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+Generalised (non-singular) entropy functions with applications to cosmology and
+black holes
+Sergei D. Odintsov1,2 , Tanmoy Paul3
+1) ICREA, Passeig Luis Companys, 23, 08010 Barcelona, Spain
+2) Institute of Space Sciences (ICE, CSIC) C. Can Magrans s/n, 08193 Barcelona, Spain
+3) Department of Physics, Chandernagore College, Hooghly - 712 136, India.
+The growing interest of different entropy functions proposed so far (like the Bekenstein-Hawking,
+Tsallis, R´enyi, Barrow, Sharma-Mittal, Kaniadakis and Loop Quantum Gravity entropies) towards
+black hole thermodynamics as well as towards cosmology lead to the natural question that whether
+there exists a generalized entropy function that can generalize all these known entropies. With this
+spirit, we propose a new 4-parameter entropy function that seems to converge to the aforementioned
+known entropies for certain limits of the entropic parameters. The proposal of generalized entropy
+is extended to non-singular case, in which case , the entropy proves to be singular-free during the
+entire cosmological evolution of the universe. The hallmark of such generalized entropies is that it
+helps us to fundamentally understand one of the important physical quantities namely “entropy”.
+Consequently we address the implications of the generalized entropies on black hole thermodynamics
+as well as on cosmology, and discuss various constraints of the entropic parameters from different
+perspectives.
+I.
+INTRODUCTION
+One of the most important discoveries in theoretical physics is the black body radiation of a black hole, which
+is described by a certain temperature and by a Bekenstein-Hawking entropy function [1, 2] (see [3, 4] for extensive
+reviews). On contrary to classical thermodynamics where the entropy is proportional to volume of the system under
+consideration, the Bekenstein-Hawking entropy is proportional to the area of the black hole horizon. Such unusual
+behaviour of the black hole entropy leads to the proposals of different entropy functions, such as, the Tsallis [5],
+R´enyi [6], Barrow [7], Sharma-Mittal [8], Kaniadakis [9] and the Loop Quantum Gravity entropies [10] are well known
+entropy functions proposed so far. All of these known entropies have the common properties like – (1) they seem
+to be the monotonic increasing function with respect to the Bekenstein-Hawkinh entropy variable, (2) they obey the
+third law of thermodynamics, in particular, all of these entropies tend to zero as S → 0 (where S represents the
+Bekenstein-Hawking entropy) and (3) they converge to the Bekenstein-Hawking entropy for suitable choices of the
+respective entropic parameter, for example, the Tsallis entropy goes to the Bekenstein-Hawking entropy when the
+Tsallis exponent tends to unity. Furthermore, these entropies have rich consequences towards cosmology, particularly
+in describing the dark energy era of the universe [16–49]. The growing interest of such known entropies and due to
+their common properties lead to a natural question that whether there exists some generalized entropy function which
+is able to generalize all the known entropies proposed so far for suitable limits of the parameters.
+The entropy functions are extensively applied in the realm of black hole thermodynamics and cosmological evolution
+of the universe. Recently we showed that the entropic cosmology corresponding to different entropy functions can be
+equivalently represented by holographic cosmology where the equivalent holographic cut-offs come in terms of either
+particle horizon and its derivative or the future horizon and its derivative. One of the mysteries in today’s cosmology
+is to explain the acceleration of the universe in the high as well as in the low curvature regime, known as inflation and
+the dark energy era respectively. These eras are well described by entropic cosmology or equivalently by holographic
+cosmology [16–51, 53–60], and more interestingly, the entropic cosmology proves to be useful to unify the early inflation
+and the late dark energy era of the universe in a covariant manner [61]. Apart from the inflation, the holographic
+cosmology turns out to be useful in describing the bouncing scenario [62, 63]. In regard to the bounce scenario, the
+energy density sourced from the holographic principle or from some entropy function under consideration helps to
+violate the null energy condition at a finite time, which in turn triggers a non-singular bouncing universe. However
+here it deserves mentioning that all the known entropies mentioned above (like Tsallis , R´enyi, Barrow, Sharma-
+Mittal, Kaniadakis and the Loop Quantum Gravity entropies) become singular (or diverge) at a certain cosmological
+evolution of the universe, particularly in the context of bounce cosmology. Actually such entropies contain a factor
+that is proportional to 1/H2 (where H is the Hubble parameter), and thus they diverge at the instant when the
+Hubble parameter vanishes, i.e, at the instant of a bounce in bouncing cosmology. This makes such known entropies
+ill-defined in describing a non-singular bounce scenario.
+Based on the above arguments, the questions that naturally arise are following:
+• Does there exist a generalized entropy function that generalizes all the known entropies proposed so far ?
+arXiv:2301.01013v1  [gr-qc]  3 Jan 2023
+
+2
+• If so, then what is its implications on black hole thermodynamics as well as on cosmology ?
+• Similar to the known entropies, is the generalized entropy becomes singular at the instant when the Hubble
+parameter of the universe vanishes, for instance, in the bounce cosmology ? If so, then does there exist an
+entropy function that generalizes all the known entropies, and at the same time, also proves to be singular-free
+during the entire cosmic evolution of the universe ?
+The present article, based on some of our previous works [50–52], gives a brief review in answering the above
+questions. The notations or conventions in this article are following: we will follow the (−, +, +, +) signature of the
+spacetime metric, and κ2 = 8πG =
+1
+M 2
+Pl where G is the Newton’s constant or MPl denotes the four dimensional Planck
+mass. In regard to the cosmological evolution, a(t) and H(t) are the scale factor and the Hubble parameter of the
+universe respectively, N being the e-folding number, an overprime will denote
+d
+dη where η is the conformal time, an
+overdot will symbolize
+d
+dt with t being the cosmic time, otherwise an overprime with some argument will represent
+the derivative of the function with respect to that argument.
+II.
+POSSIBLE GENERALIZATIONS OF KNOWN ENTROPIES
+Here we will propose a generalized four-parameter entropy function which can lead to various known entropy
+functions proposed so far for suitable choices of the parameters.
+Let us start with the Bekenstein-Hawking entropy, the very first proposal of thermodynamical entropy of black hole
+physics [1, 2],
+S = A
+4G ,
+(1)
+where A = 4πr2
+h is the area of the horizon and rh is the horizon radius. Consequently, different entropy functions have
+been introduced depending on the system under consideration. Let us briefly recall some of the entropy functions
+proposed so far:
+• For the systems with long range interactions where the Boltzmann-Gibbs entropy is not applied, one needs to
+introduce the Tsallis entropy which is given by [5],
+ST = A0
+4G
+� A
+A0
+�δ
+,
+(2)
+where A0 is a constant and δ is the exponent.
+• The R´enyi entropy is given by [6],
+SR = 1
+α ln (1 + αS) ,
+(3)
+where S is identified with the Bekenstein-Hawking entropy and α is a parameter.
+• The Barrow entropy is given by [7],
+SB =
+� A
+APl
+�1+∆/2
+,
+(4)
+where A is the usual black hole horizon area and APl = 4G is the Planck area. The Barrow entropy describes
+the fractal structures of black hole that may generate from quantum gravity effects.
+• The Sharma-Mittal entropy is given by [8],
+SSM = 1
+R
+�
+(1 + δ S)R/δ − 1
+�
+,
+(5)
+where R and δ are two parameters. The Sharma-Mittal entropy can be regarded as a possible combination of
+the Tsallis and R´enyi entropies.
+
+3
+• The Kaniadakis entropy function is of the following form [9]:
+SK = 1
+K sinh (KS) ,
+(6)
+where K is a phenomenological parameter.
+• In the context of Loop Quantum Gravity, one may get the following entropy function [10]:
+Sq =
+1
+(1 − q)
+�
+e(1−q)Λ(γ0)S − 1
+�
+,
+(7)
+where q is the exponent and Λ(γ0) = ln 2/
+�√
+3πγ0
+�
+with γ0 being the Barbero-Immirzi parameter. The γ0
+generally takes either γ0 =
+ln 2
+π
+√
+3 or γ0 =
+ln 3
+2π
+√
+2. However with γ0 =
+ln 2
+π
+√
+3, Λ(γ0) becomes unity and Sq resembles
+with the Bekenstein-Hawking entropy for q → 1.
+All the above entropies – (1) obeys the generalized third law of thermodynamics, i.e the entropy function(s) vanishes
+at the limit S → 0; (2) monotonically increases with respect to the Bekenstein-Hawking variable and (3) converges to
+the Bekenstein-Hawking entropy for suitable limit of the entropic parameter, for example, the Tsallis entropy tends
+to S at δ = 1.
+In [50, 51], we proposed two different entropy functions containing 6-parameters and 4-parameters respectively,
+which can generalize all the known entropies mentioned from Eq.(2) to Eq.(7). In particular, the generalized entropies
+are given by,
+6 parameter entropy :
+S6 [α±, β±, γ±] =
+1
+α+ + α−
+��
+1 + α+
+β+
+Sγ+
+�β+
+−
+�
+1 + α−
+β−
+Sγ−
+�−β−�
+,
+(8)
+4 parameter entropy :
+Sg [α+, α−, β, γ] = 1
+γ
+��
+1 + α+
+β
+S
+�β
+−
+�
+1 + α−
+β
+S
+�−β�
+,
+(9)
+where the respective parameters are given in the argument and they are assumed to be positive.
+Here S is the
+Bekenstein-Hawking entropy. Below we prove the generality of the above generalized entropy functions, in particular,
+we show that both the generalized entropies reduce to the known entropies mentioned in Eqs. (2), (3), (4), (5), (6),
+and (7) for suitable choices of the respective parameters. Here we establish it particularly for the 4-parameter entropy
+function, while the similar calculations hold for the 6-parameter entropy as well [50].
+• For α+ → ∞ and α− = 0, one gets
+Sg = 1
+γ
+�α+
+β
+�β
+Sβ .
+If we further choose γ = (α+/β)β, then the generalized entropy reduces to
+Sg = Sβ .
+Therefore with β = δ or β = 1 + ∆, the generalized entropy resembles with the Tsallis entropy or with the
+Barrow entropy respectively.
+• For α− = 0, β → 0 and α+
+β → finite – Eq. (9) leads to,
+Sg = 1
+γ
+��
+1 + α+
+β
+S
+�β
+− 1
+�
+= 1
+γ
+�
+exp
+�
+β ln
+�
+1 + α+
+β
+S
+��
+− 1
+�
+≈
+1
+(γ/β) ln
+�
+1 + α+
+β
+S
+�
+.
+Further choosing γ = α+ and identifying α+
+β = α, we can write the above expression as,
+Sg = 1
+α ln (1 + α S) ,
+(10)
+i.e., Sg reduces to the R´enyi entropy.
+
+4
+• In the case when α− = 0, the generalized entropy becomes,
+Sg = 1
+γ
+��
+1 + α+
+β
+S
+�β
+− 1
+�
+.
+(11)
+Thereby identifying γ = R, α+ = R and β = R/δ, the generalized entropy function Sg gets similar to the
+Sharma-Mittal entropy.
+• For β → ∞, α+ = α− = γ
+2 = K, we may write Eq. (9) as,
+Sg = 1
+2K lim
+β→∞
+��
+1 + K
+β S
+�β
+−
+�
+1 + K
+β S
+�−β�
+= 1
+2K
+�
+eKS − e−KS�
+= 1
+K sinh (KS) → Kaniadakis entropy .
+(12)
+• Finally, with α− = 0, β → ∞ and γ = α+ = (1 − q), Eq. (9) immediately yields,
+Sg =
+1
+(1 − q)
+�
+e(1−q)S − 1
+�
+,
+which is the Loop Quantum Gravity entropy with Λ(γ0) = 1 or equivalently γ0 =
+ln 2
+π
+√
+3.
+Furthermore, the generalized entropy function in Eq. (9) shares the following properties: (1) Sg → 0 for S → 0.
+(2) The entropy Sg [α+, α−, β, γ] is a monotonically increasing function with S because both the terms
+�
+1 + α+
+β S
+�β
+and −
+�
+1 + α−
+β
+S
+�−β
+present in the expression of Sg increase with S. (3) Sg [α+, α−, β, γ] seems to converge to the
+Bekenstein-Hawking entropy at certain limit of the parameters. In particular, for α+ → ∞, α− = 0, γ = (α+/β)β
+and β = 1, the generalized entropy function in Eq. (9) becomes equivalent to the Bekenstein-Hawking entropy.
+Here it deserves mentioning that beside the entropy function proposed in Eq. (9) which contains four parameters,
+one may consider a three parameter entropy having the following form:
+S3[α, β, γ] = 1
+γ
+��
+1 + α
+β S
+�β
+− 1
+�
+,
+(13)
+where α, β and γ are the parameters. The above form of S3[α, β, γ] satisfies all the properties, like – (1) S3[α, β, γ] → 0
+for S → 0, (2) S3 is an increasing function with S and (3) S3 has a Bekenstein-Hawking entropy limit for the choices:
+α → ∞, γ = (α/β)β and β = 1 respectively. However S3[α, β, γ] is not able to generalize all the known entropies
+mentioned from Eq. (2) to Eq. (7), in particular, S3[α, β, γ] does not reduce to the Kaniadakis entropy for any
+possible choices of the parameters.
+Conjecture - I: Based on our findings, we propose the following postulate in regard to the generalized entropy
+function – “The minimum number of parameters required in a generalized entropy function that can generalize all
+the known entropies mentioned from Eq. (2) to Eq. (7) is equal to four”.
+Below we will address the possible implications of such generalized entropies on black hole thermodynamics as well
+as on cosmology.
+III.
+BLACK HOLE THERMODYNAMICS WITH 3-PARAMETER GENERALIZED ENTROPY
+It is interesting to see what happens when the generalized entropy (13) is ascribed to the prototypical black hole,
+given by the Schwarzschild geometry [50]
+ds2 = −f(r) dt2 + dr2
+f(r) + r2dΩ2
+(2) ,
+f(r) = 1 − 2GM
+r
+,
+(14)
+
+5
+where M is the black hole mass and dΩ2
+(2) = dϑ2 + sin2 ϑ dϕ2 is the line element on the unit two-sphere. The black
+hole event horizon is located at the Schwarzschild radius
+rH = 2GM .
+(15)
+Studying quantum field theory on the spacetime with this horizon, Hawking discovered that the Schwarzschild black
+hole radiates with a blackbody spectrum at the temperature
+TH =
+1
+8πGM .
+(16)
+As explained in general below, if we assume that the mass M coincides with the thermodynamical energy, then the
+temperature obtained from the thermodynamical law is different from the Hawking temperature, a contradiction for
+observers detecting Hawking radiation. Alternatively, if the Hawking temperature TH is identified with the physical
+black hole temperature, the obtained thermodynamical energy differs from the Schwarzschild mass M even for the
+Tsallis entropy or the R´enyi entropy, which seems to imply a breakdown of energy conservation.
+If the mass M coincides with the thermodynamical energy E of the system due to energy conservation, as in,
+in order for this system to be consistent with the thermodynamical equation dSG = dE/T one needs to define the
+generalized temperature TG as
+1
+TG
+≡ dSG
+dM
+(17)
+which is, in general, different from the Hawking temperature TH. For example, in the case of the entropy (13), one
+has
+1
+TG
+= α
+γ
+�
+1 + α
+β S
+�β−1 dS
+dM = α
+γ
+�
+1 + α
+β S
+�β−1 1
+TH
+,
+(18)
+where
+S = A
+4G = 4πGM 2 =
+1
+16πGTH
+2 .
+(19)
+Because α
+γ
+�
+1 + α
+β S
+�β−1
+̸= 1, it is necessarily TG ̸= TH. Since the Hawking temperature (16) is the temperature
+perceived by observers detecting Hawking radiation, the generalized temperature TG in (18) cannot be a physically
+meaningful temperature.
+In Eq. (17), assuming that the thermodynamical energy E is the black hole mass M leads to an unphysical result.
+As an alternative, assume that the thermodynamical temperature T coincides with the Hawking temperature TH
+instead of assuming E = M. This assumption leads to
+dEG = TH dSG = dSG
+dS
+dS
+√
+16πGS
+(20)
+which, in the case of Eq. (13), yields
+dEG = α
+γ
+�
+1 + α
+β S
+�β−1
+dS
+√
+16πGS
+=
+α
+γ
+√
+16πG
+�
+S−1/2 + α (β − 1)
+β
+S1/2 + O
+�
+S3/2��
+.
+(21)
+The integration of Eq. (21) gives
+EG =
+α
+γ
+√
+16πG
+�
+2S1/2 + 2α (β − 1)
+3β
+S3/2 + O
+�
+S5/2��
+= α
+γ
+�
+M + 4πGα (β − 1)
+3β
+M 3 + O
+�
+M 5��
+,
+(22)
+where the integration constant is determined by the condition that EG = 0 when M = 0. Even when α = γ, due to
+the correction 4πGα(β−1)
+3β
+M 3, the expression (22) of the thermodynamical energy ER obtained differs from the black
+hole mass M, EG ̸= E, which seems unphysical.
+
+6
+IV.
+COSMOLOGY WITH THE 4-PARAMETER GENERALIZED ENTROPY
+Here we consider the 4-parameter generalized entropy (9), which is indeed more generalized compared to the 3-
+parameter entropy function of Eq.(13), to describe the cosmological behaviour of the universe [51]. In particular, we
+examine whether the 4-parameter entropy function results to an unified scenario of early inflation and the late dark
+energy era of the universe.
+The Friedmann-Lemaˆıtre-Robertson-Walker space-time with flat spacial part will serve our purpose, in particular,
+ds2 = −dt2 + a2(t)
+�
+i=1,2,3
+�
+dxi�2 .
+(23)
+Here a(t) is called as a scale factor.
+The radius rH of the cosmological horizon is given by
+rH = 1
+H ,
+(24)
+with H = ˙a/a is the Hubble parameter of the universe. Then the entropy contained within the cosmological horizon
+can be obtained from the Bekenstein-Hawking relation [65]. Furthermore the flux of the energy E, or equivalently,
+the increase of the heat Q in the region comes as
+dQ = −dE = −4π
+3 r3
+H ˙ρdt = − 4π
+3H3 ˙ρ dt = 4π
+H2 (ρ + p) dt ,
+(25)
+where, in the last equality, we use the conservation law: 0 = ˙ρ + 3H (ρ + p). Then from the Hawking temperature
+[66]
+T =
+1
+2πrH
+= H
+2π ,
+(26)
+and by using the first law of thermodynamics TdS = dQ, one obtains ˙H = −4πG (ρ + p). Integrating the expression
+immediately leads to the first FRW equation,
+H2 = 8πG
+3
+ρ + Λ
+3 ,
+(27)
+where the integration constant Λ can be treated as a cosmological constant.
+Instead of the Bekenstein-Hawking entropy of Eq. (1), we may use the generalized entropy in Eq. (9), in regard to
+which, the first law of thermodynamics leads to the following equation:
+˙H
+�∂Sg
+∂S
+�
+= −4πG (ρ + p) .
+(28)
+With the explicit form of Sg from Eq. (9), the above equation turns out to be,
+1
+γ
+�
+α+
+�
+1 + πα+
+βGH2
+�β−1
++ α−
+�
+1 + πα−
+βGH2
+�−β−1�
+˙H = −4πG (ρ + p)
+(29)
+where we use S = A/(4G) = π/(GH2). Using the conservation relation of the matter fields, i.e., ˙ρ + 3H (ρ + p) = 0,
+Eq. (29) can be written as,
+2
+γ
+�
+α+
+�
+1 + πα+
+βGH2
+�β−1
++ α−
+�
+1 + πα−
+βGH2
+�−β−1�
+H dH =
+�8πG
+3
+�
+dρ ,
+on integrating which, we obtain,
+GH4β
+πγ
+�
+1
+(2 + β)
+�GH2β
+πα−
+�β
+2F1
+�
+1 + β, 2 + β, 3 + β, −GH2β
+πα−
+�
++
+1
+(2 − β)
+�GH2β
+πα+
+�−β
+2F1
+�
+1 − β, 2 − β, 3 − β, −GH2β
+πα+
+��
+= 8πGρ
+3
++ Λ
+3 ,
+(30)
+where Λ is the integration constant (known as the cosmological constant) and 2F1(arguments) denotes the Hypergeo-
+metric function. Eq. (29) and Eq. (30) represent the modified Friedmann equations corresponding to the generalized
+entropy function Sg. In the next section, we aim to study the cosmological implications of the modified Friedmann
+Eq. (29) and Eq. (30).
+
+7
+A.
+Early universe cosmology from the 4-parameter generalized entropy
+During the early stage of the universe we consider the matter field and the cosmological constant (Λ) to be absent,
+i.e., ρ = p = Λ = 0. During the early universe, the cosmological constant is highly suppressed with respect to the
+entropic energy density and thus we can safely neglect the Λ in studying the early inflationary scenario of the universe.
+Therefore during the early universe, Eq. (30) becomes,
+�
+1
+(2 + β)
+�GH2β
+πα−
+�β
+2F1
+�
+1 + β, 2 + β, 3 + β, −GH2β
+πα−
+�
++
+1
+(2 − β)
+�GH2β
+πα+
+�−β
+2F1
+�
+1 − β, 2 − β, 3 − β, −GH2β
+πα+
+��
+= 0 .
+(31)
+Here it may be mentioned that the typical energy scale during early universe is of the order ∼ 1016GeV (= 10−3MPl
+where recall that MPl is the Planck mass and MPl = 1/
+√
+16πG). This indicates that the condition GH2 ≪ 1 holds
+during the early phase of the universe. Owing to such condition, we can safely expand the Hypergeometric function
+of Eq. (31) as the Taylor series with respect to the argument containing GH2, and as a result, the above equation
+provides a constant Hubble parameter as the solution:
+H = 4πMPl
+�α+
+β
+�
+(3 − β)
+(2 − β)(1 − β)
+�
+.
+(32)
+For α+
+β ∼ 10−6 and β ≲ O(1), the constant Hubble parameter can be fixed at H ∼ 10−3MPl which can be identified
+with typical inflationary energy scale. Therefore the entropic cosmology corresponding to the generalized entropy
+function Sg leads to a de-Sitter inflationary scenario during the early universe. However, a de-Sitter inflation has no
+exit mechanism, and moreover, the primordial curvature perturbation gets exactly scale invariant in the context of a
+de-Sitter inflation, which is not consistent with the recent Planck data [75] at all. This indicates that the constant
+Hubble parameter obtained in Eq. (32) does not lead to a good inflationary scenario of the universe. Thus in order
+to achieve a viable quasi de-Sitter inflation in the present context, we consider the parameters of Sg to be slowly
+varying functions with respect to the cosmic time. In particular, we consider the parameter γ to vary and the other
+parameters (i.e., α+, α− and β) remain constant with t. In particular,
+γ(N) =
+�
+γ0 exp
+�
+−
+� Nf
+N
+σ(N) dN
+�
+; N ≤ Nf
+γ0
+; N ≥ Nf ,
+(33)
+where γ0 is a constant and N denotes the inflationary e-folding number with Nf being the total e-folding number of
+the inflationary era. The function σ(N) has the following form,
+σ(N) = σ0 + e−(Nf −N) ,
+(34)
+where σ0 is a constant. The second term in the expression of σ(N) becomes effective only when N ≈ Nf, i.e., near
+the end of inflation. The term e−(Nf −N) in Eq. (34) is actually considered to ensure an exit from inflation era and
+thus proves to be an useful one to make the inflationary scenario viable. In such scenario where γ varies with N, the
+Friedmann equation turns out to be,
+−
+�2π
+G
+�
+�
+��
+α+
+�
+1 + α+
+β S
+�β−1
++ α−
+�
+1 + α−
+β
+S
+�−β−1
+�
+1 + α+
+β S
+�β
+−
+�
+1 + α−
+β
+S
+�−β
+�
+�� H′(N)
+H3
+= σ(N) .
+(35)
+By using S = π/(GH2), or equivalently, 2HdH = −
+π
+GS2 dS, one can integrate Eq.(35) to get H(N) as,
+H(N) = 4πMPl
+�α+
+β
+�
+����
+21/(2β) exp
+�
+− 1
+2β
+� N
+0 σ(N)dN
+�
+�
+1 +
+�
+1 + 4 (α+/α−)β exp
+�
+−2
+� N
+0 σ(N)dN
+��1/(2β)
+�
+���� .
+(36)
+
+8
+The above solution of H(N) allows an exit from inflation at finite e-fold number which can be fixed at Nf = 58 for
+suitable choices of the entropic parameters [51]. Moreover we determine the spectral index for curvature perturbation
+(ns) and the tensor-to-scalar ratio (r) in the present context of entropic cosmology, and they are given by [51]:
+ns = 1 −
+2σ0
+�
+1 + 4 (α+/α−)β exp [−2 (1 + σ0Nf)]
+(1 + σ0)
+�
+1 + 4 (α+/α−)β
+− 8σ0 (α+/α−)β
+1 + 4 (α+/α−)β ,
+(37)
+and
+r =
+16σ0
+�
+1 + 4 (α+/α−)β exp [−2 (1 + σ0Nf)]
+(1 + σ0)
+�
+1 + 4 (α+/α−)β
+(38)
+respectively. It turns out that the theoretical expectations of ns and r get simultaneously compatible with the Planck
+data for the following ranges of the parameters:
+σ0 = [0.013, 0.017] ,
+(α+/α−)β ≥ 7.5 ,
+β = (0, 0.4] and (α+/β) ≈ 10−6 ,
+(39)
+for Nf = 58. The consideration of α+
+β ∼ 10−6 leads to the energy scale at the onset of inflation as H ∼ 10−3MPl.
+B.
+Dark energy era from the 4-parameter generalized entropy
+In this section we will concentrate on late time cosmological implications of the generalized entropy function (Sg),
+where the cosmological constant Λ is considered to be non-zero. During the late time, the parameter γ becomes
+constant, in particular γ = γ0, as we demonstrated in Eq. (33). As a result, the entropy function at the late time
+takes the following form,
+Sg = 1
+γ0
+��
+1 + α+
+β
+S
+�β
+−
+�
+1 + α−
+β
+S
+�−β�
+,
+(40)
+with S = π/(GH2). Consequently, the energy density and pressure corresponding to the Sg are given by,
+ρg = 3H2
+8πG
+�
+1 −
+α+
+γ0(2 − β)
+�GH2β
+πα+
+�1−β�
+,
+pg = −
+˙H
+4πG
+�
+1 − α+
+γ0
+�GH2β
+πα+
+�1−β
+−
+�α+
+γ0
+� �α+
+α−
+�β �GH2β
+πα+
+�1+β�
+− ρg .
+(41)
+Therefore the dark energy density (ρD) is contributed from the entropic energy density (ρg) as well as from the
+cosmological constant. In particular
+ρD = ρg +
+3
+8πG
+�Λ
+3
+�
+,
+ρD + pD = ρg + pg .
+(42)
+Consequently, the dark energy EoS parameter comes with the following expression:
+ωD = pD/ρD = −1 −
+�
+2 ˙H
+3H2
+� �
+��
+1 − α+
+γ0
+�
+GH2β
+πα+
+�1−β
+−
+�
+α+
+γ0
+� �
+α+
+α−
+�β �
+GH2β
+πα+
+�1+β
+1 −
+α+
+γ0(2−β)
+�
+GH2β
+πα+
+�1−β
++
+Λ
+3H2
+�
+�� .
+(43)
+In presence of the cosmological constant, the Friedmann equations are written as,
+H2 = 8πG
+3
+(ρm + ρD) = 8πG
+3
+(ρm + ρg) + Λ
+3 ,
+
+9
+˙H = −4πG [ρm + (ρD + pD)] = −4πG [ρm + (ρg + pg)] .
+(44)
+As usual, the fractional energy density of the pressureless matter and the dark energy satisfy Ωm + ΩD = 1 which
+along with ρm = ρm0
+� a0
+a
+�3 (with ρm0 being the present matter energy density) result to the Hubble parameter in
+terms of the red shift factor (z) as follows,
+H(z) = H0
+�
+Ωm0(1 + z)3
+√1 − ΩD
+.
+(45)
+Plugging the expression of ρg from Eq. (41) into ΩD =
+� 8πG
+3H2
+�
+ρg + Λ
+3 , and using the above form of H(z), we obtain,
+ΩD(z) = 1 −
+�
+α+
+γ0(2−β)
+�
+1
+2−β �
+GH2
+0β
+πα+
+Ωm0(1 + z)3� 1−β
+2−β
+�
+1 +
+Λ
+3H2
+0Ωm0(1+z)3
+�1/(2−β)
+.
+(46)
+By using the above expressions, we determine the DE EoS parameter from Eq. (43) as follows (see [51]),
+ωD(z) = −1 +
+1
+(2 − β)
+�
+1 +
+Λ
+3H2
+0Ωm0(1+z)3
+�
+�N
+D
+�
+,
+(47)
+where N (the numerator) and D (the denominator) have the following forms,
+N = 1 − Ωm0(2 − β) (1 + z)
+3(1−β)
+(2−β)
+� �
+1 +
+Λ
+3H2
+0Ωm0
+[f(Λ, Ωm0, H0, z)]1−β
+�
++
+�α+
+α−
+�β
+[Ωm0(2 − β)γ0/α+]
+2β
+1−β (1 + z)
+6β
+2−β
+×
+�
+�
+�
+�
+1 +
+Λ
+3H2
+0Ωm0
+�(1+β)/(1−β)
+[f(Λ, Ωm0, H0, z)]1+β
+�
+�
+�
+�
+,
+and
+D = 1 − Ωm0 (1 + z)
+3(1−β)
+(2−β)
+�
+1 +
+Λ
+3H2
+0Ωm0
+[f(Λ, Ωm0, H0, z)]1−β
+�
++
+Λ
+3H2
+0
+�f(Λ, Ωm0, H0, z)
+(1 + z)3/(2−β)
+�
+(48)
+respectively.
+Therefore ωD depends on the parameters: β, (α+/α−)β, γ0 and α+.
+Recall that the inflationary
+quantities are found to be simultaneously compatible with the Planck data if some of the parameters like α+, α− and
+β get constrained according to Eq. (39), while the parameter γ0 remains free from the inflationary requirement. With
+the aforementioned ranges of α+, α− and β, ωD(0) becomes compatible with the Planck observational data, provided
+γ0 lies within a small window as follows,
+1.5 × 10−4 ≤
+γ0
+(8πGH2
+0)1−β ≤ 2 × 10−4 .
+(49)
+Furthermore the deceleration parameter (symbolized by q) at present universe is obtained as,
+q = −1 +
+3
+2(2 − β)
+�
+1 +
+Λ
+3H2
+0Ωm0
+� .
+(50)
+Therefore for γm = [1.5×10−4, 2×10−4], the theoretical expression of q lies within q = [−0.56, −0.42] which certainly
+contains the observational value of q = −0.535 from the Planck data [76]. In particular, q = −0.535 occurs for
+γm = 1.8 × 10−4. Considering this value of γm and by using Eq.(47), we give the plot of ωD(z) vs. z, see Fig. 1.
