
+2 +1, k +2) on a smooth quadric. Also, in both cases, TX(k) is an Ulrich
+line bundle, hence the bound is sharp for every even k ≥ 0. Moreover, if X has general moduli, then
+k ≤ 4.
+As far as we know, only curves show this kind of behavior, meaning that k is not bounded in terms
+of the dimension (a somewhat bad bound can also be given in terms of the degree, see Lemma 4.7). As
+supporting evidence, we prove the following
+* Research partially supported by PRIN “Advances in Moduli Theory and Birational Classification”, GNSAGA-INdAM
+and the MIUR grant Dipartimenti di Eccellenza 2018-2022.
+** Research partially supported by a Simons Postdoctoral Fellowship from the Fields Institute for Research in Mathe-
+matical Sciences.
+Mathematics Subject Classification : Primary 14J60. Secondary 14J35, 14J40.
+1
+
+2
+A.F. LOPEZ, D. RAYCHAUDHURY
+Theorem 2.
+Let X ⊆ PN be a smooth irreducible variety of dimension n such that 2 ≤ n ≤ 12. If TX(k) is an
+Ulrich vector bundle, then k ≤ n + 1.
+We should point out that, for n ≥ 2, we know no examples with k ≥ 2 and only one example with
+k = 1. As a matter of fact, the case k = 1 can be completely characterized, as follows
+Theorem 3.
+Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1. Then TX(1) is an Ulrich vector
+bundle if and only if (X, OX(1)) = (S5, −2KS5), where S5 is a Del Pezzo surface of degree 5.
+On the other hand, for k = 2, we have
+Theorem 4.
+Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 4. Then TX(2) is not an Ulrich vector
+bundle.
+We do not know what happens for k = 2, n = 3, even though some evidence suggests that it might
+not be possible. Also, for surfaces, the cases k = 2, 3 point out to the possible existence, that needs to be
+further investigated, of some minimal surfaces of general type, as shown in Lemma 6.1 and Proposition
+6.2.
+Finally, in any dimension, another interesting case is the one in which ωX and OX(1) are numerically
+proportional. This is dealt with in Theorem 4.10, Corollaries 4.11 and 4.12.
+2. Notation
+Throughout the paper we work over the complex numbers. Moreover we henceforth establish the
+following
+Notation 2.1.
+• X is a smooth irreducible variety of dimension n ≥ 1.
+• H is a very ample divisor on X.
+• For any sheaf G on X we set G(l) = G(lH).
+• d = Hn is the degree of X.
+• C is a general curve section of X under the embedding given by H.
+• S is a general surface section of X under the embedding given by H, when n ≥ 2
+• g = g(C) = 1
+2[KXHn−1 + (n − 1)d] + 1 is the sectional genus of X.
+• For 1 ≤ i ≤ n − 1, let Hi ∈ |H| be general divisors and set Xn := X and Xi = H1 ∩ · · · ∩ Hn−i.
+3. Generalities on Ulrich bundles
+We collect some well-known facts, to be used sometimes later.
+Definition 3.1. Let E be a vector bundle on X. We say that E is an Ulrich vector bundle for (X, H)
+if Hi(E(−p)) = 0 for all i ≥ 0 and 1 ≤ p ≤ n.
+We have
+Lemma 3.2. Let E be a rank r Ulrich vector bundle for (X, H). Then
+(i) c1(E)Hn−1 = r
+2[KX + (n + 1)H]Hn−1,
+(ii) If n ≥ 2, then c2(E)Hn−2 = 1
+2[c1(E)2 − c1(E)KX]Hn−2 + r
+12[K2
+X + c2(X) − 3n2+5n+2
+2
+H2]Hn−2.
+(iii) χ(E(m)) = rd
+n!(m + 1) · · · (m + n).
+(iv) Hn(E(m)) = 0 if and only if m ≥ −n.
+(v) E∗(KX + (n + 1)H) is also an Ulrich vector bundle for (X, H).
+(vi) E is globally generated.
+(vii) h0(E) = rd.
+(viii) E is arithmetically Cohen-Macaulay (aCM), that is Hi(E(j)) = 0 for 0 < i < n and all j ∈ Z.
+(ix) E|Y is Ulrich on a smooth hyperplane section Y of X.
+
+ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE
+3
+Proof. We have
+(3.1)
+KXi = (KX + (n − i)H)|Xi , 1 ≤ i ≤ n.
+By [CH, Lemma 2.4(iii)] we have that
+c1(E)Hn−1 = deg(E|C) = r(d + g − 1)
+and using (3.1) on C = X1 we have
+KXHn−1 = 2(g − 1) − (n − 1)d
+thus giving (i). To see (ii) observe that the exact sequences, for 1 ≤ i ≤ n − 1,
+0 → TXi → (TXi+1)|Xi → H|Xi → 0
+and (3.1) give by induction that
+(3.2)
+c2(S) = c2(X2) = c2(X)Hn−2 + (n − 2)KXHn−1 +
+�n − 1
+2
+�
+d.
+It follows from [C1, Prop. 2.1(2.2)], (3.1), and Noether’s formula 12χ(OS) − K2
+S = c2(S) that
+c2(E)Hn−2
+= 1
+2[c1(E)2 − c1(E)(KX + (n − 2)H)]Hn−2 − r
+�
+Hn − [KX+(n−2)]2Hn−2+c2(S)
+12
+�
+=
+= 1
+2[c1(E)2 − c1(E)KX]Hn−2 − n−2
+2 c1(E)Hn−1 − r
+�
+Hn − [KX+(n−2)H]2Hn−2+c2(S)
+12
+�
+Now (ii) follows from the above equation by using (i) and (3.2). Next, (iii) is [CH, Lemma 2.6]. To see
+(iv) observe that E is 0-regular, hence it is q-regular for every q ≥ 0 and therefore Hn(E(q − n)) = 0,
+that is (iv). Also, (v) follows by definition and Serre duality, while (vi) follows by definition, since E is
+0-regular, and [Laz, Thm. 1.8.5]. For (vii), (viii) and (ix) see [ES, Prop. 2.1] (or [Be1, (3.1)]) and [Be1,
+(3.4)].
+□
+4. TX(k) Ulrich in any dimension
+We start by drawing some consequences on (X, H, k), of cohomological and numerical type, when
+TX(k) is an Ulrich vector bundle.
+Lemma 4.1. Let (X, H) = (Pn, OPn(1)), n ≥ 1. Then TX(k) is an Ulrich vector bundle if and only if
+n = 1 and k = −2.
+Proof. The assertion is obvious if (X, H, k) = (P1, OP1(1), −2). Vice versa suppose that TX(k) is an
+Ulrich vector bundle.
+If (X, H) = (Pn, OPn(1)), it follows by [ES, Prop. 2.1] (or [Be1, Thm. 2.3])
+that TPn(k) ∼= O⊕n
+Pn , hence 0 = det(TPn(k)) = OPn(nk + n + 1), so that 1 = −n(k + 1), giving
+n = 1, k = −2.
+□
+Lemma 4.2. (cohomological conditions)
+Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1. If TX(k) is an Ulrich vector bundle
+we have:
+(i) Either (X, H, k) = (P1, OP1(1), −2), or k ≥ 0.
+(ii) If n ≥ 2, then TX is aCM, that is Hi(TX(j)) = 0 for 1 ≤ i ≤ n − 1 and for every j ∈ Z. In
+particular Hi(TX) = 0 for 1 ≤ i ≤ n − 1.
+(iii) If k ≥ 1, then H0(TX) = 0, hence X has discrete automorphism group.
+(iv) If n ≥ 2, then X is infinitesimally rigid, that is H1(TX) = 0.
+(v) H0(KX + (n − k − 2)H) = 0 and, if n ≥ 2, also H0(KX + (n − k − 1)H) = 0.
+(vi) If q(X) ̸= 0 then H0(KX + (n − k)H) = 0.
+(vii) If k ≤ n − 1, then pg(X) = 0.
+(viii) Let a(X, H) = min{l ∈ Z : lH − KX ≥ 0}. Then k ≤ a(X,H)(n+2)
+2n
++ n+1
+2 .
+Moreover H0((⌈n(2k−n−1)
+n+2
+⌉ − 1)H − KX) = 0.
+(ix) KX − kH is not big.
+
+4
+A.F. LOPEZ, D. RAYCHAUDHURY
+Proof. Since TX(k) is an Ulrich vector bundle, it is globally generated by Lemma 3.2(vi).
+Now if
+k ≤ −1 we would have that 0 ̸= H0(TX(k)) ⊆ H0(TX(−1)).
+But then the Mori-Sumihiro-Wahl’s
+theorem [MS, Thm. 8], [W, Thm. 1] implies that (X, H) = (P1, OP1(2)), (Pn, OPn(1)).
+In the first
+case we have that 0 = Hi(TP1(k − 1)) = Hi(OP1(2k)) = 0 for i ≥ 0, a contradiction. In the second
+case apply Lemma 4.1. This proves (i). Now (ii) follows by Lemma 3.2(viii). If k ≥ 1 we have that
+H0(TX) ⊆ H0(TX(k − 1)) = 0, hence (iii). (iv) is implied by (ii). As for (v), recall that, as is well
+known, Ω1
+X(2) is globally generated. Now if H0(KX + (n − k − 2)H) ̸= 0 then we get the contradiction
+0 ̸= H0(Ω1
+X(2)) ⊆ H0(Ω1
+X(KX + (n − k)H)) = Hn(TX(k − n))∗ = 0.
+This gives the first part of (v). Similarly, if q(X) ̸= 0 and H0(KX + (n − k)H) ̸= 0 then we get the
+contradiction
+0 ̸= H0(Ω1
+X) ⊆ H0(Ω1
+X(KX + (n − k)H)) = Hn(TX(k − n))∗ = 0.
+This gives (vi). Now if n ≥ 2, consider Y ∈ |H| smooth. Then TX(k)|Y is an Ulrich vector bundle on
+Y by Lemma 3.2(ix), hence Hn−1(TX(k − n + 1)|Y ) = 0. Now the exact sequence
+0 → TY (k − n + 1) → TX(k − n + 1)|Y → OY (k − n + 2) → 0
+implies that Hn−1(OY (k − n + 2)) = 0. Hence, setting L = KX + (n − k − 1)H, we get by Serre’s
+duality that
+H0(L|Y ) = H0(KY + (n − k − 2)H|Y ) = 0.
+Therefore H0(L(−l)|Y ) = 0 for every l ≥ 0 and the exact sequences
+0 → L(−l − 1) → L(−l) → L(−l)|Y → 0
+show that h0(L(−l − 1)) = h0(L(−l)) for every l ≥ 0. Since this is zero for l ≫ 0, we get that they
+are all zero, hence H0(KX + (n − k − 1)H) = 0. This proves the second part of (v). Now, to see (vii),
+suppose that k ≤ n − 1. If n ≥ 2, we see that (v) gives H0(KX) ⊆ H0(KX + (n − k − 1)H) = 0,
+hence (vii). If n = 1 we have that k ≤ 0, hence X = P1 by (i) and Lemma 4.3(i). Observe that
+a(X, H)H − KX ≥ 0, hence (a(X, H)H − KX)Hn−1 ≥ 0 and using Lemma 4.3(ii), we get
+a(X, H) ≥ n(2k − n − 1)
+n + 2
+This gives (viii) since, by its own definition, H0((a(X, H) − 1)H − KX) = 0. Finally assume that
+KX − kH is big. Then Serre’s duality gives H0(TX(k)) = Hn(Ω1
+X(KX − kH))∗ = 0 by Bogomolov-
+Sommese vanishing [Bo, Thm. 4], contradicting Lemma 3.2(vi).
+□
+Lemma 4.3. (numerical conditions)
+Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1. If TX(k) is an Ulrich vector bundle
+we have:
+(i) d = (n+2)(g−1)
+nk−1
+. In particular either (X, H, k) = (P1, OP1(1), −2), or g = k = 0, or g ≥ 2.
+(ii) k = n+1
+2
++
+� n+2
+2nd
+�
+KXHn−1; equivalently KXHn−1 = n(2k−n−1)
+n+2
+d.
+(iii) If k < n+1
+2 , then X is rationally connected and Hi(OX) = 0 for every i ≥ 1.
+(iv) If k > n+1
+2 , then −KX is not pseff.
+(v) TX is semistable.
+(vi) If n ≥ 2, then K2
+XHn−2 ≤
+2n
+n−1c2(X)Hn−2.
+(vii) If n ≥ 2, then
+(12kn − 12k2 + 12k − 3n2 − 5n − 2)nd + 2(n + 12)K2
+XHn−2 + 2(n − 12)c2(X)Hn−2 = 0.
+Proof. Since c1(TX(k)) = −KX + nkH, we get by Lemma 3.2(i) that
+(−KX + nkH)Hn−1 = n
+2
+�
+KXHn−1 + (n + 1)d
+�
+and this gives (ii). Also, using KXHn−1 = 2(g − 1) − (n − 1)d, we get that
+(nk − 1)d = (n + 2)(g − 1).
+Now if nk − 1 = 0 then n = k = g = 1, but then TX(k) = OX(1) is not Ulrich. Therefore nk − 1 ̸= 0
+and d = (n+2)(g−1)
+nk−1
+. Hence g ̸= 1 and if g = 0 then either k = 0 or k ̸= 0 and in the latter case we
+
+ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE
+5
+have that nk < 1, hence (X, H, k) = (P1, OP1(1), −2) by Lemma 4.2(i). This proves (i). Next, (v)
+follows since Ulrich vector bundles are semistable by [CH, Thm. 2.9], hence Bogomolov’s inequality
+gives (vi). To see (iii), suppose that k < n+1
+2 . If n ≥ 2, then (ii) gives that KXHn−1 < 0, hence X is
+rationally connected by (v) and [BMQ, Main Thm.] (see also [CP, Thm. 1.1]). Hence, as is well known,
+Hi(OX) = 0 for every i ≥ 1. If n = 1 then k ≤ 0 and X = P1 by (i). Thus we get (iii). If k > n+1
+2 ,
+then either n = 1 and g ≥ 2 by (i), so that −KX is not pseff, or n ≥ 2 and (ii) gives that KXHn−1 > 0,
+hence again −KX is not pseff and we get (iv). To see (vii), observe that
+(4.1)
+c2(TX(k))Hn−2 = c2(X)Hn−2 − k(n − 1)KXHn−1 +
+�n
+2
+�
+k2d.
+From Lemma 3.2(ii), we get
+(4.2)
+c2(TX(k))Hn−2 =
+�n2k2
+2
+− n
+24
+�
+3n2 + 5n + 2
+��
+d−3nk
+2 KXHn−1+
+�
+1 + n
+12
+�
+K2
+XHn−2+ n
+12c2(X)Hn−2.
+Combining (4.1), (4.2) and (ii), we obtain (vii).
+□
+Definition 4.4. For n ≥ 1 we denote by Qn a smooth quadric in Pn+1.
+Lemma 4.5. Let (X, H) = (Qn, OQn(1)), n ≥ 1. Then TX(k) is not an Ulrich vector bundle for any
+integer k.
+Proof. Since g = 0, it follows by Lemma 4.3(i) that k = 0 and 2 = d = n + 2, a contradiction.
+□
+We will use the nef value of (X, H):
+(4.3)
+τ(X, H) = min{t ∈ R : KX + tH is nef}.
+We observe that in [BS, Def. 1.5.3] the nef value is defined only when KX is not nef. On the other
+hand, it makes sense and it will be used, throughout this paper, also when KX is nef.
+A very useful observation is the following.
+Lemma 4.6. Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1. If TX(k) is an Ulrich
+vector bundle, then Ω1
+Y (KX |Y +(n+1−k)H|Y ) is globally generated for any smooth subvariety Y ⊆ X.
+Moreover:
+(i) If ±(KX + n(n+1−2k)
+n+2
+H) is pseff, then KX ≡ n(2k−n−1)
+n+2
+H.
+(ii) τ(X, H) ≥ n(n+1−2k)
+n+2
+.
+(iii) τ(X, H) ≤ n −
+nk
+n+1. In particular, if KX is not nef, then k ≤ n.
+(iv) If k ≥ n + 1, then KX is ample.
+Proof. Note that Ω1
+X(KX + (n + 1 − k)H) is Ulrich and globally generated by Lemma 3.2(v) and (vi).
+Since Ω1
+X(KX + (n + 1 − k)H) surjects onto Ω1
+Y (KX |Y + (n + 1 − k)H|Y ), the latter is also globally
+generated. Moreover so is det(Ω1
+X(KX+(n+1−k)H) = (n+1)KX+n(n+1−k)H, hence we get (iii) and,
+if k ≥ n+2, we also deduce that KX is ample. On the other hand, if k = n+1, then Ω1
+X(KX) is Ulrich,
+and we claim that det(Ω1
+X(KX)) = (n + 1)KX is ample. In fact, if not, then [LS, Thm. 1] implies that
+there is a line L ⊂ X such that Ω1
+X(KX)|L is trivial. Hence (n + 1)KX · L = deg(Ω1
+X(KX)|L) = 0. But
+then we have a surjection Ω1
+X(KX)|L → Ω1
+L, contradicting the fact that Ω1
+L is not globally generated.
+This proves (iv). As for (i) and (ii), set q = n(n+1−2k)
+n+2
+, so that (KX + qH)Hn−1 = 0 by Lemma 4.3(ii).
+Now if ±(KX + qH) is pseff, then KX + qH ≡ 0 by [FL2, Cor. 3.15] (see also [FL1, Prop. 3.7]), thus
+proving (i). Also, (i) implies that either KX ≡ −qH and then τ(X, H) = q, or KX + qH is not pseff,
+hence not nef. Therefore, in the latter case, τ(X, H) > q, proving (ii).
+□
+Lemma 4.7. Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1. If TX(k) is an Ulrich
+vector bundle we have that
+k ≤ (n + 2)(d − 4) + 4
+4n
+.
+Proof. We have X ⊂ PH0(H) = PN. If N = n then (X, H) = (Pn, OPn(1)) and Lemma 4.1 gives that
+n = 1 and k = −2. Since d = 1 we have that k = −2 ≤ − 5
+4 = (n+2)(d−4)+4
+4n
+.
+
+6
+A.F. LOPEZ, D. RAYCHAUDHURY
+We now show that it cannot be that N = n + 1. Assume that N = n + 1, so that d ≥ 2. If n = 1 we
+have that KX = (d − 3)H, g =
+�d−1
+2
+�
+and k = 3(d−3)
+2
++ 1 by Lemma 4.3(i). Now 0 = H0(TX(k − 1)) =
+H0((−d + 2 + k)H) and therefore −d + 2 + k ≤ −1, giving the contradiction d ≤ 1. Hence n ≥ 2 and
+since C ⊂ P2 we have that g − 1 = d(d−3)
+2
+and Lemma 4.3(i) implies that d = 2(nk−1)
+n+2
++ 3. On the other
+hand Lemma 4.2(v) gives that
+0 = H0(KX + (n − k − 1)H) = H0((d − k − 3)H)
+and therefore
+2(nk − 1)
+n + 2
+− k ≤ −1
+that is k(n − 2) + n ≤ 0, a contradiction.
+Therefore N ≥ n + 2 and C ⊂ PN−n+1 can be projected isomorphically to a non-degenerate smooth
+irreducible curve in P3. Then Castelnuovo’s bound gives that g − 1 ≤ d(d−4)
+4
+and Lemma 4.3(ii) implies
+the required bound on k.
+□
+A nice consequence of the above lemmas is the following.
+Proposition 4.8. There does not exist any (X, H, k) with TX(k) an Ulrich vector bundle, when:
+(i) KX ≡ 0.
+(ii) ±KX is pseff and k = n+1
+2 .
+Proof. Under hypothesis (ii), we get from Lemma 4.6(i) that KX ≡ 0. Thus we will be done if we prove
+(i). Assume next that KX ≡ 0, so that k = n+1
+2
+by Lemma 4.3(ii) and n ≥ 3 by Lemma 4.3(i). Since
+H − KX is ample, it follows by Kodaira vanishing that Hi(H) = Hi(KX + H − KX) = 0 for i > 0,
+hence
+h0(KX + H) = χ(KX + H) = χ(H) = h0(H) ̸= 0.
+On the other hand, Lemma 4.2(v) gives that h0(KX + n−3
+2 H) = 0, whence, if n ≥ 5, we get the
+contradiction h0(KX + H) = 0.
+It remains to consider the case n = 3, k = 2. Note that pg(X) = 0 by Lemma 4.2(vii) and q(X) = 0,
+for otherwise Lemma 4.2(vi) gives that h0(KX + H) = 0. Therefore χ(OX) ≥ 1. On the other hand
+χ(OX) =
+1
+24c1(X)c2(X) = 0, a contradiction.
+□
+We now prove Theorem 2.
+Proof of Theorem 2. By the Hodge index theorem we have that H2
+|SK2
+S ≤ (H|SKS)2, that is
+(4.4)
+dK2
+XHn−2 ≤ (KXHn−1)2.
+Using Lemma 4.3(vi), (vii) and (4.4) we obtain that
+0 = (12kn − 12k2 + 12k − 3n2 − 5n − 2)nd + 2(n + 12)K2
+XHn−2 + 2(n − 12)c2(X)Hn−2 ≤
+≤ (12kn − 12k2 + 12k − 3n2 − 5n − 2)nd + 3n2 + 11n + 12
+n
+K2
+XHn−2 ≤
+≤ (12kn − 12k2 + 12k − 3n2 − 5n − 2)nd + 3n2 + 11n + 12
+nd
+(KXHn−1)2
+which, using Lemma 4.3(ii) becomes
+4nk2 − 4n(n + 1)k − 3n2 − 7n − 4 ≤ 0
+giving
+k ≤ n2 + n +
+√
+n4 + 5n3 + 8n2 + 4n
+2n
+< n + 2.
+□
+The case k = 0 is known:
+Theorem 4.9. ([BMPT, Prop. 4.1, Thm. 4.5])
+Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1. Then TX is an Ulrich vector bundle
+if and only if (X, H) = (P1, OP1(3)), (P2, OP2(2)).
+
+ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE
+7
+We will give a quick alternative proof in section 8.
+Next we study the case when KX and H are proportional.
+Theorem 4.10. Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1. Suppose that the
+numerical classes of H and KX are proportional and that either
+(i) 1 ≤ n ≤ 11 and either k ≤ n+1
+2
+or k ≥ n + 2; or
+(ii) n = 12, or
+(iii) n ≥ 13 and k ̸∈ {n + 2, n + 3}.
+Then TX(k) is Ulrich if and only if (X, H, k) is one of the following:
+(1) (P1, OP1(1), −2),
+(2) (P1, OP1(3), 0),
+(3) (P2, OP2(2), 0),
+(4) (S5, −2KS5, 1), where S5 is a Del Pezzo surface of degree 5.
+Proof. In the cases (1)-(4) we have that TX(k) is Ulrich by Lemma 4.3(i), Theorem 4.9 and Theorem
+6.3.
+Vice versa, suppose that the numerical classes of H and KX are proportional and that we are under
+one of hypotheses (i), (ii) or (iii) and that TX(k) is Ulrich.
+Observe that, since N 1(X) is a torsion free finitely generated abelian group, we can find an ample
+primitive divisor A and some r, s ∈ Z such that s > 0, H ≡ sA and KX ≡ −rA. In particular, Lemma
+4.3(ii) gives
+(4.5)
+r(n + 2) = n(n + 1 − 2k)s.
+If k ≤ 0 we are in cases (1)-(3) by Lemma 4.2(i) and Theorem 4.9. Hence assume that k ≥ 1. Note
+that Proposition 4.8 shows that k ̸= n+1
+2 .
+If (ii) or (iii) holds, since the numerical classes of H and KX are proportional, Lemma 4.3(vi), (ii)
+and (vii) imply that
+4nk2 − 4n(n + 1)k − 3n2 − 7n − 4 ≥ 0
+so that k > n + 1. This is a contradiction under hypothesis (ii) by Theorem 2. Under hypothesis (iii),
+we get that k ≥ n + 4. Then it follows by (4.5) that
+−r − ks = (n − 2)k − n2 − n
+n + 2
+s ≥ n − 8
+n + 2s > 0
+hence KX − kH = (−r − ks)A is ample, contradicting Lemma 4.2(ix). Thus it remains to consider
+hypothesis (i).
+Now assume (i), so that n ≥ 2 and Theorem 2 implies that it cannot be that k ≥ n + 2. Hence
+k < n+1
+2
+and Lemma 4.3(ii) implies that X is Fano. Consequently, the numerical and linear equivalence
+for divisors coincide on X and then KX = −rA and H = sA. We can also assume that r ≤ n − 1, for
+otherwise, as is well known, (X, H) = (Pn, OPn(1)), (Qn, OQn(1)), contradicting Lemmas 4.1 and 4.5.
+Next, set PA(t) := χ(KX + tA), so that PA(t) = h0(KX + tA) whenever t ≥ 1 is an integer. By
+Riemann-Roch (see for example [Ho, eq. (1), p. 2], we have
+(4.6)
+PA(t) = An
+n! tn + An−1KX
+2(n − 1)!tn−1 + An−2(K2
+X + c2(X))
+12(n − 2)!
+tn−2 + · · · + (−1)nχ(OX).
+Note that n is even, for otherwise n and n + 2 are coprime, and consequently n divides r by (4.5),
+hence r ≥ n, a contradiction.
+Set n = 2m, where 1 ≤ m ≤ 5.
+If m = 1 we have that n = 2, k = 1 and we are in case (4) by Theorem 6.3.
+We will now exclude the remaining cases for m.
+Note that we can rewrite (4.5) as
+(4.7)
+(m + 1)r = m(2m − 2k + 1)s.
+Case 1: m = 2. In this case k ≤ 2 and r ≤ 3.
+(1a): k = 1. We see that r = 2s. Thus (r, s) = (2, 1), contradicting Lemma 4.2(v).
+(1b): k = 2. We see that 2s = 3r. Thus (r, s) = (2, 3) and Lemma 4.2(v) shows that h0(A) = 0.
+This contradicts [A, Lemma 2] or [K, Thm. 5.1].
+
+8
+A.F. LOPEZ, D. RAYCHAUDHURY
+Case 2: m = 3. In this case k ≤ 3 and r ≤ 5.
+(2a): k = 1. We see that 4r = 15s. Thus, 15 divides r which is clearly impossible.
+(2b): k = 2. We see that 9s = 4r. Thus, 9 divides r which is a contradiction.
+(2c): k = 3. We see that 3s = 4r. Thus (r, s) = (3, 4) and Lemma 4.2(v) shows that h0(KX+8A) = 0.
+This is a contradiction by [GL, Thm. 1.2], as KX + 8A is base-point-free.
+Case 3: m = 4. In this case k ≤ 4 and r ≤ 7.
+(3a): k = 1. We see that 5r = 28s. Thus, 28 divides r which is clearly impossible.
+(3b): k = 2. We see that 4s = r. Thus (r, s) = (4, 1) and Lemma 4.2(v) shows h0(KX + 5A) =
+h0(H) = 0, a contradiction.
+(3c): k = 3. We see that 12s = 5r. Thus, 12 divides r which is absurd.
+(3d): k = 4. We see that 4s = 5r. Thus, (r, s) = (4, 5).
+We have KX = −4A and H = 5A. Hence H0(KX + 15A) = 0 by Lemma 4.2(v). Then
+PA(1) = PA(2) = PA(3) = PA(5) = PA(10) = PA(15) = 0,
+PA(0) = PA(4) = 1
+and
+PA(t) = A8
+8! (t − 1)(t − 2)(t − 3)(t − 5)(t − 10)(t − 15)(t2 + at + b).
+Therefore
+(4.8)
+1 = PA(0) = A8
+8! (4500b) =⇒ A8
+8! b =
+1
+4500
+and calculating the coefficient of t7 in (4.6) we get
+(4.9)
+A8
+8! (a − 36) = A7KX
+2(7!)
+=⇒ a = 20.
+We also know that PA(4) = 1 and that gives us
+−A8
+8! (396)(16 + 4a + b) = 1.
+We simplify the above using (4.8) and (4.9) to obtain
+−38016A8
+8! = 1 + 396
+4500
+which is clearly absurd.
+Case 4: m = 5. In this case k ≤ 5 and r ≤ 9.
+(4a): k = 1. We see that 2r = 15s. Thus, 15 divides r which is impossible.
+(4b): k = 2. We see that 35s = 6r. Thus, 35 divides r which is also impossible.
+(4c): k = 3. We see that 25s = 6r. Thus, 25 divides r which is also impossible.
+(4d): k = 4. We see that 5s = 2r. Thus (r, s) = (5, 2) and Lemma 4.2(v) shows that h0(KX +10A) =
+0. Then
+PA(1) = PA(2) = PA(3) = PA(4) = PA(6) = PA(8) = PA(10) = 0,
+PA(0) = PA(5) = 1
+so that we obtain
+PA(t) = A10
+10! (t − 1)(t − 2)(t − 3)(t − 4)(t − 6)(t − 8)(t − 10)(t3 + at2 + bt + c).
+Therefore
+(4.10)
+1 = PA(0) = −A10
+10! (11520c) = 1 =⇒ A10
+10! c = −
+1
+11520.
+and calculating the coefficient of t9 in (4.6) we get
+(4.11)
+A10
+10! (a − 34) = A9KX
+2(9!)
+=⇒ a = 9.
+We also know that PA(5) = 1 and that gives us
+−A10
+10! (360)(125 + 25a + 5b + c) = 1.
+
+ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE
+9
+We simplify the above using (4.10) and (4.11) to obtain
+(4.12)
+A10
+10! b =
+1
+5(11520) −
+1
+5(360) − 70A10
+10! .
+Finally, calculating the coefficient of t8 in (4.6) we get
+(4.13)
+A10
+10! (b − 34a + 463) = A8(K2
+X + c2(X))
+12(8!)
+.
+We simplify (4.13) using (4.11) and (4.12) to obtain
+(4.14)
+67A10 + 5A8c2(X) + 1302 = 0.
+On the other hand, Lemma 4.3(vii) shows that
+115A10 = A8c2(X)
+and combining with (4.14), we get that A10 is negative, which is clearly impossible.
+(4e): k = 5. We see that 5s = 6r. Thus (r, s) = (5, 6). We have KX = −5A and H = 6A. Again
+H0(KX + 24A) = 0 by Lemma 4.2(v). Then
+PA(1) = PA(2) = PA(3) = PA(4) = PA(6) = PA(12) = PA(18) = PA(24) = 0,
+PA(0) = PA(5) = 1.
+Thus, we obtain
+PA(t) = A10
+10! (t − 1)(t − 2)(t − 3)(t − 4)(t − 6)(t − 12)(t − 18)(t − 24)(t2 + at + b).
+Then
+(4.15)
+1 = PA(0) = A10
+10! (746496b) =⇒ A10
+10! b =
+1
+746496.
+calculating the coefficient of t9 in (4.6) we get
+(4.16)
+A10
+10! (a − 70) = A9KX
+2(9!)
+=⇒ a = 45.
+We also know that PA(5) = 1 and that gives us
+A10
+10! (41496)(25 + 5a + b) = 1.
+We simplify the above using (4.15) and (4.16) to obtain
+A10 =
+�705000
+746496
+�
+(10!)
+which is clearly absurd since A10 is an integer.
+□
+Corollary 4.11. Suppose that KX = eH, e ∈ Z (hence, in particular, if Pic(X) = ZH). Then TX(k)
+is Ulrich if and only if (X, H, k) = (P1, OP1(1), −2).
+Proof. This follows by [Lop, Prop. 4.1(i)]. We give another proof. We have that e = n(2k−n−1)
+n+2
+by
+Lemma 4.3(ii). If k ≤ n+1
+2
+it follows by Theorem 4.10 that (X, H, k) = (P1, OP1(1), −2). Now assume
+that k > n+1
+2 , so that e ≥ 1. If n ≥ 2, Lemma 4.2(v) gives that k ≥ n + e, hence k(n − 2) + n ≤ 0, a
+contradiction. Then n = 1 and e = 2(k−1)
+3
+. But 0 = H0(TX(k − 1)) = H0((k − 1 − e)H), hence e ≥ k,
+so that k ≤ −2, contradicting k > 1.
+□
+Corollary 4.12. Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 2 with TX(k) is Ulrich.
+Suppose that there is an ample line bundle A on X such that KX = rA, H = sA for some r, s ∈ Z
+(hence, in particular, if Pic(X) ∼= Z). Let m(H, A) := min{m ≥ 0 : H0(mH + qA) ̸= 0 for all q ≥ 1}.
+Then:
+(i) m(H, A) > (n−2)k−2
+n+2
+and, if n ≥ 3, then k < (n+2)m(H,A)+2
+n−2
+.
+(ii) If A is effective, then n = 2.
+(iii) If m(H, A) ≤ n − 3, then n ≤ 11.
+
+10
+A.F. LOPEZ, D. RAYCHAUDHURY
+Proof. Observe first that, if n ≥ 3, then (n−2)k−2 ≥ k−2 ≥ 0: In fact if k ≤ 1 we have a contradiction
+by Theorem 4.10. Now Lemma 4.3(ii) implies that
+(4.17)
+r(n + 2) = n(2k − n − 1)s
+and Lemma 4.2(v) gives
+(4.18)
+0 = H0(KX + (n − k − 1)H) = H0((nk − 2k − 2)s
+n + 2
+A).
+To see (i), notice that it is obvious for n = 2, for m(H, A) ≥ 0 by definition. If n ≥ 3 we see by (4.18)
+that (nk−2k−2)s
+n+2
+∈ Z and we can write (nk−2k−2)s
+n+2
+= as+b for some a, b ∈ Z with a ≥ 0, 0 ≤ b < s. Since
+H0(aH + bA) = 0 by (4.18), we get that
+(n − 2)k − 2
+n + 2
+− 1 < a ≤ m(H, A) − 1
+giving (i). Now suppose that A is effective. If n ≥ 3 we know that (n − 2)k − 2 ≥ 0, contradicting
+(4.18). This proves (ii). To see (iii), notice that if n ≥ 12, then k ≥ n + 2 by Theorem 4.10(ii) and (iii).
+Hence (n−2)k−2
+n+2
+> n − 3 and (4.18) gives that
+0 = H0((nk − 2k − 2)s
+n + 2
+A) = H0((n − 3)H + qA)
+for some q ≥ 1, contradicting the hypothesis m(H, A) ≤ n − 3.
+□
+5. Curves
+Throughout this section we will have that X ⊆ PN is a smooth irreducible curve.
+It follows by Lemma 4.3(i) and Theorem 4.9 that if n = 1 and TX(k) is an Ulrich line bundle, then
+(X, H, k) = (P1, OP1(1), −2), (P1, OP1(3), 0) or k ≥ 2 and g ≥ 2.
+We will give below examples with k = 2, 3, essentially on any curve. Then we will give a sharp bound
+on k depending on the genus.
+The case k = 2 can be characterized. Note that g ≥ 3 when k = 2 by Lemma 4.3(i).
+Lemma 5.1. Let X ⊆ PN be a smooth irreducible curve. Then TX(2) is Ulrich if and only if there
+exists M ∈ Pic(X) such that Hi(M) = 0 for i ≥ 0 and H = KX + M is very ample. This occurs if and
+only if g ≥ 3.
+Proof. If TX(2) is Ulrich, set M = H −KX. Then Hi(M) = Hi(TX(1)) = 0 for i ≥ 0 and KX +M = H
+is very ample. Vice versa let M ∈ Pic(X) be such that Hi(M) = 0 for i ≥ 0 and H = KX + M is very
+ample. Then Hi(TX(1)) = Hi(M) = 0 for i ≥ 0, so that TX(2) is Ulrich.
+Suppose that g ≥ 3 and let M ∈ Pic(X) be such that Hi(M) = 0 for i ≥ 0.
+We claim that
+H := KX + M is very ample. In fact deg M = g − 1 by Riemann-Roch, hence deg H = 3g − 3. If g ≥ 4,
+then deg H ≥ 2g + 1, hence H is very ample. If g = 3 we have that deg H = 2g and, as is well known,
+H is very ample unless H = KX + P + Q for two points P, Q ∈ X. But then M = P + Q is effective,
+a contradiction. Instead, if g = 2 we have that deg(KX + M) = 3 > 2g − 2, hence h0(KX + M) = 2
+by Riemann-Roch.
+Let P + Q + R be an effective divisor linearly equivalent to KX + M.
+Then
+KX + M − P − Q ∼ R, hence h0(KX + M − P − Q) = 1 and therefore KX + M is not very ample.
+□
+Example 5.2. The case k = 3 occurs on any curve X with (necessarily) odd genus g ≥ 9. This was
+suggested to us by E. Sernesi, whom we thank.
+Proof. Let d = 3(g−1)
+2
+. We claim that a general H ∈ Picd(X) is very ample. In fact, first observe that
+H1(H) = 0, for otherwise KX − H ≥ 0. But KX − H is a general line bundle of degree g−1
+2
+≤ g − 1,
+hence h0(KX − H) = 0. Now, if H were not very ample, there will be two points p, q ∈ X such that
+h0(H − p − q) ≥ h0(H) − 1. But this can be rewritten, by Riemann-Roch, as h1(H − p − q) ≥ 1, that
+is KX − H + p + q ≥ 0. Hence there are some points p1, . . . , p g+3
+2
+∈ X such that
+KX − H + p + q ∼ p1 + . . . + p g+3
+2
+that is
+H ∼ KX − p1 − . . . − p g+3
+2
++ p + q.
+
+ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE
+11
+This means that H is in the image of the morphism h : X
+g+7
+2
+→ Picd(X) sending (p1, . . . , p g+3
+2 , p, q) to
+KX − p1 − . . . − p g+3
+2
++ p + q. But dim Imh ≤ g+7
+2
+< g, contradicting that H is general. This proves
+that there is a non-empty open subset W of Picd(X) such that any H ∈ W is very ample.
+Consider the surjective morphism ψ : Picd(X) → Pic3g−3(X) given by ψ(L) = 2L and the isomor-
+phism ϕ : Pic3g−3(X) → Picg−1(X) given by ϕ(L) = L − KX. Let U be the non-empty open subset
+of Picg−1(X) such that Hi(M) = 0 for i ≥ 0 for any M ∈ U. Now let H ∈ ψ−1(ϕ−1(U)) ∩ V ∩ W.
+Then H is very ample and 2H = KX + M. In the embedding given by H we have that Hi(TX(2)) =
+Hi(−KX + 2H) = Hi(M) = 0 for i ≥ 0, hence TX(3) is Ulrich.
+□
+Remark 5.3. If k ≥ 1 and g − 1 is a prime number, then k ∈ {2, 4}.
+Proof. By Lemma 4.3(i) we get that (k − 1)d = 3(g − 1) and g ≥ 2, hence d ≥ 4. If 3 does not divide
+k − 1 we get that 3 divides d and (k − 1)d
+3 = g − 1, so that k = 2. If 3 divides k − 1 we get that
+k−1
+3 d = g − 1, so that k = 4.
+□
+Example 5.4. Every odd k ≥ 3 occurs.
+Proof. Let E be an elliptic curve, let D be a divisor of degree 3 on E and let S = E × P1 with two
+projections π1 : S → E, π2 : S → P1. Set C0 = π∗
+2(OP1(1)). Then H = C0 + π∗
+1D is very ample on S.
+Let M ∈ Pic0(E) be not 2-torsion and let B = k−1
+2 D + M. Again H1 := (k + 2)C0 + π∗
+1B is very ample
+on S. Set
+L = −KS + (k − 1)H = (k + 1)C0 + π∗
+1(2B − 2M)
+so that
+L − H1 = −C0 + π∗
+1(B − 2M)
+while
+L − 2H1 = −(k + 3)C0 + π∗
+1(−2M)
+and it is easily seen by the K¨unneth formula, that Hi(L − pH1) = 0 for i ≥ 0, 1 ≤ p ≤ 2. Hence, if
+X ∈ |H1| is a smooth irreducible curve, the exact sequence
+0 → L − 2H1 → L − H1 → TX(k − 1) → 0
+shows that Hi(TX(k − 1)) = 0 for i ≥ 0, that is TX(k) is an Ulrich line bundle.
+□
+We now give a bound for k.
+We first analyze a special case.
+We use the notation (a; b1, b2, b3, b4, b5, b6) ∈ Z7 for the divisor
+aε∗L − �6
+i=1 biEi on a smooth cubic W ⊂ P3, where ε : W → P2 is the blow up in six points, no three
+collinear and not on a conic, with exceptional divisors Ei and L is a line in P2.
+Lemma 5.5. Let X ⊂ P3 be a smooth irreducible curve of genus 3 and degree 6 lying on a smooth cubic
+W. Then TX(2) is Ulrich if and only if X is linearly equivalent to one of the following divisors on W:
+(5.1)
+(4; 1, 1, 1, 1, 1, 1), (5; 2, 2, 2, 1, 1, 1), (6; 3, 2, 2, 2, 2, 1), (7; 3, 3, 3, 2, 2, 2), (8; 3, 3, 3, 3, 3, 3).
+Proof. Let D = −KW .
+We have that D · X = 6 and X2 = 10.
+Thus Riemann-Roch gives that
+χ(2D − X) = 0. Further, D(2D − X) = 0, whence H0(2D − X) = 0, for otherwise X ∼ 2D and then
+X2 = 12, a contradiction. Also, D(−3D + X) = −3, whence H0(−3D + X) = 0, which, be Serre’s
+duality, is H2(2D − X) = 0. Thus, also H1(2D − X) = 0. Now the exact sequence
+0 → 2D − 2X → 2D − X → TX(1) → 0
+gives that
+h1(TX(1)) = h2(2D − 2X) = h0(3KW + 2X)
+by Serre’s duality. Since deg TX(1) = 2, we deduce by Riemann-Roch, that TX(2) is Ulrich if and only
+if H1(TX(1)) = 0, hence
+(5.2)
+TX(2) is Ulrich if and only if H0(3KW + 2X) = 0.
+Let (a; b1, · · · , b6) with b1 ≥ b2 ≥ · · · ≥ b6 ≥ 0 be the class of X. It follows from the assumption on
+degree and genus that (A.7) holds, whence X is as in (5.1) by Lemma A.2. Using (5.2), it remains to
+show that H0(3KW + 2X) = 0 in all of these cases.
+
+12
+A.F. LOPEZ, D. RAYCHAUDHURY
+In case (4; 1, 1, 1, 1, 1, 1), we have that 3KW + 2X = (−1; 1, 1, 1, 1, 1, 1) is clearly not effective.
+In case (5; 2, 2, 2, 1, 1, 1), we have that 3KW + 2X = (1; −1, −1, −1, 1, 1, 1). If it were effective, then
+so would be (1; 0, 0, 0, 1, 1, 1), a contradiction since no three blown-up points are collinear.
+In case (6; 3, 2, 2, 2, 2, 1), we have that 3KW + 2X = (3; 3, 1, 1, 1, 1, −1). Assume it is effective. In-
+tersecting with (1; 1, 1, 0, 0, 0, 0) we see that (2; 2, 0, 1, 1, 1, −1) must be effective. Intersecting the latter
+with (1; 1, 0, 1, 0, 0, 0) we conclude that (1; 1, 0, 0, 1, 1, −1) must be effective, hence also (1; 1, 0, 0, 1, 1, 0),
+a contradiction since no three blown-up points are collinear.
+In case (7; 3, 3, 3, 2, 2, 2), we have that 3KW + 2X = (5; 3, 3, 3, 1, 1, 1). Assume it is effective. In-
+tersecting with (1; 1, 1, 0, 0, 0, 0) we see that (4; 2, 2, 3, 1, 1, 1) must be effective. Intersecting the latter
+with (1; 1, 0, 1, 0, 0, 0) we conclude that (3; 1, 2, 2, 1, 1, 1) must be effective. Finally, intersecting with
+(1; 0, 1, 1, 0, 0, 0) we conclude that (2; 1, 1, 1, 1, 1, 1) must be effective, a contradiction since the blown-
+up points do not lie on a conic.
+In case (8; 3, 3, 3, 3, 3, 3), we have that 3KW +2X = (7; 3, 3, 3, 3, 3, 3). Observe that D(3KW +2X) = 3.
+Thus, if 3KW + 2X were effective, it would contain a divisor Γ that is either irreducible, or is a union
+of three lines, or is a union of a line and a conic. Now pa(Γ) = −3, hence Γ is not irreducible. On the
+other hand, since the first coefficient of a line in W is at most 2 and of a conic at most 3, we see that
+Γ, whose first coefficient is 7, is not a union of three lines nor of a line and a conic. This contradiction
+shows that 3KW + 2X is not effective.
+□
+Now we give the sharp bound.
+Proof of Theorem 1. First assume that g = 0. Then Lemma 4.3(i) gives that k ≤ 0, that is the required
+bound, and if equality holds, then Theorem 4.9 shows that X is a curve of type (1, 2) on a smooth
+quadric.
+Thus, by Lemma 4.3(i), we can now assume that g ≥ 2.
+Observe that h0(H) ≥ 4. In fact, the only possibility remaining is that h0(H) = 3. But then KX =
+(d−3)H, g =
+�d−1
+2
+�
+and k = 3(d−3)
+2
++1 by Lemma 4.3(i). Now 0 = H0(TX(k −1)) = H0((−d+2+k)H)
+and therefore −d + 2 + k ≤ −1, giving the contradiction d ≤ 1.
+Now, if X has general moduli, since it has a g3
+d, the Brill-Noether theorem implies that ρ(g, 3, d) ≥ 0,
+that is d ≥ 3g+12
+4
+. By Lemma 4.3(i) we get that 3(g−1)
+k−1
+≥ 3g+12
+4
+, that gives k ≤ 4. This proves the last
+assertion of the theorem.
+Turning to the first assertion, let X ⊆ PN be a smooth irreducible curve of genus g ≥ 2 such that
+TX(k) is an Ulrich line bundle and assume that
+k ≥
+√8g + 1 − 1
+2
+.
+Using Lemma 4.3(i), the above inequality can be rephrased as
+(5.3)
+g ≥ 2
+9d2 − d + 1.
+Consider a general projection X′ of X to P3. Note that X′ ∼= X, hence TX′(k) is Ulrich. We first observe
+that X′ cannot be a complete intersection (hence, in particular, X′ is nondegenerate), for otherwise
+TX′(k) = lH for some l ∈ Z. Now TX′(k), being Ulrich, is globally generated by Lemma 3.2(vi), hence
+l ≥ 0. Also 0 = H0(TX′(k − 1)) = H0((l − 1)H) and therefore l = 0. Hence Lemma 3.2(vii) gives that
+d = h0(TX′(k)) = 1, a contradiction.
+Using Lemma 4.3(i) and Castelnuovo’s bound, we get that either (d, g, k) = (6, 3, 2) or d ≥ 7.
+Suppose that d ≥ 7.
+We aim to show that X′ must lie on a smooth quadric.
+To this end, observe that (5.3) and Lemma 4.3(i) imply that
+(5.4)
+g >
+�
+1
+6d(d − 3) + 1
+if d ≡ 0 (mod 3)
+1
+6d(d − 3) + 1
+3
+if d ≡ 1, 2 (mod 3)
+unless d = 9 and g = 10. But in the latter case it is easy to show that if X′ does not lie on a quadric,
+then it is a complete intersection of two cubics, a contradiction. Therefore (5.4) and [Ha2, Thm. 3.2]
+give that X′ lies on a quadric Q. Moreover Q is smooth, for otherwise it must be a cone, d = 2b + 1 is
+
+ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE
+13
+odd and g = b2 − b by [Ha1, Ex. V.2.9]. But then Lemma 4.3(i) gives that 4(k − 1) = 6b − 9 −
+3
+2b+1,
+and therefore b = 1, a contradiction.
+Thus X′ is a curve of type (a, b) on Q, with 2 ≤ a ≤ b. In particular X′ is linearly normal, hence
+X = X′. In the exact sequence
+0 → OQ(k + 1 − 2a, k + 1 − 2b) → OQ(k + 1 − a, k + 1 − b) → TX(k − 1) → 0
+since H0(TX(k − 1)) = 0, we get that
+H0(OQ(k + 1 − 2a, k + 1 − 2b)) = H0(OQ(k + 1 − a, k + 1 − b))
+hence k + 1 − b ≤ −1, for otherwise k + 1 − a ≥ k + 1 − b ≥ 0, but then X is a base-component of
+|OQ(k + 1 − a, k + 1 − b)|, contradicting the fact that this linear system is base-point-free. Therefore
+b ≥ k + 2. Moreover Lemma 4.3(i) can be rewritten now as
+(a + b)(k − 1) = 3((a − 1)(b − 1) − 1)
+that is
+a =
+b(k + 2)
+3b − k − 2
+and it is readily seen that b ≥ k + 2 is equivalent to a ≤ k
+2 + 1. Therefore b ≥ 2a. But the maximum
+genus of a curve of type (a, b) with b ≥ 2a and degree d is attained when b = 2
+3d. Therefore
+g ≤ (1
+3d − 1)(2
+3d − 1) = 2
+9d2 − d + 1.
+This shows that the inequality in (5.3) cannot be strict, and therefore g ≤ 2
+9d2−d+1, which is equivalent
+to (1.1). Moreover, if equality holds in (1.1), then it holds in (5.3) and therefore X is a curve of type
+(a, b) with b = 2
+3d, hence b = 2a and 2a = b ≥ k + 2 ≥ 2a, so that k is even, a = k
+2 + 1 and b = k + 2.
+Next consider the only remaining case, (d, g, k) = (6, 3, 2).
+Again X′ is linearly normal, hence X = X′. Also we have equality in (5.3) and if X lies on a quadric,
+then it must be of type (2, 4) and we are done in this case. Suppose therefore that X does not lie on
+a quadric. Then it is easily seen that JX/P3(3) is 0-regular, hence globally generated, and we get that
+X is contained in a smooth cubic. Therefore X is one of the curves (5.1) by Lemma 5.5 and TX(2) is
+Ulrich.
+Finally, to show that the bound (1.1) is sharp for every even k ≥ 0, let X be a curve of type
+(k
+2 + 1, k + 2) on a smooth quadric Q ⊂ P3, so that k =
+√8g+1−1
+2
+. It remains to show that TX(k) is
+Ulrich. Set k = 2c. We have
+TX(k − 1) = −KX + (k − 1)H = OQ(c, −1)|X
+and the exact sequence
+0 → OQ(−1, −2c − 3) → OQ(c, −1) → OQ(c, −1)|X → 0
+shows that Hi(OQ(c, −1)|X) = 0 for i ≥ 0, since Hi(OQ(c, −1)) = Hi(OQ(−1, −2c − 3)) = 0 for i ≥ 0.
+Hence TX(k) is Ulrich.
+□
+6. Surfaces
+Throughout this section we will have that X ⊆ PN is a smooth irreducible surface.
+We start by a characterization.
+Lemma 6.1. Let X ⊆ PN be a smooth irreducible surface. Then TX(k) is an Ulrich vector bundle if
+and only if
+(i) d = 4(g−1)
+2k−1
+(ii) HKX = (2k−3)d
+2
+.
+(iii) K2
+X = 5χ(OX) + (k−1)(k−2)d
+2
+.
+(iv) H0(TX(k − 1)) = 0.
+(v) H2(TX(k − 2)) = 0.
+Proof. Note that (i) and (ii) are equivalent, since HKX = 2(g − 1) − d. Now (ii) and (iii) are the
+conditions (2.2) in [C1, Prop. 2.1]. Hence the lemma follows by loc. cit.
+□
+
+14
+A.F. LOPEZ, D. RAYCHAUDHURY
+Now we show the possible cases.
+Proposition 6.2. Let X ⊆ PN be a smooth irreducible surface. If TX(k) is an Ulrich vector bundle,
+the following hold:
+(i) 0 ≤ k ≤ 3.
+Moreover, either
+(ii) k = 0 and (X, H) = (P2, OP2(2)), or
+(iii) k = 1 and X is a Del Pezzo surface of degree 5, or
+(iv) k = 2, q = 0 and X is a minimal surface of general type, or
+(v) k = 3, X is a minimal surface of general type with 2KX ≡ 3H, K2
+X = 9d
+4 , χ(OX) = d
+4. Moreover
+X is a ball quotient.
+Proof. We have that k ≥ 0 by Lemma 4.2(i).
+Now H1(TX) = 0 by Lemma 4.2(iv), that is X is
+infinitesimally rigid and [BC, Thm. 1.3] implies that either X is a minimal surface of general type or
+X is a Del Pezzo surface of degree j ≥ 5. In the latter case we have that HKX < 0 hence either k = 0
+and we get (ii) by Theorem 4.9, or k = 1 and K2
+X = 5 by Lemma 6.1(ii),(iii). This gives (iii). On the
+other hand, if X is a minimal surface of general type then HKX > 0, hence k ≥ 2 by Lemma 6.1(ii).
+Next, the Hodge index theorem H2K2
+X ≤ (HKX)2 can be rewritten, using Lemma 6.1(ii),(iii) as
+χ(OX) ≤ (2k2 − 6k + 5)d
+20
+.
+Similarly, the Bogomolov-Miyaoka-Yau inequality K2
+X ≤ 9χ(OX) can be rewritten as
+χ(OX) ≥ (k2 − 3k + 2)d
+8
+.
+Combining we get that
+(k2 − 3k + 2)d
+8
+≤ (2k2 − 6k + 5)d
+20
+and this gives that k ≤ 3 and moreover that, if k = 3, then equality holds in both inequalities. Hence,
+when k = 3 we have, as is well known, that X is a ball quotient and that H2KX ≡ (HKX)H, that is
+2KX ≡ 3H. Then K2
+X = 9d
+4 and χ(OX) = d
+4. Thus (i) and (v) are proved. Alternatively (i) follows
+by Theorem 2. To see (iv) observe that since k = 2 we have by the above that X is a minimal surface
+of general type. Now if pg = 0 then q = 0 by [Be2, Lemma VI.1 and Prop. X.1]. If pg ̸= 0 we have
+an inclusion H0(Ω1
+X) ⊆ H0(Ω1
+X(KX)) hence q = h0(Ω1
+X) ≤ h0(Ω1
+X(KX)) = h2(TX) = 0 since TX(2) is
+Ulrich. This proves (iv).
+□
+We now characterize the case k = 1 for surfaces.
+Theorem 6.3. Let X ⊆ PN be a smooth irreducible surface. Then TX(1) is an Ulrich vector bundle if
+and only if X is a Del Pezzo surface of degree 5 and H = −2KX. Moreover in the latter case TX(1) is
+very ample.
+Proof. If TX(1) is an Ulrich vector bundle, then Proposition 6.2 implies that X is a Del Pezzo surface
+of degree 5 and H2 + 2HKX = 0. Let ε : X → P2 be the blow-up map, with exceptional divisors Ei
+over the points Pi ∈ P2, 1 ≤ i ≤ 4 and let L be a line in P2. Then we can write
+H ∼ aε∗L −
+4
+�
+i=1
+biEi
+and, as H is very ample, we have, without loss of generality,
+b1 ≥ b2 ≥ b3 ≥ b4 ≥ 1, a ≥ b1 + b2 + 1
+and H2 + 2HKX = 0 is
+a2 − 6a + 4 =
+4
+�
+i=1
+(bi − 1)2.
+Setting ci = bi − 1, we get by Lemma A.1 the following possibilities:
+(a; b1, b2, b3, b4) ∈ {(6; 3, 1, 1, 1), (6; 2, 2, 2, 2), (7; 4, 2, 2, 1), (9; 4, 4, 4, 3)}.
+
+ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE
+15
+In the case (6; 2, 2, 2, 2) we have that H = −2KX. We now exclude the other cases.