+The figure reveals that that the theoretical expectation of the DE EoS parameter at present time acquires the value:
+ωD(0) = −0.950 which is well consistent with the Planck observational data [76].
+As a whole, we may argue that the entropic cosmology from the generalized entropy function Sg can unify the early
+inflation to the late dark energy era of the universe, for suitable ranges of the parameters given by:
+σ0 = [0.013, 0.017] ,
+(α+/α−)β ≥ 7.5 ,
+
+10
+-1.0
+-0.5
+0.0
+0.5
+1.0
+-3.0
+-2.5
+-2.0
+-1.5
+-1.0
+z
+ωD(z)
+FIG. 1: ωD(z) vs. z for a particular set of values of the parameters from their viable ranges as per Eq.(39) and Eq.(49), say
+β = 0.35, (α+/α−)β = 10, α+/β = 10−6 and γm = 1.8 × 10−4.
+β = (0, 0.4] and γm = [1.5 × 10−4, 2 × 10−4] .
+(51)
+Despite these successes, here it deserves mentioning that the entropy function Sg seems to be plagued with singularity
+for certain cosmological evolution of the universe, in particular, in the context of bounce cosmology. Due to the reason
+that the Bekenstein-Hawking entropy can be expressed as S = π/
+�
+GH2�
+, the generalized entropy Sg contains factor
+that is proportional to 1/H2 which diverges at H = 0, for instance at the instant of bounce in the context of bounce
+cosmology. Therefore in a bounce scenario, the generalized entropy function shown in Eq.(9) is not physical, and
+thus, we need to search for a different generalized entropy function which can lead to various known entropy functions
+for suitable choices of the parameters, and at the same time, proves to be non-singular for the entire cosmological
+evolution of the universe even at H = 0.
+V.
+SEARCH FOR A SINGULAR-FREE GENERALIZED ENTROPY
+With this spirit, we propose a new singular-free entropy function given by [52],
+Sns [α±, β, γ, ϵ] = 1
+γ
+� �
+1 + 1
+ϵ tanh
+�ϵα+
+β
+S
+��β
+−
+�
+1 + 1
+ϵ tanh
+�ϵα−
+β
+S
+��−β �
+,
+(52)
+where α±, β, γ and ϵ are the parameters which are considered to be positive, S symbolizes the Bekenstein-Hawking
+entropy and the suffix ’ns’ stands for ’non-singular’. In regard to the number of parameters, we propose a conjecture at
+the end of this section. First we demonstrate that the above entropy function remains finite, and thus is non-singular,
+during the whole cosmological evolution of a bouncing universe. In particular, the Sg takes the following form at the
+instant of bounce:
+Sns [α±, β, γ, ϵ] = 1
+γ
+� �
+1 + 1
+ϵ
+�β
+−
+�
+1 + 1
+ϵ
+�−β �
+.
+(53)
+Having demonstrated the non-singular behaviour of the entropy function, we now show that Sns of Eq.(52), for suitable
+choices of the parameters, reduces to various known entropies proposed so far.
+• For ϵ → 0, α+ → ∞ and α− = 0 along with the identification γ = (α+/β)β, Sns converges to the Tsallis entropy
+or to the Barrow entropy respectively.
+• The limit ϵ → 0, α− = 0, β → 0 and α+
+β → finite results to the similarity between the non-singular generalized
+entropy Sg and the R´enyi entropy.
+
+11
+• For ϵ → 0 and α− → 0, the non-singular generalized entropy converges to the following form,
+Sns = 1
+γ
+��
+1 + α+
+β
+S
+�β
+− 1
+�
+(54)
+Therefore with γ = R, α+ = R and β = R/δ, the above form of Sns becomes similar to the Sharma-Mittal
+entropy.
+• For ϵ → 0, β → ∞, α+ = α− = γ
+2 = K – the generalized entropy converges to the form of Kaniadakis entropy,
+• Finally, ϵ → 0, α− → 0, β → ∞ and γ = α+ = (1 − q), the generalized entropy of Eq. (52) gets resemble with
+the Loop Quantum Gravity entropy.
+Furthermore, the generalized entropy function in Eq. (52) shares the following properties: (1) the non-singular
+generalized entropy satisfies the generalized third law of thermodynamics.
+(2) Sns [α±, β, γ, ϵ] turns out to be a
+monotonically increasing function of S. (3) Sns [α±, β, γ, ϵ] proves to converge to the Bekenstein-Hawking entropy at
+certain limit of the parameters.
+At this stage it deserves mentioning that we have proposed two different generalized entropy functions in Eq.(9)
+and in Eq.(52) respectively – the former entropy function contains four independent parameters while the latter
+one has five parameters.
+Furthermore both the entropies are able to generalize the known entropies for suitable
+choices of the respective parameters. However as mentioned earlier that the entropy with four parameters becomes
+singular at H = 0 (for instance, in a bounce scenario when the Hubble parameter vanishes at the instant of bounce),
+while the entropy function having five parameters proves to be singular-free during the whole cosmological evolution
+of the universe. Based on these findings, we give a second conjecture regarding the number of parameters in the
+non-singular generalized entropy function:
+Conjecture - II: “The minimum number of parameters required in a generalized entropy function that can gen-
+eralize all the known entropies, and at the same time, is also singular-free during the universe’s evolution – is equal
+to five”.
+VI.
+COSMOLOGY WITH THE NON-SINGULAR GENERALIZED ENTROPY
+Applying the thermodynamic laws to the non-singular generalized entropy function Sns and by following the same
+procedure as of Sec.[IV], one gets the cosmological field equations corresponding to the Sgns [52]:
+1
+γ
+�
+α+ sech2
+� ϵπα+
+βGH2
+� �
+1 + 1
+ϵ tanh
+� ϵπα+
+βGH2
+��β−1
++ α− sech2
+� ϵπα−
+βGH2
+� �
+1 + 1
+ϵ tanh
+� ϵπα−
+βGH2
+��−β−1 �
+˙H = −4πG (ρ + p)
+.
+(55)
+Owing to the conservation equation of matter fields, in particular ˙ρ + 3H (ρ + p) = 0, the above expression can be
+integrated to get
+f (H; α±, β, γ, ϵ) = 8πGρ
+3
++ Λ
+3 .
+(56)
+Here the integration constant is symbolized by Λ and the function f has the following form:
+f (H; α±, β, γ, ϵ) = 2
+γ
+�
+�
+α+ sech2
+� ϵπα+
+βGH2
+� �
+1 + 1
+ϵ tanh
+� ϵπα+
+βGH2
+��β−1
++ α− sech2
+� ϵπα−
+βGH2
+� �
+1 + 1
+ϵ tanh
+� ϵπα−
+βGH2
+��−β−1 �
+H dH .
+(57)
+In regard to the functional form of f (H; α±, β, γ, ϵ), we would like to mention that the integration in Eq.(57) may not
+be performed in a closed form, unless certain conditions are imposed. For example, we consider GH2 ≪ 1 which is, in
+
+12
+fact, valid during the universe’s evolution (i.e the Hubble parameter is less than the Planck scale). With GH2 ≪ 1,
+the functional form of f turns out to be,
+f (H; α±, β, γ, ϵ) = 4
+γ H2
+�
+α+
+�
+1 + 1
+ϵ
+�β−1 �
+exp
+�
+−2ϵπα+
+βGH2
+�
++
+�2ϵπα+
+βGH2
+�
+Ei
+�
+−2ϵπα+
+βGH2
+��
++ α−
+�
+1 + 1
+ϵ
+�−β−1 �
+exp
+�
+−2ϵπα−
+βGH2
+�
++
+�2ϵπα−
+βGH2
+�
+Ei
+�
+−2ϵπα−
+βGH2
+�� �
+.
+(58)
+Therefore as a whole, Eq. (55) and Eq. (56) are the cosmological field equations corresponding to the generalized
+entropy Sg.
+A.
+Non-singular entropy on bounce cosmology
+In this section, we will address the implications of the generalized entropy Sns on non-singular bounce cosmology,
+in particular, we will investigate whether the entropic energy density can trigger a viable bounce during the early
+stage of the universe that is consistent with the observational constraints. For this purpose, we take the matter field
+and the cosmological constant to be absent, i.e., ρ = p = Λ = 0. In effect, Eq. (55) becomes,
+1
+γ
+�
+α+ sech2
+� ϵπα+
+βGH2
+� �
+1 + 1
+ϵ tanh
+� ϵπα+
+βGH2
+��β−1
++ α− sech2
+� ϵπα−
+βGH2
+� �
+1 + 1
+ϵ tanh
+� ϵπα−
+βGH2
+��−β−1 �
+˙H = 0 .
+(59)
+The parameters (α±, β, γ, ϵ) are positive, and thus the solution of the above equation is given by: ˙H = 0 or equivalently
+H = constant. Clearly H = constant does not lead to the correct evolution of the universe. Thus similar to the
+previous case, we consider the parameters of Sns[α±, β, γ, ϵ] vary with time. In particular, we consider the parameter
+γ to vary with time, and all the other parameters remain fixed, i.e.
+γ = γ(N) ,
+(60)
+with N being the e-fold number of the universe. In such scenario where γ(N) varies with time, the Friedmann equation
+corresponds to Sns[α±, β, γ, ϵ] gets modified compared to Eq.(59), and is given by:
+�
+��
+α+ sech2 �
+ϵα+
+β S
+� �
+1 + 1
+ϵ tanh
+�
+ϵα+
+β S
+��β−1
++ α− sech2 �
+ϵα−
+β S
+� �
+1 + 1
+ϵ tanh
+�
+ϵα−
+β S
+��−β−1
+�
+1 + 1
+ϵ tanh
+�
+ϵα+
+β S
+��β
+−
+�
+1 + 1
+ϵ tanh
+�
+ϵα−
+β S
+��−β
+�
+�� dS = γ′(N)
+γ(N) dN (61)
+where an overprime denotes
+d
+dη. Eq.(61) can be integrated to get,
+tanh
+� ϵπα
+βGH2
+�
+=
+�
+γ(N) +
+�
+γ2(N) + 4
+2
+�1/β
+− 1 .
+(62)
+where we take α+ = α− = α (say, without losing any generality) in order to extract an explicit solution of H(N).
+Due to the appearance of quadratic power of H, Eq.(62) allows a positive branch as well as a negative branch of the
+Hubble parameter. This leads to a natural possibility of symmetric bounce in the present context of singular free
+generalized entropic cosmology. Moreover Eq.(62) also demonstrates that the explicit evolution of H(N) does depend
+on the form of γ(N). In the following, we will consider two cases where we will determine the form of γ(N) such that
+it gives two different symmetric bounce scenarios respectively.
+1. The exponential bounce described by the scale factor,
+a(t) = exp
+�
+a0t2�
+.
+(63)
+This results to a symmetric bounce at t = 0. Here a0 is a constant having mass dimension [+2] – this constant
+is related with the entropic parameters of Sns and thus, without losing any generality, we take a0 = ϵπα
+4Gβ . Such
+an exponential bounce can be achieved from singular free entropic cosmology provided the γ(N) is given by,
+γ(N) =
+�
+1 + 1
+ϵ tanh
+� 1
+N
+��β
+−
+�
+1 + 1
+ϵ tanh
+� 1
+N
+��−β
+.
+(64)
+
+13
+2. The quasi-matter bounce is described by, In this case, the scale factor is,
+a(t) =
+�
+1 + a0
+� t
+t0
+�2�n
+(65)
+which is symmetric about t = 0 when the bounce happens. The n, a0 and t0 considered in the scale factor are
+related to the entropic parameters, and we take it as follows:
+n = √α
+,
+a0 = π
+4β
+and
+t0 =
+�
+G/ϵ ,
+(66)
+with G being the gravitational constant. The relation between (n, a0, t0) with the entropic parameters can be
+considered in a different way compared to the Eq.(66), however for a simplified expression of γ(N) we consider
+the relations as of Eq.(66). Consequently the γ(N) which leads to such quasi-matter bounce, comes as,
+γ(N) =
+�
+1 + 1
+ϵ tanh
+�
+e−N/√α �
+eN/√α − 1
+� 1
+2 ��β
+−
+�
+1 + 1
+ϵ tanh
+�
+e−N/√α �
+eN/√α − 1
+� 1
+2 ��−β
+.
+(67)
+Here it deserves mentioning that in the case of exponential bounce, the comoving Hubble radius asymptotically goes
+to zero and thus the perturbation modes remain at the super-Hubble regime at the distant past. This may results to
+the “horizon problem” in the exponential bounce scenario. On contrary, the comoving Hubble radius in the case of
+quasi-matter bounce asymptotically diverges to infinity at both sides of the bounce, and thus the perturbation modes
+lie within the deep sub-Hubble regime at the distant past – this resolves the horizon issue. Based on this arguments,
+we will concentrate on the quasi-matter bounce to perform the perturbation analysis.
+In regard to the perturbation analysis, we represent the present entropic cosmology with the ghost free Gauss-
+Bonnet (GB) theory of gravity proposed in [67]. The motivation of such representation is due to the rich structure of
+the Gauss-Bonnet theory in various directions of cosmology [68–71]. The action for f(G) gravity is given by [67],
+S =
+�
+d4x√−g
+� 1
+2κ2 R + λ
+�1
+2∂µχ∂µχ + µ4
+2
+�
+− 1
+2∂µχ∂µχ + h (χ) G − V (χ)
+�
+,
+(68)
+where µ is a constant having mass dimension [+1], λ represents the Lagrange multiplier, χ is a scalar field and V (χ)
+is its potential. Moreover G = R2 − 4RµνRµν + RµναβRµναβ is the Gauss-Bonnet scalar and h(χ) symbolizes the
+Gauss-Bonnet coupling with the scalar field. Moreover we consider such class of Gauss-Bonnet coupling functions
+that satisfy ¨h = ˙hH. This condition actually leads to the speed of the gravitational wave as unity in the context of
+GB theory and makes the model compatible with the GW170817 event. For a certain γ(N) in the context of entropic
+cosmology, there exists an equivalent set of GB parameters in the side of Gauss-Bonnet cosmology that results to the
+same cosmological evolution as of the generalized entropy. In particular, the equivalent forms of ˜V (χ) and λ(t) for a
+certain γ(N) turn out to be,
+˜V (χ) = −8πG F1 [γ(N), γ′(N)]
+� 1
+κ2 + 8h0a(t)H(t)
+� ����
+t=χ/µ2 ,
+(69)
+µ4λ(t) = −8πG F2 [γ(N), γ′(N)]
+� 1
+κ2 − 8h0a(t)H(t)
+�
+,
+(70)
+where the functions F1 [γ(N), γ′(N)] and F2 [γ(N), γ′(N)] are given by,
+F1 [γ(N), γ′(N)] = −
+� 3ϵα
+4βG2
+�
+�
+����ln
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+2
+�
+2
+γ(N)+√
+γ2(N)+4
+�1/β
+− 1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+����
+−1
++ H4
+� γ′(N)
+8π2γ(N)
+�
+×
+�
+�
+1 + 1
+ϵ tanh
+�
+ϵπα
+βGH2
+��β
+−
+�
+1 + 1
+ϵ tanh
+�
+ϵπα
+βGH2
+��−β
+α sech2 �
+ϵπα
+βGH2
+� ��
+1 + 1
+ϵ tanh
+�
+ϵπα
+βGH2
+��β−1
++
+�
+1 + 1
+ϵ tanh
+�
+ϵπα
+βGH2
+��−β−1�
+�
+
+14
+and
+F2 [γ(N), γ′(N)] = H4
+� γ′(N)
+8π2γ(N)
+�
+�
+���
+�
+1 + 1
+ϵ tanh
+�
+ϵπα
+βGH2
+��β
+−
+�
+1 + 1
+ϵ tanh
+�
+ϵπα
+βGH2
+��−β
+α sech2 �
+ϵπα
+βGH2
+� ��
+1 + 1
+ϵ tanh
+�
+ϵπα
+βGH2
+��β−1
++
+�
+1 + 1
+ϵ tanh
+�
+ϵπα
+βGH2
+��−β−1�
+�
+���
+respectively. Based on Eq.(69) and Eq.(70), we may argue that the entropic cosmology of Sns can be equivalently
+represented by Gauss-Bonnet cosmology.
+As mentioned earlier that we consider the quasi-matter bounce scenario described by the scale factor (65) to analyze
+the perturbation, where the perturbation modes generate during the contracting phase deep in the sub-Hubble regime,
+which in turn ensures the resolution of the horizon problem. The important quantities that we will need are,
+Qa = −8˙hH2 = −4n2(1 + 2n)
+�
+�R
+πG
+� �R
+R0
+� 1
+2 −n
+,
+Qb = −16˙hH = 4n(1 + 2n)
+πG
+� �R
+R0
+� 1
+2 −n
+,
+Qc = Qd = 0
+,
+Qe = −32˙h ˙H = 8n(1 + 2n)
+�
+�R
+πG
+� �R
+R0
+� 1
+2 −n
+,
+Qf = 16
+�
+¨h − ˙hH
+�
+= 0 ,
+(71)
+respectively, where R0 =
+1
+t2
+0 and �R(t) =
+R(t)
+12n(1−4n). In regard to curvature perturbation, the Mukhanov-Sasaki (MS)
+equation in Fourier mode comes as,
+d2vk(η)
+dη2
++
+�
+k2 − σ
+η2
+�
+vk(η) = 0 ,
+(72)
+here η symbolizes the conformal time coordinate and v(k, η) is the scalar MS variable. Moreover σ is given by,
+σ = ξ(ξ − 1)
+�
+�1 + 24
+�
+1 − 4n2�
+� �R
+R0
+� 1
+2 −n�
+� ,
+(73)
+which is approximately a constant during the generation era of the perturbation modes in the sub-Hubble regime
+during the contracting phase, due to the condition n < 1/2 (required to solve the horizon problem). In effect of which
+and considering the Bunch-Davies initial condition, the scalar power spectrum PΨ(k, η) in the super-horizon scale
+becomes,
+PΨ(k, η) =
+�� 1
+2π
+�
+1
+z |η|
+Γ(ν)
+Γ(3/2)
+�2 �k|η|
+2
+�3−2ν
+,
+(74)
+In regard to the tensor perturbation, the Mukhanov-Sasaki equation takes the following form,
+d2vT (k, η)
+dη2
++
+�
+k2 − σT
+η2
+�
+vT (k, η) = 0 ,
+(75)
+where vT (k, η) being the Fourier mode for the tensor MS variable, and σT has the following form,
+σT = ξ(ξ − 1)
+�
+�1 − 16(1 − 4n2)
+� �R
+R0
+� 1
+2 −n�
+� .
+(76)
+Due to n < 1/2, the quantity σT can be safely considered to be a constant during the generation era of the perturbation
+modes at the contracting phase of the universe. Here it may be mentioned that both the tensor polarization modes
+(+ and × polarization modes) obey the same evolution Eq.(75) – this means that the two polarization modes equally
+contribute to the energy density of the tensor perturbation variable, and thus we will multiply by the factor ’2’ in the
+final expression of the tensor power spectrum. Similar to the curvature perturbation variable, the tensor perturbation
+initiates from the Bunch-Davies vacuum at the distant past, i.e. vT (k, η), i.e limk|η|≫1 vT (k, η) =
+1
+√
+2ke−ikη. With
+such initial condition, we obtain the tensor power spectrum for kth mode in the super-Hubble regime as,
+PT (k, τ) = 2
+� 1
+2π
+1
+zT |η|
+Γ(θ)
+Γ(3/2)
+�2 �k|η|
+2
+�3−2θ
+,
+(77)
+
+15
+where θ =
+�
+σT + 1
+4. Having obtained the scalar and tensor power spectra, we determine ns and r, and they are
+given by (the suffix ’h’ with a quantity represents the quantity at the instant of horizon crossing),
+ns = 4 −
+√
+1 + 4σh ,
+r = 2
+� z(ηh)
+zT (ηh)
+Γ(θ)
+Γ(ν)
+�2
+(k |ηh|)2(ν−θ) ,
+(78)
+where the quantities have the following forms,
+ν =
+�
+σh + 1
+4 ;
+σh = ξ(ξ − 1)
+�
+�1 + 24
+�
+1 − 4n2�
+� �Rh
+R0
+� 1
+2 −n�
+� ,
+θ =
+�
+σT,h + 1
+4 ;
+σT,h = ξ(ξ − 1)
+�
+�1 − 16(1 − 4n2)
+� �Rh
+R0
+� 1
+2 −n�
+� ,
+z(ηh) = −
+1
+√n
+�
+an
+0
+κ �Rn
+h
+� �
+�1 − 24n(1 + 2n)
+� �Rh
+R0
+� 1
+2 −n�
+� ,
+zT (ηh) = 1
+√
+2
+�
+an
+0
+κ �Rn
+h
+� �
+�1 + 16n(1 + 2n)
+� �Rh
+R0
+� 1
+2 −n�
+� .
+(79)
+0.960 0.962 0.964 0.966 0.968 0.970
+0.01177
+0.01178
+0.01179
+0.01180
+0.01181
+0.01182
+ns
+r
+FIG. 2: Parametric plot of ns (along x-axis) vs. r (along y-axis) with respect to n. Here we take α = [0.0938, 0.0939] and
+β =
+π
+16.
+Here �Rh represents the Ricci scalar at the horizon crossing, and using the horizon crossing condition kηh =
+2n
+1−2n,
+it comes as,
+�Rh =
+�
+1
+26nan
+0
+�2/(1−2n)
+By−2 .
+(80)
+Therefore it is clear that ns and r in the present context depends on the parameters n and a0. Here we need to recall
+that n and a0 are related to the entropic parameters as n = √α and a0 = π/ (4β) respectively. It turns out that the
+theoretical predictions for ns and r get simultaneously compatible with the recent Planck data for a small range of
+the entropic parameters given by: α = [0.0938, 0.0939] and β = π
+16, see Fig.[2].
+
+16
+VII.
+CONCLUSION
+In this short review article, we have proposed generalized entropic function(s) and have addressed their implications
+on black hole thermodynamics as well as on cosmology. In the first half of the paper, a 4-parameter and a 3-parameter
+generalized entropy functions are shown, which are able to generalize the known entropies proposed so far, like the
+Tsallis, R´enyi, Barrow, Sharma-Mittal, Kaniadakis and Loop Quantum Gravity entropies for suitable choices of the
+respective entropic parameters. However the 4-parameter entropy functions proves to be more general compared to
+the 3-parameter entropy function, in particular, the 3-parameter entropy does not converge to the Kaniadakis entropy
+for any choices of the parameters, unlike to the entropy having 4 parameters which generalizes all the known entropies
+including the Kaniadakis one. Thus regarding to the number of parameters in a generalized entropy function, we have
+provided a conjecture – “The minimum number of parameters required in a generalized entropy function that can
+generalize all the known entropies mentioned above is equal to four”. Consequently the interesting implications of
+3-parameter entropy on black hole thermodynamics and the 4-parameter entropy on cosmology have been addressed.
+It turns out that the entropic cosmology corresponding to the 4-parameter generalized entropy results to an unified
+cosmological scenario of early inflation and the late dark energy era of the universe, where the observable quantities
+are found to be compatible with the recent Planck data for certain viable ranges of the entropic parameters.
+Despite these successes, here it deserves mentioning that the 4-parameter entropy function (Sg) seems to be plagued
+with singularity for certain cosmological evolution of the universe. In particular, Sg diverges at the instant when the
+Hubble parameter vanishes, for instance at the instant of bounce in the context of bounce cosmology. With this
+spirit, we have proposed a singular-free 5-parameter entropy function (Sns) which converges to all the known entropy
+functions for particular limits of the entropic parameters, and at the same time, also proves to be non-singular for the
+entire cosmological evolution of the universe even at H = 0 (where H represents the Hubble parameter). Regarding
+to the non-singular entropy, a second conjecture has been given : “The minimum number of parameters required in
+a generalized entropy function that can generalize all the known entropies, and at the same time, is also singular-free
+during the universe’s evolution – is equal to five”. Such non-singular behaviour of Sns proves to be useful in describing
+the bounce cosmology, in particular, the entropic cosmology corresponding to Sns naturally allows symmetric bounce
+universe. With the perturbation analysis in the context of entropic bounce, it has been shown that the observable
+quantities like the spectral tilt and the tensor-to-scalar ratio are simultaneously compatible with the Planck data in
+the background of symmetric quasi-matter bounce scenario.
+Finally we would like to mention that the proposals of generalized entropy functions (Sg or Sns) opens a new
+directions in theoretical physics, and its vast consequences may hint some unexplored directions of black hole thermo-
+dynamics as well as of cosmology. For example, it will be of utmost interest to study the aspects of the generalized
+entropy functions on primordial black hole formation or primordial gravitational wave or the recently found astro-
+physical black holes as well. With the recent and future advancements of different detectors (like the GW detectors
+or regarding the black hole detection), we hope that these study can indirectly quantify the viable ranges of entropic
+parameters.
+Acknowledgments
+This work was supported by MINECO (Spain), project PID2019-104397GB-I00 and also partially supported by the
+program Unidad de Excelencia Maria de Maeztu CEX2020-001058-M, Spain (SDO). This research was also supported
+in part by the International Centre for Theoretical Sciences (ICTS) for the online program - Physics of the Early
+Universe (code: ICTS/peu2022/1) (TP).
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+1
+Realtime Safety Control for Bipedal Robots to
+Avoid Multiple Obstacles via CLF-CBF Constraints
+Jinze Liu∗, Minzhe Li∗, Jiunn-Kai Huang, and Jessy W. Grizzle
+Abstract—This paper presents a reactive planning system that
+allows a Cassie-series bipedal robot to avoid multiple non-
+overlapping obstacles via a single, continuously differentiable
+control barrier function (CBF). The overall system detects an
+individual obstacle via a height map derived from a LiDAR
+point cloud and computes an elliptical outer approximation,
+which is then turned into a CBF. The QP-CLF-CBF formalism
+developed by Ames et al. is applied to ensure that safe trajectories
+are generated. Liveness is ensured by an analysis of induced
+equilibrium points that are distinct from the goal state. Safe
+planning in environments with multiple obstacles is demonstrated
+both in simulation and experimentally on the Cassie biped.
+THIS IS AN INITIAL DRAFT
+While the paper is not yet polished, it allows the co-
+first authors to highlight their research skills while they
+are seeking a PhD position. The full autonomy videos are
+upload to our YouTube channel and the video for this
+particular paper can be viewed here. This draft has been
+approved by Huang and Grizzle.
+I. INTRODUCTION AND CONTRIBUTIONS
+Bipedal robots are typically conceived to achieve agile-
+legged locomotion over irregular terrains, and maneuver in
+cluttered environments [1]–[3]. To explore safely in such
+environments, it is critical for robots to generate quick, yet
+smooth responses to any changes in the obstacles, map, and
+environment. In this paper, we propose a means to design and
+compose control barrier functions (CBFs) for multiple non-
+overlapping obstacles and evaluate the system on a 20-degree-
+of-freedom (DoF) bipedal robot.
+In an autonomous system, the task of avoiding obstacles
+is usually handled by a planning algorithm because it has
+access to the map of an entire environment. Given the map,
+the planning algorithm is then able to design a collision-free
+path from the robot’s current position to a goal. If the map
+is updated due to a change in the environment, the planner
+then needs to update the planned path, so-called replanning,
+to accommodate the new environment. Such maps are typically
+large and contain rich information such as semantics, terrain
+characteristics, and uncertainty, and thus are slow to update.
+This raises a concern when obstacles either move into the
+planned path but the map has not been updated or a robot’s
+new pose allows the detection of previously unseen obstacles.
+The slow update rate of the map leads to either collision or
+∗ equal contribution.
+Jinze Liu, Minzhe Li, Jiunn-Kai Huang, and Jessy W. Grizzle are with the
+Robotics Institute, University of Michigan, Ann Arbor, MI 48109, USA. {
+minzlee, jzliu, bjhuang, grizzle}@umich.edu.
+Fig. 1: In the top figure, Cassie Blue autonomously avoids multiple obstacles
+via the developed CLF-CBF-QP obstacle avoidance system, comprised of
+an intermediate goal selector, obstacle detection, and a CLF-CBF quadratic
+programming solver. The bottom figure is the elevation map built in real time.
+The blue and cyan blobs are obstacles that Cassie detects and avoids in real
+time. A gantry is used in the experiments because the lab-built perception
+system that has been added to the robot is unprotected in case of falls.
+abrupt maneuvers to avoid collisions. The non-smooth aspects
+arising from the map updates or changes in the perceived
+environment can be detrimental to the stability of the overall
+system.
+Research on obstacle avoidance has been studied for sev-
+eral decades as pioneered in classic probabilistic roadmap
+approaches (PRM) [4] and cell decomposition [5, Chapter
+arXiv:2301.01906v1  [cs.RO]  5 Jan 2023
+
+6]. However, the omission of robot dynamics and the extra
+computation for map discretization make these methods hard
+to use in real-time applications. Artificial potential fields [6]–
+[15] stand out for their simplicity, extendability, and efficiency,
+leading to their wide adoption for real-time obstacle avoidance
+planning problems. A drawback of potential field methods
+is that they require the entire map of an environment to be
+available when designing a potential function that will render
+attractive one or more goal points in the map. Moreover, un-
+wanted local minima and oscillations in the potential field have
+limited their deployment in the field. A control barrier function
+(CBF) [16], on the other hand, enables real-time controller
+synthesis with provable safety for mobile robots operating in a
+continuous (non-discretized) space and can work with a partial
+(or incomplete) map. A control Lyapunov function (CLF) is a
+(candidate) positive definite function for a closed-loop system
+where at any given time instance there exists a control input
+that renders the derivative of the Lyapunov function along
+the system dynamics negative definite. The CLF is typically
+designed to vanish at a desired goal state or pose.
+The main theme of [16] is that a real-time quadratic program
+(QP) can be used to combine a CLF and a CBF in such a way
+that closed-loop trajectories induced by the CLF are minimally
+modified to provide provable safety, that is, non-collision with
+obstacles. This design philosophy has been explored in [16]–
+[18].
+One means of avoiding obstacles is to come to a complete
+stop, though it is at the cost of not reaching the goal state.
+The papers [19]–[22] showed that such behavior can be an
+unintended outcome of the CLF-CBF-QP design approach
+of [16]. Specifically, the inequality constraints (of the QP)
+associated with the derivatives of the control Lyapunov and
+control barrier functions can induce equilibria in the closed-
+loop system that are distinct from the equilibrium of the
+CLF. Reference [19] characterizes these equilibria via the
+KKT conditions associated with the QP, while reference [20]
+emphasizes that if an induced equilibrium is unstable, then
+“natural noise” in the environment will avoid the robot being
+deadlocked at an unstable equilibrium.
+Inspired by the above-cited works on CLF-CBF-QPs for
+planning and control, we incorporate high-bandwidth obstacle
+avoidance into the CLF-RRT* reactive planner of [1]. The
+CLF in [1] takes into account features specific to bipeds, such
+as the limited lateral leg motion that renders lateral walking
+more laborious than sagittal plane walking. This paper seeks
+to utilize the CLF designed specifically for bipedal robots
+in tandem with a CBF to avoid multiple, non-overlapping
+obstacles in a smooth fashion, while ensuring progress to a
+goal state.
+The main contributions of the new proposed CLF-CBF
+system are the following:
+1) We propose a novel CLF-CBF-QP obstacle avoidance
+system specifically adapted for bipedal robots locomoting
+in the presence of multiple non-overlapping obstacles.
+The full system provides for real-time obstacle detection,
+CBF design, and safe control input generation through a
+QP.
+2) We mathematically prove the validity of the proposed
+CBF for both single and multiple obstacles. We also
+analytically analyze the existence of spurious equilibrium
+points induced by the CLF-CBF constraints on the QP.
+3) We provide simulations that support the mathematical
+analysis for obstacle avoidance while reaching a goal.
+4) The overall reactive planning system is demonstrated
+experimentally on a 20-degree-of-freedom Cassie-series
+bipedal robot.
+5) We
+open-source
+the
+implementations
+of
+the
+system
+in
+C++
+with
+Robot
+Operating
+System
+(ROS)
+[23]
+and
+associated
+videos
+of
+the
+experiments;
+see
+https://github.com/UMich-
+BipedLab/multi_object_avoidance_via_clf_cbf.
+The rest of the paper is organized as follows. Section II
+overviews related work. The design and validation of the
+proposed CBF is presented in Sec. III. We analyze equilibrium
+points of the proposed CBF in Appendix A. Section IV
+proposes a novel and simple method to combine CBFs for non-
+overlapping obstacles. Simulation and experimental results are
+given in Sec. V.
+II. RELATED WORK ON CONTROL WITH SAFETY
+A continuously differentiable, proper, positive definite func-
+tion V (x) that vanishes at a single point is called a candidate
+Lyapunov function [24]. If the derivative of V (x) along the
+trajectories of a control system can be rendered negative
+definite by proper choice of the control input, it is called a
+control Lyapunov Function, or CLF for short [25]–[27]. CLFs
+are widely used in the design of controllers to asymptotically
+drive a system to a goal state. Safety involves steering a control
+system to a goal state while avoiding self-collisions, obstacles,
+or other undesirable states, collectively referred to as unsafe
+states. The set complement of the unsafe states is the set of
+safe states.
+A. Artificial Potential Fields and Navigation Functions
+The first systematic method for real-time control and ob-
+stacle avoidance was introduced by Khatib in [28]. Called
+the method of Artificial Potential Functions, it revolutionized
+feedback control for manipulators in that hard constraints
+could be enforced in both the robot’s task space and joint space
+in real time. Prior to this seminal work, obstacle avoidance,
+or more generally the generation of safe paths, was relegated
+to a path planner operating at a much slower time scale. A
+survey of the method of potential functions can be found in
+[29].
+Potential functions seek to construct “repulsive fields”
+around obstacles that are active throughout the entire state
+space of the robot’s dynamical system, without destroying
+the presence of an attractive field steering the system to a
+goal state. It has been recognized that superimposed attracting
+and repelling fields can create undesired spurious equilibria,
+which prevent a robot from reaching its goal state [30]. In
+addition, potential fields have been observed to introduce tra-
+jectory oscillations as a robot passes near obstacles. Heuristic
+2
+
+modifications have been proposed to avoid local minima [11]–
+[13], while potential fields have been combined with other
+gradient-based functions to reduce oscillations [14], [15].
+The method of Navigation Functions by Koditschek and
+Rimon [31] sought to design a single function whose gradient
+produces trajectories avoiding multiple obstacles while asymp-
+totically converging to a single goal state from almost all
+initial conditions [32]–[35]; specifically, all equilibria except
+the goal state should be unstable. Because the design of a
+navigation function takes into account the global topology
+of the method of navigation functions is not appropriate for
+problems requiring the online identification and avoidance of
+obstacles; in addition, there are topological restrictions to the
+existence of navigation functions.
+B. Control Barrier Functions and Control Lyapunov Func-
+tions
+Barrier Functions provide Lyapunov-like conditions for
+proving a given set of safe states is forward invariant, meaning
+that trajectories starting in the safe set remain in the safe
+set. The natural extension of a barrier function to a system
+with control inputs is a Control Barrier Function or CBF
+for short, first proposed by [36]. CBFs parallel the extension
+of Lyapunov functions to CLFs, in that the key point is to
+impose inequality constraints on the derivative of a candidate
+CBF (resp., CLF) to establish entire classes of controllers
+that render a given set forward invariant (resp., asymptotically
+stable).
+Importantly, barrier functions and CBFs focus solely on
+safety and do not seek to simultaneously steer a system
+to any particular point in the safe set. This allows CBFs
+to be combined with other “goal-oriented” control methods
+as a (maximally permissive) supervisor that only modifies
+a trajectory when it is in conflict with the safety criteria
+established by the CBF. The papers [37], [38] introduced the
+notion of using a real-time quadratic program (QP) to combine
+a CBF with a CLF to achieve convergence to a goal state
+while avoiding unsafe states. The overall method goes by the
+acronym CLF-CBF-QP.
+For control systems that are affine in the control variable,
+CLF-CBF-QPs have proven to be enormously popular in and
+out of robotics applications [16]–[18], [39]–[43]. To highlight
+just a few example, reference [17] uses a CLF-CBF-QP to
+achieve stable walking for bipedal robots, while trajectory
+planning under spatiotemporal and control input constraints
+is presented in [18], [39], [40]. Applications to obstacle
+avoidance are addressed in [41]–[43].
+The recent paper [44] shows that CBFs are a strict general-
+ization of artificial potential functions and that in a practical
+example, a CLF-CBF-QP has reduced issues with oscillations
+as a robot passes near obstacles and improved liveness, mean-
+ing the ability to reach the goal state. Hence, we use the
+method of CLF-CBF-QPs in this paper.
+C. Combining Multiple CBFs
+Usually, a control barrier function is designed for a single
+obstacle. When there are multiple obstacles in the control
+system, the barrier functions for each obstacle must be com-
+bined in some manner to provide safety guarantees. Reference
+[45] shows that if the intersection of the set of “allowable
+controls” of individual CBFs is non-empty, then the CLF-
+CBF-QP method can be extended to several obstacles; the
+reference does not show how to check this condition online (in
+real time). Multiple CBF functions have also been combined
+to obtain a single CBF so that existing methods can be
+applied. Reference [46] combines several CBFs into an overall
+CBF using max-min operations. The resulting CBF is non-
+differentiable and hence this technique is not used here. Ref-
+erence [47] combines multiple CBFS for disjoint unsafe sets
+with a single CLF to produce a new CLF that simultaneously
+provides asymptotic stability and obstacle avoidance. This
+work is therefore related to the method navigation functions
+reviewed above and suffers from the same drawbacks; how-
+ever, a key technique used in this reference to combine the
+CBFs before merging them with a CLF will be exploited in the
+current paper, namely a continuously differentiable saturation
+function.
+D. CLF-CBF-QPs and Unwanted Equilibrium Points
+The presence of multiple stable equilibrium points intro-
+duces “deadlock” in a control system. Reference [19] shows
+that the use of real-time QPs to combine safety and goal-
+reaching in navigation problems can lead to unwanted equi-
+librium points. With this awareness, the authors of [21] modify
+the cost function in the quadratic program to remove the
+unwanted equilibria. The modification induces a rotational
+motion in the closed-loop system that steers it around the
+obstacle, something a bipedal robot can do naturally. Hence,
+here we only exploit their analysis method for finding the
+unwanted equilibria and show that our method introduces
+at most one undesired equilibrium point when obstacles are
+disjoint. Moreover, we do not need to remove the unwanted
+equilibrium using the methods in [22], [48] by transforming
+the system’s state space into a convex manifold, or by increas-
+ing the complexity of the system’s state space.
+E. Summary
+The presence of multiple obstacles is common in practice.
+While existing works can treat disjoint obstacles, they are not
+appropriate for use where obstacles are identified in real-time
+via an onboard perception system. In this work, for a biped-
+appropriate planning model, we propose a simple means to
+combine CBFs for disjoint obstacles so that the complexity
+of the real-time CLF-CBF-QP remains constant and induced
+equilibrium points are easy to characterize and avoid.
+III. CONSTRUCTION OF CONTROL LYAPUNOV FUNCTION
+AND CONTROL BARRIER FUNCTION
+This section introduces the CLF proposed in [1] and ana-
+lyzes its trajectories when combined with a quadratic CBF
+through a real-time QP. The goal is to ensure the closed-
+loop system is able to reach a goal state while smoothly
+avoiding a single obstacle. This section lays the foundation
+for considering multiple obstacles in the next section.
+3
+
+Fig. 2: The red line is the distance between the obstacle and the robot.
+A. State Representation
+Denote P = (xr, yr, θ) the robot pose, G = (xt, yt) the
+goal position in the world frame. We simplify an obstacle O
+as a circle (and hence convex) described as its center (xo, yo)
+and its radius ro. We define the robot state as
+x =
+�
+�
+r
+δ
+θ
+�
+� ,
+(1)
+where r =
+�
+(xt − xr)2 + (yt − yr)2, θ is the heading angle
+of the robot, and δ is the angle between θ and the line of sight
+from the robot to the goal, as shown in Fig. 2.
+The dynamics of the control system is defined as
+˙x = f(x) + g(x)u
+=
+�
+�
+0
+0
+0
+�
+� +
+�
+�
+− cos(δ)
+− sin(δ)
+0
+sin(δ)
+r
+− cos(δ)
+r
+1
+0
+0
+−1
+�
+�
+�
+�
+vx
+vy
+ω
+�
+� ,
+(2)
+where we view u =
+�vx,
+vy,
+ω�T as the control variables
+in the robot frame, as shown in Fig. 2.
+B. Design of CLF and CBF for Bipedal Robots
+The control Lyapunov function leveraged in the reactive
+planner proposed in [1] takes into account features specific
+to bipeds, such as the limited lateral leg motion that renders
+lateral walking more laborious than sagittal plane walking.
+Therefore, we also define the CLF as
+V (x) = r2 + γ2 sin2(βδ)
+2
+,
+(3)
+where γ is the weight on the orientation, and β controls the
+size of the field of view (FoV). Given P and G, we have a
+closed-form solution for control u in (2),
+ωref = r cos(δ) [rvδ cos(δ) − vr sin(δ)]
+α + r2 cos2(δ)
+vref
+y
+= α(vr sin(δ) − rvδ cos(δ))
+r2cos(δ)2 + α
+vref
+x
+= vr cos(δ)r2 + αvδ sin(δ)r + αvr cos(δ)
+r2cos(δ)2 + α
+;
+(4)
+where vr and vδ are defined as:
+vr = kr1
+r
+kr2 + r
+vδ = − 2
+β kδ1
+r
+kδ2 + r sin(2βδ).
+(5)
+In (4) and (5), α, β, kr1, kr2, kδ1, kδ2 are positive constants.
+See [1] for more details.
+Next, we introduce a candidate CBF as
+B(x) =
+� xr − xo
+yr − yo
+�⊤
+Q
+� xr − xo
+yr − yo
+�
+− r2
+o,
+(6)
+where (xo, yo) gives the center of the obstacle, ro specifies the
+“radius” of the obstacle, and Q is positive definite. We next
+verify that (6) is a valid CBF.
+C. Proof of CBF Validity
+Following [49], we define the sets
+D := {x ∈ R3 | B(x) ̸= −r2
+o, and r ̸= 0}
+C := {x ∈ D | B(x) ≥ 0}
+(7)
+associated with the candidate CBF (6) and note that Int(C) ̸=
+∅ and Int(C) = C. From [49], for (6) to be a valid CBF
+function of (2), there must exist some η > 0, such that,
+∀x ∈ D, ∃u ∈ R3, ˙B(x, u) + ηB(x) ≥ 0,
+(8)
+where ˙B(x, u) := LfB(x) + LgB(x)u is the time derivative
+of B(x) along the dynamics of (2), η > 0 sets the repulsive
+effort of the CBF, and
+LfB(x) := ∂B(x)
+∂x
+f(x)
+(9)
+LgB(x) := ∂B(x)
+∂x
+g(x).
+(10)
+Because the drift term f(x) in (2) is identically zero, the
+zero control u ≡ 0 satisfies (8) for x ∈ C. Hence, we need to
+show that (8) can be met for x ∈∼ C, the set complement of
+C. Direct application of the chain rule gives that
+LgB(x) = a(x)b(x)g(x),
+where
+a(x) := 2 [ xt − r cos(δ + θ) − xo,
+yt − r sin(δ + θ) − yo ] Q
+= 2
+� xr − xo,
+yr − yo
+�
+Q
+b(x) :=
+� − cos(δ + θ)
+r sin(δ + θ)
+r sin(δ + θ)
+− sin(δ + θ)
+−r cos(δ + θ)
+−r cos(δ + θ)
+�
+g(x) =
+�
+���
+− cos(δ)
+− sin(δ)
+0
+sin(δ)
+r
+− cos(δ)
+r
+1
+0
+0
+−1
+�
+��� .
+(11)
+Moreover, a(x) only vanishes at the center of an obstacle,
+the rows of b(x) are linearly independent for all r > 0, and
+det (g(x)) = − 1
+r ̸= 0 for all 0 < r < ∞. It follows that for
+all x ∈ D, LgB(x) ̸= 0 and hence (8) is satisfied, proving
+that (6) is a valid CBF.
+4
+
+V
+W
+O =(x
+X
+X
+JD. Quadratic Program of the Proposed CLF-CBF System
+A quadratic program (QP) is set up to optimize the control
+u with the slack variable s while enforcing both the CLF and
+CBF constraints. Let L(x, u, s) be the CLF constraints
+L(x, u, s) := LfV (x) + LgV (x)u + µV (x) − s ≤ 0,
+(12)
+where Lpq(x) := ∇q(x) · p(x) is the Lie derivative, µ serves
+as a decay rate of the upper bound of V (x). Next, we denote
+B(x, u) the CBF constraints
+B(x, u) := −LfB(x) − LgB(x)u − ηB(x) ≤ 0,
+(13)
+where η serves as a decay rate of the lower bound of B(x).
+Finally, the QP for the control values is formulated as
+u∗, s∗ = arg min
+L(x,u,s)≤0
+B(x,u)≤0
+J(u, s),
+(14)
+where the cost function J(u, s) is defined as
+J(u, s) := 1
+2(u − uref)T H(u − uref) + 1
+2ps2,
+(15)
+the positive definite, diagonal matrix H := diag([h1, h2, h3])
+weights the control variables, uref :=
+�vref
+x
+vref
+y
+ωref�T is
+the control vector from the CLF (3) without obstacles, and
+p ≥ 0 is the weight of the slack variable, s.
+In the proposed CLF-CBF-QP system, uref is the closed-
+form solution obtained from the CLF without obstacles, and H
+assigns weights for different control variables. The proposed
+CLF-CBF-QP cost function captures inherent features of a
+Cassie-series robot, such as the low-cost of longitudinal move-
+ment and high-cost of lateral movement, while guaranteeing
+safety. We next look at liveness, that is, the ability of the
+system to reach the desired goal.
+E. Analysis for Unwanted Equilibria
+Paper [19] points out very clearly that the CLF-CBF-QP
+formulation of Sec. III-D can introduce unwanted equilibria
+that may prevent the robot from reaching a goal state. The
+paper [20] also considered this problem and noted that if the
+equilibria are unstable, then liveness is preserved for almost
+all initial conditions. In Appendix A, we follow the KKT-
+analysis of the CLF-CBF-QP presented in [19] and show that
+Fig. 3: Illustration of a case when the robot directly faces the obstacle and the
+target creates an equilibrium in the continuous-time system. In a simulation
+with discrete-time control updates, the robot walks back and forth at the
+obstacle boundary.
+Fig.
+4:
+Illustration
+of
+breaking
+the
+equilibrium
+by
+using
+uref
+2 :
+�vref
+x
+vref
+y
+ωref + ϵ�T when δ = 0. The robot successfully reaches the
+target position without colliding with the obstacle.
+only one equilibrium point is created by the QP. Moreover, the
+equilibrium occurs at an obstacle boundary for δ = 0, dy =
+0, dx > 0, in other words, when the robot’s heading faces
+directly to the obstacle and the target, as shown in Fig. 3. The
+robot will move directly toward the obstacle and stop at the
+obstacle boundary.
+Remark 1. When the robot encounters the above equilibrium
+state, we can add a constant ϵ > 0 to uref in (14) such that
+uref =
+�vref
+x
+vref
+y
+ωref + ϵ�T. As is shown in Fig. 4, the
+robot breaks its equilibrium state, avoids the obstacle, and
+reaches the target position. This is related to, but distinct
+from, the method presented in [19] for resolving unwanted
+equilibria.
+IV. COMBINING CBFS FOR MULTIPLE OBSTACLES
+So far, we have assumed there is only one obstacle perceived
+by the robot. In this section, we will discuss how to handle
+multiple obstacles in the environment when each obstacle is a
+positive distance apart from the others [47]. Specifically, for
+i ∈ {1, 2, . . . , M}, suppose that
+Bi(x) :=
+� xr − xo,i
+yr − yo,i
+�⊤
+Qi
+� xr − xo,i
+yr − yo,i
+�
+− r2
+o,i
+Di := {x ∈ R3 | Bi(x) ̸= −r2
+o,i, and r ̸= 0}
+Ci := {x ∈ Di | Bi(x) ≥ 0}
+(16)
+are valid CBF functions for the dynamics (2). For i ̸= j, the
+obstacles corresponding to Bi : R3 → R and Bj : R3 → R are
+a positive distance apart if
+∆ij :=
+inf
+x ∈∼ Ci
+y ∈∼ Cj
+||x − y|| > 0.
+(17)
+A key innovation with respect to [46] is that we will
+compose the associated CBFs in a smooth (C1) manner.
+A potential drawback with respect to [46] is that we will
+assume the obstacles giving rise to the CBFs are a positive
+distance apart. Similar to [47], we saturate standard quadratic
+CBFs before seeking to combine them. Distinct from [47], we
+multiply the saturated CBFs instead of creating a weighted
+5
+
+-2
+-4
+-6
+-8
+-10E
+-6
+-4
+-2
+0
+2
+4
+6
+X
+m-2
+-4
+目
+-6
+-8
+-10
+-6
+-4
+-2
+0
+2
+4
+6
+X
+msum. This greatly simplifies the analysis of the composite CBF
+with respect to all previous works.
+A. Smooth Saturation Function
+We introduce a continuously differentiable saturation func-
+tion that will allow us to compose in a simple manner CBFs
+corresponding to obstacles that are a positive distance apart.
+Consider σ : R → R by
+σ(s) :=
+�
+�
+�
+�
+�
+s
+s ≤ 0
+s(1 + s − s2)
+0 < s < 1
+1
+s ≥ 1.
+(18)
+Then straightforward calculations show that for all s ∈ R,
+dσ(s)
+ds
+exists and satisfies
+dσ(s)
+ds
+:=
+�
+�
+�
+�
+�
+1
+s ≤ 0
+1 + 2s − 3s2
+0 < s < 1
+0
+s ≥ 1.
+(19)
+Upon noting that for all 0 < s < 1, 0 < dσ(s)
+ds
+< 1, it follows
+that σ : R → R is continuously differentiable and monotonic.
+Remark 2. For 0 ≤ s ≤ 1, σ is constructed from a degree-
+three Bézier polynomial p : [0, 1] → R such that p(0) = 0,
+dp(0)
+ds
+= 1, p(1) = 1, dp(1)
+ds
+= 0. Moreover, for 0 < s < 1,
+dp(s)
+ds
+> 0.
+Definition 1. For κ > 0, we define σκ : R → R by
+σκ(s) := σ( s
+κ).
+(20)
+Proposition 1. Suppose that κ > 0 and B : D → R is a
+candidate CBF with D and C defined as in (7). Then σκ ◦ B :
+D → R is a valid CBF for the system (2) if, and only if,
+B : D → R is a valid CBF.
+Proof. For x ∈ C, σκ ◦ B(x) > 0 and hence satisfies (8) for
+u = 0. For x ∈∼ C, by the chain rule and the construction of
+σ : R → R,
+∂σκ ◦ B(x)
+∂x
+= dσ(s)
+ds
+����
+s= B(x)
+κ
+∂B(x)
+∂x
+= 1
+κ
+∂B(x)
+∂x
+.
+(21)
+Hence, the proof of Sect. III-C applies.
+■
+Proposition 2. Suppose for 1 ≤ i ≤ M, the CBFs Bi(x) :
+R3 → R are a positive distance apart. Then there exist κ1 > 0,
+κ2 > 0, . . ., κM > 0, such that for all i ̸= j,
+{x ∈ R3 | σκi ◦ Bi(x) < 1} ∩ {x ∈ R3 | Bj(x) < 0} = ∅.
+(22)
+Proof. By the disjointness property, ∆i := min
+j̸=i
+∆ij > 0.
+For S ⊂ R3 and x ∈ R3, define the distance from x to S
+by
+d(x, S) := inf
+y∈S ||x − y||.
+(23)
+Then, because (i) Bi is continuous, (ii) the set complement of
+Ci is bounded, and (iii) d(x, ∼ Ci) > 0 =⇒ Bi(x) > 0, it
+follows that
+m∗
+i :=
+sup
+d(x,∼Ci)≤∆i
+Bi(x)
+(24)
+is a finite positive number. Therefore, for all 0 < κi < m∗
+i ,
+{x ∈ R3 | σκi ◦ Bi(x) < 1} ⊂ {x ∈ R3 | d(x, ∼ Ci) ≤ ∆i},
+(25)
+and hence (22) holds.
+■
+B. Multiplication Property of Smooth Saturated CBFs
+For M ≥ 2 CBFs corresponding to disjoint obstacles, define
+the sets
+DM :=
+M
+�
+i=1
+Di
+CM :=
+M
+�
+i=1
+{x ∈ DM | Bi(x) ≥ 0}
+=
+M
+�
+i=1
+Ci.
+(26)
+Theorem 1. Under the assumed disjointness property, the
+product of smoothly saturated valid CBFs,
+BM(x) :=
+M
+�
+i=1
+σκi ◦ Bi(x),
+(27)
+is a valid CBF for DM, CM, and the dynamic system (2).
+Proof. For x ∈ CM, the zero control u ≡ 0 satisfies (8)
+because the drift term f(x) is zero. We show that for x ̸∈ CM,
+(8) can be satisfied.
+By the disjoint property of the assumed CBF functions,
+when BM(x) < 0, we have ∃i, such that σκi ◦ Bi(x) =
+Bi(x) < 0, and σκj ◦ Bj(x) = 1 for j ̸= i. Hence,
+BM(x) = Bi(x). Because Bi(x) is assumed to be a valid
+CBF function, and both DM ⊂ Di and CM ⊂ Ci hold, the
+CBF property holds for BM(x).
+■
+Remark 3. Due to the way we have constructed the multi-
+obstacle CBF, the equilibrium analysis for a single obstacle
+carries over here without changes. This is because, when
+the robot is at a boundary of an obstacle, the values of the
+saturated CBFs for the other obstacles will all be one.
+V. SIMULATION RESULTS WITH SINGLE AND MULTIPLE
+OBSTACLES
+In this section, we first use simulation to study the behavior
+and liveness of the proposed CLF-CBF system with a single
+obstacle. Next, we run the system on several synthetic envi-
+ronments with 20 obstacles in Robot Operating System (ROS)
+[23] with C++.
+Remark 4. For the CBF in (6), we take Q = I and in
+Prop. 1, we take κ1 = · · · = κM = min{∆2
+i }M
+i=1, which
+is the minimum of the square of the distance between any of
+the obstacles.
+6
+
+A. Robot Model in Simulation
+In MATLAB and ROS, the bipedal robot is represented
+by the Angular momentum Linear Inverted Pendulum (ALIP)
+model [2]. The ALIP robot takes piece-wise constant inputs
+from the CLF-CBF-QP system. Let g, H, τ be the gravitational
+constant, the robot’s center of mass height, and the time
+interval of a swing phase, respectively. The motion of an ALIP
+model on the x-axis satisfies
+�xk+1
+˙xk+1
+�
+=
+� cosh(ξ)
+1
+ρ sinh(ξ)
+ρ sinh(ξ)
+cosh(ξ)
+� �xk
+˙xk
+�
++
+�1 − cosh(ξ)
+−ρ sinh(ξ)
+�
+px,
+(28)
+where xk and ˙xk are the contact position and velocity of the
+swing foot on the x-axis, px is the center of mass (CoM)
+position on the x-axis of the robot, ξ = ρτ and ρ =
+�
+g/H.
+The motion of the robot on the y-axis can be similarly defined.
+Fig. 5: Illustration of how the trajectories vary as a function of differ-
+ent obstacle positions. The target (marked in black) and the robot pose
+(−15, −15, −15◦) are fixed throughout all of the simulations. The different
+colors denote different simulations with only one obstacle present at a time.
+The red trajectory is generated without any obstacle present from the QP only
+containing CLF constraint.
+B. Behavior Study with Single Obstacle in MATLAB
+The optimal control command of the robot is the solution of
+the CLF-CBF-QP problem defined in (14). The time interval
+of a swing phase is set to τ = 0.3s. The robot updates its pose
+based on the ALIP model and the optimal control command.
+The updated pose is then fed back to the CLF-CBF-QP system
+to compute the optimal control for the next iteration. This
+process continues until the robot reaches the target or collides
+with an obstacle.
+Figure 5 shows how the trajectories vary as a function of
+a single obstacle’s position with a fixed initial robot pose
+of (−15, −15, −15◦), marked as the magenta arrow. The
+red trajectory is the nominal trajectory without any obstacles
+present. Each colored trajectory and matching circle represent
+a distinct simulation result. The robot successfully avoids the
+obstacle in all cases. In Fig. 6, we show how the trajectories
+vary as a function of different robot orientations with a fixed
+obstacle location.
+Remark 5. When the robot is within an obstacle, there is also
+a valid solution that pushes the robot outside of the obstacle.
+Fig. 6: Illustration of how the trajectories vary as a function of different robot
+orientations with a fixed obstacle location. The target (marked in cyan) and
+obstacle at (−4, −4) are fixed through out all the simulations. A different
+color stands for a different robot orientation.
+Consider the CBF constraint (13),
+LfB(x) + LgB(x)u + ηB(x) ≥ 0.
+(29)
+When the robot is withing an obstacle, B(x) < 0 and the QP
+selects u such that LfB(x) + LgB(x)u ≥ −ηB(x), causing
+the robot to leave the obstacle.
+Fig. 7: Liveness analysis for the CLF-CBF system. The initial pose is
+(−15, −15, −15◦), and the target is located at (0, 0). Each dot in the figure
+represents the center of an object with radius (r = 1). The interval between
+each center dots are 0.2 meter in both x and y direction. Note that all the red
+points either originally collide with the robot or the target.
+C. Liveness Analysis in MATLAB
+We analyze the liveness by placing an obstacle with a
+fixed radius (r = 1) at different locations. The robot starts
+at (−15, −15, −15◦) and the target is located at (0, 0). The
+obstacle is placed at every 0.2 meter. If the robot successfully
+reaches the target without collision, the obstacle location is
+marked in green otherwise in red, as shown in Fig. 7. All the
+red points either originally collide with the robot or the target.
+D. Multi-Obstacle Simulation with ROS in C++
+In this simulation, we implement a local map centering at
+robot position with a fixed size and a sub-goal selector to place
+a target within the local map to achieve long-term planning
+as not all the obstacles are perceived by the robot at the
+beginning in practice. Even though the global map is available
+in simulation but it is not available in practice, therefore, only
+7
+
+0
+-2
+-4
+-6
+-10
+-12
+-14
+-16
+-15
+-10
+-5
+00
+-1
+-2
+-3
+-4
+-5
+-6
+-7
+-8
+-9
+-10
+-10
+-8
+9-
+-4
+-2
+0
+x m0
+success
+hit
+-2
+-4
+-6
+-10
+-12
+-14
+-16
+-16
+-14
+-12
+-10
+-8
+9-
+-4
+-2
+0
+2
+x [m](a)
+(b)
+(c)
+(d)
+Fig. 8: Trajectories of 39 obstacles in noise-free (top two) and 20 obstacles in noisy (bottom two) synthetic maps with the size of 50 × 30 meters. The
+highlighted areas are the local map at that specific timestamp. The dark blue circles are the obstacles. Different colors represent different runs in the map.
+the information within the local map at the specific timestamp
+is provided to the robot. The robot model is the same ALIP
+model in Sec. V-A.
+In Fig. 8, we generate two noise-free and two noisy syn-
+thetic maps with the size of 50×30 meters. Each map contains
+20 obstacles marked as blue circles. We run six different initial
+poses and final goals for each map. Different colors represent
+different runs in the map. The highlighted area is the local
+map at that specific timestamp. An intermediate goal is chosen
+at the intersection between the boundary of the local map
+and the line connecting the robot and the final goal at the
+current timestamp. If the intermediate goal collides with an
+obstacle, it is moved back along the line. The intermediate
+goal is updated when it is reached or becomes inside of an
+obstacle due to the update of the local map. The robot with
+ALIP model successfully reaches the goals in all 6 × 4 = 24
+runs.
+VI. EXPERIMENTAL RESULTS ON A BIPEDAL ROBOT
+We perform several experiments of the proposed CLF-
+CBF-QP system on Cassie Blue, a bipedal robot with 20
+degrees of freedom. The entire system integrates elevation
+mapping, intermediate goal selection, and the low-level CLF-
+CBF obstacle avoidance system.
+(a)
+(b)
+Fig. 9: The left shows the sensor suite with different sensors, and the right
+shows the sensor suite mounted on Cassie Blue.
+A. Autonomy System Integration
+The following is summarized from [1] for the completeness
+of the paper. To allow the robot to perceive its surroundings
+under different lighting conditions and environments, we de-
+signed a perception suite that consists of an RGB-D camera
+8
+
+(a)
+(b)
+(c)
+(d)
+Fig. 10: Autonomy experiments with Cassie Blue on the first floor of FRB. The green arrow is Cassie’s pose and the green lines are the resulting trajectories.
+The blue sphere is the selected target position. The map is colored by height and the highlighted area is the local map.
+(Intel RealSense™ D435) and a 32-Beam Velodyne ULTRA
+Puck LiDAR, as shown in Fig. 9. The sensor calibrations are
+performed via [50]–[53]. The invariant extended Kalman filter
+(InEKF) [54] estimates the pose of Cassie at 2k Hz. The raw
+point cloud is motion compensated by the InEKF and then
+used to build an elevation map.