+Let H = 6ε∗L − 3E1 − E2 − E3 − E4. We will prove that h2(TX(−1)) = h0(Ω1
+X(H + KX)) ̸= 0, so
+that TX(1) cannot be an Ulrich vector bundle. To this end observe that, since ε∗Ω1
+P2 ⊂ Ω1
+X, we will be
+done in this case if we prove that
+H0(ε∗Ω1
+P2(H + KX)) ̸= 0.
+Now H + KX = 3ε∗L − 2E1, hence
+H0(ε∗Ω1
+P2(H + KX)) ∼= H0(IZ ⊗ Ω1
+P2(3))
+where Z ⊂ P2 is the 0-dimensional subscheme of length 2 supported on P1. Finally
+h0(IZ ⊗ Ω1
+P2(3)) ≥ h0(Ω1
+P2(3)) − 6 = 2 > 0
+and we are done in this case.
+Consider now the exact sequences, for any 1 ≤ i ≤ 4,
+0 → OEi(−Ei) → Ω1
+X|Ei → Ω1
+Ei → 0
+that is
+0 → OP1(1) → Ω1
+X|Ei → OP1(−2) → 0
+from which we get, for any 1 ≤ i ≤ 4, that
+(6.1)
+h1(Ω1
+X|Ei) = 1
+and
+(6.2)
+H1(Ω1
+X |Ei ⊗ OP1(2)) = 0.
+In the two remaining cases we will prove that h1(TX(−1)) = h1(Ω1
+X(H + KX)) ̸= 0.
+Let H = 7ε∗L − 4E1 − 2E2 − 2E3 − E4.
+Note that H + KX − E4 + E1 = 4ε∗L − 2E1 − E2 − E3 − E4 is very ample by [DR, Cor. 4.6], hence
+H2(Ω1
+X(H + KX − E4 + E1)) = 0
+by Bott vanishing [T, Thm. 2.1]. Then the exact sequence
+0 → Ω1
+X(H + KX − E4) → Ω1
+X(H + KX − E4 + E1) → Ω1
+X|E1(H + KX − E4 + E1) → 0
+and (6.2) imply that H2(Ω1
+X(H + KX − E4)) = 0. Now the exact sequence
+0 → Ω1
+X(H + KX − E4) → Ω1
+X(H + KX) → Ω1
+X|E4(H + KX) → 0
+and (6.1) imply that
+h1(Ω1
+X(H + KX)) ≥ h1(Ω1
+X |E4(H + KX)) = h1(Ω1
+X|E4) = 1.
+Let H = 9ε∗L − 4E1 − 4E2 − 4E3 − 3E4.
+Let C ∈ |ε∗L − E2 − E3| be the strict transform of a line through P2 and P3. Note that H + KX −
+C + E1 = 5ε∗L − 2E1 − 2E2 − 2E3 − 2E4 is very ample by [DR, Cor. 4.6], hence
+H2(Ω1
+X(H + KX − C + E1)) = 0
+by Bott vanishing [T, Thm. 2.1]. Then the exact sequence
+0 → Ω1
+X(H + KX − C) → Ω1
+X(H + KX − C + E1) → Ω1
+X|E1(H + KX − C + E1) → 0
+and (6.2) imply that H2(Ω1
+X(H + KX − C)) = 0. Now the exact sequence
+0 → OC(−C) → Ω1
+X|C → Ω1
+C → 0
+that is
+0 → OP1(1) → Ω1
+X|C → OP1(−2) → 0
+gives that h1(Ω1
+X |C) = 1. Finally, from the exact sequence
+0 → Ω1
+X(H + KX − C) → Ω1
+X(H + KX) → Ω1
+X|C(H + KX) → 0
+using that (H + KX)C = 0, we get that
+h1(Ω1
+X(H + KX)) ≥ h1(Ω1
+X|C(H + KX)) = h1(Ω1
+X|C) = 1.
+
+16
+A.F. LOPEZ, D. RAYCHAUDHURY
+This completes the proof under the assumption that TX(1) is an Ulrich vector bundle.
+Suppose now that X is a Del Pezzo surface of degree 5 and H = −2KX. Setting k = 1 in Lemma
+6.1, we have that d = 4(g − 1) and, in order to verify that TX(1) is an Ulrich vector bundle, we need to
+check that H0(TX) = H2(TX(−1)) = 0. The first vanishing is well known. As for the second, we first
+observe that for i < 2 we have
+hi(TX(−1)) = h2−i(Ω1
+X(H + KX)) = h2−i(Ω1
+X(−KX)) = 0
+by Bott vanishing [T, Thm. 2.1]. Therefore h2(TX(−1)) = χ(TX(−1)) = d − 4(g − 1) = 0. Finally, as
+X does not contain lines in the embedding given by H = −2KX, we have that TX(1) is very ample by
+[LS, Thm. 1]
+□
+7. Properties of complete intersections
+We collect some properties inherited by the complete intersections Xi of X (as in Notation 2.1),
+when TX(k) is an Ulrich vector bundle.
+Lemma 7.1. Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1. Then q(X) = q(Xi) for
+2 ≤ i ≤ n.
+Proof. By Kodaira vanishing we have that H1(OXi+1(−1)) = H2(OXi+1(−1)) = 0 as long as 2 ≤ i ≤
+n − 1. Then the exact sequences
+0 → OXi+1(−1) → OXi+1 → OXi → 0
+imply that h1(OXi+1) = h1(OXi) for every 2 ≤ i ≤ n − 1, hence q(X) = q(Xi) for 2 ≤ i ≤ n.
+□
+Lemma 7.2. Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1. Suppose that k ≤ n − 2
+and that TX(k) is Ulrich. Then Hi(OXi) = 0 for all i such that max{1, k + 1} ≤ i ≤ n − 1.
+Proof. Assume that max{1, k + 1} ≤ i ≤ n − 1. Since TX|Xi(k) is Ulrich, it follows by Lemma 3.2(iv)
+that Hi(TX |Xi(k + m)) = 0 for all m ≥ −i, hence Hi(TX |Xi(−1)) = 0. Now the exact sequence
+0 → TXi(−1) → TX|Xi(−1) → O⊕(n−i)
+Xi
+→ 0
+implies that Hi(OXi) = 0.
+□
+Lemma 7.3. Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1. Suppose that k ≤ n − 1
+and that TX(k) is Ulrich. Assume that Hi(OX) = 0 for all i ≥ 1. Then Hi(OXj) = 0 for all i ≥ 1 and
+for all j such that max{1, k + 1} ≤ j ≤ n.
+Proof. Assume that i ≥ 1 and max{1, k + 1} ≤ j ≤ n. We prove the lemma by induction on n − j ≥ 0.
+If n − j = 0 then Xj = Xn = X and Hi(OX) = 0 for all i ≥ 1 just by our assumption.
+Next suppose that n − j ≥ 1, so that max{1, k + 1} ≤ j ≤ n − 1, hence, in particular k ≤ n − 2.
+Consider the exact sequence
+0 → OXj+1(−1) → OXj+1 → OXj → 0.
+If j = i, we have that Hj(OXj) = 0 by Lemma 7.2. Also, we have by induction that Hi(OXj+1) = 0.
+Now Hi+1(OXj+1(−1)) = 0 by Kodaira vanishing if i+1 < j+1 and by dimension reasons if i+1 > j+1.
+Thus Hi(OXj) = 0 if i ̸= j and we are done.
+□
+We now collect some properties of the Xi’s that hold when TX(1) is Ulrich.
+Lemma 7.4. Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1. Suppose that n ≥ 2 and
+that TX(1) is an Ulrich vector bundle. Then:
+(i) H1(OXi) = 0 for 2 ≤ i ≤ n.
+(ii) H2(OXi) = 0 for 1 ≤ i ≤ n.
+(iii) H1(OXi(1)) = 0 for 1 ≤ i ≤ n.
+(iv) H2(OXi(1)) = 0 for 1 ≤ i ≤ n.
+(v) h0(OXi(1)) = d − g + i for 1 ≤ i ≤ n.
+(vi) d ≥ n + 3.
+
+ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE
+17
+Proof. We have Hi(OX) = 0 for i ≥ 1 by Lemma 4.3(iii). Now (i) follows by Lemma 7.1 and (ii) follows
+by Lemma 7.3. To see (iii) observe that, if i = 1 we have that X1 = C and TX(1)|C is an Ulrich vector
+bundle on C by Lemma 3.2(ix), hence H1(TX |C) = 0. Then the exact sequence
+0 → TC → TX|C → OC(1)⊕(n−1) → 0
+shows that H1(OC(1)) = 0. If i ≥ 2, since H1(OXi) = 0 by (i), the exact sequences
+(7.1)
+0 → OXi → OXi(1) → OXi−1(1) → 0
+imply by induction that H1(OXi(1)) = 0 and we get (iii). Now (iv) is obvious for i = 1, while, for
+i ≥ 2, the exact sequences (7.1) and (ii) show by induction that H2(OXi(1)) = 0. This proves (iv).
+Note that (v) follows for i = 1 by Riemann-Roch and (iii). For i ≥ 2, the exact sequences (7.1) and
+(i) show by induction that h0(OXi(1)) = 1 + h0(OXi−1(1)) = d − g + i, that is (v). Finally, to see (vi),
+observe that g − 1 = n−1
+n+2d by Lemma 4.3(i), hence g ≥ 2 and (v) gives that
+3d
+n+2 = h0(OC(1)) ≥ 3, so
+that d ≥ n + 2. Moreover, if equality holds, we get that g = n and h0(OX(1)) = n + 2 by (v), hence
+X ⊂ PH0(H) = Pn+1 is a hypersurface of degree n + 2, so that KX = 0, contradicting Lemma 4.3(ii).
+Hence (vi) is proved.
+□
+8. TX(k) Ulrich and special varieties in adjunction theory
+In this section we exclude some special varieties frequently arising in adjunction theory, under the
+hypothesis that TX(k) is an Ulrich vector bundle. The cases (X, H) = (Pn, OPn(1)), (Qn, OQn(1)) have
+been already treated in Lemmas 4.1 and 4.5.
+We start by recalling the following (see [BS, I]).
+Definition 8.1. Let E be an effective divisor on (X, H). The divisor E is called exceptional
+(i) of type 1 if (E, H|E) ∼= (Pn−1, OPn−1(1)) and NE/X ∼= OPn−1(−1),
+(ii) of type 2 if (E, H|E) ∼= (Pn−1, OPn−1(1)) and NE/X ∼= OPn−1(−2),
+(iii) of type 3 if (E, H|E) ∼= (Qn−1, OQn−1(1)) and NE/X ∼= OQn−1(−1),
+(iv) of type 4 if (E, H|E) is a linear Pn−2-bundle over a smooth curve B and (NE/X)|F ∼= OPn−2(−1),
+where F is a fiber of the structure morphism E → B.
+Often these exceptional divisors will not be present under the condition that TX(k) is Ulrich. To see
+this we first prove
+Lemma 8.2. Let W be a variety of dimension s ≥ 1 and let OW (1) be a very ample line bundle. Then
+Ω1
+W(1) is not globally generated if:
+(i) (W, OW (1)) ∼= (Ps, OPs(1)).
+(ii) (W, OW (1)) is a (possibly singular) quadric hypersurface in Ps+1 and s ≥ 2.
+(iii) (W, OW (1)) is a smooth Del Pezzo variety, s ≥ 2 and (W, OW (1)) ̸∈ {(P2, OP2(3)), (Q2, OQ2(2)),
+(P3, OP3(2))}.
+Proof. (i) follows from det(Ω1
+Ps(1)) = OPs(−1). To see (ii), observe that the restricted Euler sequence
+0 → Ω1
+Ps+1|W (1) → H0(OW (1)) ⊗ OW → OW (1) → 0
+implies that H0(Ω1
+Ps+1|W(1)) = 0. Now the exact sequence
+0 → OPs+1(−3) → OPs+1(−1) → OW (−1) → 0
+implies that H1(OW (−1)) = 0 and the dual normal bundle sequence
+0 → OW (−1) → Ω1
+Ps+1|W(1) → Ω1
+W(1) → 0
+gives that H0(Ω1
+W (1)) = 0, hence Ω1
+W(1) is not globally generated. Next, to see (iii), observe that,
+from the classification of Del Pezzo varieties [IP, Thm. 3.3.1] it follows, for the surface section W2, that
+(W2, OW2(1)) ̸∈ {(P2, OP2(3)), (Q2, OQ2(2)). Hence, as is well known, W2, and hence W, contains a line
+L. But now the surjection Ω1
+W(1) → Ω1
+L(1) = OP1(−1) gives that Ω1
+W(1) is not globally generated.
+□
+Now
+
+18
+A.F. LOPEZ, D. RAYCHAUDHURY
+Lemma 8.3. Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1. Assume that TX(k) is
+an Ulrich vector bundle. We have:
+(i) If k ≥ 1 and n ≥ 2, then (X, H) does not contain any exceptional divisor of type 1.
+(ii) If k ≥ 2 and n ≥ 2, then (X, H) does not contain any exceptional divisor of type 2.
+(iii) If k ≥ 2 and n ≥ 3, then (X, H) does not contain any exceptional divisors of types 3 or 4.
+Proof. Let E be an exceptional divisor. It follows from Lemma 4.6 that Ω1
+E(KX |E + (n + 1 − k)H|E) is
+globally generated. Now
+Ω1
+E(KX |E + (n + 1 − k)H|E) ∼=
+
+
+
+
+
+Ω1
+Pn−1(2 − k)
+if E is of type 1;
+Ω1
+Pn−1(3 − k)
+if E is of type 2;
+Ω1
+Qn−1(3 − k)
+if E is of type 3.
+Further, when E is of type 4, let F be a fiber of the structure morphism of E. Again it follows from
+Lemma 4.6 that Ω1
+F (KX |F + (n + 1 − k)H|F ) ∼= Ω1
+Pn−2(3 − k) is globally generated. Consequently, we
+draw the conclusions from Lemma 8.2.
+□
+We now recall
+Definition 8.4. We say that (X, H) is a linear Pk-bundle over a smooth variety B if (X, H) ∼=
+(P(F), OP(F)(1)), where F is a very ample vector bundle on B of rank k + 1.
+We say that (X, H) as above is a scroll (respectively a quadric fibration; respectively a Del Pezzo
+fibration) over a normal variety Y of dimension m if there exists a surjective morphism with connected
+fibers φ : X → Y such that KX +(n−m+1)H = φ∗L (respectively KX +(n−m)H = φ∗L; respectively
+KX + (n − m − 1)H = φ∗L), with L ample on Y .
+We now use the fibration to exclude several varieties as above, when TX(k) is an Ulrich vector bundle.
+Lemma 8.5. Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 1. Assume that TX(k) is
+an Ulrich vector bundle. Let f : X → B be a fibration onto a normal variety B of dimension m ≥ 1,
+with general fiber F. Then:
+(i) If m ≤ min{n − 1, k + 1}, then (F, H|F ) ̸= (Pn−m, OPn−m(1)).
+(ii) If m ≤ min{n − 2, k}, then (F, H|F) ̸= (Qn−m, OQn−m(1)).
+(iii) if m ≤ min{n − 2, k − 1}, then (F, H|F) is not a Del Pezzo variety, unless (F, H|F) ∈
+{(P2, OP2(3)), (Q2, OQ2(2)), (P3, OP3(2))}.
+Proof. We have that
+Ω1
+F(KF + (n + 1 − k)H|F) ∼=
+
+
+
+
+
+Ω1
+Pn−m(m − k)
+if (F, H|F) = (Pn−m, OPn−m(1));
+Ω1
+Qn���m(m − k + 1)
+if (F, H|F) = (Qn−m, OQn−m(1));
+Ω1
+F(m − k + 2)
+if (F, H|F) is a Del Pezzo variety.
+Now Ω1
+F(KF + (n + 1 − k)H|F ) is globally generated by Lemma 4.6. Hence, Lemma 8.2 gives that, in
+each of the three cases, the inequality in m, n, k is not satisfied.
+□
+We get a very useful consequence.
+Lemma 8.6. Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 2. If TX(k) is an Ulrich
+vector bundle, then KX +(n−1)H is nef and H0(KX +(n−1)H) ̸= 0, unless (X, H, k) = (P2, OP2(2), 0)
+(the latter case actually occurs, see Theorem 4.9).
+Proof. Recall that H0(KX + (n − 1)H) ̸= 0 if and only if KX + (n − 1)H is nef by [BS, Cor. 7.2.8].
+Now if KX + (n − 1)H is not nef, it follows by [BS, Prop.’s 7.2.2, 7.2.3 and 7.2.4] that (X, H) is either
+(Pn, OPn(1)), (Qn, OQn(1)), a linear Pn−1-bundle over a smooth curve or (P2, OP2(2)). The first three
+cases are excluded by Lemmas 4.1, 4.5 and 8.5(i), while in the fourth case we have g = 0, hence k = 0
+by Lemma 4.3(i).
+□
+We can now prove Theorem 4.9.
+
+ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE
+19
+Proof of Theorem 4.9. The assert is clear if either (X, H) = (P1, OP1(3)) or (P2, OP2(2)). Vice versa
+assume that TX is Ulrich for H. If n = 1, since TX = −KX is globally generated by Lemma 3.2(vi),
+we have that X is either P1 or an elliptic curve. Now the latter is excluded by Lemma 4.3(i), while in
+the former case TX = OP1(2) Ulrich implies that H = OP1(3). Now assume that n ≥ 2. Then Lemma
+8.6 gives that either (X, H) = (P2, OP2(2)), or KX + (n − 1)H is nef, leading, by Lemma 4.3(ii), to the
+contradiction
+0 ≤ (KX + (n − 1)H)Hn−1 = − 2d
+n + 2.
+□
+The following result will also be useful.
+Lemma 8.7. Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 2. Suppose that k ≥ 1 and
+that TX(k) is an Ulrich vector bundle. Assume that X ∼= P(F) is a projective bundle over a normal
+projective variety B of dimension 1 ≤ m ≤ n − 1. Then B is smooth and F is simple. In particular, if
+m = 1, then q(X) ̸= 0.
+Proof. Let π : X ∼= P(F) → B be the structure morphism and let ξ be the tautological bundle of
+P(F). By twisting F with a sufficiently ample line bundle we can assume that ξ is ample. Then [BS,
+Prop. 3.2.1] implies that B is smooth. Since H0(TX) = 0, the cohomology of the exact sequence
+0 → TX/B → TX → π∗TB → 0
+gives that H0(TX/B) = 0. Now the cohomology of the exact sequence
+0 → OX → π∗F∗ ⊗ ξ → TX/B → 0
+implies that
+h0(F ⊗ F∗) = h0(π∗F∗ ⊗ ξ) = h0(OX) = 1.
+Now if m = 1 and q(X) = 0 we have that B ∼= P1, hence F cannot be simple since rk F = n ≥ 2.
+□
+Next we prove three results for k = 2.
+For the first one, in order to apply the results of T. Fujita in [F1], we give the following definition,
+that coincides with the one in [F1] when B is smooth.
+Definition 8.8. Let f : X → B be a fibration over a curve, L an ample line bundle on X such that on
+the general fiber F we have that KF = −(n − 2)L|F . We say that f is minimal if there is a line bundle
+L on B such that KX + (n − 2)L = f ∗L.
+Then we have
+Lemma 8.9. Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 4. Suppose that k ≥ 2 and
+that k = 2 if n = 4. Moreover assume that TX(k) is an Ulrich vector bundle. Then:
+(i) (X, H) is not a Del Pezzo fibration over a smooth curve.
+(ii) If n = 4 and KX + 2H is ample, then (X, KX + 2H) is not a minimal (P3, OP3(2))-fibration
+over a smooth curve.
+Proof. For the sake of contradiction, let L be H in case (i) and KX + 2H in case (ii). Assume that we
+have a fibration f : X → B over a smooth curve B such that (X, L) is a Del Pezzo fibration in case (i)
+(see Definition 8.4) and (X, L) is a minimal (P3, OP3(2))-fibration in case (ii). Note that f is minimal
+also in case (i) by Definition 8.4.
+Let F be a general fiber of f. In case (i) we have that F is a smooth variety of dimension n − 1 and
+KF = KX|F = −(n − 2)L|F, hence F is a Del Pezzo variety. Since L = H, Lemma 8.5 implies that
+(F, H|F) = (P3, OP3(2)), hence n = 4. Thus (F, L|F ) is the same in both cases.
+We now claim that every fiber of f is irreducible. Indeed, if not, let F0 be a reducible fiber. Since
+F0 is connected, it must be singular, hence we can apply [F1, Table (2.20)]. It follows that we are in
+case (2.17) of [F1, Table (2.20)], the degree of F0 is 8 and, if D is an irreducible component of F0, then
+(D, L|D) is a scroll over P1 and KX |D = −2L|D. Denoting a fiber of the structure morphism D → P1
+by F ′ ∼= P2, we obtain L|F ′ = OP2(1) and KX|F ′ = −2L|F ′ = OP2(−2). Set H|F ′ = OP2(a). In case (ii)
+we have that
+OP2(1) = L|F ′ = (KX + 2H)|F ′ = OP2(−2 + 2a)
+
+20
+A.F. LOPEZ, D. RAYCHAUDHURY
+a contradiction. In case (i) we have that L = H and a = 1. But now Lemma 4.6 gives that Ω1
+P2(KX |F ′ +
+3H|F ′) ∼= Ω1
+P2(1) is globally generated, contradicting Lemma 8.2(i). Thus every fiber of f is irreducible.
+Now [F1, (4.8)] implies that every fiber of f is P3. Since B is a smooth curve, it follows, as is well
+known, that X is a projective bundle over B. On the other hand, since n = 4, we have that k = 2 and
+q(X) = 0 by Lemma 4.3(iii), contradicting Lemma 8.7.
+□
+Lemma 8.10. Let X ⊆ PN be a smooth irreducible variety of dimension 4. Suppose that KX + 2H is
+ample and that TX(2) is an Ulrich vector bundle. Then (X, KX + 2H) is not a quadric fibration over
+a smooth curve.
+Proof. Suppose that (X, KX + 2H) is a quadric fibration π : X → B over a smooth curve. Since
+χ(OX) = 1 by Lemma 4.3(iii), it follows from [Lan, Sect. 1.1, eq. (8)] that B ∼= P1. Moreover, [Lan,
+Sect. 0.1] gives that if π∗(KX + 2H) ∼=
+4�
+i=0
+OP1(ai) and e = �4
+i=0 ai, then there is b ∈ Z such that
+(KX + 2H)4 = 2e − b
+by [Lan, Sect. 1.1, eq. (3)] and
+Ki
+X(KX + 2H)4−i = (−3)i2e + (−3)i−1(−4i + 2ie + (3 − 2i)b) for 1 ≤ i ≤ 4
+by [Lan, Sect. 1.1, eq. (4)]. Solving these five equations we obtain
+KXH3 = 4e − 28b − 104 and d = H4 = 16b + 64
+and therefore Lemma 4.3(iii) gives
+(8.1)
+13d = 48(2 + e).
+Since TX(2) is Ulrich we have that H4(TX(−2)) = 0 and the exact sequence
+0 → TX/P1(−2) → TX(−2) → (π∗TP1)(−2) → 0
+implies that H4((π∗TP1)(−2)) = 0. Hence, by Serre duality
+0 = h4((π∗TP1)(−2)) = h0((π∗OP1(−2))(KX+2H)) = h0(π∗(KX+2H)⊗OP1(−2)) =
+4
+�
+i=0
+h0(OP1(ai−2))
+and therefore ai ≤ 1 for 0 ≤ i ≤ 4. But then e ≤ 5 and (8.1) gives that 1 ≤ e + 2 ≤ 7 is divisible by 13,
+a contradiction.
+□
+Lemma 8.11. Let X ⊆ PN be a smooth irreducible variety of dimension 4. Suppose that KX + 2H is
+ample and that TX(2) is an Ulrich vector bundle. Then (X, KX + 2H) is not a linear P2-bundle over
+a smooth surface.
+Proof. Assume by contradiction that we have a P2-bundle structure π : X ∼= P(F) → B onto a smooth
+surface B, with KX + 2H = ξ, the tautological bundle, where F is a rank 3 vector bundle on B. Then
+H = aξ − π∗M for some a ∈ Z and M ∈ Pic(B), so that
+ξ = KX + 2H = (2a − 3)ξ + π∗(KB + c1(F) − 2M)
+giving a = 2 and 2M = KB + c1(F), thus
+(8.2)
+H ≡ 2ξ − 1
+2π∗(KB + c1(F)).
+We will also use Grothendieck’s relation
+3�
+j=0
+(−1)jξ3−jπ∗cj(F) = 0, that is
+(8.3)
+ξ3 = ξ2π∗c1(F) − ξπ∗c2(F).
+Since ξ2f = 1 for every fiber f of π, we get from (8.3) that
+(8.4)
+ξ3π∗c1(F) = c1(F)2, ξ3π∗KB = KBc1(F) and ξ4 = c1(F)2 − c2(F).
+We first collect some invariants of X and B.
+Claim 8.12. We have:
+
+ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE
+21
+(i) KXH3 = − 2
+3d.
+(ii) χ(OX) = 1.
+(iii) χ(OX(H)) = 2 + χ(OS) − d
+6.
+(iv) h0(KX + 2H) = χ(OS) − 1.
+(v) χ(OB) = 1.
+Proof. (i) is obtained by Lemma 4.3(ii). Now Lemma 4.3(iii) gives that Hi(OX) = 0 for i ≥ 1, hence
+Hi(OB) = 0 for i ≥ 1, giving (ii) and (v). Next, to see (iii), consider the exact sequences
+0 → OXi → OXi(H) → OXi−1(H) → 0
+for i = 4, 3. They give χ(OX(H)) = 1+χ(OX3(H)) = 2+χ(OS(H)) and (iii) follows by Riemann-Roch
+since H2
+|S = d and H|SKS = (KX + 2H)H3 = 4
+3d by (i). To see (iv), observe that, since Rjπ∗(−ξ) = 0
+for every j ≥ 0, we have that Hi(KX + H) = Hi(−ξ + π∗(KB + c1(F) − M)) = 0 for every i. Hence
+the exact sequence
+0 → KX + H → KX + 2H → KX3 + H|X3 → 0
+implies that
+(8.5)
+h0(KX + 2H) = h0(KX3 + H|X3).
+Now we have q(S) = 0 by Lemma 7.1 and Hi(KX3) = 0 for i = 0, 1 by Serre duality and Lemma 7.3.
+Hence the exact sequence
+0 → KX3 → KX3 + H|X3 → KS → 0
+shows that
+χ(OS) − 1 = pg(S) = h0(KS) = h0(KX3 + H|X3)
+and we get (iv) by (8.5).
+□
+We continue the proof of the lemma.
+Next, we collect some relations among the invariants related to ξ, KB and the Chern classes of F.
+Claim 8.13. The following identities hold:
+(i) d − 6c1(F)2 + 16c2(F) − 6K2
+B + 4KBc1(F) = 0.
+(ii) KBc1(F) − 8 + 2χ(OS) − c1(F)2 + 2c2(F) = 0.
+(iii) K2
+B − 1 + c1(F)2 − 3c2(F) = 0.
+(iv) 3K2
+B + c1(F)2 − 2c2(F) − 30 = 0.
+(v) 4 − KBc1(F) + 9
+4K2
+B + 7
+4c1(F)2 − 5c2(F) − χ(OS) + 1
+6d = 0.
+Proof. We have by (8.2) and (8.4) that
+d = H4 = (2ξ − 1
+2π∗(KB + c1(F)))4 = 16(c1(F)2 − c2(F)) + 6K2
+B − 4KBc1(F) − 10c1(F)2
+that is (i). To see (ii) observe that, since π∗ξ = F and Rjπ∗ξ = 0 for j > 0 we have Hi(F) = Hi(ξ) =
+Hi(KX + 2H) = 0 for i > 0 by Kodaira vanishing. Also, by Claim 8.12(iv), (v) and Riemann-Roch we
+get
+χ(OS) − 1 = h0(KX + 2H) = h0(ξ) = h0(F) = χ(F) = 3 − 1
+2KBc1(F) + 1
+2c1(F)2 − c2(F)
+that is (ii). Next, consider the exact sequences
+(8.6)
+0 → TX/B → TX → π∗TB → 0
+and
+0 → OX → π∗F∗(ξ) → TX/B → 0.
+Since TX(2H) is Ulrich, we have that χ(TX) = 0, hence, using Claim 8.12(ii) we get
+(8.7)
+χ(TB) = χ(π∗TB) = −χ(TX/B) = −χ(π∗F∗(ξ)) + 1 = −χ(F ⊗ F∗) + 1.
+On the other hand, by Riemann-Roch and Claim 8.12(v), χ(TB) = 2K2
+B − 10 and χ(F ⊗ F∗) =
+9 + 2c1(F)2 − 6c2(F). Replacing in (8.7) gives (iii).
+
+22
+A.F. LOPEZ, D. RAYCHAUDHURY
+Finally, to see (iv), we first compute c1(S2(F)) = 4c1(F) and c2(S2(F)) = 5c1(F)2 + 5c2(F), so that
+c1(S2(F)(−M)) = −3KB + c1(F), c2(S2(F)(−M)) = 15
+4 K2
+B − 5
+2KBc1(F) − 5
+4c1(F)2 + 5c2(F).
+Now Riemann-Roch gives
+(8.8)
+χ(S2(F)(−M)) = 6 − KBc1(F) + 9
+4K2
+B + 7
+4c1(F)2 − 5c2(F).
+On the other hand, χ(S2(F)(−M)) = χ(2ξ − π∗M) = χ(OX(H)) = 2 + χ(OS) − d
+6 by Claim 8.12(iii).
+Using (8.8) we get (v).
+□
+We now conclude the proof of the lemma.
+Solving the five equations in Claim 8.13 we get
+(8.9)
+K2
+B = − 7
+48d + 7 and KBc1(F) = − 5
+48d + 9.
+In particular d ≥ 48. On the other hand, using Claim 8.12(i), we get
+µ(TX) = −KXH3
+4
+= 1
+6d
+and, using (8.2)
+µ(π∗TB) = c1(π∗TB)H3
+2
+= −π∗KB
+�
+2ξ − 1
+2π∗(KB + c1(F))
+�3
+2
+= −8ξ3π∗KB − 6ξ2π∗(K2
+B + KBc1(F))
+2
+.
+Now (8.4) and (8.9) give
+µ(π∗TB) = −KBc1(F) + 3K2
+B = −1
+3d + 12.
+Since TX is semistable by Lemma 4.3(v), we deduce by (8.6) that
+1
+6d ≤ −1
+3d + 12
+that is d ≤ 24, a contradiction.
+□
+9. TX(1) Ulrich in any dimension
+We study the case k = 1 in any dimension. We start analyzing the properties of the curve section C
+and of the surface section S.
+Lemma 9.1. Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 3. If TX(1) is an Ulrich
+vector bundle, then d ≥ 9 except, possibly, when d = 8, g = 5, n = 4 and h0(OC(1)) = 4.
+Proof. By Lemma 4.3(i) we know that g ≥ 2 and that
+(9.1)
+(n − 1)d = (n + 2)(g − 1).
+By Lemma 7.4(v) we have d − g + 1 = h0(OC(1)) ≥ 3, hence g ≤ d − 2.
+Also, if equality holds,
+then h0(OC(1)) = 3, so that d − 2 = g =
+�d−1
+2
+�
+, thus d = 3 and g = 1, a contradiction. Therefore
+2 ≤ g ≤ d − 3, hence d ≥ 5. But if d ≤ 8 the only possibility given by (9.1) is d = 8, g = 5, n = 4 and
+h0(OC(1)) = 4.
+□
+Lemma 9.2. Let X ⊆ PN be a smooth irreducible variety of dimension n ≥ 3. If TX(1) is an Ulrich
+vector bundle we have:
+(i) KSH|S = n−4
+n+2d.
+(ii) q(S) = pg(S) = 0.
+(iii) K2
+S = − 3(n−2)
+2(n+2)d − n−12
+2
+.
+(iv) S is rational.
+
+ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE
+23
+Proof. (i) follows by Lemma 4.3(ii), while (ii) follows by Lemma 4.3(iii), Lemma 7.1 and Lemma 7.4(ii).
+Note now that the equation in Lemma 4.3(vii) can be rewritten as
+3(n − 2)d + 2(n + 2)K2
+S + (n + 2)(n − 12) = 0
+giving (iii).
+Finally assume that S is not ruled, so that κ(S) ≥ 0.
+Then Lemma 8.6 gives that
+KS + H|S = (KX + (n − 1)H)|S is nef, hence KS(KS + H|S) ≥ 0, that is K2
+S ≥ −KSH|S = − n−4
+n+2d by
+(i). Then (iii) gives
+−3(n − 2)
+2(n + 2)d − n − 12
+2
+= K2
+S ≥ −n − 4
+n + 2d
+so that
+(n + 2)(d + n − 12) ≤ 0.
+Since n ≥ 3 it follows that d ≤ 9, and using Lemma 9.1 we deduce that either d = 9, n = 3 or
+d = 8, g = 5, n = 4 and h0(OC(1)) = 4. In the first case we get a contraction by Lemma 4.3(i), while in
+the second case d + n − 12 = 0, hence K2
+S = 0. As C ⊂ P3 we deduce that S ⊂ P4. But this contradicts
+the well-known formula for the invariants of a surface in P4. Therefore S is ruled, hence rational by (ii)
+and (iv) is proved.
+□
+We are now ready to prove Theorem 3.
+Proof of Theorem 3. If n = 1 we know by Lemma 4.3(i) that TX(1) is not an Ulrich vector bundle. If
+n = 2 this is Theorem 6.3. Suppose next that n ≥ 3. Note that H0(TX) = 0 by Lemma 4.2(iii), hence
+X is neither Pn nor Qn. Also q(X) = 0 by Lemma 4.3(iii). We have that (X, H) is not:
+(1) A projective bundle over a smooth curve by Lemma 8.7.
+(2) A Del Pezzo manifold by Lemma 4.3(i), since otherwise g = 1.
+(3) A hyperquadric fibration over a smooth curve (in the sense of [I]), by Lemma 8.5(ii).
+(4) A linear Pn−2-bundle over a smooth surface, by Lemma 8.5(i).
+Also observe that X does not contain any exceptional divisor of type 1 by Lemma 8.3(i). Hence (X, H)
+is isomorphic to its reduction (X′, H′) (see [I, (0.11)]). It follows by [I, Thm. (1.7)] that KX +(n−2)H
+is nef. Hence S is minimal and rational by Lemma 9.2(iv), a contradiction since a minimal rational
+surface does not have nef canonical bundle. Thus the case n ≥ 3 does not occur and the theorem is
+proved.
+□
+10. TX(2) Ulrich in any dimension
+We prove Theorem 4.
+Proof Theorem 4. It follows by Lemma 4.2(iii) that H0(TX) = 0, hence X is neither Pn nor Qn. Note
+that Hi(OX) = 0 for i ≥ 1 by Lemma 4.3(iii) and KX is not nef, since Lemma 4.3(ii) gives that
+KXHn−1 = n(3−n)
+n+2 d < 0.
+We divide the proof according to the value of τ(X, H) (see (4.3)). We will also use the notions of
+first and second reduction of (X, H), as defined in [BS, Defs. 7.3.3 and 7.5.7].
+Case A: τ(X, H) ≥ n − 1.
+This case does not occur since Lemma 4.6(iii) implies that τ(X, H) ≤ n −
+2n
+n+1 < n − 1.
+Case B: n − 2 ≤ τ(X, H) < n − 1.
+Then KX +(n−1)H is ample, hence the first reduction exists and is isomorphic to (X, H). Therefore
+[BS, Thm. 7.3.4] implies that τ(X, H) = n − 2 and then [BS, Thm. 7.5.3] gives that (X, H) is one of
+the following:
+(B.1) a Mukai variety,
+(B.2) a Del Pezzo fibration over a smooth curve,
+(B.3) a quadric fibration over a normal surface,
+(B.4) a scroll over a normal threefold,
+(B.5) (X, H) contains an exceptional divisor of type 2, 3, or 4.
+
+24
+A.F. LOPEZ, D. RAYCHAUDHURY
+Now, the case (B.1) is ruled out by Corollary 4.11. Case (B.2) is excluded for n = 4 by Lemma 8.9(i)
+and for n ≥ 5 by Lemma 8.5(iii). Also the cases (B.3) and (B.4) are ruled out by Lemma 8.5(ii) and
+(i). Finally the case (B.5) is excluded by Lemma 8.3(ii) and (iii).
+Thus also Case B does not occur.
+Case C: τ(X, H) < n − 2.
+Then the first and second reductions exist and are both isomorphic to (X, H), since KX + (n − 2)H
+is ample.
+We first claim that KX3 is not nef. In fact, assume that KX3 is nef. On the one hand, χ(OX3) = 1
+by Lemma 7.3. On the other hand, 3c2(X3) − c1(X3)2 is pseff by [M, Thm. 1.1], hence 3c2(X3)KX3 ≥
+K3
+X3 ≥ 0. But then Riemann-Roch gives χ(OX3) = − 1
+24c2(X3)KX3 ≤ 0, a contradiction.
+Hence KX3 is not nef and [BS, Prop. 7.9.1] gives the following cases:
+(C.1) n = 5 and (X, KX + 3H) is a linear P4-bundle over a smooth curve.
+(C.2) n = 4 and (X, KX + 2H) is a Del Pezzo.
+(C.3) n = 4 and (X, KX + 2H) is a quadric fibration over a smooth curve.
+(C.4) n = 4 and (X, KX + 2H) is a scroll over a normal surface.
+(C.5) n = 4 and (X, H) contains an exceptional divisor of type 2.
+(C.6) n = 4 and (X, KX + 2H) is a (P3, OP3(2))-fibration over a curve.
+In case (C.1) we have a contradiction by Lemma 8.7. In case (C.2) we have 4KX + 6H = 0, hence
+4KXH3+6H4 = 0 and Lemma 4.3(ii) gives the contradiction d = 0. Cases (C.3) and (C.5) do not occur
+by Lemmas 8.10 and 8.3(ii). In Case (C.6) we observe that the fibration is obtained in [F2, (4.6.1)] by
+contracting an extremal ray, hence it is minimal (see Definition 8.8) and the image is a normal, hence
+smooth, curve. Thus this case is excluded by Lemma 8.9(ii).
+Hence we are left with case (C.4). We have a surjective morphism π : X → B and denoting by F a
+general fiber, we have (F, (KX + 2H)|F ) ∼= (P2, OP2(1)). Now all fibers of π are 2-dimensional by [BS,
+Thm. 14.1.1], hence we get by [BS, Prop. 3.2.1] that B is a smooth surface and (X, KX + 2H) is a
+linear P2-bundle over B. But this case is excluded by Lemma 8.11.
+This concludes the proof of the theorem.
+□
+References
+[A]
+F. Ambro. Ladders on Fano varieties. Algebraic geometry, 9. J. Math. Sci. (New York) 94 (1999), no. 1,
+1126-1135. 7
+[Be1]
+A. Beauville. An introduction to Ulrich bundles. Eur. J. Math. 4 (2018), no. 1, 26-36. 1, 3
+[Be2]
+A. Beauville. Complex algebraic surfaces. London Mathematical Society Lecture Note Series, 68. Cambridge
+University Press, Cambridge, 1983. iv+132 pp. 14
+[BC]
+I. Bauer, F. Catanese. On rigid compact complex surfaces and manifolds. Adv. Math. 333 (2018), 620-669. 14
+[BMQ]
+F. Bogomolov, M. McQuillan. Rational curves on foliated varieties. Foliation theory in algebraic geometry,
+21–51, Simons Symp., Springer, Cham, 2016. 5
+[BMPT]
+V. Benedetti, P. Montero, Y. Prieto Monta˜nez, S. Troncoso. Projective manifolds whose tangent bundle is
+Ulrich. Preprint 2021, arXiv:2108.13944. 1, 6
+[Bo]
+F. A. Bogomolov. Unstable vector bundles and curves on surfaces. Proceedings of the International Congress
+of Mathematicians (Helsinki, 1978), pp. 517-524, Acad. Sci. Fennica, Helsinki, 1980. 4
+[BS]
+M. C. Beltrametti, A. J. Sommese. The adjunction theory of complex projective varieties. De Gruyter Exposi-
+tions in Mathematics, 16. Walter de Gruyter & Co., Berlin, 1995. 5, 17, 18, 19, 23, 24
+[C1]
+G. Casnati. Special Ulrich bundles on non-special surfaces with pg = q = 0. Internat. J. Math. 28 (2017), no.
+8, 1750061, 18 pp. 3, 13
+[C2]
+G. Casnati. Tangent, cotangent, normal and conormal bundles are almost never instanton bundles. Preprint
+2022, in preparation. 1
+[CH]
+M. Casanellas, R. Hartshorne. Stable Ulrich bundles. With an appendix by F. Geiss, F.-O. Schreyer. Internat.
+J. Math. 23 (2012), no. 8, 1250083, 50 pp. 3, 5
+[CMRPL] L. Costa, R. M. Mir´o-Roig, J. Pons-Llopis. Ulrich bundles. De Gruyter Studies in Mathematics, 77, De Gruyter
+2021. 1
+[CP]
+F. Campana, M. P˘aun. Foliations with positive slopes and birational stability of orbifold cotangent bundles.
+Publ. Math. Inst. Hautes ´Etudes Sci. 129 (2019), 1-49. 5
+[DR]
+S. Di Rocco. k-very ample line bundles on del Pezzo surfaces. Math. Nachr. 179 (1996), 47-56. 15
+[ES]
+D. Eisenbud, F.-O. Schreyer. Resultants and Chow forms via exterior syzygies. J. Amer. Math. Soc. 16 (2003),
+no. 3, 537-579. 1, 3
+[F1]
+T. Fujita. On del Pezzo fibrations over curves. Osaka J. Math. 27 (1990), no. 2, 229-245. 19, 20
+
+ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE
+25
+[F2]
+T. Fujita. On Kodaira energy and adjoint reduction of polarized manifolds. Manuscripta Math. 76 (1992), no.
+1, 59?84. 24
+[FL1]
+M. Fulger, B. Lehmann. Morphisms and faces of pseudo-effective cones. Proc. Lond. Math. Soc. 112 (2016),
+no. 4, 651-676. 5
+[FL2]
+M. Fulger, B. Lehmann. Positive cones of dual cycle classes. Algebr. Geom. 4 (2017), no. 1, 1-28. 5
+[GL]
+L. Ghidelli, J. Lacini. Logarithmic bounds on Fujita’s conjecture. Preprint 2021, arXiv:2107.11705. 8
+[Ha1]
+R. Hartshorne. Algebraic geometry. Graduate Texts in Mathematics 52. Springer-Verlag, New York-Heidelberg,
+1977. 13
+[Ha2]
+R. Hartshorne. On the classification of algebraic space curves. Vector bundles and differential equations (Proc.
+Conf., Nice, 1979), pp. 83-112, Progr. Math., 7, Birkh¨auser, Boston, Mass., 1980 12
+[Ho]
+A. H¨oring. On a conjecture of Beltrametti and Sommese. J. Algebraic Geom. 21 (2012), no. 4, 721-751. 7
+[I]
+P. Ionescu. Generalized adjunction and applications. Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 3,
+457-472. 17, 23
+[IP]
+V. A. Iskovskikh, Yu. Prokhorov. Fano varieties. In: Algebraic geometry, V, 1-247, Encyclopaedia Math. Sci.
+47, Springer-Verlag, Berlin, 1999. 17
+[K]
+Y. Kawamata. On effective non-vanishing and base-point-freeness. Kodaira’s issue. Asian J. Math. 4 (2000),
+no. 1, 173-181. 7
+[Lan]
+A. Lanteri. Hilbert curves of quadric fibrations. Internat. J. Math. 29 (2018), no. 10, 1850067, 20 pp. 20
+[Lop]
+A. F. Lopez. On varieties with Ulrich twisted normal bundles. Preprint 2022, arXiv:2205:06602. 1, 9
+[Laz]
+R. Lazarsfeld. Positivity in algebraic geometry, I. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge
+48, Springer-Verlag, Berlin, 2004. 3
+[LS]
+A. F. Lopez, J. C. Sierra. A geometrical view of Ulrich vector bundles. Preprint 2021, arXiv:2105.05979. To
+appear on Int. Math. Res. Not. IMRN. 5, 16
+[M]
+Y. Miyaoka. The Chern classes and Kodaira dimension of a minimal variety. In: Algebraic geometry, Sendai,
+1985, 449-476, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987. 24
+[MS]
+S. Mori, H. Sumihiro. On Hartshorne’s conjecture. J. Math. Kyoto Univ. 18 (1978), no. 3, 523-533. 4
+[T]
+B. Totaro. Bott vanishing for algebraic surfaces. Trans. Amer. Math. Soc. 373 (2020), no. 5, 3609-3626. 15, 16
+[W]
+J. M. Wahl. A cohomological characterization of Pn. Invent. Math. 72 (1983), no. 2, 315-322. 4
+Appendix A. Some numerical lemmas
+Lemma A.1. Let (a; c1, c2, c3, c4) ∈ Z5 be such that c1 ≥ c2 ≥ c3 ≥ c4 ≥ 0, a ≥ c1 + c2 + 3 and
+(A.1)
+a2 − 6a + 4 = c2
+1 + c2
+2 + c2
+3 + c2
+4.
+Then (a; c1, c2, c3, c4) ∈ {(6; 2, 0, 0, 0), (6; 1, 1, 1, 1), (7; 3, 1, 1, 0), (9; 3, 3, 3, 2)}.
+Proof. We have
+(A.2)
+a − 3 ≥ c1 + c2.
+Now, (A.1) and (A.2) imply that (a − 3)2 − 5 = c2
+1 + c2
+2 + c2
+3 + c2
+4 ≥ (c1 + c2)2 − 5, that is
+(A.3)
+c2
+3 + c2
+4 ≥ 2c1c2 − 5.
+But 2c2
+3 ≥ c2
+3 + c2
+4 and 2c1c2 ≥ 2c2
+2. Consequently, we get
+(A.4)
+5 ≥ 2(c2 − c3)(c2 + c3).
+Thus, one of the following should happen:
+(α) c2 = c3.
+(β) c2 = c3 + 1.
+First assume that case (β) holds.
+Then (A.4) yields 2c3 ≤ 1 which gives c3 = 0, c2 = 2 and hence c4 = o. The (A.1) gives
+(a + c1 − 3)(a − c1 − 3) = 6.
+Thus we have one of the following possibilities
+a + c1 = 4, a − c1 = 9, or a + c1 = 5, a − c1 = 6, or a + c1 = 6, a − c1 = 5, or a + c1 = 9, a − c1 = 4
+but none of them have integer solutions.
+Assume now that case (α) holds.
+Set c2 = c3 = c. From (A.3), we obtain c2 + c2
+4 ≥ 2c1c − 5. Since c ≥ c4, we get
+(A.5)
+5 ≥ 2c(c1 − c).
+The above implies one of the following happens:
+
+26
+A.F. LOPEZ, D. RAYCHAUDHURY
+(α1) c2 = c3 = c4 = 0.
+(α2) c1 = c2 = c3.
+(α3) c2 = c3 = 1. This case has two sub-cases, namely c1 = 2, 3.
+(α4) c2 = c3 = 1, c1 = 3.
+Suppose we are in case (α1).
+Then (A.1) gives (a − 3)2 − 5 = c2
+1, so that (a + c1 − 3)(a − c1 − 3) = 5.
+In this case, either
+a + c1 = 4, a − c1 = 8, giving the contradiction c1 = −2, or a + c1 = 8, a − c1 = 4, giving a = 6, c1 = 2
+and the solution (6; 2, 0, 0, 0).
+Suppose we are in case (α2).
+Using (A.3) we conclude
+(A.6)
+5 ≥ (c − c4)(c + c4).
+As before, we obtain the following cases:
+(α21) c = c1 = c2 = c3 = c4.
+(α22) c = c4 + 1.
+(α23) c = c4 + 2.
+We first deal with (α21). In this case, from (A.1), we obtain
+(a + 2c − 3)(a − 2c − 3) = 5.
+Thus, we have either a + 2c = 4, a − 2c = 8, giving the contradiction c = −1, or a + 2c = 8, a − 2c = 4,
+giving the solution (6; 1, 1, 1, 1).
+We now deal with (α22).
+From (A.6) we obtain c + c4 − 2 ≤ 5, hence c4 ≤ 2.
+Thus (c, c4) ∈
+{(3, 2), (2, 1), (1, 0)} and using (A.1) we see that it has no integer solutions except in the first case,
+giving the solution (9; 3, 3, 3, 2).
+We now deal with (α23). As before, in this case we have c4 ≤ 0. This implies c = 2, c4 = 0. But
+then (A.1) does not have any integer solution.
+This concludes case (α2).
+Suppose we are in case (α3).
+We know that (c1, c2, c3, c4) ∈ {(1, 1, 1, 1), (2, 1, 1, 0), (3, 1, 1, 1), (3, 1, 1, 0)}. Using (A.1) we see that
+we have no integer solutions except in the last case, giving (7; 3, 1, 1, 0).
+Suppose we are in case (α4).
+Then (c1, c2, c3, c4) ∈ {(3, 2, 2, 2), (3; 2, 2, 1), (3; 2, 2, 0)} and (A.1) has no integer solutions.
+This concludes case (α) and the proof.
+□
+Lemma A.2. Let z = (a; b1, b2, b3, b4, b5, b6) ∈ Z7 with b1 ≥ b2 ≥ b3 ≥ b4 ≥ b5 ≥ b6 satisfying the
+following
+(A.7)
+a2 −
+6
+�
+i=1
+b2
+i = 10,
+3a −
+6
+�
+i=1
+bi = 6.
+Then z ∈ {(4; 1, 1, 1, 1, 1, 1), (5; 2, 2, 2, 1, 1, 1), (6; 3, 2, 2, 2, 2, 1), (7; 3, 3, 3, 2, 2, 2), (8; 3, 3, 3, 3, 3, 3)}.
+Proof. We first use the Cauchy-Scwartz’s inequality (�6
+i=1 bi)2 ≤ 6(�6
+i=1 b2
+i ) to obtain (a2−12a+32) ≤
+0 whence 4 ≤ a ≤ 8. We further observe that
+(A.8)
+6
+�
+i=1
+(b2
+i − bi) = a2 − 3a − 4.
+Also, (b2
+i − bi) ≥ 0 for all i ≥ 1, and b1 > 0 as 3a − 6 > 0 for a ≥ 4.
+Case 1: a = 4. We have �6
+i=1(b2
+i − bi) = 0 whence |bi| ≤ 1 for all i. Since �6
+i=1 bi = 6, we have bi = 1
+for all i.
+Case 2: a = 5. We have �6
+i=1(b2
+i − bi) = 6 whence |bi| ≤ 3. Also, �6
+i=1 b2
+i = 15 and �6
+i=1 bi = 9.
+Subcase 2.1) b1 = 3. In this case �6
+i=2(b2
+i − bi) = 0 whence |bi| ≤ 1 for all i ≥ 2. Consequently
+�6
+i=1 bi ≤ 8 which is a contradiction.
+Subcase 2.2) b1 ≤ 2. In this case we must have b1 = b2 = b3 = 2. Consequently �6
+i=4(b2
+i − bi) = 0
+whence |bi| ≤ 1 for all i ≥ 4 whence the only solution is z = (5; 2, 2, 2, 1, 1, 1).
+
+ON VARIETIES WITH ULRICH TWISTED TANGENT BUNDLE
+27
+Case 3: a = 6. We have �6
+i=1(b2
+i − bi) = 14 whence |bi| ≤ 4. Also �6
+i=1 b2
+i = 26 and �6
+i=1 bi = 12.
+Subcase 3.1) b1 = 4. Then �6
+i=2(b2
+i − bi) = 2 whence |bi| ≤ 2 for i ≥ 2. Consequently b2 = b3 = b4 = 2.
+But then �6
+i=1 b2
+i ≥ 28 which is a contradiction.
+Subcase 3.2) b1 = 3.
+3.2.1) b2 = 3. In this case �6
+i=3(b2
+i − bi) = 2 whence |bi| ≤ 2 for i ≥ 3. Consequently, b3 = b4 = 2.
+Thus b5 + b6 = 2 and b2
+5 + b2
+6 which is a contradiction.
+3.2.2) b2 ≤ 2. In this case we have the only solution z = (6; 3, 2, 2, 2, 2, 1).
+Subcase 3.3) b1 ≤ 2. In this case bi = 2 for all i whence �6
+i=1 b2
+i = 24 which is a contradiction.
+Case 4: a = 7. We have �6
+i=1(b2
+i − bi) = 24 whence |bi| ≤ 5. Also, �6
+i=1 b2
+i = 39 and �6
+i=1 bi = 15.
+Subcase 4.1) b1 = 5. Then �6
+i=2(b2
+i − bi) = 4 whence |bi| ≤ 2 for all i ≥ 2. Consequently bi = 2 for all
+i, thus �6
+i=1 b2
+i = 45 which is a contradiction.
+Subcase 4.2) b1 = 4.
+4.2.1) b2 = 4. Then �6
+i=3(b2
+i − bi) = 0 whence |bi| ≤ 1 for all i ≥ 3. Consequently �6
+i=1 bi ≤ 12
+which is a contradiction.
+4.2.2) b2 = 3.
+4.2.2.1) b3 = 3. Then �6
+i=4(b2
+i − bi) = 0 whence |bi| ≤ 1 for i ≥ 4. Consequently �6
+i=1 bi ≤ 13
+which is a contradiction.
+4.2.2.2) b3 ≤ 2. Then bi = 2 for all i ≥ 3 whence �6
+i=1 b2
+i = 41 which is a contradiction.
+4.2.3) b2 ≤ 2. Then �6
+i=1 bi ≤ 14 which is a contradiction.
+Subcase 4.3) b1 = 3. Then b2 = b3 = 3.
+4.3.1) b4 = 3. Then �6
+i=5(b2
+i − bi) = 0 whence |bi| ≤ 1 for i ≥ 5. Consequently �6
+i=1 bi ≤ 14 which
+is a contradiction.
+4.3.2) b4 ≤ 2. Then b4 = b5 = b6 = 2. We get only one solution z = (7; 3, 3, 3, 2, 2, 2).
+Subcase 4.4) b1 ≤ 2. Then �6
+i=1 bi ≤ 12 which is a contradiction.
+Case 5: a = 8. We have �6
+i=1(b2
+i − bi) = 36 whence |bi| ≤ 6. Also, �6
+i=1 b2
+i = 54 and �6
+i=1 bi = 18.
+Subcase 5.1) b1 = 6. Then �6
+i=2(b2
+i − bi) = 6 whence |bi| ≤ 3 for i ≥ 2.
+5.1.1) b2 = 3. In this case �6
+i=3(b2
+i − bi) = 0 whence |bi| ≤ 1 for i ≥ 3. Consequently �6
+i=1 bi ≤ 13
+which is a contradiction.
+5.1.2) b2 ≤ 2. In this case �6
+i=1 bi ≤ 16 which is a contradiction.
+Subcase 5.2) b1 = 5. Then �6
+i=2(b2
+i − bi) = 16 whence |bi| ≤ 4.
+5.2.1) b2 = 4. Then �6
+i=3(b2
+i − bi) = 4 whence |bi| ≤ 2 for i ≥ 3. Thus �6
+i=1 bi ≤ 17 which is a
+contradiction.
+5.2.2) b2 = 3 which implies b3 = 3. Then �6
+i=4(b2
+i − bi) = 4 whence |bi| ≤ 2 for i ≥ 4. Consequently
+�6
+i=1 bi ≤ 17 which is a contradiction.
+5.2.3) b2 ≤ 2. Then �6
+i=1 bi ≤ 15 which is a contradiction.
+Subcase 5.3) b1 = 4.
+5.3.1) b2 = 4.
+5.3.1.1) b3 = 4. Then �6
+i=4(b2
+i − bi) = 0 whence |bi| ≤ 1 for i ≥ 4. Consequently �6
+i=1 bi ≤ 15
+which is a contradiction.
+5.3.1.2) b3 = 3 which implies b4 = 3. Thus �6
+i=5(b2
+i −bi) = 0 whence |bi| ≤ 1 for i ≥ 5. Consequently
+�6
+i=1 bi ≤ 16 which is a contradiction.
+5.3.1.3) b3 ≤ 2. Then �6
+i=1 bi ≤ 16 which is a contradiction.
+5.3.2) b2 = 3. Then b3 = b4 = b5 = 3 and b6 = 2. Consequently �6
+i=1 b2
+i = 56 which is a contradiction.
+5.3.3) b2 ≤ 2. Then �6
+i=1 bi ≤ 14 which is a contradiction.
+Subcase 5.4) b1 ≤ 3. In this case we have the only solution z = (8; 3, 3, 3, 3, 3, 3).
+□
+Angelo Felice Lopez, Dipartimento di Matematica e Fisica, Universit`a di Roma Tre, Largo San Leonardo
+Murialdo 1, 00146, Roma, Italy. e-mail [email protected]
+Debaditya Raychaudhury, Department of Mathematics, University of Toronto, Bahen Centre, 40 St.
+George St., Room 6290, Toronto, ON M5S 2E4, Canada. email: [email protected]
+