+B. Autonomy Experiment on Cassie Blue
+We conducted several indoor experiments with Cassie Blue
+on the first floor of the Ford Robotics Building (FRB) where
+tables and chairs are considered obstacles. To detect obstacles
+in the environment, an occupancy grid map is updated in real-
+time using the timestamped elevation map. Grids with heights
+greater than 0.2 meters are considered occupied. An occupied
+grid is defined as the boundary of obstacles if there is an
+unoccupied grid in its neighborhood. The Breadth First Search
+(BFS) algorithm [55] is utilized to find the separated obstacles
+in the map. Next, we apply the Gift Wrapping Algorithm [56]
+to the boundary grids of obstacles to find the convex hulls of
+the obstacles. Finally, the minimum bounding ball algorithm
+[57] is applied to the convex hulls to find the minimum
+bounding circles of the obstacles. The circles are used to
+represent obstacles in the CBF function (6). The target position
+is selected by clicking a point in the global map. If the final
+target is not within the current local map, an intermediate goal
+will be selected within the local map. When an intermediate
+goal is reached by Cassie or becomes invalid because of the
+update of the local map, it is updated. In the experiments,
+Cassie successfully avoids all the obstacles and reaches the
+target position, as shown in Fig. 10.
+VII. CONCLUSION
+This paper presented a reactive planning system that al-
+lows a Cassie-series bipedal robot to avoid multiple non-
+overlapping obstacles via a single, continuously differentiable
+control barrier function (CBF). The overall system detects an
+individual obstacle via a height map derived from a LiDAR
+point cloud and computes an elliptical outer approximation,
+which is then turned into a quadratic CBF. A continuously
+differentiable saturation function is presented that preserves
+the CBF property of a quadratic CBF while allowing the
+9
+
+saturated CBFs for individual obstacles to be turned into a
+single CBF. The CLF-CBF-QP formalism developed by Ames
+et al. can then be applied to ensure that safe trajectories are
+generated in the presence of multiple obstacles. Liveness is
+ensured by an analysis of induced equilibrium points that are
+distinct from the goal state. Safe planning in environments
+with multiple obstacles is demonstrated both in simulation and
+experimentally on the Cassie bipedal robot.
+ACKNOWLEDGMENT
+Toyota Research Institute provided funds to support this work.
+Funding for J. Grizzle was in part provided by NSF Award
+No. 1808051. This article solely reflects the opinions and conclusions
+of its authors and not the funding entities.
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+APPENDIX A
+EQUILIBRIUM ANALYSIS OF MULTI-OBSTACLE SYSTEMS
+We give the complete analysis for a single obstacle, following
+the work of [19]. Because the drift term of our model is zero, any
+equilibrium points are where the optimal control is the zero vector.
+To avoid this undesirable situation, we seek to find all equilibrium
+points E = {x|u∗ = [0, 0, 0]T, r > 0, and δ, θ ∈ (−π, π]}, where
+u∗ is the optimal control variable. Recall thatf =
+�0
+0
+0�T in (2),
+which leads to LfV (x) = LfB(x) = 0. We denote the following to
+re-write the CLF and the CBF constraints:
+�dx
+dy
+0�
+= −LgB(x)
+�ax
+ay
+aω
+�
+= LgV (x),
+(30)
+where
+ax = −r cos(δ) + βγ2 sin(2βδ) sin(δ)
+2r
+ay = −r sin(δ) − βγ2 sin(2βδ) cos(δ)
+2r
+aω = βγ sin(2βδ)
+2
+.
+(31)
+The constraints become
+L(x, u, s) = axvx + ayvy + aωω − s + µV (x),
+(32)
+B(x, u) = dxvx + dyvy − ηB(x),
+(33)
+and the cost function (15) of the QP can then be re-written as:
+J(u, s) = 1
+2h1(vx − vref
+x )2 + 1
+2h2(vy − vref
+y )2
++ 1
+2h3(ω − ωref)2 + 1
+2ps2,
+(34)
+where {hi}3
+i=1 are the diagonal elements of H in (15), the weights
+of control variables [vx, vy, ω] with hi > 0.
+The KKT conditions [58] of this quadratic program are:
+∂L
+∂u = Hu∗ − Huref + λ1LgV T − λ2LgBT = 0
+(35)
+∂L
+∂s = ps − λ1 = 0
+(36)
+0 = λ1(LfV + LgV u∗ + µV − s)
+(37)
+0 = λ2(−LfB − LgBu∗ − ηB)
+(38)
+0 ≥ LfV + LgV u∗ + µV − s
+(39)
+0 ≥ −LfB − LgBu∗ − ηB
+(40)
+0 ≤ λ1, λ2,
+(41)
+where λ1, λ2 ∈ R, and L is the Lagrangian function and defined as
+L(u, s, λ1, λ2) = J(u, s) + λ1L(x, u, s) + λ2B(x, u).
+(42)
+Next, we analyze equilibrium points (if any) via four cases
+depending on whether each CLF or CBF constraint is active or
+inactive following [19].
+A. Both CLF and CBF are inactive
+When both constraints are inactive, we have
+λ1 = 0
+λ2 = 0
+0 > LfV + LgV u∗ + µV − s
+0 > −LfB − LgBu∗ − ηB.
+(43)
+With (35) and (36), u∗ and s∗ in this case are
+u∗ = uref
+s∗ = 0.
+(44)
+From (4), as long as the goal is not reached, uref is not a zero vector.
+Hence, there is no equilibrium point in this case.
+B. CLF constraint inactive and CBF constraint active
+We prove that there is no equilibrium point in this case by
+contradiction. When the CLF constraint is inactive and the CBF
+constraint is active, we have
+λ1 = 0
+λ2 ≥ 0
+0 > LfV + LgV u∗ + µV − s
+0 = −LfB − LgBu∗ − ηB.
+(45)
+With (35) and (36), u∗, s∗ and λ2 in this case are
+u∗ = uref + λ2H−1LgBT
+s∗ = 0
+λ2 = −ηB + LfB + LgBuref
+LgBH−1LgBT
+.
+(46)
+11
+
+If there is an equilibrium point, then u∗ is the zero vector. Hence, at
+the equilibrium point, by LfV (x) = 0, u∗ = 0 and s∗ = 0, we have
+LfV + LgV u∗ + µV − s∗ = µV > 0,
+(47)
+which conflicts with (45). Therefore, there is no equilibrium point in
+this case.
+C. CLF constraint active and CBF constraint inactive
+When the CLF constraint is active and the CBF constraint is
+inactive, we have
+λ1 ≥ 0
+λ2 = 0
+0 = LfV + LgV u∗ + µV − s
+0 > −LfB − LgBu∗ − ηB.
+(48)
+With (35) and (36), u∗, s∗ and λ1 in this case are
+u∗ = uref − λ1H−1LgV T
+s∗ = λ1
+p
+λ1 = pµV + pLfV + pLgV uref
+pLgV H−1LgV T + 1
+.
+(49)
+Using the variables defined in (30), u∗ can be rewritten as:
+u∗ =
+�
+�
+v∗
+x
+v∗
+y
+ω∗
+�
+� =
+�
+��
+vref
+x
+− λ1ax
+h1
+vref
+y
+− λ1ay
+h2
+ωref − λ1aω
+h3
+�
+�� .
+(50)
+We know from (31) that
+(ay = 0 & aω = 0) ⇐⇒ δ = 0.
+(51)
+In addition, we know from (4) that
+(vref
+y
+= 0 & ωref = 0) ⇐⇒ δ = 0.
+(52)
+Therefore, we split this case into three cases based on the value of
+δ.
+1) δ = 0 (Case I): Substituting δ = 0 to (30), we have ax =
+−r < 0, ay = 0, aω = 0, and to (4), we have vref
+x
+> 0, vref
+y
+=
+0, ωref = 0. Finally, with (41), the optimal control command (50)
+can be simplified as:
+u∗ =
+�
+�
+v∗
+x
+v∗
+y
+ω∗
+�
+� =
+�
+�
+vref
+x
++ λ1r
+h1 > 0
+0
+0
+�
+� .
+(53)
+The optimal control command is not a zero vector, and hence there
+is no equilibrium point in this case.
+2) δ > 0 (Case II): When δ > 0, by the definitions in (31), we
+have ay < 0, aω > 0, and by (4), we have vref
+y
+> 0, ωref < 0. With
+(41) and (50), we have
+v∗
+y = vref
+y
+− λ1ay
+h2
+> 0
+ω∗ = ωref − λ1aω
+h3
+< 0.
+(54)
+The optimal control command is not a zero vector in this case.
+Therefore, there is no equilibrium points in this case either.
+3) δ < 0 (Case III): Similarly, by (30) and (4), we have ay >
+0, aω < 0 and vref
+y
+< 0, ωref > 0. With (41) and (50), we have
+v∗
+y = vref
+y
+− λ1ay
+h2
+< 0
+ω∗ = ωref − λ1aω
+h3
+> 0
+(55)
+The optimal control command is not a zero vector in this case; there
+is, thus, no equilibrium point in this case.
+In summary, there is no equilibrium point when the CLF constraint
+is active and the CBF constraint is inactive.
+D. Both CLF and CBF constraint are active
+When the CLF constraint is active and the CBF constraint is active,
+we have
+λ1 ≥ 0
+λ2 ≥ 0
+0 = LfV + LgV u∗ + µV − s
+0 = −LfB − LgBu∗ − ηB.
+(56)
+We can rewrite (35) and (36) as:
+u∗ = uref − λ1H−1LgV T + λ2H−1LgBT
+s∗ = λ1
+p
+(57)
+Using the variables defined in (30), u∗ can be rewritten as:
+u∗ =
+�
+�
+v∗
+x
+v∗
+y
+ω∗
+�
+� =
+�
+��
+vref
+x
+− λ1ax
+h1
+− λ2dx
+h1
+vref
+y
+− λ1ay
+h2
+− λ2dy
+h2
+ωref − λ1aω
+h3
+�
+�� .
+(58)
+When the robot is at an equilibrium point, u∗ is the zero vector.
+By (56) and LfB = 0, u∗ = 0, we have B = 0, which implies that
+the robot is at the boundary of an obstacle. In the following proof of
+Sec. A-D, we will assume the robot is at the boundary of obstacles.
+The property of B = 0 leads to an immediate proposition which
+is helpful in finding the equilibrium point in the system when one of
+the components of the optimal control is 0.
+Proposition 3. dy = 0 =⇒ v∗
+x = 0.
+Proof. By the proof in III-C, we have LgB(x) = ∇B(x) · g(x) ̸= 0
+for x ∈ D. Therefore, when dy = 0, we have dx ̸= 0. Then, we can
+further have LgB(x)u∗ = 0 =⇒ v∗
+x = 0.
+■
+In addition, with the properties (51) and (52), we split this case
+into four cases based on whether δ and dy are zero.
+1) dy = δ = 0 (Case I): Substituting to (30), we have ax =
+−r < 0, ay = 0, aω = 0, and to (4), we have vref
+x
+> 0, vref
+y
+=
+0, ωref = 0. Finally, with Proposition 3, in this case the optimal
+control command (58) can be written as:
+u∗ =
+�
+�
+v∗
+x
+v∗
+y
+ω∗
+�
+� =
+�
+�
+vref
+x
+− λ1ax
+h1
+− λ2dx
+h1
+0
+0
+�
+� =
+�
+�
+0
+0
+0
+�
+� .
+(59)
+λ1 and λ2 can be obtained by (59), (56) and (57):
+λ1 = pµV > 0
+λ2 = h1vref
+x
+− pµV ax
+dx
+(60)
+By (41) and (60), we have
+∵ vref
+x
+> 0, ax < 0, h1vref
+x
+− pµV ax
+dx
+≥ 0 −→ dx > 0.
+(61)
+Hence, there is an equilibrium point when B = 0, dy = δ = 0 and
+dx > 0.
+2) dy ̸= 0, δ = 0 (Case II): When δ = 0, by (30) and (4),
+we have ax = −r < 0, ay = 0, aω = 0 and vref
+x
+> 0, vref
+y
+=
+0, ωref = 0. Finally, with (41), the optimal control command (58)
+can be simplified as:
+u∗ =
+�
+�
+v∗
+x
+v∗
+y
+ω∗
+�
+� =
+�
+�
+vref
+x
+− λ1ax
+h1
+− λ2dx
+h1
+− λ2dy
+h2
+̸= 0
+0
+�
+� .
+(62)
+Because v∗
+y ̸= 0, the optimal command is not a zero vector in this
+case. Equilibrium points don’t exist when dy ̸= and δ = 0.
+12
+
+3) δ > 0 (Case III): When δ > 0, by (54), ω∗ < 0. Hence, the
+optimal command is not a zero vector and there are no equilibrium
+points in this case.
+4) δ < 0 (Case IV): When δ < 0, by (55), ω∗ > 0. Hence, the
+optimal command is not a zero vector and there are no equilibrium
+points in this case.
+13
+

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@@ -0,0 +1,704 @@
+Confinement of fractional excitations in a triangular lattice antiferromagnet
+L. Facheris,1, ∗ S. D. Nabi,1 A. Glezer Moshe,2 U. Nagel,2 T. R˜o˜om,2
+K. Yu. Povarov,1, 3 J. R. Stewart,4 Z. Yan,1 and A. Zheludev1, †
+1Laboratory for Solid State Physics, ETH Z¨urich, 8093 Z¨urich, Switzerland
+2National Institute of Chemical Physics and Biophysics, Akadeemia tee 23, 12618 Tallinn, Estonia
+3Present address: Dresden High Magnetic Field Laboratory
+(HLD-EMFL) and W¨urzburg-Dresden Cluster of Excellence ct.qmat,
+Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany
+4ISIS Neutron and Muon Source, Rutherford Appleton Laboratory, Didcot, OX11 0QX, United Kingdom
+(Dated: February 1, 2023)
+High-resolution neutron and THz spectroscopies are used to study the magnetic excitation spec-
+trum of Cs2CoBr4, a distorted-triangular-lattice antiferromagnet with nearly XY-type anisotropy.
+What was previously thought of as a broad excitation continuum [Phys. Rev. Lett. 129, 087201
+(2022)] is shown to be a series of dispersive bound states reminiscent of “Zeeman ladders” in quasi-
+one-dimensional Ising systems. At wave vectors where inter-chain interactions cancel at the Mean
+Field level, they can indeed be interpreted as bound finite-width kinks in individual chains. Else-
+where in the Brillouin zone their true two-dimensional structure and propagation are revealed.
+In conventional magnetic insulators the dynamic re-
+sponse is typically dominated by coherent single-particle
+S = 1 excitations, aka magnons or spin waves. In many
+low-dimensional and highly frustrated quantum spin sys-
+tems elementary excitations carry fractional quantum
+numbers, be they spinons in Heisenberg spin chains [1–4]
+or Majorana fermions in the now-famous Kitaev model
+[5–7].
+The physical excitation spectrum, such as that
+measured by neutron spectroscopy, is then dominated by
+broad multi-particle continua [8–11]. In addition to the
+continuum, fractional excitations may also form bound
+states due to attractive interactions between them.
+A
+spectacular new phenomenon emerges when interactions
+are confining, i.e. do not fall off with distance, much like
+strong forces that bind quarks in hadrons [12]. This pro-
+duces an entire series of bound states inside the resulting
+potential well. An example is the sequence of domain wall
+(kink) bound states in quasi-one-dimensional Ising spin
+chains [13–15].
+The confining potential for this model
+is linear and results from 3-dimensional couplings, which
+generate an effective field acting on individual chains [14].
+The binding energies are, in supreme mathematical ele-
+gance, spaced according to the negative zeros of the Airy
+function [13, 15].
+The best-known experimental examples of such “Zee-
+man ladder” spectra are the quasi-one-dimensional Ising
+ferromagnet CoNb2O6 [16] and antiferromagnet (AF)
+BaCo2V2O8
+[17], as well as the isostructural com-
+pound SrCo2V2O8 [18], where as many as 8 consecutive
+bound states are observed. Shorter sequences have been
+found in another prototypical Ising spin chain material,
+RbCoCl3 [19].
+In the present work we report the ob-
+servation of a somewhat similar phenomenon in an en-
+tirely different type of system, namely in a quasi-two-
+dimensional distorted-triangular-lattice AF where the
+effective magnetic anisotropy is predominantly of XY,
+rather than Ising character.
+That the quintessentially
+one-dimensional physics of bound kinks survives in two
+dimensions is remarkable. We argue that it is “rescued”
+at certain special wave vectors by the intrinsic frustration
+in triangular lattice geometry. Elsewhere in the Brillouin
+zone the bound states are no longer restricted to sin-
+gle chains and are to be viewed as 2-dimensional objects
+propagating on the entire triangular plane.
+The material in question, Cs2CoBr4 (space group
+Pnma, a = 10.19, b = 7.73, c = 13.51 ˚A), is a very
+interesting J − J′ model distorted-triangular-lattice AF
+[20, 21].
+Despite a prominent triangular motif in its
+structure, it demonstrates certain one-dimensional fea-
+tures such as a field-induced incommensurate spin den-
+sity wave with Tomonaga-Luttinger spin liquid type dy-
+namics and a propagation vector controlled by a one-
+dimensional nesting in the spinon Fermi sea.
+Its true
+2-dimensional nature is manifest in the presence of a ro-
+bust m = 1/3 magnetization plateau, typical of a trian-
+gular AF. The model magnetic Hamiltonian is described
+in detail in Refs. [20, 21]. The key structural features
+are chains of Co2+ ions that run along the crystallo-
+graphic b axis of the orthorhombic lattice (see Fig.
+1
+in Ref. [20]). The chains are coupled in the (bc) plane in
+a zigzag fashion to form a distorted triangular network
+(inset of Fig. 1(d)). Easy-plane single-ion anisotropy en-
+sures that the low-energy physics of the spin-3/2 Co2+
+ions can be described in terms of effective S = 1/2
+pseudo-spins. The components of the effective exchange
+coupling constants are subject to restrictions imposed
+by the pseudo-spin projection.
+A simplistic spin-wave
+analysis of previous inelastic neutron data provided a
+rough estimate for the nearest-neighbor in-chain AF ex-
+change tensor components:
+JXX ∼ J, JY Y
+∼ 1.1J,
+JZZ ∼ 0.25J, J = 0.8 meV [21]. Here Y is chosen along
+the b crystallographic direction, and X and Z alternate
+between adjacent chains, where anisotropy planes are al-
+most orthogonal. Note that this is practically a planar
+arXiv:2301.13596v1  [cond-mat.str-el]  31 Jan 2023
+
+2
+(b)
+(c)
+(d)
+(a)
+ground state
+J
+J'
+bound state m3  
+FIG. 1.
+(a)-(b) Neutron scattering intensity (solid sym-
+bols) measured at T = 40 mK versus energy transfer at
+the one-dimensional AF zone-centers q = (0, 0.5, 0.5) and
+q = (0, 1, 0.5), respectively.
+The data are integrated fully
+along h direction and in ±0.025 r.l.u. and ±0.25 r.l.u. along
+k and l, respectively. Solid lines are fits to a series of Gaus-
+sian peaks.
+Dashed Gaussians represent the calculated ex-
+perimental energy resolution. Black dotted lines indicate the
+fitted flat background.
+(c) Measured terahertz absorption
+(solid line) versus absorbed photon energy for light propa-
+gating along the c axis at 0.2 K. Dashed areas highlight the
+individual components that find counterparts in the neutron
+spectra. (d) Measured excitation energy plotted versus the
+value of negative roots of the Airy function. The solid line is
+a linear fit as described in the text. The blue area highlights
+the points used for the fit. Inset: cartoons of the magnetic
+ground state and a representative m = 3 2-kink bound state.
+exchange anisotropy, with only a tiny in-plane Ising com-
+ponent to account for the ∆ ∼ 0.4 meV spectral gap
+found in this system.
+The frustrated inter-chain cou-
+pling J′ is significant, of the order of 0.45J, and is of
+predominantly Ising (Y Y ) character. Inter-plane inter-
+actions J′′ are not frustrated. The material orders mag-
+netically in a colinear stripe-type structure, with an or-
+dering wavevector (0, 1/2, 1/2) (see inset in Fig. 1(d)).
+The N´eel temperature TN = 1.3 K allows us to esti-
+mate J′′. If this were the only coupling between chains
+with no additional frustration due to J′, we could expect
+kBTN ∼ 2∆/ ln(∆/J′′) [22]. The actual value of J′′ must
+be larger than thus obtained, as the in-plane frustration
+interferes with the emerging magnetic structure. A cer-
+tain upper estimate is given by the mean field picture
+where kBTN ∼ 2J′′S(S + 1). This leads us to conclude
+that 3 · 10−4 meV ≲ J′′ ≲ 0.075 meV ≪ J, confirming
+the quasi-2-dimensional character of the material.
+Our previous inelastic neutron scattering experiments
+indicated that the excitation spectrum in zero applied
+field is a gapped continuum of states, with intensity con-
+centrated on its lower bound, and a strong dispersion
+along the chain axis [21].
+The central finding of the
+present work is that this “continuum” is actually a se-
+quence of at least 9 sharp bound states that previously
+could not be observed due to poor experimental energy
+resolution. New neutron data were collected at the LET
+time-of-flight spectrometer at ISIS (UK), using 2.35 meV
+incident energy neutrons in repetition-rate-multiplication
+mode [23].
+We used the same 1.16 g single crystal as
+in [21] mounted on a 3He-4He dilution refrigerator. All
+measurements were performed at a base temperature of
+40 mK. In the experiment the sample was rotated 180◦
+around the a axis in steps of 1◦. The spectra were mea-
+sured for ∼ 10 minute counting time at each sample po-
+sition.
+We first focus on the one-dimensional AF zone-centers
+(qb = 0, π), where inter-chain interactions within the tri-
+angular planes cancel out at the Mean Field-RPA level,
+and where spin wave theory predicts no transverse disper-
+sion or intensity modulation of excitations. Fig. 1(a),(b)
+show constant-q cuts through the data at wave vectors
+q = (0, 0.5, 0.5) and q = (0, 1, 0.5), respectively. A se-
+quence of sharp peaks is clearly apparent in both cases.
+A fit to the data using empirical Gaussian profiles yields
+an accurate measure of the peak positions and shows
+that their widths are essentially resolution-limited.
+In
+Fig. 1(a),(b) this is emphasized by the shaded Gaussians
+representing the computed experimental resolution [24].
+Corroborative evidence is also obtained by THz spec-
+troscopy. The experiment was performed with a Martin-
+Puplett-type interferometer and a 3He-4He dilution re-
+frigerator with base temperature of 150 mK using a 3He-
+cooled Si bolometer at 0.3 K. The sample was a circu-
+lar plate approximately 1 mm thick in c direction and
+4 mm in diameter. THz radiation propagating along the
+
+3
+crystal c axis was unpolarized and the apodized instru-
+mental resolution was 0.025 meV. The THz absorption
+spectrum is shown in Fig. 1(c). It is calculated as a dif-
+ference of spectra measured at 0.2 K and 2 K, i.e. in
+the magnetically ordered phase and above TN. The THz
+spectrum appears to have some features absent in the
+neutron spectrum, but all peaks found in the latter are
+also present here. The positions of these peaks were de-
+termined in Gaussian fits (shaded peaks) in a narrow
+range ±0.025 meV near each peak value.
+The spacing between the excitation peaks present in
+both measurements corresponds to confinement in an ap-
+proximately linear one-dimensional potential. To demon-
+strate this, we plot the excitation energies deduced from
+neutron spectra at several wave vectors, as well as the
+positions of corresponding THz peaks, versus the neg-
+ative roots zi of the Airy function in Fig. 1(d). For a
+precise linear attractive potential λ|x| between the dis-
+persive particles, near the minimum ϵ(k) = m0+ℏ2k2/2µ
+we expect the excitation energies to be [15, 16]
+mi = 2m0 + (ℏλ)2/3µ−1/3zi with i = 1, 2, . . . .
+(1)
+In the actual data, the linear dependence is appar-
+ent for all but the first few points.
+As will be ad-
+dressed in more detail below, this slight deviation in-
+dicates that the confining force increases somewhat at
+short distances. From a linear fit to the higher-energy
+peaks we can immediately extract the slope 0.072(3) meV
+and the energy of a single particle m0 = 0.18(1) meV
+(half-intercept).
+Using the single-particle kinetic mass
+ℏ2/µ = 0.39 meV×b2 [24], we estimate the confining force
+constant λ = 0.031(2) meV/b [25].
+The next point that we make is that the observed
+bound states at the one-dimensional AF zone-center are
+essentially one-dimensional objects. This is concluded by
+analyzing the neutron spectra shown in Figs. 2(a),(b).
+The bound states do not propagate in either transverse
+direction and thus have an essentially flat dispersion.
+Moreover, their intensity shows no modulation trans-
+verse to the chains, as shown for the first two modes in
+Figs. 2(e),(f). The measured transverse wave vector de-
+pendencies are entirely accounted for (solid lines) by the
+combined effects of i) the magnetic form factor of Co2+
+and ii) a neutron polarization factor for spin components
+perpendicular to the chain axis (to the direction of or-
+dered moments in the ground state). This implies that
+these excitations do not involve cross-chain correlations
+and are confined to a single chain.
+This consideration prompts a simple interpretation
+of the observed behavior.
+Similarly to the situation
+in CoNb2O6 and BaCo2V2O8, the observed modes are
+bound states of two kinks (domain walls) in individual
+chains.
+Such an excitation is illustrated by the car-
+toon in the inset of Fig. 1(d).
+Since the ordered mo-
+ments are along the b crystallographic axis, they are po-
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+(g)
+(h)
+FIG. 2. (a)-(d) False color plot of neutron scattering inten-
+sity measured at T = 40 mK plotted versus energy transfer
+and momentum transfer transverse to the crystallographic b-
+axis. Gray areas mask regions of elastic-incoherent scattering.
+Background subtraction has been performed as described in
+[24].
+The orange regions represent energy-integration win-
+dows used to extract the cuts in panels below.
+(e)-(h)
+Intensity-momentum cuts (solid symbols) for the first two
+modes in the Zeeman ladder. The blue line shows the product
+of calculated neutron polarization factor for excitations polar-
+ized perpendicular to the direction of the ordered moment and
+the magnetic form-factor-squared for Co2+.
+larized transverse to that direction [21], in agreement
+with the measurement.
+The energy m0 is to be asso-
+ciated with that of a single domain wall. As a consis-
+tency check, we can compare that to the computed en-
+ergy of a domain wall in a classical spin chain. Using
+JY Y /JXX ∼ 1.1 as estimated for Cs2CoBr4, with a triv-
+ial numerical classical-energy minimization procedure we
+get m0 ∼ 0.9JS2 = 0.18 meV, in excellent agreement
+with the measured value.
+Geometric frustration ensures that at the magnetic
+zone-center these strings of flipped spins within a single
+chain incur no energy cost due to interactions with adja-
+cent chains within the triangular lattice. Moreover, any
+transverse dispersion is suppressed. At the same time,
+the interaction energy due to unfrustrated inter-layer
+coupling is proportional to the string length, resulting
+in confinement. In this simplistic picture, the confining
+force is λ = 2J′′S2/b. This yields an inter-layer coupling
+constant J′′ = 0.062(4) meV, inside the possible range
+deduced from TN. The first lowest-energy bound state
+with energy m1 corresponds to a single spin flip in the
+chain, in other words to a single-magnon excitation. The
+i-th higher-energy states are two domain walls separated
+by a length-i string of spins that are aligned opposite to
+the ground state AF spin configuration.
+
+4
+The deviation from linear-potential behavior at low en-
+ergies is also readily explained by this picture. Since the
+material is almost planar, the domain walls are not con-
+fined to a single bond as in the ideal Ising case, but have
+a characteristic size l [26]. We can estimate that quan-
+tity in a classical spin chain using the above-mentioned
+anisotropy parameters: l ∼ 2b. The energy of the first
+few bound states is thus modified due to a physical over-
+lap of the two bounding domain walls. Experimentally,
+the bound state energy is reduced, which corresponds to
+an additional attractive interaction between kinks. Once
+the kinks are separated by a distance of more than ∼ l,
+this interaction becomes negligible and the confinement
+potential becomes linear, originating only from inter-
+layer interactions.
+Away from the one-dimensional AF zone-centers, the
+excitations are considerably more complex. This is very
+clear in the longitudinal dispersion of the bound states
+shown in Fig. 3(a),(b). Other than at qb = 0, π (k =
+0, 1/2) the m1 mode splits into two branches, each with
+an asymmetric dispersion relation. In fact, the m1 state
+at qb = π seems to be continuously connected to the m2
+excitations at qb = 2π (k = 1) and vice versa. Fitting
+the dispersion of the strongest low-energy mode in the
+vicinity of qb = π to a Lorentz-invariant relativistic form
+(ℏωq)2 = ℏ2∆ (qb)2 /µ + ∆2,
+(2)
+yields the value of kinetic mass quoted above.
+A look at the intensities reveals that other than at the
+special wave vectors, the bound states can no longer be
+seen as strings in a single chain, but are “dressed” with
+correlations extending to several neighboring chains in
+the triangular plane.
+This conclusion is reached from
+Fig. 2(c),(d), that show a transverse cut of the spectrum
+at qb = 5π/4 and qb = 3π/2, respectively.
+As plot-
+ted in Fig. 2(g),(h), the measured intensity of the first
+two modes now shows a much steeper transverse wave
+vector dependence than computed from just the polar-
+ization and form factors (solid line). The second mode
+even seems to show signs of intensity oscillations.
+Our data reveal that away from the special wave vec-
+tors the bound states also propagate in two dimensions,
+albeit with a small bandwidth. Indeed, in Fig. 2(d) one
+can see that at qb = 3π/2 the bound states develop a
+non-zero dispersion along the c∗ direction, in contrast to
+what is seen at qb = 0, π. Although the bandwidth of
+transverse dispersion, 0.08 meV, is at the limit of our
+experimental resolution, qualitatively one can say that
+qc = 0, 4π are dispersion minima for the m1 mode, while
+the maximum is at qc = 2π. That periodicity is consis-
+tent with having two chains per unit cell along the c-axis
+direction in the crystal structure.
+Overall, the differences between our results and spectra
+of Ising spin chains [16, 17] are striking. In the latter,
+all bound states, including the first one, are much less
+dispersive than the lower edge of the entire spectrum,
+(a)
+(b)
+FIG. 3. (a)-(b) False color plot of neutron scattering inten-
+sity measured at T = 40 mK plotted versus energy transfer
+and momentum transfer along q = (0, k, 0.5) and q = (0, k, 1)
+respectively. The data were fully integrated along h, and in
+the range ±0.25 r.l.u. along l around the central value. The
+gray areas mask regions where the incoherent scattering domi-
+nates the signal. Background subtraction has been performed
+as described in [24].
+which approximately corresponds to the lower edge of the
+two-kink continuum in the absence of long-range order.
+As a result, each bound state persists only in a restricted
+area in the Brillouin zone. In contrast, in Cs2CoBr4 a
+few of the lower-energy bound states are highly dispersive
+and span across the entire zone.
+In summary,
+we demonstrate that “Zeeman lad-
+ders” of confined fractional excitations can exist in a
+bona fide quasi-two-dimensional system.