BdE3T4oBgHgl3EQfUAoU/content/tmp_files/2301.04446v1.pdf.txt

@@ -0,0 +1,1117 @@
+An Efficient Approach to the Online Multi-Agent Path Finding Problem
+by Using Sustainable Information
+1, 3Mingkai TANG, 1, 6Boyi LIU, 1, 4Yuanhang LI, 1, 2, 5Hongji LIU, 1, 2, 7Ming LIU, 1, 8Lujia WANG
+1The Hong Kong University of Science and Technology
+2The Hong Kong University of Science and Technology (Guangzhou)
+{3mtangag, 4yliog, 5hliucq}@connect.ust.hk, [email protected], {7eelium, 8eewanglj}@ust.hk
+Abstract
+Multi-agent path finding (MAPF) is the problem of mov-
+ing agents to the goal vertex without collision. In the on-
+line MAPF problem, new agents may be added to the envi-
+ronment at any time, and the current agents have no infor-
+mation about future agents. The inability of existing online
+methods to reuse previous planning contexts results in redun-
+dant computation and reduces algorithm efficiency. Hence,
+we propose a three-level approach to solve online MAPF uti-
+lizing sustainable information, which can decrease its redun-
+dant calculations. The high-level solver, the Sustainable Re-
+plan algorithm (SR), manages the planning context and sim-
+ulates the environment. The middle-level solver, the Sustain-
+able Conflict-Based Search algorithm (SCBS), builds a con-
+flict tree and maintains the planning context. The low-level
+solver, the Sustainable Reverse Safe Interval Path Planning
+algorithm (SRSIPP), is an efficient single-agent solver that
+uses previous planning context to reduce duplicate calcula-
+tions. Experiments show that our proposed method has sig-
+nificant improvement in terms of computational efficiency. In
+one of the test scenarios, our algorithm can be 1.48 times
+faster than SOTA on average under different agent number
+settings.
+1
+Introduction
+The multi-agent path finding problem (MAPF) is finding
+paths for a set of agents to move from their starting ver-
+tex to the goal vertex without collision. MAPF has a wide
+practical application, such as aircraft towing vehicles (Mor-
+ris et al. 2016), warehouse robots (Wurman, D’Andrea, and
+Mountz 2008), video games (Ma et al. 2017b) and urban
+road networks (Choudhury et al. 2022).
+For MAPF, most works assume that the environment can
+be fully captured before the system runs (Salzman and Stern
+2020; Stern 2019; Ma et al. 2017a). Under this assumption,
+the solution can be calculated in advance, and the agent only
+needs to take actions along the pre-calculated plan at run-
+time. These path finding problems are referred to as offline
+MAPF. However, in practice, the assumption is not always
+guaranteed. During the running of a system, new agents
+might appear in the system suddenly, and agents need to
+do replanning to fit the new situation. Recently, the online
+MAPF problem (ˇSvancara et al. 2019) was proposed. It is
+assumed that new agents may be added to the environment
+at any time, and the current agents have no information about
+Figure 1: An example of two online MAPF instances. In
+both instances, a1 appears A4 at time point 0 and needs to
+travel to D1. P1 and P2 are two paths with equal costs for
+a1. In the first instance, a2 appears in C1 at time point 1,
+and its goal is A2. In the second instance, a′
+2 appears in D2
+at time point 1, and its goal is C4.
+future agents. All agents in the environment need to do re-
+planning to fit the new situation.
+Online MAPF is a problem of great practical importance.
+For example, in a real-world warehouse system, the robot
+frequently enters and exits the work area due to factors
+such as charging or malfunction. Moreover, the time point
+at which the robot re-enters the work area is unpredictable.
+The agents in the warehouse need to adjust their path due to
+the appearance of a new agent.
+Optimally solving the offline MAPF problem is NP-hard
+(Yu and LaValle 2013; Ma et al. 2016). Compared with the
+offline MAPF, which can calculate the paths before the sys-
+tem runs, the online MAPF further needs to calculate high-
+quality paths for all agents in real-time when new agents
+appear. ˇSvancara et al. proposed several methods to solve
+the online MAPF problem. The Replan Single algorithm
+searches an optimal path for each new agent when they
+appear, while all paths of the old agents remain the same.
+The Replan Single Group algorithm jointly plans for all new
+agents appearing at the same time, and the new agents’ path
+plan cannot affect the old agents’ path plan. These two al-
+arXiv:2301.04446v1  [cs.MA]  11 Jan 2023
+
+P1
+α1
+α2
+P2
+α2
+α2
+α2
+α1gorithms can execute fast, but the solutions are not optimal.
+The Replan All algorithm replan for all agents without con-
+sidering existing path plans when new agents appear, and
+it can get high-quality solutions. However, each iteration of
+planning in the Replan All algorithm has a high computa-
+tional complexity when the number of agents becomes large.
+Considering reusing the information in previous planning
+iterations to reduce the running time, we propose an efficient
+algorithm. In this paper, we refer to this kind of information
+as sustainable information or planning context. Our method
+consists of three levels of algorithms. We name the high-
+level algorithm as the Sustainable Replan algorithm(SR). It
+simulates the environment and maintains the whole plan-
+ning context. The middle-level algorithm, the Sustainable
+Conflict-Based Search Algorithm (SCBS), is called by SR
+for searching the multi-agent path planning solution based
+on current information. SCBS uses the Sustainable Reverse
+Safe Interval Path Planning algorithm (SRSIPP), which is
+the low-level solver, for single-agent planning. Given that
+each planning iteration for a single agent has the same goal
+point, but different starting points, SRSIPP searches the path
+in the backward direction (from goal point to starting point)
+to reuse the previous planning information.
+The main contributions of this paper are as follows.
+1. We propose a three-level approach for reusing previous
+planning context to reduce the running time for the online
+MAPF.
+2. We prove the completeness and the snapshot optimality
+of our approaches.
+3. We performed detailed algorithm performance compar-
+ison experiments with SOTA. The average acceleration
+rate relative to the SOTA can reach up to 1.48.
+2
+Problem Definition
+The definition of the online multi-agent path finding prob-
+lem is that given a directed graph G(V, E), and a set of k
+agents a1, a2, a3 ... ak, find a collision-free path for each
+agent. The agent ai is described by the triplet (ts
+i, vs
+i , vg
+i ),
+meaning agent ai appears in the starting vertex vs
+i ∈ V in
+time point ts
+i and its goal is the vertex vg
+i ∈ V . In this paper,
+we call agent i starts at ts
+i. Without loss of generality, we
+assume 0 ≤ ts
+1 ≤ ts
+2 ≤ ... ≤ ts
+k. Specially, agents whose
+start time point is 0 can be seen as agents already in the
+scene before the environment starts to run, and we refer to
+the planning for these agents as the offline part of the on-
+line MAPF problem. In contrast, we refer to the planning
+after the system starts as the online part. In the beginning,
+all agents plan their path from their own start vertex to the
+goal vertex, while they do not know any information about
+the agents that will start in the future. After that, agents fol-
+low their own plan at each time. When it comes to the time
+point when new agents start, all agents need to replan their
+paths considering the new input of the online MAPF prob-
+lem. Let m be the number of time points when new agents
+start, and tnew
+1
+, tnew
+2
+... tnew
+m
+be the corresponding time point
+sequence. m might be smaller than k because there may be
+more than one agent starting at the same time point. The so-
+lution to an online MAPF problem is defined as a sequence
+of valid plans Π =
+�
+π0, π1, π2...πm�
+, where πj is a col-
+lection of all path plans at tnew
+j
+. Let pj
+i be the path plan of
+agent i in πj, and pj
+i[t] be the vertex of agent i in time point
+t in πj. We define pj
+i[tl : tr] as the concatenation of the
+path plan of agent i from time point tl to time point tr, i.e.
+pj
+i[u : v] = pj
+i[u]◦pj
+i[u+1]◦...◦pj
+i[v−1], where ◦ is the con-
+catenation operator. The execute plan of agent i is defined as
+Exi[Π] = p1
+i [tnew
+1
+: tnew
+2
+]◦p2
+i [tnew
+2
+: tnew
+3
+]◦...◦pm−1
+j
+[tnew
+n−1 :
+tnew
+n
+] ◦ pn
+j [tnew
+n
+: ∞], showing the actual path of agent i.
+In this paper, we focus on the variant:
+• We assume that the agent starts in the garage, which
+means that the new agent can choose to enter the start ver-
+tex at the start time or later. Before they enter, they wait
+in the garage and do not conflict with other agents. In ad-
+dition, we use the setting of disappearing at the goal ver-
+tex. Under these two assumptions, the problem is always
+solvable if the offline part is solvable and each agent has a
+path from its initial location to its goal location, as proved
+in Proposition 2 in (ˇSvancara et al. 2019). Although we
+use these two assumptions, our proposed method can be
+easily extended to other assumptions at the start and goal.
+• We only consider vertex conflict and edge conflict. Two
+agents collide iff they occupy the same vertex or cross
+the same edge in opposite directions at the same time.
+Two objectives are commonly used for offline MAPF prob-
+lems, minimizing makespan and minimizing sum-of-cost
+(SOC). Makespan is the maximum complete time above all
+agents. However, minimizing makespan is improper for the
+online MAPF problem because new agents will continu-
+ously be added to the environment, and the later added agent
+will more likely affect the objective. SOC is the summation
+of the cost of all agents’ path plans. However, two online
+MAPF solvers are not comparable in SOC directly because
+SOC cannot measure the exact quality of the two solvers.
+For example, two solvers s1 and s2 are used to solve the
+instances in Figure 1. At time point 0, a1 starts in A4. It
+has two paths with the same cost to its goal D1. Assume
+s1 choose P1 and s2 choose P2. Now considering for s1, a1
+goes to B4 at time point 1. In the first instance, a2 appears,
+the path of a1 will not be affected, and it will continuously
+follow the path [B4, C4, D4, D3, D2, D1] with length 6.
+However, in another instance, a′
+2 appears, the path of a1 will
+make a detour [B4, A4, A3, A2, A1, B1, C1, D1] with
+length 8. The symmetrical situation will appear on s2. We
+cannot say s1 is better than s2 or not because the actual cost
+depends on the future agents, which is unpredictable at early
+time points. Using SOC directly can not judge the quality of
+the solver. We define a solver as a snapshot optimal solver if
+the solver can get optimal paths in terms of SOC, assuming
+no new agent will appear in the future. A snapshot optimal
+solver is better than a non-optimal solver in solution quality.
+3
+Methodology
+Our approach is a three-level method. Figure 2 shows the
+architecture of the method. The high-level solver is the Sus-
+tainable Replan algorithm (SR), which simulates the envi-
+ronment and maintains the planning context of all agents.
+
+Figure 2: The architecture of the three-level approach. The SR algorithm is the high-level solver, simulating the environment
+and managing the planning context. The SCBS algorithm is the middle-level solver, which builds a conflict tree and extracts the
+individual planning context. The low-level solver, the SRSIPP algorithm, uses backward search on the TIS state for single-agent
+path planning.
+The Sustainable Conflict-Based Search algorithm (SCBS),
+the middle-level solver, plans the optimal path for multi-
+agents and manages the planning context. The low-level
+solver, Sustainable Reverse Safe Interval Path Planning (SR-
+SIPP), solves a single-agent problem under a set of con-
+straints.
+3.1
+Sustainable Replan Algorithm
+SR algorithm is the highest solver. It can simulate the scene
+and maintain the planning context sustainably. When one or
+more agents appear, the algorithm will do replanning for all
+agents. Figure 2 shows an example for the SR algorithm. The
+’W’ in the figure means the action is waiting in the garage.
+We define pc as the planning context, a two-level hash
+table, to save all the planning context. Its keys are the agent’s
+id and all constraints on this agent, while its values can be
+determined by its lower-level solvers. We will describe the
+specific planning context in the later subsections.
+The pseudo-code is shown in Algorithm 1. Let tc be the
+current time point, vc be the current vertex of the agent lo-
+cated, A be the agent set that has started, A+ be the new
+agent set appearing in time point tc, and Ex be the execu-
+tion plan. In lines 1-8, the environment is simulated to time
+point tc. If an agent a reaches its goal before time point tc,
+all related elements in A and pc will be removed. Otherwise,
+the current vertex vc of the agent a will be obtained from the
+previous execute plan Ex. In line 10, the SCBS algorithm
+calculates the optimal path plan p. In line 11, the execution
+plan is updated by the p, deleting the path from time point tc
+and concatenating the new path plan to the execution plan.
+Algorithm 1: Sustainable Replanning
+Input: original agent set A, new agent set A+, execute plan
+Ex, current time point tc, planning context pc
+1: for agent a in A do
+2:
+if a reach goal before time tc then
+3:
+A ← A \ {a}
+4:
+Remove all planning context of a in pc.
+5:
+else
+6:
+Update a.vc by Ex.
+7:
+end if
+8: end for
+9: A ← A ∪ A+.
+10: p, pc ← SCBS(A, tc, pc) // Algorithm 2
+11: Update Ex by p.
+12: return A, Ex, pc
+3.2
+Sustainable Conflict-Based Search Algorithm
+The Sustainable Conflict-Based Search algorithm (SCBS)
+calculates the multi-agent path plan and maintains the plan-
+ning context sustainably. It is extended from the Conflict-
+Based Search algorithm (CBS) (Sharon et al. 2015). The
+main difference between SCBS and CBS is the processing
+of the planning context.
+An example of using the SCBS algorithm is shown in Fig-
+ure 2. Before each low-level search, the planning context
+directly related to the low-level search will be extracted ac-
+cording to the agent and its related constraints. We name
+this part of the planning context as individual planning con-
+text and use ipc to represent it. During the low-level search,
+
+SR
+Planning Context
+ai
+con1
+con2
+a2
+a2
+CLOSED, OPEN
+CLOSED, OPEN
+a3
+a2
+......
+a2
+...
+con2
+con1
+Time Point 1
+Time Point 3
+CLOSED, OPEN
+CLOSED, OPEN
+SCBS
+Paths: .....
+Cons: @
+a3
+Icon1
+i con2
+ Individual Planning Context
+Paths: ..
+Paths: ....
+i CLOSED, OPEN
+CLOSED, OPEN
+Cost:
+Cost: ..
+CLOSED, OPEN
+Cons: ((t4, a1, V4))
+Cons: [(t4, a4, V4))
+··
+a4
+Icon1
+con2
+SRSIPP
+CLOSED, OPEN
+CLOSED, OPEN
+.....
+Path Building
+Backward Search on TIS State(a) Forward search on TS states.
+(b) Backward search on TS states.
+(c) Backward search on TIS states. The number
+in the block is the cost to vg.
+(d) Path building on TIS states.
+Figure 3: (a)-(c) are three different search methods on the same graph and constraints. (d) is based on (c). The number near the
+block shows the time point or the time interval.
+ipc is modified to fit the new instance. After the low-level
+search, the new ipc will be put back into pc for usage in
+later iterations.
+Algorithm 2 shows the pseudo-code. In lines 5-7 and 25-
+27, ipc is filtered from pc by the function GetIPC. After
+using the low-level solver, the new ipc is placed back to pc.
+3.3
+Sustainable Reverse Safe Interval Path
+Planning Algorithm
+We now introduce the Sustainable Reverse Safe Interval
+Path Planning algorithm (SRSIPP). The SRSIPP is a single-
+agent solver based on A* (Hart, Nilsson, and Raphael
+1968) and SIPP (Phillips and Likhachev 2011), designed for
+reusing the previous individual planning context to minimize
+the complexity of searching. We omit the agent index i and
+the current number of the planning iteration j in all sym-
+bols when discussing the SRSIPP algorithm, e.g. ts
+i,j, tg
+i,j,
+vs
+i , vg
+i to ts, tg. Let vc be the agent’s current vertex, and
+sc = (tc, vc) be the agent’s current state.
+In the online MAPF, agents may replan while executing
+their plan. Although the starting vertex of each planning is
+different, the ending vertex is invariant. SRSIPP uses this
+property to achieve the target of reusing the previous plan-
+ning context. For some single-agent solvers, the agent is
+planned from its current state to its goal through the edges.
+These search methods are called forward search. The plan-
+ning can also search from the goal to its current state through
+the reverse edges. These search methods are called backward
+search. The SRSIPP is a backward search algorithm.
+In the MAPF problem, most single-agent search methods
+are forward search on the time-space (TS) state. An example
+is shown in Figure 3(a). However, since the entire search tree
+is rooted at the start vertex, which changes with each search,
+the forward search cannot reuse previous planning informa-
+tion. Observing that the goal vertex is invariant for the same
+agent, we can set the goal vertex as the root of the search
+tree to reuse this tree in the following search. However, we
+cannot predict the arrival time before the search starts. For
+the optimality of the algorithm, all states that reach the goal
+earlier must be fully searched before states that arrive later,
+resulting in a large amount of additional computation. Fig-
+ure 3(b) shows an example.
+To speed up the calculation, we propose to search on the
+time-interval-space (TIS) state. Figure 3(c) shows an exam-
+ple. Let ([tl, tr], v) be the TIS state where [tl, tr] is a time
+interval and v is the vertex, and (t, v) be a TS state where t
+is a time point. The TIS state ([tl, tr], v) contains a collection
+of TS states {(t, v)|t ∈ [tl, tr]}. Let gT S(s) and gT IS(s) be
+the cost from a TS and TIS state to vg. All TS states in the
+same TIS state can use the same vertices sequence as the
+shortest path to the goal. Formally, we have
+∀t ∈ [tl, tr], gT IS(([tl, tr], v)) = gT S((t, v)).
+(1)
+We refer to the function value of gT S(s) and gT IS(s) as the
+g value of the TS state and the TIS state, respectively. A TIS
+state is valid if and only if it does not cover any constrained
+
+22
+V1
+2522
+21
+V51
+2
+0
+V2
+3
+V5
+2
+1
+V3
+V4Algorithm 2: SCBS
+Input: agents A, current time point tc, planning context pc
+1: OPEN ← ∅
+2: R ← new node
+3: R.cons ← ∅
+4: for each agent ai do
+5:
+ipc ← GetIPC(pc,ai,R.cons[ai])
+6:
+R.path[ai], ipc ←
+SRSIPP(ai, NULL, tc, ipc) // Algorithm 3
+7:
+Update pc by ipc.
+8: end for
+9: R.cost ← calculate the SOC of P.paths
+10: OPEN ← OPEN ∪ {R}
+11: while OPEN ̸= ∅ do
+12:
+N ← minimum cost node from OPEN.
+13:
+OPEN ← OPEN\{N}
+14:
+L ← the earliest collision in N
+15:
+if L is None then
+16:
+return N.paths, pc
+17:
+end if
+18:
+C ← Get constraints from L
+19:
+for constraint c in C do
+20:
+P ← new node
+21:
+P.cons ← N.cons
+22:
+P.paths ← N.paths
+23:
+a ← c.agent
+24:
+Insert c in P.cons[a].
+25:
+ipc ← GetIPC(pc, a, P.cons[a])
+26:
+P.paths[a], ipc ←
+SRSIPP(a, P.cons[a], tc, ipc) // Algorithm 3
+27:
+Update pc by ipc.
+28:
+if P.path[a] is not NULL then
+29:
+P.cost ← calculate the SOC of P.paths
+30:
+OPEN ← OPEN ∪ {P}
+31:
+end if
+32:
+end for
+33: end while
+TS states or cover TS states with a time point less than ts. A
+maximum TIS state is a valid TIS state whose time interval
+is not a subset of other valid TIS states on the same vertex.
+Before searching a vertex, all the maximum TIS states in
+the vertex will be created. Their g values are set to infinity,
+except that the g values of TIS states on vg are set to 0.
+The procedure of expanding a state for backward search
+is described as follows. The actions of the search include
+moving to a neighbor vertex through reverse edges and stay-
+ing in the same vertex. We use ([tl, tr], v) to represent the
+TIS state that needs to expand. We define a TIS state s′
+as a dummy son of another TIS s iff all TS states in s′
+can take one action to one of TS states in s in the for-
+ward direction. The cost of the dummy son will be one
+more than the original state, i.e., gT IS(s′) = gT IS(s) + 1.
+Let N(v) = {v′|((v′, v) ∈ E) ∨ (v′ = v)} be the neighbor-
+hood of v, and ds(([tl, tr], v)) = {([t′
+l, t′
+r], v′)|v′ ∈ N(v)}
+be the dummy son set of ([tl, tr], v). For the dummy son
+state ([t′
+l, t′
+r], v′) ∈ ds(([tl, tr], v)). We set
+t′
+l = max(ts, tl − 1)
+t′
+r = tr − 1
+(2)
+The dummy son is used to update all current TIS states
+on the same vertex. If the entire TIS state can be improved,
+the cost of the TIS state can be modified directly. Suppose
+only part of the TIS state can be improved due to the time
+interval coverage. In that case, the TIS state will be split into
+several TIS states according to the time interval coverage,
+and only the state completely covered by the dummy son’s
+time interval will be updated. Figure 4 shows an example.
+We use A* for the backward search. Let hT IS(s) be the
+heuristic function of the cost estimation from the TIS state s
+to sc, and hv(v) be the heuristic function of the path length
+estimation from vertex v and vc. We define hT IS(s) as
+hT IS(([tl, tr], v)) = max(max(tl − tc, 0), hv(v))
+(3)
+where tl ≥ ts. The states, where tr < ts, will not be
+searched, and the heuristic function for these states is un-
+defined. In the 4-neighbor grid, hv(v) is usually defined by
+the Manhattan distance to the goal point, i.e.,
+hv(v) = |vx − vg
+x| + |vy − vg
+y|
+(4)
+where (vx, vy) is the coordination of v and (vg
+x, vg
+y) is the
+coordination of vg. The evaluation function of a TIS state is
+defined as follows.
+fT IS(s) = gT IS(s) + hT IS(s)
+(5)
+Let the function value of hT IS(s) and fT IS(s) be the h
+value and the f value of a TIS state s, respectively.
+In the SRSIPP, the individual planning context includes
+the open list and the closed list. We use OPEN and
+CLOSED to represent them. At the beginning of the new
+search, we adjust the individual planning context to fit the
+new planning. Specifically, we recalculate the h value and
+the f value of all states in the OPEN, according to the new
+current state and Equations (3, 5). The CLOSED can be
+used directly without modification. After the adjustment, the
+new search can reuse the OPEN and the CLOSED of the
+previous search.
+The pseudo-code is shown in Algorithm 3. In the code,
+a.vc and a.vg are the current vertex and the goal vertex of
+the agent a. In addition, s.tl and s.tr are the endpoints of the
+time interval in s. The g value, h value and f value of state
+s are saved in s.g, s.h and s.f, respectively. A state is termi-
+nal state if the state is the optimal final state for the search.
+We use ts to save the terminal state and cts to save all can-
+didate terminal states. More specifically, cts saves all closed
+states in the vertex a.vc, which is reachable for the agent,
+i.e., tc ≤ s.tr. In line 1, the individual planning context is
+extracted. If the current state is invalid, return directly (lines
+2-4). In line 5, we update all states’ h value and f value in
+OPEN. In line 6, if the TIS states on a.vg have not been
+created, create all maximum TIS states and put them into
+the OPEN. In lines 7-12, the function StopCheck is used
+to check whether the search can stop. If yes, build the path
+
+Figure 4: An example to show the process of updating the cost of TIS states. The blocks indicate the time interval of the state,
+and the number in the block shows the cost to vg. The first row and the second row show the TIS state ([t0, t1], v) and its dummy
+son state ([t′
+0, t′
+1], v). The third row indicates all valid TIS states in v before being updated. The fourth line shows the updated
+TIS states. The first state cannot be improved, while for the second state, the whole state can be improved. For the third state,
+only part of the time interval is covered by [t′
+0, t′
+1]. The state is split into two states, and only the cost of the covered state is
+updated. The time interval of the fourth state is not covered, so it is not affected. The grey block shows the improved TIS states.
+by the function BuildPath and return directly (We will dis-
+cuss the detail of StopCheck and BuildPath later). Oth-
+erwise, the search starts. In each iteration, get the state with
+minimum f value in the OPEN (line 14). If the time inter-
+val cannot cover any time point after tc, the state is useless
+for the current and later searches. We ignore it and go to the
+next iteration of the while loop (lines 15-17). In lines 18-27,
+dummy sons are generated to update the states. In line 29,
+we update the OPEN, the CLOSED, and the cts. Finally,
+we check whether the search can stop (lines 30-32). If no,
+go to the next iteration.
+The StopCheck algorithm checks whether the search can
+stop and finds the best terminal state. When finding a better
+goal state out of cps is impossible, we stop searching. There
+are two different scenarios for the stop. If the agent is in
+the scene, the search stops when a state in cps covers tc.
+Otherwise, the agent can choose a time point to enter the
+scene. Supposed a state ([tl, tr], v) is selected as the terminal
+state, the best enter time point is max(tl, tc), and the total
+cost from agents’ current TS state (tc, vc) to vertex vg is
+max(tl − tc, 0) + gT IS(([tl, tr], v)). If the minimum total
+cost of choosing a state in cps is less than or equal to the f
+value of the current expanded state, no better solution can be
+found, and the search can stop.
+The pseudo-code of the StopCheck algorithm is shown
+in Algorithm 4. In lines 1-3, if there is no element in cts,
+the search can not stop. In lines 4-10, if the agent is in the
+scene, the search can stop only when a state in cts covers tc.
+In lines 12-18, if the agent is not in the scene, find the state
+with minimum total cost. If the cost is not higher than the
+minimum f value of all nodes in the OPEN, the search can
+stop and return the best terminal state.
+The BuildPath function in Algorithm 3 builds the final
+TS state path. After getting to the terminal state, we back-
+track to get a TIS path. Based on it, we build the TS path as
+the solution. Specially, if the agent is not in the scene and the
+current time point is earlier than any time point in the termi-
+nal state’s time interval, the agent will wait until it reaches
+the earliest time point of the target TIS state and then enter
+the scene. Figure 3(d) shows an example of the path build-
+ing, selecting ([3, ∞], v1) as the terminal state.
+4
+Theoretical Analysis
+Theorem 1. If hv(v) is admissible and satisfies the con-
+sistency assumption, when the first return value of the
+StopCheck function is true, it is impossible to have a better
+terminal state out of cts.
+Proof. The proof is given in the appendix.
+Theorem 2. If hv(v) is admissible and satisfies the con-
+sistency assumption, the SRISPP algorithm is complete and
+optimal.
+Proof. If hv(v) is admissible and satisfies the consistency
+assumption, then hT IS(s) is also admissible and satisfies the
+consistency assumption. It is proved in the appendix.
+If it is the first planning for the configuration of a and
+cons, the OPEN and CLOSED are empty initially. It is a
+standard A*, and the search’s completeness and optimality
+are satisfied.
+If the OPEN and CLOSED are not empty at the be-
+ginning of the search, the state in the CLOSED will not be
+reopened, and the g value of the state is already the smallest
+cost to the goal vertex. For the state in the OPEN, we up-
+date their h value and f value to fit the new situation. The
+scenario can be seen starting from a snapshot in the standard
+A*, except that some states are closed in advance. However,
+the early closed nodes will not affect the completeness and
+optimality of the algorithm because all their unclosed neigh-
+bors are in the OPEN.
+Combining theorem 1 and the above discussions, the the-
+orem is proved.
+Corollary 1. If hv(v) is admissible and satisfies the consis-
+tency assumption, SCBS is complete and optimal.
+Corollary 2. If hv(v) is admissible and satisfies the consis-
+tency assumption, SR is complete and snapshot optimal.
+These two corollaries can be proved according to the
+property of CBS, and RA algorithm (Sharon et al. 2015;
+ˇSvancara et al. 2019). Because Theorem 2 is proved and the
+operations on planning context in SCBS and SR will not af-
+fect completeness and optimality, the corollaries are proved.
+
+3
+4
+4-1
+5
+61
+5
+:6
+4
+4
+5Algorithm 3: SRSIPP
+Input:agent a, constraints cons, current time point tc, indi-
+vidual planning context ipc
+1: OPEN, CLOSED ← ipc
+2: if a is in the scene and (tc, a.vc) ∈ cons then
+3:
+return false, ipc
+4: end if
+5: Update h value and f value of states in OPEN.
+6: Create maximum TIS states in v′ by cons if the states
+are uncreated, and put them into OPEN.
+7: fmin ← the smallest f value in OPEN
+8: cps ← {([tl, tr], v) ∈ CLOSED|a.vc = v∧
+a.ts ≤ tr)}
+9: stop, ts ← StopCheck(cps, fmin, a) // Algorithm 4
+10: if stop then
+11:
+return BuildPath(ts, a), {OPEN, CLOSED}
+12: end if
+13: while OPEN is not empty do
+14:
+s ← the TIS state with minimum f value in OPEN
+15:
+if s.tr < tc then
+16:
+continue
+17:
+end if
+18:
+for each vertex v′ in N(s.v) do
+19:
+ds ← the dummy son of s in v′
+20:
+Create maximum TIS states in v′ by cons if the
+states are uncreated, and put them into OPEN.
+21:
+for each TIS state s′ in v′ do
+22:
+if s′ can be improved by ds then
+23:
+S′
+new ← New states after improving s′
+24:
+OPEN ← OPEN\{s′} ∪ S′
+new
+25:
+end if
+26:
+end for
+27:
+end for
+28:
+stop, ts ← StopCheck(cps, s.f, a) // Algorithm 4
+29:
+Update OPEN, CLOSED and cps by s.
+30:
+if stop then
+31:
+return BuildPath(ts, a), {OPEN, CLOSED}
+32:
+end if
+33: end while
+5
+Experiment
+The purpose of our experiments is to evaluate the computa-
+tional efficiency of the proposed approach. Two 4-neighbor
+grid map datasets are used with settings similar to (ˇSvancara
+et al. 2019), which is a small grid map dataset and a large
+grid map dataset, respectively. We perform online MAPF in
+these grid maps. Specifically, we move each of the agents
+from one cell to another. During this period, there will be
+new agents starting at any time.
+We use success rate and average running time as the met-
+rics. The running time limit is 30 seconds. If the algorithm
+run exceeds the time limit, the experimental instance is un-
+successful and uses the time limit as the running time. We
+make statistics based on the number of agents. For each
+number of agents, it has 100 instances in both datasets.
+The experiments assume no agents are in the scene at the
+beginning. On the one hand, the offline parts of algorithms
+Algorithm 4: StopCheck
+Input: candidate terminal state set cts, the minimum esti-
+mated function value in the open list fmin, the agent a
+1: if cts is empty then
+2:
+return false, NULL
+3: end if
+4: if a is in the scene then
+5:
+s ← the state which covers tc
+6:
+if s is NULL then
+7:
+return false, NULL
+8:
+else
+9:
+return true, s
+10:
+end if
+11: else
+12:
+smin ← argmins∈cts(max(s.tl − tc, 0) + s.g)
+13:
+cmin ← mins∈cts(max(s.tl − tc, 0) + s.g)
+14:
+if cmin ≤ fmin then
+15:
+return true, smin
+16:
+else
+17:
+return false, NULL
+18:
+end if
+19: end if
+(a) Small grid maps.
+(b) Large grid map.
+Figure 5: Grids for experiments from (ˇSvancara et al. 2019).
+are the same. If some instances fail in the offline part, the
+online algorithm is not executed. These test cases are use-
+less for comparison. All randomly generated instances are
+always solvable if there are no agents in the scene in the be-
+ginning. On the other hand, all compared methods use iden-
+tical offline MAPF solvers. Ignoring it does not affect the
+comparison.
+We run the algorithms on an AMD R7-5800X CPU with
+4.40 GHz and 16GB RAM. Four baselines are selected for
+comparison:
+• RA+CBS+A*(A1): This algorithm uses the Replan All
+algorithm to solve the online MAPF problem. When new
+agents start, it uses the CBS algorithm to calculate the
+multi-agent path plan, whose low-level solver is A*.
+• RA+CBS+RSIPP(A2): This algorithm removes all sus-
+tainable operations from our proposed approach, i.e., all
+solvers do not maintain or use the planning context.
+• SR+SCBS+RSIPP(A3): For this algorithm, the low-
+level solver can not reuse the planning context. In the
+middle-level solver, if the agent strictly follows the pre-
+vious path in the node of the conflict tree, the low-level
+
+k
+A1
+A2
+A3
+A4
+10
+96%
+96%
+96%
+96%
+12
+87%
+84%
+86%
+87%
+15
+78%
+74%
+79%
+80%
+17
+65%
+63%
+63%
+64%
+20
+49%
+47%
+47%
+49%
+22
+36%
+35%
+37%
+37%
+25
+27%
+26%
+28%
+29%
+Table 1: Table of the success rate in small grids. k is the
+agent number.
+k
+A1
+A2
+A3
+A4
+10
+1.68(-)
+2.2(0.77)
+1.72(0.98)
+1.58(1.06)
+12
+5.65(-)
+5.96(0.95)
+5.35(1.06)
+4.77(1.18)
+15
+8.05(-)
+8.67(0.93)
+7.96(1.01)
+7.27(1.11)
+17
+12.26(-)
+12.61(0.97)
+11.96(1.03)
+11.54(1.06)
+20
+17.01(-)
+17.18(0.99)
+16.6(1.02)
+16.23(1.05)
+22
+21.08(-)
+22.02(0.96)
+20.83(1.01)
+20.15(1.05)
+25
+22.53(-)
+22.39(1.01)
+22.15(1.02)
+21.9(1.03)
+Table 2: Table of the running time and the speedup ratio
+in small grids. The first number in the cell shows the run-
+ning time(sec), and the number in parentheses indicates the
+speedup relative to A1. k is the agent number.
+solver won’t be called, and a suffix from the results of the
+previous planning is used as the results of this planning.
+• SR+SCBS+SRSIPP(A4): The proposed method.
+5.1
+Small Grid Map
+In the first dataset, 4 small grid maps are used, as shown
+in Figure 5(a). The start and goal points are randomly
+sampled in two cells of the opposing sides of the grids,
+and the start time point is uniformly sampled from [1, 30].
+Let k be the number of agents, which is in the range
+{10, 12, 15, 17, 20, 22, 25}. In each grid map, we generate
+25 experimental instances for each setting of the agent num-
+ber. For each number of agents, it has 25∗4 = 100 instances.
+Table 1 shows the result of success rate. Except when the
+agent number is 17, the success rate of A4 is larger than or
+equal to A1. Table 2 shows the running time of all baselines
+and the speedup ratio relative to A1 of other baselines. In
+the result, A2 did not obtain significant improvement, while
+A3 can speed up in most cases. The best performance comes
+from A4. It shows an improvement in average runtime rela-
+tive to A1 under all agent number settings and achieves the
+maximum speedup ratio in all baselines.
+5.2
+Large Grid Map
+In the second dataset, we use the large grid map shown
+in Figure 5(b). We generate 100 experimental instances for
+each setting of the agent number k in the range of [90, 100].
+The start time points are randomly sampled from [1, 100].
+An agent’s start point and goal point are sampled from two
+different sides.
+Table 3 and Table 4 show the result of the success rate,
+the average running time, and the speedup ratio relative to
+A1 of different baselines. A2 and A3 perform better than A1
+k
+A1
+A2
+A3
+A4
+90
+84%
+85%
+88%
+91%
+92
+82%
+84%
+86%
+89%
+94
+90%
+89%
+91%
+92%
+96
+75%
+77%
+81%
+84%
+98
+72%
+76%
+82%
+83%
+100
+54%
+69%
+72%
+77%
+Table 3: Table of the success rate in large grids. k is the agent
+number.
+k
+A1
+A2
+A3
+A4
+90
+10.09(-)
+8.98(1.12)
+7.48(1.35)
+6.06(1.67)
+92
+9.69(-)
+9.61(1.01)
+8.13(1.19)
+6.8(1.43)
+94
+8.04(-)
+8.27(0.97)
+6.59(1.22)
+5.52(1.46)
+96
+13.05(-)
+13.35(0.98)
+11.25(1.16)
+9.84(1.33)
+98
+13.37(-)
+12.4(1.08)
+10.5(1.27)
+9.21(1.45)
+100
+18.06(-)
+14.78(1.22)
+13.02(1.39)
+11.58(1.56)
+Table 4: Table of the running time and the speedup ratio
+in large grids. The first number in the cell shows the run-
+ning time(sec), and the number in parentheses indicates the
+speedup relative to A1. k is the agent number.
+in most cases, while A4 runs the fastest among all methods.
+In all instances, the average speedup ratio of A4 relative to
+A1 reaches 1.48.
+The running time of 92 and 94 agents are shorter than
+90 agents. It is because when the number of agents is large,
+the running time is not significantly influenced by the small
+increment of the agent number but is dominated by some
+hard-to-solve cases.
+5.3
+Discussion
+The improvement in the large grids is more obvious than in
+the small grids. We believe it is because of the path length.
+In the small grid maps, the path of agents is short. Sustain-
+able information can only be used a few times. In the large
+map, the path is much longer. Sustainable information is
+used more frequently, making A4 get a higher speedup ratio.
+6
+Conclusion
+We proposed a three-level algorithm to solve the online
+MAPF problem. Three levels are responsible for simulating
+the multi-agent online environment, solving the multi-agent
+path planning, and using the historical planning informa-
+tion to assist in solving the single-agent path planning. We
+proved the completeness and the snapshot optimality of our
+approach. The experiment shows that our proposed method
+runs faster than the SOTA algorithm. During the experiment,
+we also found that the performance in large grids is much
+better than in small grids. This is because the agent has a
+longer path in a larger grid so that the planning context can
+be reused more times. It shows that the longer the path, the
+better the acceleration effect of our method.
+In the future, more aspects of using sustainable informa-
+tion, such as building the conflict tree, will be explored to
+improve the efficiency of the algorithm further.
+
+References
+Choudhury, S.; Solovey, K.; Kochenderfer, M.; and Pavone,
+M. 2022. Coordinated Multi-Agent Pathfinding for Drones
+and Trucks over Road Networks. In Proceedings of the 21st
+International Conference on Autonomous Agents and Multi-
+agent Systems, 272–280.
+Hart, P. E.; Nilsson, N. J.; and Raphael, B. 1968. A for-
+mal basis for the heuristic determination of minimum cost
+paths. IEEE transactions on Systems Science and Cybernet-
+ics, 4(2): 100–107.
+Ma, H.; Koenig, S.; Ayanian, N.; Cohen, L.; H¨onig, W.; Ku-
+mar, T.; Uras, T.; Xu, H.; Tovey, C.; and Sharon, G. 2017a.
+Overview: Generalizations of multi-agent path finding to
+real-world scenarios. arXiv preprint arXiv:1702.05515.
+Ma, H.; Tovey, C.; Sharon, G.; Kumar, T.; and Koenig, S.
+2016. Multi-agent path finding with payload transfers and
+the package-exchange robot-routing problem. In Proceed-
+ings of the AAAI Conference on Artificial Intelligence, vol-
+ume 30.
+Ma, H.; Yang, J.; Cohen, L.; Kumar, T. S.; and Koenig, S.
+2017b. Feasibility study: Moving non-homogeneous teams
+in congested video game environments. In Thirteenth Artifi-
+cial Intelligence and Interactive Digital Entertainment Con-
+ference.
+Morris, R.; Pasareanu, C. S.; Luckow, K.; Malik, W.; Ma, H.;
+Kumar, T. S.; and Koenig, S. 2016. Planning, scheduling and
+monitoring for airport surface operations. In Workshops at
+the Thirtieth AAAI Conference on Artificial Intelligence.
+Phillips, M.; and Likhachev, M. 2011. Sipp: Safe interval
+path planning for dynamic environments. In 2011 IEEE In-
+ternational Conference on Robotics and Automation, 5628–
+5635. IEEE.
+Salzman, O.; and Stern, R. 2020. Research challenges and
+opportunities in multi-agent path finding and multi-agent
+pickup and delivery problems. In Proceedings of the 19th
+International Conference on Autonomous Agents and Multi-
+Agent Systems, 1711–1715.
+Sharon, G.; Stern, R.; Felner, A.; and Sturtevant, N. R. 2015.
+Conflict-based search for optimal multi-agent pathfinding.
+Artificial Intelligence, 219: 40–66.
+Stern, R. 2019. Multi-agent path finding–an overview. Arti-
+ficial Intelligence, 96–115.
+ˇSvancara, J.; Vlk, M.; Stern, R.; Atzmon, D.; and Bart´ak,
+R. 2019. Online multi-agent pathfinding. In Proceedings
+of the AAAI conference on artificial intelligence, volume 33,
+7732–7739.
+Wurman, P. R.; D’Andrea, R.; and Mountz, M. 2008. Co-
+ordinating hundreds of cooperative, autonomous vehicles in
+warehouses. AI magazine, 29(1): 9–9.
+Yu, J.; and LaValle, S. M. 2013. Structure and intractability
+of optimal multi-robot path planning on graphs. In Twenty-
+Seventh AAAI Conference on Artificial Intelligence.
+

BtE1T4oBgHgl3EQfpgUG/content/2301.03331v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:bb5adf65f9209e6861f9b34a35ce7359c9211a6f0638a4dea5401a9fa6a4619c
+size 3167479

BtE1T4oBgHgl3EQfpgUG/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ddf6302b5de2373ebf0ba07a106e71d6153723d4019d59b30b2df4e37d3dae38
+size 91187

CNE1T4oBgHgl3EQf9wYh/content/tmp_files/2301.03559v1.pdf.txt

@@ -0,0 +1,1544 @@
+Color Me Intrigued: Quantifying Usage of Colors in Fiction
+Siyan Li
+Stanford University
+[email protected]
+Abstract
+We present preliminary results in quantitative analyses of
+color usage in selected authors’ works from LitBank. Using
+Glasgow Norms, human ratings on 5000+ words, we measure
+attributes of nouns dependent on color terms. Early results
+demonstrate a significant increase in noun concreteness over
+time. We also propose future research directions for compu-
+tational literary color analytics. 1
+1
+Introduction
+All great writers are great colourists, Virginia Woolf once
+stated (Woolf 1934). Analyzing colors in literary work
+across time and authors has fascinated the field of literature,
+philosophy, and psychology (Skard 1946).
+Most literary analyses of colors focus on only one author,
+one work, or one historical era. There has been very few
+large-scale analyses of color usage shifts. Recently, natural
+language processing (NLP) has progressed in fields poten-
+tially relevant to literary color analyses, such as dependency
+parsing (Qi et al. 2020) and named entity recognition (Li
+et al. 2020). Leveraging these tools expedites localization of
+spans of interest, increasing efficiency and ease of larger-
+scale literary analyses.
+What makes literary color analyses interesting for nat-
+ural language processing? Authors utilize colors in nu-
+merous ways, and NLP tools should capture this variety.
+While Goethe uses colors as a backdrop of his narratives,
+only using them to emphasize the plastic shapes of objects
+(Bruns 1928), Dante’s coloring in his Divine Comedy dis-
+plays more symbolic undertones. The colors on the three
+faces of Dante’s Lucifer can be related to the three horses
+of the Apocalypse (Skard 1946). The sudden absence of
+green in Heavenly Paradise may stem from green’s associ-
+ation with hope, and Dante’s Paradise eliminates the need
+for hope since it fulfills all wishes (Austin 1933). In con-
+trast, James Joyce’s green can be interpreted to symbolize
+absinthe (Earle 2003) and the author’s frustration with the
+Irish Catholic Church (Xie 2015). For more contemporary
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+1All code and data used are available at https://github.com/
+siyan-sylvia-li/ColorLit.
+writers, Virginia Woolf’s blue in To the Lighthouse accom-
+panies Mrs. Ramsay for her Madonna role and her mix-
+ture of radiance and somberness (Stewart 1985). The same
+blue manifests cholera and illness in Edgar Allan Poe’s The
+Masque of the Red Death (Yoon 2021). Despite some sub-
+jectivity in these interpretations, the existence of differences
+in color usages are absolute. We want to examine whether
+current NLP tools “understand” these differences.
+We propose a novel line of research using word embed-
+dings and pre-trained language models to quantify color us-
+ages in literature. Specifically, we measure the attributes of
+nouns dependent on color adjectives according to the Glas-
+gow Norms (Scott et al. 2019). Preliminary results demon-
+strate statistically significant trends over time for certain col-
+ors’ Glasgow Norm attributes. We present future research
+directions and plausible experiments.
+Our proposed framework can supplement literary color
+analyses research and provide additional insight for color
+usages comparisons. Looking at literature and creativity
+through the lens of colors is informative because of the
+prevalence of color terms in literature. Color terms can serve
+as anchoring points of comparison between authors, and po-
+tentially between humans and language models.
+2
+Related Work
+The most similar work to ours would be Rabinovich and
+Carmeli (2022), a study of color term usage by both non-
+color-blind and color-blind individuals on Reddit. The au-
+thors discover significant differences in certain color terms.
+They then concentrate on the nouns that are modified by
+color words, using dependency parsing to obtain NOUN
+words in an AMOD dependency pair with an ADJ color
+term. The authors identify significant discrepancies between
+the two populations in imageability (Scott et al. 2019) values
+of color-modified nouns. Our preliminary work is method-
+ologically similar, but we study literature instead. Addition-
+ally, our work more extensively leverages labels from the
+Glasgow Norms by using three dimensions instead of one.
+Word embeddings play a crucial role in computational
+social science. Garg et al. (2018) leverages Word2vec
+(Mikolov et al. 2013) to reflect changes in relationships be-
+tween the embedding representing women and different ad-
+jectives as potentially a result of the feminist movement. A
+similar work, Bailey, Williams, and Cimpian (2022), show-
+arXiv:2301.03559v1  [cs.CL]  9 Jan 2023
+
+cases that people = man by comparing distances be-
+tween word embeddings of trait words of people, men, and
+women respectively. We are interested in similar techniques
+in a literary color analysis context.
+3
+Dataset
+We use LitBank (Bamman, Popat, and Shen 2019), a Euro-
+centric collection of 100 English fictions from 75 authors.
+We conduct a scrape of Project Gutenberg using LitBank’s
+Gutenberg ID’s to obtain the full text of each work. The gen-
+res consist primarily of realistic novels, with few exceptions
+of science fiction (H.G. Wells, Mary Shelley), fantasy (Bram
+Stoker, Oscar Wilde), and horror (Edgar Allan Poe).
+4
+Methodology
+4.1
+Extracting Modified Nouns
+Colors and Synonyms
+We select common colors and cu-
+rate a list of their synonyms. The colors include “red”,
+“green”, “black”, “white”, “blue”, “brown”, “gray”, “yel-
+low”, “pink”, and “purple”. All color terms and their syn-
+onyms are in Appendix A. Each set of sentences from a
+Project Gutenberg E-book are split into lemmatized words.
+We choose sentences containing either our specified color
+adjectives or their synonyms for dependency parsing.
+Dependency Parsing
+Although Rabinovich and Carmeli
+(2022) strictly studies nouns modified by color terms
+through the AMOD dependency, this would be limiting in
+lyrical writing. For instance, “she has eyes of sapphire” de-
+scribes blue eyes and should be included in our analysis, but
+dependency parsing would categorize “eyes” and “saphire”
+to linked by NMOD instead of AMOD. Therefore, we ex-
+pand upon our pool of nouns by including all nouns with
+a dependency link to our color terms. We employ Stanza’s
+Dependency Parser (Qi et al. 2020). Upon obtaining depen-
+dencies on a sentence, we perform a filtering process to re-
+tain the relevant head-dependent pairs. The specific filter-
+ing process is as follows. For each head-dependent pair:
+(1) Lemmatize both the head and the dependent. (2) Iter-
+ate through all color words and their synonyms; if none of
+them is present in either the head or the dependent, prune
+out this pair. (3) If the other word in the dependency pair is
+not a noun or a proper noun, prune out this pair.
+4.2
+Glasgow Norm Models
+The Glasgow Norms are a list of 5,553 words with corre-
+sponding normative human ratings on different psycholin-
+guistic dimensions. Our ongoing work concentrates on: (1)
+Imageability (IMAG), the ease of summoning a mental im-
+age from a word; (2) Concreteness (CNC), the extent to
+which words can be experienced to our senses; and (3) Va-
+lence (VAL), how positive or negative a word’s value is. We
+hypothesize that different authors differ on the imageabili-
+ty/concreteness/valence values of color-dependent nouns.
+Although the Glasgow Norms vocabulary is extensive,
+we still hope to handle unseen words. FastText embeddings
+(Joulin et al. 2016) reduced to 100 dimensions are used to
+train separate 1-layer Multi-Layer Perceptron (MLP) models
+Color
+# of Occurrences
+Color
+# of Occurrences
+red
+2888
+green
+1839
+black
+3325
+white
+4990
+blue
+1622
+brown
+1206
+gray
+1575
+yellow
+848
+pink
+648
+purple
+545
+Table 1: The total numbers of occurrences of our selected
+color terms in the 100 LitBank novels. These are instances
+where the color terms act as either a dependent or a depen-
+dency head.
+to predict these values. We choose FastText for its adaptabil-
+ity for unseen words. Prior to training, all scores are normal-
+ized to the 0 to 1 range for better interpretability, consistent
+with Rabinovich and Carmeli (2022). Three neural networks
+with sigmoid activations are trained on these data, and eval-
+uated on a held-out test set with an 8:1:1 split. We use Pear-
+son’s correlation between predictions and ground truths as
+our metric. Rabinovich and Carmeli (2022) reports a Pear-
+son’s correlation of 0.76 for their IMAG model on a random
+held-out set, while ours achieves 0.79 on the test set. We un-
+derstand that the held-out test set may be different, but this
+indicates that our IMAG model should as potent as the prior
+model. Our CNC model and VAL model achieve correlation
+scores of 0.83 and 0.76, respectively.
+To prevent repeated occurrences of a word affecting the
+average Glasgow Norm values, the dependent nouns are
+deduplicated when computing the averages.
+5
+Preliminary Results
+Despite our analyses on LitBank yield statistically signifi-
+cant results when analyzed across time, this could stem from
+an imbalance in the distribution of publication time in Lit-
+Bank. This paper aims to establish a preliminary framework
+for studying color usages in literature, and current results
+would need corroboration from additional texts from differ-
+ent eras and genres.
+5.1
+Color-dependent Nouns
+We conduct both quantitative and qualitative analyses on
+color-dependent nouns in LitBank through both comput-
+ing average Glasgow Norm values and through inspection
+of most frequently associated nouns. Additional analyses of
+color term frequencies are in Appendix B.
+Out of all unique nouns, 1299 are within the Glas-
+gow Norm vocabulary, and 1924 are OOV. We use our
+trained MLPs to infer Glasgow Norm scales of the out-
+of-vocabulary nouns. After recognizing an upward trend in
+IMAG and CNC, we compute Pearson’s correlations be-
+tween publication year and average IMAG and CNC val-
+ues for all color terms in novels where the color is present
+(Table 2, Figure 1). The IMG and CNC values increase sig-
+nificantly over time for black, white, yellow, and pink. This
+indicates that the nouns associated with these color terms
+
+Figure 1: Imageability plots for the color terms with signif-
+icant results. We omit the concreteness plots here because
+IMAG and CNC are highly correlated, but we provide the
+full set of plots in Appendix C.
+become increasingly concrete easier to conjure mental im-
+ages of. This is consistent with some conclusions from Skard
+(1946) that early color usages are obscure and abstract.
+Nouns in Individual Works
+We observe interesting data
+points in our plots (the full sets of figures are available in
+Appendix C). For instance, we notice a significantly lower
+valence for red in Edgar Allan Poe’s The Masque of the
+Red Death, because the only nouns associated with red are
+“death”, “stain”, and “horror”. Similarly, an abnormally low
+green valence arises in Henry Fielding’s History of Tom
+Jones, a Foundling, because green is attached to “slut”,
+“witch”, and “monster”.
+Nouns over Publication Time
+We divide our fictions into
+pre-1800’s, 1800 - 1900, and post-1900 based on patterns
+in our data. We then deduplicate dependent nouns in each
+work so that we can measure most frequent nouns in each
+era without thematic motifs biasing our analyses. A list of
+selected color terms and their most dependent nouns in each
+of the eras are in Table 3.
+From the table we observe a significant shift in frequently
+used nouns across time. Pre-1800 dependent nouns are more
+abstract and complex compared to post-1800 nouns depen-
+dent on the same colors, while there is no significant differ-
+ence between 1800 - 1900 and post-1900. This is a possible
+explanation for the increase in imageability and concrete-
+ness over time among LitBank works.
+5.2
+Inter-Author Differences
+We plot the Word2vec embeddings of nouns dependent on
+the same colors from different authors to decipher how
+the color terms are used. Out-of-vocabulary nouns are dis-
+carded. This serves as a crude visualization of different top-
+ics associated with these color terms. For instance, when
+comparing nouns modified by yellow in works of Fitzger-
+ald and Joyce in LitBank, the topic of facial hair (hair, beard,
+IMAG Results
+Color
+Pearson’s r
+Color
+Pearson’s r
+red
+-0.095*
+green
+0.059
+black
+0.257**
+white
+0.303***
+pink
+0.534***
+yellow
+0.340**
+CNC Results
+Color
+Pearson’s r
+Color
+Pearson’s r
+black
+0.237**
+white
+0.239**
+pink
+0.517***
+yellow
+0.318***
+VAL Results
+Color
+Pearson’s r
+Color
+Pearson’s r
+green
+0.273***
+purple
+0.210*
+Table 2: Pearson’s correlations between published years and
+Glasgow Norm values of the 100 fictions for color terms
+with sig. results. *** p < 0.001, ** p < 0.05, and * p < 0.1.
+Full results are in Appendix C.
+Color
+Era
+Frequent Nouns
+pink
+Pre-1800
+shame, guilt, folly,
+ribbon, indignation
+1800 - 1900
+cheek, face, ribbon, rose, lip
+Post-1900
+cheek, face, rose, paper, bud
+black
+Pre-1800
+color, eye, grain, hair, wave
+1800 - 1900
+hair, eye, shadow, dress, face
+Post-1900
+hair, eye, dress, figure, man
+white
+Pre-1800
+face, cheek, countenance,
+cliff, cave
+1800 - 1900
+face, cheek, hand, hair, man
+Post-1900
+face, hand, eye, hair, man
+yellow
+Pre-1800
+appearance, complexion,
+blossom, lustre, mist
+1800 - 1900
+hair, light, face, glove, skin
+Post-1900
+hair, light, eye, flower, house
+Table 3: Most frequently color-modified nouns from each
+era for selected color terms. Example sentences are in Ap-
+pendix D
+pompadour) only manifests in Fitzgerald’s, while food items
+(soup, cheese) appear in Joyce’s. Additional examples are in
+Appendix D.
+6
+Proposal of Future Work
+6.1
+Further Analyses
+Fine-grained Timeline Analyses
+Similar to Garg et al.
+(2018), we can train separate Word2vec models on each
+decade of literature in our collection for fine-grained anal-
+yses. Preliminary results indicate that certain colors become
+
+black
+0.9
+IMAG
+0.8
+0.7
+0.6
+1750
+1800
+1850
+1900
+Yearwhite
+0.9
+0.8
+IMAG
+0.7
+0.6
+1750
+1800
+1850
+1900
+Yearyellow
+1.0
+0.8
+IMAG
+0.6
+0.4
+1750
+1800
+1850
+1900
+Yearpink
+0.8
+IMAG
+0.6
+0.4
+0.2
+1750
+1800
+1850
+1900
+YearFigure 2: Word2vec embeddings of nouns modified by yel-
+low in novels by F. Scott Fitzgerald (red points) and James
+Joyce (blue points).
+increasingly associated with concrete descriptions (pink as-
+sociated with cheeks and face). We can compute cosine sim-
+ilarities between Word2vec embeddings of certain colors
+with words such as “face” and “lips”; this should increase
+over time as we observe increasing presence of colors in
+character descriptions. A similar metric can further quantize
+inter-author differences as well.
+Additional Clustering
+We demonstrate the prowess of
+Word2vec for visualizing color-related topics in this pro-
+posal, but word embeddings fail to account for contexts.
+Further clustering analyses can include embeddings from
+context-dependent pre-trained language models such as Sen-
+tenceBERT (Reimers and Gurevych 2019) and BERT (De-
+vlin et al. 2018).
+6.2
+RQs: From a Literature Perspective
+How do colors differ across genres?
+We chose LitBank
+for its ease of access and thorough documentations, but one
+emerging issue is that LitBank skews heavily towards re-
+alistic fictions. Due to this imbalance, we cannot compare
+color usages meaningfully across genres. We will include
+more books in future analyses. If we observe significant dif-
+ferences in frequencies of different color words or in the
+concepts and objects associated with the colors, we can con-
+clude that there exists cross-genre differences in color usage.
+We already observe such differences. Bram Stoker’s
+Dracula (Stoker 1897), a pioneering work in vampire liter-
+ature, features numerous descriptions of pale maidens with
+their red lips, as well as of scarlet blood and crimson eyes.
+Therefore, we notice much more frequent usage of the color
+red in this work, compared to H. G. Wells’s The War of the
+Worlds (Wells 1897), where red most commonly modifies
+weeds. We want to inspect whether the same general pattern
+would persist on larger-scale analyses.
+How do colors differ across literary forms?
+Literary
+color analyses often separate novelists and poets. Compar-
+isons are often drawn only between two poets or two nov-
+elists, but rarely both. Our hypothesis is that colors usage
+in poetry differs significantly from color usage in prose and
+novels. Such differences can manifest as a discrepancy in
+concreteness (e.g. in poetry, colors can more often associate
+with a concept instead of a concrete object). Expanding our
+research to include poets, preferably those contemporary to
+our novelists, should enable us to address this systematically.
+6.3
+RQs: From a Social Science Perspective
+How do colors differ across cultures and classes?
+Dif-
+ferent cultures can have different associations with the same
+color; for instance, white is often a symbol of purity and a
+staple at Western weddings, whereas the same color is used
+more traditionally in Chinese funerals. These associations
+may reflect socio-economic classes as well (e.g. white-collar
+and blue-collar jobs) through colors frequently co-occurring
+with characters from different cultures and societal classes.
+To analyze this, we will utilize named entity recognition to
+link colors to characters, operating with more context. Pur-
+suing this research direction would involve translated work,
+instead of using only Euro-centric collections of literature.
+The social class of a character can either be looked up on-
+line or inferred by a model to automate the process. We can
+then cluster character colors by cultures and classes.
+How do colors affect biases and stereotypes?
+Current-
+day color associations, such as pink with girls and blue with
+boys, can fuel biases. For instance, boys who enjoy wearing
+pink may be regarded as “girly” and overly feminine. Cer-
+tain colors also contain associations with LGBTQ+ commu-
+nities. We are interested in identifying how these color-based
+biases (going beyond race) manifest in literature and online
+communities. We can study this by finding colors associated
+with characters of a demographic group of interest. Tracing
+through past literature may also shed light upon the evolu-
+tion of color associations.
+7
+Discussion
+Our work serves as a step towards more systematic analy-
+ses of color usages in literature using natural language pro-
+cessing tools. Following prior work, we propose using The
+Glasgow Norms and word embeddings as tools for quan-
+tifying color usage differences. We demonstrate significant
+increasing trends in imageability and concreteness in color-
+dependent nouns over time.
+One limitation is the range of language we are capable of
+handling is constrained by the language models we employ.
+Our current collection does not have many pre-1800 pieces.
+While it is possible to increase the representation of pre-
+1800s literature, the domain shift in English style and word
+conventions may require different word embedding and pre-
+trained models to embed narratives (e.g. Chaucer often uses
+“red” in place of “read”, standard of his time, but causes am-
+biguity when processing texts on a large scale). Given such
+shifts in vocabulary and sentence structures, we may fail to
+provide meaningful insights into earlier literature, since the
+word embeddings may have disjoint vocabularies, and mod-
+els such as SentenceBERT are trained on more modern texts.
+
+yellow
+bhaird
+pompadour
+4 
+2
+wax
+0
+sheet
+square
+tiewh
+streak
+journal
+knee
+gingry
+boot
+sol
+dadchild
+-2
+dwaistebtnd
+cap
+itaivgri
+reflestiartle
+-4
+sobbing
+twilight
+Jight
+sun
+gw
+jnsoence
+paese
+Mkeen
+melon
+-6
+flower
+2
+3
+5
+6
+7
+8
+9
+4References
+Austin, H. D. 1933. Heavenly Gold; A Study of the Use of
+Color in Dante. Philological Quarterly, 12: 44.
+Bailey, A. H.; Williams, A.; and Cimpian, A. 2022. Based
+on billions of words on the internet, people= men. Science
+Advances, 8(13): eabm2463.
+Bamman, D.; Popat, S.; and Shen, S. 2019. An annotated
+dataset of literary entities. In Proceedings of the 2019 Con-
+ference of the North American Chapter of the Association
+for Computational Linguistics: Human Language Technolo-
+gies, Volume 1 (Long and Short Papers), 2138–2144.
+Bruns, F. 1928. Auge und Ohr in Goethes Lyrik. The Journal
+of English and Germanic Philology, 27(3): 325–360.
+Devlin, J.; Chang, M.-W.; Lee, K.; and Toutanova, K. 2018.
+Bert: Pre-training of deep bidirectional transformers for lan-
+guage understanding. arXiv preprint arXiv:1810.04805.
+Earle, D. M. 2003. ”Green Eyes, I See You. Fang, I Feel”:
+The Symbol of Absinthe in ”Ulysses”. James Joyce Quar-
+terly, 40(4): 691–709.
+Garg, N.; Schiebinger, L.; Jurafsky, D.; and Zou, J. 2018.
+Word embeddings quantify 100 years of gender and ethnic
+stereotypes. Proceedings of the National Academy of Sci-
+ences, 115(16): E3635–E3644.
+Joulin, A.; Grave, E.; Bojanowski, P.; Douze, M.; J´egou, H.;
+and Mikolov, T. 2016. FastText.zip: Compressing text clas-
+sification models. arXiv preprint arXiv:1612.03651.
+Li, J.; Sun, A.; Han, J.; and Li, C. 2020. A survey on deep
+learning for named entity recognition. IEEE Transactions
+on Knowledge and Data Engineering, 34(1): 50–70.
+Mikolov, T.; Chen, K.; Corrado, G.; and Dean, J. 2013. Ef-
+ficient estimation of word representations in vector space.
+arXiv preprint arXiv:1301.3781.
+Qi, P.; Zhang, Y.; Zhang, Y.; Bolton, J.; and Manning,
+C. D. 2020.
+Stanza: A Python natural language process-
+ing toolkit for many human languages.
+arXiv preprint
+arXiv:2003.07082.
+Rabinovich, E.; and Carmeli, B. 2022. Exploration of the
+Usage of Color Terms by Color-blind Participants in Online
+Discussion Platforms. arXiv preprint arXiv:2210.11905.
+Reimers, N.; and Gurevych, I. 2019. Sentence-bert: Sen-
+tence embeddings using siamese bert-networks.
+arXiv
+preprint arXiv:1908.10084.
+Scott, G. G.; Keitel, A.; Becirspahic, M.; Yao, B.; and
+Sereno, S. C. 2019. The Glasgow Norms: Ratings of 5,500
+words on nine scales. Behavior research methods, 51(3):
+1258–1270.
+Skard, S. 1946. The Use of Color in Literature: A Survey
+of Research.
+Proceedings of the American Philosophical
+Society, 90(3): 163–249.
+Stewart, J. F. 1985. Color in To the Lighthouse. Twentieth
+Century Literature, 31(4): 438–458. Publisher: [Duke Uni-
+versity Press, Hofstra University].
+Stoker, B. 1897. Dracula: 1897.
+Wells, H. G. 1897. The war of the worlds. Broadview Press.
+Color Word
+Synonyms
+red
+cardinal, coral, crimson, maroon,
+burgundy, flaming, scarlet, fuchsia
+green
+emerald, olive, aquamarine, beryl,
+jade, lime
+black
+ebony, jet, obsidian, onyx, inky
+white
+alabaster, ashen, blanched, bleached,
+cadaverous, doughy, pale, pallid,
+pasty, ivory, pearly, beige
+blue
+azure, indigo, sapphire, cerulean,
+cobalt, turquoise, teal
+brown
+amber, khaki, tan, umber, hazel
+gray
+grey
+yellow
+N/A
+pink
+rosy, blush, magenta
+purple
+lavender, lilac, mauve, periwinkle,
+plum, violet, amethyst
+Table 4: List of color words used and their synonyms
+Woolf, V. 1934. Walter Sickert: A Conversation. L. and
+Virginia Woolf at the Hogarth Press.
+Xie, Y. 2015. Color as Metaphor-A Study of Joyce’s Use of”
+Black” and” Green” in Dubliners and A Portrait of the Artist
+as a Young Man. English Language and Literature Studies,
+5(4): 61.
+Yoon, S. 2021. Color Symbolisms of Diseases: Edgar Al-
+lan Poe’s “The Masque of the Red Death”. The Explicator,
+79(1-2): 21–24.
+A
+Color terms and their synonyms used
+We list the color terms used in this work, along with their
+corresponding synonyms obtained through manual inspec-
+tion on Thesaurus.com entries, in Table 4.
+B
+Normalized Frequencies Over Time
+After filtering through sentences in LitBank for relevant de-
+pendence pairs, the absolute frequencies of each color term
+are listed in Table 1. We also calculate the normalized fre-
+quencies (absolute frequency divided by the total number of
+words) for the 100 fictions. We notice an increasing trend in
+the normalized frequencies over time for LitBank, and com-
+pute Pearson’s correlations between normalized frequencies
+of color terms and time to verify this trend (Table 5).
+We present the scatter plots of normalized frequencies in
+Figure 3.
+C
+Glasgow Norm Plots of Modified Nouns
+Over Time
+The plots corresponding to how color-modified nouns
+change over time with respect to their Glasgow Norm
+
+Color
+Pearson’s r
+Color
+Pearson’s r
+red
+0.173*
+green
+0.134
+black
+0.139
+white
+0.168*
+blue
+0.148
+brown
+0.202**
+gray
+0.162
+yellow
+0.184*
+pink
+0.173*
+purple
+0.150
+Table 5: Pearson’s correlations between published years and
+normalized frequencies of the 100 fictions for each color
+term. ** indicates p < 0.05, and * p < 0.1.
+IMAG, CNC, and VAL values are presented here in Fig-
+ures 4, 5, and 6. We also include the full list of Pearson
+correlation results for all color terms in Table 6.
+IMAG Results
+Color
+Pearson’s r
+Color
+Pearson’s r
+red
+-0.095*
+green
+0.059
+black
+0.257**
+white
+0.303***
+blue
+-0.163
+brown
+0.129
+gray
+-0.081
+yellow
+0.340**
+pink
+0.534***
+purple
+0.162
+CNC Results
+Color
+Pearson’s r
+Color
+Pearson’s r
+red
+-0.054
+green
+0.160
+black
+0.237**
+white
+0.239**
+blue
+-0.133
+brown
+0.175
+gray
+-0.044
+yellow
+0.318***
+pink
+0.517***
+purple
+0.055
+VAL Results
+Color
+Pearson’s r
+Color
+Pearson’s r
+red
+-0.025
+green
+0.273***
+black
+0.059
+white
+0.161
+blue
+-0.104
+brown
+-0.148
+gray
+0.072
+yellow
+-0.01
+pink
+0.081
+purple
+0.210*
+Table 6: Full list of Pearson’s correlations between published
+years and Glasgow Norm values of the 100 fictions for each
+color term. *** p < 0.001, ** p < 0.05, and * p < 0.1.
+D
+Example of Color Term Usages
+In the following usage examples, the color terms are un-
+derlined, bolded and colored, while the dependent noun is
+bolded.
+D.1
+Pink
+HENRY FIELDING, 1749
+Adorned with all the charms in which nature can array
+her; bedecked with beauty, youth, sprightliness, innocence,
+modesty, and tenderness, breathing sweetness from her rosy
+lips, and darting brightness from her sparkling eyes, the
+lovely Sophia comes!
+FANNY BURNEY, 1778
+We were sitting in this manner, he conversing with all
+gaiety, I looking down with all foolishness, when that fop
+who had first asked me to dance, [...] I interrupted him I
+blush for my folly, with laughing; yet I could not help it;
+for, added to the man’s stately foppishness, [...] that I could
+not for my life preserve my gravity.
+ANN WARD RADCLIFFE, 1794
+As they descended, they saw at a distance, on the right,
+one of the grand passes of the Pyrenees into Spain, gleaming
+with its battlements and towers to the splendour of the set-
+ting rays, yellow tops of woods colouring the steeps below,
+while far above aspired the snowy points of the mountains,
+still reflecting a rosy hue.
+Conscious innocence could not prevent a blush from
+stealing over Emily’s cheek; she trembled, and looked con-
+fusedly under the bold eye of Madame Cheron, who blushed
+also; but hers was the blush of triumph, such as sometimes
+stains the countenance of a person, congratulating himself
+on the penetration which had taught him to suspect another,
+and who loses both pity for the supposed criminal, and
+indignation of his guilt, in the gratification of his own vanity
+CHARLOTTE BRONTE, 1847
+Her beauty, her pink cheeks and golden curls, seemed to
+give delight to all who looked at her, and to purchase indem-
+nity for every fault.
+Its garden, too, glowed with flowers: hollyhocks had
+sprung up tall as trees, lilies had opened, tulips and roses
+were in bloom; the borders of the little beds were gay with
+pink thrift and crimson double daisies; the sweetbriars gave
+out, morning and evening, their scent of spice and apples;
+and these fragrant treasures were all useless for most of the
+inmates of Lowood, except to furnish now and then a hand-
+ful of herbs and blossoms to put in a coffin.
+She pulled out of her box, about ten minutes ago, a
+little pink silk frock; rapture lit her face as she unfolded
+it; coquetry runs in her blood, blends with her brains, and
+seasons the marrow of her bones.
+ARTHUR CONAN DOYLE, 1892
+That trick of staining the fishes’ scales of a delicate pink
+is quite peculiar to China.
+Her violet eyes shining, her lips parted, a pink flush upon
+her cheeks, all thought of her natural reserve lost in her over-
+powering excitement and concern.
+As we approached, the door flew open, and a little blonde
+woman stood in the opening, clad in some sort of light
+
+Figure 3: Log-scale scatter plots of normalized color term frequencies for each publication in LitBank.
+
+blue
+103
+Frequency (x 1E-5)
+102
+101
+100
+1750
+1800
+1850
+1900
+Yearbrown
+Frequency (x 1E-5)
+102
+101
+100
+1750
+1800
+1850
+1900
+Yeargray
+103
+Frequency (x 1E-5)
+102
+101
+100
+1750
+1800
+1850
+1900
+Yearyellow
+Frequency (x 1E-5)
+102
+101
+100
+1750
+1800
+1850
+1900
+Yearpink
+102
+Frequency (x 1E-5)
+101
+100
+1750
+1800
+1850
+1900
+Yearpurple
+102
+Frequency (x 1E-5)
+101
+100
+1750
+1800
+1850
+1900
+Yearred
+Frequency (x 1E-5)
+102
+101
+100
+1750
+1800
+1850
+1900
+Yeargreen
+Frequency (x 1E-5)
+102
+101
+100
+1750
+1800
+1850
+1900
+Yearblack
+103
+LE-5)
+1
+X
+102
+Fregquency
+101
+1750
+1800
+1850
+1900
+Yearwhite
+103
+1E-5)
+Frequency (x 1
+102
+101
+1750
+1800
+1850
+1900
+YearFigure 4: IMAG scatter plots of nouns modified by color terms for each publication in LitBank.
+
+black
+0.9
+IMAG
+0.8
+0.7
+0.6
+1750
+1800
+1850
+1900
+Yearred
+0.9
+IMAG
+0.8
+0.7
+0.6
+1750
+1850
+1800
+1900
+Yeargreen
+0.9
+0.8
+IMAG
+0.7
+0.6
+0.5
+1750
+1800
+1850
+1900
+Yearblue
+1.0
+0.9
+IMAG
+0.8
+0.7
+0.6
+1750
+1800
+1850
+1900
+Yearbrown
+0.9
+0.8
+IMAG
+0.7
+0.6
+0.5
+1750
+1800
+1850
+1900
+Yearwhite
+0.9
+0.8
+IMAG
+0.7
+0.6
+1750
+1800
+1850
+1900
+Yeargray
+0.9
+IMAG
+0.8
+0.7
+0.6
+1750
+1800
+1850
+1900
+Yearpurple
+1.0
+0.8
+IMAG
+0.6
+0.4
+1750
+1800
+1850
+1900
+Yearyellow
+1.0
+0.8
+IMAG
+0.6
+0.4
+1750
+1800
+1850
+1900
+Yearpink
+0.8
+IMAG
+0.6
+0.4
+0.2
+1750
+1800
+1850
+1900
+YearFigure 5: CNC scatter plots of nouns modified by color terms for each publication in LitBank.
+
+red
+0.9
+0.8
+CNC
+0.7
+0.6
+0.5
+1750
+1800
+1850
+1900
+Yeargreen
+0.9
+0.8
+CNC
+0.7
+0.6
+0.5
+1750
+1800
+1850
+1900
+Yearblack
+1.0
+0.9
+CNC
+0.8
+0.7
+0.6
+1750
+1800
+1850
+1900
+Yearwhite
+0.8
+CNC
+0.7
+0.6
+1750
+1800
+1850
+1900
+Yearblue
+0.9
+0.8
+CNC
+0.7
+0.6
+0.5
+1750
+1800
+1850
+1900
+Yearbrown
+0.9
+0.8
+CNC
+0.7
+0.6
+0.5
+1750
+1800
+1850
+1900
+Yeargray
+0.9
+0.8
+CNC
+0.7
+0.6
+1750
+1800
+1850
+1900
+Yearyellow
+1.0
+0.8
+CNC
+0.6
+0.4
+0.2
+1750
+1800
+1850
+1900
+Yearpink
+0.8
+0.4
+1750
+1800
+1850
+1900
+Yearpurple
+0.9
+0.8
+CNC
+0.7
+0.6
+0.5
+1750
+1800
+1850
+1900
+YearFigure 6: VAL scatter plots of nouns modified by color terms for each publication in LitBank.
+
+red
+0.7
+0.6
+0.5
+VAL
+0.4
+0.3
+0.2-
+1750
+1800
+1850
+1900
+Yeargreen
+0.8
+0.7
+0.6
+VAL
+0.5
+0.4
+0.3
+1750
+1800
+1850
+1900
+Yearblack
+0.8
+0.7
+≤0.6
+0.5
+1750 1800
+1850
+1900
+Yearwhite
+0.7.
+M0.6
+0.5
+1750
+1800
+1850
+1900
+Yearblue
+0.7
+M0.6
+0.5
+1750
+1800
+1850
+1900
+Yearbrown
+0.7
+0.5
+1750
+1800
+1850
+1900
+Yeargray
+0.70
+0.65
+0.55
+0.50
+1750
+1800
+1850
+1900
+Yearyellow
+0.7
+0.6
+0.5
+VAL
+0.4
+0.3
+0.2
+1750
+1800
+1850
+1900
+Yearpink
+0.8
+0.7
+≤0.6
+0.5
+0.4
+1750
+1800
+1850
+1900
+Yearpurple
+0.8
+0.6
+VAL
+0.4
+0.2
+1750
+1800
+1850
+1900
+Yearmousseline de soie, with a touch of fluffy pink chiffon at
+her neck and wrists.
+H.G. WELLS, 1897
+It was the fact that all his forehead above his blue glasses
+was covered by a white bandage, and that another covered
+his ears, leaving not a scrap of his face exposed excepting
+only his pink, peaked nose.
+P. G. WODEHOUSE, 1919
+Jimmy turned into that drug store at the top of the Hay-
+market at which so many Londoners have found healing and
+comfort on the morning after, and bought the pink drink for
+which his system had been craving since he rose from bed.
+The clerk had finished writing the ticket, and was pressing
+labels and a pink paper on him.
+F. SCOTT FITZGERALD, 1920:
+Myra sprang up, her cheeks pink with bruised vanity, the
+great bow on the back of her head trembling sympathetically.
+D.2
+White
+FANNY BURNEY, 1778
+She told her niece, that she had been indulging in fanciful
+sorrows, and begged she would have more regard for deco-
+rum, than to let the world see that she could not renounce an
+improper attachment; at which Emily’s pale cheek became
+flushed with crimson, but it was the blush of pride, and she
+made no answer.
+MARY WOLLSTONECRAFT, 1788
+Henry started at the sight of her altered appearance; the
+day before her complexion had been of the most pallid hue;
+but now her cheeks were flushed, and her eyes enlivened
+with a false vivacity, an unusual fire.
+JANE AUSTEN, 1813
+Her pale face and impetuous manner made him start, and
+before he could recover himself to speak, she, in whose mind
+every idea was superseded by Lydia’s situation, hastily ex-
+claimed, “I beg your pardon, but I must leave you.
+From his garden, Mr. Collins would have led them round
+his two meadows; but the ladies, not having shoes to en-
+counter the remains of a white frost, turned back
+ELIZABETH CLEGHORN GASKELL, 1855
+She lay curled up on the sofa in the back drawing room in
+Harley Street looking very lovely in her white muslin and
+blue ribbons.
+She found out that the water in the urn was cold, and or-
+dered up the great kitchen tea kettle; the only consequence
+of which was that when she met it at the door, and tried to
+carry it in, it was too heavy for her, and she came in pouting,
+with a black mark on her muslin gown, and a little round
+white hand indented by the handle, which she took to show
+to Captain Lennox, just like a hurt child, and, of course, the
+remedy was the same in both cases.
+But he had the same large, soft eyes as his daughter, eyes
+which moved slowly and almost grandly round in their or-
+bits, and were well veiled by their transparent white eyelids.
+CHARLES DICKENS, 1861
+I lighted my fire, which burnt with a raw pale flare at that
+time of the morning, and fell into a doze before it.
+She’s all in white,’ he says, ‘wi’ white flowers in her hair,
+and she’s awful mad, and she’s got a shroud hanging over her
+arm, and she says she’ll put it on me at five in the morning.’
+When we was put in the dock, I noticed first of all what
+a gentleman Compeyson looked, wi’ his curly hair and his
+black clothes and his white pocket handkercher, and what
+a common sort of a wretch I looked.
+ELIZABETH VON ARNIM, 1898
+There are so many bird cherries round me, great trees with
+branches sweeping the grass, and they are so wreathed just
+now with white blossoms and tenderest green that the gar-
+den looks like a wedding.
+When that time came, and when, before it was over, the
+acacias all blossomed too, and four great clumps of pale, sil-
+very pink peonies flowered under the south windows, I felt
+so absolutely happy, and blest, and thankful, and grateful,
+that I really cannot describe it.
+D.3
+Black
+JONATHAN SWIFT, 1726
+In his right waistcoat pocket we found a prodigious bun-
+dle of white thin substances, folded one over another, about
+the bigness of three men, tied with a strong cable, and
+marked with black figures; which we humbly conceive to
+be writings, every letter almost half as large as the palm of
+our hands.
+In the left pocket were two black pillars irregularly
+shaped: we could not, without difficulty, reach the top of
+them, as we stood at the bottom of his pocket.
+In one of these cells were several globes, or balls, of a
+most ponderous metal, about the bigness of our heads, and
+requiring a strong hand to lift them: the other cell contained a
+heap of certain black grains, but of no great bulk or weight,
+for we could hold above fifty of them in the palms of our
+hands.
+About two or three days before I was set at liberty, as
+I was entertaining the court with this kind of feat, there
+arrived an express to inform his majesty, that some of his
+subjects, riding near the place where I was first taken up,
+had seen a great black substance lying on the ground, [...]
+they would undertake to bring it with only five horses.
+FANNY BURNEY, 1778
+You can’t think how oddly my head feels; full of powder
+and black pins, and a great cushion on the top of it.
+if you’ll say that, you’ll say anything: however, if you
+swear till you’re black in the face, I sha’n’t believe you;
+for nobody sha’n’t persuade me out of my senses, that I’m
+resolved.
+Ridiculous, I told him, was a term which he would find
+no one else do him the favour to make use of, in speaking of
+the horrible actions belonging to the old story he made so
+light of; ’actions’ continued I, ’which would dye still deeper
+the black annals of Nero or Caligula.
+
+JANE AUSTEN, 1813
+The ladies were somewhat more fortunate, for they had
+the advantage of ascertaining from an upper window, that
+he wore a blue coat and rode a black horse.
+CHARLOTTE BRONTE, 1847
+And then she had such a fine head of hair; raven black
+and so becomingly arranged: a crown of thick plaits behind,
+and in front the longest, the glossiest curls I ever saw.
+Afternoon arrived: Mrs. Fairfax assumed her best black
+satin gown, her gloves, and her gold watch; for it was her
+part to receive the company,- to conduct the ladies to their
+rooms, Adele, too, would be dressed: though I thought she
+had little chance of being introduced to the party that day at
+least.
+Fluttering veils and waving plumes filled the vehicles; two
+of the cavaliers were young, dashing looking gentlemen; the
+third was Mr. Rochester, on his black horse, Mesrour, Pilot
+bounding before him; at his side rode a lady, and he and she
+were the first of the party.
+The noble bust, the sloping shoulders, the graceful neck,
+the dark eyes and black ringlets were all there; - but her
+face?
+Mrs. Fairfax was summoned to give information respect-
+ing the resources of the house in shawls, dresses, draperies
+of any kind; and certain wardrobes of the third storey were
+ransacked, and their contents, in the shape of brocaded
+and hooped petticoats, satin sacques, black modes, lace
+lappets, etc, were brought down in armfuls by the abi-
+gails; then a selection was made, and such things as were
+chosen were carried to the boudoir within the drawing room.
+NATHANIEL HAWTHORNE, 1850
+Doubtless, however, either of these stern and black
+browed Puritans would have thought it quite a sufficient ret-
+ribution for his sins that, after so long a lapse of years, the
+old trunk of the family tree, with so much venerable moss
+upon it, should have borne, as its topmost bough, an idler
+like myself.
+The Custom House marker imprinted it, with a stencil and
+black paint, on pepper bags, and baskets of anatto, and cigar
+boxes, and bales of all kinds of dutiable merchandise, in
+testimony that these commodities had paid the impost, and
+gone regularly through the office.
+Before this ugly edifice, and between it and the wheel
+track of the street, was a grass plot, much overgrown
+with burdock, pig weed, apple pern, and such unsightly
+vegetation, which evidently found something congenial in
+the soil that had so early borne the black flower of civilised
+society, a prison.
+CHARLES DICKENS, 1861
+I still held her forcibly down with all my strength, like a
+prisoner who might escape; and I doubt if I even knew who
+she was, or why we had struggled, or that she had been in
+flames, or that the flames were out, until I saw the patches of
+tinder that had been her garments no longer alight but falling
+in a black shower around us.
+The sudden exclusion of the night, and the substitution of
+black darkness in its place, warned me that the man had
+closed a shutter.
+He had a boat cloak with him, and a black canvas bag;
+and he looked as like a river pilot as my heart could have
+wished.
+The marshes were just a long black horizontal line then,
+as I stopped to look after him; and the river was just another
+horizontal line, not nearly so broad nor yet so black; and the
+sky was just a row of long angry red lines and dense black
+lines intermixed.
+G.K. CHESTERTON, 1908
+The tall hat and long frock coat were black; the face, in
+an abrupt shadow, was almost as dark.
+He wore an old fashioned black chimney pot hat; he
+was wrapped in a yet more old fashioned cloak, black and
+ragged; and the combination gave him the look of the early
+villains in Dickens and Bulwer Lytton.
+A long, lean, black cigar, bought in Soho for twopence,
+stood out from between his tightened teeth, and altogether he
+looked a very satisfactory specimen of the anarchists upon
+whom he had vowed a holy war.
+He had a black French beard cut square and a black
+English frock coat cut even squarer.
+SOMERSET W. MAUGHAM, 1915
+She wore a black dress, and her only ornament was a gold
+chain, from which hung a cross.
+It was a large black stove that stood in the hall and was
+only lighted if the weather was very bad and the Vicar had a
+cold.
+And the poor lady, so small in her black satin, shrivelled
+up and sallow, with her funny corkscrew curls, took the little
+boy on her lap and put her arms around him and wept as
+though her heart would break.
+D.4
+Yellow
+DANIEL DEFOE, 1719
+The colour of his skin was not quite black, but very tawny;
+and yet not an ugly, yellow, nauseous tawny, as the Brazil-
+ians and Virginians, and other natives of America are, but of
+a bright kind of a dun olive colour, that had in it something
+very agreeable, though not very easy to describe.
+JONATHAN SWIFT, 1726
+The projector of this cell was the most ancient student of
+the academy; his face and beard were of a pale yellow; his
+hands and clothes daubed over with filth.
+The hair of both sexes was of several colours, brown, red,
+black, and yellow.
+I forgot another circumstance (and perhaps I might
+have the reader’s pardon if it were wholly omitted), that
+while I held the odious vermin in my hands, it voided its
+filthy excrements of a yellow liquid substance all over my
+clothes; but by good fortune there was a small brook hard
+by, where I washed myself as clean as I could; although
+I durst not come into my master’s presence until I were
+sufficiently aired.
+
+WILLIAM MAKEPEACE THACKERAY, 1848
+Joseph still continued a huge clattering at the poker and
+tongs, puffing and blowing the while, and turning as red as
+his yellow face would allow him.
+Why, he had the yellow fever three times; twice at Nas-
+sau, and once at St.
+At one end of the hall is the great staircase all in black oak,
+as dismal as may be, and on either side are tall doors with
+stags’ heads over them, leading to the billiard room and the
+library, and the great yellow saloon and the morning rooms.
+He’s never content unless he gets my yellow sealed wine,
+which costs me ten shillings a bottle, hang him!
+JAMES JOYCE, 1914
+I liked the last best because its leaves were yellow.
+Breakfast was over in the boarding house and the table
+of the breakfast room was covered with plates on which lay
+yellow streaks of eggs with morsels of bacon fat and bacon
+rind.
+An immense scarf of peacock blue muslin was wound
+round her hat and knotted in a great bow under her chin;
+and she wore bright yellow gloves, reaching to the elbow.
+His face, shining with raindrops, had the appearance of
+damp yellow cheese save where two rosy spots indicated
+the cheekbones.
+While the point was being debated a tall agile gentleman
+of fair complexion, wearing a long yellow ulster, came from
+the far end of the bar.
+I saw that he had great gaps in his mouth between his
+yellow teeth.
+On the closed square piano a pudding in a huge yellow
+dish lay in waiting and behind it were three squads of bot-
+tles of stout and ale and minerals, drawn up according to
+the colours of their uniforms, the first two black, with brown
+and red labels, the third and smallest squad white, with trans-
+verse green sashes.
+A dull yellow light brooded over the houses and the river;
+and the sky seemed to be descending.
+F. SCOTT FITZGERALD, 1920
+The Gothic halls and cloisters were infinitely more myste-
+rious as they loomed suddenly out of the darkness, outlined
+each by myriad faint squares of yellow light.
+His face was cast in the same yellow wax as in the cafe,
+neither the dull, pasty color of a dead man—rather a sort of
+virile pallor—nor unhealthy, you’d have called it; but like a
+strong man who’d worked in a mine or done night shifts in a
+damp climate.
+The two hours’ ride were like days, and he nearly cried
+aloud with joy when the towers of Princeton loomed up be-
+side him and the yellow squares of light filtered through the
+blue rain.
+Browsing in her library, Amory found a tattered gray book
+out of which fell a yellow sheet that he impudently opened.
+There was that shade of glorious yellow hair, the desire
+to imitate which supports the dye industry.
+F. SCOTT FITZGERALD, 1922
+That this feeble, unintelligent old man was possessed of
+such power that, yellow journals to the contrary, the men in
+the republic whose souls he could not have bought directly
+or indirectly would scarcely have populated White Plains,
+seemed as impossible to believe as that he had once been a
+pink and white baby.
+He bulges in other places his paunch bulges, propheti-
+cally, his words have an air of bulging from his mouth, even
+his dinner coat pockets bulge, as though from contamination,
+with a dog eared collection of time tables, programmes, and
+miscellaneous scraps on these he takes his notes with great
+screwings up of his unmatched yellow eyes and motions of
+silence with his disengaged left hand.
+It was a crackling dusk when they turned in under the
+white fac¸ade of the Plaza and tasted slowly the foam and
+yellow thickness of an egg nog.
+He had fixed his aunt with the bright yellow eye, giving
+her that acute and exaggerated attention that young males
+are accustomed to render to all females who are of no further
+value.
+On the appointed Wednesday in February Anthony had
+gone to the imposing offices of Wilson, Hiemer and Hardy
+and listened to many vague instructions delivered by an en-
+ergetic young man of about his own age, named Kahler, who
+wore a defiant yellow pompadour, and in announcing him-
+self as an assistant secretary gave the impression that it was
+a tribute to exceptional ability.
+Joe Hull had a yellow beard continually fighting through
+his skin and a low voice which varied between basso pro-
+fundo and a husky whisper.
+