+These states
+are inherently related to those in the one-dimensional
+model, as revealed at special wave vectors where two-
+dimensional interactions are canceled by geometric frus-
+tration.
+However, elsewhere in reciprocal space their
+true 2-dimensional character is manifest.
+Once again,
+the distorted triangular lattice model provides a link be-
+tween one- and two-dimensional quantum magnetism.
+This work was supported by a MINT grant of the Swiss
+National Science Foundation. We acknowledge support
+by the Estonian Research Council grants PRG736 and
+MOBJD1103, and by European Regional Development
+Fund Project No. TK134. Experiments at the ISIS Neu-
+tron and Muon Source were supported by beamtime allo-
+cation RB2210048 from the Science and Technology Fa-
+cilities Council [27].
+∗ [email protected]
+
+AA5
+† [email protected]; http://www.neutron.ethz.ch/
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+L. Facheris,
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+Z. Yan, S. Gvasaliya, and A. Zheludev, Magnetization
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+magnet Cs2CoBr4, Phys. Rev. Research 2, 043384 (2020).
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+J. Lass, B. Roessli, E. Ressouche, Z. Yan, S. Gvasaliya,
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+Magnetization Plateau in a Triangular Antiferromagnet,
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+[23] R. Bewley, J. Taylor, and S. Bennington., LET, a cold
+neutron multi-disk chopper spectrometer at ISIS, Nuclear
+Instruments and Methods in Physics Research Section A:
+Accelerators, Spectrometers, Detectors and Associated
+Equipment 637, 128 (2011).
+[24] See Supplemental Material for detailed discussion of the
+resolution calculations, additional inelastic neutron scat-
+tering data, background subtraction procedure, and esti-
+mate of the kinetic mass for a kink.
+[25] This force of ∼ 6 fN corresponds to the gravity pull be-
+tween two average humans at a separation of 8 km.
+[26] We define the domain wall width in a spin-S chain as the
+distance over which the z-axis spin component changes
+from S/2 to −S/2 near its center.
+[27] L. Facheris, et al.;
+(2022):
+Spin-density wave dy-
+namics in a 2D distorted triangular lattice antifer-
+romagnet,
+STFC
+ISIS
+Neutron
+and
+Muon
+Source,
+https://doi.org/10.5286/ISIS.E.RB2210048 .
+
+Supplemental Material for “Confinement of fractional excitations in a triangular
+lattice antiferromagnet”
+L. Facheris,1, ∗ S. D. Nabi,1 A. Glezer Moshe,2 U. Nagel,2 T. R˜o˜om,2
+K. Yu. Povarov,1, 3 J. R. Stewart,4 Z. Yan,1 and A. Zheludev1, †
+1Laboratory for Solid State Physics, ETH Z¨urich, 8093 Z¨urich, Switzerland
+2National Institute of Chemical Physics and Biophysics, Akadeemia tee 23, 12618 Tallinn, Estonia
+3Present address: Dresden High Magnetic Field Laboratory
+(HLD-EMFL) and W¨urzburg-Dresden Cluster of Excellence ct.qmat,
+Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany
+4ISIS Neutron and Muon Source, Rutherford Appleton Laboratory, Didcot, OX11 0QX, United Kingdom
+(Dated: January 31, 2023)
+This Supplemental Material provides further details supporting the main text that may be of
+interest to the specialized reader.
+In particular, the resolution calculations, additional inelastic
+data, the background subtraction for the neutron spectroscopic measurements, and estimate for the
+kink’s kinetic mass are presented.
+CONTENTS
+I. Determination of energy resolution for the LET
+experiment
+1
+II. Additional cuts used for Fig. 1(d)
+1
+III. Background subtraction procedure for LET data
+1
+IV. Estimating a kink’s kinetic mass µ.
+2
+References
+2
+I.
+DETERMINATION OF ENERGY
+RESOLUTION FOR THE LET EXPERIMENT
+The neutron scattering data presented in the main text
+were obtained on the direct-geometry time-of-flight LET
+spectrometer at ISIS (UK) [1]. The instrument was op-
+erated in the high-flux mode, with a chopper resolution
+frequency of 210 Hz and a pulse remover frequency of
+140 Hz. A phase delay time for chopper 2 of 87000 µs
+was introduced to avoid contamination on the main in-
+coming channel Ei = 2.35 meV by slower neutrons. The
+resolution calculations were performed with the PyChop
+interface of Mantid Workbench [2]. The obtained resolu-
+tion profile is shown in SUPP. FIG. 1.
+The widths of the shaded Gaussian profiles in Fig.
+1(a),(b) of the main text were calculated based on the
+fitted peak positions and the data in SUPP. FIG. 1.
+∗ [email protected]
+† [email protected]; http://www.neutron.ethz.ch/
+SUPP. FIG. 1. Calculated energy resolution (solid line) ver-
+sus neutron energy transfer for the spectrometer settings
+listed in the text.
+Dotted lines mark the positions mi at
+q = (0, 0.5, 0.5) as obtained from Fig. 1(a) of the main text.
+II.
+ADDITIONAL CUTS USED FOR FIG. 1(d)
+The additional cuts at q = (0, 0.5, 1) and q = (0, 1, 1)
+(not shown in the main text) are displayed in SUPP. FIG.
+2. The fit is performed in full analogy to Fig. 1(a),(b) as
+described in the main text. The extracted peak positions
+from SUPP. FIG. 2 (a),(b) are plotted in Fig. 1(d) of the
+main text.
+III.
+BACKGROUND SUBTRACTION
+PROCEDURE FOR LET DATA
+The inelastic neutron scattering data presented in Fig.
+2 and Fig. 3 of the main text are background subtracted.
+Although the dataset was rather clean, a background
+subtraction similar to that in [3] was nonetheless per-
+formed. In this section the model adopted to describe
+the background is outlined. The analysis was performed
+using the Horace software package [4].
+SUPP. FIG. 3 shows raw data corresponding to Fig. 3
+of the main text. Strong sharp lines at the edges of the
+
+2
+(a)
+(b)
+SUPP. FIG. 2.
+(a)-(b) Neutron scattering intensity (solid
+symbols) measured at T = 40 mK versus energy transfer at
+q = (0, 0.5, 1) and q = (0, 1, 1), respectively. The data are
+integrated fully along h direction and in ±0.025 r.l.u. and
+±0.25 r.l.u. along k and l, respectively. Solid lines are fits to
+a series of Gaussian peaks. Dashed Gaussians represent the
+calculated experimental energy resolution. Black dotted lines
+indicate the fitted flat background.
+dataset below 0.4 meV are known spurious originating
+from scattering from the sample environment employed.
+The total background was modeled assuming no mag-
+netic scattering below the gap and above the top of the
+spectrum. Thus, the background dataset is identical to
+original data for ℏω ≤ 0.34 meV and ℏω ≥ 1.28 meV
+(see dashed horizontal lines in SUPP. FIG. 3 for the
+background regions projected on these particular cuts).
+In the intermediate energy region, momentum-dependent
+boxes were constructed as shown in SUPP. FIG. 3 and
+numerically interpolated over the total explored (q, ℏω)-
+space. The so-obtained background was then point-to-
+point subtracted from the original data.
+IV.
+ESTIMATING A KINK’S KINETIC MASS µ.
+Near it’s minimum at a one-dimensional wave vector
+k0 = π
+b , the dispersion relation for a single kink can be
+approximated as
+ϵk = m0 + ℏ2
+2µ(k − k0)2.
+(S.1)
+The parameter µ is the kinetic “mass” of this quasiparti-
+cle. We can access it from the experimentally measured
+spectrum of two-kink excitations. For a two-kink state,
+energy-momentum conservation dictates
+ℏω(2−kink)
+q
+= ϵk+ϵq−k = 2m0+ ℏ2
+2µ
+�
+(k − k0)2 + (q − k + k0)2�
+.
+(S.2)
+Minimizing (S.2) with respect to the “hidden” quasi-
+momentum k, we find that the lower boundary of the
+two-particle continuum lies at k = q. Thus, the lowest
+magnon-like dispersion is given by:
+ℏωq = 2m0 + ℏ2
+2µ
+�
+(q − k0)2 + k2
+0
+�
+(S.3)
+Near the minimum wavevector q0 = k0 → π/b, we find
+that the curvature of the parabola-like dispersion is ac-
+tually the same for a single kink and the lowest bound
+state.
+(a)
+(b)
+SUPP. FIG. 3. (a)-(b) False color plot of raw neutron scat-
+tering intensity measured at T = 40 mK plotted versus en-
+ergy transfer and momentum transfer along q = (0, k, 0.5)
+and q = (0, k, 1) respectively. The data were fully integrated
+along h, and in the range ±0.25 r.l.u.
+along l around the
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+Optical Flow for Autonomous Driving: Applications, Challenges
+and Improvements
+Shihao Shen 1, Louis Kerofsky 2 and Senthil Yogamani 3
+1Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.
+2Qualcomm Technologies, Inc., San Diego, California, U.S.
+3Automated Driving, QT Technologies Ireland Limited.
+ABSTRACT
+Optical flow estimation is a well-studied topic for automated
+driving applications. Many outstanding optical flow estimation
+methods have been proposed, but they become erroneous when
+tested in challenging scenarios that are commonly encountered.
+Despite the increasing use of fisheye cameras for near-field sens-
+ing in automated driving, there is very limited literature on optical
+flow estimation with strong lens distortion. Thus we propose and
+evaluate training strategies to improve a learning-based optical
+flow algorithm by leveraging the only existing fisheye dataset with
+optical flow ground truth. While trained with synthetic data, the
+model demonstrates strong capabilities to generalize to real world
+fisheye data. The other challenge neglected by existing state-of-
+the-art algorithms is low light. We propose a novel, generic semi-
+supervised framework that significantly boosts performances of
+existing methods in such conditions. To the best of our knowl-
+edge, this is the first approach that explicitly handles optical flow
+estimation in low light.
+I
+INTRODUCTION
+Advancement in the field of computer vision has enabled the
+rapid development of perception systems for autonomous vehicles
+(AV) in recent years. Optical flow estimation, known as the study
+of how to estimate per-pixel 2D motion between two temporally
+consecutive frames, is one of the fundamental problems in com-
+puter vision that are widely used in autonomous driving. Specif-
+ically, optical flow estimation helps vehicles perceive the tempo-
+ral continuity of the surrounding environment and hence it plays
+significant roles in time-series-based tasks such as object track-
+ing [1, 2], visual odometry [3], semantic segmentation [4], motion
+segmentation [5], and SLAM systems [6], to point out a few. Horn
+and Schunck [7] introduce the first method to compute optical
+flow through energy minimization and many excellent methods
+obtain better results based on it. However, the optimizing problem
+of a complex objective is usually computationally expensive in
+terms of real-time applications such as AV. To achieve faster and
+more reliable performance, end-to-end neural networks are pro-
+posed [8, 9, 10, 11]. These data-driven learning-based methods
+are more efficient and robust against challenges, such as occlu-
+sions, large displacement and motion blur, that break the bright-
+ness constancy and small motion assumptions traditional methods
+are built upon. Nevertheless, there are still a few unique chal-
+lenges in AV applications that have been neglected by existing
+state-of-the-art methods. In this paper, we investigate two com-
+Figure 1: Erroneous optical flow estimation by feeding fisheye images into
+off-the-shelf RAFT [11]. From left to right in each row: current frame,
+next frame, color coded result, sparse vector overlay plots for better visu-
+alization. Note how the estimated flow vectors on the ground are either
+missing or inconsistent with the vehicle motion.
+monly encountered challenges among them and propose the solu-
+tions respectively: lens distortion and low-light scenes.
+Near-field sensing is a ubiquitous topic for automated driv-
+ing.
+Some primary use cases are automated parking systems
+and traffic jam assistance systems. Near-field sensing is usually
+achieved by building a surround-view system with a number of
+wide-angle cameras that come with strong radial distortion. For
+example, fisheye cameras offer a significantly wider field-of-view
+(FoV) than standard pinhole cameras, and in practice four fish-
+eye cameras located at the front, rear, and on each wing mirror
+are sufficient to build a surround-view system for a full-size ve-
+hicle [12]. Although such fisheye systems are widely deployed,
+to the best of our knowledge, there is no previous work explic-
+itly handling optical flow estimation on images with strong lens
+distortion, such as fisheye imagery. As shown in Figure 1, one
+of the current state-of-the-art methods [11] shows erroneous re-
+sults when taking in fisheye images from WoodScape [13] due to
+its focus on narrow field-of-view cameras with mild radial distor-
+tion only. An intuitive way to solve this is to correct the distor-
+tion in the input images as a preprocessing step before passing
+through the neural network. However, this inevitably leads to re-
+duced field-of-view and resampling distortion artifacts at the pe-
+riphery [14]. Without rectification, building an automotive dataset
+is the major bottleneck in optical flow estimation on fisheye im-
+agery. Very few synthetic datasets provide optical flow ground
+truth associated with fisheye images [15], whereas no real-world
+dataset exists with optical flow ground truth. This is due to the
+fact that per-pixel motion between every two consecutive frames
+is extremely difficult to be manually labelled. Simulators [16, 17]
+can readily generate background motion but dynamic foreground
+objects need to be explicitly taken care of. In this paper, we inves-
+arXiv:2301.04422v1  [cs.CV]  11 Jan 2023
+
+tigate and boost the performance of RAFT on strongly distorted
+inputs by making use of the only existing dataset with optical flow
+groundtruth, SynWoodScape [15].
+Most AV applications are expected to operate not only dur-
+ing the day but also at night. Cameras become unreliable and
+camera-based computations are prone to failure under low-light
+conditions due to its susceptibility to noise and inconsistent expo-
+sure. Alternatively, LiDAR sensors can perform robustly in low-
+light autonomous driving [18] because active sensors that measure
+the time-of-flight of the emitted lasers are independent of illumi-
+nation. However, LiDAR is bulky, costly, and requires much more
+computation as well as memory resources to process the output,
+which makes it inferior to cameras if the latter can provide equiv-
+alently reliable results in low light. Thermal cameras [19] provide
+robust low light performance but they are not commonly used in
+recent automated driving systems. Current optical flow methods
+show poor capabilities of dealing with low-light data because low
+light is a complex scenario coming with low signal-to-noise ratio,
+motion blur and local illumination changes brought by multiple
+light sources. In addition, current optical flow datasets [20, 21, 22]
+are dominated by daytime images.
+In this paper, we propose
+a novel, generic architecture that facilitates learning nighttime-
+robust representations in a semi-supervised manner, without the
+help of any extra data or sacrificing the daytime performance. To
+the best of our knowledge, this is the first learning-based method
+that explicitly handles optical flow estimation in low light. The
+main contributions of this paper are:
+1. Introduction and investigation of two challenges in optical
+flow estimation for AV applications: strong lens distortion
+and low-light scenes.
+2. Implementation and improvement of a baseline optical flow
+algorithm on fisheye inputs and experimental evaluation.
+3. Implementation of an effective but also generic framework
+of novel strategies to learn nighttime-robust representations
+for learning-based optical flow algorithms.
+The paper is organized as follows. Section II discusses re-
+lated work on optical flow estimation in the automotive industry
+and existing attempts to solve the two aforementioned challenges.
+Section III describes the implementation of our proposed flow es-
+timation algorithms for fisheye and low-light inputs respectively,
+as well as presents the experimental evaluation and results anal-
+ysis. Finally, Section IV discusses the remaining challenges for
+flow estimation in AV applications and concludes the paper.
+II
+RELATED WORK
+Optical Flow Estimation: Traditional solutions have been stud-
+ied and adapted for decades [7, 23]. In order to be robust against
+more challenging open world problems including lack of features,
+motions in different scales, and occlusions, recent learning-based
+methods outperform traditional ones. Dosovitskiy et al. [8] pro-
+pose FlowNetS and FlowNetC, which is a pioneer work in show-
+ing the feasibility of directly estimating optical flow given im-
+ages. Sun et al. [9] design PWC-Net, a much more efficient solu-
+tion based on pyramidal processing, warping and the use of a cost
+volume. RAFT [11], proposed by Teed and Deng, demonstrates
+notable improvement by building multi-scale 4D correlation vol-
+umes for all pairs of pixels and iteratively updating flow estimates
+through refinement module based on gated recurrent units (GRU).
+All these methods are fully supervised and trained using imagery
+from a standard pinhole camera. The training data are also col-
+lected during the day with sufficient brightness. None of them
+pays attention to the performance of optical flow in more chal-
+lenging AV applications such as strong lens distortion and driving
+at night, which leads to errors and even catastrophic failures.
+Strong Lens Distortion: There is very limited work on percep-
+tion tasks for strongly distorted images such as fisheye images.
+Popular approaches include rectifying the radial distortion before
+passing images into any regular perception pipeline. However,
+this will inevitably bring reduced field-of-view and resampling
+distortion artifacts especially at the image borders [14].
+Spa-
+tially variant distortion that makes closer objects appear larger
+also poses scaling problems and complexity to geometric percep-
+tion tasks. Additionally, Rashed et al. [24] show that the com-
+mon use of bounding boxes for object detection no longer fit well
+for rectangular objects in distorted images. More sophisticated
+representations for detected objects, such as a curved bounding
+box exploiting the known radial distortion, are explored in [25].
+Although there is some literature using distorted images with-
+out rectification on other perception tasks, such as depth estima-
+tion [26, 27], soiling [28], visual odometry [29] and multi-task
+models [30, 31], there is no previous work estimating optical flow
+due to the difficulty in labeling ground truth. WoodScape [13],
+KITTI 360 [32] and Oxford RobotCar [33] are some well-known
+autonomous driving datasets containing strongly distorted im-
+ages such as fisheye images, but none of them has optical flow
+ground truth. In this paper, we take advantage of the synthetic
+fisheye dataset published recently, SynWoodScape [15], which is
+the first dataset providing optical flow for both foreground and
+background motions by computing it analytically using other data
+modalities extracted from the simulator. We train our network
+using synthetic data from SynWoodScape and evaluate it on real-
+world fisheye data from WoodScape.
+Low-Light Scenes: Similar to optical flow estimation on strongly
+distorted images, there is some work handling low light in a few
+perception tasks [18, 34, 35] but none of them has proposed an
+optical flow estimation algorithm that is robust against low-light
+scenes. Very related to ours, Zheng et al. [36] propose a method
+to synthesize low-light optical flow data by simulating the noise
+model on dark raw images, which is then used to finetune an off-
+the-shelf network. However, their method is not able to synthe-
+size more realistic characteristics of real-world low-light scenes
+one would observe in AV applications, such as the motion blur
+and local illumination changes brought by multiple light sources.
+Their improvement is also very limited due to the off-the-shelf
+network is not designed nor trained to learn nighttime-robust rep-
+resentations. In addition, a variety of techniques have been de-
+veloped for low-light image enhancement [37, 38] and image-to-
+image translation [39, 40]. The former can preprocess inputs to
+a flow estimation network during inference by brightening up a
+given low-light image, whereas the latter can translate a daytime
+image into its nighttime counterpart so as to complement the lack
+of optical flow datasets in low light [18]. But neither approach fa-
+cilitates the network training in that the processed data bring in ex-
+tra complexities such as additional artificial noise, overexposure,
+or inconsistent image translation across frames. Finally, semi-
+supervised learning is a common approach to tackling the lack of
+
+Figure 2: Optical flow estimation (color coded) on real-world automotive data from WoodScape [13]. Input frames are from the fisheye cameras of front
+view, right-side view, and left-side view respectively.
+optical flow data in particular scenarios, where a set of predefined
+transformations are applied to the original labeled data and the
+output of the perturbed data are enforced to agree with the out-
+puts of the original data [41]. For example, Jeong et al. [42] use
+a semi-supervised setup to impose translation and rotation con-
+sistency equivariance for optical flow estimation. Yan et al. [43]
+synthesize foggy images from clean and labelled images in or-
+der to avoid flow estimation errors caused in dense foggy scenes.
+Similar to these semi-supervised methods, we incorporate low-
+light consistency that facilitates learning explicit nighttime-robust
+representations without additional labeling.
+III
+PROPOSED ALGORITHMS AND RESULTS
+In this section, we describe the two proposed optical flow es-
+timation algorithms for strongly distorted inputs and low-light in-
+puts respectively. We also present the corresponding experimental
+evaluation and results analysis.
+III. A
+Strong Lens Distortion
+The limited availability of datasets with strong lens distortion
+is the bottleneck that prevents recent methods from generalizing
+to more distorted inputs. With the help of SynWoodScape [15],
+the first fisheye dataset providing optical flow ground truth for
+both foreground and background motions, we are able to train
+an optical flow model, using RAFT [11] as the backbone, that
+generalizes well on strongly distorted lenses without sacrificing
+its original performance on pinhole cameras.
+We run the off-the-shelf RAFT on real-world fisheye auto-
+motive dataset, e.g. WoodScape [13] and we find sharp and incon-
+sistent optical flow estimation, which is especially illustrated on
+the ground plane in Figure 1. To solve this, we provide two base-
+lines and their qualitative as well as quantitative evaluation. One
+is to finetune the pretrained RAFT using SynWoodScape, follow-
+ing the training schedule in Table 1a. The other is to jointly train
+RAFT on both SynWoodScape and images from pinhole cam-
+era that are regularly used in learning-based optical flow meth-
+ods [8, 20, 21, 22, 44]. The jointly training baseline follows the
+training schedule in Table 1b.
+Table 1: Details of the training schedule. Column header abbreviations:
+LR: learning rate, BS: batch size, WD: weight decay, CS: crop size. Train-
+ing dataset abbreviations: C: FlyingChairs, W: SynWoodScape, S: Sintel,
+T: FlyingThings3D, K: KITTI-2015, H: HD1K.
+(a) Finetuning baseline. During the Sintel stage, the dataset distribution is
+S(.67), T(.12), K(.13), H(.08).
+Stage
+Weights
+Dataset
+LR
+BS
+WD
+CS
+Chairs
+-
+C
+4e-4
+6
+1e-4
+[368, 496]
+Things
+Chairs
+T
+1.2e-4
+3
+1e-4
+[400, 720]
+Sintel
+Things
+S+T+K+H
+1.2e-4
+3
+1e-5
+[368, 768]
+Finetune
+Sintel
+W
+1e-4
+3
+1e-5
+[600, 800]
+(b) Jointly training baseline. During the Joint stage, the dataset distribu-
+tion is W(.65), S(.17), T(.13), K(.03), H(.02).
+Stage
+Weights
+Dataset
+LR
+BS
+WD
+CS
+Chairs
+-
+C
+4e-4
+6
+1e-4
+[368, 496]
+Things
+Chairs
+T
+1.2e-4
+3
+1e-4
+[400, 720]
+Joint
+Things
+W+S+T+K+H
+1e-4
+3
+1e-5
+[368, 768]
+
+Finetuned on SynWoodScape
+Current Frame
+Pretrained on Sintel
+Jointly TrainedFigure 3: Overview of our proposed framework. During training, the framework takes two consecutive frames as input and passes them through a set
+of low-light-specific data augmentations as well as applies a random illumination mask. Then the optical flow estimator estimates flow on two pairs of
+augmented frames in parallel. The network is supervised by two losses: the conventional optical flow loss and the novel brightness consistency loss.
+During inference, the input frames are directly passed into the estimator which outputs optical flow, as is the standard way in the existing state of the art.
+We then show the quantitative results in Table 2. We use
+the endpoint error (EPE) as the metric, which is the standard er-
+ror measure for optical flow estimation. It is the Euclidean dis-
+tance between the estimated flow vector and the ground truth,
+averaged over all pixels. We evaluate the two baselines (Stages
+”Finetune” and ”Joint”) described above along with the pretrained
+model (Stage ”Sintel”) provided by the author on four hold-out
+test sets from SynWoodScape, Sintel (clean and final passes), and
+KITTI. SynWoodScape is the only test set of strongly distorted
+inputs, while the other three assume a pinhole camera model with
+very little distortion. Although the pretrained model gives out-
+standing performance on pinhole cameras, its performance sig-
+nificantly drops on fisheye inputs. Our first baseline, the one fine-
+tuned on fisheye images, gives the best result on SynWoodScape
+but has very poor performance on the others. This matches our
+expectation because both the pretrained and the finetuned mod-
+els are trained to the best for pinhole camera and fisheye camera
+respectively, without taking generalization into account. On the
+other hand, our second baseline, the jointly trained model, keeps
+the second best while being very close to the best score on all four
+datasets. Therefore, jointly training provides a straightforward yet
+strong baseline that generalizes well over lenses with distinct dis-
+tortions.
+Table 2: Endpoint-error results on datasets with diverse lens distortion.
+Stage
+SynWoodScape
+Sintel - Clean
+Sintel - Final
+KITTI
+Sintel
+5.12
+1.94
+3.18
+5.10
+Finetune
+1.40
+5.44
+10.32
+14.34
+Joint
+1.48
+2.44
+4.14
+7.31
+In Figure 2, we further show their qualitative results on
+WoodScape that support the improvements we obtain by jointly
+training RAFT on a mixture of lens distortions. In the front view
+case, note how the jointly trained model is able to consistently
+estimate the flow on the ground as is the major failure of recent
+methods shown in Figure 1. The results on side-view cameras also
+show the jointly trained model captures finer details than its fine-
+tuned counterpart. For example in the right-side view, not only the
+inconsistency on the ground is solved, but optical flow associated
+with the bicycle wheel in the upper right corner is also clearly
+estimated. In the left-side view, the finetuned model misses the
+flow associated with the vehicle’s front wheel, which is captured
+by the pretrained model, but the jointly trained model ”regains”
+such detailed estimations. In other words, the finetuned model es-
+timates more consistent optical flow, which poses a challenge to
+the pretrained model due to markedly different projection geome-
+tries between fisheye and pinhole cameras, but in return, it loses
+some details observed by the pretrained model because interesting
+local features become much less significant given the strong lens
+distortion. However, the jointly trained model achieves a great
+trade-off among the previous two: it re-captures the details lo-
+cally while maintaining good performance globally across differ-
+ent camera views.
+III. B
+Low-Light Scenes
+We propose a novel and generic semi-supervised framework
+that significantly boosts performances of existing state-of-the-art
+methods in low light conditions. Figure 3 shows the architec-
+ture of the framework. The benefit of our framework is three-
+fold. First, it is independent from the design of the existing meth-
+ods, so one can apply it generically to an estimator of his choice
+(e.g. [8, 9, 11, 45]) and augment its nighttime performance out of
+the box. Second, semi-supervised learning does not require any
+extra data as the labeling cost for nighttime optical flow datasets is
+immense. Lastly, it maintains the estimator’s competitive perfor-
+mance on the original daytime data without making any trade-off.
+We first break down the root causes of failures in optical flow
+estimation under low light and then describe our proposed strate-
+gies in the framework that address these root causes accordingly:
+1. the complex noise model of images captured at night,
+2. severe motion blur caused by longer exposure time,
+3. inconsistent local brightness brought by multiple indepen-
+dent light sources in the scene.
+Images captured in low light tend to have more complex
+noises than those captured with sufficient ambient light. Such
+noises are never synthesized in the data augmentation step by ex-
+isting methods, which is the first reason why the optical flow esti-
+mators fail in low light. Similar to [36], we decompose the noise
+model in low light as an aggregate of the photon shot noise and
+
+Ground Truth Flow
+Photometric
+Motion Blur
+Input Pair w/o Brightness Mask
+Original Input
+Low-light Noise
+It+1
+Occlusion
+Estimator
+Estimated Flow
+Supervised Loss
+Spatial
+Ls
+V
+Shared Weights
+Data
+Augmentation
+Input Pair w/ Brightness Mask
+I!
+Local Brightness
+Estimator
+Estimated Flow
+RE
+Consistency Loss
+Lb
+Random MaskFigure 4: Effects of low-light noise augmentation and motion blur aug-
+mentation.
+thermal noise. The former is due to the changing amount of pho-
+tons hitting the sensor with different exposure levels and pixel lo-
+cations. The photon shot noise is approximated by a Poisson dis-
+tribution. Thermal noise refers to the noise in readout circuitry in
+the sensor and is approximated by a Gaussian distribution. There-
+fore, we synthesize the low-light noise onto the input frames as
+one extra data transform in the data augmentation step. Specif-
+ically, we sample Poisson and Gaussian parameters, (a,b), from
+ranges observed in real-world low-light images, formulate it into
+a single heteroscedastic Gaussian (Equation 1), and apply it to an
+input frame I. With probability 0.5, the low-light noise augmen-
+tation is performed on each pair of consecutive frames.
+I(x) = N
+�
+µ = x,σ2 = ax+b
+�
+(1)
+Motion blur is another root cause we need to address when
+estimating optical flow in low light. In order to mimic the blurring
+effects caused by longer exposure length, we generate authentic
+motion blur kernels using Point Spread Functions (PSF) at dif-
+ferent kernel sizes and intensities. The intensity determines how
+non-linear and shaken the motion blur looks. Similar to low-light
+noise, we apply the authentic blurring to a pair of input frames
+as one extra data augmentation, with probability 0.6. An illustra-
+tion of the two introduced data augmentation strategies is shown
+in Figure 4.
+Inconsistent local brightness is the last but not least root
+cause. This is due to multiple independent lighting sources exist-
+ing in a low-light scene (street light, headlight, moonlight, etc.),
+which leads to uneven bright areas in an image. For example,
+the ground plane in the original input in Figure 3 is illuminated
+only in front of vehicles’ headlights but remains dark elsewhere.
+Unlike in the daytime where sun is the dominant lighting source,
+images captured at night have inconsistent local brightness even
+on the same object. Because optical flow is estimated by match-
+ing pixels across two images, such inconsistencies cause exist-
+ing methods to fail easily. For instance, in Figure 5, the first
+row shows the catastrophic failure of RAFT when a pedestrian
+walks from the dark into the vehicle headlight and his illumina-
+tion changes drastically across frames. In order to resolve this, we
+resort to semi-supervised learning. Similar to [42], we also adopt
+the cow-mask [46] to create sufficiently random yet locally con-
+nected illumination patterns as the inconsistent local brightness
+occurs in any size, shape and position in images while exhibiting
+Figure 5: Optical flow estimation on low-speed sequences from CU-
+Lane [47].
+locally explainable structures, depending on the driving environ-
+ment and the time. We apply the same binary mask to the original
+pair of input frames and randomly adjust the brightness of pix-
+els according to the mask. The true area of the binary mask is
+uniformly sampled from 40% to 70% of the image. With a proba-
+bility of 0.5, we increase the absolute brightness of the true area,
+whereas in the remaining time we increase the brightness of the
+false area. Finally, we introduce the local brightness consistency
+regularization. We use (I′t,I′
+t+1) and (It,It+1) to denote the input
+pair after data augmentation with and without applying a random
+brightness mask. Both passes in Figure 3 are independent ex-
+cept that the spatial transform is shared in order to keep the same
+cropped areas for consistency loss calculation. The local bright-
+ness consistency loss is calculated as follows
+Lb =
+��Estimator(It,It+1)−Estimator
+�
+I′
+t,I′
+t+1
+���2
+2 .