CNE5T4oBgHgl3EQfTg-d/content/tmp_files/2301.05537v1.pdf.txt

@@ -0,0 +1,2268 @@
+Almost Surely
+√
+T Regret Bound for Adaptive LQR
+Yiwen Lu and Yilin Mo
+Abstract—The Linear-Quadratic Regulation (LQR) problem
+with unknown system parameters has been widely studied, but it
+has remained unclear whether ˜O(
+√
+T) regret, which is the best
+known dependence on time, can be achieved almost surely. In
+this paper, we propose an adaptive LQR controller with almost
+surely
+˜O(
+√
+T) regret upper bound. The controller features a
+circuit-breaking mechanism, which circumvents potential safety
+breach and guarantees the convergence of the system parameter
+estimate, but is shown to be triggered only finitely often and
+hence has negligible effect on the asymptotic performance of
+the controller. The proposed controller is also validated via
+simulation on Tennessee Eastman Process (TEP), a commonly
+used industrial process example.
+I. INTRODUCTION
+Adaptive control, the study of decision-making under the
+parametric uncertainty of dynamical systems, has been pur-
+sued for decades [1]–[4]. Although early research mainly
+focused on the the aspect of convergence and stability, recent
+years have witnessed significant advances in the quantitative
+performance analysis of adaptive controllers, especially for
+multivariate systems. In particular, in the adaptive Linear-
+Quadratic Regulation (LQR) setting considered in this paper,
+the controller attempts to solve the stochastic LQR problem
+without access to the true system parameters, and its perfor-
+mance is evaluated via regret, the cumulative deviation from
+the optimal cost over time. This adaptive LQR setting has
+been widely studied in the past decade [5]–[11], but it has
+remained unclear whether adaptive controllers for LQR can
+achieve ˜O(
+√
+T) regret, whose asymptotic dependence on the
+time T is the best known (up to poly-logarithmic factors) 1,
+almost surely.
+Existing regret upper bounds on adaptive LQR, summarized
+in Table I, are either weaker than almost-sure in terms of
+the type of probabilistic guarantee, or suboptimal in terms of
+asymptotic dependence on time. By means of optimism-in-
+face-of-uncertainty [7], Thompson sampling [6], or ϵ-greedy
+algorithm [9], an ˜O(
+√
+T) regret can be achieved with proba-
+bility at least 1 − δ. In other words, these algorithms may not
+converge to the optimal one or could even be destablizing
+with a non-zero probability δ. In practice, such a failure
+probability may hinder the application of these algorithms in
+safety-critical scenarios. From a theoretical perspective, we
+argue that it is highly difficult to extend these algorithms
+to provide stronger performance guarantees. To be specific,
+The
+authors
+are
+with
+the
+Department
+of
+Automa-
+tion
+and
+BNRist,
+Tsinghua
+University,
+Beijing,
+P.R.China.
+Emails:
+[email protected],
+[email protected].
+1It is known that
+√
+T is the optimal L1 rate of regret [9], and we suspect
+that it is also the optimal almost sure rate.
+despite different exploration strategies, the aforementioned
+methods all compute the control input using a linear feedback
+gain synthesized from a least-squares estimate of system
+parameters. Due to Gaussian process noise, the derived linear
+feedback gain may be destabilizing, regardless of the amount
+of data collected. For these algorithms, the probability of such
+catastrophic event is bounded by a positive δ > 0, which is a
+predetermined design parameter and cannot be changed during
+online operation.
+Alternative methods that may preclude the above de-
+scribed failure probability include introducing additional sta-
+bility assumptions, using parameter estimation algorithms with
+stronger convergence guarantees, and adding a layer of safe-
+guard around the linear feedback gain. Faradonbeh et al. [10]
+achieves ˜O(
+√
+T) regret almost surely under the assumption
+that the closed-loop system remains stable all the time, based
+on a stabilization set obtained from adaptive stabilization [12].
+Since the stabilization set is estimated from finite data and
+violates the desired property with a nonzero probability, this
+method essentially has a nonzero failure probability as well.
+Guo [13] achieves sub-linear regret almost surely by adopting
+a variant of ordinary least squares with annealing weight
+assigned to recent data, a parameter estimation algorithm
+convergent even with unstable trajectory data. However, the
+stronger convergence guarantee may come at the cost of less
+sharp asymptotic rate, and it is unclear whether the regret
+of this method can achieve
+˜O(
+√
+T) dependence on time.
+Wang et al. [11] achieves ˜O(
+√
+T) regret with a convergence-
+in-probability guarantee, via the use of a switched, rather
+than linear feedback controller, which falls back to a known
+stabilizing gain on the detection of large states. However,
+the controller design of this work does not rule out the
+frequent switching between the learned and fallback gains,
+a typical source of instability in switched linear systems [14],
+which restricts the correctness of their results to the case
+of commutative closed-loop system matrices. Moreover, the
+regret analysis in this work is not sufficiently refined to lead
+to almost sure guarantees.
+TABLE I: Comparison with selected existing
+works on adaptive LQR
+Work
+Rate
+Type of guarantee
+[13]
+Not provided
+Almost sure
+[6], [7], [9]
+˜
+O(
+√
+T)
+Probability 1 − δ
+[10]
+˜
+O(
+√
+T)
+Almost sure*
+[11]
+˜
+O(
+√
+T)
+Convergence in probability*
+This work
+˜
+O(
+√
+T)
+Almost sure
+* Requires non-standard assumptions; please refer to the
+main text for details.
+arXiv:2301.05537v1  [math.OC]  13 Jan 2023
+
+In this paper, we present an adaptive LQR controller with
+˜O(
+√
+T) regret almost surely, only assuming the availability
+of a known stabilizing feedback gain, which is a common
+assumption in the literature. This is achieved by a “circuit-
+breaking” mechanism motivated similarly as [11], which cir-
+cumvents safety breach by supervising the norm of the state
+and deploying the feedback gain when necessary. In contrast
+to [11], however, by enforcing a properly chosen dwell time on
+the fallback mode, the stability of the closed-loop system under
+our proposed controller is unaffected by switching. Another
+insight underlying our analysis is that the above mentioned
+circuit-breaking mechanism is triggered only finitely often.
+This fact implies that the conservativeness of the proposed
+controller, which prevents the system from being destabilized
+in the early stage and hence ensures the convergence of the
+system parameter estimates, may have negligible effect on
+the asymptotic performance of the system. Although similar
+phenomena have also been observed in [11], [15], we derive
+an upper bound on the time of the last trigger (Theorem 3,
+item 7)), a property missing from pervious works that paves
+the way to our almost sure regret guarantee.
+Outline
+The remainder of this manuscript is organized as follows:
+Section II introduces the problem setting. Section III describes
+the proposed controller. Section IV states and proves the
+theoretical properties of the closed-loop system under the pro-
+posed controller, and establishes the main regret upper bound.
+Section V validates the theoretical results using a numerical
+example. Finally, Section VI summarizes the manuscript and
+discusses directions for future work.
+Notations
+The set of nonnegative integers is denoted by N, and the
+set of positive integers is denoted by N∗. The n-dimensional
+Euclidean space is denoted by Rn, and the n-dimensional
+unit sphere is denoted by Sn. The n × n identity matrix is
+denoted by In. For a square matrix M, ρ(M) denotes the
+spectral radius of M, and tr(M) denotes the trace of M. For
+a real symmetric matrix M, M ≻ 0 denotes that M is positive
+definite. For any matrix M, M † denotes the Moore-Penrose
+inverse of M. For two vectors u, v ∈ Rn, ⟨u, v⟩ denotes
+their inner product. For a vector v, ∥v∥ denotes its 2-norm,
+and ∥v∥P = ∥P 1/2v∥ for P ≻ 0. For a matrix M, ∥M∥
+denotes its induced 2-norm, and ∥M∥F denotes its Frobenius
+norm. For a random vector x, x ∼ N(µ, Σ) denotes x is
+Gaussian distributed with mean µ and covariance Σ. For a
+random variable X, X ∼ χ2(n) denotes X has a chi-squared
+distribution with n degrees of freedom. P(·) denotes the
+probability operator, and E[·] denotes the expectation operator.
+For non-negative quantities f, g, which can be deterministic
+or random, we say f ≲ g to denote that f ≤ C1g+C2 for some
+universal constants C1 > 0 and C2 > 0, and f ≳ g to denote
+that g ≲ f. For a random function f(T) and a deterministic
+function g(T), we say f(T) = O(g(T)) to denote that
+lim supT →∞ f(T)/g(T) < ∞, and f(T) =
+˜O(g(T)) to
+denote that f(T) = O(g(T)(log(T))α) for some α > 0.
+II. PROBLEM FORMULATION
+This paper considers a fully observed discrete-time linear
+system with Gaussian process noise specified as follows:
+xk+1 = Axk + Buk + wk,
+k ∈ N∗,
+x1 = 0,
+(1)
+where xk ∈ Rn is the state, uk ∈ Rm is the control input,
+and wk
+i.i.d.
+∼
+N(0, W) is the process noise, where W ≻ 0.
+It is assumed without loss of generality that W = In, but all
+the conclusions apply to general positive definite W up to the
+scaling of constants. It is also assumed that the system and
+input matrices A, B are unknown to the controller, but (A, B)
+is controllable, and that the system is open-loop stable, i.e.,
+ρ(A) < 1. Consequently, there exists P0 ≻ 0 that satisfies the
+discrete Lyapunov equation
+A⊤P0A − P0 + Q = 0,
+(2)
+and there exists a scalar 0 < ρ0 < 1 such that
+A⊤P0A ≺ ρ0P0.
+(3)
+Remark 1. It has been commonly assumed in the liter-
+ature [9], [11] that (A, B) being stabilizable by a known
+feedback gain K0, i.e., ρ(A + BK0) < 1. In such case, the
+system can be rewritten as
+xk+1 = A′xk + Bu′
+k + wk,
+(4)
+where A′ = A + BK0, u′
+k = uk − K0xk, which reduces the
+problem to the case of open-loop stable systems. Therefore,
+we may assume without loss of generality that the system is
+open-loop stable, i.e., K0 = 0, for the simplicity of notations.
+The following average infinite-horizon quadratic cost is
+considered:
+J = lim sup
+T →∞
+1
+T E
+� T
+�
+k=1
+x⊤
+k Qxk + u⊤
+k Ruk
+�
+,
+(5)
+where Q ≻ 0, R ≻ 0 are fixed weight matrices specified by the
+system operator. It is well known that the optimal control law
+is the linear feedback control law of the form u(x) = K∗x,
+where the optimal feedback gain K∗ can be specified as:
+K∗ = −
+�
+R + B⊤P ∗B
+�−1 B⊤P ∗A,
+(6)
+and P ∗ is the unique positive definite solution to the discrete
+algebraic Riccati equation
+P ∗ = A⊤P ∗A − A⊤P ∗B
+�
+R + B⊤P ∗B
+�−1 B⊤P ∗A + Q.
+(7)
+The corresponding optimal cost is
+J∗ = tr
+�
+E
+�
+wkw⊤
+k
+�
+P ∗�
+= tr(WP ∗) = tr(P ∗).
+(8)
+The matrix P ∗ also satisfies the discrete Lyapunov equation
+(A + BK∗)⊤P ∗(A + BK∗) − P ∗ + Q + K⊤RK = 0, (9)
+which implies that there exists a scalar 0 < ρ∗ < 1 such that
+(A + BK∗)⊤P ∗(A + BK∗) ≺ ρ∗P ∗.
+(10)
+Since A, B are unknown to the controller in the considered
+setting, it is not possible to directly compute the optimal
+
+control law from (6) and (7). Instead, the controller learns the
+optimal control law online, whose performance is measured
+via the regret defined as follows:
+R(T) =
+T
+�
+k=1
+(x⊤
+k Qxk + u⊤
+k Ruk) − TJ∗.
+(11)
+The goal of the controller is to minimize the asymptotic growth
+of R(T).
+III. CONTROLLER DESIGN
+The logic of the proposed controller is presented in Algo-
+rithm 1. It can also be illustrated by the block diagram in
+Fig. 1. The remainder of this section would be devoted to
+explaining the components of the controller:
+Algorithm 1 Proposed controller
+1: ˆK0 ← 0
+2: ξ ← 0
+3: for k = 1, 2, . . . do
+4:
+Update parameter estimates ˆAk, ˆBk using (12)
+5:
+if ( ˆAk, ˆBk) is controllable then
+6:
+Update ˆKk from (6)-(7), replacing (A, B) with
+( ˆAk, ˆBk).
+7:
+else
+8:
+ˆKk ← 0
+9:
+uce
+k ← ˆKkxk
+10:
+if ξ = 0 then
+11:
+if ∥uce
+k ∥ > Mk := log(k) then
+12:
+ξ ← tk := ⌊log(k)⌋
+13:
+ucb
+k ← 0
+14:
+else
+15:
+ucb
+k ← uce
+k
+16:
+else
+17:
+ucb
+k ← 0
+18:
+ξ ← ξ − 1
+19:
+upr
+k ← k−1/4vk, where vk ∼ N(0, Im)
+20:
+Apply uk ← ucb
+k + upr
+k
+The proposed controller is a variant of the certainty equiv-
+alent controller [16], where the latter applies the input uce
+k =
+ˆKkxk, where the feedback gain ˆKk is calculated from (6)-
+(7) by treating the current estimates of the system parameters
+ˆAk, ˆBk as the true values. The differences of the proposed
+controller compared to the standard certainty equivalent con-
+troller is that i) it includes a “circuit-breaking” mechanism,
+which replace uce
+k
+with zero in certain circumstances; ii) it
+superposes the control input at each step with a probing noise
+upr
+k .
+The circuit-breaking mechanism replaces uce
+k
+with zero
+for the subsequent tk steps when the norm of the certainty
+equivalent control input ∥uce
+k ∥ exceeds a threshold Mk. The
+intuition behind this design is that a large certainty equivalent
+control input is indicative of having applied a destabilizing
+feedback gain recently, and circuit-breaking may prevent the
+state from exploding by leveraging the innate stability of the
+system, and hence help with the convergence of the parameter
+Circuit-
+breaking
+logic
+ξ > 0
+0
+ˆK
+uce
+ucb
++
+Plant
+(eq. (1))
+Estimator
+(eq. (7))
+ξ
+w
+upr
+u
+x
+ξ = 0
+Fig. 1: Block diagram of the closed-loop system under the
+proposed controller. The control input is the superposition of
+a deterministic input ucb and a random probing input upr. The
+deterministic part ucb is normally the same as the certainty
+equivalent input uce, but takes the value zero when circuit-
+breaking is triggered, where ξ is a counter for circuit-breaking.
+The certainty equivalent gain is updated using the parameter
+estimator in the meantime.
+estimator and the asymptotic performance of the controller.
+The threshold Mk is increased with the time index k, so
+that the “false alarm rate” of circuit-breaking, caused by the
+occasional occurrence of large noise vectors, decays to zero.
+The dwell time tk is also increased with time, in order to
+circumvent the potential oscillation of the system caused by
+the frequent switching between ˆK and 0 (c.f. [17]). Both Mk
+and tk are chosen to grow logarithmically with k, which would
+support our technical guarantees.
+Similarly to [9], [11], a probing noise upr
+k is superposed to
+the control input at each step to provide sufficient excitation
+to the system, which is required for the estimation of system
+parameters. Specifically, the probing noise is chosen to be
+upr
+k
+= k−1/4vk, where vk
+i.i.d.
+∼
+N(0, Im), which would
+correspond to the optimal rate of regret.
+The estimates of the system parameters ˆAk, ˆBk are up-
+dated using an ordinary least squares estimator. Denote Θ =
+[A
+B], ˆΘk
+= [ ˆAk
+ˆBk], and zk
+= [x⊤
+k
+u⊤
+k ]⊤, then
+according to xk+1 = Θzk, the ordinary least squares estimator
+can be specified as
+ˆΘk =
+�k−1
+�
+t=1
+xtz⊤
+t
+� �k−1
+�
+t=1
+ztz⊤
+t
+�†
+.
+(12)
+Remark 2. In the presented algorithm, the certainty equivalent
+gain is updated at every step k (see line 6 of Algorithm 1), but
+it may also be updated “logarithmically often” [11] (e.g., at
+steps k = 2i, i ∈ N∗), which is more computationally efficient
+in practice. It can be verified that all our theoretical results
+also apply to the case of logarithmically often updates.
+IV. MAIN RESULTS
+Underlying our analysis of the closed-loop system under the
+proposed controller are two random times, defined below:
+
+Tstab := inf
+�
+T
+���
+�
+Atk�⊤ P ∗Atk < ρP ∗,
+�
+A + B ˆKk
+�⊤
+P ∗ �
+A + B ˆKk
+�
+< ρP ∗, ∀k ≥ T
+�
+,
+(13)
+where ρ = (1 + ρ∗)/2, and tk is the dwell time defined in
+line 12 of Algorithm 1. And
+Tnocb := inf
+�
+T | ucb
+k ≡ uce
+k , ∀k ≥ T
+�
+.
+(14)
+With the above two random times, the evolution of the system
+can be divided into three stages:
+1) From the beginning to Tstab, the adaptive controller grad-
+ually refines its estimate of system parameters, and hence
+improves the performance of the certainty equivalent
+feedback gain, until the system becomes stabilized in the
+sense that there is a common Lyapunov function between
+the two modes under circuit-breaking as indicated in (13).
+2) From Tstab to Tnocb, the closed-loop system is stable as is
+ensured by the aforementioned common Lyapunov func-
+tion, and under mild regularity conditions on the noise,
+an upper bound on the magnitude certainty equivalent
+control input ∥uce
+k ∥ eventually drops below the circuit-
+breaking threshold Mk.
+3) From Tnocb on, circuit-breaking is not triggered any more,
+and the system behaves similarly as the closed-loop
+system under the optimal controller, with only small
+perturbations on the feedback gain which stem from the
+parameter estimation error and converge to zero.
+In the following theorem, we state several properties of the
+closed-loop system, based on which the above two random
+times are quantified:
+Theorem 3. Let n, m be the dimensions of the state and input
+vectors respectively, and P0, ρ0, P ∗, ρ∗ be defined in (2), (3),
+(7), (10) respectively. Then the following properties hold:
+1) For 0 < δ ≤ 1/2, the event
+Enoise(δ) :=
+�
+max{∥wk∥, ∥vk∥} ≤
+2
+√
+n + 1
+�
+log(k/δ), ∀k ∈ N∗�
+(15)
+occurs with probability at least 1 − 2δ.
+2) For 0 < δ ≤ 1/(8n2), the event
+Ecov(δ) :=
+������
+k
+�
+i=1
+(wiw⊤
+i − In)
+����� ≤
+7n
+√
+k log(8n2k/δ), ∀k ∈ N∗
+�
+(16)
+occurs with probability at least 1 − δ.
+3) On the event Enoise(δ), it holds
+∥xk∥ ≤ Cx log(k/δ), ∀k ∈ N∗,
+(17)
+where
+Cx = (∥B∥ + 1)(2√n + 1 + 1)∥P0∥∥P −1
+0
+∥
+1 − ρ1/2
+0
+.
+(18)
+4) For 0 < δ ≤ 1/6, the event
+Ecross(δ) :=
+������
+k
+�
+i=1
+w⊤
+i P ∗(Axi + Bucb
+i )
+����� ≤
+Ccross
+√
+k(log(k/δ))2, ∀k ∈ N∗
+�
+(19)
+occurs with probability at least 1 − 6δ, where
+Ccross = 4
+√
+n + 1∥P ∗∥(∥A∥Cx + ∥B∥).
+(20)
+5) For δ satisfying
+0 < δ < min
+�
+(800CV )−1, exp
+�
+24(m + n)1/3��
+,
+(21)
+the event
+Eest(δ) :=
+����ˆΘk − Θ
+���
+2
+≤ CΘk−1/2 log(k/δ),
+∀k ≥ k0
+�
+,
+(22)
+occurs with probability at least 1 − 6δ, where
+k0 = ⌈600(m + n) log(1/δ) + 5400⌉,
+(23)
+CV = Cx + 2
+√
+n + 1 + 1,
+(24)
+CΘ = (3200n/9)(5n/2 + 2).
+(25)
+6) On the event Eest(δ), it holds
+Tstab ≲ (log(1/δ))2.
+(26)
+7) On the event Enoise(δ) ∩ Eest(δ), for any α > 0, it holds
+Tnocb ≲ (1/δ)α.
+(27)
+Remark 4. The conclusions of Theorem 3 can be explained
+as follows:
+Items 1) and 2) defines two high-probability events on the
+regularity of noise. Item 3) bounds the state norm under regular
+noise, based on which item 4) bounds the growth of a cross
+term between noise and state that would be useful in regret
+analysis. Item 5) bounds the parameter estimation error, based
+on which item 6) bounds Tstab. Finally, item 7) states that
+the circuit-breaking mechanism is triggered only finitely, and
+bounds Tnocb, the time after which the circuit-breaking is not
+triggered any more.
+The following corollary characterizes the tail probabilities
+of Tstab and Tnocb, which follows directly from items 6) and
+7) of Theorem 3:
+Corollary 5. For Tstab defined in (13) and Tnocb defined
+in (14), as T → ∞, it holds
+1)
+P(Tstab ≥ T) = O(exp(−c
+√
+T)),
+(28)
+where c > 0 is a system-dependent constant.
+
+2) For any α > 0,
+P(Tnocb ≥ T) = O(T −α).
+(29)
+Building upon the properties stated in Theorem 3, a high-
+probability bound on the regret under the proposed controller
+can be ensured:
+Theorem 6. Given a failure probability δ, there exists a
+constant T0 ≲ (1/δ)1/4, such that for any fixed T > T0,
+it holds with probability at least 1 − δ that the regret defined
+in (11) satisfies
+R(T) ≲ (1/δ)1/4 +
+√
+T(log(T/δ))3.
+(30)
+As corollaries of Theorem 6, one can obtain a bound on the
+tail probability of R(T) (Theorem 7), and the main conclusion
+of this work, i.e., an almost sure bound on R(T) (Theorem 8):
+Theorem 7. For sufficiently large T, it holds
+P
+�
+R(T) ≥ CR
+√
+T(log(T))3�
+≤ 1
+T 2 ,
+(31)
+where CR is a system-dependent constant.
+Proof. The conclusion follows from invoking Theorem 6 with
+δ = 1/T 2.
+Theorem 8. It holds almost surely that
+R(T) = ˜O
+�√
+T
+�
+.
+(32)
+Proof. By Theorem 7, we have
+∞
+�
+T =1
+P
+�
+R(T) ≥ CR
+√
+T(log(T))3�
+< +∞.
+(33)
+By Borel-Cantelli lemma, it holds almost surely that the event
+�
+R(T) ≥ CR
+√
+T(log(T))3�
+(34)
+occurs finitely often, i.e.,
+R(T) = ˜O
+�√
+T
+�
+,
+a.s.
+(35)
+The remainder of this section is dedicated to proving
+Theorems 3 and 6.
+A. Proof of Theorem 3, item 1)
+Proof. Since wk ∼ N(0, In), it holds ∥wk∥ ∼ χ2(n). Apply-
+ing the Chernoff bound, for any a > 0, and any 0 < t < 1/2,
+it holds
+P(X ≥ a) ≤ E
+�
+etX/eta�
+= (1 − 2t)−n/2 exp(−ta).
+(36)
+Choosing t = 1/4, we have
+P(∥wk∥ ≥ a) ≤ 2n/2 exp(−a2/4)
+(37)
+for any k ∈ N∗ and a > 0. Invoking (37) with ak =
+2(log(ck2/δ))1/2, where c = 2n/2π2/6, we have
+∞
+�
+k=1
+P(∥wk∥ ≥ ak) ≤
+∞
+�
+k=1
+2n/2c−1δ/k2 = δ,
+(38)
+i.e., it holds with probability at least 1 − δ that
+∥wk∥ ≤ 2(log(ck2/δ))1/2 ≤ 2
+√
+n + 1
+�
+log(k/δ), ∀k ∈ N∗.
+(39)
+Similarly, it also holds with probability at least 1 − δ that
+∥vk∥ ≤ 2
+√
+n + 1
+�
+log(k/δ), ∀k ∈ N∗.
+(40)
+Combining (39) and (40) leads to the conclusion.
+B. Proof of Theorem 3, item 2)
+We start with a concentration bound on the sum of product
+of Gaussian random variables:
+Lemma 9. Let Xi
+i.i.d.
+∼
+N(0, 1), Yi
+i.i.d.
+∼
+N(0, 1), and
+{Xi}, {Yi} be mutually independent, then
+1) With probability at least 1 − 4δ,
+�����
+k
+�
+i=1
+X2
+i − k
+����� ≤ 7
+√
+k log(k/δ), ∀k ∈ N∗.
+(41)
+2) With probability at least 1 − 8δ,
+�����
+k
+�
+i=1
+XiYi
+����� ≤ 5
+√
+k log(k/δ), ∀k ∈ N∗.
+(42)
+Proof. Since �k
+i=1 X2
+i ∼ χ2(k), according to [18, Lemma 1],
+for any a > 0 and any k ∈ N∗, it holds
+P
+������
+k
+�
+i=1
+X2
+i − k
+����� ≥ 2√na + 2a
+�
+≤ 2 exp(−a).
+(43)
+Fix k and choose a = log(k2/δ), and it follows that with
+probability at least 1 − 2δ/k2,
+�����
+k
+�
+i=1
+X2
+i − k
+����� ≤ 2
+�
+k log(k2/δ) + 2 log(k2/δ)
+≤ 7
+√
+k log(k/δ).
+(44)
+Taking the union bound over k ∈ N∗, one can show that (41)
+holds with probability at least 1 − 2δ �∞
+k=1(1/k2) > 1 − 4δ,
+and hence claim 1) is proved.
+Since XiYi = (Xi+Yi)2/4+(Xi−Yi)2/4, and Xi+Yi
+i.i.d.
+∼
+N(0, 2), Xi−Yi
+i.i.d.
+∼ N(0, 2), claim 2) follows from applying
+claim 1) to {(Xi +Yi)/
+√
+2} and {(Xi −Yi)/
+√
+2} respectively
+and taking the union bound.
+Theorem 3, item 2) follows from the above lemma:
+Proof. Applying Lemma 9, item 1) to the diagonal elements,
+and Lemma 9, item 2) to the off-diagonal elements of
+�k
+i=1(wiw⊤
+i − I), and taking the union bound, one can show
+that with probability at least 1 − 8n2δ,
+k
+�
+i=1
+(wiw⊤
+i − In) ≤ 7
+√
+k log(k/δ),
+(45)
+
+where the inequality holds component-wise. Hence, with prob-
+ability at least 1 − 8n2δ,
+�����
+k
+�
+i=1
+(wiw⊤
+i − In)
+�����
+2
+≤
+�����
+k
+�
+i=1
+(wiw⊤
+i − In)
+�����
+2
+F
+≤ n2(7
+√
+k log(k/δ)),
+(46)
+and scaling the failure probability leads to the conclusion.
+C. Proof of Theorem 3, item 3)
+Proof. Notice
+xk = Ak−2d1 + Ak−3d2 + · · · + dk−1,
+(47)
+where dk = B(ucb
+k + upr
+k ) + wk. On Enoise(δ), it holds
+∥dk∥ ≤ ∥B∥ log(k) + 2(∥B∥ + 1)
+√
+n + 1
+�
+log(k/δ)
+≤ (∥B∥ + 1)(2
+√
+n + 1 + 1) log(k/δ).
+(48)
+Furthermore, from (2) and (47), it holds
+∥xk∥P ≤ ρ(k−2)/2
+0
+∥d1∥P + ρ(k−3)/2
+0
+∥d2∥P + · · · + ∥dk−1∥P
+≤
+1
+1 − ρ1/2
+0
+∥dk∥P ,
+(49)
+from which the conclusion follows.
+D. Proof of Theorem 3, item 4)
+This result is a corollary of a time-uniform version of
+Azuma-Hoeffding inequality [19], stated below:
+Lemma 10. Let {φk}k≥1 be a martingale difference sequence
+adapted to the filtration {Fk} satisfying |φk| ≤ dk a.s., then
+with probability at least 1 − 4δ, it holds
+�����
+k
+�
+i=1
+φi
+����� ≤ 2
+�
+�
+�
+�
+k
+�
+i=1
+d2
+i log(k/δ), ∀k.
+(50)
+Proof. By Azuma-Hoeffding inequality [19], for a fixed k, it
+holds with probability at least 1 − 2δ/k2 that
+�����
+k
+�
+i=1
+φi
+����� ≤
+�
+�
+�
+�2
+k
+�
+i=1
+d2
+i log(k2/δ) ≤ 2
+�
+�
+�
+�
+k
+�
+i=1
+d2
+i log(k/δ).
+(51)
+Taking the union bound over k ∈ N∗, one can prove that (50)
+holds with probability at least 1 − 2δ �∞
+k=1(1/k2) > 1 −
+4δ.
+Theorem 3, item 4) follows from the above lemma:
+Proof. According to Theorem 3, item 1), we only need to
+prove
+P(Ecross(δ) | Enoise(δ)) ≥ 1 − 4δ.
+(52)
+Therefore, we condition the remainder of the proof upon the
+event Enoise(δ). Let Fk be the σ-algebra generated by v1, w1,
+v2, w2, . . . , vk−1, wk−1, vk. Since xk ∈ Fk−1, ucb
+k ∈ Fk−1,
+E[wk | Fk−1] = 0 due to symmetry, it holds
+E
+�
+w⊤
+k P ∗ �
+Axk + Bucb
+k
+� ��Fk
+�
+=E [wk | Fk−1]⊤ P ∗ �
+Axk + Bucb
+k
+�
+= 0,
+(53)
+i.e., {w⊤
+k P ∗(Axk+Bucb
+k )} is a martingale difference sequence
+adapted to the filtration {Fk}. Furthermore, by Theorem 3,
+item 3, we have
+��w⊤
+k P ∗ �
+Axk + Bucb
+k
+���
+≤∥wk∥∥P ∗∥
+�
+∥A∥∥xk∥ + ∥B∥
+��ucb
+k
+���
+≤1
+2Ccross log(k/δ).
+(54)
+Hence, the conclusion follows from applying Lemma 10.
+E. Proof of Theorem 3, item 5)
+This subsection is devoted to deriving the time-uniform
+upper bound on estimation error stated in Theorem 3, item 5).
+Throughout this subsection, we denote Θ = [A
+B], ˆΘk =
+[ ˆAk
+ˆBk], zk = [x⊤
+k
+u⊤
+k ]⊤, and Vk = �k−1
+i=1 ziz⊤
+i .
+The proof can be split into three parts: firstly, we character-
+ize the estimation error ∥ˆΘk − Θ∥ in terms of the maximum
+and minimum eigenvalues of the regressor covariance matrix
+Vk, using a result in martingale least squares [5]. Secondly,
+an upper bound on the ∥Vk∥, which is a consequence of the
+non-explosiveness of states, can be derived as an corollary
+of Theorem 3, item 3). Finally, an upper bound on ∥V −1
+k
+∥,
+or equivalently a lower bound on the minimum eigenvalue
+of Vk, which is a consequence of sufficient excitation of the
+system, can be proved using an anti-concentration bound on
+block martingale small-ball (BMSB) processes [20]. The three
+parts would be discussed respectively below.
+1) Upper bound on least squares error, in terms of Vk:
+Lemma 11 ( [5, Corollary 1 of Theorem 3]). Let
+Sk =
+k
+�
+i=1
+ηimi−1, Uk =
+k
+�
+i=1
+mi−1m⊤
+i−1,
+(55)
+where {Fk}k∈N∗ is a filtration, {ηk}k∈N∗ is a random scalar
+sequence with ηk | Fk being conditionally σ2-sub-Gaussian,
+and {mk}k∈N∗ is a random vector sequence with mk ∈ Fk.
+Then with probability at least 1 − δ,
+∥Sk∥(U0+Uk)−1 ≤ 2σ2·
+log
+�
+det(U0)−1/2 det(U0 + Uk)1/2/δ
+�
+,
+∀k ∈ N∗, (56)
+where U0 ≻ 0 is an arbitrarily chosen constant positive semi-
+definite matrix.
+Proposition 12. With probability at least 1 − δ,
+���ˆΘk − Θ
+���
+2
+≤ 2n
+��V −1
+k
+��
+�
+log
+�n
+δ
+�
++ n
+2 log(1 + ∥Vk∥)
+�
+,
+∀k ≥ m + n + 1.
+(57)
+
+Proof. Let Fk be the σ-algebra generated by v1, w1, v2, w2,
+. . . , vk−1, wk−1, vk. From xk+1 = Θzk + wk, we have
+ˆΘk − Θ =
+�k−1
+�
+i=1
+wiz⊤
+i
+� �k−1
+�
+i=1
+ziz⊤
+i
+�†
+,
+(58)
+where wk|Fk ∼ N(0, In), zk ∈ Fk. With Vk = �k−1
+i=1 ziz⊤
+i ,
+we have Vi ≻ 0 a.s. for k ≥ m + n + 1. Now we can apply
+Lemma 11 to each row of ˆΘk − Θ: for each of j = 1, . . . , n,
+let Sj,k = �k−1
+i=1 (e⊤
+j wi)zi, where ej is the j-th standard unit
+vector. By invoking Lemma 11 with Uk = Vk and U0 = Im+n,
+we have: with probability at least 1 − δ,
+���e⊤
+j (ˆΘk − Θ)
+���
+2
+=
+��V −1
+k
+Sj,k
+�� ≤
+��V −1
+k
+�� ∥Sj,k∥(I+Vk)−1
+≤ 2
+��V −1
+k
+�� log
+�
+det(I + Vk)1/2/δ
+�
+≤ 2
+��V −1
+k
+��
+�
+log
+�1
+δ
+�
++ n
+2 log(1 + ∥Vk∥)
+�
+.
+(59)
+Taking the union bound over j = 1, . . . , n, we have: with
+probability at least 1 − nδ,
+���ˆΘk − Θ
+���
+2
+≤
+���ˆΘk − Θ
+���
+2
+F
+≤
+n
+�
+j=1
+���e⊤
+j (ˆΘk − Θ)
+���
+2
+≤2n
+��V −1
+k
+��
+�
+log
+�1
+δ
+�
++ n
+2 log(1 + ∥Vk∥)
+�
+.
+(60)
+Scaling the failure probability results in the conclusion.
+2) Upper bound on ∥Vk∥:
+Proposition 13. On the event Enoise(δ) defined in (15), it holds
+∥Vk∥ ≤ CV k(log(k/δ))2,
+(61)
+where CV is defined in (24).
+Proof. On Enoise(δ), we have
+∥uk∥ ≤ log(k) + 2
+√
+n + 1
+�
+log(k/δ),
+(62)
+and by Theorem 3, item 3), we have
+∥xk∥ ≤ Cx log(k/δ).
+(63)
+Hence,
+∥zk∥ ≤ ∥xk∥ + ∥uk∥ ≤
+�
+CV log(k/δ),
+(64)
+which implies
+∥Vk∥ ≤
+k−1
+�
+i=1
+∥zk∥2 ≤ CV k(log(k/δ))2.
+(65)
+3) Upper bound on ∥V −1
+k
+∥: We shall borrow the techniques
+of analyzing BMSB processes from Simchowitz et al. [20] to
+bound ∥V −1
+k
+∥. The BMSB process is defined as follows:
+Definition 14 ( [20, Definition 2.1]). Suppose that {φk}k∈N∗ is
+a real-valued stochastic process adapted to the filtration {Fk}.
+We say the process {φk} satisfies the (l, ν, p) block martingale
+small-ball (BMSB) condition if:
+1
+l
+l
+�
+i=1
+P (|φk+i| ≥ ν | Ft) ≥ p, ∀k ∈ N∗.
+(66)
+The following lemma verifies that {zk}, projected along an
+arbitrary direction, is BMSB:
+Lemma 15. For any µ ∈ Sm+n, the process {⟨zi, µ⟩}k−1
+i=1
+satisfies the (1, k−1/4, 3/10) BMSB condition.
+Proof. Let Fk be the σ-algebra generated by v1, w1, v2, w2,
+. . . , vk−1, wk−1, vk. Since
+zi =
+�xi
+ui
+�
+=
+�
+xi
+ucb
+i + upr
+i
+�
+,
+(67)
+and xi+1 = Aixt + Bupr
+i
++ wi, where Ai takes value from
+{A, A + B ˆKi} and belongs to Fi, we have
+|⟨zi+1, µ⟩| = |⟨xi+1, µ1⟩ + ⟨ucb
+i+1, µ2⟩ + ⟨upr
+i+1, µ2⟩|
+≥ |⟨Aixi + Bupr
+i + wi, µ1⟩ + ⟨upr
+i+1, µ2⟩|
+=
+����
+� �Aixi + Bupr
+i + wi
+upr
+i+1
+�
+, µ
+����� ,
+(68)
+where µ1 = [In
+0]µ, µ2 = [0
+Im]µ. Therefore, we only
+need to verify
+P
+�����
+� �Aixi + Bupr
+i + wi
+upr
+i+1
+�
+, µ
+����� ≥ k−1/4���Fi
+�
+≥ 3
+10. (69)
+Since Aixi ∈ Fi, and upr
+i
+| Fi, wi | Fi, upr
+i+1 | Fi are all
+Gaussian,
+� �Aixi + Bupr
+i + wi
+upr
+i+1
+�
+, µ
+�
+,
+(70)
+as an affine function of the above terms, is also Fi-
+conditionally Gaussian, whose mean and variance are:
+E
+�� �Aixi + Bupr
+i + wi
+upr
+i+1
+�
+, µ
+����� Fi
+�
+= ⟨Aixi, µ1⟩,
+E
+��� �Aixi + Bupr
+i + wi
+upr
+i+1
+�
+, µ
+�
+− ⟨Aixi, µ1⟩
+�2����� Fi
+�
+= E
+�� �Bupr
+i + wi
+upr
+i+1
+�
+, µ
+�2���� Fi
+�
+= µT
+�
+i−1/2BB⊤ + I
+0
+0
+(i + 1)−1/2
+�
+µ
+≥ µT
+�I
+0
+0
+k−1/2
+�
+µ ≥ k−1/2.
+(71)
+The conclusion (69) then follows from the fact that for any
+X ∼ N(µ, σ2), it holds P(|X| ≥ σ) ≥ P(|X − µ| ≥ σ) ≥
+3/10.
+An upper bound on ∥V −1
+k
+∥ can be obtained by applying an
+anti-concentration property of BMSB, along with a covering
+argument in [20]:
+
+Lemma 16. For any fixed k ≥ 2 and 0 < δk ≤ 1/2, it holds
+P
+���V −1
+k
+�� ≥ 1600
+9
+k−1/2
+�
+≤
+δk + e−
+9
+800 k+(m+n) log(800CV k1/2(log(k/δk))2),
+(72)
+where CV is defined in Theorem 3, item 5).
+Proof. Firstly, for any fixed µ ∈ Sm+n, applying [20, Prop.
+2.5] to the process {⟨zi, µ⟩}k−1
+i=1 , which is (1, k−1/4, 3/10)-
+BMSB by Lemma 15, we have
+P
+�k−1
+�
+i=1
+⟨zi, µ⟩2 ≤ k−1/2(3/10)2
+8
+k
+�
+≤ e− (3/10)2k
+8
+,
+(73)
+i.e.,
+P
+�
+µ⊤Vkµ ≤
+9
+800k1/2
+�
+≤ e−
+9
+800 k.
+(74)
+Next, we shall choose multiple µ’s and using a covering
+argument to lower bound the minimum eigenvalue of Vk, and
+hence to upper bound ∥V −1
+k
+∥: let Γ = CV k(log(k/δk))2Im+n,
+and Γ = (9/800)k1/2Im+n. Let T be a minimal 1/4-net of
+SΓ in the norm ∥ · ∥Γ, then by [20, Lemma D.1], we have
+log |T | ≤ (m + n) log(9) + log det(ΓΓ−1)
+≤ (m + n) log(800CV k1/2(log(k/δk))2).
+(75)
+According to (74) and (75), we have
+P
+�
+µ⊤Vkµ ≤
+9
+800k1/2, ∀µ ∈ T
+�
+≤ |T |e−
+9
+800 k
+≤ e−
+9
+800 k+(m+n) log(800CV k1/2(log(k/δk))2).
+(76)
+On the other hand, according to Proposition 13 and Theorem 3,
+item 1), we have
+P
+�
+∥Vk∥ ≤ CV k(log(k/δk))2�
+≤ δk,
+(77)
+From (76), (77) and [20, Lemma D.1], it follows that
+P (Vk ≻ Γ/2) is no greater than the RHS of (72), which is
+equivalent to the conclusion.
+The next proposition converts Lemma 16 into a time-
+uniform bound:
+Proposition 17. For δ satisfying (21) and k0 defined in (23),
+it holds
+P
+���V −1
+k
+�� ≤ 1600
+9
+k−1/2, ∀k ≥ k0
+�
+≤ 3δ.
+(78)
+Proof. For k ≥ k0, with δk = δ/k2, it holds
+k ≥ (1300(m + n))4/3 ,
+(79)
+and hence,
+(m + n) log(800CV k1/2(log(k/δk))2)
+≤(m + n)
+�
+3 log(1/δ) + 13
+2 log(k)
+�
+≤ 1
+200k + 13
+2 (m + n)k1/4 ≤
+1
+100k.
+(80)
+Substituting (80) into (72), we have
+P
+���V −1
+k
+�� ≥ 1600
+9
+k−1/2
+�
+≤ δ
+k2 + e−
+1
+800 k.
+(81)
+Taking the union bound over k = k0, k0 + 1, . . ., we have
+P
+���V −1
+k
+�� ≥ 1600
+9
+k−1/2, ∀k ≥ k0
+�
+≤π2
+6 δ + 801e−
+1
+800 k0 ≤ 3δ.
+(82)
+Theorem 3, item 5) follows from Propositions 12, 13 and
+17.
+F. Proof of Theorem 3, item 6)
+Proof. Let
+T1 = inf
+�
+T
+���
+�
+Atk�⊤ P ∗Atk < ρP ∗, ∀k ≥ T
+�
+,
+(83)
+T2 = inf
+�
+T
+����
+�
+A + B ˆKk
+�⊤
+P ∗ �
+A + B ˆKk
+�
+< ρP ∗,
+∀k ≥ T
+�
+.
+(84)
+We shall bound T1 and T2 respectively:
+1) From the assumption ρ(A) < 1, it holds
+lim
+k→∞
+�
+Ak�⊤ PAk = 0,
+(85)
+which, together with tk = ⌊log(k)⌋, implies T1 is a finite
+constant independent of δ, i.e., T1 ≲ 1.
+2) Since (A + BK)⊤P ∗(A + BK) is a continuous function
+of K, from (10), there exists a system-dependent con-
+stant ϵK, such that (A + BK)⊤P ∗(A + BK) < ρP ∗
+whenever ∥K − K∗∥ < ϵK. On the other hand, since
+ˆKk is a continuous function of ˆΘk [9, Proposition 6],
+there exists a system-dependent constant ϵΘ, such that
+∥ ˆK − K∗∥ < ϵK as long as ∥ˆΘk − Θ∥ < ϵΘ. It follows
+from Theorem 3, item 5) that ∥ˆΘk − Θ∥ < ϵΘ whenever
+k ≥ (9C2
+Θ/ϵ2
+Θ)(log(1/δ))2, and hence T2 ≲ (log(1/δ))2.
+In summary, it holds Tstab = max{T1, T2} ≲ (log(1/δ))2.
+G. Proof of Theorem 3, item 7)
+This subsection is dedicated to bounding the time after
+which the circuit-breaking is not triggered any more. The
+outline of the proof is stated as follows: firstly, we define
+a subsequence notation to deal with the dwell time tk of
+circuit breaking. Secondly, an upper bound on the state and
+the certainty equivalent input after Tstab is derived, which is
+shown to be asymptotically smaller than the circuit-breaking
+threshold Mk = log(k). Based on the above upper bound on
+the certainty equivalent input, we can finally bound Tnocb, i.e.,
+the time it takes for the certainty equivalent input to stay below
+the threshold Mk, which leads to the desired conclusion.
+
+1) A subsequence notation: Consider the subsequence of
+states and inputs, where steps within the circuit-breaking
+period are skipped, defined below:
+i(1) = 1, i(k + 1) =
+�
+i(k) + 1
+ucb
+i(k) ̸= 0
+i(k) + ti(k)
+ucb
+i(k) = 0 ,
+(86)
+˜xk = xi(k), ˜uce
+k = uce
+i(k), ˜ucb
+k = ucb
+i(k).
+(87)
+Consider Tstab defined in (13). We can define ˜Tstab as the
+first index in the above subsequence for which the stabilization
+condition is satisfied, i.e.,
+˜Tstab = inf{T | i(T) ≥ Tstab}.
+(88)
+2) Upper bound on state and certainty equivalent input
+after Tstab:
+Proposition 18. On the event Enoise(δ) ∩ Eest(δ), it holds
+��˜x ˜Tstab+k
+�� ≲ ρk/2 log(1/δ) +
+�
+log(k/δ),
+(89)
+���˜uce
+˜Tstab+k
+��� ≲ ρk/2 log(1/δ) +
+�
+log(k/δ),
+(90)
+where ρ = (1 + ρ∗)/2.
+Proof. We can expand ˜x ˜Tstab+k as:
+˜x ˜Tstab+k = ˜A ˜Tstab+k−1 ˜A ˜Tstab+k−2 · · · ˜A ˜Tstab ˜x ˜Tstab+
+˜A ˜Tstab+k−1 ˜A ˜Tstab+k−2 · · · ˜A ˜Tstab+1 ˜w ˜Tstab+
+· · · +
+˜w ˜Tstab+k−1,
+(91)
+where:
+• ˜Aj ∈ {A + B ˆKi(j), Ati(j)}, and must satisfy ˜A⊤
+j P ˜Aj <
+ρP for j ≥ ˜Tstab, by definition of ˜Tstab in (13);
+• ˜wj ∈ {wi(j), �ti(j)−1
+τ=0
+Aτwi(j−1)+τ}, and on the event
+Enoise(δ), it must satisfy ∥ ˜wj∥
+≲
+A
+�
+log(j/δ)
+≲
+�
+log(j/δ) for any j, where A = �∞
+τ=0 ∥Aτ∥.
+Combining the above two items, we have
+��˜x ˜Tstab+k
+�� ≲ ρk/2 ��˜x ˜Tstab
+�� +
+�
+log
+�
+i
+�
+˜Tstab + k
+�
+/δ
+�
+. (92)
+We shall next bound
+��˜x ˜Tstab
+�� and i( ˜Tstab + k) respectively:
+1) According to the definition of ˜Tstab and i(·) in (86)
+and (88), we have
+i
+�
+˜Tstab
+�
+≤ Tstab + log(Tstab).
+(93)
+On Eest(δ), according to Theorem 3, item 6, we have
+Tstab ≲ (log(1/δ))2, and hence
+i
+�
+˜Tstab
+�
+≲ (log(1/δ))2 + log((log(1/δ))2)
+≲ (log(1/δ))2.
+(94)
+Furthermore, according to Theorem 3, item 3, on
+Enoise(δ), we have ∥xk∥ ≲ log(k/δ) for any k, and hence
+��˜x ˜Tstab
+�� =
+���xi( ˜Tstab)
+��� ≲ log
+�(log(1/δ))2
+δ
+�
+= log(1/δ) + log((log(1/δ))2) ≲ log(1/δ).
+(95)
+2) By definition of i(·) in (86), we have
+i
+�
+˜Tstab + k + 1
+�
+≤ ˜Tstab +k+log
+�
+˜Tstab + k
+�
+, ∀k ∈ N∗.
+(96)
+Applying induction to (93) and (96), we can obtain
+i
+�
+˜Tstab + k
+�
+≲ k log(Tstabk) + Tstab.
+(97)
+Substituting Tstab ≲ (log(k/δ))2, which holds on Eest(δ)
+according to Theorem 3, item 6), into (97), we have
+i
+�
+˜Tstab + k
+�
+≲ k(log(k/δ))2.
+(98)
+Hence, inequality (89) follows from substituting (95) and
+(98) into (92). Moreover, since
+��� ˆKi( ˜Tstab+k)
+��� is uniformly
+bounded by definition of Tstab in (13), we have
+���˜uce
+˜Tstab+k
+��� ≤
+��� ˆKi( ˜Tstab+k)
+���
+��˜x ˜Tstab+k
+�� ≲
+��˜x ˜Tstab+k
+�� ,
+(99)
+which implies (90).
+3) Upper bound on Tnocb: Now we are ready to prove
+Theorem 3, item 7):
+Proof. For any ϵ > 0, according to (98), we have i( ˜Tstab+k) ≲
+(1/δ)α+ϵ as long as k ≲ (1/δ)α−ϵ. Hence, we only need to
+prove
+ucb
+k ≡ uce
+k ,
+∀k ≥ i
+�
+˜Tstab + k0
+�
+,
+(100)
+for some k0 ≲ (1/δ)α−ϵ.
+Notice that (100) is equivalent to
+˜ucb
+˜Tstab+k ≡ ˜uce
+˜Tstab+k,
+∀k ≥ k0,
+(101)
+which is then equivalent to
+���˜uce
+˜Tstab+k
+��� ≤ Mk = log(k),
+∀k ≥ k0.
+(102)
+According to Proposition 18, we only need to verify
+ρk/2 log(1/δ) +
+�
+log(k/δ) ≲ log(k),
+(103)
+whenever k ≳ (1/δ)α−ϵ. In such case, we have
+ρk/2 log(1/δ) +
+�
+log(k/δ)
+≲ log(1/δ) +
+�
+log(k) + log(1/δ)
+≲ log(k) +
+�
+log(k) ≲ log(k),
+(104)
+from which the conclusion follows.
+H. Proof of Theorem 6
+In this subsection, we first decompose the regret of the
+proposed controller into multiple terms, then derive upper
+bounds on the terms respectively to obtain the high-probability
+regret bound stated in Theorem 6.
+
+1) Regret decomposition:
+We first state a supporting
+lemma:
+Lemma 19. Let K∗, P ∗ be defined in (6), (7) respectively,
+and K = K∗ + ∆K, then
+Q + K⊤RK + (A + BK)⊤ P ∗ (A + BK) − P ∗
+=∆K⊤(R + B⊤P ∗B)∆K.
+(105)
+Proof. Substituting the Lyapunov equation (9) into the LHS
+of (105), we have
+Q + K⊤RK + (A + BK)⊤P ∗(A + BK) − P ∗
+=K⊤RK − (K∗)⊤ RK∗ + (A + BK)⊤P ∗(A + BK)−
+(A + BK∗)⊤ P ∗ (A + BK∗)
+=∆K⊤ �
+R + B⊤P ∗B
+�
+∆K + G + G⊤,
+(106)
+where
+G = ∆K⊤ �
+(R + B⊤P ∗B)K∗ + B⊤P ∗A
+�
+.
+(107)
+By definition of K∗ in (6), we have G = 0, and hence the
+conclusion holds.
+Now we are ready to state the decomposition of regret:
+Proposition 20. The regret of the proposed controller defined
+in (11) can be decomposed as
+R(T) =
+7
+�
+i=1
+Ri(T),
+(108)
+with the terms Ri(T) defined as:
+R1(T) =
+T
+�
+k=1
+x⊤
+k (Kk − K∗)⊤(R + BT P ∗B)(Kk − K∗)xk,
+(109)
+R2(T) = 2
+T
+�
+k=1
+(upr
+k )⊤ B⊤P ∗(A + BKk)xk,
+(110)
+R3(T) = 2
+T
+�
+k=1
+w⊤
+k P ∗(A + BKk)xk,
+(111)
+R4(T) =
+T
+�
+k=1
+(s⊤
+k P ∗sk − w⊤
+k P ∗wk),
+(112)
+R5(T) =
+T
+�
+k=1
+w⊤
+k P ∗wk − TJ∗,
+(113)
+R6(T) = x⊤
+1 P ∗x1 − x⊤
+T +1P ∗xT +1,
+(114)
+R7(T) =
+T
+�
+k=1
+2 (upr
+k )⊤ Rucb
+k + (upr
+k )⊤ Rupr
+k ,
+(115)
+where Kk, sk are defined as:
+Kk =
+� ˆKk
+ucb
+k = uce
+k
+0
+otherwise , sk = Bupr
+k + wk.
+(116)
+Proof. From uk = ucb
+k + upr
+k = Kkxk + upr
+k , it holds
+R(T) =
+T
+�
+k=1
+�
+x⊤
+k Qxk + u⊤
+k Ruk
+�
+− TJ∗
+=
+T
+�
+k=1
+�
+x⊤
+k
+�
+Q + K⊤
+k RKk
+�
+xk + 2 (upr
+k )⊤ Rucb
+k +
+(upr
+k )⊤ Rupr
+k
+�
+− TJ∗
+=
+T
+�
+k=1
+�
+x⊤
+k
+�
+Q + K⊤
+k RKk
+�
+xk + x⊤
+k+1P ∗xk+1−
+x⊤
+k P ∗xk
+�
+− TJ∗ + R6(T) + R7(T).
+(117)
+We can further expand the summands in the RHS of the above
+inequality: from xk+1 = (A + BKk)xk + sk, we have
+x⊤
+k
+�
+Q + K⊤
+k RKk
+�
+xk + x⊤
+k+1P ∗xk+1 − x⊤
+k P ∗xk
+=x⊤
+k
+�
+Q + K⊤
+k RKk + (A + BKk)⊤P ∗(A + BKk) − P ∗�
+xk
++ 2s⊤
+k P ∗(A + BKk)xk + s⊤
+k P ∗sk
+=x⊤
+k (Kk − K∗)⊤(R + B⊤P ∗B)(Kk − K∗)xk+
+2s⊤
+k P ∗(A + BKk)xk + s⊤
+k P ∗sk,
+(118)
+where the last equality follows from Lemma 19. It follows
+from simple algebra that
+T
+�
+k=1
+�
+x⊤
+k
+�
+Q + K⊤
+k RKk
+�
+xk + x⊤
+k+1P ∗xk+1 − x⊤
+k P ∗xk
+�
+− TJ∗
+=
+5
+�
+i=1
+Ri(T),
+(119)
+and hence the conclusion follows.
+2) Upper bound on regret terms: Next we shall bound the
+terms Ri(T) (i = 1, . . . , 8) respectively:
+Proposition 21. The regret terms defined in (109)-(115) can
+be bounded as follows:
+1) On the event Enoise(δ) ∩ Eest(δ), for T > Tnocb, it holds
+R1(T) ≲ (1/δ)1/4(log(1/δ))4+
+√
+T(log(T/δ))3. (120)
+2) On the event Enoise(δ), it holds
+|R2(T)| ≲
+√
+T(log(T/δ))3/2.
+(121)
+3) On the event Ecross(δ), it holds
+|R3(T)| ≲
+√
+T(log(T/δ))2.
+(122)
+4) On the event Enoise(δ), it holds
+|R4(T)| ≤
+√
+T log(T/δ).
+(123)
+5) On the event Ecov(δ), it holds
+|R5(T)| ≲
+�
+T log(1/δ).
+(124)
+6) On the event Enoise(δ), it holds
+|R6(T)| ≲ (log(T/δ))2.
+(125)
+7) On the event Enoise(δ), it holds
+|R7(T)| ≲
+√
+T log(T/δ).
+(126)
+
+Proof.
+1) Let
+r1k = x⊤
+k (Kk − K∗)⊤ �
+R + B⊤P ∗B
+�
+(Kk − K∗)xk.
+(127)
+We shall next bound �Tnocb
+k=1 r1k and �T
+k=Tnocb+1 r1k re-
+spectively:
+a) For k ≤ Tnocb, we have ∥xk∥ ≲ log(k/δ) by The-
+orem 3, item 3), and ∥Kkxk∥ = ∥ucb
+k ∥ ≤ log(k).
+Therefore, it holds
+r1k ≲ (log(k/δ))2,
+(128)
+and hence,
+Tnocb
+�
+k=1
+r1k ≲ Tnocb(log(Tnocb/δ))2.
+(129)
+Invoking Theorem 3, item 7 with α = 1/4, we get
+Tnocb
+�
+k=1
+r1k ≲ (1/δ)1/4(log(1/δ))4.
+(130)
+b) For k ≤ Tnocb, by definition of Tnocb, we have Kk =
+ˆKk. Hence, by definition of Eest and the fact that ˆKk
+is a continuous function of ˆΘk [9, Proposition 6], we
+have
+∥Kk − K∗∥ =
+��� ˆKk − K∗���
+≲
+���ˆΘk − Θ
+��� ≲ k−1/4(log(k/δ))1/2.
+(131)
+Furthermore, by Theorem 3, item 3, we have ∥xk∥ ≲
+log(k/δ), and hence,
+r1k ≲ ∥Kk−K∗∥2∥xk∥2 ≲ k−1/2(log(k/δ))3. (132)
+Therefore,
+T
+�
+k=Tnocb+1
+r1k ≲
+√
+T(log(T/δ))3.
+(133)
+Summing up (130) and (133) leads to (120).
+2) Let
+r2k = (upr
+k )⊤ B⊤P ∗(A + BKk)xk,
+(134)
+whose factors can be bounded as follows:
+• ∥upr
+k ∥ = k1/2∥vk∥ ≲ k−1/2(log(k/δ))1/2 by defini-
+tion of Enoise;
+• ∥xk∥ ≲ log(k/δ) by Theorem 3, item 3);
+• ∥Kkxk∥ =
+��ucb
+k
+�� ≤ Mk = log(k) according to the
+proposed controller.
+Hence,
+|r2k| ≲ k−1/2(log(k/δ))3/2,
+(135)
+and summing up (135) from k = 1 to T leads to (121).
+3) The inequality (122) follows directly from the definition
+of Ecross(δ) in (19).
+4) Let
+r4k = s⊤
+k P ∗sk − w⊤
+k P ∗wk.
+(136)
+From sk = wk + Bupr
+k = wk + k−1/2Bvk, we have
+r4k = 2k−1/2w⊤
+k P ∗Bvk + k−1v⊤
+k B⊤P ∗Bvk.
+(137)
+Hence, by definition of Enoise(δ), we have
+|r4k| ≲ k−1/2 log(k/δ).
+(138)
+Summing up (138) from k = 1 to T leas to (123).
+5) Since J∗ = tr(P ∗) (see (8)), we have
+|R5(T)| =
+�����
+T
+�
+k=1
+tr(wkw⊤
+k P) − T tr(P)
+�����
+=
+�����tr
+�� T
+�
+k=1
+(wkw⊤
+k − In)
+�
+P
+������
+≲
+�����
+T
+�
+k=1
+(wkw⊤
+k − In)
+����� ≲
+�
+T log(1/δ), (139)
+where the last inequality follows from the definition of
+Ecov(δ), which proves (124).
+6) The inequality (125) is a direct corollary of Theorem 3,
+item 3).
+7) Let
+r7k = 2 (upr
+k )⊤ Rucb
+k + (upr
+k )⊤ Rupr
+k ,
+(140)
+where upr
+k and ucb
+k satisfy:
+• ∥upr
+k ∥ = k−1/2∥vk∥ ≲ k−1/2(log(k/δ))1/2 by defini-
+tion of Enoise(δ);
+•
+��ucb
+k
+�� ≤ Mk = log(k) according to the proposed
+controller.
+Hence,
+|r7k| ≲ k−1/2 log(k/δ),
+(141)
+and summing up (141) from k = 1 to T leads to (126).
+Theorem 6 follows from combining Propositions 20 and 21.
+V. SIMULATION
+In this section, the proposed controller is validated on
+the Tennessee Eastman Process (TEP) [21]. In particular, we
+consider a simplified version of TEP similar to the one in [22],
+with full state feedback. The system is open-loop stable, and
+has state dimension n = 8 and input dimension m = 4. The
+process noise distribution is chosen to be wk
+i.i.d.
+∼
+N(0, In).
+The weight matrices of LQR are chosen to be Q = In and
+R = Im. The plant under the proposed controller is simulated
+for 10000 independent trials, each with T = 3 × 108 steps.
+As mentioned in Remark 2, the certainty equivalent gain ˆKk
+is updated only at steps k = 2i, i ∈ N∗ for the sake of fast
+computation.
+The evolution of regret against time is plotted in Fig. 2.
+For the ease of observation, we plot the relative average
+regret R(T)/(TJ∗) against the total time step T, where J∗
+is the optimal cost. Fig. 2 shows 5 among the 10000 trials,
+from which one can observe a 1/
+√
+T convergence rate of the
+relative average regret (i.e., a 1 order-of-magnitude increase
+in T corresponds to a 0.5 order-of-magnitude decrease in
+R(T)/(TJ∗)), which matches the
+√
+T theoretical growth rate
+of regret. To inspect the statistical properties of all the trials,
+we sort them by the average regret at the last step, and plot
+
+the worst, median and mean cases in Fig. 2b. One can observe
+that the average regret converge to zero even in the worst case,
+which validates the almost-sure guarantee in Theorem 8.
+102
+103
+104
+105
+106
+107
+108 3 × 108
+10−2.5
+10−2.0
+10−1.5
+10−1.0
+10−0.5
+100.0
+100.5
+Total time step T
+R(T)/(TJ∗)
+(a) Five random sample paths
+102
+103
+104
+105
+106
+107
+108 3 × 108
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+Total time step T
+R(T)/(TJ∗)
+Worst
+Median
+Best
+(b) Worst, median and best cases among all sample paths
+Fig. 2: Double-log plot of average regret against time step
+An insight to behold on the performance of the proposed
+controller is that the circuit-breaking mechanism is triggered
+only finitely, and the time of the last trigger Tnocb, as stated
+in Corollary 5, has a super-polynomial tail. This insight is
+also empirically validated: among all the 10000 trials, circuit-
+breaking is never triggered after step 1.4×106, and a histogram
+of Tnocb is shown in Fig. 3, from which one can observe that
+the empirical distribution of Tnocb has a fast decaying tail.
+2
+4
+6
+8
+10
+12
+14
+0
+200
+400
+600
+800
+1000
+1200
+Tnocb(×105)
+Frequency
+Fig. 3: Histogram of Tnocb among all sample paths
+VI. CONCLUSION
+In this paper, we propose an adaptive LQR controller that
+can achieve ˜O(
+√
+T) regret almost surely. A key underlying
+the controller design is a circuit-breaking mechanism, which
+ensures the convergence of the parameter estimate, but is
+triggered only finitely often and hence has negligible effect
+on the asymptotic performance. A future direction would be
+extending such circuit-breaking mechanism to the partially
+observed LQG setting.
+REFERENCES
+[1] K. J. Astrom, “Adaptive control around 1960,” IEEE Control Systems
+Magazine, vol. 16, no. 3, pp. 44–49, 1996.
+[2] K. J. ˚Astr¨om and B. Wittenmark, “On self tuning regulators,” Automat-
+ica, vol. 9, no. 2, pp. 185–199, 1973.
+[3] A. Morse, “Global stability of parameter-adaptive control systems,”
+IEEE Transactions on Automatic Control, vol. 25, no. 3, pp. 433–439,
+1980.
+[4] T. Lai and C.-Z. Wei, “Extended least squares and their applications to
+adaptive control and prediction in linear systems,” IEEE Transactions
+on Automatic Control, vol. 31, no. 10, pp. 898–906, 1986.
+[5] Y. Abbasi-Yadkori, D. P´al, and C. Szepesv´ari, “Online least squares
+estimation with self-normalized processes: An application to bandit
+problems,” arXiv preprint arXiv:1102.2670, 2011.
+[6] M. Abeille and A. Lazaric, “Improved regret bounds for thompson sam-
+pling in linear quadratic control problems,” in International Conference
+on Machine Learning.
+PMLR, 2018, pp. 1–9.
+[7] A. Cohen, T. Koren, and Y. Mansour, “Learning linear-quadratic regu-
+lators efficiently with only
+√
+t regret,” in International Conference on
+Machine Learning.
+PMLR, 2019, pp. 1300–1309.
+[8] S. Dean, H. Mania, N. Matni, B. Recht, and S. Tu, “Regret bounds for
+robust adaptive control of the linear quadratic regulator,” Advances in
+Neural Information Processing Systems, vol. 31, 2018.
+[9] M. Simchowitz and D. Foster, “Naive exploration is optimal for online
+lqr,” in International Conference on Machine Learning.
+PMLR, 2020,
+pp. 8937–8948.
+[10] M. K. S. Faradonbeh, A. Tewari, and G. Michailidis, “On adaptive
+linear–quadratic regulators,” Automatica, vol. 117, p. 108982, 2020.
+[11] F. Wang and L. Janson, “Exact asymptotics for linear quadratic adaptive
+control.” J. Mach. Learn. Res., vol. 22, pp. 265–1, 2021.
+[12] M. K. S. Faradonbeh, A. Tewari, and G. Michailidis, “Finite-time adap-
+tive stabilization of linear systems,” IEEE Transactions on Automatic
+Control, vol. 64, no. 8, pp. 3498–3505, 2018.
+[13] L. Guo, “Self-convergence of weighted least-squares with applications
+to stochastic adaptive control,” IEEE transactions on automatic control,
+vol. 41, no. 1, pp. 79–89, 1996.
+[14] H. Lin and P. J. Antsaklis, “Stability and stabilizability of switched linear
+systems: a survey of recent results,” IEEE Transactions on Automatic
+control, vol. 54, no. 2, pp. 308–322, 2009.
+[15] L. Guo and H. Chen, “Convergence rate of an els-based adaptive
+tracker,” System Sciece and Mathematical Sciences, vol. 1, no. 2, p.
+131, 1988.
+[16] H. Mania, S. Tu, and B. Recht, “Certainty equivalence is efficient for
+linear quadratic control,” Advances in Neural Information Processing
+Systems, vol. 32, 2019.
+[17] Y. Lu and Y. Mo, “Ensuring the safety of uncertified linear state-
+feedback controllers via switching,” arXiv preprint arXiv:2205.08817,
+2022.
+[18] B. Laurent and P. Massart, “Adaptive estimation of a quadratic functional
+by model selection,” Annals of Statistics, pp. 1302–1338, 2000.
+[19] K. Azuma, “Weighted sums of certain dependent random variables,”
+Tohoku Mathematical Journal, Second Series, vol. 19, no. 3, pp. 357–
+367, 1967.
+[20] M. Simchowitz, H. Mania, S. Tu, M. I. Jordan, and B. Recht, “Learning
+without mixing: Towards a sharp analysis of linear system identifica-
+tion,” in Conference On Learning Theory.
+PMLR, 2018, pp. 439–473.
+[21] J. J. Downs and E. F. Vogel, “A plant-wide industrial process control
+problem,” Computers & chemical engineering, vol. 17, no. 3, pp. 245–
+255, 1993.
+[22] H. Liu, Y. Mo, J. Yan, L. Xie, and K. H. Johansson, “An online approach
+to physical watermark design,” IEEE Transactions on Automatic Control,
+vol. 65, no. 9, pp. 3895–3902, 2020.
+

D9E0T4oBgHgl3EQfQgCE/content/tmp_files/2301.02194v1.pdf.txt

@@ -0,0 +1,1419 @@
+arXiv:2301.02194v1  [cs.PL]  5 Jan 2023
+Builtin Types viewed as Inductive Families
+Guillaume Allais
+January 6, 2023
+Abstract
+State of the art optimisation passes for dependently typed languages
+can help erase the redundant information typical of invariant-rich data
+structures and programs. These automated processes do not dramatically
+change the structure of the data, even though more efficient representa-
+tions could be available.
+Using Quantitative Type Theory, we demonstrate how to define an
+invariant-rich, typechecking time data structure packing an efficient run-
+time representation together with runtime irrelevant invariants. The com-
+piler can then aggressively erase all such invariants during compilation.
+Unlike other approaches, the complexity of the resulting representation
+is entirely predictable, we do not require both representations to have the
+same structure, and yet we are able to seamlessly program as if we were
+using the high-level structure.
+1
+Introduction
+Dependently typed languages have empowered users to precisely describe their
+domain of discourse by using inductive families [Dyb94]. Programmers can bake
+crucial invariants directly into their definitions thus refining both their func-
+tions’ inputs and outputs. The constrained inputs allow them to only consider
+the relevant cases during pattern matching, while the refined outputs guaran-
+tee that client code can safely rely on the invariants being maintained. This
+programming style is dubbed ‘correct by construction’.
+However, relying on inductive families can have a non-negligible runtime
+cost if the host language is compiling them na¨ıvely. And even state of the art
+optimisation passes for dependently typed languages cannot make miracles: if
+the source code is not efficient, the executable will not be either.
+A state of the art compiler will for instance successfully compile length-
+indexed lists to mere lists thus reducing the space complexity from quadratic
+to linear in the size of the list. But, confronted with a list of booleans whose
+length is statically known to be less than 64, it will fail to pack it into a single
+machine word thus spending linear space when constant would have sufficed.
+In section 2, we will look at an optimisation example that highlights both
+the strengths and the limitations of the current state of the art when it comes to
+removing the runtime overheads potentially incurred by using inductive families.
+1
+
+In section 3 we will give a quick introduction to Quantitative Type Theory,
+the expressive language that grants programmers the ability to have both strong
+invariants and, reliably, a very efficient runtime representation.
+In section 4 we will look at an inductive family that we use in a performance-
+critical way in the TypOS project [AAM+22] and whose compilation suffers
+from the limitations highlighted in section 2. Our current and unsatisfactory
+approach is to rely on the safe and convenient inductive family when experi-
+menting in Agda and then replace it with an unsafe but vastly more efficient
+representation in our actual Haskell implementation.
+Finally in section 5, we will study the actual implementation of our efficient
+and invariant-rich solution implemented in Idris 2. We will also demonstrate
+that we can recover almost all the conveniences of programming with inductive
+families thanks to smart constructors and views.
+2
+An Optimisation Example
+The prototypical examples of the na¨ıve compilation of inductive families being
+inefficient are probably the types of vectors (Vect) and finite numbers (Fin).
+Their interplay is demonstrated by the lookup function.
+Let us study this
+example and how successive optimisation passes can, in this instance, get rid of
+the overhead introduced by using indexed families over plain data.
+A vector is a length-indexed list. The type Vect is parameterised by the
+type of values it stores and indexed over a natural number corresponding to its
+length. More concretely, its Nil constructor builds an empty vector of size Z
+(i.e. zero), and its (::) (pronounced ‘cons’) constructor combines a value of
+type a (the head) and a subvector of size n (the tail) to build a vector of size (S
+n) (i.e. successor of n).
+data Vect : Nat -> Type -> Type where
+Nil : Vect Z a
+(::) : a -> Vect n a -> Vect (S n) a
+The size n is not explicitly bound in the type of (::). In Idris 2, this means
+that it is automatically generalised over in a prenex manner reminiscent of the
+handling of free type variables in languages in the ML family. This makes it
+an implicit argument of the constructor.
+Consequently, given that Nat is a
+type of unary natural numbers, a na¨ıve runtime representation of a (Vect n a)
+would have a size quadratic in n. A smarter representation with perfect sharing
+would still represent quite an overhead as observed by Brady, McBride, and
+McKinna [BMM03].
+A finite number is a number known to be strictly smaller than a given natural
+number. The type Fin is indexed by said bound. Its Z constructor models 0 and
+is bound by any non-zero bound, and its S constructor takes a number bound
+by n and returns its successor, bound by (1 + n). A na¨ıve compilation would
+here also lead to a runtime representation suffering from a quadratic blowup.
+2
+
+data Fin : Nat -> Type where
+Z : Fin (S n)
+S : Fin n -> Fin (S n)
+This leads us to the definition of the lookup function. Provided a vector of
+size n and a finite number k bound by this same n, we can define a total function
+looking up the value stored at position k in the vector. It is guaranteed to return
+a value. Note that we do not need to consider the case of the empty vector in the
+pattern matching clauses as all of the return types of the Fin constructors force
+the index to be non-zero and, because the vector and the finite number talk
+about the same n, having an empty vector would automatically imply having a
+value of type (Fin 0) which is self-evidently impossible.
+lookup : Vect n a -> Fin n -> a
+lookup (x :: _) Z = x
+lookup (_ :: xs) (S k) = lookup xs k
+Thanks to our indexed family, we have gained the ability to define a function
+that cannot possibly fail, as well as the ability to only talk about the pattern
+matching clauses that make sense. This seemed to be at the cost of efficiency but
+luckily for us there has already been extensive work on erasure to automatically
+detect redundant data [BMM03] or data that will not be used at runtime [Tej20].
+2.1
+Optimising Vect, Fin, and lookup
+An analysis in the style of Brady, McBride, and McKinna’s [BMM03] can solve
+the quadratic blowup highlighted above by observing that the natural number
+a vector is indexed by is entirely determined by the spine of the vector. In
+particular, the length of the tail does not need to be stored as part of the
+constructor: it can be reconstructed as the predecessor of the length of the
+overall vector. As a consequence, a vector can be adequately represented at
+runtime by a pair of a natural number and a list. Similarly a bounded number
+can be adequately represented by a pair of natural numbers. Putting all of this
+together and remembering that the vector and the finite number share the same
+n, lookup can be compiled to a function taking two natural numbers and a list.
+In Idris 2 we would write the optimised lookup as follows (we use the partial
+keyword because this transformed version is not total at that type).
+partial
+lookup : (n : Nat) -> List a -> Nat -> a
+lookup (S n) (x :: _) Z = x
+lookup (S n) (_ :: xs) (S k) = lookup n xs k
+We can see in the second clause that the recursive call is performed on the tail
+of the list (formerly vector) and so the first argument to lookup corresponding
+to the vector’s size is decreased by one. The invariant, despite not being explicit
+anymore, is maintained.
+3
+
+A Tejiˇsˇc´ak-style analysis [Tej20] can additionally notice that the lookup func-
+tion never makes use of the bound’s value and drop it entirely. This leads to
+the lookup function on vectors being compiled to its partial-looking counterpart
+acting on lists.
+partial
+lookup : List a -> Nat -> a
+lookup (x :: _) Z = x
+lookup (_ :: xs) (S k) = lookup xs k
+Even though this is in our opinion a pretty compelling example of erasing
+away the apparent complexity introduced by inductive families, this approach
+has two drawbacks.
+Firstly, it relies on the fact that the compiler can and will automatically
+perform these optimisations. But nothing in the type system prevents users from
+inadvertently using a value they thought would get erased, thus preventing the
+Tejiˇsˇc´ak-style optimisation from firing. In performance-critical settings, users
+may rather want to state their intent explicitly and be kept to their word by
+the compiler in exchange for predictable and guaranteed optimisations.
+Secondly, this approach is intrinsically limited to transformations that pre-
+serve the type’s overall structure: the runtime data structures are simpler but
+very similar still. We cannot expect much better than that. It is so far unre-
+alistic to expect e.g. a change of representation to use a balanced binary tree
+instead of a list in order to get logarithmic lookups rather than linear ones.
+2.2
+No Magic Solution
+Even if we are able to obtain a more compact representation of the inductive
+family at runtime through enough erasure, this does not guarantee runtime
+efficiency. As the Coq manual [CDT22] reminds its users, extraction does not
+magically optimise away a user-defined quadratic multiplication algorithm when
+extracting unary natural numbers to an efficient machine representation. In
+a pragmatic move, Coq, Agda, and Idris 2 all have ad-hoc rules to replace
+convenient but inefficiently implemented numeric functions with asymptotically
+faster counterparts in the target language.
+However this approach is not scalable: if we may be willing to extend our
+trusted core to a high quality library for unbounded integers, we do not want to
+replace our code only proven correct thanks to complex invariants with a wildly
+different untrusted counterpart purely for efficiency reasons.
+In this paper we use Quantitative Type Theory [McB16, Atk18] as imple-
+mented in Idris 2 [Bra21] to bridge the gap between an invariant-rich but in-
+efficient representation based on an inductive family and an unsafe but effi-
+cient implementation using low-level primitives. Inductive families allow us to
+view [Wad87, MM04] the runtime relevant information encoded in the low-level
+and efficient representation as an information-rich compile time data structure.
+Moreover the quantity annotations guarantee that this additional information
+4
+
+will be erased away during compilation.
+3
+Some Key Features of Idris 2
+Idris 2 implements Quantitative Type Theory, a Martin-L¨of type theory enriched
+with a semiring of quantities classifying the ways in which values may be used.
+In a type, each binder is annotated with the quantity by which its argument
+must abide.
+3.1
+Quantities
+A value may be runtime irrelevant, linear, or unrestricted.
+Runtime irrelevant values (0 quantity) cannot possibly influence control flow
+as they will be erased entirely during compilation. This forces the language
+to impose strong restrictions on pattern-matching over these values. Typical
+examples are types like the a parameter in (List a), or indices like the natural
+number n in (Vect n a). These are guaranteed to be erased at compile time.
+The advantage over a Tejiˇsˇc´ak-style analysis is that users can state their intent
+that an argument ought to be runtime irrelevant and the language will insist
+that it needs to be convinced it indeed is.
+Linear values (1 quantity) have to be used exactly once. Typical examples
+include the %World token used by Idris 2 to implement the IO monad `a la Haskell,
+or file handles that cannot be discarded without first explicitly closing the file.
+At runtime these values can be updated destructively. We will not use linearity
+in this paper.
+Last, unrestricted values (denoted by no quantity annotation) can flow into
+any position, be duplicated or thrown away. They are the usual immutable
+values of functional programming.
+The most basic of examples mobilising both the runtime irrelevance and
+unrestricted quantities is the identity function.
+id : {0 a : Type} -> (x : a) -> a
+id x = x
+Its type starts with a binder using curly braces. This means it introduces
+an implicit variable that does not need to be filled in by the user at call sites
+and will be reconstructed by unification. The variable it introduces is named
+a and has type Type. It has the 0 quantity annotation which means that this
+argument is runtime irrelevant and so will be erased during compilation.
+The second binder uses parentheses. It introduces an explicit variable whose
+name is x and whose type is the type a that was just bound. It has no quantity
+annotation which means it will be an unrestricted variable.
+Finally the return type is the type a bound earlier. This is, as expected, a
+polymorphic function from a to a. It is implemented using a single clause that
+binds x on the left-hand side and immediately returns it on the right-hand side.
+5
+
+If we were to try to annotate the binder for x with a 0 quantity to make
+it runtime irrelevant then Idris 2 would rightfully reject the definition.
+The
+following failing block shows part of the error message complaining that x
+cannot be used at an unrestricted quantity on the right-hand side.
+failing "x is not accessible in this context."
+id : {0 a : Type} -> (0 x : a) -> a
+id x = x
+3.2
+Proof Search
+In Idris 2, Haskell-style ad-hoc polymorphism [WB89] is superseded by a more
+general proof search mechanism.
+Instead of having blessed notions of type
+classes, instances and constraints, the domain of any dependent function type
+can be marked as auto. This signals to the compiler that the corresponding
+argument will be an implicit argument and that it should not be reconstructed
+by unification alone but rather by proof search. The search algorithm will use
+the appropriate user-declared hints as well as the local variables in scope.
+By default, a datatype’s constructors are always added to the database of
+hints. And so the following declaration brings into scope both an indexed family
+So of proofs that a given boolean is True, and a unique constructor Oh that is
+automatically added as a hint.
+data So : Bool -> Type where
+Oh : So True
+As a consequence, we can for instance define a record type specifying what
+it means for n to be an even number by storing its half together with a proof
+that is both runtime irrelevant and filled in by proof search. Because (2 * 3 ==
+6) computes to True, Idris 2 is able to fill-in the missing proof in the definition
+of even6 using the Oh hint.
+record Even (n : Nat) where
+constructor MkEven
+half : Nat
+{auto 0 prf : So (2 * half == n)}
+even6 : Even 6
+even6 = MkEven { half = 3 }
+We will use both So and the auto mechanism in section 5.3.
+3.3
+Application: Vect, as List
+We can use the features of Quantitative Type Theory to give an implementa-
+tion of Vect that is guaranteed to erase to a List at runtime independently of
+the optimisation passes implemented by the compiler. The advantage over the
+optimisation passes described in section 2 is that the user has control over the
+6
+
+runtime representation and does not need to rely on these optimisations being
+deployed by the compiler.
+The core idea is to make the slogan ‘a vector is a length-indexed list’ a
+reality by defining a record packing together the encoding as a list and a proof
+its length is equal to the expected Nat index. This proof is marked as runtime
+irrelevant to ensure that the list is the only thing remaining after compilation.
+record Vect (n : Nat) (a : Type) where
+constructor MkVect
+encoding : List a
+0 valid : length encoding === n
+Smart constructors
+Now that we have defined vectors, we can recover the
+usual building blocks for vectors by defining smart constructors, that is to say
+functions Nil and (::) that act as replacements for the inductive family’s data
+constructors.
+Nil : Vect Z a
+Nil = MkVect [] Refl
+The smart constructor Nil returns an empty vector. It is, unsurprisingly,
+encoded as the empty list ([]). Because (length []) statically computes to Z,
+the proof that the encoding is valid can be discharged by reflexivity.
+(::) : a -> Vect n a -> Vect (S n) a
+x :: MkVect xs eq = MkVect (x :: xs) (cong S eq)
+Using (::) we can combine a head and a tail of size n to obtain a vector of
+size (S n). The encoding is obtained by consing the head in front of the tail’s
+encoding and the proof this is valid (cong S eq) uses the fact that propositional
+equality is a congruence and that (length (x :: xs)) computes to (S (length
+xs)).
+View
+Now that we know how to build vectors, we demonstrate that we can
+also take them apart using a view.
+A view for a type T , in the sense of Wadler [Wad87], and as refined by
+McBride and McKinna [MM04], is an inductive family V indexed by T together
+with a total function mapping every element t of T to a value of type (V t). This
+simple gadget provides a powerful, user-extensible, generalisation of pattern-
+matching. Patterns are defined inductively as either a pattern variable, a forced
+term (i.e. an arbitrary expression that is determined by a constraint arising
+from another pattern), or a data constructor fully applied to subpatterns. In
+contrast, the return indices of an inductive family’s constructors can be arbitrary
+expressions.
+In the case that interests us, the view allows us to emulate ‘matching’ on
+which of the two smart constructors Nil or (::) was used to build the vector
+being taken apart.
+7
+
+data View : Vect n a -> Type where
+Nil : View Nil
+(::) : (x : a) -> (xs : Vect n a) -> View (x :: xs)
+The inductive family View is indexed by a vector and has two constructors
+corresponding to the two smart constructors. We use Idris 2’s overloading capa-
+bilities to give each of the View’s constructors the name of the smart constructor
+it corresponds to. By pattern-matching on a value of type (View xs), we will be
+able to break xs into its constitutive parts and either observe it is equal to Nil
+or recover its head and its tail.
+view : (xs : Vect n a) -> View xs
+view (MkVect [] Refl) = Nil
+view (MkVect (x :: xs) Refl) = x :: MkVect xs Refl
+The function view demonstrates that we can always tell which constructor
+was used by inspecting the encoding list. If it is empty, the vector was built
+using the Nil smart constructor. If it is not then we got our hands on the
+head and the tail of the encoding and (modulo some re-wrapping of the tail)
+they are effectively the head and the tail that were combined using the smart
+constructor.
+3.3.1
+Application: map
+We can then use these constructs to implement the function map on vectors
+without ever having to explicitly manipulate the encoding.
+The maximally
+sugared version of map is as follows:
+map : (a -> b) -> Vect n a -> Vect n b
+map f xs@_ with (view xs)
+_ | [] = []
+_ | hd :: tl = f hd :: map f tl
+On the left-hand side the view lets us seamlessly pattern-match on the input
+vector. Using the with keyword we have locally modified the function defini-
+tion so that it takes an extra argument, here the result of the intermediate
+computation (view xs). Correspondingly, we have two clauses matching on this
+extra argument; the symbol | separates the original left-hand side (here elided
+using _ because it is exactly the same as in the parent clause) from the addi-
+tional pattern. This pattern can either have the shape [] or (hd :: tl) and,
+correspondingly, we learn that xs is either [] or (hd :: tl).
+On the right-hand side the smart constructors let us build the output vec-
+tor. Mapping a function over the empty vector yields the empty vector while
+mapping over a cons node yields a cons node whose head and tail have been
+appropriately modified.
+This sugared version of map is equivalent to the following more explicit one:
+8
+
+map : (a -> b) -> Vect n a -> Vect n b
+map f xs with (view xs)
+map f .([]) | [] = []
+map f .(hd :: tl) | hd :: tl = f hd :: map f tl
+In the parent clause we have explicitly bound xs instead of merely introduc-
+ing an alias for it by writing (xs@ ) and so we will need to be explicit about the
+ways in which this pattern is refined in the two with-clauses.
+In the with-clauses, we have explicitly repeated the refined version of the
+parent clause’s left-hand side. In particular we have used dotted patterns to
+insist that xs is now entirely forced by the match on the result of (view xs).
+We have seen that by matching on the result of the (view xs) call, we get to
+‘match’ on xs as if Vect were an inductive type. This is the power of views.
+3.3.2
+Application: lookup
+The type (Fin n) can similarly be represented by a single natural number and
+a runtime irrelevant proof that it is bound by n. We leave these definitions
+out, and invite the curious reader to either attempt to implement them for
+themselves or look at the accompanying code.
+Bringing these definitions together, we can define a lookup function which
+is similar to the one defined in section 2.
+lookup : Vect n a -> Fin n -> a
+lookup xs@_ k@_ with (view xs) | (view k)
+_ | hd :: _ | Z = hd
+_ | _ :: tl | S k’ = lookup tl k’
+We are seemingly using view at two different types (Vect and Fin respec-
+tively) but both occurrences actually refer to separate functions: Idris 2 lets us
+overload functions and performs type-directed disambiguation.
+For pedagogical purposes, this sugared version of lookup can also be ex-
+panded to a more explicit one that demonstrates the views’ power.
+lookup : Vect n a -> Fin n -> a
+lookup xs k with (view xs) | (view k)
+lookup .(hd :: tl) .(Z) | hd :: tl | Z = hd
+lookup .(hd :: tl) .(S k’) | hd :: tl | S k’ = lookup tl k’
+The main advantage of this definition is that, based on its type alone, we
+know that this function is guaranteed to be processing a list and a single natural
+number at runtime. This efficient runtime representation does not rely on the
+assumption that state of the art optimisation passes will be deployed.
+We have seen some of Idris 2’s powerful features and how they can be lever-
+aged to empower users to control the runtime representation of the inductive
+families they manipulate. This simple example only allowed us to reproduce
+the performance that could already be achieved by compilers deploying state of
+the art optimisation passes. In the following sections, we are going to see how
+9
+
+we can use the same core ideas to compile an inductive family to a drastically
+different runtime representation while keeping good high-level ergonomics.
+4
+Thinnings, cooked two ways
+We experienced a major limitation of compilation of inductive families during
+our ongoing development of TypOS [AAM+22], a domain specific language to
+define concurrent typecheckers and elaborators.
+Core to this project is the defi-
+nition of actors manipulating a generic notion of syntax with binding. Internally
+the terms of this syntax with binding are based on a co-de Bruijn representa-
+tion (an encoding we will explain below) which relies heavily on thinnings. A
+thinning (also known as an Order Preserving Embedding [Cha09]) between a
+source and a target scope is an order preserving injection of the smaller scope
+into the larger one. They are usually represented using an inductive family.
+The omnipresence of thinnings in the co-de Bruijn representation makes their
+runtime representation a performance critical matter.
+Let us first remind the reader of the structure of abstract syntax trees in a
+named, a de Bruijn, and a co-de Bruijn representation. We will then discuss two
+representations of thinnings: a safe and convenient one as an inductive family,
+and an unsafe but efficient encoding as a pair of arbitrary precision integers.
+4.1
+Named, de Bruijn, and co-de Bruijn syntaxes
+In this section we will use the S combinator (λg.λf.λx.gx(fx)) as a running
+example and represent terms using a syntax tree whose constructor nodes are
+circles and variable nodes are squares.
+To depict the S combinator we will
+only need λ-abstraction and application (rendered $) nodes. A constructor’s
+arguments become its children in the tree. The tree is laid out left-to-right and
+a constructor’s arguments are displayed top-to-bottom.
+Named syntax
+The first representation is using explicit names. Each binder
+has an associated name and each variable node carries a name. A variable refers
+to the closest enclosing binder which happens to be using the same name.
+λg.
+λf.
+λx.
+$
+$
+g
+x
+$
+f
+x
+To check whether two terms are structurally equivalent (α-equivalence) po-
+tentially requires renaming bound names. In order to have a simple and cheap
+α-equivalence check we can instead opt for a nameless representation.
+10
+
+De Bruijn syntax
+An abstract syntax tree based on de Bruijn indices [dB72]
+replaces names with natural numbers counting the number of binders separating
+a variable from its binding site. The S combinator is now written (λ λ λ 2 0 (1 0)).
+You can see in the following graphical depiction that λ-abstractions do not
+carry a name anymore and that variables are simply pointing to the binder
+that introduced them.
+We have left the squares empty but in practice the
+various coloured arrows would be represented by a natural number. For instance
+the dashed magenta one corresponds to 1 because you need to ignore one λ-
+abstraction (the orange one) on your way towards the root of the tree before
+you reach the corresponding magenta binder.
+λ.
+λ.
+λ.
+$
+$
+$
+To check whether a subterm does not mention a given set of variables (a
+thickening test, the opposite of a thinning which extends the current scope with
+unused variables), you need to traverse the whole term.
+In order to have a
+simple cheap thickening test we can ensure that each subterms knows precisely
+what its support is and how it embeds in its parent’s.
+Co-de Bruijn syntax
+In a co-de Bruijn representation [McB18] each subterm
+selects exactly the variables that stay in scope for that term, and so a variable
+constructor ultimately refers to the only variable still in scope by the time it is
+reached. This representation ensures that we know precisely what the scope of
+a given term currently is.
+In the following graphical rendering, we represent thinnings as lists of full
+(•) or empty (◦) discs depending on whether the corresponding variable is either
+kept or discarded. For instance the thinning represented by ◦•• throws the blue
+variable away, and keeps both the magenta and orange ones.
+λ.
+λ.
+λ.
+$
+$
+$
+•
+••
+•••
+◦••
+•◦•
+•◦
+◦•
+•◦
+◦•
+11
+
+We can see that in such a representation, each node in the tree stores one
+thinning per subterm. This will not be tractable unless we have an efficient
+representation of thinnings.
+4.2
+The Performance Challenges of co-de Bruijn
+Using the co-de Bruijn approach, a term in an arbitrary context is repre-
+sented by the pairing of a term in co-de Bruijn syntax with a thinning from
+its support into the wider scope. Having such a precise handle on each term’s
+support allows us to make operations such as thinning, substitution, unification,
+or common sub-expression elimination more efficient.
+Thinning a term does not require us to traverse it anymore. Indeed, embed-
+ding a term in a wider context will not change its support and so we can simply
+compose the two thinnings while keeping the term the same.
+Substitution can avoid traversing subterms that will not be changed. Indeed,
+it can now easily detect when the substitution’s domain does not intersect with
+the subterm’s support.
+Unification requires performing thickening tests when we want to solve a
+metavariable declared in a given context with a terms seemingly living in a
+wider one. We once more do not need to traverse the term to perform this test,
+and can simply check whether the outer thinning can be thickened.
+Common sub-expression elimination requires us to identify alpha-equivalent
+terms potentially living in different contexts. Using a de Bruijn representation,
+these can be syntactically different: a variable represented by the natural num-
+ber v in Γ would be (1+v) in Γ, σ but (2+v) in Γ, τ, ν. A co-de Bruijn represen-
+tation, by discarding all the variables not in the support, guarantees that we can
+once more use syntactic equality to detect alpha-equivalence. This encoding is
+used for instance (albeit unknowingly) by Maziarz, Ellis, Lawrence, Fitzgibbon,
+and Peyton-Jones in their ‘Hashing modulo alpha-equivalence’ work [MEL+21].
+For all of these reasons we have, as we mentioned earlier, opted for a co-de
+Bruijn representation in the implementation of TypOS [AAM+22].
+And so it
+is crucial for performance that we have a compact representation of thinnings.
+4.2.1
+Thinnings in TypOS
+We first carefully worked out the trickier parts of the implementation in Agda
+before porting the resulting code to Haskell. This process highlighted a glaring
+gap between on the one hand the experiments done using a strongly typed
+inductive representation of thinnings and on the other hand their more efficient
+but unsafe encoding in Haskell.
+Agda
+The Agda-based experiments use inductive families that make the key
+invariants explicit which helps tracking complex constraints and catches design
+flaws at typechecking time. The indices guarantee that we always transform the
+12
+
+thinnings appropriately when we add or remove bound variables. In Idris 2, the
+inductive family representation of thinnings would be written:
+data Thinning : (sx, sy : SnocList a) -> Type where
+Done : Thinning [<] [<]
+Keep : Thinning sx sy -> (0 x : a) -> Thinning (sx :< x) (sy :< x)
+Drop : Thinning sx sy -> (0 x : a) -> Thinning sx (sy :< x)
+The Thinning family is indexed by two scopes (represented as snoclists i.e. lists
+that are extended from the right, just like contexts in inference rules): sx the
+tighter scope and sy the wider one.
+The Done constructor corresponds to a
+thinning from the empty scope to itself ([<] is Idris 2 syntactic sugar for the
+empty snoclist), and Keep and Drop respectively extend a given thinning by
+keeping or dropping the most local variable (:< is the ‘snoc’ constructor, a sort
+of flipped ‘cons’). The ‘name’ (x of type a) is marked with the quantity 0 to
+ensure it is erased at compile time (cf. section 3).
+During compilation, Idris 2 would erase the families’ indices as they are
+forced (in the sense of Brady, McBride, and McKinna [BMM03]), and drop the
+constructor arguments marked as runtime irrelevant. The resulting inductive
+type would be the following simple data type.
+data Thinning = Done | Keep Thinning | Drop Thinning
+At runtime this representation is therefore essentially a linked list of booleans
+(Done being Nil, and Keep and Drop respectively (True ::) and (False ::)).
+Haskell
+The Haskell implementation uses this observation and picks a packed
+encoding of this list of booleans as a pair of integers. One integer represents the
+length n of the list, and the other integer’s n least significant bits encode the
+list as a bit pattern where 1 is Keep and 0 is Drop.
+Basic operations on thinnings are implemented by explicitly manipulating
+individual bits. It is not indexed and thus all the invariant tracking has to be
+done by hand. This has led to numerous and hard to diagnose bugs.
+4.2.2
+Thinnings in Idris 2
+Idris 2 is a self-hosting language whose core datatype is currently based on a
+well-scoped de Bruijn representation. This precise indexing of terms by their
+scope helped entirely eliminate a whole class of bugs that plagued Idris 1’s
+unification machinery.
+If we were to switch to a co-de Bruijn representation for our core language
+we would want, and should be able, to have the best of both worlds: a safe and
+efficient representation!
+Thankfully Idris 2 implements Quantitative Type Theory (QTT) which gives
+us a lot of control over what is to be runtime relevant and what is to be erased
+during compilation. This should allow us to insist on having a high-level in-
+terface that resembles an inductive family while ensuring that everything but
+13
+
+a pair of integers is erased at compile time. We will exploit the key features of
+QTT presented in section 3 to have our cake and eat it.
+5
+An Efficient Invariant-Rich Representation
+We can combine both approaches highlighted in section 4.2 by defining a record
+parameterised by a source (sx) and target (sy) scopes corresponding to the two
+ends of the thinnings, just like we would for the inductive family. This record
+packs two numbers and a runtime irrelevant proof.
+Firstly, we have a natural number called bigEnd corresponding to the size of
+the big end of the thinning (sy). We are happy to use a (unary) natural number
+here because we know that Idris 2 will compile it to an unbounded integer.
+Secondly, we have an integer called encoding corresponding to the thinning
+represented as a bit vector stating, for each variable, whether it is kept or
+dropped.
+We only care about the integer’s bigEnd least significant bits and
+assume the rest is set to 0.
+Thirdly, we have a runtime irrelevant proof invariant that encoding is in-
+deed a valid encoding of size bigEnd of a thinning from sx to sy. We will explore
+the definition of the relation Invariant later on in section 5.3.
+record Th {a : Type} (sx, sy : SnocList a) where
+constructor MkTh
+bigEnd : Nat
+encoding : Integer
+0 invariant : Invariant bigEnd encoding sx sy
+The first sign that this definition is adequate is our ability to construct any
+valid thinning. We demonstrate it is the case by introducing functions that act
+as smart constructor analogues for the inductive family’s data constructors.
+5.1
+Smart Constructors for Th
+The first and simplest one is done, a function that packs a pair of 0 (the size of
+the big end, and the empty encoding) together with a proof that it is an adequate
+encoding of the thinning from the empty scope to itself. In this instance, the
+proof is simply the Done constructor.
+done : Th [<] [<]
+done = MkTh { bigEnd = 0, encoding = 0, invariant = Done }
+To implement both keep and drop, we are going to need to perform bit-level
+manipulations. These are made easy by Idris 2’s Bits interface which provides us
+with functions to shift the bit patterns left or right (shiftl, shiftr), set or clear
+bits at specified positions (setBit, clearBit), take bitwise logical operations like
+disjunction (.|.) or conjunction (.&.), etc.
+14
+
+In both keep and drop, we need to extend the encoding with an additional
+bit. For this purpose we introduce the cons function which takes a bit b and an
+existing encoding bs and returns the new encoding bs·b.
+cons : Bool -> Integer -> Integer
+cons b bs = let bs0 = bs ‘shiftL‘ 1 in
+if b then (bs0 ‘setBit‘ 0) else bs0
+No matter what the value of the new bit is, we start by shifting the encoding
+to the left to make space for it; this gives us bs0 which contains the bit pattern
+bs·0. If the bit is True then we need to additionally set the bit at position 0
+to obtain bs·1. Otherwise if the bit is False, we can readily return the bs·0
+encoding obtained by left shifting. The correctness of this function is backed by
+two lemma: testing the bit at index 0 after consing amounts to returning the
+cons’d bit, and shifting the cons’d encoding to the right takes us back to the
+unextended encoding.
+testBit0Cons : (b : Bool) -> (bs : Integer) ->
+testBit (cons b bs) 0 === b
+consShiftR : (b : Bool) -> (bs : Integer) ->
+(cons b bs) ‘shiftR‘ 1 === bs
+The keep smart constructor demonstrates that from a thinning from sx to
+sy and a runtime irrelevant variable x we can compute a thinning from the
+extended source scope (sx :< x) to the target scope (sy :< x) where x was kept.
+keep : Th sx sy -> (0 x : a) -> Th (sx :< x) (sy :< x)
+keep th x = MkTh
+{ bigEnd = S (th .bigEnd)
+, encoding = cons True (th .encoding)
+, invariant =
+let 0 b = eqToSo $ testBit0Cons True (th .encoding) in
+Keep (rewrite consShiftR True (th .encoding) in th.invariant) x
+}
+The outer scope has grown by one variable and so we increment bigEnd. The
+encoding is obtained by cons-ing the boolean True to record the fact that this
+new variable is kept. Finally, we use the two lemmas shown above to convince
+Idris 2 the invariant has been maintained.
+Similarly the drop function demonstrates that we can compute a thinning
+getting rid of the variable x freshly added to the target scope.
+15
+
+drop : Th sx sy -> (0 x : a) -> Th sx (sy :< x)
+drop th x = MkTh
+{ bigEnd = S (th .bigEnd)
+, encoding = cons False (th .encoding)
+, invariant =
+let 0 prf = testBit0Cons False (th .encoding)
+0 nb = eqToSo $ cong not prf in
+Drop (rewrite consShiftR False (th .encoding) in th .invariant) x
+}
+We once again increment the bigEnd, use cons to record that the variable is
+being discarded and use the lemmas ensuring its correctness to convince Idris 2
+the invariant is maintained.
+We can already deploy these smart constructors to implement functions pro-
+ducing thinnings. We use which as our example. It is a filter-like function that
+returns a dependent pair containing the elements that satisfy a boolean predi-
+cate together with a proof that there is a thinning embedding them back into
+the input snoclist.
+which : (a -> Bool) -> (sy : SnocList a) ->
+(sx : SnocList a ** Th sx sy)
+which p [<] = ([<] ** done)
+which p (sy :< y) =
+let (sx ** th) = which p sy in
+if p y then (sx :< y ** keep th y)
+else (sx ** drop th y)
+If the input snoclist is empty then the output shall also be, and done builds
+a thinning from [<] to itself. If it is not empty we can perform a recursive call
+on the tail of the snoclist and then depending on whether the predicates holds
+true of the head we can either keep or drop it.
+We are now equipped with these smart constructors that allow us to seam-
+lessly build thinnings.
+To recover the full expressive power of the inductive
+family, we also need to be able to take these thinnings apart. We are now going
+to tackle this issue.
+5.2
+Pattern Matching on Th
+The View family is a sum type indexed by a thinning. It has one data constructor
+associated to each smart constructor and storing its arguments.
+data View : Th sx sy -> Type where
+Done : View done
+Keep : (th : Th sx sy) -> (0 x : a) -> View (keep th x)
+Drop : (th : Th sx sy) -> (0 x : a) -> View (drop th x)
+The accompanying view function witnesses the fact that any thinning arises
+as one of these three cases.
+16
+
+view : (th : Th sx sy) -> View th
+We show the implementation of view in its entirety but leave out the tech-
+nical auxiliary lemma it invokes. The interested reader can find them in the
+accompanying material. We will however inspect the code view compiles to af-
+ter erasure in section 5.5 to confirm that these auxiliary definitions do not incur
+any additional runtime cost.
+We first start by pattern matching on the bigEnd of the thinning. If it is 0
+then we know the thinning has to be the empty thinning. Thanks to an inversion
+lemma called isDone, we can collect a lot of equality proofs: the encoding bs has
+to be 0, the source and target scopes sx and sy have to be the empty snoclists,
+and the proof prf of the invariant has to be of a specific shape. Rewriting by
+these equalities changes the goal type enough for the typechecker to ultimately
+see that the thinning was constructed using the done smart constructor and so
+we can use the view’s Done constructor.
+view (MkTh 0 bs prf) =
+let 0 eqs = isDone prf in
+rewrite bsIsZero eqs in
+rewrite fstIndexIsLin eqs in
+rewrite sndIndexIsLin eqs in
+rewrite invariantIsDone eqs in
+Done
+In case the thinning is non-empty, we need to inspect the 0-th bit of the
+encoding to know whether it keeps or discards its most local variable. This is
+done by calling the choose function which takes a boolean b and returns a value
+of type (Either (So b) (So (not b)) i.e. we not only inspect the boolean but also
+record which value we got in a proof using the So family introduced in section 3.
+view (MkTh (S i) bs prf) = case choose (testBit bs Z) of
+If the bit is set then we know the variable is kept. And so we can invoke an
+inversion lemma that will once again provide us with a lot of equalities that we
+immediately deploy to reshape the goal’s type. This ultimately lets us assemble
+a sub-thinning and use the view’s Keep constructor.
+Left so =>
+let 0 eqs = isKeep prf so in
+rewrite fstIndexIsSnoc eqs in
+rewrite sndIndexIsSnoc eqs in
+rewrite invariantIsKeep eqs in
+rewrite isKeepInteger bs so in
+let th : Th eqs.fstIndexTail eqs.sndIndexTail
+th = MkTh i (bs ‘shiftR‘ 1) eqs.subInvariant in
+cast $ Keep th eqs.keptHead
+If the bit is not set then we learn that the thinning was constructed using
+17
+
+drop. We can once again use an inversion lemma to rearrange the goal and
+finally invoke the view’s Drop constructor.
+Right soNot =>
+let 0 eqs = isDrop prf soNot in
+rewrite sndIndexIsSnoc eqs in
+rewrite invariantIsDrop eqs in
+rewrite isDropInteger bs soNot in
+let th : Th sx eqs.sndIndexTail
+th = MkTh i (bs ‘shiftR‘ 1) eqs.subInvariant in
+cast $ Drop th eqs.keptHead
+We can readily use this function to implement pattern matching functions
+taking a thinning apart. We can for instance define kept, the function that
+counts the number of keep smart constructors used when manufacturing the
+input thinning and returns a proof that this is exactly the length of the source
+scope sx.
+kept : Th sx sy -> (n : Nat ** length sx === n)
+kept th = case view th of
+Done
+=> (0 ** Refl)
+Keep th x => let (n ** eq) = kept th in
+(S n ** cong S eq)
+Drop th x => kept th
+We proceed by calling the view function on the input thinning which im-
+mediately tells us that we only have three cases to consider. The Done case is
+easily handled because the branch’s refined types inform us that both sx and
+sy are the empty snoclist [<] whose length is evidently 0. In the Keep branch
+we learn that sx has the shape (_ :< x) and so we must return the successor of
+whatever the result of the recursive call gives us. Finally in the Drop case, sx
+is untouched and so a simple recursive call suffices. Note that the function is
+correctly detected as total because the target scope sy is indeed getting struc-
+turally smaller at every single recursive call. It is runtime irrelevant but it can
+still be successfully used as a termination measure by the compiler.
+5.3
+The Invariant Relation
+We have shown the user-facing Th and have claimed that it is possible to define
+smart constructors done, keep, and drop, as well as a view function. This should
+become apparent once we show the actual definition of Invariant.
+5.3.1
+Definition of Invariant
+The relation maintains the invariant between the record’s fields bigEnd (a Nat)
+and encoding (an Integer) and the index scopes sx and sy. Its definition can
+favour ease-of-use of runtime efficiency because we statically know that all of
+18
+
+the Invariant proofs will be erased during compilation.
+data Invariant : (i : Nat) -> (bs : Integer) ->
+(sx, sy : SnocList a) -> Type where
+Done : Invariant Z 0 [<] [<]
+Keep : Invariant i (bs ‘shiftR‘ 1) sx sy -> (0 x : a) ->
+{auto 0 b
+: So (testBit bs Z)} ->
+Invariant (S i) bs (sx :< x) (sy :< x)
+Drop : Invariant i (bs ‘shiftR‘ 1) sx sy -> (0 x : a) ->
+{auto 0 nb : So (not (testBit bs Z))} ->
+Invariant (S i) bs sx (sy :< x)
+As always, the Done constructor is the simplest. It states that the thinning
+of size Z and encoded as the bit pattern 0 is the empty thinning.
+The Keep constructor guarantees that the thinning of size (S i) and encoding
+bs represents an injection from (sx :< x) to (sy :< x) provided that the bit at
+position Z of bs is set, and that the rest of the bit pattern (obtained by a right
+shift on bs) is a valid thinning of size i from sx to sy.
+The Drop constructor is structured the same way, except that it insists the
+bit at position Z should not be set.
+We can readily use this relation to prove that some basic encoding are valid
+representations of useful thinnings.
+5.3.2
+Examples of Invariant proofs
+For instance, we can always define a thinning from the empty scope to an
+arbitrary scope sy.
+none : (sy : SnocList a) -> Th [<] sy
+none sy = MkTh (length sy) 0 (none sy)
+The encoding of this thinning is 0 because every variable is being discarded
+and its bigEnd is the length of the outer scope sy. The proof that this encoding
+is valid is provided by the none lemma proven below. We once again use Idris 2’s
+overloading to give the same to functions that play similar roles but at different
+types.
+none : (sy : SnocList a) -> Invariant (length sy) 0 [<] sy
+none [<] = Done
+none (sy :< y) = Drop (none sy) y
+The proof proceeds by induction over the outer scope sy. If it is empty,
+we can simply use the constructor for the empty thinning. Otherwise we can
+invoke Drop on the induction hypothesis. This all typechecks because (testBit
+0 Z) computes to False and so the nb proof can be constructed automatically
+by Idris 2’s proof search (cf. section 3.2), and (0 ‘shiftR‘ 1) evaluates to 0
+which means the induction hypothesis has exactly the right type.
+19
+
+The definition of the identity thinning is a bit more involved. For a scope of
+size n, we are going to need to generate a bit pattern consisting of n ones. We
+define it in two steps. First, cofull defines a bit pattern of k zeros followed by
+infinitely many ones by shifting k places to the left a bit pattern of ones only.
+Then, we obtain full by taking the complement of cofull.
+cofull : Nat -> Integer
+cofull n = oneBits ‘shiftL‘ n
+full : Nat -> Integer
+full n = complement (cofull n)
+We can then define the identity thinning for a scope of size n by pairing
+(full n) as the encoding and n as the bigEnd.
+ones : (sx : SnocList a) -> Th sx sx
+ones sx = let n : Nat; n = length sx in MkTh n (full n) (ones sx)
+The bulk of the work is once again in the eponymous lemma proving that
+this encoding is valid.
+ones : (sx : SnocList a) ->
+let n = length sx in Invariant n (full n) sx sx
+ones [<] = Done
+ones (sx :< x) =
+let 0 nb = eqToSo (testBitFull (S (length sx)) Z) in
+Keep (rewrite shiftRFull (length sx) in ones sx) x
+This proof proceeds once more by induction on the scope. If the scope is
+empty then once again the constructor for the empty thinning will do. In the
+non-empty case, we first appeal to an auxiliary lemma (not shown here) to con-
+struct a proof nb that the bit at position Z for a non-zero full integer is known
+to be True. We then need to use another lemma to cast the induction hypothesis
+which mentions (full (length sx)) so that it may be used in a position where
+we expect a proof talking about (full (length (sx :< x)) ‘shiftR‘ 1).
+5.3.3
+Properties of the Invariant relation
+This relation has a lot of convenient properties.
+First, it is proof irrelevant: any two proofs that the same i, bs, sx, and sy
+are related are provably equal. Consequently, equality on Th values amounts to
+equality of the bigEnd and encoding values. In particular it is cheap to test
+whether a given thinning is the empty or the identity thinning.
+Second, it can be inverted [CT95] knowing only two bits: whether the natural
+number is empty and what the value of the bit at position Z of the encoding
+is. This is what allowed us to efficiently implement the view function by using
+these two checks and then inverting the Invariant proof to gain access to the
+proof that the remainder of the thinning’s encoding is valid. We will see in
+section 5.5 that this leads to efficient runtime code for the view.
+20
+
+5.4
+Choose Your Own Abstraction Level
+Access to both the high-level View and the internal Invariant relation means
+that programmers can pick the level of abstraction at which they want to work.
+They may need to explicitly manipulate bits to implement key operators that
+are used in performance-critical paths but can also stay at the highest level for
+more negligible operations, or when proving runtime irrelevant properties.
+In the previous section we saw simple examples of these bit manipulations
+when defining none (using the constant 0 bit pattern) and ones using bit shifting
+and complement to form an initial segment of 1s followed by 0s.
+Other natural examples include the meet and join of two thinnings sharing
+the same wider scope.
+The join can for instance be thought of either as a
+function defined by induction on the first thinning and case analysis on the
+second, emitting a Keep constructor whenever either of the inputs does. Or we
+can observe that the bit pattern in the join is exactly the disjunction of the
+inputs’ respective bit patterns and prove a lemma about the Invariant relation
+instead. This can be visualised as follows. In each column, the meet is a •
+whenever either of the inputs is.
+◦◦••◦
+∨ •◦◦••
+•◦•••
+The join is of particular importance because it appears when we convert
+an ‘opened’ view of a term into its co-de Bruijn counterpart. As we mentioned
+earlier, co-de Bruijn terms in an arbitrary scope are represented by the pairing of
+a term indexed by its precise support with a thinning embedding this support
+back into the wider scope.
+When working with such a representation, it is
+convenient to have access to an ‘opened’ view where the outer thinning has
+been pushed inside therefore exposing the term’s top-level constructor, ready
+for case-analysis.
+The following diagram shows the correspondence between an ‘opened’ ap-
+plication node using the view (the diamond ‘$’ node) with two subterms both
+living in the outer scope and its co-de Bruijn form (the circular ‘$’ node) with
+an outer thinning selecting the term support.
+$
+t1
+t2
+◦◦••◦
+•◦◦••
+$
+t1
+t2
+•◦•••
+◦••◦
+•◦••
+The outer thinning of the co-de Bruijn term is obtained precisely by com-
+puting the join of the respective outer thinnings of the ‘opened’ application’s
+function and argument.
+These explicit bit manipulations will be preserved during compilation and
+thus deliver more efficient code.
+21
+
+5.5
+Compiled Code
+The following code block shows the JavaScript code that is produced when
+compiling the view function. We chose to use the JavaScript backend rather
+than e.g. the ChezScheme one because it produces fairly readable code. We
+have modified the backend to also write comments reminding the reader of the
+type of the function being defined and the data constructors the natural number
+tags correspond to. These changes are now available to all in Idris 2’s current
+development version.
+The only manual modifications we have performed are the inlining of a func-
+tion corresponding to a case block, renaming variables and property names to
+make them human-readable, introducing the $tail definitions to make lines
+shorter, and slightly changing the layout.
+/* Thin.Smart.view : (th : Th sx sy) -> View th */
+function Thin_Smart_view($th) {
+switch($th.bigEnd) {
+case 0n: return {h: 0 /* Done */};
+default: {
+const $predBE = ($th.bigEnd-1n);
+const $test = choose(notEq(($th.encoding&1n), 0n)));
+switch($test.tag) {
+case 0: /* Left */ {
+const $tail = $th.encoding>>1n;
+return { tag: 1 /* Keep */
+, val: {bigEnd: $predBE, encoding: $tail}}; }
+case 1: /* Right */ {
+const $tail = $th.encoding>>1n;
+return { tag: 2 /* Drop */
+, val: {bigEnd: $predBE, encoding: $tail}}; }
+}}}}
+Readers can see that the compilation process has erased all of the indices
+and the proofs showing that the invariant tying the efficient runtime represen-
+tation to the high-level specification is maintained. A thinning is represented
+at runtime by a JavaScript object with two properties corresponding to Th’s
+runtime relevant fields: bigEnd and encoding. Both are storing a JavaScript
+bigInt (one corresponding to the Nat, the other to the Integer). For instance
+the thinning [01101] would be at runtime { bigEnd: 5n, encoding: 13n }.
+The view proceeds in two steps. First if the bigEnd is 0n then we know the
+thinning is empty and can immediately return the Done constructor. Otherwise
+we know the thinning to be non-empty and so we can compute the big end of its
+tail ($predBE) by subtracting one to the non-zero bigEnd. We can then inspect
+the bit at position 0 to decide whether to return a Keep or a Drop constructor.
+This is performed by using a bit mask to 0-out all the other bits ($th.bigEnd&1n)
+and checking whether the result is zero. If it is not equal to 0 then we emit
+22
+
+Keep and compute the $tail of the thinning by shifting the original encoding
+to drop the 0th bit. Otherwise we emit Drop and compute the same tail.
+By running view on this [01101] thinning, we would get back (Keep [0110]),
+that is to say { tag: 1, val: { bigEnd: 4n, encoding: 6n } }.
+Thanks to Idris 2’s implementation of Quantitative Type Theory we have
+managed to manufacture a high level representation that can be manipulated
+like a classic inductive family using smart constructors and views without giving
+up an inch of control on its runtime representation.
+The remaining issues such as the fact that we form the view’s constructors
+only to immediately take them apart thus creating needless allocations can be
+tackled by reusing Wadler’s analysis (section 12 of [Wad87]).
+6
+Conclusion
+We have seen that inductive families provide programmers with ways to root out
+bugs by enforcing strong invariants. Unfortunately these families can get in the
+way of producing performant code despite existing optimisation passes erasing
+redundant or runtime irrelevant data. This tension has led us to take advantage
+of Quantitative Type Theory in order to design a library combining the best of
+both worlds: the strong invariants and ease of use of inductive families together
+with the runtime performance of explicit bit manipulations.
+6.1
+Related Work
+For historical and ergonomic reasons, idiomatic code in Coq tends to center
+programs written in a subset of the language quite close to OCaml and then
+prove properties about these programs in the runtime irrelevant Prop fragment.
+This can lead to awkward encodings when the unrefined inputs force the user
+to consider cases which ought to be impossible. Common coping strategies in-
+volve relaxing the types to insert a modicum of partiality e.g. returning an
+option type or taking an additional input to be used as the default return value.
+This approach completely misses the point of type-driven development.
+We
+benefit a lot from having as much information as possible available during in-
+teractive editing.
+This information not only helps tremendously getting the
+definitions right by ensuring we always maintain vital invariants thus making
+invalid states unrepresentable, it also gives programmers access to type-driven
+tools and automation. Thankfully libraries such as Equations [Soz10, SM19]
+can help users write more dependently typed programs, by taking care of the
+complex encoding required in Coq. A view-based approach similar to ours but
+using Prop instead of the zero quantity ought to be possible. We expect that
+the views encoded this way in Coq will have an even worse computational be-
+haviour given that Equations uses a sophisticated elaboration process to encode
+dependent pattern-matching into Gallina. However Coq does benefit from good
+automation support for unfolding lemmas, inversion principles, and rewriting
+23
+
+by equalities which may compensate for the awkwardness introduced by the
+encoding.
+Prior work on erasure [Tej20] has the advantage of offering a fully automated
+analysis of the code. The main inconvenience is that users cannot state explic-
+itly that a piece of data ought to be runtime irrelevant and so they may end
+up inadvertently using it which would prevent its erasure. Quantitative Type
+Theory allows us users to explicitly choose what is and is not runtime relevant,
+with the quantity checker keeping us true to our word. This should ensure that
+the resulting program has a much more predictable complexity.
+A somewhat related idea was explored by Brady, McKinna, and Hammond
+in the context of circuit design [BMH07]. In their verification work they index
+an efficient representation (natural numbers as a list of bits) by its meaning
+as a unary natural number. All the operations are correct by construction as
+witnessed by the use of their unary counterparts acting as type-level specifica-
+tions. In the end their algorithms still process the inductive family instead of
+working directly with binary numbers. This makes sense in their setting where
+they construct circuits and so are explicitly manipulating wires carrying bits.
+By contrast, in our motivating example we really want to get down to actual
+(unbounded) integers rather than linked lists of bits.
+6.2
+Limitations and Future Work
+Overall we found this case study using Idris 2, a state of the art language based
+on Quantitative Type Theory, very encouraging. The language implementation
+is still experimental (see for instance appendix B for some of the bugs we found)
+but none of the issues are intrinsic limitations. We hope to be able to push
+this line of work further, tackling the following limitations and exploring more
+advanced use cases.
+6.2.1
+Limitations
+Unfortunately it is only propositionally true that (view (keep th x)) computes
+to (Keep th x) (and similarly for done/Done and drop/Drop). This means that
+users may need to manually deploy these lemmas when proving the properties
+of functions defined by pattern matching on the result of calling the view func-
+tion. This annoyance would disappear if we had the ability to extend Idris 2’s
+reduction rules with user-proven equations as implemented in Agda and formally
+studied by Cockx, Tabareau, and Winterhalter [CTW21].
+In this paper’s case study, we were able to design the core Invariant relation
+making the invariants explicit in such a way that it would be provably proof
+irrelevant. This may not always be possible given the type theory currently im-
+plemented by Idris 2. Adding support for a proof-irrelevant sort of propositions
+(see e.g. Altenkirch, McBride, and Swierstra’s work [AMS07]) could solve this
+issue once and for all.
+The Idris 2 standard library thankfully gave us access to a polished pure
+interface to explicitly manipulate an integer’s bits. However these built-in oper-
+24
+
+ations came with no built-in properties whatsoever. And so we had to postulate
+a (minimal) set of axioms (see appendix A) and prove a lot of useful corollar-
+ies ourselves. There is even less support for other low-level operations such as
+reading from a read-only array, or manipulating pointers.
+We also found the use of runtime irrelevance (the 0 quantity) sometimes
+somewhat frustrating.
+Pattern-matching on a runtime irrelevant value in a
+runtime relevant context is currently only possible if it is manifest for the
+compiler that the value could only arise using one of the family’s construc-
+tors.
+In non-trivial cases this is unfortunately only merely provable rather
+than self-evident. Consequently we are forced to jump through hoops to ap-
+pease the quantity checker, and end up defining complex inversion lemmas to
+bypass these limitations. This could be solved by a mix of improvements to
+the typechecker and meta-programming using prior ideas on automating inver-
+sion [CT95, McB96, Mon10].
+6.2.2
+Future work
+We are planning to explore more memory-mapped representations equipped
+with a high level interface.
+We already have experimental results demonstrating that we can use a read-
+only array as a runtime representation of a binary search tree. Search can be
+implemented as a proven-correct high level decision procedure that is seem-
+ingly recursively exploring the ”tree”. At runtime however, this will effectively
+execute like a classic search by dichotomy over the array.
+More generally, we expect that a lot of the work on programming on serialised
+data done in LoCal [VKR+19] thanks to specific support from the compiler can
+be done as-is in a QTT-based programming language.
+Indeed, QTT’s type
+system is powerful enough that tracking these invariants can be done purely in
+library code.
+In the short term, we would like to design a small embedded domain specific
+language giving users the ability to more easily build and take apart products
+and sums efficiently represented in the style we presented here. Staging would
+help here to ensure that the use of the eDSL comes at no runtime cost. There
+are plans to add type-enforced staging to Idris 2, thus really making it the ideal
+host language for our project.
+Our long term plan is to go beyond read-only data and look at imperative
+programs proven correct using separation logic and see how much of this after-
+the-facts reasoning can be brought back into the types to enable a high-level
+correct-by-construction programming style that behaves the same at runtime.
+Acknowledgements
+We are grateful to Conor McBride for discussions per-
+taining to the fine details of the unsafe encoding used in TypOS, as well as James
+McKinna, Fredrik Nordvall Forsberg, Ohad Kammar, and Jacques Carette for
+providing helpful comments and suggestions on early versions of this paper.
+25
+
+References
+[AAM+22] Guillaume Allais, Malin Altenm¨uller,
+Conor McBride, Georgi
+Nakov, Fredrik Nordvall Forsberg, and Craig Roy. TypOS: An oper-
+ating system for typechecking actors. In 28th International Confer-
+ence on Types for Proofs and Programs, TYPES 2022, June 20-25,
+2022, Nantes, France, 2022.
+[AMS07]
+Thorsten Altenkirch, Conor McBride, and Wouter Swierstra. Ob-
+servational equality, now! In Aaron Stump and Hongwei Xi, editors,
+Proceedings of the ACM Workshop Programming Languages meets
+Program Verification, PLPV 2007, Freiburg, Germany, October 5,
+2007, pages 57–68. ACM, 2007.
+[Atk18]
+Robert Atkey. Syntax and semantics of quantitative type theory. In
+Anuj Dawar and Erich Gr¨adel, editors, Proceedings of the 33rd An-
+nual ACM/IEEE Symposium on Logic in Computer Science, LICS
+2018, Oxford, UK, July 09-12, 2018, pages 56–65. ACM, 2018.
+[BMH07]
+Edwin C. Brady, James McKinna, and Kevin Hammond. Construct-
+ing correct circuits: Verification of functional aspects of hardware
+specifications with dependent types.
+In Marco T. Moraz´an, edi-
+tor, Proceedings of the Eighth Symposium on Trends in Functional
+Programming, TFP 2007, New York City, New York, USA, April
+2-4. 2007, volume 8 of Trends in Functional Programming, pages
+159–176. Intellect, 2007.
+[BMM03]
+Edwin C. Brady, Conor McBride, and James McKinna. Inductive
+families need not store their indices.
+In Stefano Berardi, Mario
+Coppo, and Ferruccio Damiani, editors, Types for Proofs and Pro-
+grams, International Workshop, TYPES 2003, Torino, Italy, April
+30 - May 4, 2003, Revised Selected Papers, volume 3085 of Lecture
+Notes in Computer Science, pages 115–129. Springer, 2003.
+[Bra21]
+Edwin C. Brady. Idris 2: Quantitative type theory in practice. In
+Anders Møller and Manu Sridharan, editors, 35th European Confer-
+ence on Object-Oriented Programming, ECOOP 2021, July 11-17,
+2021, Aarhus, Denmark (Virtual Conference), volume 194 of LIPIcs,
+pages 9:1–9:26. Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik,
+2021.
+[CA20]
+Jesper Cockx and Andreas Abel. Elaborating dependent (co)pattern
+matching: No pattern left behind. J. Funct. Program., 30:e2, 2020.
+[CDT22]
+The Coq Development Team. The Coq Proof Assistant Reference
+Manual, version 8.15.2, May 2022.
+[Cha09]
+James Maitland Chapman. Type checking and normalisation. PhD
+thesis, University of Nottingham, July 2009.
+26
+
+[CT95]
+Cristina Cornes and Delphine Terrasse. Automating inversion of in-
+ductive predicates in coq. In Stefano Berardi and Mario Coppo,
+editors, Types for Proofs and Programs, International Workshop
+TYPES’95, Torino, Italy, June 5-8, 1995, Selected Papers, volume
+1158 of Lecture Notes in Computer Science, pages 85–104. Springer,
+1995.
+[CTW21]
+Jesper Cockx, Nicolas Tabareau, and Th´eo Winterhalter. The tam-
+ing of the rew: a type theory with computational assumptions. Proc.
+ACM Program. Lang., 5(POPL):1–29, 2021.
+[dB72]
+Nicolaas Govert de Bruijn. Lambda calculus notation with nameless
+dummies, a tool for automatic formula manipulation, with appli-
+cation to the Church-Rosser theorem. Indagationes Mathematicae
+(Proceedings), 75(5):381–392, 1972.
+[Dyb94]
+Peter Dybjer.
+Inductive families.
+Formal Aspects Comput.,
+6(4):440–465, 1994.
+[McB96]
+Conor McBride. Inverting inductively defined relations in LEGO.
+In Eduardo Gim´enez and Christine Paulin-Mohring, editors, Types
+for Proofs and Programs, International Workshop TYPES’96, Aus-
+sois, France, December 15-19, 1996, Selected Papers, volume 1512 of
+Lecture Notes in Computer Science, pages 236–253. Springer, 1996.
+[McB16]
+Conor McBride.
+I got plenty o’ nuttin’.
+In Sam Lindley, Conor
+McBride, Philip W. Trinder, and Donald Sannella, editors, A List of
+Successes That Can Change the World - Essays Dedicated to Philip
+Wadler on the Occasion of His 60th Birthday, volume 9600 of Lec-
+ture Notes in Computer Science, pages 207–233. Springer, 2016.
+[McB18]
+Conor McBride.
+Everybody’s got to be somewhere.
+In Robert
+Atkey and Sam Lindley, editors, Proceedings of the 7th Workshop on
+Mathematically Structured Functional Programming, MSFP@FSCD
+2018, Oxford, UK, 8th July 2018, volume 275 of EPTCS, pages 53–
+69, 2018.
+[MEL+21] Krzysztof Maziarz, Tom Ellis, Alan Lawrence, Andrew W. Fitzgib-
+bon, and Simon Peyton Jones. Hashing modulo alpha-equivalence.
+In Stephen N. Freund and Eran Yahav, editors, PLDI ’21: 42nd
+ACM SIGPLAN International Conference on Programming Lan-
+guage Design and Implementation, Virtual Event, Canada, June
+20-25, 2021, pages 960–973. ACM, 2021.
+[MM04]
+Conor McBride and James McKinna. The view from the left. J.
+Funct. Program., 14(1):69–111, 2004.
+[Mon10]
+Jean-Fran¸cois Monin. Proof Trick: Small Inversions. In Yves Bertot,
+editor, Second Coq Workshop, Edinburgh, United Kingdom, July
+2010. Yves Bertot.
+27
+
+[SM19]
+Matthieu Sozeau and Cyprien Mangin. Equations reloaded: high-
+level dependently-typed functional programming and proving in coq.
+Proc. ACM Program. Lang., 3(ICFP):86:1–86:29, 2019.
+[Soz10]
+Matthieu Sozeau. Equations: A dependent pattern-matching com-
+piler. In Matt Kaufmann and Lawrence C. Paulson, editors, Inter-
+active Theorem Proving, First International Conference, ITP 2010,
+Edinburgh, UK, July 11-14, 2010. Proceedings, volume 6172 of Lec-
+ture Notes in Computer Science, pages 419–434. Springer, 2010.
+[Tej20]
+Mat´uˇs Tejiˇsˇc´ak.
+A dependently typed calculus with pattern
+matching and erasure inference.
+Proc. ACM Program. Lang.,
+4(ICFP):91:1–91:29, 2020.
+[VKR+19] Michael Vollmer, Chaitanya Koparkar, Mike Rainey, Laith Sakka,
+Milind Kulkarni, and Ryan R. Newton. Local: a language for pro-
+grams operating on serialized data. In Kathryn S. McKinley and
+Kathleen Fisher, editors, Proceedings of the 40th ACM SIGPLAN
+Conference on Programming Language Design and Implementation,
+PLDI 2019, Phoenix, AZ, USA, June 22-26, 2019, pages 48–62.
+ACM, 2019.
+[Wad87]
+Philip Wadler. Views: A way for pattern matching to cohabit with
+data abstraction. In Conference Record of the Fourteenth Annual
+ACM Symposium on Principles of Programming Languages, Mu-
+nich, Germany, January 21-23, 1987, pages 307–313. ACM Press,
+1987.
+[WB89]
+Philip Wadler and Stephen Blott. How to make ad-hoc polymor-
+phism less ad-hoc. In Conference Record of the Sixteenth Annual
+ACM Symposium on Principles of Programming Languages, Austin,
+Texas, USA, January 11-13, 1989, pages 60–76. ACM Press, 1989.
+A
+Postulated lemmas for the Bits interface
+It is often more convenient to reason about integers in terms of their bits. We
+define the notion of bitwise equality as the pointwise equality according to the
+testBit.
+(˜˜˜) : Integer -> Integer -> Type
+bs ˜˜˜ cs = (i : Nat) -> testBit bs i === testBit cs i
+Our first postulate is a sort of extensionality principle stating that bitwise
+equality implies propositional equality.
+extensionally : {bs, cs : Integer} -> bs ˜˜˜ cs -> bs === cs
+28
+
+This gives us a powerful reasoning principle once combined with axioms
+explaining the behaviour of various primitives at the bit level.
+This is why
+almost all of the remaining axioms are expressed in terms of testBit calls.
+A.1
+Logical operations
+Our first batch of axioms relates logical operations on integers to their boolean
+counterparts. This is essentially stating that these operations are bitwise.
+testBitAnd : (bs, cs : Integer) -> (i : Nat) ->
+testBit (bs .&. cs) i === (testBit bs i && testBit cs i)
+testBitOr : (bs, cs : Integer) -> (i : Nat) ->
+testBit (bs .|. cs) i === (testBit bs i || testBit cs i)
+testBitComplement : (bs : Integer) -> (i : Nat) ->
+testBit (complement bs) i === not (testBit bs i)
+Together with the extensionality principle mentioned above this already al-
+lows us to prove for instance that the binary operators are commutative and
+associative, that the de Morgan laws hold, or that conjunction distributes over
+disjunction.
+A.2
+Bit Shifting
+The second set of axiom describes the action of left and right shifting on bit
+patterns.
+A right shift of size k will drop the k least significant bits; consequently
+testing the bit i on the right-shifted integer amounts to testing the bit (k + i)
+on the original integer.
+testBitShiftR : (bs : Integer) -> (k : Nat) ->
+(i : Nat) -> testBit (bs ‘shiftR‘ k) i === testBit bs (k + i)
+A left shift will add k new least significant bits initialised at 0; consequently
+testing a bit i on the left-shifted integer will either return False if i is strictly
+less than k, or the bit at position (i - k) in the original integer.
+For simplicity we state these results without mentioning the ‘strictly less
+than’ relation, by considering on the one hand the effect of a non-zero left shift,
+and on the other the fact that a left-shift by 0 bits is the identity function.
+testBit0ShiftL : (bs : Integer) -> (k : Nat) ->
+testBit (bs ‘shiftL‘ S k) Z === False
+testBitSShiftL : (bs : Integer) -> (k : Nat) -> (i : Nat) ->
+testBit (bs ‘shiftL‘ S k) (S i) === testBit (bs ‘shiftL‘ k) i
+29
+
+shiftL0 : (bs : Integer) -> (bs ‘shiftL‘ 0) === bs
+A.3
+Bit testing
+The last set of axioms specifies what happens when a bit is set.
+Testing a bit other than the one that was set amounts to testing it on the
+original integer.
+testSetBitOther : (bs : Integer) -> (i, j : Nat) -> Not (i === j) ->
+testBit (setBit bs i) j === testBit bs j
+Finally, we have an axiom stating that the integer (bit i) (i.e. 2i) is non-
+zero.
+bitNonZero : (i : Nat) -> (bit i == 0) === False
+B
+Current Limitations of Idris 2
+This challenge, suggested by Jacques Carette, highlights some of the current
+limitations of Idris 2.
+B.1
+Problem statement
+The goal is to use the Vect type defined in section 3.3 and define a view that
+un-does vector-append. This is a classic exercise in dependently-typed program-
+ming, the interesting question being whether we can implement the function just
+as seamlessly with our encoding.
+Vector append can easily be defined by induction over the first vector.
+(++) : Vect m a -> Vect n a -> Vect (m + n) a
+xs@_ ++ ys with (view xs)
+_ | [] = ys
+_ | hd :: tl = hd :: (tl ++ ys)
+If the first vector is empty we can readily return the second vector. If it
+is cons-headed, we can return the head and compute the tail by performing a
+recursive call.
+Equipped with this definition, we can declare the view type which we call
+SplitAt by analogy with its weakly typed equivalent processing lists. It states
+that a vector xs of length p can be split at m if p happens to be (m + n) and xs
+happens to be (pref ++ suff) where pref and suff’s respective lengths are m
+and n.
+30
+
+data SplitAt : (m : Nat) -> (xs : Vect p a) -> Type where
+MkSplitAt : (pref : Vect m a) -> (suff : Vect n a) ->
+SplitAt m (pref ++ suff)
+The challenge is to define the function proving that a vector of size (m + n)
+can be split at m.
+B.2
+Failing attempts
+The proof will necessarily go by induction on m, followed by a case analysis on
+the input vector and a recursive call in the non-zero case.
+Our first failing attempt successfully splits the natural number, calls the view
+on the vector xs to take it apart but then fails when performing the recursive
+call to splitAt.
+failing "tl is not accessible in this context"
+splitAt : (m : Nat) -> (xs : Vect (m + n) a) -> SplitAt m xs
+splitAt 0 xs = MkSplitAt [] xs
+splitAt (S m) xs@_ with (view xs)
+_ | hd :: tl@_ with (splitAt m tl)
+_ | res = ?a
+This reveals an issue in Idris 2’s handling of the interplay between @-patterns
+and quantities: the compiler arbitrarily decided to make the alias tl runtime
+irrelevant only to then complain that tl is not accessible when we want to
+perform the recursive call (splitAt m tl)!
+In order to work around this limitation, we decided to let go of @-patterns
+and write the fully explicit clause ourselves, using dotted patterns to mark the
+forced expressions.
+failing "Can’t match on ?postpone [no locals in scope] (User dotted)"
+splitAt : (m : Nat) -> (xs : Vect (m + n) a) -> SplitAt m xs
+splitAt 0 xs = MkSplitAt [] xs
+splitAt (S m) xs@_ with (view xs)
+_ | hd :: tl with (splitAt m tl)
+splitAt (S m) .(hd :: (pref ++ suff))
+| hd :: .(pref ++ suff)
+| MkSplitAt pref suff = ?a
+The left-hand side now typechecks but the case tree builder fails with a
+perplexing error. This reveals a bug in Idris 2’s implementation of elaboration of
+pattern-matching functions to case trees. Instead of ignoring dotted expressions
+when building the case tree (these expressions are forced and so the variables
+they mention will have necessarily been bound in another pattern), it attempts
+to use them to drive the case-splitting strategy. This is a well-studied problem
+and should be fixable by referring to Cockx and Abel’s work [CA20].
+31
+
+B.3
+Working Around Idris 2’s Limitations
+This leads us to our working solution. Somewhat paradoxically, working around
+these Idris 2 bugs led us to a more principled solution whereby the pattern-
+matching step needed to adjust the result returned by the recursive call is ab-
+stracted away in an auxiliary function whose type clarifies what is happening.
+From an m split on xs, we can easily compute an (S m) split on (x :: xs) by
+cons-ing x on the prefix.
+(::) : (x : a) -> SplitAt m xs -> SplitAt (S m) (x :: xs)
+x :: MkSplitAt pref@(MkVect _ Refl) suff
+= MkSplitAt (x :: pref) suff
+In this auxiliary function, xs is clearly runtime irrelevant and so the case-
+splitter will not attempt to inspect it, thus generating the correct case tree.
+We are forced to match further on pref (in particular by making the equality
+proof Refl) so that just enough computation happens at the type level for the
+typechecker to see that things do line up. A proof irrelevant type of propositional
+equality would have helped us here.
+We can put all of these pieces together and finally get our splitAt view.
+splitAt : (m : Nat) -> (xs : Vect (m + n) a) -> SplitAt m xs
+splitAt 0 xs = MkSplitAt [] xs
+splitAt (S m) xs@_ with (view xs)
+_ | hd :: tl = hd :: splitAt m tl
+We do want to reiterate that these limitations are not intrinsic limitations of
+the approach, there are just flaws in the current experimental implementation
+of the Idris 2 language and can and should be remedied.
+32
+