+(2)
+This regularization explicitly constrains the network to output
+consistent optical flow on (I′t,I′
+t+1) as on (It,It+1), which enforces
+illumination invariance between the estimated optical flow for the
+original pair the estimated optical flow for the randomly trans-
+formed pair. Note how this semi-supervised approach is different
+from simply adjusting brightness randomly as another data aug-
+mentation scheme, which expand training samples without impo-
+sition of a sophisticated consistency loss during training.
+We choose RAFT [11] as the estimator and we supervise our
+network on the aggregated loss L = Ls +Lb. Ls is the l1 distance
+between the predicted flow ˜f i(It,It+1) and ground truth flow ft
+over all iterations i, as in [11]:
+Ls =
+N
+∑
+i=1
+γN−i ��� ˜f i(It,It+1)− ft
+���
+1 .
+(3)
+Due to the lack of nighttime data with optical flow ground
+truth, we are restricted to qualitatively evaluating our approach,
+which we call RAFT-Dark for short. We use CULane [47], a large
+automotive dataset containing a lot of challenging real-world low
+light sequences. In Figure 5, we show the comparison between
+vanilla RAFT and RAFT-Dark on some low-speed sequences.
+RAFT-Dark demonstrates superior performance to RAFT. In the
+
+Original ImagePair
+After Low-Light Noise Augmentation
+Original ImagePair
+AfterMotionBlurAugmentationInput
+RAFT
+RAFT-Dark (Ours)Figure 6: Optical flow estimation on high-speed sequences from CU-
+Lane [47].
+first, third and fourth rows, RAFT-Dark is able to detect motions
+associated with pedestrians and vehicles that either experience
+some drastic illumination change or appear to be too dark and
+noisy. In the other cases, note how RAFT-Dark gives a signif-
+icantly better estimation on the ground plane as well as the di-
+rections and magnitudes that are consistent with the ego vehi-
+cle’s motion. For convenience, a color coding wheel to visualize
+per-pixel optical flow vectors is attached to the top right corner:
+color denotes direction of the flow vector while intensity denotes
+length of the displacement. Since the ego vehicle always heads
+forward, the ground truth optical flow vectors in the front cam-
+era’s image should intuitively point to the image boundaries and
+away from the image center. And due to the motion parallax, one
+should expect larger magnitudes of flow vectors toward the im-
+age boundaries and small magnitudes around the image center.
+In other words, although we have no access to numerical ground
+truth flow, we know the color coded ground truth should exhibit
+the same pattern as the color wheel: bluish or greenish on the
+left side while reddish or yellowish on the right side of the image.
+With this in mind, RAFT fails to estimate correct optical flow con-
+sistent with the vehicle’s motion, especially in background areas
+such as the ground plane. On the other hand, RAFT-Dark not only
+performs well on these areas but also learns to separate the dark
+sky in some cases and to capture details such as the street light
+in the second row. Such improvements are further illustrated in
+high-speed sequences from CULane in Figure 6.
+Our framework of learning strategies enables RAFT to im-
+prove estimation accuracy by more than 50% on average (based
+on visual observations), and even solves some catastrophic fail-
+ures. Although we show our results based on RAFT as the esti-
+mator, our framework is generic and one can replace RAFT with
+any existing state-of-the-art method of one’s choice.
+III. C
+Discussion
+The goal of this work is to emphasize the importance of
+addressing optical flow challenges which are not well explored
+in automated driving. We investigate two of them and propose
+our solutions accordingly, but the others require further research.
+Lack of data tends to be the major bottleneck for most data-driven
+optical flow algorithms. We are able to leverage synthetic data to
+improve existing methods’ adaptation of various lens distortions
+but the sim-to-real gap still exists when these methods are evalu-
+ated on real world fisheye data. Optical flow in low light cannot
+be addressed in the same way without any synthetic data avail-
+able. We experiment image enhancement prior to the network
+inference, but it leads to even worse results because enhancement
+happens per frame rather than per pair of frames and temporal
+consistency is easily broken. Without any extra data, our approach
+takes full advantage of publicly available data and simulates three
+root causes through novel data augmentation schemes and semi-
+supervised learning. However, low light is merely one of many
+scenarios that make optical flow estimation harder. Others in-
+clude foggy, rainy or snowy weather [48]. A unified and robust
+approach aiming for all these cases is encouraged and we see it
+also as an opportunity for further investigation by the community.
+IV
+Conclusion
+Both lens distortion and low light are important problems for
+higher levels of automated driving, but they are not explored in
+detail in the optical flow community as there is no public dataset
+available. Thus we propose our approaches to these two respec-
+tively. We implement and improve a state-of-the-art optical flow
+algorithm by training it on synthetic fisheye data and demonstrat-
+ing its adaptation to real-world distorted images as well as gen-
+eralizability over various lens distortions. We implement a novel,
+generic framework that facilitates learning nighttime-robust rep-
+resentations in a semi-supervised manner, which shows superior
+performance to the existing state of the art. In future work, we
+plan to integrate our current solutions into higher-level pipelines
+as well as explore other unique challenges of optical flow estima-
+tion in the context of automated driving.
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+AUTHORS BIOGRAPHY
+Shihao Shen is a second-year graduate student in the Robotics
+Institute at Carnegie Mellon University and expects to receive his
+M.Sc. in Robotic Systems Development in 2023. He worked as
+an Interim Engineering Intern in the Multimedia Research and
+Development department at Qualcomm in summer 2022 and this
+is his work done during his internship. His main research focus
+is machine learning with applications in computer vision as well
+as simultaneous localization and mapping (SLAM).
+Louis Kerofsky is researcher in video compression, video
+processing and display. He received M.S. and Ph.D. degrees in
+Mathematics from the University of Illinois, Urbana-Champaign
+(UIUC). He has over 20 years of experience in research and al-
+gorithm development and standardization of video compression.
+He has served as an expert in the ITU and ISO video compression
+standards committees. He is an author of over 40 publications
+which have over 5000 citations. He is an inventor on over 130
+issued US patents. He is a senior member of IEEE, member of
+Society for Information Display.
+Senthil Yogamani is an artificial intelligence architect for au-
+tonomous driving and holds a principal engineer position at
+Qualcomm. He leads the research and design of AI algorithms
+for various modules of autonomous driving systems. He has over
+17 years of experience in computer vision and machine learn-
+ing including 14 years of experience in industrial automotive sys-
+tems. He is an author of 110+ publications which have 4000+
+citations and 150+ inventions with 85 filed patent families. He
+serves on the editorial board of various leading IEEE automotive
+conferences including ITSC and IV and advisory board of various
+industry consortia including Khronos, Cognitive Vehicles and IS
+Auto. He is a recipient of the best associate editor award at ITSC
+2015 and best paper award at ITST 2012.
+

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+EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
+CERN-EP-2023-005
+23 January 2023
+© 2023 CERN for the benefit of the ALICE Collaboration.
+Reproduction of this article or parts of it is allowed as specified in the CC-BY-4.0 license.
+Exploring the non-universality of charm hadronisation through the
+measurement of the fraction of jet longitudinal momentum carried by Λ+
+c
+baryons in pp collisions
+ALICE Collaboration
+Abstract
+Recent measurements of charm-baryon production in hadronic collisions have questioned the univer-
+sality of charm-quark fragmentation across different collision systems. In this work the fragmentation
+of charm quarks into charm baryons is probed, by presenting the first measurement of the longitudinal
+jet momentum fraction carried by Λ+
+c baryons, 𝑧ch
+|| , in hadronic collisions. The results are obtained
+in proton–proton (pp) collisions at √𝑠 = 13 TeV at the LHC, with Λ+
+c baryons and track-based jets
+reconstructed in the transverse momentum intervals of 3 ≤ 𝑝Λ+
+c
+T < 15 GeV/𝑐 and 7 ≤ 𝑝jet ch
+T
+< 15
+GeV/𝑐, respectively. The 𝑧ch
+|| distribution is compared to a measurement of D0-tagged charged jets in
+pp collisions as well as to PYTHIA 8 simulations. The data hints that the fragmentation of charm
+quarks into charm baryons is softer with respect to charm mesons, as predicted by hadronisation
+models which include colour correlations beyond leading-colour in the string formation.
+arXiv:2301.13798v1  [nucl-ex]  31 Jan 2023
+
+ALICEIn-jet Λ+
+c production in pp collisions at √𝑠 = 13 TeV
+ALICE Collaboration
+Heavy-flavour hadrons are produced in high-energy particle collisions through the fragmentation of heavy
+(charm and beauty) quarks, which typically originate in hard scattering processes in the early stages of
+the collisions. The most common theoretical approach to describe heavy-flavour production in hadronic
+collisions is based on the quantum chromodynamics (QCD) factorisation approach [1], and consists of a
+convolution of three independent terms: the parton distribution functions of the incoming hadrons, the
+cross sections of the partonic scattering producing the heavy quarks, and the fragmentation functions that
+parametrise the evolution of a heavy quark into given species of heavy-flavour hadrons. As the transition
+of quarks to hadrons cannot be described in perturbation theory, the fragmentation functions cannot be
+calculated and must be extracted from data.
+Fragmentation functions of charm quarks to charm baryons and mesons have been constrained in e+e−
+and e−p collisions [2–5], using a variety of different observables, such as the hadron momentum as a
+fraction of its maximum possible momentum, as dictated by the centre-of-mass energy of the collision.
+Another method to probe the fragmentation of quarks to hadrons is to parametrise the hadron momentum
+in relation to the momentum of jets, which are collimated bunches of hadrons giving experimental access
+to the properties of the scattered quark. Recently, the production of charm mesons in jets, probed via the
+fractional longitudinal momentum of the jet carried by the D meson, was measured in pp collisions at the
+Large Hadron Collider (LHC) [6–8] and appears consistent with Monte Carlo (MC) simulations tuned
+on e+e− data. These measurements support the assumption of fragmentation universality across collision
+systems in the charm-meson sector. This assumption underpins theoretical calculations describing the
+production of heavy-flavour hadrons in hadronic collisions, which make use of fragmentation functions
+tuned on e+e− and e−p data.
+Measurements of the production cross sections of baryons in pp collisions have questioned the hypothesis
+of fragmentation universality across collision systems [9]. In the charm sector, which provides a clean
+probe of hadronisation phenomena due to the large mass of the charm quark, recent measurements
+performed by the ALICE Collaboration [10–18] in pp collisions have shown that the ratio of the Λ+
+c (and
+other charm baryons) and D0 production cross sections measured at low 𝑝T (≲ 12 GeV/𝑐) is significantly
+larger than the value expected from MC simulations in which the charm fragmentation is tuned on e+e−
+and e−p measurements, such as PYTHIA 8 [19] with the Monash tune [20] or HERWIG 7 [21]. A recent
+measurement of the Λ+
+c/D0 ratio in pp collisions, performed by the ALICE Collaboration in intervals
+of charged-particle multiplicity, also points to a substantial increase of the Λ+
+c/D0 ratio with increasing
+multiplicity, with respect to e+e− collisions, starting at very low multiplicities [14].
+The study of charm-baryon production in jets can provide more differential insights into hadronisation
+mechanisms in pp collisions, compared to 𝑝T-differential cross sections and yield ratios of heavy-flavour
+hadrons, allowing for a more accurate study of the dynamical properties of baryon production. In this
+letter, the first measurement of the longitudinal momentum fraction of the jet carried by Λ+
+c baryons, 𝑧ch
+|| , is
+presented. The measurement is performed in pp collisions at √𝑠 = 13 TeV in the interval 0.4 ≤ 𝑧ch
+|| ≤ 1.0.
+The 𝑧ch
+|| distribution, fully corrected to particle level, is presented for prompt (charm-quark initiated)
+Λ+
+c-tagged jets with 7 ≤ 𝑝jet ch
+T
+< 15 GeV/𝑐 and 3 ≤ 𝑝Λ+
+c
+T < 15 GeV/𝑐. The results are then compared
+to PYTHIA 8 simulations [19, 22], including a version where mechanisms beyond the leading-colour
+approximation are considered in string formation processes during hadronisation [20], and to an analogous
+measurement of the 𝑧ch
+|| distribution of D0 mesons, performed by the ALICE Collaboration [6].
+A full description of the ALICE setup and apparatus can be found in Refs. 23, 24. The main detectors
+used in this analysis are the Inner Tracking System (ITS), which is used for vertex reconstruction and
+tracking; the Time Projection Chamber (TPC), which is used for tracking and particle identification (PID);
+and the Time-Of-Flight (TOF) detector, which is used for PID. These detectors cover a pseudorapidity
+interval of |𝜂| < 0.9. The analysis was performed on pp collisions at √𝑠 = 13 TeV, collected using a
+minimum-bias (MB) trigger during the years 2016, 2017, and 2018. The trigger condition required
+2
+
+In-jet Λ+
+c production in pp collisions at √𝑠 = 13 TeV
+ALICE Collaboration
+coincident signals in the two scintillator arrays of the V0 detector, with background events originating
+from beam–gas interactions removed offline using timing information from the V0. To mitigate against
+pile-up effects, events with multiple reconstructed primary vertices were rejected. To ensure uniform
+acceptance, only events with a primary-vertex position along the beam axis direction of |𝑧vtx| < 10 cm
+around the nominal interaction point were accepted. After the selections described above, the data sample
+consisted of 1.7×109 events, corresponding to an integrated luminosity of Lint = 29 nb−1 [25].
+The Λ+
+c candidates and their charge conjugates were reconstructed via the hadronic Λ+
+c → pK0
+S → pπ+π−
+decay channel with a total branching ratio of (1.10 ± 0.06)% [26], in the Λ+
+c transverse-momentum
+interval of 3 ≤ 𝑝Λ+
+c
+T < 15 GeV/𝑐. Only tracks with |𝜂| < 0.8 and 𝑝T > 0.4 GeV/𝑐, which fulfilled the track
+quality selections described in Ref. 13, were considered for the Λ+
+c reconstruction. The Λ+
+c candidates
+themselves were reconstructed in the |𝑦Λ+c | < 0.8 rapidity interval.
+The Λ+
+c-candidate selection was
+performed using a multivariate technique based on the Boosted Decision Tree (BDT) algorithm provided
+by the XGBoost package [27]. The features considered in the optimisation include the PID signal for
+the proton track, the invariant mass of the K0
+S-meson candidate, and topological variables that exploit the
+kinematic properties of the displaced K0
+S-meson decay vertex. The training was performed in intervals
+of Λ+
+c-candidate 𝑝T, considering prompt signal candidates from PYTHIA 8 events with the Monash
+tune [19, 20], transported through a realistic description of the detector geometry and material budget
+using GEANT 3 [28]. Background candidates were extracted from the sidebands of the invariant-mass
+distributions in data. The probability thresholds of the BDT selections were tuned, using MC simulations,
+to maximise the statistical significance for the signal. Further details on the Λ+
+c-candidate reconstruction
+and machine learning procedure are provided in Ref. 14, where the same reconstruction and BDT model
+were employed.
+For the events where at least one selected Λ+
+c candidate was identified, a jet-finding procedure was
+performed, using the FastJet package [29]. Prior to jet clustering, the Λ+
+c-candidate daughter tracks were
+replaced by the reconstructed Λ+
+c-candidate four-momentum vector. Track-based jet finding was carried
+out on charged tracks with |𝜂| < 0.9 and 𝑝T > 0.15 GeV/𝑐, using the anti-𝑘T algorithm [30], with a
+resolution parameter of 𝑅 = 0.4. Tracks were combined using the 𝐸-scheme recombination [31], with the
+jet transverse momentum limited to the interval of 5 ≤ 𝑝jet ch
+T
+< 35 GeV/𝑐. The full jet cone was required
+to be within the ALICE central barrel acceptance, limiting the jet axis to the interval |𝜂jet| < 0.5. Only
+jets tagged via the presence of a reconstructed Λ+
+c candidate amongst their constituents were considered
+for the analysis. For events where more than one Λ+
+c candidate was found, the jet finding and tagging
+pass was performed independently for each candidate, with only the daughters of that particular candidate
+replaced by the corresponding Λ+
+c four-vector each time. In mechanisms of hadronisation that include
+colour correlations beyond the leading-colour approximation [20], which have been shown to be relevant
+in hadronic collisions at LHC energies [9], hadrons can be formed in processes that combine quarks from
+the parton shower with those from the underlying event [32]. As such, the underlying event is not well
+defined with respect to the measured hadron distributions. Therefore no underlying event correction is
+implemented in this work.
+The fragmentation of charm quarks to Λ+
+c baryons is probed by measuring the fraction of the jet momentum
+carried by the Λ+
+c along the direction of the jet axis, 𝑧ch
+|| . This is calculated for each jet using
+𝑧ch
+|| = 𝒑jet · 𝒑Λ+c
+𝒑jet · 𝒑jet
+,
+(1)
+where 𝒑jet and 𝒑Λ+c are the jet and Λ+
+c three-momentum vectors, respectively.
+The 𝑧ch
+|| distributions of true Λ+
+c-tagged jets were extracted in intervals of Λ+
+c 𝑝T and 𝑝jet ch
+T
+using a sideband
+subtraction procedure. To enact this subtraction, the invariant-mass (𝑚inv) distributions of Λ+
+c candidates,
+obtained for each Λ+
+c 𝑝T and 𝑝jet ch
+T
+interval, were fitted with a function comprising a Gaussian for the signal
+3
+
+In-jet Λ+
+c production in pp collisions at √𝑠 = 13 TeV
+ALICE Collaboration
+and an exponential for the background. The fit parameters were then used to define signal (containing the
+majority of true signal candidates) and sideband (entirely composed of background candidates) regions,
+defined by |𝑚inv − 𝜇fit| < 2𝜎fit and 4𝜎fit < |𝑚inv − 𝜇fit| < 9𝜎fit, respectively, where 𝜇fit and 𝜎fit represent
+the mean and sigma of the fitted Gaussian distributions. The 𝑧ch
+|| (𝑝Λ+
+c
+T ,𝑝jet ch
+T
+) distributions were extracted
+in the signal and sideband regions, with the sideband distribution scaled by the ratio of the background
+function integrals in the signal and sideband regions. The sideband distribution was then subtracted from
+the signal one, with the resulting distribution scaled to account for the fact that the 2𝜎fit width of the
+signal region only encompasses approximately 95% of the total signal, to obtain the sideband subtracted
+𝑧ch
+|| yield in each 𝑝Λ+
+c
+T and 𝑝jet ch
+T
+interval.
+To account for the reconstruction and selection efficiency of the Λ+
+c-tagged jet signal, the sideband
+subtracted 𝑧ch
+|| distributions in each 𝑝Λ+
+c
+T and 𝑝jet ch
+T
+interval, 𝑁(𝑧ch
+|| , 𝑝Λ+
+c
+T , 𝑝jet ch
+T
+), were scaled by the recon-
+struction efficiency of prompt Λ+
+c-tagged jets, 𝜖prompt, and summed over the entire 𝑝Λ+
+c
+T interval to obtain
+the efficiency-corrected 𝑧ch
+|| yield of Λ+
+c-tagged jets, 𝑁corr(𝑧ch
+|| , 𝑝jet ch
+T
+), given by
+𝑁corr(𝑧ch
+|| , 𝑝jet ch
+T
+) =
+∑︁
+𝑝Λ+c
+T
+𝑁(𝑧ch
+|| , 𝑝Λ+
+c
+T , 𝑝jet ch
+T
+)
+𝜖prompt(𝑝Λ+c
+T )
+.
+(2)
+The 𝜖prompt(𝑝Λ+
+c
+T ) efficiency is strongly dependent on 𝑝Λ+
+c
+T , ranging from about 20% at 3 < 𝑝Λ+
+c
+T < 4 GeV/𝑐
+to 40% at 12 < 𝑝Λ+
+c
+T < 24 GeV/𝑐, and was calculated using PYTHIA 8 simulations with the Monash tune
+containing prompt Λ+
+c-tagged jets, transported through the detector using GEANT 3. This efficiency does
+not exhibit a 𝑝jet ch
+T
+dependence.
+In order to isolate the 𝑁corr(𝑧ch
+|| , 𝑝jet ch
+T
+) distribution of prompt Λ+
+c-tagged jets, a feed-down subtraction
+was employed to remove the non-prompt (beauty-quark initiated) contribution. The non-prompt cross
+section was obtained from particle level POWHEG [33] + PYTHIA 6 [34] + EvtGen [35] simulations, as
+a function of 𝑝jet ch
+T
+, 𝑝Λ+
+c
+T and 𝑧ch
+|| , and was scaled according to the integrated luminosity of the analysed
+data sample and the branching ratio of the Λ+
+c → pK0
+S → pπ+π− decay channel. The resulting particle-
+level yield was multiplied by the ratio of the non-prompt to prompt Λ+
+c-tagged jet reconstruction and
+selection efficiency in intervals of 𝑝Λ+c
+T
+and integrated over the 𝑝Λ+c
+T
+range. The simulated non-prompt
+results were then folded to reconstructed level, using a four-dimensional response matrix generated using
+non-prompt Λ+
+c-tagged jets in PYTHIA 8 with the Monash tune, transported through a simulation of
+the ALICE detector using GEANT 3. The response matrix was constructed as a function of 𝑝jet ch
+T
+and
+𝑧ch
+|| at generator and reconstruction levels. The folded results were then subtracted from the measured
+𝑁corr(𝑧ch
+|| , 𝑝jet ch
+T
+) distribution in data, removing the non-prompt contribution. The estimated fraction of
+Λ+
+c-tagged jets coming from b-quark fragmentation is found to be about 5%, with no significant 𝑧ch
+||
+dependence.
+A two-dimensional Bayesian unfolding procedure [36] was performed to correct for detector effects and
+obtain the 𝑧ch
+|| distribution for prompt Λ+
+c-tagged jets at particle level.
+A four-dimensional response
+matrix as a function of 𝑝jet ch
+T
+and 𝑧ch
+|| , at generator and reconstruction levels, was populated with prompt
+Λ+
+c-tagged jets, obtained with PYTHIA 8 simulations with the Monash tune, passed through a simulation
+of the ALICE detector using GEANT 3. The measured data and response matrix were provided in the
+intervals of 5 ≤ 𝑝jet ch
+T
+< 35 GeV/𝑐 and 0.4 ≤ 𝑧ch
+|| ≤ 1.0, with the final unfolded results reported in the
+intervals 7 ≤ 𝑝jet ch
+T
+< 15 GeV/𝑐 and 0.4 ≤ 𝑧ch
+|| ≤ 1.0. Corrections accounting for migrating entries in
+and out of the response matrix ranges, as modelled by the same MC simulation, were also applied. The
+corrected 𝑧ch
+|| distribution is normalised to the total number of Λ+
+c-tagged jets in the reported 𝑧ch
+|| and 𝑝jet ch
+T
+interval.
+4
+
+In-jet Λ+
+c production in pp collisions at √𝑠 = 13 TeV
+ALICE Collaboration
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+ch
+z
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+ch
+z
+/d
+N
+) d
+jet
+N
+(1/
+-tagged jets
++
+c
+Λ
+data
+Monash
+CR-BLC Mode 2
+PYTHIA 8:
+ = 13 TeV
+s
+, pp, 
+ALICE
+ = 0.4
+R
+, 
+T
+k
+charged jets, anti-
+ 0.5
+≤
+ 
+jet
+η
+, 
+c
+ < 15 GeV/
+jet ch
+T
+p
+ 
+≤
+7 
+ 0.8
+≤
+ 
++
+c
+Λ
+y
+, 
+c
+ < 15 GeV/
++
+c
+Λ
+T
+p
+ 
+≤
+3 
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+ch
+z
+1
+1.5
+2
+MC/data
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+ch
+z
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+ch
+z
+/d
+N
+) d
+jet
+N
+(1/
+-tagged jets
++
+c
+Λ
+-tagged jets
+0
+D
+ = 13 TeV
+s
+, pp, 
+ALICE
+ = 0.4
+R
+, 
+T
+k
+charged jets, anti-
+ 0.5
+≤
+ 
+jet
+η
+, 
+c
+ < 15 GeV/
+jet ch
+T
+p
+ 
+≤
+7 
+ 0.8
+≤
+ 
+h
+y
+, 
+c
+ < 15 GeV/
+h
+T
+p
+ 
+≤
+3 
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+ch
+z
+0.5
+1
+1.5
+2
+0
+/D
++
+c
+Λ
+data
+PYTHIA 8 Monash
+PYTHIA 8 CR-BLC Mode 2
+Figure 1: (Left) Fully corrected 𝑧ch
+|| distribution of Λ+
+c-tagged track-based jets (black open circles) measured in the
+7 ≤ 𝑝jet ch
+T
+< 15 GeV/𝑐 and 3 ≤ 𝑝Λ+
+c
+T < 15 GeV/𝑐 intervals in pp collisions at √𝑠 = 13 TeV, compared with predictions
+from different PYTHIA 8 tunes [19, 20, 22] (red-dotted and green-dashed lines). The ratios of the MC simulations
+to the data are shown in the bottom panel. (Right) Comparison of the measured 𝑧ch
+|| distribution of Λ+
+c-tagged jets
+and the previously measured 𝑧ch
+|| distribution of D0-tagged jets [6], obtained in the same kinematic interval. The
+ratio of the 𝑧ch
+|| distribution of Λ+
+c-tagged and D0-tagged jets is shown in the bottom panel for both the data and the
+different PYTHIA tunes.
+The systematic uncertainties affecting the measurement were evaluated, in each 𝑧ch
+|| interval, by modifying
+the strategy adopted at various steps of the analysis procedure and assessing the impact on the unfolded
+𝑧ch
+|| distribution. The total systematic uncertainty includes contributions from multiple sources. The
+first considered source is the sideband subtraction procedure (ranging from 3.7% to 7.6% depending
+on the 𝑧ch
+|| inteval), whose contribution was estimated by varying the invariant-mass fit parameters as
+well as the invariant-mass intervals of the signal and sideband regions. The contribution from the BDT
+selection of Λ+
+c candidates (from 7.3% to 19%) was estimated by varying the BDT probability thresholds
+to induce a 25% variation in the Λ+
+c-tagged jet reconstruction and selection efficiency. The uncertainty
+from the jet energy resolution (from 4.5% to 19%) was estimated by recalculating the response matrix
+used for unfolding with a 4% reduced tracking efficiency. The reduction in the tracking efficiency was
+evaluated by varying the track-selection criteria and propagating the ITS–TPC track-matching efficiency
+uncertainty. The uncertainty on the feed-down subtraction (< 2%) was estimated by varying the choice
+of POWHEG parameters considered to generate the feed-down cross section, including the factorisation
+and renormalisation scales, as well as the mass of the beauty quark, which were varied according to
+theoretical prescriptions [37]. Finally the contribution from the unfolding procedure (from 1.1% to 2.7%)
+was estimated by altering the choice of prior, regularisation parameter, and ranges of the response matrix.
+For each of the aforementioned categories, several variations were made and the root-mean-square of
+the resulting distributions was considered. The exceptions are related to the contribution associated to
+the choice of parameters of the POWHEG calculations, where only the largest deviation from the central
+result, in each direction, was considered, as well as the uncertainty on the jet energy resolution where
+the variation with respect to the central result was taken as the uncertainty. All uncertainties (other than
+from the feed-down subtraction) were then symmetrised. The uncertainties were combined in quadrature
+to obtain the total systematic uncertainty on the measurement, which ranges from 13% to 28%.
+5
+
+In-jet Λ+
+c production in pp collisions at √𝑠 = 13 TeV
+ALICE Collaboration
+The fully corrected 𝑧ch
+|| distribution of prompt Λ+
+c-tagged track-based jets in the intervals of 7 ≤ 𝑝jet ch
+T
+<
+15 GeV/𝑐 and 3 ≤ 𝑝Λ+
+c
+T < 15 GeV/𝑐 is presented in the left-hand panel of Fig. 1 and compared to PYTHIA 8
+simulations with two different tunes. In PYTHIA 8 the Lund string model of fragmentation is employed,
+where endpoints are confined by linear potentials encoded in strings. For the case of heavy quarks, the
+Lund fragmentation function is modified to account for the slower propagation of the massive endpoints
+compared to their massless counterparts. The Monash tune (red-dotted line) [19], in which the charm
+fragmentation is tuned on e+e− measurements, predicts a harder fragmentation than the measurement.
+An evaluation of the 𝜒2/ndf between the measured data points and the model was performed, combining
+the statistical and systematic uncertainties on the data in quadrature and assuming the uncertainties are
+uncorrelated across the 𝑧ch
+|| intervals. This exercise determines that there is a 0.4% probability that the
+model describes the data. A better agreement is achieved by the PYTHIA 8 with the CR-BLC Mode 2
+tune, that includes colour reconnection mechanisms beyond the leading-colour approximation [22] (green-
+dashed line). In this model, the minimisation of the string potential is implemented considering the SU(3)
+multiplet structure of QCD in a more realistic way than in the leading-colour approximation, allowing
+for the formation of “baryonic” configurations where for example two colours can combine coherently to
+form an anti-colour. The same 𝜒2/ndf approach results in a 78% probability that the model describes the
+data. The simulation with PYTHIA 8 with the CR-BLC Mode 2 tune also provides a much more accurate
+description of the Λ+
+c/D0 cross section ratio, previously measured in pp collisions at the LHC [10–14, 38].
+In the right-hand panel of Fig. 1, a comparison of the 𝑧ch
+|| distribution of Λ+
+c-tagged jets and the 𝑧ch
+|| distri-
+bution previously measured for D0-tagged jets [6] is presented. The latter is consistent with PYTHIA 8
+simulations using both the Monash and CR-BLC Mode 2 tunes. The ratio of the two distributions is
+also presented in the bottom panel. The uncertainty from the jet energy resolution was considered to
+be correlated between the Λ+
+c-tagged jet and D0-tagged jet measurements and was evaluated directly on
+the ratio of the distributions. The remaining uncertainties were considered uncorrelated when taking
+the ratio and were then combined in quadrature with the uncertainty of the jet energy resolution. The
+uncertainties were considered uncorrelated across the 𝑧ch
+|| intervals. The same 𝜒2/ndf exercise described
+above determines that there is a 12% probability that the measured ratio is described by a flat distribution
+at unity, hinting at a softer fragmentation of charm quarks into charm baryons than charm mesons. The
+ratio is better described by the PYTHIA 8 simulations with the CR-BLC Mode 2 compared to the ones
+with the Monash tune, with the former describing the data with 88% probability compared to a 0.03%
+probability for the latter.
+In summary the first measurement in hadronic collisions of the longitudinal momentum fraction of the
+jet carried by Λ+
+c baryons was presented for pp collisions at √𝑠 = 13 TeV. The result is fully corrected to
+particle level and obtained in the jet and Λ+
+c transverse-momentum intervals of 7 ≤ 𝑝jet ch
+T
+< 15 GeV/𝑐
+and 3 ≤ 𝑝Λ+c
+T < 15 GeV/𝑐, respectively. The measurement presented in this Letter hints that charm quarks
+have a softer fragmentation into Λ+
+c baryons compared to D0 mesons, in the measured kinematic interval.
+One possible explanation is that charm-baryon production is favoured in the presence of higher particle
+multiplicity originating from both the jet fragmentation and the underlying event, which could be tested
+with future measurements of the in-jet multiplicity of Λ+
+c-tagged jets. The fragmentation of charm quarks
+into Λ+
+c baryons in hadronic collisions exhibits tension with simulations tuned on e+e− data that employ
+a leading-colour formalism of hadronisation, such as in the Monash tune of PYTHIA 8. This occurs
+despite their successful description of the fragmentation of charm quarks into D0 mesons. However, the
+inclusion of mechanisms sensitive to the surrounding partonic density that feature colour reconnection
+beyond the leading-colour approximation results in a better agreement with data. This result also partially
+explains the 𝑝T shape of the prompt Λ+
+c/D0 cross section ratio [10–14, 38], which shows a peak at low
+𝑝T (≈ 3 GeV/𝑐) and is also described within uncertainties by PYTHIA 8 with the CR-BLC Mode 2 tune.
+The 𝑝T trend of this ratio is driven by the fact that the Λ+
+c baryons produced from the fragmenting charm
+quark carry a significantly lower fraction of the charm-quark transverse momentum than the D0 mesons
+6
+
+In-jet Λ+
+c production in pp collisions at √𝑠 = 13 TeV
+ALICE Collaboration
+produced in a similar way.
+References
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+[11] ALICE Collaboration, S. Acharya et al., “Λ+
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+9
+

9dFST4oBgHgl3EQfbDi0/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf,len=455
+page_content='EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH  CERN-EP-2023-005  23 January 2023  © 2023 CERN for the benefit of the ALICE Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  Reproduction of this article or parts of it is allowed as specified in the CC-BY-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='0 license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  Exploring the non-universality of charm hadronisation through the  measurement of the fraction of jet longitudinal momentum carried by Λ+  c  baryons in pp collisions  ALICE Collaboration  Abstract  Recent measurements of charm-baryon production in hadronic collisions have questioned the univer-  sality of charm-quark fragmentation across different collision systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' In this work the fragmentation  of charm quarks into charm baryons is probed, by presenting the first measurement of the longitudinal  jet momentum fraction carried by Λ+  c baryons, 𝑧ch  || , in hadronic collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The results are obtained  in proton–proton (pp) collisions at √𝑠 = 13 TeV at the LHC, with Λ+  c baryons and track-based jets  reconstructed in the transverse momentum intervals of 3 ≤ 𝑝Λ+  c  T < 15 GeV/𝑐 and 7 ≤ 𝑝jet ch  T  < 15  GeV/𝑐, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The 𝑧ch  || distribution is compared to a measurement of D0-tagged charged jets in  pp collisions as well as to PYTHIA 8 simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The data hints that the fragmentation of charm  quarks into charm baryons is softer with respect to charm mesons, as predicted by hadronisation  models which include colour correlations beyond leading-colour in the string formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='13798v1  [nucl-ex]  31 Jan 2023  ALICEIn-jet Λ+  c production in pp collisions at √𝑠 = 13 TeV  ALICE Collaboration  Heavy-flavour hadrons are produced in high-energy particle collisions through the fragmentation of heavy  (charm and beauty) quarks, which typically originate in hard scattering processes in the early stages of  the collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The most common theoretical approach to describe heavy-flavour production in hadronic  collisions is based on the quantum chromodynamics (QCD) factorisation approach [1], and consists of a  convolution of three independent terms: the parton distribution functions of the incoming hadrons, the  cross sections of the partonic scattering producing the heavy quarks, and the fragmentation functions that  parametrise the evolution of a heavy quark into given species of heavy-flavour hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' As the transition  of quarks to hadrons cannot be described in perturbation theory, the fragmentation functions cannot be  calculated and must be extracted from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  Fragmentation functions of charm quarks to charm baryons and mesons have been constrained in e+e−  and e−p collisions [2–5], using a variety of different observables, such as the hadron momentum as a  fraction of its maximum possible momentum, as dictated by the centre-of-mass energy of the collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  Another method to probe the fragmentation of quarks to hadrons is to parametrise the hadron momentum  in relation to the momentum of jets, which are collimated bunches of hadrons giving experimental access  to the properties of the scattered quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' Recently, the production of charm mesons in jets, probed via the  fractional longitudinal momentum of the jet carried by the D meson, was measured in pp collisions at the  Large Hadron Collider (LHC) [6–8] and appears consistent with Monte Carlo (MC) simulations tuned  on e+e− data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' These measurements support the assumption of fragmentation universality across collision  systems in the charm-meson sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' This assumption underpins theoretical calculations describing the  production of heavy-flavour hadrons in hadronic collisions, which make use of fragmentation functions  tuned on e+e− and e−p data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  Measurements of the production cross sections of baryons in pp collisions have questioned the hypothesis  of fragmentation universality across collision systems [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' In the charm sector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' which provides a clean  probe of hadronisation phenomena due to the large mass of the charm quark,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' recent measurements  performed by the ALICE Collaboration [10–18] in pp collisions have shown that the ratio of the Λ+  c (and  other charm baryons) and D0 production cross sections measured at low 𝑝T (≲ 12 GeV/𝑐) is significantly  larger than the value expected from MC simulations in which the charm fragmentation is tuned on e+e−  and e−p measurements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' such as PYTHIA 8 [19] with the Monash tune [20] or HERWIG 7 [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' A recent  measurement of the Λ+  c/D0 ratio in pp collisions, performed by the ALICE Collaboration in intervals  of charged-particle multiplicity, also points to a substantial increase of the Λ+  c/D0 ratio with increasing  multiplicity, with respect to e+e− collisions, starting at very low multiplicities [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  The study of charm-baryon production in jets can provide more differential insights into hadronisation  mechanisms in pp collisions, compared to 𝑝T-differential cross sections and yield ratios of heavy-flavour  hadrons, allowing for a more accurate study of the dynamical properties of baryon production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' In this  letter, the first measurement of the longitudinal momentum fraction of the jet carried by Λ+  c baryons, 𝑧ch  || , is  presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The measurement is performed in pp collisions at √𝑠 = 13 TeV in the interval 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='4 ≤ 𝑧ch  || ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  The 𝑧ch  || distribution, fully corrected to particle level, is presented for prompt (charm-quark initiated)  Λ+  c-tagged jets with 7 ≤ 𝑝jet ch  T  < 15 GeV/𝑐 and 3 ≤ 𝑝Λ+  c  T < 15 GeV/𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The results are then compared  to PYTHIA 8 simulations [19, 22], including a version where mechanisms beyond the leading-colour  approximation are considered in string formation processes during hadronisation [20], and to an analogous  measurement of the 𝑧ch  || distribution of D0 mesons, performed by the ALICE Collaboration [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  A full description of the ALICE setup and apparatus can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' 23, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The main detectors  used in this analysis are the Inner Tracking System (ITS), which is used for vertex reconstruction and  tracking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' the Time Projection Chamber (TPC), which is used for tracking and particle identification (PID);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  and the Time-Of-Flight (TOF) detector, which is used for PID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' These detectors cover a pseudorapidity  interval of |𝜂| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The analysis was performed on pp collisions at √𝑠 = 13 TeV, collected using a  minimum-bias (MB) trigger during the years 2016, 2017, and 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The trigger condition required  2  In-jet Λ+  c production in pp collisions at √𝑠 = 13 TeV  ALICE Collaboration  coincident signals in the two scintillator arrays of the V0 detector, with background events originating  from beam–gas interactions removed offline using timing information from the V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' To mitigate against  pile-up effects, events with multiple reconstructed primary vertices were rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' To ensure uniform  acceptance, only events with a primary-vertex position along the beam axis direction of |𝑧vtx| < 10 cm  around the nominal interaction point were accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' After the selections described above, the data sample  consisted of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='7×109 events, corresponding to an integrated luminosity of Lint = 29 nb−1 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  The Λ+  c candidates and their charge conjugates were reconstructed via the hadronic Λ+  c → pK0  S → pπ+π−  decay channel with a total branching ratio of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='06)% [26], in the Λ+  c transverse-momentum  interval of 3 ≤ 𝑝Λ+  c  T < 15 GeV/𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' Only tracks with |𝜂| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='8 and 𝑝T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='4 GeV/𝑐, which fulfilled the track  quality selections described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' 13, were considered for the Λ+  c reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The Λ+  c candidates  themselves were reconstructed in the |𝑦Λ+c | < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='8 rapidity interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  The Λ+  c-candidate selection was  performed using a multivariate technique based on the Boosted Decision Tree (BDT) algorithm provided  by the XGBoost package [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The features considered in the optimisation include the PID signal for  the proton track, the invariant mass of the K0  S-meson candidate, and topological variables that exploit the  kinematic properties of the displaced K0  S-meson decay vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The training was performed in intervals  of Λ+  c-candidate 𝑝T, considering prompt signal candidates from PYTHIA 8 events with the Monash  tune [19, 20], transported through a realistic description of the detector geometry and material budget  using GEANT 3 [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' Background candidates were extracted from the sidebands of the invariant-mass  distributions in data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The probability thresholds of the BDT selections were tuned, using MC simulations,  to maximise the statistical significance for the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' Further details on the Λ+  c-candidate reconstruction  and machine learning procedure are provided in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' 14, where the same reconstruction and BDT model  were employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  For the events where at least one selected Λ+  c candidate was identified, a jet-finding procedure was  performed, using the FastJet package [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' Prior to jet clustering, the Λ+  c-candidate daughter tracks were  replaced by the reconstructed Λ+  c-candidate four-momentum vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' Track-based jet finding was carried  out on charged tracks with |𝜂| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='9 and 𝑝T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='15 GeV/𝑐, using the anti-𝑘T algorithm [30], with a  resolution parameter of 𝑅 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' Tracks were combined using the 𝐸-scheme recombination [31], with the  jet transverse momentum limited to the interval of 5 ≤ 𝑝jet ch  T  < 35 GeV/𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The full jet cone was required  to be within the ALICE central barrel acceptance, limiting the jet axis to the interval |𝜂jet| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' Only  jets tagged via the presence of a reconstructed Λ+  c candidate amongst their constituents were considered  for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' For events where more than one Λ+  c candidate was found, the jet finding and tagging  pass was performed independently for each candidate, with only the daughters of that particular candidate  replaced by the corresponding Λ+  c four-vector each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' In mechanisms of hadronisation that include  colour correlations beyond the leading-colour approximation [20], which have been shown to be relevant  in hadronic collisions at LHC energies [9], hadrons can be formed in processes that combine quarks from  the parton shower with those from the underlying event [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' As such, the underlying event is not well  defined with respect to the measured hadron distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' Therefore no underlying event correction is  implemented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  The fragmentation of charm quarks to Λ+  c baryons is probed by measuring the fraction of the jet momentum  carried by the Λ+  c along the direction of the jet axis, 𝑧ch  || .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' This is calculated for each jet using  𝑧ch  || = 𝒑jet · 𝒑Λ+c  𝒑jet · 𝒑jet  ,  (1)  where 𝒑jet and 𝒑Λ+c are the jet and Λ+  c three-momentum vectors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  The 𝑧ch  || distributions of true Λ+  c-tagged jets were extracted in intervals of Λ+  c 𝑝T and 𝑝jet ch  T  using a sideband  subtraction procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' To enact this subtraction, the invariant-mass (𝑚inv) distributions of Λ+  c candidates,  obtained for each Λ+  c 𝑝T and 𝑝jet ch  T  interval, were fitted with a function comprising a Gaussian for the signal  3  In-jet Λ+  c production in pp collisions at √𝑠 = 13 TeV  ALICE Collaboration  and an exponential for the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The fit parameters were then used to define signal (containing the  majority of true signal candidates) and sideband (entirely composed of background candidates) regions,  defined by |𝑚inv − 𝜇fit| < 2𝜎fit and 4𝜎fit < |𝑚inv − 𝜇fit| < 9𝜎fit, respectively, where 𝜇fit and 𝜎fit represent  the mean and sigma of the fitted Gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The 𝑧ch  || (𝑝Λ+  c  T ,𝑝jet ch  T  ) distributions were extracted  in the signal and sideband regions, with the sideband distribution scaled by the ratio of the background  function integrals in the signal and sideband regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The sideband distribution was then subtracted from  the signal one, with the resulting distribution scaled to account for the fact that the 2𝜎fit width of the  signal region only encompasses approximately 95% of the total signal, to obtain the sideband subtracted  𝑧ch  || yield in each 𝑝Λ+  c  T and 𝑝jet ch  T  interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  To account for the reconstruction and selection efficiency of the Λ+  c-tagged jet signal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' the sideband  subtracted 𝑧ch  || distributions in each 𝑝Λ+  c  T and 𝑝jet ch  T  interval,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' 𝑁(𝑧ch  || ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' 𝑝Λ+  c  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' 𝑝jet ch  T  ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' were scaled by the recon-  struction efficiency of prompt Λ+  c-tagged jets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' 𝜖prompt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' and summed over the entire 𝑝Λ+  c  T interval to obtain  the efficiency-corrected 𝑧ch  || yield of Λ+  c-tagged jets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' 𝑁corr(𝑧ch  || ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' 𝑝jet ch  T  ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' given by  𝑁corr(𝑧ch  || ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' 𝑝jet ch  T  ) =  ∑︁  𝑝Λ+c  T  𝑁(𝑧ch  || ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' 𝑝Λ+  c  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' 𝑝jet ch  T  )  𝜖prompt(𝑝Λ+c  T )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  (2)  The 𝜖prompt(𝑝Λ+  c  T ) efficiency is strongly dependent on 𝑝Λ+  c  T , ranging from about 20% at 3 < 𝑝Λ+  c  T < 4 GeV/𝑐  to 40% at 12 < 𝑝Λ+  c  T < 24 GeV/𝑐, and was calculated using PYTHIA 8 simulations with the Monash tune  containing prompt Λ+  c-tagged jets, transported through the detector using GEANT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' This efficiency does  not exhibit a 𝑝jet ch  T  dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  In order to isolate the 𝑁corr(𝑧ch  || , 𝑝jet ch  T  ) distribution of prompt Λ+  c-tagged jets, a feed-down subtraction  was employed to remove the non-prompt (beauty-quark initiated) contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The non-prompt cross  section was obtained from particle level POWHEG [33] + PYTHIA 6 [34] + EvtGen [35] simulations, as  a function of 𝑝jet ch  T  , 𝑝Λ+  c  T and 𝑧ch  || , and was scaled according to the integrated luminosity of the analysed  data sample and the branching ratio of the Λ+  c → pK0  S → pπ+π− decay channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The resulting particle-  level yield was multiplied by the ratio of the non-prompt to prompt Λ+  c-tagged jet reconstruction and  selection efficiency in intervals of 𝑝Λ+c  T  and integrated over the 𝑝Λ+c  T  range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The simulated non-prompt  results were then folded to reconstructed level, using a four-dimensional response matrix generated using  non-prompt Λ+  c-tagged jets in PYTHIA 8 with the Monash tune, transported through a simulation of  the ALICE detector using GEANT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The response matrix was constructed as a function of 𝑝jet ch  T  and  𝑧ch  || at generator and reconstruction levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The folded results were then subtracted from the measured  𝑁corr(𝑧ch  || , 𝑝jet ch  T  ) distribution in data, removing the non-prompt contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The estimated fraction of  Λ+  c-tagged jets coming from b-quark fragmentation is found to be about 5%, with no significant 𝑧ch  ||  dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  A two-dimensional Bayesian unfolding procedure [36] was performed to correct for detector effects and  obtain the 𝑧ch  || distribution for prompt Λ+  c-tagged jets at particle level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  A four-dimensional response  matrix as a function of 𝑝jet ch  T  and 𝑧ch  || , at generator and reconstruction levels, was populated with prompt  Λ+  c-tagged jets, obtained with PYTHIA 8 simulations with the Monash tune, passed through a simulation  of the ALICE detector using GEANT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The measured data and response matrix were provided in the  intervals of 5 ≤ 𝑝jet ch  T  < 35 GeV/𝑐 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='4 ≤ 𝑧ch  || ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='0, with the final unfolded results reported in the  intervals 7 ≤ 𝑝jet ch  T  < 15 GeV/𝑐 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='4 ≤ 𝑧ch  || ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' Corrections accounting for migrating entries in  and out of the response matrix ranges, as modelled by the same MC simulation, were also applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The  corrected 𝑧ch  || distribution is normalised to the total number of Λ+  c-tagged jets in the reported 𝑧ch  || and 𝑝jet ch  T  interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  4  In-jet Λ+  c production in pp collisions at √𝑠 = 13 TeV  ALICE Collaboration  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='9  1  ch  z  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='5  ch  z  /d  N  ) d  jet  N  (1/  tagged jets  +  c  Λ  data  Monash  CR-BLC Mode 2  PYTHIA 8:  = 13 TeV  s  , pp,  ALICE  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='4  R  ,  T  k  charged jets, anti-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='5  ≤  jet  η  ,  c  < 15 GeV/  jet ch  T  p  ≤  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='8  ≤  +  c  Λ  y  ,  c  < 15 GeV/  +  c  Λ  T  p  ≤  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='9  1  ch  z  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='5  2  MC/data  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='9  1  ch  z  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='5  5  ch  z  /d  N  ) d  jet  N  (1/  tagged jets  +  c  Λ  tagged jets  0  D  = 13 TeV  s  , pp,  ALICE  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='4  R  ,  T  k  charged jets, anti-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='5  ≤  jet  η  ,  c  < 15 GeV/  jet ch  T  p  ≤  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='8  ≤  h  y  ,  c  < 15 GeV/  h  T  p  ≤  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='9  1  ch  z  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='5  2  0  /D  +  c  Λ  data  PYTHIA 8 Monash  PYTHIA 8 CR-BLC Mode 2  Figure 1: (Left) Fully corrected 𝑧ch  || distribution of Λ+  c-tagged track-based jets (black open circles) measured in the  7 ≤ 𝑝jet ch  T  < 15 GeV/𝑐 and 3 ≤ 𝑝Λ+  c  T < 15 GeV/𝑐 intervals in pp collisions at √𝑠 = 13 TeV, compared with predictions  from different PYTHIA 8 tunes [19, 20, 22] (red-dotted and green-dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The ratios of the MC simulations  to the data are shown in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' (Right) Comparison of the measured 𝑧ch  || distribution of Λ+  c-tagged jets  and the previously measured 𝑧ch  || distribution of D0-tagged jets [6], obtained in the same kinematic interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The  ratio of the 𝑧ch  || distribution of Λ+  c-tagged and D0-tagged jets is shown in the bottom panel for both the data and the  different PYTHIA tunes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  The systematic uncertainties affecting the measurement were evaluated, in each 𝑧ch  || interval, by modifying  the strategy adopted at various steps of the analysis procedure and assessing the impact on the unfolded  𝑧ch  || distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The total systematic uncertainty includes contributions from multiple sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The  first considered source is the sideband subtraction procedure (ranging from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='7% to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='6% depending  on the 𝑧ch  || inteval), whose contribution was estimated by varying the invariant-mass fit parameters as  well as the invariant-mass intervals of the signal and sideband regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The contribution from the BDT  selection of Λ+  c candidates (from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='3% to 19%) was estimated by varying the BDT probability thresholds  to induce a 25% variation in the Λ+  c-tagged jet reconstruction and selection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The uncertainty  from the jet energy resolution (from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='5% to 19%) was estimated by recalculating the response matrix  used for unfolding with a 4% reduced tracking efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The reduction in the tracking efficiency was  evaluated by varying the track-selection criteria and propagating the ITS–TPC track-matching efficiency  uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The uncertainty on the feed-down subtraction (< 2%) was estimated by varying the choice  of POWHEG parameters considered to generate the feed-down cross section, including the factorisation  and renormalisation scales, as well as the mass of the beauty quark, which were varied according to  theoretical prescriptions [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' Finally the contribution from the unfolding procedure (from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='1% to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='7%)  was estimated by altering the choice of prior, regularisation parameter, and ranges of the response matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  For each of the aforementioned categories, several variations were made and the root-mean-square of  the resulting distributions was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The exceptions are related to the contribution associated to  the choice of parameters of the POWHEG calculations, where only the largest deviation from the central  result, in each direction, was considered, as well as the uncertainty on the jet energy resolution where  the variation with respect to the central result was taken as the uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' All uncertainties (other than  from the feed-down subtraction) were then symmetrised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The uncertainties were combined in quadrature  to obtain the total systematic uncertainty on the measurement, which ranges from 13% to 28%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  5  In-jet Λ+  c production in pp collisions at √𝑠 = 13 TeV  ALICE Collaboration  The fully corrected 𝑧ch  || distribution of prompt Λ+  c-tagged track-based jets in the intervals of 7 ≤ 𝑝jet ch  T  <  15 GeV/𝑐 and 3 ≤ 𝑝Λ+  c  T < 15 GeV/𝑐 is presented in the left-hand panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' 1 and compared to PYTHIA 8  simulations with two different tunes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' In PYTHIA 8 the Lund string model of fragmentation is employed,  where endpoints are confined by linear potentials encoded in strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' For the case of heavy quarks, the  Lund fragmentation function is modified to account for the slower propagation of the massive endpoints  compared to their massless counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The Monash tune (red-dotted line) [19], in which the charm  fragmentation is tuned on e+e− measurements, predicts a harder fragmentation than the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  An evaluation of the 𝜒2/ndf between the measured data points and the model was performed, combining  the statistical and systematic uncertainties on the data in quadrature and assuming the uncertainties are  uncorrelated across the 𝑧ch  || intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' This exercise determines that there is a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='4% probability that the  model describes the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' A better agreement is achieved by the PYTHIA 8 with the CR-BLC Mode 2  tune, that includes colour reconnection mechanisms beyond the leading-colour approximation [22] (green-  dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' In this model, the minimisation of the string potential is implemented considering the SU(3)  multiplet structure of QCD in a more realistic way than in the leading-colour approximation, allowing  for the formation of “baryonic” configurations where for example two colours can combine coherently to  form an anti-colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The same 𝜒2/ndf approach results in a 78% probability that the model describes the  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The simulation with PYTHIA 8 with the CR-BLC Mode 2 tune also provides a much more accurate  description of the Λ+  c/D0 cross section ratio, previously measured in pp collisions at the LHC [10–14, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  In the right-hand panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' 1, a comparison of the 𝑧ch  || distribution of Λ+  c-tagged jets and the 𝑧ch  || distri-  bution previously measured for D0-tagged jets [6] is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The latter is consistent with PYTHIA 8  simulations using both the Monash and CR-BLC Mode 2 tunes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The ratio of the two distributions is  also presented in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The uncertainty from the jet energy resolution was considered to  be correlated between the Λ+  c-tagged jet and D0-tagged jet measurements and was evaluated directly on  the ratio of the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The remaining uncertainties were considered uncorrelated when taking  the ratio and were then combined in quadrature with the uncertainty of the jet energy resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The  uncertainties were considered uncorrelated across the 𝑧ch  || intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The same 𝜒2/ndf exercise described  above determines that there is a 12% probability that the measured ratio is described by a flat distribution  at unity, hinting at a softer fragmentation of charm quarks into charm baryons than charm mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The  ratio is better described by the PYTHIA 8 simulations with the CR-BLC Mode 2 compared to the ones  with the Monash tune, with the former describing the data with 88% probability compared to a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='03%  probability for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  In summary the first measurement in hadronic collisions of the longitudinal momentum fraction of the  jet carried by Λ+  c baryons was presented for pp collisions at √𝑠 = 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The result is fully corrected to  particle level and obtained in the jet and Λ+  c transverse-momentum intervals of 7 ≤ 𝑝jet ch  T  < 15 GeV/𝑐  and 3 ≤ 𝑝Λ+c  T < 15 GeV/𝑐, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The measurement presented in this Letter hints that charm quarks  have a softer fragmentation into Λ+  c baryons compared to D0 mesons, in the measured kinematic interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  One possible explanation is that charm-baryon production is favoured in the presence of higher particle  multiplicity originating from both the jet fragmentation and the underlying event, which could be tested  with future measurements of the in-jet multiplicity of Λ+  c-tagged jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' The fragmentation of charm quarks  into Λ+  c baryons in hadronic collisions exhibits tension with simulations tuned on e+e− data that employ  a leading-colour formalism of hadronisation, such as in the Monash tune of PYTHIA 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' This occurs  despite their successful description of the fragmentation of charm quarks into D0 mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' However, the  inclusion of mechanisms sensitive to the surrounding partonic density that feature colour reconnection  beyond the leading-colour approximation results in a better agreement with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content=' This result also partially  explains the 𝑝T shape of the prompt Λ+  c/D0 cross section ratio [10–14, 38], which shows a peak at low  𝑝T (≈ 3 GeV/𝑐) and is also described within uncertainties by PYTHIA 8 with the CR-BLC Mode 2 tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
+page_content='  The 𝑝T trend of this ratio is driven by the fact that the Λ+  c baryons produced from the fragmenting charm  quark carry a significantly lower fraction of the charm-quark transverse momentum than the D0 mesons  6  In-jet Λ+  c production in pp collisions at √𝑠 = 13 TeV  ALICE Collaboration  produced in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9dFST4oBgHgl3EQfbDi0/content/2301.13798v1.pdf'}
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9tFQT4oBgHgl3EQf6DZi/content/tmp_files/2301.13437v1.pdf.txt

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+Astronomy & Astrophysics manuscript no. aanda
+©ESO 2023
+February 1, 2023
+KiDS-1000: cross-correlation with Planck cosmic microwave
+background lensing and intrinsic alignment removal with
+self-calibration
+Ji Yao1, 2, 3⋆ , Huanyuan Shan1⋆⋆ , Pengjie Zhang2, 3, 4⋆⋆⋆, Xiangkun Liu5, Catherine Heymans6, 7, Benjamin
+Joachimi8, Marika Asgari9, Maciej Bilicki10, Hendrik Hildebrandt6, Konrad Kuijken11, Tilman Tröster6, Jan Luca van
+den Busch8, 12, Angus Wright7, and Ziang Yan7
+1 Shanghai Astronomical Observatory (SHAO), Nandan Road 80, Shanghai 200030, China
+2 Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai, 200240, China
+3 Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai 200240, China
+4 Tsung-Dao Lee Institute, Shanghai, 200240, China
+5 South-Western Institute for Astronomy Research, Yunnan University, Kunming, 650500, China
+6 Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK
+7 Ruhr-Universität Bochum, Astronomisches Institut, German Centre for Cosmological Lensing (GCCL), Universitätsstr. 150,
+44801, Bochum, Germany
+8 Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK
+9 E.A Milne Centre, University of Hull, Cottingham Road, Hull, HU6 7RX, United Kingdom
+10 Center for Theoretical Physics, Polish Academy of Sciences, al. Lotników 32/46, 02-668 Warsaw, Poland
+11 Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, the Netherlands
+12 Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
+Received January 30, 2023; accepted ?
+ABSTRACT
+Context. Galaxy shear - cosmic microwave background (CMB) lensing convergence cross-correlations contain additional informa-
+tion on cosmology to auto-correlations. While being immune to certain systematic effects, they are affected by the galaxy intrinsic
+alignments (IA). This may be responsible for the reported low lensing amplitude of the galaxy shear × CMB convergence cross-
+correlations, compared to the standard Planck ΛCDM (cosmological constant and cold dark matter) cosmology prediction.
+Aims. In this work, we investigate how IA affects the Kilo-Degree Survey (KiDS) galaxy lensing shear - Planck CMB lensing
+convergence cross-correlation and compare it to previous treatments with or without IA taken into consideration.
+Methods. More specifically, we compare marginalization over IA parameters and the IA self-calibration (SC) method (with additional
+observables defined only from the source galaxies) and prove that SC can efficiently break the degeneracy between the CMB lensing
+amplitude Alens and the IA amplitude AIA. We further investigate how different systematics affect the resulting AIA and Alens, and
+validate our results with the MICE2 simulation.
+Results. We find that by including the SC method to constrain IA, the information loss due to the degeneracy between CMB lensing
+and IA is strongly reduced. The best-fit values are Alens = 0.84+0.22
+−0.22 and AIA = 0.60+1.03
+−1.03, while different angular scale cuts can affect
+Alens by ∼ 10%. We show that appropriate treatment of the boost factor, cosmic magnification, and photometric redshift modeling is
+important for obtaining the correct IA and cosmological results.
+Key words. cosmology – weak lensing – CMB lensing – intrinsic alignment – self-calibration
+1. Introduction
+Weak lensing due to the distortion of light by gravity is a power-
+ful probe of the underlying matter distribution and the encoded
+secrets of cosmological physics such as dark matter, dark energy,
+and the nature of gravity (Refregier 2003; Mandelbaum 2018).
+The auto-correlation statistics have been widely used in the anal-
+ysis, both for galaxy lensing shear, e.g. “cosmic shear” (Hilde-
+brandt et al. 2017; Hamana et al. 2020; Hikage et al. 2019; As-
+gari et al. 2021; Secco et al. 2022; Amon et al. 2022), and CMB
+lensing convergence (Planck Collaboration et al. 2020c; Omori
+et al. 2017). Furthermore, cross-correlations between galaxy
+⋆ e-mail: [email protected]
+⋆⋆ e-mail: [email protected]
+⋆⋆⋆ e-mail: [email protected]
+shear and CMB lensing have been measured extensively (Hand
+et al. 2015; Chisari et al. 2015; Liu & Hill 2015; Kirk et al. 2016;
+Harnois-Déraps et al. 2016; Singh et al. 2017a; Harnois-Déraps
+et al. 2017; Omori et al. 2019; Namikawa et al. 2019; Marques
+et al. 2020; Robertson et al. 2021). Cross-correlation statistics
+contain highly complementary information to auto-correlations,
+both for cosmology and the cross-check of systematics. They
+partly reveal the hidden redshift information in CMB lensing
+and are more sensitive to structure growth at redshifts between
+the epochs probed by galaxy shear and CMB lensing. Cross-
+correlations are also immune to additive errors in shear measure-
+ment and provide an external diagnosis of multiplicative errors
+(Schaan et al. 2017).
+Most existing cross-correlation measurements have found a
+lower CMB lensing amplitude than the prediction of their as-
+Article number, page 1 of 15
+arXiv:2301.13437v1  [astro-ph.CO]  31 Jan 2023
+
+A&A proofs: manuscript no. aanda
+sumed ΛCDM cosmology (Hand et al. 2015; Liu & Hill 2015;
+Kirk et al. 2016; Harnois-Déraps et al. 2016, 2017; Singh et al.