DNE2T4oBgHgl3EQf9Qm6/content/tmp_files/2301.04227v1.pdf.txt

@@ -0,0 +1,3169 @@
+arXiv:2301.04227v1  [hep-th]  10 Jan 2023
+Induced energy-momentum tensor in de Sitter scalar QED and its implication for
+induced current
+Omid Gholizadeh Meimanat and Ehsan Bavarsad∗
+Department of Physics, University of Kashan, 8731753153 Kashan, Iran
+The aim of this research is to investigate the vacuum energy-momentum tensor of a quantized,
+massive, nonminimally coupled scalar field induced by a uniform electric field background in a four-
+dimensional de Sitter spacetime (dS4). We compute the expectation value of the energy-momentum
+tensor in the in-vacuum state and then regularize it using the adiabatic subtraction procedure.
+The correct trace anomaly of the induced energy-momentum tensor that confirmed our results is
+significant. The nonconservation equation for the induced energy-momentum tensor imposes the
+renormalization condition for the induced electric current of the scalar field. The findings of this
+research indicate that there are significant differences between the two induced currents which are
+regularized by this renormalization condition and the minimal subtraction condition.
+I.
+INTRODUCTION
+In the past several decades quantum field theory in curved spacetime has played a major role in the study of
+the effects of gravity on quantum fields; for an pedagogical introduction see [1–3]. A precise understanding of the
+quantum effects in curved spacetime was acquired by the late 1960s by Parker [4–6] and followed by investigations of
+others. Indeed, these early investigations focused primarily on the physical consequences of particle creation in the
+cosmological spacetimes. It has been illustrated that a time-varying gravitational field creates elementary particles
+from the vacuum. Parker discovered that this particle creation process can be analyzed by using the Bogoliubov
+transformations method [5]. Formulating a general framework of quantum field theory in curved spacetime involves
+nontrivial questions.
+The problem of particle concept is a deep one, and it is associated with one of the most
+fundamental difficulties of quantum field theory in curved spacetime, that there is no an unambiguous or unique
+vacuum state; a detailed discussion can be found in [1, 2]. This ambiguity is reflected in the theory by the absence of
+an unambiguous or unique preferred mode solutions of the field equation, which is in turn a consequence of symmetry
+group of the spacetime. Since in a general nonstatic curved spacetime there is generically no any timelike Killing
+vector field, it is not possible to classify modes as positive-frequency or negative-frequency. Indeed, it is possible to
+construct a compleat set of modes, however the problem is that there are many of such sets and we will not have
+any criterion available to select a unique privileged choice of the modes. One of the key lessens learned from the
+development of this subject is that the notion of particle number does not generally have universal significant. Hence,
+the expectation value of the energy-momentum tensor in a suitable vacuum state is perhaps more physically relevant
+object to probe the structure of quantum fields in curved spacetime rather than particle number quantity [1]. Part of
+reason is that the expectation value of the energy-momentum tensor in a fixed vacuum state transforms according to
+the usual tensor transformation law. In particular, if the vacuum expectation value of the energy-momentum tensor
+vanishes for an observer, it will vanish for all observers. On the contrary, the expectation of particle number is an
+essential observer-dependent quantity. The energy-momentum tensor is also important, because it is directly relevant
+to explore the consequences of the quantum field dynamics for the geometry of the spacetime through the Einstein
+equation. Hence, it is interesting to investigate the expectation value of the energy-momentum tensor in a suitable
+vacuum state.
+Since the mid-seventies, significant advances have been made in the computation of the energy-momentum tensor of
+the quantum fields as well as its applications in the cosmological context. An essential feature of these computations
+is that they all involve the ultraviolet divergencies.
+In fact, such divergencies occur naturally in quantum field
+theory calculations. Various prescriptions have been developed for rendering the energy-momentum tensor finite.
+Among these prescriptions, Pauli-Villars regularization [7, 8], dimensional regularization [9–11], and zeta-function
+regularization [11–14] were originally developed for use in Minkowski spacetime. Several sets of fictitious fields may
+be necessary to remove all of the divergences, making Pauli-Villars regularization rather complicated. In [13], it was
+pointed out that dimensional regularization procedure is ambiguous in curved spacetime due to the fact that in some
+classes of spacetimes, there is no natural way to generalize the dimensionality of the spacetime and the answer would
+be different in different extensions to D dimensions. This method also suffers from limitation that one needs, in
+∗ [email protected]
+
+2
+principle, to solve the field equations exactly. Another often used regularization procedure is point-splitting technique
+[15–18] which is intrinsically designed to be used in the position-space representation of the composite operators.
+This regularization is implemented by placing the two quantum fields at distant points separated by an infinitesimal
+distance in a nonnull direction and then the divergencies that arise in the coincidence limit are absorbed into the
+renormalized parameters. Although point-splitting technique is the most applicable regularization scheme, it involves
+considerable technical complication. An alternative regularization prescription is adiabatic subtraction [19–23] which
+was originally invented by Parker [5] to obtain the average density of the scalar particles created in a spatially flat
+Friedmann-Lemaitre-Robertson-Walker (FLRW) universe. This approach is only applicable in spacetimes with slowly
+varying curvature, and in fact an adiabatic expansion is an expansion in number derivatives of the spacetime metric.
+Hence, it is especially useful for studies that involve numerical techniques and problems of cosmological interest
+[22]. In order to arrive a sensible semiclassical backreaction theory of quantum fields in curved spacetime, Wald has
+propounded [24] a minimal set of properties that the renormalized energy-momentum tensor should satisfy. These
+properties which are often called the Wald axioms, in the weaker set of axioms, can be expressed as follows [1, 3]:
+(1) The expectation value of the energy-momentum tensor in any state at a point p in spacetime is covariant under
+general coordinate transformations (diffeomorphisms) and is independent of spacetime geometry at any point q ̸= p.
+(2) The off-diagonal matrix element of energy-momentum tensor between any orthogonal pairs of states is finite and
+unambiguous. (3) For all states, the expectation value of the energy-momentum tensor is covariantly conserved.
+(4) The expectation value of the energy-momentum tensor vanishes in the relevant vacuum state in the Minkowski
+spacetime. It was shown in [18] that the fifth axiom of Ref. [24] cannot be satisfied. Wald then proved a remarkable
+result, i.e., the uniqueness theorem, which states that if the expectation value of the energy-momentum tensor satisfies
+the Wald axioms (1)-(3), then it is unique up to the addition of a local conserved tensor [24]. Hence, the final results
+for the renormalized energy-momentum tensor do not depend on the employed regularization method [1, 3]. Using
+these regularization techniques, the regularized and renormalized energy-momentum tensor of a neutral scalar field
+has been widely investigated in FLRW universes [19–22, 25–33] and its physical implications for cosmological issues
+have been explored [34–36].
+It is thought that the very early universe can be approximately described by de Sitter spacetime (dS) [2], which
+motivates us to study the quantum field theory in this spacetime. Gibbons and Hawking [37] originally discovered
+that an inertial observer with a particle detector at rest perceives the de Sitter invariant vacuum state as a bath of
+thermal radiation which apparently comes from the cosmological event horizon. In this context, a relatively simple
+model problem is that of a neutral scalar field with no self-interactions whose regularized and renormalized energy-
+momentum tensor has been analyzed [11, 38–45]. In [39–41], it was subsequently shown that the vacuum state of the
+quantum field in dS is unstable to particle creation. Furthermore, the study of semiclassical backreaction effect of
+the quantum corrections onto the Hubble rate in [42] indicated that these contributions can potentially result in a
+superacceleration phase, i.e., a phase when the Hubble rate increases as the dS expands. The spontaneous creation
+of pairs of charged particles from the vacuum by a strong electric field background in Minkowski spacetime is a
+well-known nonperturbative feature of quantum field theory [46–48]. Since the clearest version of this effect was
+worked out by Schwinger, it is named the Schwinger effect; reviews of this subject can be found in, e.g., [49, 50]. The
+phenomenon of particle creation has revealed the close analogy between quantum field theory in dS and a constant,
+uniform electric field in Minkowski spacetime [40, 41, 51, 52]. This analogy motivates us to explore the combined
+implications of the dS Gibbons-Hawking effect and the Schwinger effect. Aside from this analogy, there are arguments
+and evidences that require the existence of strong electromagnetic fields in the early universe [53, 54]. This logical
+possibility provides a further reason for studying quantum field theory in the presence of an electric field in dS.
+There has been numerous studies to investigate the Schwinger pair creation by a uniform electric field background
+with the constant energy density in a dS. Seminal contributions have been made by [55, 56].
+The Bogoliubov
+transformation method has been used to analyze the Schwinger effect for the charged scalar field in two- [55–60],
+four- [61], and arbitrary-dimensional [62] de Sitter spacetimes. The Bogoliubov transformation method has been
+used to investigate the Schwinger effect for the Dirac field in dS [63–67], in a manner similar to that used for the
+corresponding charged scalar field. In this method of analysis, which bring its own insights, the expectation value for
+the number of particles is related to the Bogoliubov coefficients which, in turn, can be determined by specifying the
+in-vacuum and out-vacuum states. A reasonable definition of the out-vacuum state requires that the parameters mass
+of the quantum field and the electric field strength to satisfy the semiclassical condition. For a created semiclassical
+particle, this condition implies that either or both the mass and the magnitude of its electric potential energy acroses
+one Hubble radius must be much greater than the energy scale determined by the spacetime curvature [60, 61]. An
+immediate consequence of the semiclassical condition is that the entire physical ranges of the parameters mass and
+electric field strength cannot be probed by this method. A useful alternative approach for studying the Schwinger
+effect in dS was introduced in Ref. [60]. In this approach the expectation values of the objects such as the electric
+current and the energy-momentum tensor of the quantum field in the in-vacuum state are investigated. The choose
+of the in-state as the vacuum is justified form various viewpoints; see [55]. Regarding the regularity properties, it is a
+
+3
+Hadamard and adiabatic state [56, 60]. The regularized expectation value of the electric current (also called induced
+current) in the in-vacuum state of the charged scalar field coupled to a constant, uniform electric field background
+has been evaluated in two- [60], three- [62], and four-dimensional [61, 68] de Sitter spacetimes. In those analyses the
+behavior of the induced current can be probed in the infrared regime for which the quantum field mass is smaller than
+the magnitude of the electric potential energy acroses one Hubble radius which in turn is smaller than the energy scale
+determined by the spacetime curvature. The authors found that, although the induced current has been computed
+using different regularization method, in the infrared regime the induced current increases with deceasing electric field
+strength [60–62, 68]. This peculiar behavior was first observed in [60] and is called the infrared hyperconductivity. In
+[69], the authors gave an alternative derivation of the infrared hyperconductivity phenomenon in dS4 using the uniform
+asymptotic approximation method. For a charged scalar field coupled to a uniform electromagnetic field background
+with a constant energy density electric field parallel to a conserved flux magnetic field in dS4, the Schwinger effect
+using Bogoliubov transformation method [70–72] and the induced current [70, 71] in the in-vacuum state have been
+investigated; and a period of infrared hyperconductivity was found. The in-vacuum induced current of the Dirac field
+coupled to a constant, uniform electric field background in two- [65] and four-dimensional [73] de Sitter spacetimes
+has been analyzed in a manner similar to that used for the corresponding scalar field. It was subsequently found that,
+in contrast to the case of corresponding scalar field, the infrared hyperconductivity phenomenon does not occur in the
+induced current of the Dirac field. Another peculiar behavior of the induced current occurs when the direction of the
+induced current is opposite the direction of applied electric field background. This phenomenon which is called the
+negative current, has been reported for both the scalar field [61, 68] with essentially small mass and the Dirac field
+[73] with any mass in the four-dimensional de Sitter spacetime. For the scalar field case the negative current occurs in
+a finite interval of the electric field strength, and for the Dirac field case it occurs below a certain value of the electric
+field strength which depends on the mass. In Refs. [74, 75], some notable attempts have been made to explain and
+remedy these peculiarities of the induced current. Beside the induced current, the regularized expectation value of the
+energy-momentum tensor (also called the induced energy-momentum tensor) in the in-vacuum state of the charged
+scalar [76–78] and Dirac [79] quantum fields coupled to a constant, uniform electric field background has been analyzed
+in different dimensions of dS. In two-dimensional dS, the induced energy-momentum tensor of the charged scalar field
+has been analyzed in [76], and subsequently the nonperturbative, regularized, one-loop effective Lagrangian of scalar
+QED has been constructed; the induced energy-momentum tensor of the corresponding Dirac field has been derived
+in [79], and then applied to study of the gravitational backreaction effect. The trace of the induced energy-momentum
+tensor of the charged, massive scalar field conformally coupled to the Ricci scalar curvature of three- [77] and four-
+dimensional [78] de Sitter spacetimes has been computed; and to examine the evolution of the Hubble constant, the
+induced energy-momentum tensor has been obtained from the trace, along with the assumption that the created pairs
+act like a perfect fluid with a vacuum equation of state. By applying the Bogoliubov transformation method, the
+energy-momentum tensor of the Schwinger scalar pairs created by a constant, uniform electric field background in an
+arbitrary dimensional dS has been computed under the two limiting conditions, i.e., the heavy scalar field [62] and
+the strong electric field [80]. In both these cases, it was found that the Hubble constant decays as a consequence
+of the Schwinger pair creation. Before closing the present section, it is worthwhile to mention that the Schwinger
+effect has been studied in FLRW spacetimes [81] and in the framework of cosmological models [69, 82–92]. The role
+of strong electromagnetic fields in astrophysics and cosmology was reviewed in [93]. The objectives of this article
+are to investigate the induced energy-momentum tensor of the charged scalar field coupled to a uniform electric field
+background in dS4, and to analyze its nonconservation equation. The importance and originality of this study are that
+it calculates the induced energy-momentum tensor and explores a relation between the induced energy-momentum
+tensor and the induced current which leads to new insights into the regularization and behavior of the induced current.
+The remaining part of the article proceeds as follows: In Sec. II, we define and construct the basic elements of
+the formalism that will be necessary in our subsequent discussions. We then carry out explicit computation of the
+induced energy-momentum tensor in Sec. III. The properties of the induced energy-momentum tensor are explored
+in detail in Sec. IV. In Sec. V, we analyze the nonconservation equation of the induced energy-momentum tensor,
+this investigation yields a relation between the induced energy-momentum tensor and the induced current. Then the
+properties of the resulting induced current are discussed in detail. In Sec. VI, we present the findings of the research.
+In the Appendix we present further supplementary data associated with the calculation of the expectation values of
+the components of the energy-momentum tensor.
+II.
+BASIC DEFINITIONS AND CONSTRUCTIONS
+In this section we shall define and construct the basic elements of the formalism that will be necessary in our
+subsequent discussions.
+
+4
+A.
+Specification of the model
+We have already mentioned that we consider a massive complex scalar field ϕ(x), coupled to the electromagnetic
+vector potential Aµ(x), which describes a uniform electric field background with a constant energy density in the
+conformal Poincar´e patch of dS4. To represent this region of dS4 which is conformally related to a region of Minkowski
+spacetime, we choose the coordinates xµ = (τ, x) that the ranges of the conformal time τ, and the comoving spatial
+coordinates x are given by
+τ ∈
+�
+− ∞, 0
+�
+,
+x ∈ R3.
+(1)
+In terms of these coordinates the metric of the spacetime takes the form
+gµνdxµdxν = Ω2(τ)
+�
+dτ 2 − dx · dx
+�
+,
+(2)
+with the conformal scale factor
+Ω(τ) = − 1
+Hτ ,
+(3)
+where H is the Hubble constant. The nonzero Christoffel symbols for the metric (2) are given by
+Γ0
+00 =
+˙Ω
+Ω,
+Γ0
+ij =
+˙Ω
+Ωδij,
+Γi
+0j =
+˙Ω
+Ωδi
+j,
+(4)
+where the roman indices i, j denote only three spatial components and run from 1 to 3. We use the overdot to denote
+differentiation with respect to the conformal time τ. From Eq. (4) the Ricci tensor and hence the Ricci scalar can be
+calculated
+Rµν = 3H2gµν,
+R = 12H2.
+(5)
+We put a uniform electric field background with a constant energy density on the Poincar´e patch represented in
+Eq. (2). Without loss of generality, we choose our coordinates so that this electric field to point in the x1 direction.
+Thus the nonzero components of the electromagnetic field tensor are
+F01 = −F10 = Ω2(τ)E,
+(6)
+where E is a constant. It is convenient to express this electromagnetic field tensor in terms of a vector potential in
+the Coulomb gauge [60–62] as
+Aµ(τ) = − E
+H2τ δ1
+µ.
+(7)
+The compleat action Stot of this theory can be represented as sum of a pure gravitational piece, an electromagnetic
+piece, and a scalar field piece
+Stot = Sgr + Sem + S.
+(8)
+Here Sgr is the Einstein-Hilbert action that only includes the gravitation piece of the compleat action, and is given by
+Sgr =
+1
+16πG
+�
+d4x√−g
+�
+R − 2Λc
+�
+,
+(9)
+where G is Newton’s gravitational constant, g is the determinant of the metric, R is the Ricci scalar of the spacetime,
+and Λc is the cosmological constant. The electromagnetic piece of the compleat action is expressed in terms of the
+electromagnetic field tensor as
+Sem = −1
+4
+�
+d4x√−gFµνF µν.
+(10)
+The dynamics of the complex scalar field ϕ(x) of mass m coupled to the electromagnetic vector potential (7) with
+coupling e is governed by the last piece of the compleat action which can be written as
+S =
+�
+d4x√−g
+�
+gµν�
+∂µ + ieAµ
+�
+ϕ
+�
+∂ν − ieAν
+�
+ϕ∗ − (m2 + ξR)ϕϕ∗�
+,
+(11)
+
+5
+where ξ is a dimensionless nonminimal coupling constant which describes the strength of the coupling ϕ to the Ricci
+scalar (5). The Euler-Lagrange equations of motion give the Klein-Gordon equation for the scalar field
+1
+√−g ∂µ
+�√−ggµν∂νϕ
+�
++ 2iegµνAµ∂νϕ − e2gµνAµAνϕ + (m2 + ξR)ϕ = 0,
+(12)
+and the Maxwell equation for the electromagnetic field
+∇νF νµ = jµ,
+(13)
+where ∇ denotes the covariant derivative operator, and jµ is the electric current of the scalar field caused by the
+electric field background (6) which is defined by
+jµ(x) = iegµν��
+∂νϕ + ieAνϕ
+�
+ϕ∗ − ϕ
+�
+∂νϕ∗ − ieAνϕ∗��
+.
+(14)
+It is straightforward to verify that ∇µjµ = 0, i.e., the electric current is conserved.
+B.
+Preliminary definition of the energy-momentum tensor
+The classical Einstein equation can be derived from the compleat action (8) by demanding the invariance of Stot
+under infinitesimal variation of the metric δgµν, or equivalently, infinitesimal variation of the inverse metric δgµν.
+This condition requires that
+2
+√−g
+δStot
+δgµν =
+2
+√−g
+� δSgr
+δgµν + δSem
+δgµν + δS
+δgµν
+�
+= 0.
+(15)
+The Variation of expression (9) with respect to δgµν leads to
+2
+√−g
+δSgr
+δgµν =
+1
+8πG
+�
+Rµν − 1
+2Rgµν + Λcgµν
+�
+.
+(16)
+The variations of the electromagnetic action Sem, and the scalar field action S, with respect to δgµν define the
+energy-momentum tensor of the electromagnetic field T (em)
+µν
+, and the energy-momentum tensor of the scalar field Tµν,
+respectively, as
+2
+√−g
+δSem
+δgµν = T (em)
+µν
+,
+(17)
+2
+√−g
+δS
+δgµν = Tµν.
+(18)
+Plugging expressions (10) and (11) into Eqs. (17) and (18), respectively, and following the standard calculus of
+variations procedure yields the energy-momentum tensor of the electromagnetic field
+T (em)
+µν
+= 1
+4gµνFρσF ρσ + gρσFµρFσν,
+(19)
+and a preliminary expression for the energy-momentum tensor of the scalar field
+Tµν =
+��
+4ξ − 1
+�
+gρσ�
+∂ρ + ieAρ
+�
+ϕ
+�
+∂σ − ieAσ
+�
+ϕ∗ +
+�
+1 − 4ξ
+�
+m2ϕϕ∗ +
+�1
+2 − 4ξ
+�
+ξRϕϕ∗�
+gµν
++
+�
+1 − 2ξ
+��
+∂µϕ∂νϕ∗ + ∂νϕ∂µϕ∗�
++ ieAµ
+�
+ϕ∂νϕ∗ − ∂νϕϕ∗�
++ ieAν
+�
+ϕ∂µϕ∗ − ∂µϕϕ∗�
++ 2e2AµAνϕϕ∗ + 2ξΓρ
+µν
+�
+ϕ∂ρϕ∗ + ∂ρϕϕ∗�
+− 2ξ
+�
+∂µ∂νϕϕ∗ + ϕ∂µ∂νϕ∗�
+.
+(20)
+Plugging three expressions (16)-(18) into Eq. (15) gives the Einstein equation
+Rµν − 1
+2Rgµν + Λcgµν = −8πG
+�
+T (em)
+µν
++ Tµν
+�
+.
+(21)
+
+6
+An important property of the Einstein equation is that both sides of Eq. (21) have identically vanishing covariant
+divergences, which implies
+∇µT µν = −∇µT (em)µν.
+(22)
+Using the Klein-Gordon Eq. (12), it is straightforward to show that the covariant divergence of expression (20) is
+given by
+∇µT µν = −jµF µν,
+(23)
+Apparently, the energy-momentum tensor of the scalar field is not covariantly conserved in the presence of the
+electromagnetic field, as the consequence of the electromagnetic interactions. We show that the nonconservation of
+Tµν is compatible with the nonconservation of T (em)
+µν
+so that the relation (22) is satisfied, and hence the total energy
+momentum tensor is covariantly conserved. By taking the covariant divergence of the expression (19) and using the
+Maxwell equation (13), it is seen that
+∇µT (em)µν = jµF µν.
+(24)
+As a result of Eqs. (23) and (24), it is evident that the relation (22) is satisfied. Hence, the total energy-momentum
+tensor Tµν + T (em)
+µν
+is manifestly conserved.
+It will be important to state that we will treat the classical gravitational field (2) and the classical electromagnetic
+field (7) as fixed field configurations which they are unaffected by the dynamics of the quantum complex scalar field
+ϕ(x) in response to these backgrounds. In fact, the Einstein equation (21) describes the backreaction effects on the
+gravitational field, and the Maxwell equation (13) and Eq. (24) describe the backreaction effects on the electromagnetic
+field. Indeed, due to the conceptual importance of Eq. (23) in our subsequent discussions, we have presented Eqs. (13),
+(21), and (24) to provide a fairly detailed discussion of its derivation. It is then clear that we do not discuss these
+equations further in this article.
+C.
+Quantizing the complex scalar field
+Quantizing the complex scalar field ϕ(x), in the classical gravitational (2) and electromagnetic (7) field backgrounds
+is completely straightforward and follows exactly the same route as that of a complex scalar field in Minkowski
+spacetime. We will solve the Kline-Gordon Eq. (12), and then adopting canonical quantization method, we can define
+the in-vacuum state to obtain the expectation value of the energy-momentum tensor operator.
+Taking account of the fact that the gravitational (2) and electromagnetic (7) backgrounds are invariant under
+spatial translations, it is convenient to write the mode solution of Eq. (12) as
+Uk(x) = Ω−1(τ)eik·xf(τ),
+(25)
+where k is the comoving momentum. Plugging Eqs. (2), (7), and (25) into Eq. (12), we can put the differential
+equation in a standard form by the change of variable z = 2ikτ so that k = |k|. Then it becomes
+d2f
+dz2 +
+�
+− 1
+4 + κ
+z + 1/4 − γ2
+z2
+�
+f(z) = 0,
+(26)
+where the dimensionless parameters are defined through
+µ = m
+H ,
+λ = − eE
+H2 ,
+¯ξ = ξ − 1
+6,
+r = kx
+k ,
+κ = −iλr,
+γ =
+�
+1
+4 − λ2 − µ2 − 12¯ξ.
+(27)
+Two values of ξ are of particular interest that are the minimally coupled case ξ = 0, and conformally coupled
+case ξ = 1/6 which implies ¯ξ = 0. The variable kx which appears in the definition of the parameter r, denotes
+the component of the comoving momentum k along the electric field background. Equation (26) is the Whittaker
+equation, and the solutions are called Whittaker functions; see, e.g., [94]. We define the in-vacuum state |in⟩ so
+that in the remote past (τ → −∞) where the spacetime is asymptotically Minkowskian, an inertial observer there
+would identify this state with a physical vacuum. This vacuum state may be represented by a mode solution of the
+Whittaker Eq. (26) which behaves like a mode function in Minkowski spacetime in the limit of τ → −∞. Hence, the
+
+7
+solution of Eq. (26) with the desired asymptotic form in the limit of |z| → ∞ which can be represented in terms of a
+Mellin-Barnes integral [94] is
+Wκ,γ(z) =
+e− z
+2
+Γ
+� 1
+2 + γ − κ
+�
+Γ
+� 1
+2 − γ − κ
+�
+� +i∞
+−i∞
+ds
+2πiΓ
+�1
+2 + γ + s
+�
+Γ
+�1
+2 − γ + s
+�
+Γ
+�
+− κ − s
+�
+z−s,
+(28)
+with the condition that the phase of the variable z and the values of the parameters γ and κ must satisfy the following
+inequalities
+��ph(z)
+�� < 3π
+2 ,
+1
+2 ± γ − κ ̸= 0, −1, −2, · · · .
+(29)
+In expression (28), the factors denoted by Γ are the gamma functions. The contour of the Mellin-Barnes integral (28)
+consists of a straight vertical line from minus infinity to infinity, parallel to imaginary axis in the complex plane, and
+of a semicircle at infinity with indentations if necessary to avoid poles of the integrand in a way that separates the
+poles of Γ
+� 1
+2 + γ + s
+�
+and Γ
+� 1
+2 − γ + s
+�
+from the poles of Γ
+�
+− κ − s
+�
+. The normalized mode functions which behave
+like the positive frequency Minkowski mode functions in the remote past are given by [61, 62],
+Uk(x) =
+1
+√
+2k
+e
+iπκ
+2 Ω−1(τ)eik·xWκ,γ
+�
+2ikτ
+�
+.
+(30)
+Besides, the normalized mode functions which behave like the negative frequency Minkowski mode functions in the
+remote past are found to be [61, 62],
+Vk(x) =
+1
+√
+2k
+e− iπκ
+2 Ω−1(τ)e−ik·xWκ,γ
+�
+− 2ikτ
+�
+.
+(31)
+We have conventionally normalized the mode functions (30) and (31) such that their Wronskian to be
+Uk ˙U ∗
+k − U ∗
+k ˙Uk = V ∗
+k ˙Vk − Vk ˙V ∗
+k = iΩ−2(τ).
+(32)
+These mode functions will be orthonormal with respect to the conserved scalar product integrated over the constant
+τ hypersurface [62]. The usual canonical quantization will proceed by introducing the creation a†
+k, and annihilation
+ak operators for each of mode functions Uk, and similarly the creation b†
+k, and annihilation bk operators for each of
+mode functions Vk. The creation and annihilation operators satisfy the commutation rules
+�
+ak, a†
+k′
+�
+=
+�
+bk, b†
+k′
+�
+= (2π)3δ
+�
+k − k′�
+,
+(33)
+with all other commutators vanishing. Then, the complex scalar field operator can be expanded in terms of the
+creation and annihilation operators in the standard manner as
+ϕ(x) =
+�
+d3k
+(2π)3
+�
+akUk(x) + b†
+kVk(x)
+�
+.
+(34)
+The in-vacuum state |in⟩ is characterized by the fact that it is annihilated by each of ak and bk operators
+ak
+��in
+�
+= bk
+��in
+�
+= 0,
+∀k.
+(35)
+III.
+CONSTRUCTION OF THE INDUCED ENERGY-MOMENTUM TENSOR
+Having built a foundation for our discussion and presented the definition for the energy-momentum tensor of the
+scalar field in Eq. (20), we will proceed to present the regularized in-vacuum expectation value of the energy-momentum
+tensor of the scalar field.
+A.
+Unregularized expectation values in the in-vacuum state
+Integral representations for the in-vacuum expectation values of the components of the energy-momentum tensor
+can be obtained by substituting the mode expansion (34) for the quantum scalar field ϕ(x) into the definition (20),
+
+8
+and then calculating the expectation values in the in-vacuum state using relations (33) and (35). By using Eq. (12)
+and some algebra, we have the following integral expressions in terms of the positive frequency mode function (30)
+for the expectation values of the components. The integral expression of the timelike component is given by
+�
+in
+��T00
+��in
+�
+=
+�
+d3k
+(2π)3
+�
+˙Uk ˙U ∗
+k − 6ξτ −1�
+Uk ˙U ∗
+k + ˙UkU ∗
+k
+�
++ τ −2�
+k2τ 2 + 2λrkτ + λ2 + µ2 + 6ξ
+�
+UkU ∗
+k
+�
+.
+(36)
+For the diagonal spacelike components we get
+�
+in
+��T11
+��in
+�
+=
+�
+d3k
+(2π)3
+��
+1 − 4ξ
+� ˙Uk ˙U ∗
+k − 2ξτ −1�
+Uk ˙U ∗
+k + ˙UkU ∗
+k
+�
++ τ −2��
+4ξ − 1 + 2r2�
+k2τ 2 + 2
+�
+4ξ + 1
+�
+λrkτ
++
+�
+4ξ + 1
+�
+λ2 +
+�
+4ξ − 1
+�
+µ2 + 6ξ
+�
+8ξ − 1
+��
+UkU ∗
+k
+�
+,
+(37)
+and
+�
+in
+��T22
+��in
+�
+=
+�
+in
+��T33
+��in
+�
+=
+�
+d3k
+(2π)3
+��
+1 − 4ξ
+� ˙Uk ˙U ∗
+k − 2ξτ −1�
+Uk ˙U ∗
+k + ˙UkU ∗
+k
+�
++ τ−2��
+4ξ − 1
+�
+k2τ 2
++ 2k2
+zτ 2 + 2
+�
+4ξ − 1
+�
+λrkτ +
+�
+4ξ − 1
+�
+λ2 +
+�
+4ξ − 1
+�
+µ2 + 6ξ
+�
+8ξ − 1
+��
+UkU ∗
+k
+�
+.
+(38)
+The only nonvanishing in-vacuum expectation values of the off-diagonal components can be expressed as
+�
+in
+��T01
+��in
+�
+=
+�
+in
+��T10
+��in
+�
+= iτ −1
+�
+d3k
+(2π)3
+�
+rkτ + λ
+��
+Uk ˙U ∗
+k − ˙UkU ∗
+k
+�
+.
+(39)
+Using the asymptotic expansion of the Whittaker function (28) for large values of its argument [94], it can be shown
+that the mode function (30) is proportional to k− 1
+2 −iλr in the limit of k → ∞. Then, inspection of integrals in
+Eqs. (36)-(39) shows that these expressions are ultraviolet divergent. Thus, we first regulate them by cutting them
+of at a large momentum K. Further description of the calculation of the expressions (36)-(38) is available in the
+Appendix. We find the final expression for the unregularized in-vacuum expectation value of the timelike component
+�
+in
+��T00
+��in
+�
+= Ω2(τ) H4
+8π2
+�
+Λ4 +
+�
+µ2 − 6¯ξ + 2λ2
+3
+�
+Λ2 −
+�µ4
+2 + 6¯ξµ2 − λ2
+6
+�
+log
+�
+2Λ
+�
++ 54¯ξ2 − 2λ4
+15 + µ2
+4 + 6¯ξµ2 + µ4
+8
+− 19λ2
+72
+− 2λ2¯ξ − λ2µ2
+2
++
+γ
+24π
+�45
+π2 − 6 − 96¯ξ + 4λ2 − 26µ2
+�cosh
+�
+2πλ
+�
+sin
+�
+2πγ
+� −
+γ
+48π2λ
+� 45
+π2 − 6 − 96¯ξ + 64λ2 − 26µ2
+�
+× sinh
+�
+2πλ
+�
+sin
+�
+2πγ
+� + i csc
+�
+2πγ
+� � +1
+−1
+C0r
+��
+e2πλr + e2iπγ�
+ψ
+�1
+2 + γ + iλr
+�
+−
+�
+e2πλr + e−2iπγ�
+ψ
+�1
+2 − γ + iλr
+��
+dr
++ iπ
+12
+�
+3µ4 + 36¯ξµ2 − λ2��
+,
+(40)
+where the coefficient C0r is given by
+C0r = −5
+8λ4r4 + 1
+8
+�
+6µ2 + 6λ2 + 36¯ξ + 1
+�
+λ2r2 − 1
+8
+�
+µ2 + λ2��
+µ2 + λ2 + 12¯ξ
+�
+.
+(41)
+The final results of our evaluation of the unregularized in-vacuum expectation values of the diagonal spacelike com-
+ponents are
+�
+in
+��T11
+��in
+�
+= Ω2(τ) H4
+8π2
+�Λ4
+3 +
+�
+2¯ξ − µ2
+3 + 14λ2
+15
+�
+Λ2 +
+�µ4
+2 + 6¯ξµ2 − λ2
+6
+�
+log
+�
+2Λ
+�
+− 54¯ξ2 − 26λ4
+105 − 7µ4
+24 − 8¯ξµ2
+− 18¯ξ
+5 λ2 − µ2
+4 + 19λ2
+72
++ λ2µ2
+30
+−
+γ
+24πλ2
+�15
+π2
+�
+105π−2 − 15 − 132¯ξ + 35λ2 − 11µ2�
+− 66λ2 − 336¯ξλ2 − 4λ4
+− 46λ2µ2
+�cosh
+�
+2πλ
+�
+sin
+�
+2πγ
+� +
+γ
+48π2λ3
+�15
+π2
+�
+105π−2 − 15 − 132¯ξ + 175λ2 − 11µ2�
+− 366λ2 − 2976¯ξλ2 + 136λ4
+− 266λ2µ2
+�sinh
+�
+2πλ
+�
+sin
+�
+2πγ
+� + i csc
+�
+2πγ
+� � +1
+−1
+C1r
+��
+e2πλr + e2iπγ�
+ψ
+�1
+2 + γ + iλr
+�
+−
+�
+e2πλr + e−2iπγ�
+ψ
+�1
+2 − γ + iλr
+��
+dr − iπ
+12
+�
+3µ4 + 36¯ξµ2 − λ2��
+,
+(42)
+
+9
+where the coefficient C1r is given by
+C1r = 35
+8 λ4r6 − 1
+8
+�
+70λ4 + 30λ2µ2 + 360¯ξλ2 + 25λ2�
+r4 + 1
+8
+�
+39λ4 + 3µ4 + 30λ2µ2 + 396¯ξλ2 + 72¯ξµ2 + 20λ2
++ 6µ2 + 432¯ξ2 + 72¯ξ
+�
+r2 − 1
+8
+�
+4λ4 + 4λ2µ2 + 60¯ξλ2 + 12¯ξµ2 + 2λ2 + 2µ2 + 144¯ξ2 + 24¯ξ
+�
+,
+(43)
+we also have
+�
+in
+��T22
+��in
+�
+=
+�
+in
+��T33
+��in
+�
+= Ω2(τ) H4
+8π2
+�Λ4
+3 +
+�
+2¯ξ − µ2
+3 − 2λ2
+15
+�
+Λ2 +
+�µ4
+2 + 6¯ξµ2 + λ2
+6
+�
+log
+�
+2Λ
+�
+− 54¯ξ2 + 2λ4
+35
+− 7µ4
+24 − 8¯ξµ2 + 4¯ξ
+5 λ2 − µ2
+4 − 19λ2
+72
++ 2
+5λ2µ2 +
+γ
+16πλ2
+� 5
+π2
+�
+105π−2 − 15 − 132¯ξ + 38λ2 − 11µ2�
+− 24λ2
+− 144¯ξλ2
+�cosh
+�
+2πλ
+�
+sin
+�
+2πγ
+� −
+γ
+32π2λ3
+� 5
+π2
+�
+105π−2 − 15 − 132¯ξ + 178λ2 − 11µ2�
+− 124λ2 − 1024¯ξλ2 + 200λ4
+3
+− 220
+3 λ2µ2
+�sinh
+�
+2πλ
+�
+sin
+�
+2πγ
+� + i csc
+�
+2πγ
+� � +1
+−1
+C2r
+��
+e2πλr + e2iπγ�
+ψ
+�1
+2 + γ + iλr
+�
+−
+�
+e2πλr + e−2iπγ�
+ψ
+�1
+2 − γ + iλr
+��
+dr − iπ
+12
+�
+3µ4 + 36¯ξµ2 + λ2��
+,
+(44)
+where the coefficient C2r is given by
+C2r = −C1r
+2
+− 5
+16λ4r4 + 1
+16
+�
+6λ2 − 6µ2 + 36¯ξ + 1
+�
+λ2r2 + 1
+16
+�
+3µ4 + 2λ2µ2 + 12¯ξλ2 + 36¯ξµ2 − λ2�
+.
+(45)
+Plugging the resulting expression given in Eq. (32) for the Wronskian of the mode functions Uk into Eq. (39) and
+integrating over momentum phase space, we obtain the unregularized expression for the only nonvanishing off-diagonal
+components
+�
+in
+��T01
+��in
+�
+=
+�
+in
+��T10
+��in
+�
+= Ω2(τ)H4λ
+6π2 Λ3.
+(46)
+The expressions (40), (42), (44), and (46) clearly have ultraviolet divergences when Λ → ∞. This was expected,
+because the expectation values in the Hadamard in-vacuum state suffer from the same ultraviolet divergence properties
+as Minkowski spacetime.
+B.
+Construction of the counterterms and adiabatic subtractions
+To render the in-vacuum expectation values given by Eqs. (40), (42), (44), and (46) finite, we provide a set of
+needed adiabatic counterterms. The adiabatic regularization method consists of subtracting the appropriate adiabatic
+counterterms from the corresponding unregularized expressions. To adjust the set of appropriate counterterms, we will
+treat the conformal scale factor Ω(τ), and the electromagnetic vector potential Aµ(τ) as quantities of zero adiabatic
+order. As pointed out by Wald [18], in four dimensions, to obtain a renormalized energy-momentum tensor which
+is consistent with the Wald axioms, the subtraction counterterms should be expanded up to fourth adiabatic order;
+see also [1, 2]. Thus, to construct the expectation value of the energy-momentum tensor with the desired physical
+properties, we expand the subtraction counterterms up to fourth adiabatic order.
+To construct the appropriate
+counterterms, we need an expansion for the mode functions up to fourth adiabatic order. We use the definition
+z = 2ikτ to turn the Klein-Gordon Eq. (26) into the convenient form
+d2FA
+dτ 2
++
+�
+ω2
+0(τ) + ∆(τ)
+�
+FA = 0,
+(47)
+where FA is the positive frequency adiabatic solution and the conformal time dependent frequencies read
+ω0(τ) =
+�
+k2 + 2eA1kr + e2A2
+1 + m2Ω2� 1
+2 ,
+(48)
+∆(τ) = 12¯ξ
+� ˙Ω
+Ω
+�2
+.
+(49)
+
+10
+It is then obvious that ω0 is of zero adiabatic order and ∆ is of second adiabatic order. The adiabatic form of FA is
+the Wentzel-Kramers-Brillouin (WKB) solution of Eq. (47). This WKB solution can be written as
+FA(τ) =
+1
+�
+2W(τ)
+exp
+�
+− i
+� τ
+W(τ ′)dτ ′�
+,
+(50)
+where W satisfies the exact equation
+W2 = ω2
+0 + ∆ −
+¨
+W
+2W + 3 ˙W2
+4W2 .
+(51)
+To put the solution into the desired form, it is convenient to write W as
+W = W(0) + W(2) + W(4),
+(52)
+where the superscript numbers in parentheses denote the adiabatic order approximation to W.
+Substituting the
+expansion (52) into Eq. (51), we find the zero adiabatic order approximation
+W(0) = ω0.
+(53)
+The next iteration gives the second adiabatic order approximation
+W(2) = ∆
+2ω0
+− ¨ω0
+4ω2
+0
++ 3 ˙ω2
+0
+8ω3
+0
+.
+(54)
+Repeated iteration yields the fourth adiabatic order approximation
+W(4) = −W(2)2
+2ω0
+−
+¨
+W(2)
+4ω2
+0
++ ¨ω0W(2)
+4ω3
+0
++ 3 ˙ω0 ˙W(2)
+4ω3
+0
+− 3 ˙ω2
+0W(2)
+4ω4
+0
+.
+(55)
+It should be remarked that all terms in the adiabatic expansion of W of odd adiabatic order vanish. Assembling the
+pieces given in Eqs. (25), (50), and (52)-(55), we find the positive frequency adiabatic solution to fourth adiabatic
+order approximation
+U [4]
+k (x) = Ω−1(τ)
+1
+√2ω0
+�
+1 − W(2)
+2ω0
+− W(4)
+2ω0
++ W(2)2
+2ω2
+0
+�
+exp
+�
+ik · x − i
+� τ
+W(τ ′)dτ ′�
+.
+(56)
+We use the superscript symbol [4] to indicate that the cross terms from the field expansion products that are of
+adiabatic order greater than 4 are to be discarded. To obtain expansions of the required counterterms to adiabatic
+order four, it is only necessary to calculate (36)-(39) with Uk replaced by U [4]
+k . Doing this, we find the counterterm
+to adiabatic order four for the timelike component
+T [4]
+00 = Ω2(τ) H4
+8π2
+�
+Λ4 +
+�
+µ2 − 6¯ξ + 2λ2
+3
+�
+Λ2 −
+�µ4
+2 + 6¯ξµ2 − λ2
+6
+�
+log
+�2Λ
+µ
+�
++ 72¯ξ2 − λ4
+15 + µ2
+6 + 9¯ξµ2 + µ4
+8
+− 2λ2
+9
+− 2λ2 ¯ξ − λ2µ2
+3
+− 1
+60 +
+7λ4
+240µ4 +
+λ2
+30µ2 − 3¯ξλ2
+2µ2
+�
+.
+(57)
+We find the counterterms to adiabatic order four for the diagonal spacelike components
+T [4]
+11 = Ω2(τ) H4
+8π2
+�Λ4
+3 +
+�
+2¯ξ − µ2
+3 + 14λ2
+15
+�
+Λ2 +
+�µ4
+2 + 6¯ξµ2 − λ2
+6
+�
+log
+�2Λ
+µ
+�
+− 72¯ξ2 − 13λ4
+105 − 7µ4
+24 − 11¯ξµ2
+− 14¯ξ
+5 λ2 − µ2
+6 + 7λ2
+18 − λ2µ2
+15
++ 1
+60 −
+7λ4
+240µ4 +
+λ2
+18µ2 + 3¯ξλ2
+2µ2
+�
+,
+(58)
+and
+T [4]
+22 = T [4]
+33 = Ω2(τ) H4
+8π2
+�Λ4
+3 +
+�
+2¯ξ − µ2
+3 − 2λ2
+15
+�
+Λ2 +
+�µ4
+2 + 6¯ξµ2 + λ2
+6
+�
+log
+�2Λ
+µ
+�
+− 72¯ξ2 + λ4
+35
+− 7µ4
+24 − 11¯ξµ2 + 2¯ξ
+5 λ2 − µ2
+6 − 7λ2
+18 + λ2µ2
+5
++ 1
+60 +
+7λ4
+720µ4 −
+λ2
+45µ2 −
+¯ξλ2
+2µ2
+�
+.
+(59)
+
+11
+We obtain the counterterms to adiabatic order four for the only nonvanishing off-diagonal components
+T [4]
+01 = T [4]
+10 = Ω2(τ)H4λ
+6π2 Λ3.
+(60)
+Then, the adiabatic regularization procedure is carried out by subtracting the counterterms (57)-(60) from the corre-
+sponding unregularized in-vacuum expectation values (40), (42), (44), and (46). Thus we obtain our final expression
+for the timelike component of the regularized energy-momentum tensor
+T00 =
+�
+in
+��T00
+��in
+�
+− T [4]
+00
+= Ω2(τ) H4
+8π2
+� 1
+60 −
+7λ4
+240µ4 −
+λ2
+30µ2 + 3¯ξλ2
+2µ2 − 18¯ξ2 − λ4
+15 + µ2
+12 − 3¯ξµ2 − λ2
+24 − λ2µ2
+6
++
+γ
+24π
+�45
+π2 − 6 − 96¯ξ + 4λ2 − 26µ2
+�cosh
+�
+2πλ
+�
+sin
+�
+2πγ
+� −
+γ
+48π2λ
+�45
+π2 − 6 − 96¯ξ + 64λ2 − 26µ2
+�sinh
+�
+2πλ
+�
+sin
+�
+2πγ
+�
++ i csc
+�
+2πγ
+� � +1
+−1
+C0r
+��
+e2πλr + e2iπγ�
+ψ
+�1
+2 + γ + iλr
+�
+−
+�
+e2πλr + e−2iπγ�
+ψ
+�1
+2 − γ + iλr
+��
+dr
+−
+�µ4
+2 + 6¯ξµ2 − λ2
+6
+�
+log
+�
+µ
+�
++ iπ
+12
+�
+3µ4 + 36¯ξµ2 − λ2��
+.
+(61)
+We obtain our final expressions for the diagonal spacelike components of the regularized energy-momentum tensor
+T11 =
+�
+in
+��T11
+��in
+�
+− T [4]
+11
+= Ω2(τ) H4
+8π2
+�
+− 1
+60 +
+7λ4
+240µ4 −
+λ2
+18µ2 − 3¯ξλ2
+2µ2 + 18¯ξ2 − 13λ4
+105 + 3¯ξµ2 − 4¯ξλ2
+5
+− µ2
+12 − λ2
+8 + λ2µ2
+10
+−
+γ
+24πλ2
+�15
+π2
+�
+105π−2 − 15 − 132¯ξ + 35λ2 − 11µ2�
+− 66λ2 − 336¯ξλ2 − 4λ4 − 46λ2µ2
+�cosh
+�
+2πλ
+�
+sin
+�
+2πγ
+�
++
+γ
+48π2λ3
+� 15
+π2
+�
+105π−2 − 15 − 132¯ξ + 175λ2 − 11µ2�
+− 366λ2 − 2976¯ξλ2 + 136λ4 − 266λ2µ2
+�sinh
+�
+2πλ
+�
+sin
+�
+2πγ
+�
++ i csc
+�
+2πγ
+� � +1
+−1
+C1r
+��
+e2πλr + e2iπγ�
+ψ
+�1
+2 + γ + iλr
+�
+−
+�
+e2πλr + e−2iπγ�
+ψ
+�1
+2 − γ + iλr
+��
+dr
++
+�µ4
+2 + 6¯ξµ2 − λ2
+6
+�
+log
+�
+µ
+�
+− iπ
+12
+�
+3µ4 + 36¯ξµ2 − λ2��
+,
+(62)
+and
+T22 = T33 =
+�
+in
+��T33
+��in
+�
+− T [4]
+33
+= Ω2(τ) H4
+8π2
+�
+− 1
+60 −
+7λ4
+720µ4 +
+λ2
+45µ2 +
+¯ξλ2
+2µ2 + 18¯ξ2 + λ4
+35 + 3¯ξµ2 + 2¯ξλ2
+5
+− µ2
+12 + λ2
+8 + λ2µ2
+5
++
+γ
+16πλ2
+� 5
+π2
+�
+105π−2 − 15 − 132¯ξ + 38λ2 − 11µ2�
+− 24λ2 − 144¯ξλ2
+�cosh
+�
+2πλ
+�
+sin
+�
+2πγ
+�
+−
+γ
+32π2λ3
+� 5
+π2
+�
+105π−2 − 15 − 132¯ξ + 178λ2 − 11µ2�
+− 124λ2 − 1024¯ξλ2 + 200λ4
+3
+− 220
+3 λ2µ2
+�sinh
+�
+2πλ
+�
+sin
+�
+2πγ
+�
++ i csc
+�
+2πγ
+� � +1
+−1
+C2r
+��
+e2πλr + e2iπγ�
+ψ
+�1
+2 + γ + iλr
+�
+−
+�
+e2πλr + e−2iπγ�
+ψ
+�1
+2 − γ + iλr
+��
+dr
++
+�µ4
+2 + 6¯ξµ2 + λ2
+6
+�
+log
+�
+µ
+�
+− iπ
+12
+�
+3µ4 + 36¯ξµ2 + λ2��
+.
+(63)
+We see that the nonvanishing unregularized expectation values of the off-diagonal components, given by Eq. (46), are
+exactly cancelled by their counterterms (60),
+T01 = T10 =
+�
+in
+��T10
+��in
+�
+− T [4]
+10 = 0.
+(64)
+Thus we have arrived at the desired expressions for the regularized in-vacuum expectation values of all the components
+of the energy-momentum tensor, also called the induced energy-momentum tensor.
+
+12
+0.01
+10
+104
+107
+10-8
+100
+1012
+1022
+1032
+1042
+�
+|T0
+0|/H4
+�=0.01
+�=0.1
+�=1
+�=10
+�=100
+ξ=0
+ξ=
+1
+6
+FIG. 1. The absolute value of T 0
+0 component of the induced energy-momentum tensor is plotted in unit of H4 as a function of
+the electric field parameter λ = −eE/H2. The graphs correspond to different values of the mass parameter µ = m/H, and the
+coupling constant ξ as indicated. Both axes have logarithmic scales.
+IV.
+PROBING THE INDUCED ENERGY-MOMENTUM TENSOR
+The nonvanishing components of the induced energy-momentum tensor are given by Eqs. (61)-(63). Observe that
+all the off-diagonal components of the induced energy-momentum tensor are zero, and the components T22 and T33 are
+equal as consequences of the underlying symmetries of the backgrounds (2) and (6). Furthermore, since the electric
+field background (6) is not invariant under full symmetries of de Sitter spacetime, indeed violates the time reversal
+symmetry [51], and causes an electric current along its direction, we observe that T00, T11, and T22 are not equal
+to one another. Thus we expect that in the limit of vanishing electric field background that the in-vacuum state
+possesses the full set of de Sitter invariances, the induced energy-momentum tensor takes the maximally invariant
+form under the transformations of de Sitter group, i.e., must be proportional to the de Sitter metric. By setting λ = 0
+in Eqs. (61)-(63), which corresponds to vanishing electric field background, the induced energy-momentum tensor
+reduced to the form
+Tµν = H4
+32π2
+� 1
+15 − 72¯ξ2 − 12¯ξµ2 − 2µ2
+3
++ µ2�
+12¯ξ + µ2��
+ψ
+�3
+2 + γ0
+�
++ ψ
+�3
+2 − γ0
+�
+− log
+�
+µ2���
+gµν,
+(65)
+where γ0 =
+�
+(1/4) − µ2 − 12¯ξ is obtained by setting λ = 0 in the definition of γ. The expression (65) accords with
+the result of computing the renormalized vacuum energy-momentum tensor of a real scalar field in dS4 obtained in
+Refs. [11, 38], except for the overall factor of 2 in Eq. (65). This factor 2 is consistent with the complex scalar field
+as being made of two real scalar fields with the number of degrees of freedom doubling up.
+A.
+Behavior of the induced energy-momentum tensor
+Figures 1-3 provide some useful insight into the general behavior of the induced energy-momentum tensor. The
+absolute values of the expressions (61)-(63) as functions of the electric field parameter λ, for various values of the scalar
+field mass parameter µ, and two values of the coupling constant ξ are shown on the graphs in Figs. 1-3, respectively.
+Note especially that the scales are logarithmic on both axes, hence on the graphs zero values of the expressions, where
+the signs of the plots change, are displayed as singularities. Therefore, these figures signal that the induced energy-
+momentum tensor is analytic and varies continuously with the parameters λ, µ, and ξ, this statement consistent with
+the requirements discussed in [95]. Figures 1-3 also illustrate the outstanding qualitative features of the induced
+energy-momentum tensor. For fixed values of µ and ξ, the absolute values of the nonvanishing components of induced
+energy-momentum tensor are increasing functions of λ, but by excluding a neighbourhood of the zero value points this
+
+13
+0.01
+10
+104
+107
+10-11
+0.1
+109
+1019
+1029
+1039
+�
+|T1
+1|/H4
+μ=0.01
+μ=0.1
+μ=1
+μ=10
+μ=100
+ξ=0
+ξ=
+1
+6
+FIG. 2. The absolute value of T 1
+1 component of the induced energy-momentum tensor is plotted in unit of H4 as a function of
+the electric field parameter λ = −eE/H2. The graphs correspond to different values of the mass parameter µ = m/H, and the
+coupling constant ξ as indicated. Both axes have logarithmic scales.
+behavior is assured. For fixed values of λ and ξ, the absolute values of the nonvanishing components are decreasing
+functions of µ. For fixed values of λ and µ, the nonvanishing components do not vary significantly with the parameter
+ξ in the range 0 ≤ ξ ≪ λ, µ. The qualitative behaviors shown in Figs. 1-3 can be given quantitative treatments by
+inspection of expressions (61)-(63) in the limiting regimes. We concentrate our attention on three regimes of interest:
+(1) The strong electric field regime with the criterion λ ≫ max(1, µ, ξ). (2) The heavy scalar field regime with the
+criterion µ ≫ max(1, λ, ξ). (3) The infrared regime with the criteria µ ≪ 1, λ ≪ 1, and ξ = 0.
+1.
+Strong electric field regime
+In the strong electric field regime λ ≫ max(1, µ, ξ), it is appropriate to find approximate behavior of the induced
+energy-momentum tensor in the limit λ → ∞. By expanding expressions (61)-(63) around λ = ∞ with µ and ξ fixed,
+we find the dominant terms in the components of the induced energy-momentum tensor
+T00 = −T11 = 3T22 = 3T33 = −Ω2 H4
+8π2
+� 7λ4
+240µ4
+�
+.
+(66)
+Thus, in this regime the absolute value of the nonvanishing components of induced energy-momentum tensor increase
+monotonically with increasing λ but decrease monotonically with increasing µ, as we see on the right end of any
+graphs in Figs. 1-3.
+2.
+Heavy scalar field regime
+In the heavy scalar field regime µ ≫ max(1, λ, ξ), it is appropriate to find approximate behavior of the induced
+energy-momentum tensor in the limit µ → ∞. By expanding expressions (61)-(63) around µ = ∞ with λ and ξ fixed,
+we find the dominant terms in the components of the induced energy-momentum tensor
+T00 = Ω2 H4
+8π2
+� a
+µ2 + b
+µ4 + c0λ2
+µ4
++ O(µ−6)
+�
+,
+T11 = −Ω2 H4
+8π2
+� a
+µ2 + b
+µ4 + c1λ2
+µ4
++ O(µ−6)
+�
+,
+T22 = T33 = −Ω2 H4
+8π2
+� a
+µ2 + b
+µ4 + c2λ2
+µ4
++ O(µ−6)
+�
+,
+(67)
+
+14
+0.01
+10
+104
+107
+10-9
+10
+1011
+1021
+1031
+1041
+λ
+|T2
+2|/H4
+μ=0.01
+μ=0.1
+μ=1
+μ=10
+μ=100
+ξ=0
+ξ=
+1
+6
+FIG. 3. The absolute value of T 2
+2 component of the induced energy-momentum tensor is plotted in unit of H4 as a function of
+the electric field parameter λ = −eE/H2. The graphs correspond to different values of the mass parameter µ = m/H, and the
+coupling constant ξ as indicated. Both axes have logarithmic scales. Recall that T 2
+2 = T 3
+3 .
+where the coefficients a, b, c0, c1 and c2 are given by
+a = − 2
+315 +
+¯ξ
+5 − 72¯ξ3,
+b = − 1
+210
+�
+1 − 32¯ξ + 504¯ξ2 − 90720¯ξ4�
+,
+c0 = − 1
+315 − 7¯ξ
+15 + 12¯ξ2,
+c1 = − 22
+315 + 3¯ξ
+5 + 12¯ξ2,
+c2 =
+2
+105 −
+¯ξ
+3.
+(68)
+The asymptotic forms in Eq. (67) reveal that the induced energy-momentum tensor falls off as H2/m2 in the limit
+(m/H) → ∞. Thus, the behavior of the induced energy-momentum tensor is inconsistent with the behavior of the
+semiclassical energy-momentum tensor [39, 62] that falls of as exp(−2πm/H) in the heavy scalar field regime where
+m ≫ H. A similar feature arises for the induced electric current of both scalar [61] and Dirac [73] fields in dS4.
+Studies [74, 75] have proposed explanations in physical terms for this feature of the induced electric current.
+3.
+Infrared regime
+To find approximate behavior of the induced energy-momentum tensor in the infrared regime, where µ ≪ 1, λ ≪ 1
+and ξ = 0, it is appropriate to make Taylor series expansions of the expressions (61)-(63) around µ = 0 and λ = 0,
+and set ξ = 0. We find that the dominant terms in the expansions of (61) and (63) are given by
+T00 = Ω2 H4
+8π2
+�61
+60 − 17λ2
+60µ2 −
+7λ4
+240µ4 + O
+�
+λ2, µ2��
+,
+(69)
+T22 = T33 = −Ω2 H4
+8π2
+�61
+60 + 11λ2
+180µ2 +
+7λ4
+720µ4 + O
+�
+λ2, µ2��
+,
+(70)
+which are valid for µ ≪ λ ≪ 1 as well as λ ≪ µ ≪ 1. The expansion of expression (62) for µ ≪ λ ≪ 1 takes the form
+T11 = Ω2 H4
+8π2
+�299
+60 + 7λ2
+36µ2 +
+7λ4
+240µ4 + O
+�
+λ2, µ2��
+,
+(71)
+while for λ ≪ µ ≪ 1 it is approximated by
+T11 = −Ω2 H4
+8π2
+�61
+60 − 223λ2
+36µ2 + 1433λ4
+240µ4 + O
+�
+λ2, µ2��
+.
+(72)
+
+15
+Observe that all the asymptotic expansions (69)-(71) diverge as m−4 in the exactly massless case, i.e., m = 0. We
+can understand the origin of these infrared-divergent terms by looking at the counterterms (57)-(59). These terms
+arise from the contribution of the zero modes in the massless case to the counterterms. And therefore signal that
+for the massless case the method of adiabatic regularization cannot be used because it leads to singularities in the
+counterterms, as pointed out in [19, 20, 61]. We emphasize that the result (72) is an approximation valid only for
+λ ≪ µ ≪ 1, and hence we cannot use it in the limit of zero mass for a fixed value of λ.
+B.
+Trace anomaly
+It will be reassuring to do a consistency check, to see that whether the induced energy momentum tensor yields the
+well-known predicted trace anomaly for a free, massless, conformally invariant scalar field in dS4. Putting the metric
+(2) and components (61)-(63) together, we obtain the trace of the induced energy-momentum tensor
+T = gµνTµν = H4
+8π2
+� 1
+15 −
+7λ4
+180µ4 −
+λ2
+45µ2 + 2¯ξλ2
+µ2
+− 72¯ξ2 + µ2
+3 − 12¯ξµ2 − λ2
+6 − 2λ2µ2
+3
+−
+3µ2γ
+2π2λ sin
+�
+2πγ
+�
+×
+�
+2πλ cosh
+�
+2πλ
+�
+− sinh
+�
+2πλ
+��
++
+iµ2
+2 sin
+�
+2πγ
+�
+� +1
+−1
+�
+3λ2r2 − λ2 − µ2 − 12¯ξ
+���
+e2πλr + e2iπγ�
+× ψ
+�1
+2 + γ + iλr
+�
+−
+�
+e2πλr + e−2iπγ�
+ψ
+�1
+2 − γ + iλr
+��
+dr − µ2�
+µ2 + 12¯ξ
+�
+log
+�
+µ2�
++ iπµ2�
+µ2 + 12¯ξ
+��
+. (73)
+We see that the trace anomaly of the free, massless, conformally coupled complex scalar field is given by
+lim
+λ→0 lim
+µ→0 lim
+ξ→ 1
+6
+T =
+H4
+120π2 =
+2
+2880π2
+�1
+3R2 − RµνRµν�
+,
+(74)
+in arriving at the second equality, we have expressed H4 in terms of a combination of the Ricci tensor and the scalar
+curvature of dS4 which are given by Eq. (5), to put the result into a familiar form. The trace anomaly (74) is in
+agreement with the result of computing the trace anomaly [96] of a free, massless, conformally coupled real scalar
+field in dS4, except for the overall factor of 2 in Eq. (74) as explained below Eq. (65).
+V.
+IMPLICATIONS FOR THE INDUCED CURRENT
+The generalization of the nonconservation equation (23) for the classical energy-momentum tensor of the scalar field
+to the induced energy-momentum tensor Tµν, and the induced current jµ of the scalar field has important implications
+for the induced current. Recall that the nonvanishing components of Tµν are given by Eqs. (61)-(63), and the nonzero
+components of Fµν are given by Eq. (6). The only nontrivial relation that arises from Eq. (23) is obtained by setting
+ν = 0, which leads to
+∂0T 00 + HΩ
+�
+5T 00 + T 11 + T 22 + T 33�
+= −Ω−2Ej1,
+(75)
+where we have used Eqs. (4) and (6). The relation (75), along with the relations
+∂0T 00 = −2HΩ(τ)T 00,
+T = gµνT µν,
+−Ω−2Ej1 = HΩ−1A.j,
+(76)
+suffice to show that the timelike component of the induced energy-momentum tensor can be written in terms of the
+trace T , and the effective electromagnetic potential energy A.j as
+T00 = 1
+4Ω2�
+T + A.j
+�
+.
+(77)
+The induced current jµ is defined as the regularized expectation value of the scalar field electric current operator (14)
+in the in-vacuum state specified in Eq. (35). It can be verified that the induced current jµ is conserved and whose
+only non-vanishing component is j1. Thus, the induced current flows along the electric field background direction.
+For convenience we write the induced current as
+jµ = Ω(τ)J[n]δ1
+µ,
+(78)
+
+16
+here we use the subscript [n] to indicate the adiabatic order n of the subtracted counterterms to obtain the induced
+current J[n]. Comparison of Eqs. (61), (73), and (77) allows us to read off the effective electromagnetic potential
+energy A.j, and then we will use the last relation in Eq. (76) and definition (78) to extract J[4]. This gives
+J[4] = eH3
+8π2
+�4¯ξλ
+µ2 −
+λ
+9µ2 − 7λ3
+90µ4 + λ
+3 log
+�
+µ2�
+− iπλ
+3
+− 4λ3
+15 +
+γ
+6πλ
+�45
+π2 − 6 − 96¯ξ + 4λ2 − 8µ2�cosh
+�
+2πλ
+�
+sin
+�
+2πγ
+�
+−
+γ
+12π2λ2
+�45
+π2 − 6 − 96¯ξ + 64λ2 − 8µ2�sinh
+�
+2πλ
+�
+sin
+�
+2πγ
+� −
+iλ
+2 sin
+�
+2πγ
+�
+� +1
+−1
+�
+5λ2r4 −
+�
+1 + 36¯ξ + 6λ2 + 3µ2�
+r2
++ λ2 + µ2 + 12¯ξ
+���
+e2πλr + e2iπγ�
+ψ
+�1
+2 + γ + iλr
+�
+−
+�
+e2πλr + e−2iπγ�
+ψ
+�1
+2 − γ + iλr
+��
+dr
+�
+,
+(79)
+where the subscript [4] indicates J[4] has been derived from the expressions which have been regularized by the
+counterterms expanded up to fourth adiabatic order. This should be clear from Eqs. (61), (77), and (78). To verify
+our result (79), we have evaluated directly the induced current by calculating the expectation value of the current
+operator (14) in the in-vacuum state and then subtracting the corresponding counterterms expanded up to fourth
+adiabatic order. The expression for J[4] that follows from this direct analysis reproduces exactly the expression given
+by Eq. (79).
+In Ref. [61], the induced current of the massive, minimally coupled ξ = 0, scalar field has been evaluated by
+calculating the expectation value of the current operator (14) in the in-vacuum state which is represented by the mode
+functions given in Eqs. (30) and (31) with ξ = 0 for this case. In order to regularize the expectation value, the method
+of adiabatic subtraction was employed. Those authors expanded the required counterterm up to second adiabatic
+order that it suffices to remove the divergences and the regularized expression resulting from use of it reduces to the
+expected results in the Minkowski spacetime limit. We refer to this prescription as minimal subtraction. Furthermore,
+they argued that including the contribution of fourth adiabatic order in the counterterm spoils the expected behavior
+in the Minkowski spacetime limit. Following these restrictions and arguments, it was subsequently found in Ref. [61]
+that the induced current is given in terms of J[2] as in Eq. (78) by
+J[2] = eH3
+8π2
+�λ
+3 log
+�
+µ2�
+− iπλ
+3
+− 4λ3
+15 +
+¯γ
+6πλ
+�45
+π2 + 10 + 4λ2 − 8µ2�cosh
+�
+2πλ
+�
+sin
+�
+2π¯γ
+� −
+¯γ
+12π2λ2
+�45
+π2 + 10 + 64λ2 − 8µ2�
+× sinh
+�
+2πλ
+�
+sin
+�
+2π¯γ
+� −
+iλ
+2 sin
+�
+2π¯γ
+�
+� +1
+−1
+�
+5λ2r4 +
+�
+5 − 6λ2 − 3µ2�
+r2 + λ2 + µ2 − 2
+���
+e2πλr + e2iπ¯γ�
+× ψ
+�1
+2 + ¯γ + iλr
+�
+−
+�
+e2πλr + e−2iπ¯γ�
+ψ
+�1
+2 − ¯γ + iλr
+��
+dr
+�
+,
+(80)
+where ¯γ =
+�
+9/4 − λ2 − µ2 is obtained by setting ξ = 0 in the definition of γ, and the subscript [2] indicates J[2] has
+been regularized by the counterterms expanded up to second adiabatic order. It is important to note that there are
+two differences between expressions (79) and (80). First, the coupling constant ξ is treated as arbitrary real value
+parameter in the expression (79), whereas the expression (80) has been computed for fixed value ξ = 0. Second,
+expression (79) contains contributions from the fourth adiabatic order expansion of the counterterm. With these two
+differences, (79) is a generalization of (80). Therefore, the difference J[4]
+�
+ξ = 0
+�
+− J[2] would give only the fourth
+adiabatic order contribution of the counterterm to J[4], that is
+J[4]
+�
+ξ = 0
+�
+− J[2] = −eH3
+8π2
+� 7λ
+9µ2 + 7λ3
+90µ4
+�
+.
+(81)
+In our subsequent discussion, we demonstrate that the expected behavior of J[4] in Minkowski spacetime is not spoiled
+by these fourth adiabatic order contributions. Furthermore, we show that these contributions alter the behavior of
+the J[4] when compared to J[2], especially in the two regimes of the infrared hyperconductivity and the strong electric
+field.
+A.
+Minkowski spacetime limit
+Here we explore the behavior of the result (79) in the Minkowski spacetime limit. Note that in the limit where the
+Hubble constant tends to zero, i.e., H → 0 we recover Minkowski spacetime. As was mentioned in the discussion of
+Eq. (81), the difference between the expressions (79) and (80) comes from the contribution of fourth adiabatic order
+
+17
+0.001
+1
+1000
+10-6
+104
+1014
+1024
+λ
+μ=10-3
+μ=10-2
+μ=10-1
+μ=1
+μ=5
+μ=5
+J[4]
+J[2]
+FIG. 4. Dependence of the absolute values of the normalized induced currents |J[4]|/eH3 (solid curves) and |J[2]|/eH3 (dashed
+curves) on the electric field parameter λ = −eE/H2, for the case of minimal coupling ξ = 0. The graphs correspond to different
+values of the scalar field mass parameter µ = m/H as indicated. Both axes have logarithmic scales.
+in the adiabatic expansion of the counterterm. Comparison of the expressions (79) and (80) shows that these fourth
+adiabatic order contributions are
+δJ = eH3
+8π2
+�4¯ξλ
+µ2 −
+λ
+9µ2 − 7λ3
+90µ4
+�
+= − e
+8π2
+�eE
+m2
+��
+4¯ξH3 − H3
+9 − 7(eE)2
+90m2 H
+�
+.
+(82)
+where the second equality comes from using the definitions of λ and µ given in Eq. (27). It is clear from the second
+equality in Eq. (82) that all these terms vanish in the limit H → 0 for fixed and finite values of E and m. This
+observation will automatically insure the validity of (79) in the Minkowski spacetime limit.
+If the electric field
+background E and the scalar field mass m are regarded as fixed and finite, then by taking the limit H → 0 of the
+expressions (79), we find
+lim
+H→0 J[4] =
+e3
+12π3H E2e− πm2
+|eE| ,
+(83)
+which is exactly the same as the result obtained in Ref. [61] for the behavior of J[2], given by Eq. (80), in the
+Minkowski spacetime limit. It has been argued [61] that in the expanding dS the inverse of the Hubble constant
+H−1, in fact, is equivalent to the finite time interval between switching on and off the electric field background in
+Minkowski spacetime. By this argument, we can see that the behavior (83) of the induced current in the Minkowski
+spacetime limit agrees with the electric current of the charged scalar particles produced by the Schwinger mechanism
+in Minkowski spacetime [40, 41].
+A comparison of the result of this article for the induced current J[4] [see Eq. (79)] to the result of Ref. [61] for the
+induced current J[2] [see Eq. (80)] is shown in Fig. 4. The figure is drawn for ξ = 0, and illustrates that, for the light
+scalars µ < 1, the induced currents J[4] and J[2] differ considerably in the regime λ ≳ µ. Subsequently, we set ξ = 0
+and concentrate our attention on the regime λ ≳ µ. We examine the behaviors of J[4] and J[2] in the two regimes:
+The infrared hyperconductivity with the criterion µ < λ ≲ 1, and the strong electric field λ ≫ max(1, µ, ξ).
+B.
+Behaviors in the infrared hyperconductivity regime
+In Ref. [61], it was pointed out that, in the infrared hyperconductivity regime µ < λ ≲ 1, the absolute value of the
+induced current J[2] monotonically increases with decreasing the electric field parameter λ, as shown in Fig. 4. In this
+regime, we can approximate expression (80) by
+J[2] ≃ eH3
+8π2
+�
+6λ
+λ2 + µ2
+�
+.
+(84)
+
+18
+From Fig. 4 it is obvious that in the infrared hyperconductivity regime, the absolute value of the induced current J[4]
+reduces to zero at a certain value of the electric field parameter λ∗ which depends on the mass parameter µ. Indeed,
+the sign of J[4] changes at λ∗. This figure also indicates that the absolute value of J[4] is a decreasing function of λ
+in the interval µ < λ < λ∗ and is an increasing function for λ > λ∗. In the infrared hyperconductivity regime, we can
+approximate expression (79) by
+J[4] ≃ eH3
+8π2
+�
+− 7λ3
+90µ4 − 7λ
+9µ2 +
+6λ
+λ2 + µ2
+�
+.
+(85)
+This expression has a zero at λ∗ ≃ 2.090µ. In the range of λ > 2.090µ, the dominant contribution to J[4] comes from
+the first two terms in Eq. (85) which its absolute value increases with increasing λ. Thus, in this range, the infrared
+hyperconductivity phenomenon does not occur. On the other hand, in the interval µ < λ < 2.090µ, the dominant
+contribution to J[4] comes from the last term in Eq. (85) which increases with decreasing λ. Thus, in the interval
+µ < λ < 2.090µ, the infrared hyperconductivity phenomenon occurs. We therefore conclude that although there exist
+a period of the infrared hyperconductivity in our result for the induced current J[4], but now this phenomenon occurs
+in the more restricted interval µ < λ < 2.090µ.
+C.
+Behaviors in the strong electric field regime
+Figure 4 shows that although the absolute values of the two induced current J[2] and J[4] are increasing functions
+of λ in the strong electric field regime λ ≫ max(1, µ, ξ), the induced current J[2] does not depend on the scalar field
+mass, whereas the induced current J[4] is proportional to µ−4. We find that, in the limit λ → ∞ for a fixed µ, the
+induced current J[4] is given approximately by
+J[4] ≃ −eH3
+8π2
+� 7λ3
+90µ4
+�
+= 7He4
+720π2
+� E3
+m4
+�
+.
+(86)
+We observe that the behavior of the induced current J[4] in the Minkowski spacetime limit [see Eq. (83)] will be
+different from that in the strong electric field limit, see Eq. (86). Whereas, in [61] it was pointed out that the induced
+current J[2] has the same behavior in these two limits, indicated on the right side of Eq. (83). We note that in
+the Minkowski spacetime limit, the electric field strength E, the scalar field mass m, and the coupling constant ξ,
+are regarded as fixed and the Hubble constant H, tends to zero. Thus, in the Minkowski spacetime limit, although
+λ = −eE/H2 and µ = m/H tend to infinity, but the ratio λ/µ2 remains finite. We also note that in the strong electric
+field regime, the Hubble constant H, the scalar field mass m, and the coupling constant ξ, are regarded as fixed and
+the electric field strength E, tends to infinity. Thus, in this regime, λ goes to infinity while µ is regarded as a fixed
+and finite value.
+VI.
+CONCLUSIONS
+The aim of the present research was to examine the induced energy-momentum tensor of a massive, complex
+scalar field coupled to the electromagnetic vector potential (7) which describes a uniform electric field background
+with a constant energy density in the conformal Poincar´e patch of dS4 where the metric takes the form (2). The
+dynamics of the complex scalar field is governed by the action (11). The energy-momentum tensor of the scalar field
+is not covariantly conserved in the presence of the electromagnetic field, as the consequence of the electromagnetic
+interactions, see Eq. (23). The nonconservation of the scalar field energy-momentum tensor is compatible with the
+nonconservation of the electromagnetic field energy-momentum tensor [see Eq. (24)], and hence the total energy
+momentum tensor of the theory is covariantly conserved. We treated the classical gravitational field (2) and the
+classical electromagnetic field (7) as fixed field configurations which they are unaffected by the dynamics of the
+quantum complex scalar field in response to these backgrounds. We discussed canonical quantization of the scalar
+field. The normalized positive and negative frequency mode functions of the scalar field which behave like the positive
+and negative frequency Minkowski mode functions in the remote past are given by Eqs. (30) and (31). These mode
+functions determine the in-vacuum state of the scalar field.
+We calculated the expectation values of all the components of the energy-momentum tensor in the in-vacuum
+state of the scalar field; the nonzero expectation values have been obtained in Eqs. (40), (42), (44), and (46). It is
+expected that these expectation values contain ultraviolet divergences, because the expectation values in the Hadamard
+in-vacuum state suffer from the same ultraviolet divergence properties as Minkowski spacetime. To render these in-
+vacuum expectation values finite, we employed the adiabatic regularization method. To adjust the set of appropriate
+
+19
+counterterms, we treated the conformal scale factor and the electromagnetic vector potential as quantities of zero
+adiabatic order. As pointed out by Wald [18], in four dimensions, to obtain a renormalized energy-momentum tensor
+which is consistent with the Wald axioms, the subtraction counterterms should be expanded up to fourth adiabatic
+order. Under these assumptions, we then constructed the set of the appropriate adiabatic counterterms, which are
+given by Eqs. (57)-(60). The adiabatic regularization procedure was carried out by subtracting the counterterms
+from the corresponding unregularized in-vacuum expectation values. Thus we have arrived at our final expressions for
+all the nonvanishing components of the induced energy-momentum tensor, which are given by Eqs. (61)-(63). This
+research has shown that all the off-diagonal components of the induced energy-momentum tensor are zero, and the
+components T22 and T33 are equal as consequences of the underlying symmetries of the backgrounds (2) and (6).
+Furthermore, since the electric field background (6) is not invariant under full symmetries of dS, indeed violates the
+time reversal symmetry [51], and causes an electric current along its direction, we observe that T00, T11, and T22 are
+not equal to one another. The research has also shown that in the limit of zero electric field, our result for the induced
+energy momentum tensor reduces to the form (65). The expression (65) accords with the result of computing the
+renormalized vacuum energy-momentum tensor of a real scalar field in dS4 obtained in Refs. [11, 38], except for the
+overall factor of 2 in Eq. (65). This factor 2 is consistent with the complex scalar field as being made of two real
+scalar fields with the number of degrees of freedom doubling up. The absolute values of the expressions (61)-(63) as
+functions of the electric field parameter λ, for various values of the scalar field mass parameter µ, and two values of the
+coupling constant ξ are shown on the graphs in Figs. 1-3, respectively. These figures signal that the induced energy-
+momentum tensor is analytic and varies continuously with the parameters λ, µ, and ξ, this statement consistent with
+the requirements discussed in [95]. For fixed values of µ and ξ, the absolute values of the nonvanishing components
+of the induced energy-momentum tensor are increasing functions of λ, but by excluding a neighbourhood of the zero
+value points this behavior is assured. For fixed values of λ and ξ, the absolute values of the nonvanishing components
+are decreasing functions of µ.
+For a fixed λ and µ, the nonvanishing components do not vary significantly with
+the parameter ξ in the range 0 ≤ ξ ≪ λ, µ. The qualitative behaviors shown in Figs. 1-3 can be given quantitative
+treatments by inspection of expressions (61)-(63) in the limiting regimes. The examination of the expressions (61)-(63)
+has shown that in the strong electric field regime λ ≫ max(1, µ, ξ), the components of the induced energy-momentum
+tensor can be approximated by Eq. (66). In the heavy scalar field regime µ ≫ max(1, λ, ξ), the components can be
+approximated according to Eq. (67). We also found that in the infrared regime, where µ ≪ 1, λ ≪ 1, ξ = 0, the
+components can be approximated by Eqs. (69)-(72). To do a consistency check, we derived the trace anomaly of
+the induced energy-momentum tensor for the case of a free, massless, conformally invariant scalar field in dS4; see
+Eq. (74). This investigation shows that our result (74) is in agreement with the earlier result of computing the trace
+anomaly [96] of a free, massless, conformally coupled real scalar field in dS4, except for the overall factor of 2 in
+Eq. (74), as explained below Eq. (65).
+One of the more significant findings to emerge from this research is that the nonconservation equation (23) implies
+the relation (77) between the induced energy-momentum tensor and the induced current. This relation in turn implies
+the renormalization condition for the induced current. We derived the expression (79) for the induced current of the
+scalar field by using the expressions (61), (73), and relation (77). This result for the induced current has been derived
+from the expressions which have been regularized by the counterterms expanded up to fourth adiabatic order. In
+Ref. [61], the induced current of the massive, minimally coupled ξ = 0, scalar field has been evaluated by subtracting
+the counterterm expanded up to second adiabatic order; see Eq. (80). In the discussion of Eq. (83), we remarked that
+our result for the induced current in the Minkowski spacetime limit agrees with the electric current of the charged
+scalar particles produced by the Schwinger mechanism in Minkowski spacetime [40, 41]. A comparison of the result
+of this article for the induced current J[4] [see Eq. (79)] to the result of Ref. [61] for the induced current J[2] [see
+Eq. (80)] is shown in Fig. 4. The figure is drawn for ξ = 0, and illustrates that, for the light scalars µ < 1, the
+induced currents J[4] and J[2] differ considerably in the regime λ ≳ µ. From Fig. 4 it is obvious that in the infrared
+hyperconductivity regime µ < λ ≲ 1, the absolute value of the induced current J[4] reduces to zero at a certain
+value of the electric field parameter λ∗ which depends on the mass parameter µ. We found that in the infrared
+hyperconductivity regime, the induced current J[4] can be approximated by Eq. (85) which has a zero at λ∗ ≃ 2.090µ.
+In the interval µ < λ < 2.090µ, the dominant contribution to J[4] comes from the last term in Eq. (85) which increases
+with decreasing λ. Thus, the infrared hyperconductivity phenomenon occurs in the interval µ < λ < 2.090µ. We
+therefore conclude that although there exist a period of the infrared hyperconductivity in our result for the induced
+current J[4], but now this phenomenon occurs in the more restricted interval µ < λ < 2.090µ. Figure 4 also shows
+that the absolute value of the induced current J[4] is an increasing function of λ in the strong electric field regime
+λ ≫ max(1, µ, ξ), but decreases as µ−4 with increasing µ. We saw that in the limit λ → ∞ for a fixed µ, the induced
+current J[4] is given approximately by Eq. (86). The results of this investigation show that the behavior of the induced
+current J[4] in the Minkowski spacetime limit [see Eq. (83)] will be different from that in the strong electric field limit,
+see Eq. (86).
+This would be a fruitful area for further work. A natural progression of this work is to analyse the backreaction of
+
+20
+the induce energy-momentum tensor on the gravitational field of dS4 and the backreaction of the induce current J[4]
+on the electromagnetic field. This is also an issue for future research to explore the induced energy-momentum tensor
+of a Dirac field in the context of our discussion.
+ACKNOWLEDGMENTS
+E. B. very much appreciate the support by the University of Kashan Grant No. 1143880/1.
+Appendix: Evaluation of the momentum integrals over the mode functions
+In the appendix we present further supplementary data associated with the calculation of the expressions (36)-(38).
+It is convenient at this stage to switch to the three-dimensional spherical momentum space, and then we can perform
+the entire three-dimensional integrals in three-dimensional spherical coordinates. We introduce a transformation from
+the Cartesian momentum coordinates (kx, ky, kz) to the spherical momentum coordinates (k, θ, φ) by equations
+kx = k cosθ,
+ky = k sin θ cos φ,
+kz = k sin θ sin φ,
+(A.1)
+In this coordinates the variable r = kx/k is given by r = cos θ with range −1 ≤ r ≤ 1. The integration measure is then
+d3k = k2 sin θdφdθdk. It is useful to covert the variables of integration from the momentum k to the dimensionless
+physical momentum p = −kτ, and from the angle θ to the variable r. Thus, the previous expression for the integration
+measure may be rewritten as
+d3k = −H3Ω3(τ)dφdrp2dp.
+(A.2)
+Accordingly, we use a dimensionless physical momentum cutoff Λ, which is related to the momentum cutoff K as
+Λ = −Kτ. To begin the evaluation of (36)-(38), we change the integration measure according to (A.2) and substitute
+the expression (30) for Uk(x).
+After integration over azimuth angle φ and some simplifications, the expressions
+(36)-(38) reduce to
+�
+in
+��T00
+��in
+�
+= Ω2 H4
+8π2
+� +1
+−1
+dr
+�
+2I1 − 4λrI2 +
+�1
+4 − γ2 − 18¯ξ + λ2r2�
+I3 + 2ℑ
+�
+I4
+�
+− 2ℜ
+��
+6ξ − 1 − iλr
+�
+I5
+�
++ I6
+�
+,
+(A.3)
+�
+in
+��T11
+��in
+�
+= Ω2 H4
+8π2
+� +1
+−1
+dr
+�
+2r2I1 − 4λrI2 +
+��
+4ξ + 1
+�
+λ2 +
+�
+4ξ − 1
+�
+µ2 −
+�
+4ξ − 1
+�
+λ2r2
++
+�
+6ξ − 1
+��
+8ξ − 1
+��
+I3 − 2ℑ
+��
+4ξ − 1
+�
+I4
+�
++ 2ℜ
+��
+1 − 6ξ + iλr − 4iλrξ
+�
+I5
+�
+−
+�
+4ξ − 1
+�
+I6
+�
+,
+(A.4)
+and
+�
+in
+��T22
+��in
+�
+=
+�
+in
+��T33
+��in
+�
+= Ω2 H4
+8π2
+� +1
+−1
+dr
+��
+1 − r2�
+I1 +
+��
+4ξ − 1
+��
+λ2 + µ2 − λ2r2�
++
+�
+6ξ − 1
+��
+8ξ − 1
+��
+I3 − 2ℑ
+��
+4ξ − 1
+�
+I4
+�
++ 2ℜ
+��
+1 − 6ξ + iλr − 4iλrξ
+�
+I5
+�
+−
+�
+4ξ − 1
+�
+I6
+�
+,
+(A.5)
+where ℑ and ℜ stand for the imaginary and real parts of any expressions, respectively. In these expressions the
+momentum integrals over the Whittaker functions have been defined by
+I1 = eπλr
+� Λ
+0
+dpp3���Wκ,γ(−2ip)
+���
+2
+,
+(A.6)
+I2 = eπλr
+� Λ
+0
+dpp2���Wκ,γ(−2ip)
+���
+2
+,
+(A.7)
+
+21
+I3 = eπλr
+� Λ
+0
+dpp
+���Wκ,γ(−2ip)
+���
+2
+,
+(A.8)
+I4 =
+�1
+4 − γ2 − λ2r2 + iλr
+�
+eπλr
+� Λ
+0
+dpp2Wκ−1,γ(−2ip)W−κ,γ(2ip),
+(A.9)
+I5 =
+�1
+4 − γ2 − λ2r2 + iλr
+�
+eπλr
+� Λ
+0
+dppWκ−1,γ(−2ip)W−κ,γ(2ip),
+(A.10)
+I6 =
+���1
+4 − γ2 − λ2r2 + iλr
+���
+2
+eπλr
+� Λ
+0
+dppWκ−1,γ(−2ip)W−κ−1,γ(2ip).
+(A.11)
+The integrals in Eqs. (A.6)-(A.11) are of the same kind of those momentum integrals over the Whittaker functions
+which encountered in the calculation of the induced current of a scalar field in two- [60] and four-dimensional [61]
+de Sitter spacetimes. The first step in the evaluation of the integrals (A.6)-(A.11) by the method that explained in
+Ref. [61], is to plug the Mellin-Barnes representation (28) for the Whittaker function W and make use of the theorem
+of residues. It is rather straightforward, but rather lengthy, to show that our final results are
+I1 = Λ4
+4 + λr
+3 Λ3 + 1
+16
+�
+4γ2 + 12λ2r2 − 1
+�
+Λ2 + λr
+8
+�
+12γ2 + 20λ2r2 − 7
+�
+Λ +
+1
+128
+�
+27 + 48γ4 + 560λ4r4 − 120γ2
++ 480γ2λ2r2 − 520λ2r2�
+log
+�
+2Λ
+�
+− 7γ4
+32 + 83γ2
+64
+− 533
+96 λ4r4 + 1607
+192 λ2r2 − 59
+16γ2λ2r2 − 351
+512 −
+�55γ2
+24
++ 35λ2r2
+8
+− 355
+96
+�
+γλr
+sin
+�
+2πγ
+�
+�
+cos
+�
+2πγ
+�
++ e2πλr�
+−
+i
+256 sin
+�
+2πγ
+�
+�
+27 + 48γ4 + 560λ4r4 − 120γ2 + 480γ2λ2r2
+− 520λ2r2
+��
+π sin
+�
+2πγ
+�
++
+�
+e2πλr + e−2iπγ�
+ψ
+�1
+2 − γ + iλr
+�
+−
+�
+e2πλr + e2iπγ�
+ψ
+�1
+2 + γ + iλr
+��
+,
+(A.12)
+I2 = Λ3
+3 + λr
+2 Λ2 + 1
+8
+�
+4γ2 + 12λ2r2 − 1
+�
+Λ + λr
+8
+�
+12γ2 + 20λ2r2 − 7
+�
+log
+�
+2Λ
+�
+− 37
+12λ3r3 + 95
+48λr − 5γ2
+4 λr
+−
+�
+4γ2 + 15λ2r2 − 4
+�
+γ
+6 sin
+�
+2πγ
+�
+�
+cos
+�
+2πγ
+�
++ e2πλr�
+−
+iλr
+16 sin
+�
+2πγ
+�
+�
+12γ2 + 20λ2r2 − 7
+�
+×
+�
+π sin
+�
+2πγ
+�
++
+�
+e2πλr + e−2iπγ�
+ψ
+�1
+2 − γ + iλr
+�
+−
+�
+e2πλr + e2iπγ�
+ψ
+�1
+2 + γ + iλr
+��
+,
+(A.13)
+I3 = Λ2
+2 + λrΛ + 1
+8
+�
+4γ2 + 12λ2r2 − 1
+�
+log
+�
+2Λ
+�
+− γ2
+4 − 7λ2r2
+4
++ 5
+16 −
+3γλr
+2 sin
+�
+2πγ
+�
+�
+cos
+�
+2πγ
+�
++ e2πλr���
+−
+i
+16 sin
+�
+2πγ
+�
+�
+4γ2 + 12λ2r2 − 1
+��
+π sin
+�
+2πγ
+�
++
+�
+e2πλr + e−2iπγ�
+ψ
+�1
+2 − γ + iλr
+�
+−
+�
+e2πλr + e2iπγ�
+ψ
+�1
+2 + γ + iλr
+��
+,
+(A.14)
+I4 = − i
+16
+�
+4γ2 + 4λ2r2 − 4iλr − 1
+�
+Λ2 − 1
+8
+�
+4γ2 + 4λ2r2 − 4iλr − 1
+��
+1 + 2iλr
+�
+Λ − 3i
+128
+�
+4γ2 + 4λ2r2 − 4iλr
+− 1
+��
+4γ2 + 20λ2r2 − 20iλr − 9
+�
+log
+�
+2Λ
+�
++ 79
+32iλ4r4 + 47
+8 λ3r3 − 409
+64 iλ2r2 + 39γ2
+16 iλ2r2 − 109
+32 λr + 21γ2
+8
+λr
++ 7i
+32γ4 − 83i
+64 γ2 + 351i
+512 +
+γ
+4 sin
+�
+2πγ
+�
+�
+4γ2 + 15
+2 iλ3r3 + 15λ2r2 + 13
+2 iγ2λr − 97
+8 iλr − 4
+��
+cos
+�
+2πγ
+�
++ e2πλr�
+−
+3
+256 sin
+�
+2πγ
+�
+�
+4γ2 + 4λ2r2 − 4iλr − 1
+��
+4γ2 + 20λ2r2 − 20iλr − 9
+��
+π sin
+�
+2πγ
+�
++
+�
+e2πλr + e−2iπγ�
+ψ
+�1
+2 − γ + iλr
+�
+−
+�
+e2πλr + e2iπγ�
+ψ
+�1
+2 + γ + iλr
+��
+,
+(A.15)
+
+22
+I5 = − i
+8
+�
+4γ2 + 4λ2r2 − 4iλr − 1
+�
+Λ − 1
+8
+�
+4γ2 + 4λ2r2 − 4iλr − 1
+��
+1 + 2iλr
+�
+log
+�
+2Λ
+�
+− 2γ2 − 10λ2r2
+− 16
+3 iλ3r3 − 4iγ2λr + 20
+3 iλr + 3
+2 −
+iγ
+3 sin
+�
+2πγ
+�
+�
+8γ2 + 12λ2r2 − 18iλr − 8
+��
+cos
+�
+2πγ
+�
++ e2πλr�
++
+i
+16 sin
+�
+2πγ
+�
+�
+4γ2 + 4λ2r2 − 4iλr − 1
+��
+1 + 2iλr
+��
+π sin
+�
+2πγ
+�
++
+�
+e2πλr + e−2iπγ�
+ψ
+�1
+2 − γ + iλr
+�
+−
+�
+e2πλr + e2iπγ�
+ψ
+�1
+2 + γ + iλr
+��
+,
+(A.16)
+I6 = 1
+64
+�
+16γ4 − 8γ2 + 16λ4r4 + 32γ2λ2r2 + 8λ2r2 + 1
+�
+log
+�
+2Λ
+�
+− 3γ4
+16 + 7γ2
+32 − 25
+48λ4r4 − 7
+8γ2λ2r2 − 29
+96λ2r2
+− 11
+256 −
+γ
+48 sin
+�
+2πγ
+�
+�
+16γ4 − 8γ2 + 16λ4r4 + 32γ2λ2r2 + 8λ2r2 + 1
+��
+12λ3r3 + 20γ2λr + 7λr
+��
+cos
+�
+2πγ
+�
++ e2πλr�
+−
+i
+128 sin
+�
+2πγ
+�
+�
+16γ4 − 8γ2 + 16λ4r4 + 32γ2λ2r2 + 8λ2r2 + 1
+��
+π sin
+�
+2πγ
+�
++
+�
+e2πλr + e−2iπγ�
+ψ
+�1
+2 − γ + iλr
+�
+−
+�
+e2πλr + e2iπγ�
+ψ
+�1
+2 + γ + iλr
+��
+,
+(A.17)
+where we use log(z) to denote the natural logarithm function, and ψ(z) to denote the digamma function. By substi-
+tuting Eqs. (A.12)-(A.17) into expressions (A.3)-(A.5) and performing the integrals over r, we obtain the results (40),
+(42), and (44), respectively.
+[1] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, England,
+1982).
+[2] L. E. Parker and D. J. Toms, Quantum Field Theory in Curved Spacetime: Quantized Fields and Gravity (Cambridge
+University Press, Cambridge, England, 2009).
+[3] R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (The University of Chicago
+Press, Chicago, 1994).
+[4] L. Parker, Phys. Rev. Lett. 21, 562-564 (1968).
+[5] L. Parker, Phys. Rev. 183, 1057-1068 (1969).
+[6] L. Parker, Phys. Rev. D 3, 346-356 (1971) [erratum: Phys. Rev. D 3, 2546-2546 (1971)].
+[7] C. Bernard and A. Duncan, Annals Phys. 107, 201 (1977).
+[8] A. Vilenkin, Nuovo Cim. A 44, 441-450 (1978).
+[9] Lowell S. Brown, Phys. Rev. D 15, 1469 (1977).
+[10] P. Candelas and D. J. Raine, Phys. Rev. D 12, 965-974 (1975).
+[11] J. S. Dowker and R. Critchley, Phys. Rev. D 13, 3224 (1976).
+[12] J. S. Dowker and R. Critchley, Phys. Rev. D 16, 3390 (1977).
+[13] S. W. Hawking, Commun. Math. Phys. 55, 133 (1977).
+[14] V. Moretti, Phys. Rev. D 56, 7797-7819 (1997) [arXiv:hep-th/9705060 [hep-th]].
+[15] S. M. Christensen, Phys. Rev. D 14, 2490-2501 (1976).
+[16] S. L. Adler, J. Lieberman and Y. J. Ng, Annals Phys. 106, 279 (1977).
+[17] S. M. Christensen, Phys. Rev. D 17, 946-963 (1978).
+[18] R. M. Wald, Phys. Rev. D 17, 1477-1484 (1978).
+[19] L. Parker and S. A. Fulling, Phys. Rev. D 9, 341-354 (1974).
+[20] S. A. Fulling and L. Parker, Annals Phys. 87, 176-204 (1974).
+[21] S. A. Fulling, L. Parker, and B. L. Hu, Phys. Rev. D 10, 3905-3924 (1974); 11, 1714(E) (1975).
+[22] N. D. Birrell, Proc. Roy. Soc. Lond. A 361, 513-526 (1978).
+[23] B. L. Hu, Phys. Rev. D 18, 4460-4470 (1978).
+[24] R. M. Wald, Commun. Math. Phys. 54, 1-19 (1977).
+[25] T. S. Bunch, S. M. Christensen, and S. A. Fulling, Phys. Rev. D 18, 4435-4459 (1978).
+[26] T. S. Bunch and P. C. W. Davies, Proc. Roy. Soc. Lond. A 357, 381-394 (1977).
+[27] T. S. Bunch and P. C. W. Davies, J. Phys. A 11, 1315-1328 (1978).
+[28] P. C. W. Davies, S. A. Fulling, S. M. Christensen and T. S. Bunch, Annals Phys. 109, 108-142 (1977).
+[29] T. S. Bunch, J. Phys. A 11, 603-607 (1978).
+[30] T. S. Bunch, J. Phys. A 13, 1297-1310 (1980).
+[31] P. R. Anderson and L. Parker, Phys. Rev. D 36, 2963 (1987).
+
+23
+[32] S. Habib, C. Molina-Par´ıs and E. Mottola, Phys. Rev. D 61, 024010 (1999) [arXiv:gr-qc/9906120 [gr-qc]].
+[33] Y. Zhang, B. Wang and X. Ye, Chin. Phys. C 44, no.9, 095104 (2020) [arXiv:1909.13010 [gr-qc]].
+[34] T. Markkanen, J. Cosmol. Astropart. Phys. 05 (2018) 001 [arXiv:1712.02372 [hep-th]].
+[35] J. Sol`a Peracaula, Phil. Trans. Roy. Soc. Lond. A 380, 20210182 (2022) [arXiv:2203.13757 [gr-qc]].
+[36] C. Moreno-Pulido and J. Sol`a Peracaula, Eur. Phys. J. C 82, no.6, 551 (2022) [arXiv:2201.05827 [gr-qc]].
+[37] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738-2751 (1977).
+[38] T. S. Bunch and P. C. W. Davies, Proc. Roy. Soc. Lond. A 360, 117-134 (1978).
+[39] E. Mottola, Phys. Rev. D 31, 754 (1985).
+[40] P. R. Anderson and E. Mottola, Phys. Rev. D 89, 104038 (2014) [arXiv:1310.0030 [gr-qc]].
+[41] P. R. Anderson and E. Mottola, Phys. Rev. D 89, 104039 (2014) [arXiv:1310.1963 [gr-qc]].
+[42] T. Markkanen and A. Rajantie, J. High Energy Phys. 01 (2017) 133 [arXiv:1607.00334 [gr-qc]].
+[43] P. R. Anderson, W. Eaker, S. Habib, C. Molina-Par´ıs and E. Mottola, Phys. Rev. D 62, 124019 (2000) [arXiv:gr-qc/0005102
+[gr-qc]].
+[44] Y. Zhang, X. Ye and B. Wang, Sci. China Phys. Mech. Astron. 63, no.5, 250411 (2020) [arXiv: 1903.10115 [gr-qc]].
+[45] X. Ye, Y. Zhang and B. Wang, J. Cosmol. Astropart. Phys. 09 (2022) 020 [arXiv:2205.04761 [gr-qc]].
+[46] J. Schwinger, Phys. Rev. 82, 664-679 (1951).
+[47] W. Heisenberg and H. Euler Z. Phys. 98, 714-732 (1936) [arXiv:physics/0605038 [physics.hist-ph]].
+[48] F. Sauter, Z. Phys. 69, 742-764 (1931).
+[49] F. Gelis and N. Tanji, Prog. Part. Nucl. Phys. 87, 1-49 (2016) [arXiv:1510.05451 [hep-ph]].
+[50] G. V. Dunne, “Heisenberg-Euler effective Lagrangians: Basics and extensions,” edited by M. Shifman et al., From Fields
+to Strings (World Scientific Publishing, Singapore, 2005), Vol. 1, pp. 445–522. [arXiv:hep-th/0406216 [hep-th]].
+[51] I. Antoniadis, P. O. Mazur and E. Mottola, New J. Phys. 9, 11 (2007) [arXiv:gr-qc/0612068 [gr-qc]].
+[52] J. Martin, Lect. Notes Phys. 738, 193-241 (2008) [arXiv:0704.3540 [hep-th]].
+[53] A. Maleknejad, M. M. Sheikh-Jabbari and J. Soda, Phys. Rept. 528, 161-261 (2013) [arXiv:1212.2921 [hep-th]].
+[54] R. Durrer and A. Neronov, Astron. Astrophys. Rev. 21, 62 (2013) [arXiv:1303.7121 [astro-ph.CO]].
+[55] J. Garriga, Phys. Rev. D 49, 6327-6342 (1994) [arXiv:hep-ph/9308280 [hep-ph]].
+[56] J. Garriga, Phys. Rev. D 49, 6343-6346 (1994).
+[57] S. P. Kim and D. N. Page, Phys. Rev. D 78, 103517 (2008) [arXiv:0803.2555 [hep-th]].
+[58] R. G. Cai and S. P. Kim, J. High Energy Phys. 09 (2014) 072 [arXiv:1407.4569 [hep-th]].
+[59] B. Hamil and M. Merad, Int. J. Mod. Phys. A 33, no.30, 1850177 (2018).
+[60] M. B. Fr¨ob, J. Garriga, S. Kanno, M. Sasaki, J. Soda, T. Tanaka and A. Vilenkin, J. Cosmol. Astropart. Phys. 04 (2014)
+009 [arXiv:1401.4137 [hep-th]].
+[61] T. Kobayashi and N. Afshordi, J. High Energy Phys. 10 (2014) 166 [arXiv:1408.4141 [hep-th]].
+[62] E. Bavarsad, C. Stahl and S. S. Xue, Phys. Rev. D 94, no.10, 104011 (2016) [arXiv:1602.06556 [hep-th]].
+[63] V. M. Villalba, Phys. Rev. D 52, 3742-3745 (1995) [arXiv:hep-th/9507021 [hep-th]].
+[64] S. Haouat and R. Chekireb, Phys. Rev. D 87, no.8, 088501 (2013) [arXiv:1207.4342 [hep-th]].
+[65] C. Stahl, E. Strobel and S. S. Xue, Phys. Rev. D 93, no.2, 025004 (2016) [arXiv:1507.01686 [gr-qc]].
+[66] S. Haouat and R. Chekireb, [arXiv:1504.08201 [gr-qc]].
+[67] C. Stahl and S. Eckhard, AIP Conf. Proc. 1693, no.1, 050005 (2015) [arXiv:1507.01401 [hep-th]].
+[68] T. Hayashinaka and J. Yokoyama, J. Cosmol. Astropart. Phys. 07 (2016) 012 [arXiv:1603.06172 [hep-th]].
+[69] J. J. Geng, B. F. Li, J. Soda, A. Wang, Q. Wu and T. Zhu, J. Cosmol. Astropart. Phys. 02 (2018) 018 [arXiv:1706.02833
+[gr-qc]].
+[70] E. Bavarsad, S. P. Kim, C. Stahl and S. S. Xue, Phys. Rev. D 97, no.2, 025017 (2018) [arXiv:1707.03975 [hep-th]].
+[71] E. Bavarsad, S. P. Kim, C. Stahl and S. S. Xue, Eur. Phys. J. Web Conf. 168, 03002 (2018).
+[72] S. Moradi, Mod. Phys. Lett. A 24, 1129-1136 (2009).
+[73] T. Hayashinaka, T. Fujita and J. Yokoyama, J. Cosmol. Astropart. Phys. 07 (2016) 010 [arXiv:1603.04165 [hep-th]].
+[74] M. Banyeres, G. Dom`enech and J. Garriga, J. Cosmol. Astropart. Phys. 10 (2018) 023 [arXiv:1809.08977 [hep-th]].
+[75] T. Hayashinaka and S. S. Xue, Phys. Rev. D 97, no.10, 105010 (2018) [arXiv:1802.03686 [gr-qc]].
+[76] M. Akbari Ahmadmahmoudi and E. Bavarsad, Phys. Rev. D 103, no.10, 105009 (2021) [arXiv:2102.03407 [hep-th]].
+[77] E. Bavarsad and M. Mortezazadeh, Iran. J. Phys. Res. 18, 91 (2018).
+[78] E. Bavarsad, S. P. Kim, C. Stahl and S. S. Xue, Effect of Schwinger pair production on the evolution of the Hubble constant
+in de Sitter spacetime, edited by E. Battistelli et al., the Fifteenth Marcel Grossmann Meeting (World Scientific Publishing,
+Singapore, 2022), pp. 2041-2046, DOI:10.1142/9789811258251 0306 [arXiv:1909.09319 [hep-th]].
+[79] M. Botshekananfard and E. Bavarsad, Phys. Rev. D 101, no.8, 085011 (2020) [arXiv:1911.10588 [hep-th]].
+[80] E. Bavarsad and N. Margosian, J. Res. Many-Body Syst. 8, 1 (2018), DOI: 10.22055/jrmbs.2018.13879.
+[81] S. Haouat and R. Chekireb, Eur. Phys. J. C 72, 2034 (2012) [arXiv:1201.5738 [hep-th]].
+[82] S. Shakeri, M. A. Gorji and H. Firouzjahi, Phys. Rev. D 99, no.10, 103525 (2019) [arXiv:1903.05310 [hep-th]].
+[83] C. Stahl, Nucl. Phys. B 939, 95-104 (2019) [arXiv:1806.06692 [hep-th]].
+[84] M. Giovannini, Phys. Rev. D 97, no.6, 061301(R) (2018) [arXiv:1801.09995 [hep-th]].
+[85] R. Sharma and S. Singh, Phys. Rev. D 96, no.2, 025012 (2017) [arXiv:1704.05076 [gr-qc]].
+[86] V. Domcke, Y. Ema and K. Mukaida, J. High Energy Phys. 02 (2020) 055 [arXiv:1910.01205 [hep-ph]].
+[87] W. Tangarife, K. Tobioka, L. Ubaldi and T. Volansky, J. High Energy Phys. 02 (2018) 084 [arXiv:1706.03072 [hep-ph]].
+[88] H. Kitamoto, Phys. Rev. D 98, no.10, 103512 (2018) [arXiv:1807.03753 [hep-th]].
+[89] W. Z. Chua, Q. Ding, Y. Wang and S. Zhou, J. High Energy Phys. 04 (2019) 066 [arXiv:1810.09815 [hep-th]].
+
+24
+[90] E. V. Gorbar, A. I. Momot, O. O. Sobol and S. I. Vilchinskii, Phys. Rev. D 100, no.12, 123502 (2019) [arXiv:1909.10332
+[gr-qc]].
+[91] O. O. Sobol, E. V. Gorbar and S. I. Vilchinskii, Phys. Rev. D 100, no.6, 063523 (2019) [arXiv:1907.10443 [astro-ph.CO]].
+[92] O. O. Sobol, E. V. Gorbar, M. Kamarpour and S. I. Vilchinskii, Phys. Rev. D 98, no.6, 063534 (2018) [arXiv:1807.09851
+[hep-ph]].
+[93] S. P. Kim, [arXiv: 1905.13439 [gr-qc]].
+[94] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, NIST Handbook of Mathematical Functions (Cambridge
+University Press, Cambridge, England, 2010).
+[95] S. Hollands and R. M. Wald, Commun. Math. Phys. 223, 289-326 (2001) [arXiv:gr-qc/0103074 [gr-qc]].
+[96] M. J. Duff, Nucl. Phys. B 125, 334-348 (1977).
+

GNE3T4oBgHgl3EQftgtb/content/tmp_files/2301.04676v1.pdf.txt

@@ -0,0 +1,1889 @@
+CFTP/23-001
+A viable A4 3HDM theory of quark mass matrices
+Iris Brée∗, Sérgio Carrôlo †, Jorge C. Romão ‡, João P. Silva §,
+CFTP, Departamento de Física,
+Instituto Superior Técnico, Universidade de Lisboa,
+Avenida Rovisco Pais 1, 1049 Lisboa, Portugal
+January 13, 2023
+Abstract
+It is known that a three Higgs doublet model (3HDM) symmetric under an exact A4 sym-
+metry is not compatible with nonzero quark masses and/or non-block-diagonal CKM matrix.
+We show that a 3HDM with softly broken A4 terms in the scalar potential does allow for a
+fit of quark mass matrices. Moreover, the result is consistent with mh = 125GeV and the
+h → WW, ZZ signal. We also checked numerically that, for each point that passes all the
+constraints, the minimum is a global minimum of the potential.
+1
+Introduction
+The observation in 2012 of a scalar particle with 125GeV by the ATLAS and CMS collaborations
+[1, 2] has incentivized experimental searches for beyond the Standard Model (SM) particles at the
+LHC. On par with these experimental endeavors, theoretical efforts in the search for extra scalar
+particles have been strengthened since this discovery. A promising framework is found in N-Higgs
+doublet models (NHDM).
+Such models have many free parameters, which are often curtailed by imposing some discrete
+family symmetry. Here, we focus on the implementation of A4 in a three Higgs doublet model
+(3HDM). The A4 group is the group of even permutations on 4 elements.
+It is the smallest
+discrete group to contain a three-dimensional irreducible representation (irrep), which is ideal for
+describing the three families of quarks with a minimal number of independent Yukawa couplings.
+Thus, NHDM supplemented by the A4 discrete symmetry has long been of interest in flavour
+physics research.
+A number of early articles include: [3], mainly devoted to the leptonic sector and where the
+solution to the quark sector is briefly mentioned to include a fourth Higgs doublet and all quark
+fields in singlets (which is effectively the same as the Standard Model quark sector); [4], where A4
+is broken by dimension four Yukawa couplings, thus rendering the theory non-renormalizable or,
+alternatively, the low energy limit of a broader complete theory; [5], which requires three Higgs
+doublets in the down-type quark sector and a further two in the up-type quark sector, consisting
+∗[email protected]
+†[email protected]
+‡[email protected]
+§[email protected]
+1
+arXiv:2301.04676v1  [hep-ph]  11 Jan 2023
+
+of a 5HDM; and [6], which is devoted to the leptonic sector, but has the interesting side query
+that it might be possible to recover a realistic CKM matrix through soft-breaking of A4.
+Quark mass matrices in the context of a 3HDM with Higgs doublets in the triplet representation
+of A4 were studied in [7] and [8], with the vacuum expectation value (vev) structure (eiα, e−iα, r),
+where α and r are real constants. Unfortunately, Degee, Ivanov and Keus [9] proved in 2013 that
+such a vacuum can never be the global minimum of the A4 symmetric 3HDM. In this beautiful
+paper, geometric techniques were used in order to identify all possible global minima (thus, all
+possible viable vacua) of the A4 symmetric 3HDM. Immediately thereafter, those minima were
+used to show that all assignments of the quark fields into irreps of A4, when combined with the
+possible vevs for the exact A4 potential, yield vanishing quark masses and/or a CP conserving
+CKM matrix, both of which are forbidden by experiment. This is in fact a consequence of a much
+broader theorem, proved in [10, 11]: given any flavour symmetry group, one can obtain a physical
+CKM mixing matrix and, simultaneously, non-degenerate and non-zero quark masses only if the
+vevs of the Higgs fields break completely the full flavour group. The idea is that a symmetry will
+reduce the number of redundant Yukawa couplings present in the SM, and it might even predict
+relations among observables which turn out to be consistent with experiment.
+When studying in detail the extensions of A4 to the quark sector found by Ref. [12], we noticed
+that, in some of them, if it weren’t for the particular form of the vevs allowed by the exact A4
+3HDM potential, the Yukawa matrices could allow for massive quarks, and for a realistic CKM
+matrix. Since the A4 symmetric potential doesn’t allow for minima other than those shown in [9],
+here we consider the case where the A4 symmetry is softly broken by the addition of quadratic
+terms to the potential. Such terms do not spoil the theory’s renormalizability, but break the A4
+symmetry.
+Our article is organized as follows. We define the notation for the scalar potential in Sec. 2.1,
+discuss the Yukawa Lagrangian and the form of the possible mass matrices in Sec. 2.2, giving all
+the expressions needed for the fit in Sec. 2.3. In Sec. 3 we present our fit to the quarks mass
+matrices, while in Sec. 4 we discuss the viability of the vacuum found in the fit in terms of the
+scalar potential.
+Sec. 5 is devoted to the implementation of the theoretical constraints to be
+imposed, and in Sec. 6 we briefly discuss the constraints coming from the LHC. The results and
+conclusions are presented in Sec. 7 and 8, respectively. The Appendices contain some additional
+expressions that are needed for the fits.
+2
+Parameterization for the softly-broken A4 3HDM
+2.1
+Potential and candidates for local minimum
+The softly-broken potential of the 3HDM with an A4 symmetry is given by
+VH = V4, A4 + M2
+ij
+�
+φ†
+iφj
+�
+,
+(1)
+where V4, A4 is the quartic potential for the A4 symmetric three Higgs doublet model (3HDM),
+which is, in the notation of [9],
+2
+
+V4, A4 =Λ0
+3
+�
+φ†
+1φ1 + φ†
+2φ2 + φ†
+3φ3
+�2
++ Λ1
+��
+Re
+�
+φ†
+1φ2
+��2 +
+�
+Re
+�
+φ†
+2φ3
+��2 +
+�
+Re
+�
+φ†
+3φ1
+��2�
++ Λ2
+��
+Im
+�
+φ†
+1φ2
+��2 +
+�
+Im
+�
+φ†
+2φ3
+��2 +
+�
+Im
+�
+φ†
+3φ1
+��2�
++ Λ3
+3
+�
+(φ†
+1φ1)2 + (φ†
+2φ2)2 + (φ†
+3φ3)2 − (φ†
+1φ1)(φ†
+2φ2) − (φ†
+2φ2)(φ†
+3φ3) − (φ†
+3φ3)(φ†
+1φ1)
+�
++ Λ4
+�
+Re
+�
+φ†
+1φ2
+�
+Im
+�
+φ†
+1φ2
+�
++ Re
+�
+φ†
+2φ3
+�
+Im
+�
+φ†
+2φ3
+�
++ Re
+�
+φ†
+3φ1
+�
+Im
+�
+φ†
+3φ1
+��
+.
+(2)
+The matrix M2
+ij is a general hermitian matrix, which can be parameterized by
+(M2
+ij) =
+�
+�
+�
+m2
+11
+m2
+12eiθ12
+m2
+13eiθ13
+m2
+12e−iθ12
+m2
+22
+m2
+23eiθ23
+m2
+13e−iθ13
+m2
+23e−iθ23
+m2
+33
+�
+�
+� ,
+(3)
+where m2
+ij are real parameters with the dimension of mass squared.1
+Additionally, in the notation of [13], the exact A4 potential can be written as
+VA4 =r1 + 2r4
+3
+�
+(φ†
+1φ1) + (φ†
+2φ2) + (φ†
+3φ3)
+�2 + 2(r1 − r4)
+3
+�
+(φ†
+1φ1)2 + (φ†
+2φ2)2
++(φ†
+3φ3)2 − (φ†
+1φ1)(φ†
+2φ2) − (φ†
+2φ2)(φ†
+3φ3) − (φ†
+3φ3)(φ†
+1φ1)
+�
++ 2r7
+�
+|φ†
+1φ2|2 + |φ†
+2φ3|2 + |φ†
+3φ1|2�
++
+�
+c3
+�
+(φ†
+1φ2)2 + (φ†
+2φ3)2 + (φ†
+3φ1)2�
++ h.c.
+�
+.
+(4)
+The relation between the two notations is
+r1 = 1
+3(Λ0 + Λ3) ,
+r4 = 1
+6(2Λ0 − Λ3) ,
+r7 = 1
+4(Λ1 + Λ2) ,
+Re(c3) = 1
+4(Λ1 − Λ2) ,
+Im(c3) = −1
+4Λ4 .
+(5)
+We consider that the scalar fields can take complex vacuum expectation values (vevs), to be
+determined later. Thus, we write,
+φi =
+�
+ϕ+
+i
+|vi|eiρi
+√
+2
++
+1
+√
+2 (xi + ixi+3)
+�
+.
+(6)
+Because CP is spontaneously violated, the unrotated neutral fields have no definite CP, and for
+convenience we label them xi, i = 1, . . . , 6. We can also use the gauge freedom to absorb one of
+the phases in the vevs, that we choose to be ρ1. Therefore we have the vector of vevs defined as
+⃗v = (|v1|, |v2|eiρ2, |v3|eiρ3) .
+(7)
+1In the quadratic terms, the combination − M0
+√
+3
+�
+φ†
+1φ1 + φ†
+2φ2 + φ†
+3φ3
+�
+is also invariant under A4. But, since we
+are keeping all soft-breaking terms, we find the notation in (3) more convenient.
+3
+
+This vev contributes with four free parameters to our model, because one of the parameters is
+constrained by the mass of the gauge bosons to match the observed SM values,
+|v1|2 + |v2|2 + |v3|2 ≡ v2 ≃ (246GeV)2.
+(8)
+The vev can also be parameterised as
+⃗v = v
+�
+cos(β1) cos(β2), cos(β2) sin(β1)eip2, sin(β2)eip3�
+.
+(9)
+Of the quantities arising out of the scalar potential, the vevs are the only relevant to the quark
+mass matrices. This leads many authors to just proclaim some vevs, without checking whether
+they can indeed be the global minima of a realistic Higgs potential. We will perform this crucial
+verification below, in Section 4.
+2.2
+Yukawa Lagrangian
+As in Refs. [7, 12], we consider that the Higgs doublets are in the 3 of A4 as well as the three
+left-handed SU(2) doublets QLj of hypercharge 1/6. There are three right-handed SU(2) singlets
+nR,j of hypercharge −1/3 and three right-handed SU(2) singlets pR,j of hypercharge 2/3. Our
+assignments for the singlets are as follows
+nR1, pR1 → 1,
+nR2, pR2 → 1′,
+nR3, pR3 → 1′′ of A4 .
+(10)
+Then, the A4 transformations on the fields are generated by [7, 12]
+T :
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+φ1 → φ2 → φ3 → φ1,
+QL1 → QL2 → QL3 → QL1,
+nR1 → nR1, nR2 → ωnR2, nR3 → ω2nR3,
+pR1 → nR1, pR2 → ωpR2, pR3 → ω2pR3,
+(11)
+and
+S :
+�φ1 → φ1, φ2 → −φ2, φ3 → −φ3,
+QL1 → QL1, QL2 → −QL2, QL3 → −QL3.
+(12)
+One can easily verify that the scalar potential in Eq. (4) is invariant under the previous transfor-
+mations. Now we write the A4 invariant Yukawa Lagrangian for quarks. We have
+−LYukawa =
+√
+2 ˆa
+�
+QL1φ1 + QL2φ2 + QL3φ3
+�
+nR1
++
+√
+2ˆb
+�
+QL1φ1 + ω QL2φ2 + ω2 QL3φ3
+�
+nR2
++
+√
+2 ˆc
+�
+QL1φ1 + ω2 QL2φ2 + ω QL3φ3
+�
+nR3
++
+√
+2 ˆa′ �
+QL1 ˜φ1 + QL2 ˜φ2 + QL3 ˜φ3
+�
+pR1
++
+√
+2ˆb′ �
+QL1 ˜φ1 + ω QL2 ˜φ2 + ω2 QL3 ˜φ3
+�
+pR2
++
+√
+2 ˆc′ �
+QL1 ˜φ1 + ω2 QL2 ˜φ2 + ω QL3 ˜φ3
+�
+pR3 + h.c.,
+(13)
+4
+
+where, as usual,
+˜φj ≡ i σ2φ∗
+j ,
+(14)
+and we define
+ˆa = aei α, ˆb = bei β, ˆc = cei γ, ˆa′ = a′ei α′, ˆb′ = b′ei β′, ˆc′ = c′ei γ′ ,
+(15)
+where a, b, c, a′, b′, c′ are real and positive. This choice of invariant Lagrangian corresponds to the
+case I identified in Ref. [12] (see the next section).
+2.3
+Yukawa matrices, masses and CKM
+We aim to fit six quark masses and four CKM matrix elements to the currently accepted SM
+values for these observables. Therefore, we’re interested in softly-broken A4 symmetric models
+with up to ten parameters. Ref. [12] has studied all of the possible extensions of A4 to the fermion
+sector. Using their results, we can check which of them can accommodate non-vanishing quark
+masses, CKM mixing angles and CP violation by considering a general vev ⃗v. We take the Jarlskog
+invariant as a measure of CP violation [14]. Out of all possibilities, we are left with five of them,
+which we list in Table 1. There, A are real constants, Ω are constants in the [0, 2π[ interval,
+ω = ei 2π
+3 (ω3 = 1) and T is the transpose of the matrix.
+Case
+Md
+Mu
+I
+�
+�
+�
+aeiαv1
+beiβv1
+ceiγv1
+aeiαv2
+ωbeiβv2
+ω2ceiγv2
+aeiαv3
+ω2beiβv3
+ωceiγv3
+�
+�
+�
+�
+�
+�
+A → A′,
+A ∈ {a, b, c}
+Ω → Ω′,
+Ω ∈ {α, β, γ}
+vi → v∗
+i ,
+i ∈ {1, 2, 3}
+�
+�
+�
+II
+IT
+d
+IT
+u
+III
+�
+�
+�
+0
+(aeiα − beiβ)v3
+(aeiα + beiβ)v2
+(aeiα + beiβ)v3
+0
+(aeiα − beiβ)v1
+(aeiα − beiβ)v2
+(aeiα + beiβ)v1
+0
+�
+�
+�
+�
+�
+�
+A → A′,
+A ∈ {a, b}
+Ω → Ω′,
+Ω ∈ {α, β}
+vi → v∗
+i ,
+i ∈ {1, 2, 3}
+�
+�
+�
+IV
+Id
+IIIu
+V
+IIId
+Iu
+Table 1: Extensions of A4 to the Yukawa sector with non-vanishing determinant, and non-zero J
+for general, complex valued, vevs (v1, v2, v3). In the Table, Id stands for the matrix Md for case I
+and similarly for the other entries.
+In the Yukawa sector, there are ten observables, six masses, three mixing angles and one
+Jarlskog invariant, therefore, we would prefer to look for a case with ten parameters, or less. All
+possible neutral vevs of the 3HDM are consistent with the parameterization in Eq. (9), which
+consists of four free parameters that we can fit; two angles, and two phases. Looking at the cases
+in Table 1, we will see that it is possible to reduce the number of free parameters by performing
+both basis transformations to right-handed quarks and global U(1)Y rephasings, both of which
+have no effect on the physical predictions of the theory.
+For case I, the down quark mass matrices read
+5
+
+Md =
+�
+�
+�
+aeiαv1
+beiβv1
+ceiγv1
+aeiαv2
+ωbeiβv2
+ω2ceiγv2
+aeiαv3
+ω2beiβv3
+ωceiγv3
+�
+�
+� = DvWDaDα,
+(16)
+where (remember that the vi are complex)
+Dv = diag(v1, v2, v3) , Da = diag(a, b, c) , Dα = diag(eiα, eiβ, eiγ) , W =
+�
+�
+�
+1
+1
+1
+1
+ω
+ω2
+1
+ω2
+ω
+�
+�
+� .
+(17)
+We see that we can perform a unitary transformation to the right-handed quarks that removes all
+three phases α, β, γ. The same holds for Mu, by performing the substitution A → A′, Ω → Ω′
+and vi → v∗
+i .
+For case II, the quark mass matrices have the form
+Md = DaDαWDv and Mu = Da′Dα′WDv∗ .
+(18)
+Thus, we can remove the phases of the vev through an appropriate transformation of the right-
+handed quarks, but we can only remove α through a global rephasing QL → Q′
+L = eiαQL of
+the left-handed quarks. This rephasing, however, doesn’t remove the phase α′ in the matrix Mu,
+because, in general, it will be different from α. Now, for case III, we can only remove α and
+α′ 2 by performing the corresponding rephasings of the right-handed quarks. Finally, for the case
+IV (V), we can remove, as we did for case I, all the phases from the Yukawa couplings α, β, γ
+(corresponding primes), and only one phase α′ (corresponding unprimed). To summarise, we can
+classify the cases above according to their number of free-parameters, which we show in the Table
+below.
+case
+#θ
+#p
+#A
+#α
+#A′
+α′
+Total
+I
+2
+2
+3
+3 − 3 = 0
+3
+3 − 3 = 0
+10
+II
+2
+2 − 2 = 0
+3
+3 − 1 = 2
+3
+3
+13
+III
+2
+2
+2
+2 − 1 = 1
+2
+2 − 1 = 1
+10
+IV
+2
+2
+3
+3 − 3 = 0
+2
+2 − 1 = 1
+10
+V
+2
+2
+2
+2 − 1 = 1
+3
+3 − 3 = 0
+10
+Table 2: Number of parameters for each case of Table 1. θ and p are the angles and phases of the
+vev, A(A′) are the Yukawa couplings of the down (up) quarks, and α(α′) are the Yukawa phases
+of the down (up) quarks. The minus signs correspond to the parameters that we can remove by
+a basis transformation of the quark sector.
+In this work, we study case I, that corresponds to the Lagrangian in Eq. (13). Then, given
+that DαD†
+α = 1 and DaD†
+a = Da2 = diag(a2, b2, c2), we find
+Hd
+≡
+MdM†
+d = DvSdD†
+v ,
+Hu
+≡
+MuM†
+u = D†
+vSuDv ,
+(19)
+2We could have chosen to remove β and β′ instead, this is just one possible choice.
+6
+
+where Sd = WDa2W † and a2 → a′2 for the up quark case. This matrix can now be explicitly
+written out using appropriate parameters as
+Sd =
+�
+�
+�
+Σd
+Zdeiφd
+Zde−iφd
+Zde−iφd
+Σd
+Zdeiφd
+Zdeiφd
+Zde−iφd
+Σd
+�
+�
+� ,
+(20)
+where Σd and Zd are real, and
+Σd
+≡
+a2 + b2 + c2 ,
+Zd eiφd
+≡
+a2 + ω2b2 + ωc2 ,
+(21)
+with corresponding primes for the up case.
+For completeness, the specific forms for Hd and
+Hu found after using the parameterizations in Eqs. (9) and (21) are written in Appendix A.
+The eigenvalues of the matrices Hd and Hu will be fitted for the (square of the) quark masses,
+(m2
+d, m2
+s, m2
+b) and (m2
+u, m2
+c, m2
+t ), respectively
+We now turn to the Cabibbo-Kobayashi-Maskawa (CKM) matrix. As found by Branco and
+Lavoura [15], the absolute values of the CKM matrix can be obtained through calculating the
+traces of appropriate powers of the matrices Hu and Hd. They observe that
+Tr
+�
+Ha
+uHb
+d
+�
+≡ Lab =
+�
+k,i
+Uki(Da
+u)kk(Db
+d)ii ,
+(22)
+where Uki = |Vki|2 and V is the CKM matrix. The CKM matrix is unitary and therefore U only
+has four independent entries. Consequently, in order to compute U, it is only necessary to resort
+to
+L11 =Uki(Du)kk(Dd)ii ,
+L12 =Uki(Du)kk(D2
+d)ii ,
+L21 =Uki(D2
+u)kk(Dd)ii ,
+L22 =Uki(D2
+u)kk(D2
+d)ii .
+(23)
+These equations are linear in Uik and are, therefore, invertible for this variable. Thus, by picking
+U11, U21, U13, and U23 (respectively, Uud, Ucd, Uub, and Ucb), we are able to obtain a unique
+solution for the magnitudes of the CKM elements as a function of Lab and the quark masses.
+Namely,
+U11
+=
+�
+mb2 − ms2� �
+mc2 − mt2� a11
+det ,
+U21
+=
+�
+mb2 − ms2� �
+mu2 − mt2� a21
+det ,
+U13
+=
+�
+md2 − ms2� �
+mc2 − mt2� a13
+det ,
+U23
+=
+�
+md2 − ms2� �
+mu2 − mt2� a23
+det ,
+(24)
+where
+a11
+=
+L11
+�
+mb2 + ms2� �
+mc2 + mt2�
+− L12
+�
+mc2 + mt2�
+− L21
+�
+mb2 + ms2�
++ L22
+7
+
++m2
+b
+�
+−m2
+cm2
+t
+�
+m2
+d + m2
+s
+�
+− m2
+sm2
+u
+�
+m2
+c + m2
+t
+�
++ m2
+sm4
+u
+�
++ m2
+cm2
+dm2
+t
+�
+m2
+d − m2
+s
+�
+,(25)
+a21
+=
+−L11
+�
+mb2 + ms2� �
+mt2 + mu2�
++ L12
+�
+mu2 + mt2�
++ L21
+�
+mb2 + ms2�
+− L22
++m2
+b
+�
+m2
+cm2
+s
+�
+m2
+t + m2
+u − m2
+c
+�
++ m2
+t m2
+u
+�
+m2
+d + m2
+s
+��
++ m2
+dm2
+t m2
+u
+�
+m2
+s − m2
+d
+�
+,
+(26)
+a13
+=
+−L11
+�
+md2 + ms2� �
+mt2 + mc2�
++ L12
+�
+mc2 + mt2�
++ L21
+�
+md2 + ms2�
+− L22
++m2
+bm2
+cm2
+t
+�
+m2
+d + m2
+s − m2
+b
+�
++ m2
+dm2
+s
+�
+m2
+c
+�
+m2
+t + m2
+u
+�
++ m2
+u
+�
+m2
+t − mu2��
+,
+(27)
+a23
+=
+L11
+�
+md2 + ms2� �
+mt2 + mu2�
+− L12
+�
+mu2 + mt2�
+− L21
+�
+md2 + ms2�
++ L22
++m2
+t m2
+u
+�
+m4
+b − m2
+b
+�
+m2
+d + m2
+s
+�
+− m2
+dm2
+s
+�
++ m4
+cm2
+dm2
+s − m2
+cm2
+dm2
+s
+�
+m2
+t + m2
+u
+�
+,
+(28)
+and
+det =
+�
+mb2 − md2� �
+mc2 − mu2� �
+md2 − ms2� �
+mu2 − mt2� �
+mb2 − ms2� �
+mc2 − mt2�
+.
+(29)
+In these equations, the Lij are obtained by evaluating the left hand side of Eq. (22). Finally, we
+note that knowing these four CKM magnitudes, we can determine the Jarslkog invariant [14], up
+to its sign. Thus, given some phase convention, we are also able to determine the phases of all
+CKM matrix elements.
+3
+The fit to the quark mass matrices
+3.1
+The fitting procedure
+We have implemented a χ2 analysis of the model, through a minimization performed using the
+CERN Minuit library [16]. The observables employed in this analysis, labeled by i = 1, ..., 11 are
+specified in Table 3, where Xi represents the experimental mean value of the observable Xi and
+σi is the experimental error, which, when both left and right bounds are stated, is assumed to be
+the largest of the two. The data on the quark masses as well as for the CKM matrix elements and
+Observable
+Experimental value
+Model prediction
+mu [MeV]
+2.16 ± 0.50
+2.15
+mc [MeV]
+1270 ± 20
+1271.9
+mt [GeV]
+172.69 ± 0.30
+172.69
+md [MeV]
+4.67 ± 0.50
+4.66
+ms [MeV]
+93.4 ± 8.6
+92.08
+mb [MeV]
+4180 ± 30
+4179.74
+|V11|
+0.97435 ± 0.00016
+0.97434
+|V21|
+0.22486 ± 0.00067
+0.22479
+|V13|
+0.00369 ± 0.00011
+0.00369
+|V23|
+0.04182 ± 0.00085
+0.04175
+J
+(3.08 ± 0.15) × 10−5
+3.09 × 10−5
+Table 3: Experimental values and fit results.
+the Jarlskog invariant experimental values were obtained from [17]. As mentioned, |J| is fixed by
+|V11|, |V21|, |V13|, and |V23|. However, using it in the fit speeds the numerical convergence onto a
+good solution.
+8
+
+The χ2 function depends on the 10 parameters of our model,
+β1, β2, ρ2, ρ3, Σd, Σu, Zd, Zu, φd, φu
+(30)
+and is written as
+χ2(p) =
+11
+�
+i=1
+�
+Pi(p) − Xi
+σi
+�2
+,
+(31)
+where Pi(p) is our model’s prediction for each of the 11 (10 + J) observables. The fit is complicated
+by the fact that the masses (squared) are obtained from the eigenvalues of Hd, Hu but the elements
+of the CKM also depend on the masses, see Eq. (24). So, we start by calculating the eigenvalues
+of Hd and Hu, which depend only on the parameters in Eq. (30). Then, we evaluate the Lij
+from the left hand side of Eq. (22), and finally the CKM elements are obtained from Eq. (24). In
+Appendix A we give the explicit expressions for the matrices Hd and Hu.
+3.2
+Results of the fit
+We have found an excellent fit of our model to the data, given in the second column of Table 3.
+This fit results in χ2 = 0.058, for the parameters
+β1 =1.42608 radians ,
+β2 =1.54243 radians ,
+ρ2 =4.27865 radians ,
+ρ3 =5.37039 radians ,
+Σd =0.288824 × 10−3 ,
+Σu =0.492828 ,
+Zd =0.181571 × 10−3 ,
+Zu =0.475911 ,
+φd = − 1.73226 radians ,
+φu =0.206453 × 10−3 radians .
+(32)
+This fit also leads to the data in the third column of Table 3, as well as to the vevs
+|vi| = (1.00625, 6.90462, 245.901) (GeV).
+(33)
+We notice that the vevs obey v1 < v2 << v3. This hierarchy of vevs is related to the hierarchy of
+the quark masses. This was also obtained in Ref. [7], although their model is not consistent, as
+their vev structure is not that of [9] for the symmetric A4 potential they consider.
+4
+Viability of the vacuum found in the fit
+We start by defining the three doublets as in Eq. (6). Next we define the physical eigenstates for
+the charged Higgs as (G+, S+
+2 , S2
+3)T , and for the neutral states we have (G0, S0
+2, S0
+3, S0
+4, S0
+5, S0
+6)T ,
+identifying the would-be Goldstone bosons G+ ≡ S+
+1 and G0 ≡ S0
+1. With these conventions, and
+following the definitions in [18], we define the 3 × 3 matrix ˜U by
+ϕ+
+i ≡
+3
+�
+j=1
+˜UijS+
+j ,
+(34)
+9
+
+and the 3 × 6 matrix ˜V by
+xi + ixi+3 =
+6
+�
+j=1
+˜VijS0
+j .
+(35)
+These matrices3 are then related to the diagonalization matrices of the charged and neutral scalars,
+to which we now turn.
+4.1
+The minimization of the potential
+In our procedure we already know the values of the vevs. So, we use the stationarity equations
+to solve for the soft parameters, and leave the quartic parameters of the potential Λi as free
+parameters. In this way we can solve for m2
+11, m2
+22, m2
+33 as well as for Im(m2
+12), Im(m2
+13), leaving
+as free parameters the Λi and Re(m2
+12), Re(m2
+13), Re(m2
+23), Im(m2
+23). When evaluating the scalar
+mass matrices (see below) the conditions have to be applied to ensure that we are at the minimum.
+For completeness we write these conditions in Appendix B.
+4.2
+The charged mass matrix
+The charged mass matrix is obtained from the second derivatives at the minimum,
+M2
+C =
+∂2VH
+∂ϕ+
+i ∂ϕ−
+j
+�����
+Min
+.
+(36)
+The matrix M2
+C is an hermitian matrix, with real eigenvalues and satisfying, with our usual
+conventions,
+RchM2
+CR†
+ch = diag(0, m2
+S+
+2 , m2
+S+
+3 ) ≡ M2
+Dch ,
+(37)
+where Rch is an unitary matrix that satisfies,
+S+
+i =
+3
+�
+j=1
+(Rch)ij ϕ+
+j .
+(38)
+This can be seen from
+Lmass = − ϕ−
+i
+�
+M2
+C
+�
+ij ϕ+
+j = −ϕ−
+i
+�
+R†
+chRchM2
+CR†
+chRch
+�
+ij ϕ+
+j = −ϕ−
+i
+�
+R†
+chM2
+DchRch
+�
+ij ϕ+
+j
+= − S−
+i
+�
+M2
+Dch
+�
+ij S+
+j ,
+(39)
+where we have used Eq. (38).
+We have checked both algebraically and numerically that we have a zero eigenvalue corre-
+sponding to G+ and we require that all other masses squared are positive, a condition for a local
+minimum.
+3From the point of view of a simultaneous fit of the Yukawa and scalar sectors, it is a pity that these matrices
+˜V and ˜U have in the literature the same notation as the CKM matrix V and Uki = |Vki|2.
+10
+
+4.3
+The neutral mass matrix
+Since in our case CP is not conserved, we denote the unrotated neutral scalars by xi, i = 1, . . . , 6,
+as in Eq. (6). We therefore obtain the neutral mass matrix as,
+M2
+N =
+∂2VH
+∂xi∂xj
+�����
+Min
+.
+(40)
+This is a symmetric real matrix diagonalized by an orthogonal 6 × 6 matrix,
+RneuM2
+NRT
+neu = diag(0, m2
+S0
+2, m2
+S0
+3, m2
+S0
+4, m2
+S0
+5, m2
+S0
+6) ≡ M2
+Dneu ,
+(41)
+with
+S0
+i =
+6
+�
+j=1
+(Rneu)ij xj .
+(42)
+As for the case of the charged scalars, we have checked both algebraically and numerically that
+we have a zero eigenvalue corresponding to G0 and we require that all other masses squared are
+positive, a condition for a local minimum.
+5
+Theoretical Constraints
+After having shown that a solution exists for the vevs and parameters in the Yukawa sector that
+correctly fits the quarks masses and the CKM entries, we have to show that this is compatible
+with the scalar potential analysis. In particular we have to show that the vevs correspond to a
+local minimum of the potential and that both the theoretical constraints as well as those coming
+from LHC are satisfied. In this section we analyze the theoretical constraints.
+5.1
+Perturbative Unitarity
+This problem was already solved in [13], so we take the potential in the form of Eq. (4). From
+Ref. [13] we have the following expression for the eigenvalues λi4
+λ1 =2 (2Re(c3) + r1)
+(43)
+λ2 =2
+�√
+3 |Im(c3)| − Re(c3) + r1
+�
+(44)
+λ3 =2
+�
+−
+√
+3 |Im(c3)| − Re(c3) + r1
+�
+(45)
+λ4 =2(r4 + r7)
+(46)
+λ5 =2(r4 − r7)
+(47)
+λ6 =2(r1 + 2r7)
+(48)
+λ7 =2(r1 − r7)
+(49)
+λ8 =2(r4 + |c3|)
+(50)
+λ9 =2(r4 − |c3|)
+(51)
+λ10 =6r1 + 8r4 + 4r7
+(52)
+λ11 =6r1 − 2(2r4 + r7)
+(53)
+λ12 =6|c3| + 2r4 + 4r7
+(54)
+4We use λi instead of Λi, in order to not confuse with the notation of Eq. (2).
+11
+
+λ13 = − 6|c3| + 2r4 + 4r7
+(55)
+Perturbative unitarity is satisfied if
+|λi| < 8π,
+∀i.
+(56)
+5.2
+The BFB conditions
+For the A4 symmetric potential, the conditions for boundedness from below along the neutral
+directions (BFB-n) have been conjectured in [19], and proved to hold in [20]. These are
+Λ0 + Λ3 ≥ 0 ,
+(57)
+4
+3(Λ0 + Λ3) + 1
+2(Λ1 + Λ2) − Λ3 − 1
+2
+�
+(Λ1 − Λ2)2 + Λ2
+4 ≥ 0 ,
+(58)
+Λ0 + 1
+2(Λ1 + Λ2) + 1
+2(Λ1 − Λ2) cos (2kπ/3) + 1
+2Λ4 sin (2kπ/3) ≥ 0
+(k = 1, 2, 3) .
+(59)
+However, as shown in [21, 19], a potential which is BFB-n is not necessarily BFB along the
+charge breaking directions (BFB-c). Necessary BFB-c conditions have yet to be found for the A4
+3HDM, but sufficient conditions have been proposed in [22] following the technique developed in
+[23]. They are,
+Ad ≥ 0 ,
+Ao ≥ −Ad/2 ,
+(60)
+where
+Ad =a = 2
+3(Λ0 + Λ3) ,
+Ao =b + min(0, c) − d
+=1
+3(2Λ0 − Λ3) + 1
+2(Λ1 + Λ2) + min(0, −1
+2(Λ1 + Λ2)) − 1
+2
+�
+(Λ1 − Λ2)2 + Λ2
+4 .
+(61)
+It is important to remark that, since these are sufficient, but not necessary, conditions, some
+good points in parameter space may be excluded by this restriction.
+5.3
+The oblique parameters S, T, U
+For this we use the notation and results from [18], which require the matrices ˜U and ˜V . Comparing
+Eq. (38) with the definition in Eq. (34), we conclude that
+˜U = R†
+ch ,
+(62)
+where the matrix Rch is obtained from the numerical diagonalization of Eq. (37).
+Similarly,
+comparing Eq. (42) with the definition of ˜V in Eq. (35), we get,
+˜Vij =
+�
+RT
+neu
+�
+ij + i
+�
+RT
+neu
+�
+i+3,j .
+(63)
+Having ˜U and ˜V , we can construct the needed matrices Im
+� ˜V † ˜V
+�
+, ˜U† ˜U, ˜V † ˜V and ˜U† ˜V , and
+implement the procedure of [18].
+12
+
+5.4
+Global minimum
+After finding a set of mi,j and Λi which reproduce the vevs in Eq. (33) necessary for a good fit
+of the quark mass matrices, and after performing the previous theoretical checks on the scalar
+potential, we must still ensure that our minimum is indeed the global minimum. This step is
+almost never taken in studies of quark mass matrices, since there are no exact analytical formulae
+for it. Moreover, one must check that there are no lower minima both along the neutral directions
+and along the charge breaking directions. We follow the strategy discussed in Ref. [22]. Take a
+specific set of m2
+ij and Λi. Then we parameterize the scalar doublets as [21, 22],
+⟨φ1⟩ = √r1
+�
+0
+1
+�
+,
+⟨φ2⟩ = √r2
+�
+sin(α2)
+cos(α2)eiβ2
+�
+,
+⟨φ3⟩ = √r3eiγ
+�
+sin(α3)
+cos(α3)eiβ3
+�
+,
+(64)
+where we have already used the gauge freedom. Now we let the vevs run free, for both charge
+conserving and charge violating directions. We give one seed point and perform a minimization of
+the potential using the CERN Minuit library [16]. We obtain not only the value of the potential
+at the minimum, but also the values of ri, α2, β2, α3, β3 and γ. Then, we take one more (randomly
+generated) seed point and repeat the minimization. Finally, we take the minimum as the global
+one if it is found as the global minimum in each of 200 searches with randomly generated seed
+points. We have done this verification for every point that passed all the constraints. In all cases,
+we found that the local minimum was also a global minimum. In particular we always found that
+sin(α2) = sin(α3) = 0,
+(65)
+showing that we do not have charged breaking directions5 and, comparing with Eq. (6), we verified
+numerically that,
+|vi|
+√
+2 = √ri,
+ei ρ2 = cos(α2) ei β2,
+ei ρ3 = cos(α3) ei (β3+γ) .
+(66)
+6
+Simple LHC Constraints
+Up to now we have implemented the theoretical constraints on the model. The next step is to
+implement the LHC constraints. To do this completely one would have to implement all the decays
+of the neutral and charged Higgs as well as their branching ratios. One would also have to worry
+about the electric dipole moments (EDM) and the flavour-changing neutral couplings (FCNC), as
+the model does not have a structure of couplings of the Higgs to the fermions that automatically
+ensures vanishing FCNC [24, 25, 26]. This lies beyond the scope of the present work. Nonetheless,
+we can implement easily the constraints that come from h → WW/ZZ in the κ formalism, where
+the deviation from the coupling of the SM Higgs boson to a pair of W’s (or Z’s) is measured by
+κV . In our model,
+κV = Rneu
+21 v1 + Rneu
+22 v2 cos(ρ2) + Rneu
+23 v3 cos(ρ3) + Rneu
+25 v2 sin(ρ2) + Rneu
+26 v3 sin(ρ3),
+(67)
+where Rneu is matrix defined in Eq. (41). We take the experimental constraint from ATLAS [27],
+κW = 1.0206 +0.05172
+−0.05087,
+κZ = 0.99 +0.06136
+−0.05214 .
+(68)
+5To cross check our numerical procedure we also considered points that violated the BFB conditions. And, indeed
+for these points, our algorithm showed that the potential was not BFB and could have charge breaking directions
+as well.
+13
+
+7
+Results
+In this section we present the results of the analysis of the scalar potential after imposing that we
+have a good solution for the fit of the quarks masses and CKM entries, as explained in Section 3.
+7.1
+Scanning strategy
+We start by imposing the vevs obtained in the fit.
+v1 = 1.00625 (GeV),
+v2 = 6.90462 ei 4.27865 (GeV),
+v3 = 245.901 ei 5.37039 (GeV).
+(69)
+Now we vary the free parameters of the potential in the following ranges,
+log10 |Λi| ∈ [−3, 1],
+log10 |Im(m2
+23)| ∈ [−1, 7]GeV2,
+log10 |Re(m2
+ij)| ∈ [−1, 7]GeV2,
+(70)
+where in the last equation we use
+m2
+ij ∈
+�
+m2
+12, m2
+13, m2
+23
+�
+.
+(71)
+We randomly scan as in Eq. (70), and then:
+1. Apply the theoretical constraints that only depend on the Λi, that is BFB and perturbative
+unitarity.
+2. Then obtain the eigenvalues for the charged and neutral scalars. Verify that all the masses
+squared are positive, and that we have a zero eigenvalue corresponding to the Goldstone
+bosons, G0 and G+.
+3. Verify the S, T and U oblique parameters.
+4. Apply the LHC constraint on κV .
+5. Check numerically that the vev is indeed a global minimum.
+7.2
+The scalar spectrum
+We found that there is a strong correlation in the scalar masses.
+If we denote the masses of
+the neutral scalars by (mG0 = 0, mS0
+2, mS0
+3, mS0
+4, mS0
+5, mS0
+6), and (mG+ = 0, mH+
+1 , mH+
+2 ) for the
+charged scalars, we find numerically that
+mS0
+3 ≃ mS0
+4 ≃ mH+
+1 ,
+mS0
+5 ≃ mS0
+6 ≃ mH+
+2 .
+(72)
+This is true even if we do not require mS0
+2 = 125 GeV, and specially true after implementing
+the constraints of perturbative unitarity, BFB and STU. But, as we want to reproduce the LHC
+results, we also required that [17]
+mS0
+2 = 125.25 ± 0.17
+GeV.
+(73)
+In the following figures we show the correlation among the masses. Included in red are the
+points generated before the theoretical cuts were applied, and in green the points remaining after
+the constraints were implemented.
+14
+
+Figure 1: Left panel: Relation between mS0
+3 and mS0
+4. Right panel: Relation between mS0
+3 and
+mH+
+1 . Color conventions: No cuts (red); with cuts (green)
+Figure 2: Left panel: Relation between mS0
+5 and mS0
+6. Right panel: Relation between mS0
+5 and
+mH+
+2 . Color conventions: No cuts (red); with cuts (green)
+15
+
+2000
+1500
+1000
+500
+0
+500
+1000
+1500
+2000
+ms (GeV)2000
+1500
+mHt (GeV)
+1000
+500
+0
+500
+1000
+1500
+2000
+ms (GeV)2000
+1500
+1000
+500
+0
+500
+1000
+1500
+2000
+msg (GeV)2000
+1500
+ (GeV)
+1000
+500
+0
+500
+1000
+1500
+2000
+msg (GeV)Figure 3: Left panel: Relation between mS0
+3 and mS0
+5. Right panel: Relation between mH+
+1 and
+mH+
+2 . Color conventions: No cuts (red); with cuts (green)
+7.3
+The κV constraint
+We can now implement the κV constraint on the model. In the following figures, in red are points
+without cuts, in green with cuts but no κV constraint, and finally in blue points remaining after
+this constraint is applied. We took the ATLAS result of Eq. (68) at 2σ. While the theoretical
+constraints cut around 99% of the points, the κV constraint only cuts 9% of the remaining points.
+In Fig. 4 we show the relation between κV and Λ0,4 for the three sets of points as discussed above.
+Figure 4: Left panel: Relation between κV and Λ0. Right panel: Relation between κV and Λ4.
+Color conventions: No cuts (red), with theoretical cuts (green), and after the κV constraint (blue).
+In fact it is not obvious from Fig. 4 that the κV constraint only cuts about 9% of the points.
+This is because there is a very large number of points with |κV | ≲ 1, even without theoretical
+cuts, and this is even more so after imposing the theoretical cuts. In this figure, we have 157810
+points in the green region, but from these 145074 are in the blue region. That is, after theoretical
+16
+
+2000
+1500
+1000
+500
+500
+1000
+1500
+2000
+ms (GeV)2000
+..
+1500
+ (GeV)
+1000
+H
+500
+500
+1000
+1500
+2000
+mHt (GeV)10
+5
+.5
+-10
+0
+0.2
+0.4
+0.6
+0.8
+[Ky]10
+4
+-5
+-10
+0
+0.2
+0.4
+0.6
+0.8
+[Kvlcuts, 91% of the points also satisfy the κV constraint. In Fig. 5 we show the relation between Λ0
+and Λ3,4 for the same sets of points.
+Figure 5: Left panel: Relation between Λ0 and Λ4. Right panel: Relation between Λ0 and Λ3
+Color conventions: No cuts (red), with theoretical cuts (green), and after the κV constraint (blue).
+We see that, while for (Λ0, Λ4) there is not much difference before and after the κV constraint,
+the same is not true for (Λ0, Λ3), where the constraints impose a linear relation between those
+two parameters.
+8
+Conclusions
+It is known that the 3HDM symmetric under an exact A4 symmetry is not compatible with non-
+zero quark masses and/or non-block-diagonal CKM matrix [12]. In this work, we studied a 3HDM
+with A4 softly broken. This allows us to evade the above result, by enlarging the structure of the
+possible vacua.
+We obtained an excellent fit of the quarks mass matrices, including the CP-violating Jarlskog
+invariant. This leads to a unique solution for the vevs. We showed that, with the solution for the
+vevs obtained from the fit, it is possible to have a local minimum of the potential. We enforce
+this by imposing that all squared masses are positive. As in our scheme the scalar masses are not
+input parameters, we have to restrict one of the neutral scalars to have the mass of the known
+Higgs boson.
+We have implemented the BFB, perturbative unitarity and the oblique parameters S, T, U
+theoretical constraints.
+From LHC, we have considered the observed Higgs mass and the κV
+constraint.6 After imposing the other constraints, we found that most of the points are close to
+the alignment required to respect the experimental κV constraint. We have discovered a strong
+correlation among the masses of the scalars, even before applying the theoretical constraints,
+especially for moderate to large scalar masses.
+One important point is that we have numerically checked for all the points that pass our
+constraints, that for a given set of parameters of the potential, our minimum is the true global
+minimum.
+6The detailed study of other LHC constraints as well as those coming from FCNC and the EDM lies beyond the
+scope of the present work, and is left for a future publication.
+17
+
+0.6
+0.4
+0.2
+0
+0
+0.2
+0.4
+0.6
+No0.4
+0.2
+2
+0
+-0.2
+-0.4
+0
+0.1
+0.2
+0.3
+0.4
+NoAcknowledgments
+This work is supported in part by FCT (Fundação para a Ciência e Tecnologia) under Contracts
+CERN/FIS-PAR/0002/2021, CERN/FIS-PAR/0008/2019, UIDB/00777/2020, and UIDP/00777/2020;
+these projects are partially funded through POCTI (FEDER), COMPETE, QREN, and the EU.
+The work of I.B. was supported by a CFTP fellowship with reference BL210/2022-IST-ID and the
+work of S. C. by a CFTP fellowship with reference BL255/2022-IST-ID.
+A
+The matrices Hd and Hu
+Hd(1, 1) =Σdv2 cos2(β1) cos2(β2)
+(74)
+Hd(1, 2) =v2Zd cos(β1) cos2(β2) cos(ρ2 − φd) sin(β1)
+− i v2Zd cos(β1) cos2(β2) sin(β1) sin(ρ2 − φd)
+(75)
+Hd(1, 3) =v2Zd cos(β1) cos(β2) cos(ρ3 + φd) sin(β2)
+− i v2Zd cos(β1) cos(β2) sin(β2) sin(ρ3 + φd)
+(76)
+Hd(2, 1) =(Hd(1, 2))∗
+(77)
+Hd(2, 2) =Σdv2 cos2(β2) sin2(β1)
+(78)
+Hd(2, 3) =v2Zd cos(β2) cos(ρ2 − ρ3 + φd) sin(β1) sin(β2)
++ i v2Zd cos(β2) sin(β1) sin(β2) sin(ρ2 − ρ3 + φd)
+(79)
+Hd(3, 1) =(Hd(1, 3))∗
+(80)
+Hd(3, 2) =(Hd(2, 3))∗
+(81)
+Hd(3, 3) =Σdv2 sin2(β2)
+(82)
+Hu(1, 1) =Σuv2 cos2(β1) cos2(β2)
+(83)
+Hu(1, 2) =v2Zu cos(β1) cos2(β2) cos(ρ2 − φu) sin(β1)
+− i v2Zu cos(β1) cos2(β2) sin(β1) sin(ρ2 − φu)
+(84)
+Hu(1, 3) =v2Zu cos(β1) cos(β2) cos(ρ3 + φu) sin(β2)
+− i v2Zu cos(β1) cos(β2) sin(β2) sin(ρ3 + φu)
+(85)
+Hu(2, 1) =(Hu(1, 2))∗
+(86)
+Hu(2, 2) =Σuv2 cos2(β2) sin2(β1)
+(87)
+Hu(2, 3) =v2Zu cos(β2) cos(ρ2 − ρ3 + φu) sin(β1) sin(β2)
++ i v2Zu cos(β2) sin(β1) sin(β2) sin(ρ2 − ρ3 + φu)
+(88)
+Hu(3, 1) =(Hu(1, 3))∗
+(89)
+Hu(3, 2) =(Hu(2, 3))∗
+(90)
+Hu(3, 3) =Σuv2 sin2(β2)
+(91)
+B
+The minimization conditions
+m2
+11 = − sec(ρ2) sec(ρ3)
+24v2
+1
+�
+−12Im(m2
+23)v2v3 sin(2(ρ2 − ρ3)) + cos(ρ2 − ρ3)
+�
+4Λ0v2
+1v2
+18
+
++6Λ1v2
+1v2
+2 + 6Λ1v2
+1v2
+3 + 3Λ1v2
+2v2
+3 − 3Λ2v2
+2v2
+3 + 2Λ3v2
+1
+�
+2v2
+1 − v2
+2 − v2
+3
+��
++4Λ0v2
+1v2
+2 cos(ρ2 + ρ3) + 4Λ0v2
+1v2
+3 cos(ρ2 + ρ3) + 4Λ0v4
+1 cos(ρ2 + ρ3)
++6Λ1v2
+1v2
+2 cos(ρ2 + ρ3) + 6Λ1v2
+1v2
+3 cos(ρ2 + ρ3) − 3Λ1v2
+2v2
+3 cos(3(ρ2 − ρ3))
++3Λ2v2
+2v2
+3 cos(3(ρ2 − ρ3)) − 2Λ3v2
+1v2
+2 cos(ρ2 + ρ3) − 2Λ3v2
+1v2
+3 cos(ρ2 + ρ3)
++4Λ3v4
+1 cos(ρ2 + ρ3) + 3Λ4v2
+1v2
+2 sin(ρ2 − ρ3) + 3Λ4v2
+1v2
+2 sin(ρ2 + ρ3)
++3Λ4v2
+1v2
+3 sin(ρ2 − ρ3) − 3Λ4v2
+1v2
+3 sin(ρ2 + ρ3) − 3Λ4v2
+2v2
+3 sin(ρ2 − ρ3)
++3Λ4v2
+2v2
+3 sin(3(ρ2 − ρ3)) − 12Re(m2
+23)v2v3 cos(2(ρ2 − ρ3))
++24Re(m2
+13)v1v3 cos(ρ2) + 24Re(m2
+12)v1v2 cos(ρ3) + 12Re(m2
+23)v2v3
+�
+(92)
+m2
+22 = −
+1
+12v2
+�
+3 sec(ρ2)
+�
+−4Im(m2
+23)v3 sin(ρ3) + v2v2
+3(Λ1 − Λ2) cos(ρ2 − 2ρ3) − Λ4v2v2
+3 sin(ρ2 − 2ρ3)
++4Re(m2
+23)v3 cos(ρ3) + 4Re(m2
+12)v1
+�
++ v2
+�
+4Λ0v2 + 6Λ1v2
+1 + 3Λ1v2
+3 + 3Λ2v2
+3
+−2Λ3v2
+1 + 4Λ3v2
+2 − 2Λ3v2
+3 + 3Λ4v2
+1 tan(ρ2)
+� �
+(93)
+m2
+33 = −
+1
+12v3
+�
+3 sec(ρ3)
+�
+4Im(m2
+23)v2 sin(ρ2) + v2
+2v3(Λ1 − Λ2) cos(2ρ2 − ρ3) − Λ4v2
+2v3 sin(2ρ2 − ρ3)
++4Re(m2
+23)v2 cos(ρ2) + 4Re(m2
+13)v1
+�
++ v3
+�
+4Λ0
+�
+v2
+1 + v2
+2 + v2
+3
+�
++ 6Λ1v2
+1 + 3Λ1v2
+2
++3Λ2v2
+2 − 2Λ3v2
+1 − 2Λ3v2
+2 + 4Λ3v2
+3 − 3Λ4v2
+1 tan(ρ3)
+� �
+(94)
+Im(m2
+12) = 1
+4v1
+�
+sec(ρ2)
+�
+4Im(m2
+23)v3 cos(ρ2 − ρ3) − Λ1v2v2
+3 sin(2(ρ2 − ρ3)) − Λ1v2
+1v2 sin(2ρ2)
++Λ2v2v2
+3 sin(2(ρ2 − ρ3)) + Λ2v2
+1v2 sin(2ρ2) − Λ4v2v2
+3 cos(2(ρ2 − ρ3))
++Λ4v2
+1v2 cos(2ρ2) − 4Re(m2
+23)v3 sin(ρ2 − ρ3) − 4Re(m2
+12)v1 sin(ρ2)
+� �
+(95)
+Im(m2
+13) = − 1
+4v1
+�
+sec(ρ3)
+�
+4Im(m2
+23)v2 cos(ρ2 − ρ3) − Λ1v2
+2v3 sin(2(ρ2 − ρ3)) + Λ1v2
+1v3 sin(2ρ3)
++Λ2v2
+2v3 sin(2(ρ2 − ρ3)) − Λ2v2
+1v3 sin(2ρ3) − Λ4v2
+2v3 cos(2(ρ2 − ρ3))
++Λ4v2
+1v3 cos(2ρ3) − 4Re(m2
+23)v2 sin(ρ2 − ρ3) + 4Re(m2
+13)v1 sin(ρ3)
+� �
+(96)
+References
+[1] ATLAS collaboration, Observation of a new particle in the search for the Standard Model
+Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1 [1207.7214].
+[2] CMS collaboration, Observation of a New Boson at a Mass of 125 GeV with the CMS
+Experiment at the LHC, Phys. Lett. B 716 (2012) 30 [1207.7235].
+19
+
+[3] E. Ma and G. Rajasekaran, Softly broken a(4) symmetry for nearly degenerate neutrino
+masses, Phys. Rev. D64 (2001) 113012 [hep-ph/0106291].
+[4] E. Ma, Quark mass matrices in the a(4) model, Mod. Phys. Lett. A17 (2002) 627
+[hep-ph/0203238].
+[5] E. Ma, H. Sawanaka and M. Tanimoto, Quark Masses and Mixing with A4 Family
+Symmetry, Phys. Lett. B 641 (2006) 301 [hep-ph/0606103].
+[6] E. Ma, Suitability of a(4) as a family symmetry in grand unification, Mod. Phys. Lett. A21
+(2006) 2931 [hep-ph/0607190].
+[7] L. Lavoura and H. Kuhbock, A(4) model for the quark mass matrices, Eur. Phys. J. C 55
+(2008) 303 [0711.0670].
+[8] S. Morisi and E. Peinado, An A4 model for lepton masses and mixings, Phys. Rev. D80
+(2009) 113011 [0910.4389].
+[9] A. Degee, I.P. Ivanov and V. Keus, Geometric minimization of highly symmetric potentials,
+JHEP 02 (2013) 125 [1211.4989].
+[10] M. Leurer, Y. Nir and N. Seiberg, Mass matrix models, Nucl. Phys. B 398 (1993) 319
+[hep-ph/9212278].
+[11] R. González Felipe, I.P. Ivanov, C.C. Nishi, H. Serôdio and J.P. Silva, Constraining
+multi-Higgs flavour models, Eur. Phys. J. C 74 (2014) 2953 [1401.5807].
+[12] R. González Felipe, H. Serôdio and J.P. Silva, Models with three Higgs doublets in the triplet
+representations of A4 or S4, Phys. Rev. D 87 (2013) 055010 [1302.0861].
+[13] M.P. Bento, J.C. Romão and J.P. Silva, Unitarity bounds for all symmetry-constrained
+3HDMs, JHEP 08 (2022) 273 [2204.13130].
+[14] C. Jarlskog, Commutator of the Quark Mass Matrices in the Standard Electroweak Model
+and a Measure of Maximal CP Nonconservation, Phys. Rev. Lett. 55 (1985) 1039.
+[15] G.C. Branco and L. Lavoura, Rephasing Invariant Parametrization of the Quark Mixing
+Matrix, Phys. Lett. B 208 (1988) 123.
+[16] F. James and M. Roos, Minuit: A System for Function Minimization and Analysis of the
+Parameter Errors and Correlations, Comput. Phys. Commun. 10 (1975) 343.
+[17] Particle Data Group collaboration, Review of Particle Physics, PTEP 2022 (2022)
+083C01.
+[18] W. Grimus, L. Lavoura, O.M. Ogreid and P. Osland, A Precision constraint on
+multi-Higgs-doublet models, J. Phys. G35 (2008) 075001 [0711.4022].
+[19] I.P. Ivanov and F. Vazão, Yet another lesson on the stability conditions in multi-Higgs
+potentials, JHEP 11 (2020) 104 [2006.00036].
+[20] N. Buskin and I.P. Ivanov, Bounded-from-below conditions for A4-symmetric 3HDM, J.
+Phys. A 54 (2021) 325401 [2104.11428].
+20
+
+[21] F.S. Faro and I.P. Ivanov, Boundedness from below in the U(1) × U(1) three-Higgs-doublet
+model, Phys. Rev. D 100 (2019) 035038 [1907.01963].
+[22] S. Carrôlo, J.C. Romão and J.P. Silva, Conditions for global minimum in the A4 symmetric
+3HDM, Eur. Phys. J. C 82 (2022) 749 [2207.02928].
+[23] R. Boto, J.C. Romão and J.P. Silva, Bounded from below conditions on a class of symmetry
+constrained 3HDM, Phys. Rev. D 106 (2022) 115010 [2208.01068].
+[24] S.L. Glashow and S. Weinberg, Natural Conservation Laws for Neutral Currents, Phys. Rev.
+D 15 (1977) 1958.
+[25] P.M. Ferreira, L. Lavoura and J.P. Silva, Renormalization-group constraints on Yukawa
+alignment in multi-Higgs-doublet models, Phys. Lett. B 688 (2010) 341 [1001.2561].
+[26] K. Yagyu, Higgs boson couplings in multi-doublet models with natural flavour conservation,
+Phys. Lett. B 763 (2016) 102 [1609.04590].
+[27] ATLAS collaboration, A detailed map of Higgs boson interactions by the ATLAS
+experiment ten years after the discovery, Nature 607 (2022) 52 [2207.00092].
+21
+

I9E0T4oBgHgl3EQfiAGs/content/tmp_files/2301.02440v1.pdf.txt

@@ -0,0 +1,1178 @@
+XXX-X-XXXX-XXXX-X/XX/$XX.00 ©20XX IEEE
+An Image captioning algorithm based on the Hybrid Deep Learning
+Technique (CNN+GRU)
+Rana Adnan Ahmad
+Department of Computer Science
+Comsats university islamabad, sahiwal
+campus,Sahiwal,Pakistan
+[email protected]
+Muhammad Azhar
+Department of computer science
+Comsats university islamabad, sahiwal
+campus,Sahiwal,Pakistan
+[email protected]
+Chosun University, Republic of Korea
+[email protected]
+Hina Sattar
+Department of computer science
+Comsats university islamabad, sahiwal
+campus, Sahiwal, Pakistann
+[email protected]
+Abstract—
+Image
+captioning
+by
+the
+encoder-decoder
+framework has shown tremendous advancement in the last
+decade where CNN is mainly used as encoder and LSTM is
+used as a decoder. Despite such an impressive achievement in
+terms of accuracy in simple images, it lacks in terms of time
+complexity and space complexity efficiency. In addition to this,
+in case of complex images with a lot of information and objects,
+the
+performance
+of
+this
+CNN-LSTM
+pair
+downgraded
+exponentially due to the lack of semantic understanding of the
+scenes presented in the images. Thus, to take these issues into
+consideration,
+we
+present
+CNN-GRU
+encoder
+decode
+framework for caption-to-image reconstructor to handle the
+semantic context into consideration as well as the time
+complexity. By taking the hidden states of the decoder into
+consideration, the input image and its similar semantic
+representations is reconstructed and reconstruction scores
+from a semantic reconstructor are used in conjunction with
+likelihood during model training to assess the quality of the
+generated caption. As a result, the decoder receives improved
+semantic
+information,
+enhancing
+the
+caption
+production
+process. During model testing, combining the reconstruction
+score and the log-likelihood is also feasible to choose the most
+appropriate caption. The suggested model outperforms the
+state-of-the-art LSTM-A5 model for picture captioning in
+terms of time complexity and accuracy.
+Index Terms— Deep Learning, Image captioning, CNN, GRU
+1.
+INTRODUCTION
+Deep Learning has made great strides recently due to rapid
+growth and high utilization [1-4]. Thus, similar to Neural
+Machine Translation (NMT)[5], generating captions of the
+images through neural encoder-decoder framework has
+shown the dominance in recent years. In this process of
+image captioning, encoding of the image is done through
+encoder which is typically from the Convolutional Neural
+Networks (CNN) [6] family (like Vanilla CNN [6], Region
+based CNN [7], Fast R-CNN [8], Faster R-CNN [9] etc.) and
+decoder is from the RNN family [10] (like LSTM [11],
+BLSTM [12] etc.). In this framework of CNN-LSTM pair
+[13-16], the encoder (CNN) learns the visual features by
+making the feature maps and max-pooling during the feature
+learning stage and then detection of objects after flattening
+and applying fully connected layer. Thus it converts the
+image to vector of numbers which is learned form of the
+visual content of the image under consideration. In the
+decoder part, the vector output of the encoder is used as the
+initial input of the decoder to produce caption word by word.
+Even though Long Short Term Memory (LSTM) solves the
+issue of handling long dependency by decreasing the effect
+of
+exploding
+and
+vanishing
+gradients
+[11],
+the
+time
+complexity issue is still a major drawback in this model due
+to
+many
+gates
+residing
+in
+the
+LSTM
+unit
+for
+the
+memorization purpose. Another key issue with these kind of
+encoder-decoder models are the lack of understanding of the
+semantic context as the encoder of these models fail to
+transfer the major key visual information to the decoder.
+Because of the absence of reverse dependency checking
+(Caption-to-Image), these models do not perform well in
+case of complex images.
+Several approaches have been proposed to deal with the
+above-mentioned issues [17-20]. Some researches have
+proposed the attention mechanism to get the information
+from the key regions automatically and tried to encode that
+specific information into the context vector which then used
+by the decoder to generate the caption [17,18]. Some other
+researchers have tried to extract semantic attributes as a
+supplement of the CNN features to embed into encoder by
+various methods [19, 20].
+The major drawback of all the above mentioned methods
+was that those methods only explore the image-to-caption
+dependency but not the reverse way for the validation of the
+extracted information. Even though, Jinsong Su et.el. [21]
+have tried to use the semantic reconstructor of caption-to-
+image but still they could not
+validated the results in
+effective way. In addition to this, the time complexity issue
+was also remained due to the usage of LSTM unit.
+To resolve the above mentioned issues, we have proposed a
+hybrid deep learning technique based on the CNN-GRU
+encoder-decoder gramework with the better hyper-parameter
+tuning and with the caption-to-image validation method by
+taking the motivation from Jinsong Su et.el. [21]. This
+caption-to-image reconstructor helps to handle the semantic
+context into consideration as well as the time complexity. By
+taking the hidden states of the decoder into consideration, the
+input image and its similar semantic representations is
+reconstructed and reconstruction scores from a semantic
+reconstructor are used in conjunction with likelihood during
+model training to assess the quality of the generated caption.
+As a result, the decoder receives improved semantic
+information, enhancing the caption production process.
+During model testing, combining the reconstruction score
+and the log-likelihood is also feasible to choose the most
+appropriate caption.
+To validate our proposed method, we have used the
+benchmark MS COCO dataset [22] and the experimental
+results have proved that our method outperformed the current
+state-of-the-art methods in terms of accuracy and time
+complexity.
+2.
+RELATED WORK
+
+Inspiration for our work comes from the auto encoder [23,24]
+and how well it performs in NMT [25], which employs
+semantic production to hone the learning representation of
+input data. In this activity, we are fine-tuning the idea with
+captions to the image. Basically, related work involves
+taking after two strands. In general, NMT's common hands
+are very much based on the demonstration of source-to-target
+interpretation. Encouraged by questions about the auto
+encoder that makes reproduction more realistic and looking
+at whether the recreated inputs are more reliable than the
+original inputs [26], many analysts are committed to using
+the adaptation of dual-directed NMT conditions [27].
+Compared with NMT, most models of neural image captions
+are based on the neural encoder-decoder system [30].
+However, this engineer cannot guarantee that the image data
+can be completely converted into a decoder. To discuss this
+problem, analysts are currently accepting to take after two
+types of approaches: (1) As in NMT attention [31], a few
+analysts link part of the visual considerations to capture the
+semantic presentations of critical image regions [32,33]. (2)
+In various ways, a number of analysts are committed to
+extract semantic features or high-level concepts into images,
+which can be integrated into an LSTM-based decoder as an
+additional input [28,29]. In this way, the show will be
+directed to settings that are closely related to the theme of the
+image. Besides, You et al. [34] encompassed the two types
+of methods listed above.
+Our proposed representation is based on the CNN-LSTM
+model, in which the proposed semantic reconstructor is
+comparably compared to the LSTM, which is why it benefits
+both to display preparation and testing when the regional
+language indicate and the coding system are modeled
+independently. From Wena et al. [35] Institution devoted to
+improving
+automatically
+generated
+image
+captions
+by
+making inferences about their semantic content. However,
+the visual highlights are generally employed as the decode of
+the decoder in the current model captions, while the semantic
+elements of the image are provided exclusively to the
+decoder. As a result, we agree that visual robustness is more
+crucial than semantic characteristics. Through this research,
+we provide semantic features to neural machine translation
+as
+well
+as
+video
+captions.
+In
+conclusion,
+we
+are
+experimenting
+with
+three
+different
+methods
+for
+reconstructing
+photos
+based
+on
+fabricated
+captions.
+Additionally, our claim may be distinct from earlier studies
+due to K's extensive utilization of visually similar photos.
+3.
+PROPOSED MODEL
+This section describes the proposed hybrid deep learning
+approach based on CNN-GRU encoder-decoder framework.
+This framework has 3 major parts, 1) Encoder: which is the
+CNN. 2) Decoder: which is the GRU layer and 3) The
+Semantic validator: for validation of the caption-to-image
+information.
+A.
+Model architecture
+The three neural network modules (Encoder, Decoder, and
+Semantic validator) that make up our proposed model are
+depicted in Fig. 1. The details of each module is given below:
+
+Encoder
+In encoder, a model similar to [36] has been used where the
+image I is taken as input and the features from the image F is
+extracted by the CNN-based encoder. The feature vector F
+∈ RDv is used to represent the features extracted from the
+image I. Dv represents the diemnsions of the feature vector.
+As all the sementic information can not be extracted by one
+feature vector, thus additional semantic attributes have been
+extracted by the algorithm proposed by Yao et al. [36]. The
+extracted attribute vector is denoted by A ∈ RDa which
+shows the probabilty of each high level attribute existed in
+the caption dataset which is generated by the MIL (Multiple
+Instance Learning) model presented in [27]. MIL model
+showed the promising results in finding the semantic
+relations between the attributes of the image. Da represents
+the diemnsions of the attribute vector A.
+After extracting both feature map F and attribute vector A,
+the encoder gives these 2 outputs to decoder as an input
+which is used for the caption generation purpose.
+
+Decoder
+As we got feature vector F and the attribute vector A as an
+output of the encoder from previous network, this F and A is
+used as the input to the decoder for the caption generation.
+Yao et al. [36] proposes 5 different and diverse variants for
+the LSTM network and it is proved that the fifth one named
+LSTM-A5 works better than others, so we also used the same
+network for getting the better performance. Thus, according
+to LSTM-A5, we used the A and F vectors to calculate the log
+Probability Ɛ as mentioned in Equation (1).
+Ɛ(S|I) = Ɛ(S|F, A) =
+1
+Ns
+t=
+Ɛ(wt | F, A, w<t)
+(1)
+Where F and A represents the feature vector and attribute
+vector respectively. S is the set of words generated by the
+attribute vector F. S= {w1,w2,…wNs} and Ns is the size of
+the set S. I is the actual image.
+The log probability Ɛ(wt | F, A, w<t) is directly proportional
+to the expection of (
+T
+tw
+E (
+t
+vh + b) as shown in Equation
+(2).
+Ɛ(wt |F, A, w <t) ∝ exp (
+T
+tw
+E (
+t
+vh + b)
+(2)
+Where E represents the matrix of the word embeddings, v
+denotes the corresponding matrix while b shows the bais. ht
+is the hidden state. The hidden state calculation is discussed
+in detail in [7].
+
+Semantic validator of caption to image
+As shown in Fig. 1, the semantic redesign of the description
+to picture work to recreate the semantic demonstration of
+every single input image since its comparative captions.
+
+Figure 1: Provides an overview of our proposed model's
+architecture, which
+consists
+of
+three neural
+networks
+(Encoder, Decoder, and Semantic Reconstructor).
+Naturally, a complete semantic reconstructor must meet the
+subsequent dual requirements: On the other
+side, its
+reconstructed presentation must be precise and sufficient to
+replicate image data; on the other side, its use is not
+compiled to have the greatest impact on professionalism. In
+this case, we are referring directly to the caption S that has
+the coverings h = {h1, h2,. . ., hNs} play a significant part in
+the description era. At that point, in this framework, we are
+examining
+three
+semantic
+functions
+to
+determine
+the
+semantic demonstration of the created captions, represented
+by hc, which can help to recreate the reconstituted direction
+of the input image, labeled Ir.
+.
+
+Model Training
+The training set Dtrain = {(F, A, S)}, is used to generate the
+objective function as follows:
+O(D; θed, θdr) = arg max(θed , θdr)
+(I,S)∈D
+{Ɛ(S|I; θed ) +
+�
+λ · R({I1, . . . , IK}|S)}
+(3)
+So we have to maximize the reconstruction score based on
+θed, θdr. θed, θdr represents the encoder and decoder model
+parameters. λ is the hyper-parameters.
+
+Model Testing
+During testing, semantic reconstruction can be utilized to
+improve selected captions. As is it shown in Fig 2, we use a
+multi-stage system that combines beam search and position
+reset. Inserted image captioning techniques:
+Figure 2: A test image of our model. h, P, and R
+show
+the
+hidden
+sequence,
+log
+likelihood
+probability,
+and
+caption
+reconstruction
+score,
+respectively.
+1.
+A collection of applicant captions, log possibilities, and
+unobserved state sequences are generated using the
+standard decoder components via an initial application
+beam investigation.
+2.
+After that, we use the hidden captions of each candidate
+to reconstruct the semantic model of the merged image
+by computing the appropriate reconstructive points.
+
+<cand1,h1.p1>
+<cand2,h2,p2>
+INPUT
+Encoding
+Reconstruction
+Decoding
+Image
+<candk,hk.pk>
+<cand1,p1+gR1>
+<cand2,p2+gR2>
+Caption
+Max
+111
+<candk.pk+gRk>SmallerImage
+DataSet
+Three
+persons
+<aos>
+CNNFeature
+GRU
+GRU
+GRU
+As
+Reconstructed
+Features
+660
+<bos>
+Three
+during
+Caption (S)
+Image (0)
+Image (1)
+Pr(S/l)
+R(1/s)
+LSTM-A5 Model:
+Threepersons standingtogether
+Our Proposed CNN+GRU : Three person standing together on beach during sunset
+Ground Truth:
+Threeperson standingtogether onbeach holding each other
+hand during sunset3.
+After arranging a log and potential school rebuilding
+sites, we calculate the final outcome of each caption
+and select final captions based on the combination of
+points.
+4.
+EXPERIMENTS
+The experiments have been conducted on the most popular
+benchmark dataset COCO [30] to compare the performance
+of our image captioning proposed model with other state-of-
+the-art methods.
+
+Experimental setup
+COCO data-set was used to check the validity of our
+proposed model which contains 130000 manually annotated
+images. Each image has 4 descriptions which were used for
+the training purpose. In addition to this, 5000 images were
+used as testing dataset.
+Out of the 130000 images of the training set, 80000 images
+were used for the training purpose while 5000 images were
+used for the validation purpose. Based on these settings, the
+vocabolary has been built with 8500 unique words. For
+getting the image features, the following setting was used for
+the hyper-paramter tuning.
+Adam [32] is used as the optimizer. We employed stop-
+reading techniques [33] and pre-stop techniques, and we
+determined
+the
+following
+take-out
+hyper-parameter
+parameters: reading level beginning at 2 4, input rate as 300,
+covering layer size as -1024, mini-batch. A maximum cycle
+count of 30 is used with a scale of 1024. We used
+Word2vec's
+[34]
+pre-trained
+embeddings,
+which
+we
+optimized by setting the tradeoff parameter to 1. The
+threshold was established at 3 in our model testing.
+
+Evaluation metrices used:
+The evaluation metrices used are 1) BLEU [40] where we set
+beam size K=3 thus BLEU@1, BLEU@2, BLEU@3 and
+BLEU@4 are calculated. In addition to this, METEOR [46]
+which is shown as M in Table 1, ROUGE-L [37] which is
+shown as R and CIDEr-D [38] which is shown as C in the
+Table1. The values of these metrices were calculated by the
+COCO
+released
+code
+[39].
+BLEU,
+ROUGE-L,
+and
+METEOR were initially developed as benchmarks for
+evaluating the accuracy of machine translation. Image
+caption testing follows the same procedure as machine
+translation testing, where the phrases generated are compared
+to the actual sentences, and metrics are often utilized.
+
+Description of the compared state-of-the-art methods
+1) NIC: The decoder of NIC is based on LSTM which
+directly use features of the images as input to LSTM.
+2) ME: The distinction of this method is its language model
+that explore the mappings bidirectionally in images and their
+captions. This language model is independently built from
+the encoder-decoder framework.
+3) ATT: This model uniquely extracts the key information of
+the images by a model based on semantic attention.
+4) Soft-Attention and Hard Attention (SA and HA) models:
+This model differs from other models in terms of using CNN
+features as input to decoder. The Soft-Attention (SA) is with
+the
+normal
+Back-propagation
+method
+while
+in
+Hard-
+Attention (HA), the stochastic attention is used with re-
+inforcement learning.
+5) LRCN: It is unique in terms of taking the image feature
+and its previous caption as the input at each time-step.
+6) Sentence
+Condition
+(SE):
+In
+this
+method,
+a
+text-
+conditional attention model is used which helps decoder to
+learn the semantic information of the text.
+7) LSTM-A5: This is based on the best variant of LSTM.
+Our propsoed model is inspired by this. We have used the
+same settings of LSTM-A5 for comparison purpose as the
+dataset is also same.
+
+Test results on COCO
+The results got from the experiments by using the COCO
+dataset is shown in the Table 1. As it is obvious from the
+results, our method performed better than all other state-of-
+the-art methods. The results of the metreces BLEU@1,
+BLEU@2, BLEU@3 and BLEU@4 are all better than the
+NIC, HA, SA, ATT, ME, and other compared methods. Even
+on the metrics METEOR [37] which is shown as M in Table
+1, ROUGE-L [38] which is shown as R and CIDEr-D which
+is shown as C in the Table1, which were initially developed
+as benchmarks for evaluating the accuracy of machine
+translation, our resulting indexes are still better on the above
+metrices as compared to NIC, HA, SA, ATT, ME, and other
+compared methods.
+These results proved that the proposed CNN-GRU method
+with semantic validator of caption-to-image is working
+perfectly.
+Table 1: The performance of our proposed model against
+other state-of-the-art methods building VGG framework or
+GoogleNet framework. For clarity, B@K is for BLEU@K
+where K={1,2,3,4}, MET is used for METEOR, ROU is
+represents ROUGE-L, and CID is used for CIDER-D.
+Model
+B@1
+B@2
+B@3
+B@4
+MET
+ROU
+CID
+SA [5]
+0.700
+0.490
+0.322
+0.242
+0.238
+-
+-
+ME [26]
+0.731
+0.559
+0.429
+0.299
+0.246
+0.529
+1.001
+ATT [12]
+0.699
+0.527
+0.399
+0.299
+0.232
+-
+-
+SC [15]
+0.719
+0.540
+0.400
+0.297
+0.239
+-
+0.94
+HA[5]
+0.715
+0.503
+0.355
+0.249
+0.229
+-
+-
+NIC [6]
+0.659
+0.449
+0.399
+0.202
+-
+-
+-
+LRCN
+[41]
+0.690
+0.514
+0.379
+0.270
+0.230
+0.500
+0.830
+LSTM-
+A5
+0.729
+0.559
+0.429
+0.325
+0.253
+0.539
+1.002
+Proposed
+CNN+G
+RU
+0.751
+0.578
+0.439
+0.335
+0.259
+0.545
+1.035
+
+Test results on COCO's online test server
+Table 2: Performance comparisons on online COCO
+test server (C40). MS Captivator is a photo caption
+model suggested by Fang et al. [27].
+Model
+B@1
+B@2
+B@3
+B@4
+MET
+ROU
+CID
+SA [5]
+0.721
+0.494
+0.333
+0.251
+0.242
+0.511
+0.98
+ME [26]
+0.743
+0.561
+0.432
+0.293
+0.251
+0.534
+1.013
+ATT [12]
+0.691
+0.532
+0.393
+0.287
+0.242
+-
+-
+SC [15]
+0.729
+0.544
+0.412
+0.281
+0.241
+0.511
+0.92
+HA[5]
+0.725
+0.512
+0.358
+0.253
+0.231
+-
+-
+NIC [6]
+0.669
+0.456
+0.382
+0.218
+0.232
+-
+-
+LRCN [41]
+0.698
+0.521
+0.382
+0.281
+0.241
+0.512
+0.812
+LSTM-A5
+0.731
+0.562
+0.432
+0.331
+0.258
+0.542
+1.000
+Proposed
+0.742
+0.583
+0.441
+0.339
+0.263
+0.556
+1.012
+
+CNN+GRU
+Table 3: Performance comparisons on online COCO
+test server (C40). MS Captivator is a photo caption
+model suggested by Fang et al. [27].
+Model
+B@1
+B@2
+B@3
+B@4
+MET
+ROU
+CID
+SA [5]
+0.898
+0.800
+0.605
+0.505
+0.347
+0.686
+0.940
+ME [26]
+0.900
+0.781
+0.701
+0.575
+0.340
+0.685
+0.864
+ATT [12]
+0.897
+0.825
+0.605
+0.505
+0.345
+0.680
+0.928
+SC [15]
+0.900
+0.808
+0.705
+0.505
+0.343
+0.685
+0.919
+HA[5]
+0.899
+0.804
+0.622
+0.515
+0.345
+0.682
+0.940
+NIC [6]
+0.901
+0.809
+0.711
+0.510
+0.337
+0.680
+0.952
+LRCN [41]
+0.902
+0.814
+0.710
+0.512
+0.339
+0.680
+0.947
+LSTM-A5
+0.903
+0.816
+0.702
+0.602
+0.338
+0.686
+0.964
+Proposed
+CNN+GRU
+0.904
+0.818
+0.712
+0.603
+0.343
+0.687
+0.967
+To further confirm the validity of our model, the COCO's
+online test server was used to evaluate the performance on
+the test set. In particular, the captions made by proposed
+CNN-GRU
+were
+uploaded
+to
+the
+server
+to
+do
+the
+comparisons with the baseline models. The official test
+images, 5 (c5) reference captions, and 40 (c40) reference
+captions used in Table 2 and Table 3 are shown. It is clearly
+seen again on the results that our method outperformed all
+other state-of-the-art methods. The results of the metreces
+BLEU@1, BLEU@2, BLEU@3 and BLEU@4 are all better
+than the NIC, HA, SA, ATT, ME, and other compared
+methods. Even on the metrics METEOR which is shown as
+M in Table 2 and Table 3, ROUGE-L which is shown as R
+and CIDEr-D which is shown as C in the Table 2 and Table
+3, which were initially developed as benchmarks for
+evaluating the accuracy of machine translation, our resulting
+indexes are still better on the above metrices as compared to
+NIC, HA, SA, ATT, ME, and other compared methods.
+Fig.3 Model performance with different K.
+5.
+CONCLUSION AND FUTURE WORK
+In this paper, we have proposed CNN-GRU based hybrid
+deep learning model with better hyper-parameter tuning to
+do image captioning. Our CNN-GRU encoder decode
+framework do caption-to-image reconstruction to handle the
+semantic context into consideration as well as the time
+complexity. By taking the hidden states of the decoder into
+consideration, the input image and its similar semantic
+representations were reconstructed and reconstruction scores
+from a semantic reconstructor were used in conjunction with
+likelihood during model training to assess the quality of the
+generated
+caption. As
+a
+result, the
+decoder
+received
+improved semantic information, enhancing the caption
+production process. During model testing, combining the
+reconstruction score and the log-likelihood was also feasible
+to choose the most appropriate caption. The suggested model
+outperforms the state-of-the-art LSTM-A5 model for picture
+captioning in terms of time complexity and accuracy.
+REFERENCES
+1.
+Alahmadi, R., & Hahn, J. (2022). Improve Image
+Captioning by Estimating the Gazing Patterns from the
+Caption. In Proceedings of the IEEE/CVF Winter
+Conference on Applications of Computer Vision (pp.
+1025-1034).
+2.
+Wei, J., Li, Z., Zhu, J., & Ma, H. (2022). Enhance
+understanding
+and
+reasoning
+ability
+for
+image
+captioning. Applied Intelligence, 1-17.
+3.
+Zhao, S., Li, L., & Peng, H. (2022). Aligned visual
+semantic
+scene
+graph
+for
+image
+captioning. Displays, 74, 102210.
+4.
+Singh, A., Krishna Raguru, J., Prasad, G., Chauhan, S.,
+Tiwari, P. K., Zaguia, A., &Ullah, M. A. (2022).
+Medical Image Captioning Using Optimized Deep
+Learning
+Model.
+Computational
+Intelligence
+and
+Neuroscience, 2022.
+5.
+Bahdanau, D., Cho, K., & Bengio, Y. (2014). Neural
+machine translation by jointly learning to align and
+translate. arXiv preprint arXiv:1409.0473.
+6.
+Fukushima, K., & Miyake, S. (1982). Neocognitron: A
+self-organizing neural network model for a mechanism
+of visual pattern recognition. In Competition and
+cooperation in neural nets (pp. 267-285). Springer,
+Berlin, Heidelberg.
+7.
+He, K., Gkioxari, G., Dollár, P., & Girshick, R. (2017).
+Mask r-cnn. In Proceedings of the IEEE international
+conference on computer vision (pp. 2961-2969).
+8.
+Girshick, R. (2015). Fast r-cnn. In Proceedings of the
+IEEE international conference on computer vision (pp.
+1440-1448).
+9.
+Ren, S., He, K., Girshick, R., & Sun, J. (2015). Faster r-
+cnn: Towards real-time object detection with region
+proposal networks. Advances in neural information
+processing systems, 28.
+10.
+Ji, J., Huang, X., Sun, X., Zhou, Y., Luo, G., Cao,
+L., ...& Ji, R. (2022). Multi-Branch Distance-Sensitive
+Self-Attention Network for Image Captioning. IEEE
+Transactions on Multimedia.
+11.
+Wang, X., Wu, J., Chen, J., Li, L., Wang, Y. F., &
+Wang, W. Y. (2019). Vatex: A large-scale, high-quality
+multilingual dataset for video-and-language research.
+In
+Proceedings
+of
+the
+IEEE/CVF
+International
+Conference on Computer Vision (pp. 4581-4591).
+12.
+Cornia, M., Baraldi, L., & Cucchiara, R. (2019). Show,
+control
+and
+tell:
+A
+framework
+for
+generating
+controllable and grounded captions. In Proceedings of
+the IEEE/CVF Conference on Computer Vision and
+
+0.975
+0.950
+0.925
+D
+0060
+CID
+0.875
+0.850
+0.825
+0.800
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+K0.34
+0.32
+0.30
+0.26
+0.24
+0.22
+0.20
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+KPattern Recognition (pp. 8307-8316).
+13.
+Herdade, S., Kappeler, A., Boakye, K., & Soares, J.
+(2019). Image captioning: Transforming objects into
+words. Advances in Neural Information Processing
+Systems, 32.
+14.
+Huang, L., Wang, W., Chen, J., & Wei, X. Y. (2019).
+Attention
+on
+attention
+for
+image
+captioning.
+In
+Proceedings
+of
+the
+IEEE/CVF
+International
+Conference on Computer Vision (pp. 4634-4643).
+15.
+Ke, L., Pei, W., Li, R., Shen, X., & Tai, Y. W. (2019).
+Reflective decoding network for image captioning.
+In
+Proceedings
+of
+the
+IEEE/CVF
+International
+Conference on Computer Vision (pp. 8888-8897).
+16.
+Li, G., Zhu, L., Liu, P., & Yang, Y. (2019). Entangled
+transformer for image captioning. In Proceedings of the
+IEEE/CVF
+International
+Conference
+on
+Computer
+Vision (pp. 8928-8937).
+17.
+Vinyals, O., Toshev, A., Bengio, S., & Erhan, D. (2016).
+Show and tell: Lessons learned from the 2015 mscoco
+image captioning challenge. IEEE transactions on
+pattern analysis and machine intelligence, 39(4), 652-
+663.
+18.
+Tu, Z., Liu, Y., Shang, L., Liu, X., & Li, H. (2017,
+February).
+Neural
+machine
+translation
+with
+reconstruction. In Thirty-First AAAI Conference on
+Artificial Intelligence.
+19.
+Mao, J., Xu, W., Yang, Y., Wang, J., Huang, Z., &
+Yuille, A. (2014). Deep captioning with multimodal
+recurrent neural networks (m-rnn). arXiv preprint
+arXiv:1412.6632..
+20.
+Karpathy, A., & Fei-Fei, L. (2015). Deep visual-
+semantic alignments for generating image descriptions.
+In Proceedings of the IEEE conference on computer
+vision and pattern recognition (pp. 3128-3137)..
+21.
+Xu, K., Ba, J., Kiros, R., Cho, K., Courville, A.,
+Salakhudinov, R., ...& Bengio, Y. (2015, June). Show,
+attend and tell: Neural image caption generation with
+visual attention. In International conference on machine
+learning (pp. 2048-2057). PMLR.
+22.
+Vinyals, O., Toshev, A., Bengio, S., & Erhan, D. (2015).
+Show and tell: A neural image caption generator.
+In Proceedings of the IEEE conference on computer
+vision and pattern recognition (pp. 3156-3164).
+23.
+Yang, Z., Yuan, Y., Wu, Y., Cohen, W. W., &
+Salakhutdinov, R. R. (2016). Review networks for
+caption generation. Advances in neural information
+processing systems, 29.
+24.
+Liu, C., Mao, J., Sha, F., & Yuille, A. (2017, February).
+Attention correctness in neural image captioning.
+In Proceedings of the AAAI Conference on Artificial
+Intelligence (Vol. 31, No. 1).
+25.
+Anderson, P., He, X., Buehler, C., Teney, D., Johnson,
+M., Gould, S., & Zhang, L. (2018). Bottom-up and top-
+down
+attention
+for
+image
+captioning
+and
+visual
+question answering. In Proceedings of the IEEE
+conference
+on
+computer
+vision
+and
+pattern
+recognition (pp. 6077-6086).
+26.
+Jia, X., Gavves, E., Fernando, B., & Tuytelaars, T.
+(2015). Guiding the long-short term memory model for
+image caption generation. In Proceedings of the IEEE
+international conference on computer vision (pp. 2407-
+2415).
+27.
+Wu, Q., Shen, C., Liu, L., Dick, A., & Van Den Hengel,
+A. (2016). What value do explicit high level concepts
+have in vision to language problems? In Proceedings of
+the IEEE conference on computer vision and pattern
+recognition (pp. 203-212).
+28.
+Cheng, Y. (2019). Joint Training for Neural Machine
+Translation. Springer Nature.
+29.
+19. Gan, Z., Gan, C., He, X., Pu, Y., Tran, K., Gao,
+J., ...& Deng, L. (2017). Semantic
+compositional
+networks for visual captioning. In Proceedings of the
+IEEE conference on computer vision and pattern
+recognition (pp. 5630-5639).
+30.
+Yao, T., Pan, Y., Li, Y., Qiu, Z., & Mei, T. (2017).
+Boosting
+image
+captioning
+with
+attributes.
+In Proceedings of the IEEE international conference on
+computer vision (pp. 4894-4902).
+31.
+Lu, J., Zhou, R., & Luo, S. (2016). Design of Extended
+‘Digital
+Communication
+Theory’Hardware
+Implementation Platform. In Computer Science and
+Engineering
+Technology
+(CSET2015)
+&
+Medical
+Science
+and
+Biological
+Engineering
+(MSBE2015)
+Proceedings of the 2015 International Conference on
+CSET & MSBE (pp. 289-297).
+32.
+Karpathy, A., & Fei-Fei, L. (2015). Deep visual-
+semantic alignments for generating image descriptions.
+In Proceedings of the IEEE conference on computer
+vision and pattern recognition (pp. 3128-3137)..
+33.
+Xu, K., Ba, J., Kiros, R., Cho, K., Courville, A.,
+Salakhudinov, R., ...& Bengio, Y. (2015, June). Show,
+attend and tell: Neural image caption generation with
+visual attention. In International conference on machine
+learning (pp. 2048-2057). PMLR.
+34.
+Vinyals, O., Toshev, A., Bengio, S., & Erhan, D. (2015).
+Show and tell: A neural image caption generator.
+In Proceedings of the IEEE conference on computer
+vision and pattern recognition (pp. 3156-3164).
+35.
+Papineni, K., Roukos, S., Ward, T., & Zhu, W. J. (2002,
+July). Bleu: a method for automatic evaluation of
+machine translation. In Proceedings of the 40th annual
+meeting
+of
+the
+Association
+for
+Computational
+Linguistics (pp. 311-318).
+36.
+Banerjee, S., & Lavie, A. (2005, June). METEOR: An
+automatic metric for MT evaluation with improved
+correlation with human judgments. In Proceedings of
+the acl workshop on intrinsic and extrinsic evaluation
+measures
+for
+machine
+translation
+and/or
+summarization (pp. 65-72).
+37.
+Lin, C. Y. (2004, July). Rouge: A package for
+automatic
+evaluation
+of
+summaries.
+In
+Text
+summarization branches out (pp. 74-81).
+38.
+Vedantam, R., Lawrence Zitnick, C., & Parikh, D.
+(2015).
+Cider:
+Consensus-based
+image
+description
+evaluation. In Proceedings of the IEEE conference on
+computer vision and pattern recognition (pp. 4566-
+4575).
+39.
+Chen, X., Fang, H., Lin, T. Y., Vedantam, R., Gupta, S.,
+Dollár, P., & Zitnick, C. L. (2015). Microsoft coco
+captions: Data collection and evaluation server. arXiv
+preprint arXiv:1504.00325.
+

INAzT4oBgHgl3EQfHvu0/content/2301.01051v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:58a5da5a96cc54a51d9202c59507dfb09cceca3997434e9c6824a8950b461e56
+size 796984

INAzT4oBgHgl3EQfHvu0/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:c280ee717222f802d778819c72aab3dbe646b58d9ec2d4aa00dc637be4b8c887
+size 73031

JNAzT4oBgHgl3EQfVPyV/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:0e8a66d310e71448dd8ebf47e85892a042be96afa50efc90107423a61d0d1394
+size 7471149