+2017a; Marques et al. 2020; Robertson et al. 2021). The ra-
+tio, which is normally referred as the CMB lensing amplitude,
+Alens ∼ 0.5-0.9, although the deviation from unity is only within
+1-2σ. The low lensing amplitude is consistent across many com-
+binations of data sets and analysis methods, suggesting the ex-
+istence of a common systematic errors or a deviation from the
+best-fit Planck cosmology. This might be related to the tension
+between galaxy lensing surveys and Planck CMB observation
+(Lin & Ishak 2017; Chang et al. 2019; Heymans et al. 2021),
+and the Planck internal inconsistencies (Planck Collaboration
+et al. 2020a,b). In this paper we focus on the galaxy intrinsic
+alignment (IA), which can mimic weak lensing signals (Croft &
+Metzler 2000; Catelan et al. 2001; Crittenden et al. 2001; Lee
+& Pen 2001; Jing 2002; Hirata & Seljak 2004; Heymans et al.
+2004; Bridle & King 2007; Okumura et al. 2009; Joachimi et al.
+2013; Kiessling et al. 2015; Blazek et al. 2015; Rong et al. 2015;
+Krause et al. 2016; Blazek et al. 2019; Troxel et al. 2018; Chis-
+ari et al. 2017; Xia et al. 2017; Samuroff et al. 2019; Yao et al.
+2020a; Samuroff et al. 2021; Yao et al. 2020b). Here the CMB
+lensing convergence is expected to be anti-correlated with the
+intrinsic ellipticities of the foreground galaxy field, resulting in
+a dilution of the overall cross-correlation signal (Troxel & Ishak
+2014; Chisari et al. 2015; Kirk et al. 2015; Omori et al. 2019;
+Robertson et al. 2021). Taking IA into account can alleviate the
+tension in Alens, at the expense of a significant loss of lensing
+constraining power, because of the degeneracy between the lens-
+ing amplitude Alens and the IA amplitude AIA. Therefore, a com-
+mon compromise is to fix both the IA model and its amplitude
+AIA (Kirk et al. 2016; Harnois-Déraps et al. 2017; Omori et al.
+2019) or assume a strong prior (Robertson et al. 2021).
+Since IA is already a major limiting factor in the current
+cross-correlation analysis, its mitigation will be essential for up-
+coming measurements with significantly smaller statistical er-
+rors. We utilize the IA self-calibration (SC) method (Zhang
+2010a,b; Troxel & Ishak 2012a,b; Yao et al. 2017, 2019), which
+is a galaxy-galaxy lensing method but with a different weight-
+ing scheme, to mitigate the IA problem in the shear-convergence
+cross-correlation. It is based on the fact that the IA-galaxy cor-
+relation is insensitive to the redshift order, while it matters for
+lensing-galaxy correlation whether the lens is in front of the
+source or not. Therefore, we can isolate IA by comparing ex-
+tra observables, i.e., the galaxy shear × number density cross-
+correlation with a different weighting of the redshift pairs. This
+measurement of IA is independent of a physical model of the
+IA and requires no data external to the shear data. SC was first
+applied to KiDS450/KV450 (Yao et al. 2020a; Pedersen et al.
+2020) and DECaLS DR3 (Yao et al. 2020b) and has enabled
+significant IA detections. The detected IA signal can then be
+applied to remove IA in the lensing shear auto-correlation and
+shear-convergence cross-correlation. The IA information is ob-
+tained from a shear × number density cross-correlation within
+the same photometric redshift (photo-z) bin, more importantly,
+with different weighting schemes on the photo-z ordering, which
+is usually not used for cosmological parameter constraints. We
+find that this removal of IA losses almost no cosmological infor-
+mation.
+In previous work Yao et al. (2020b), we have demonstrated
+the importance and methodology of including certain types of
+systematics in the SC lensing-IA separation method, namely
+galaxy bias, the covariance between the separated lensing signal
+and IA signal, the IA signal drop QIg due to the photo-z selection,
+and the scale dependency of the signal drops QGg and QIg. In this
+work, we further investigate other sources of systematics, includ-
+ing the boost factor (Mandelbaum et al. 2005), photo-z modeling
+bias (Yao et al. 2020a), and cosmic magnification (Bartelmann
+1995; Bartelmann & Schneider 2001; Yang et al. 2017; Liu et al.
+2021). Interestingly, as the survey goes to higher redshift, the
+contamination to the SC method from magnification will quickly
+increase to a non-negligible level. The cosmic magnification will
+change the observed galaxy number density due to the lensing-
+magnified flux and lensing-enlarged area, therefore biasing our
+SC analysis. We investigate the proper treatments for the above
+systematics together with the cosmological study.
+This paper is organized as follows. In Sect. 2 we review the
+physics of galaxy shear × CMB convergence and how our SC
+method works to subtract the IA information. In Sect. 3 we in-
+troduce the KiDS-1000 and Planck data used in this work, and
+the MICE2 simulation (van den Busch et al. 2020; Fosalba et al.
+2015) we use to validate how the SC method is affected by differ-
+ent systematics. We show the measurements of the observables
+in Sect. 4. The results and summary are shown in Sect. 5 and 6.
+2. Methods
+We apply our self-calibration method to separate the intrinsic
+alignment and the lensing signals and show how the intrinsic
+alignment will bias the galaxy shear-CMB convergence corre-
+lation. In this section, we review the theory of lensing cross-
+correlation and the self-calibration method, with a modification
+to account for the contamination from cosmic magnification.
+2.1. Galaxy shear × CMB convergence
+The gravitational field can distort the shape of the background
+source galaxy image and introduce an extra shape that is tan-
+gentially aligned to the lens. This gravitational shear γG of the
+source galaxy contains integral information of the foreground
+overdensity along the line of sight (Bartelmann & Schneider
+2001). Similarly, the photons from the CMB are deflected, and
+the lensing convergence κ can be reconstructed from the CMB
+temperature and polarization observations (Planck Collabora-
+tion et al. 2020c). By correlating these two quantities
+�
+γGκ
+�
+,
+we probe the clustering of the underlying matter field ⟨δδ⟩. In
+harmonic space while assuming flat space (Omori et al. 2019;
+Marques et al. 2020), we have:
+CκgalκCMB(ℓ) =
+� χCMB
+0
+qgal(χ)qCMB(χ)
+χ2
+Pδ
+�
+k = ℓ + 1/2
+χ
+, z
+�
+dχ.
+(1)
+Eq. (1) is the galaxy-lensing CMB-lensing cross angular
+power spectrum, which probes the matter power spectrum
+Pδ(k, z), as well as the background geometry χ(z) if precision
+allows. Here z is the redshift, χ is the comoving distance, k is
+the wavenumber, ℓ is the angular mode, qgal(χ) and qCMB(χ) are
+the lensing efficiency functions for galaxy-lensing and CMB-
+lensing, with the analytical forms:
+qgal(χl) = 3
+2Ωm
+H2
+0
+c2 (1 + zl)
+� ∞
+χl
+n(χs)(χs − χl)χl
+χs
+dχs,
+(2)
+qCMB(χl) = 3
+2Ωm
+H2
+0
+c2 (1 + zl)(χs − χl)χl
+χs
+,
+(3)
+where χs and χl are the comoving distance to the source and
+lens, and the χs in Eq. (3) takes CMB as the source of light
+(z ∼ 1100). We note the spacial curvature Ωk = 0 is assumed
+Article number, page 2 of 15
+
+Yao et al 2022: KiDS shear × Planck lensing and IA removal
+so that the comoving angular diameter distances in Eqs. (2) and
+(3) are replaced with the comoving radial distances. Here n(χ)
+gives the source galaxy distribution as a function of comoving
+distance, and it is connected with the galaxy redshift distribution
+via n(χ) = n(z)dz/dχ. In this work, we only use one redshift bin
+due to the limit of the total S/N on the CMB lensing signal, while
+a tomographic example can be found in Harnois-Déraps et al.
+(2017). In the future with higher S/N, for example, for CMB-S4
+× LSST, tomography can be used to subtract more cosmological
+information.
+The shear-convergence cross-correlation function measured
+in real space is given by the Hankel transformation:
+wGκ(θ) = 1
+2π
+� ∞
+0
+dℓℓCκgalκCMB(ℓ)J2(ℓθ),
+(4)
+where J2(x) is the Bessel function of the first kind and order 2.
+The “G” represents the gravitational lensing shear γG, to be sep-
+arated from the intrinsic alignment γI in the following section.
+Also for the current low S/N reasons, we choose not to in-
+vestigate full cosmological constraints in this work. Instead, we
+perform a matched-filter fitting, with lensing amplitude Alens that
+suits ˆwGκ = AlenswGκ, where ˆwGκ is the measured correlation
+function, and wGκ is the theoretical model.
+2.2. Intrinsic alignment of galaxies
+The observed galaxy shear estimator contains three components:
+gravitational shear, an intrinsic alignment term, and random
+noise, namely, ˆγ = γG + γI + γN. Both the gravitational shear
+and the IA term are related to the underlying matter overdensity
+δ and are associated with the large-scale structure. This means
+that when we cross-correlate the galaxy shape and the CMB con-
+vergence, there will be contributions from both lensing and IA:
+⟨ˆγκ⟩ =
+�
+γGκ
+�
++
+�
+γIκ
+�
+.
+(5)
+Therefore the IA part of the correlation will contaminate the
+measurement and lead to a bias in the lensing amplitude Alens
+or the cosmological parameters when assuming ⟨ˆγκ⟩ =
+�
+γGκ
+�
+.
+The IA-convergence correlation function is linked to the IA-
+convergence power spectrum
+CIκCMB =
+� χCMB
+0
+n(χ)qCMB(χ)
+χ2
+Pδ,γI
+�
+k = ℓ + 1/2
+χ
+, z
+�
+dχ.
+(6)
+Here Pδ,γI is the 3D matter-IA power spectrum. The conventional
+method is to assume an IA model with some nuisance parame-
+ters, which will enter the fitting process. The most widely used
+IA model is the non-linear linear tidal alignment model (Cate-
+lan et al. 2001; Hirata & Seljak 2004; Bridle & King 2007), ex-
+pressed as:
+P��,γI = −AIA(L, z)C1ρm,0
+D(z) Pδ(k; χ),
+(7)
+which is proportional to the non-linear matter power spectrum
+Pδ, suggesting that the IA is caused by the gravitational tidal
+field. AIA is the IA amplitude, which can be redshift(z)- and
+luminosity(L)- dependent (Joachimi et al. 2011). Its redshift evo-
+lution has been measured recently in simulations (Chisari et al.
+2016; Samuroff et al. 2021) and suggestions in observations with
+low significance (Johnston et al. 2019; Yao et al. 2020b; Secco
+et al. 2022; Tonegawa & Okumura 2022). The other related
+quantities include: the mean matter density of the universe at z =
+0, expressed as ρm,0 = ρcritΩm,0; C1 = 5 × 10−14(h2Msun/Mpc−3)
+the empirical amplitude taken from Brown et al. (2002) and the
+normalized linear growth factor D(z). We note that the IA model
+in Eq. (7) can be replaced by more complicated models as in
+Krause et al. (2016); Blazek et al. (2015, 2019); Fortuna et al.
+(2021) for different samples (Yao et al. 2020b; Samuroff et al.
+2021; Zjupa et al. 2020). The self-calibration method can intro-
+duce new observables to constrain IA with additional constrain-
+ing power, and in the future when the signal-to-noise (S/N) al-
+lows, it can be extended to constrain more complicated IA mod-
+els.
+2.3. Self-calibration of intrinsic alignment
+The IA self-calibration (SC) method (Zhang 2010b; Yao et al.
+2017, 2019, 2020a,b) uses the same galaxy sample as both the
+source and the lens, which is different from most galaxy-galaxy
+lensing studies. It introduces two observables: the shape-galaxy
+correlation in the same redshift bin wγg, and a similar correlation
+wγg|S using the pairs where the photo-z of the source galaxy is
+lower than the photo-z of the lens galaxy, namely
+zP
+γ < zP
+g
+(8)
+(this will be denoted as “the SC selection”).
+In this work, we extend our methodology to include the im-
+pact from cosmic magnification (Bartelmann 1995; Bartelmann
+& Schneider 2001; Yang et al. 2017; Liu et al. 2021). Because of
+the existence of magnification, the intrinsic galaxy number den-
+sity field δg is affected by the foreground lensing convergence
+κgal, leading to a lensed galaxy overdensity
+δL
+g = δg + gmagκgal,
+(9)
+where the prefactor writes gmag = 2(α − 1) for a complete and
+flux-limited sample. It accounts for the increase in galaxy num-
+ber density due to lensing-magnified flux (α = −d ln N/d ln F,
+where N(F) denotes the galaxy number N that is brighter than
+the flux limit F) and the decrease of galaxy number density
+due to the lensing-area-enlargement (-2 in gmag). The observed
+shape-galaxy correlation is given by
+�
+ˆγδL
+g
+�
+=
+�
+(γG + γI)(δg + gmagκgal)
+�
+.
+(10)
+The two SC observables can be written as:
+wγgL
+ii (θ) = wGg
+ii (θ) + wIg
+ii (θ) + gmag
+�
+wGκgal
+ii
+(θ) + wIκgal
+ii
+(θ)
+�
+,
+(11)
+wγgL
+ii |S(θ) = wGg
+ii |S(θ) + wIg
+ii |S(θ) + gmag
+�
+wGκgal
+ii
+|S(θ) + wIκgal
+ii
+|S(θ)
+�
+,
+(12)
+where the “|S” denotes the SC selection, and i denotes the i-th
+redshift bin if tomography is applied. The lensing-galaxy wGg
+and the IA-galaxy wIg signal are affected by this SC selection, as
+quantified by the Q parameters:
+QGg
+i (θ) ≡ wGg
+ii |S(θ)
+wGg
+ii (θ)
+,
+(13)
+QIg
+i (θ) ≡ wIg
+ii |S(θ)
+wIg
+ii (θ)
+.
+(14)
+For the lensing signal to exist, the redshift of the source, zγ,
+needs to be greater than the redshift of the lens, zg: zγ > zg.
+Article number, page 3 of 15
+
+A&A proofs: manuscript no. aanda
+0.4
+0.45
+0.5
+0.55
+0.6
+zP
+g
+-0.0004
+-0.0002
+0
+0.0002
+0.0004
+wXg(zP|zP
+g = 0.5,
+)
+X=G, lensing, 
+= 1'
+X=I, IA, 
+= 1'
+X=G, lensing, 
+= 5'
+X=I, IA, 
+= 5'
+X=G, lensing, 
+= 50'
+X=I, IA, 
+= 50'
+Fig. 1. A toy model to illustrate the different redshift dependences for
+the lensing signal and the IA signal, and why the SC selection Eq. (8)
+works. We place many lens galaxies at photo-z zP
+g = 0.5 (the grey dotted
+line), while allowing the photo-z of the source galaxies zP
+γ to change (x-
+axis) to evaluate the corresponding lensing correlation function wGg or
+IA correlation function wIg at different angular separation θ. The true-z
+has a Gaussian scatter of 0.04 (this number is chosen for exhibition, so
+that the lensing/IA signals have comparable maximum/minimum val-
+ues) around the photo-z, for both source galaxies and lens galaxies. As
+the gravitational lensing shear is an optical shape that requires zg < zγ, it
+will have a non-symmetric power around zP
+g, as the positive solid curves
+show. This also demonstrate QGg ≪ 1 according to Eq. (13). As the
+IA shape is a dynamical shape, it does not have requirements on the
+relative redshifts, leading to a symmetric power around zP
+g, as the neg-
+ative dashed curves show. This also demonstrate QIg ∼ 1 according to
+Eq. (14). These relations hold for signals at different angular separa-
+tions (different colors). The different IA models (which could deviate
+from Eq. 7 and AIA = 1 being assumed) will only change the rela-
+tive amplitudes of the negative signals at different scales, but not the
+redshift-dependency around zP
+g. We note at such a redshift range, the
+magnification signal is much smaller than the IA signal.
+The SC photo-z selection zP
+γ < zP
+g largely reduces the lensing
+signal, leading to QGg ≪ 1. The IA signal does not rely on the
+ordering along the line-of-sight, with QIg ∼ 1. The lensing-drop
+QGg and the IA-drop QIg are dependent on the photo-z quality,
+as described in Zhang (2010b); Yao et al. (2017, 2020a,b). If the
+photo-z quality is perfect, the SC selection will result in no lens-
+ing signal so that QGg approaches 0. For incorrect photo-zs, the
+SC selection fails and QGg is ∼ 1. Given a photo-z distribution
+nP(zP) and the true-z distribution n(z), the lensing-drop QGg and
+IA-drop QIg can be theoretically derived, following Yao et al.
+(2020a,b), with more technical details in Appendix A. We also
+present a toy model to visualize how the SC selection works in
+Fig. 1.
+We quantitatively test the terms in Eq. (11), and they gener-
+ally follow |wIκgal| < |wGκgal| ≪ |wIg| < |wGg| for z < 0.9 data,
+therefore in previous analysis (Zhang 2010b; Yao et al. 2020a,b)
+the magnification terms were neglected. For the z ∼ 1 galax-
+ies, however, the magnification term wGκgal quickly approaches
+wIg and becomes a non-negligible source of contamination to
+the SC method. In Fig. 2 we show a theoretical comparison of
+the angular power spectra. We can write the SC selection for the
+magnification term as wGκgal|S = QGκwGκgal. The drop of the sig-
+nal QGκ ∼ QIg ∼ 1 given that these are not z-pair-dependent
+correlations, therefore the magnification signal wGκgal will con-
+10
+100
+1000
+ℓ
+10−10
+10−9
+10−8
+10−7
+C(ℓ)
+CGg, bg,eff = 0.88
+CIg, bg,eff = 0.88, AIA = 0.6
+gmagCGκgal, gmag = −0.3
+Fig. 2. A theoretical comparison between the galaxy-shear CGg(ℓ),
+galaxy-IA CIg(ℓ) and shear-magnification gmagCGκgal(ℓ) angular power
+spectra, with the best-fit of our baseline analysis and the redshift distri-
+bution n(z) from KiDS-1000 0.5 < zP < 1.2 shear catalog. The dashed
+lines represent negative signals. This figure demonstrates that the mag-
+nification contamination is important in the self-calibration method for
+the high-z KiDS source sample.
+taminate the IA signal wIg due to similar behavior, leaving the
+lensing signal wGg unaffected. We note the wIκ term is negligible
+in this work.
+After measuring the galaxy-galaxy lensing observables
+{wγgL, wγgL|S} and the drops of the signals {QGg, QIg} (see
+Eq. (13), (14) and Appendix A for more details), the corre-
+sponding lensing-galaxy correlation wGg, IA-galaxy correlation
+wIg and shear-magnification correlation wGκ can be linearly ob-
+tained:
+wGg
+ii (θ) = QIg
+i (θ)wγgL
+ii (θ) − wγgL
+ii |S(θ)
+QIg
+i (θ) − QGg
+i (θ)
+,
+(15)
+wIg
+ii (θ) + wGκgal
+ii
+(θ) = wγgL
+ii |S(θ) − QGg
+i (θ)wγgL
+ii (θ)
+QIg
+i (θ) − QGg
+i (θ)
+.
+(16)
+In previous work, the IA information was directly extracted
+in wIg. However, as shown in Fig. 2 and Eq. 16, for KiDS the
+subtracted signal suffers from the contamination from a magni-
+fication term wGκ. By constraining the measurements of {wGg,
+wIg+wGκgal, wγκCMB} together, including the covariance, will lead
+to robust constraints on both the lensing amplitude and the nui-
+sance parameters. For the current stage where the S/N for the
+measurements are not very high, we choose to ignore the pos-
+sible scale-dependent features for the effective galaxy bias bg,eff
+and IA amplitude AIA, and assume they are linear and determin-
+istic. The parameters {Alens, AIA, bg,eff, gmag} are connected to
+the observables following:
+wGg(θ) = bg,effwGm
+theory(θ),
+(17)
+wIg(θ) + wGκgal(θ) = bg,effAIAwIm
+theory(θ) + gmagwGκgal
+theory(θ),
+(18)
+wγκCMB(θ) = AlenswGκCMB
+theory (θ) + AIAwIκCMB
+theory(θ),
+(19)
+where “m” stands for matter, which is the case if one sets the ef-
+fective galaxy bias bg,eff = 1. We separate the CMB convergence
+and the galaxy convergence (due to magnification) with κCMB
+Article number, page 4 of 15
+
+Yao et al 2022: KiDS shear × Planck lensing and IA removal
+Table 1. The ΛCDM cosmological parameters adopted in this work,
+corresponding to the best-fit cosmology from Planck Collaboration
+et al. (2020a), and the KiDS-1000 multivariate maximum posterior
+(MAP) results from the two-point correlation functions ξ±, the band
+powers C(ℓ), and the COSEBIs (Complete Orthogonal Sets of E/B-
+Integrals) as in Asgari et al. (2021).
+Survey
+h0
+Ωbh2
+Ωch2
+ns
+σ8
+Planck
+0.673
+0.022
+0.120
+0.966
+0.812
+KiDS ξ±
+0.711
+0.023
+0.088
+0.928
+0.895
+KiDS C(ℓ)
+0.704
+0.022
+0.132
+0.999
+0.723
+KiDS COSEBI
+0.727
+0.023
+0.105
+0.949
+0.772
+and κgal. On the LHS of Eq. (17), (18) and (19) are the measure-
+ments, while on the RHS the correlations w(θ) are the theoreti-
+cal predictions assuming Planck cosmology (Planck Collabora-
+tion et al. 2020a), see Table 1. We note the Q values being used
+to obtain the LHS are also cosmology dependent, however, the
+sensitivity is weak as the cosmological part is mostly canceled
+when taking the ratio in Eq. (13) and (14). We tested if the fidu-
+cial cosmology is changed to any of the KiDS-1000 cosmolo-
+gies in Table 1, the Qs will change by ∼ 1%, similar to Yao et al.
+(2020b), and the resulting changes to the fitting parameters {AIA,
+bg,eff, gmag, Alens} are negligible. However, considering the RHS,
+those four fitting parameters are sensitive to the fiducial cosmol-
+ogy used to produce the wtheory values when magnification exists,
+which differs from previous analysis (Yao et al. 2020b). The the-
+oretical predictions wtheory are calculated with ccl1 (Chisari et al.
+2019) and camb2 (Lewis et al. 2000). The effective galaxy bias
+bg,eff in this work is used to separate from the true galaxy bias of
+this sample, as we will discuss later it can absorb several sources
+of systematics.
+The theoretical prediction of wGκCMB
+theory (θ) is given in Eq. (4),
+and wIκgal
+theory(θ) is obtained similarly with the Hankel transform
+from its power spectrum as in Eq. (6). The wGm
+theory, wIm
+theory and
+wGκgal
+theory terms are the Hankel transform from the following angu-
+lar power spectra:
+CGm(ℓ) =
+� zmax
+zmin
+qgal(χ)n(χ)
+χ2
+Pδ
+�
+k = ℓ + 1/2
+χ
+, z
+�
+dχ,
+(20)
+CIm(ℓ) =
+� zmax
+zmin
+n(χ)n(χ)
+χ2
+Pδ,γI
+�
+k = ℓ + 1/2
+χ
+, z
+�
+dχ,
+(21)
+CGκgal(ℓ) =
+� zmax
+zmin
+qgal(χ)qgal(χ)
+χ2
+Pδ
+�
+k = ℓ + 1/2
+χ
+, z
+�
+dχ.
+(22)
+As discussed in previous work (Yao et al. 2020b), by in-
+cluding the effective galaxy bias bg,eff, we can obtain an unbi-
+ased estimation of AIA. This information will be propagated into
+Eq. (19) to break the degeneracy between AIA and Alens. In this
+work, we further extend the fitting to include the impact from
+magnification with the nuisance parameter gmag. We will show
+later that an unbiased CMB lensing amplitude Alens can be ob-
+tained from the simultaneous fitting of Eq. (17), (18) and (19).
+3. Data
+In this section, we introduce the data we use for the
+�
+γκCMB�
+cross-correlation study. Additionally, we use mock KiDS data,
+1 Core Cosmology Library, https://github.com/LSSTDESC/CCL
+2 Code for Anisotropies in the Microwave Background, https://
+camb.info/
+based on the MICE2 simulation (see van den Busch et al. (2020)
+for details) to quantify the potential bias in the SC method due
+to magnification, photo-z modeling, and the boost factor.
+3.1. KiDS-1000 shear catalog
+We use the fourth data release of the Kilo-Degree Survey that
+covers 1006 deg2, known as KiDS-1000 (Kuijken et al. 2019). It
+has images from four optical bands ugri and five near-infrared
+bands ZYJHKs. The observed galaxies can reach a primary
+r−band median limiting 5σ point source magnitude at ∼ 25. The
+shear catalog (Giblin et al. 2021) contains ∼ 21 M galaxies and
+is divided into five tomographic bins in the range 0.1 < zB < 1.2
+based on the bpz (Benitez 2000) algorithm. The ellipticity dis-
+persion σϵ is ∼ 0.27 per component, and the shear multiplicative
+bias is generally consistent with 0.
+The KiDS data are processed by theli (Erben et al. 2013)
+and Astro-WISE (Begeman et al. 2013; de Jong et al. 2015).
+Shears are measured using lensfit (Miller et al. 2013), and pho-
+tometric redshifts are obtained from PSF-matched photometry
+and calibrated using external overlapping spectroscopic surveys
+(Hildebrandt et al. 2021).
+The application of SC requires not only an accurate redshift
+distribution n(z), but also relatively accurate photo-z for each
+galaxy, serving for the SC selection (Eq. 8). We discussed in pre-
+vious work (Yao et al. 2020a) that the quality of photo-z is very
+important for the lensing-IA separation. Therefore in this work,
+we choose to combine the three high-z bins, namely bin 3+4+5
+in KiDS-1000 data, as a large bin so that the photo-z error for
+an individual galaxy is relatively small compared to the total
+bin width. The photo-z and the SOM-calibrated redshift distri-
+butions are shown in Fig. 3. We choose to use the high-z bins be-
+cause the CMB lensing efficiency Eq. (3) peaks at z ∼ 1 to 2 (see
+lower panel of Fig. 3), while the S/N for the cross-correlation is
+very low for the two low-z bins of KiDS-1000.
+To account for the selection functions for the shape of the
+footprint (Mandelbaum et al. 2006) of the overlapped region and
+the varying galaxy number density due to observation (Johnston
+et al. 2021; Rezaie et al. 2020), we divide the region into 200
+sub-regions with a resolution of Healpix Nside = 512 (∼ 50
+arcmin2 per pixel), and generate random points with 20 times
+the number of galaxies of the KiDS-1000 shear catalog in each
+sub-region. The pixels within the same sub-region are assigned
+the same galaxy numbers. This random catalog is used for the
+SC-related galaxy-galaxy lensing calculation, while its potential
+defects will not extend to cross-correlations.
+3.2. Planck legacy lensing map
+We use the CMB lensing map κ(θ) from the Planck data release
+(Planck Collaboration et al. 2020c). The CMB lensing map is
+reconstructed with the quadratic estimator with the minimum-
+variance method combining the temperature map and the polar-
+ization map, after foreground removal with the SMICA method
+(Planck Collaboration et al. 2020a). It covers fsky = 0.671 of the
+whole sky with the maximum multiple ℓ = 4096.
+In this work we combine the footprint from the Planck lens-
+ing map and the mask of the KiDS-1000 shear catalog, leading
+to an overlapped region of ∼ 829 deg2. We include the Planck
+Wiener filter (Planck Collaboration et al. 2020c)
+ˆκWF
+ℓm =
+Cφφ,fid
+ℓ
+Cφφ,fid
+ℓ
++ Nφφ
+ℓ
+ˆκMV
+ℓm
+(23)
+Article number, page 5 of 15
+
+A&A proofs: manuscript no. aanda
+0
+0.5
+1
+1.5
+2
+2.5
+n(z)
+n(z)
+nP(zP)
+0
+0.5
+1
+1.5
+2
+z
+0
+0.2
+0.4
+0.6
+0.8
+1
+lensing efficiency
+galaxy lensing
+CMB lensing
+Fig. 3. The photo-z distribution and the SOM-reconstructed redshift dis-
+tribution of the combined galaxy sample in this work. The correspond-
+ing galaxy lensing efficiency Eq. (2) and its comparison with CMB lens-
+ing efficiency Eq. (3) are shown in the lower panel.
+to strengthen the CMB lensing signal at large scales, which will
+also lead to a suppression of the power spectrum at small scales,
+where the noise dominates (Dong et al. 2021). The Wiener filter
+is used both in the CMB lensing κ map and in the theoretical pre-
+dictions of Eq. (1) to prevent potential bias. After the application
+of the Wiener filter, we use Healpy3 (Górski et al. 2005; Zonca
+et al. 2019) to convert the κℓm to the desired κ-map, and rotate
+from the galactic coordinates of Planck to the J2000 coordinates
+of KiDS with Astropy (Astropy Collaboration et al. 2013). The
+two-point correlation functions are calculated with TreeCorr 4
+(Jarvis et al. 2004).
+3.3. MICE2 mock catalog
+Additionally, we use the MICE2 simulation gold samples (van
+den Busch et al. 2020; Fosalba et al. 2015), which highly
+mimic the KiDS-1000 shear catalog galaxies, to validate our
+SC method, concerning cosmic magnification and photo-z PDF
+model bias. MICE2 uses a simulation box width of 3.1 h−1Gpc,
+particle mass resolution of 2.9 × 1010 h−1M⊙, and a total particle
+number of ∼ 6.9 × 1010. The fiducial cosmology is flat ΛCDM
+with Ωm = 0.25, σ8 = 0.8, Ωb = 0.044, ΩΛ = 0.75 and h = 0.7.
+The halos are identified with Friends-of-Friends as in Crocce
+et al. (2015). The galaxies are populated within the halos with
+a mixture of halo abundance matching (HAM) and halo occupa-
+tion distribution (HOD) up to z ∼ 1.4 (Carretero et al. 2015).
+We note that in the MICE2 simulation that we use for the
+KiDS samples, intrinsic alignment is not yet included in the
+galaxy shapes (while an IA-included version can be found in
+Hoffmann et al. (2022), but for DES). So that we aim to get
+AIA = 0 to validate the SC method, while considering system-
+atics from cosmic magnification and photo-z model bias, in ad-
+3 https://github.com/healpy/healpy
+4 https://github.com/rmjarvis/TreeCorr
+101
+102
+103
+104
+ℓ
+0.2
+0.4
+0.6
+0.8
+1.0
+Q(ℓ) & Q(θ)
+QGg(ℓ)
+QIg(ℓ)
+100
+101
+102
+θ [arcmin]
+QGg(θ)
+QIg(θ)
+Fig. 4. The lensing-drop QGg and the IA-drop QIg as a function of ℓ and
+θ by applying the SC selection Eq. (8), see Eq. (13) and (14). These val-
+ues are adopted to obtain the separation of wGg and wIg + wGκgal, follow-
+ing Eq. (15) and (16). The left panel shows the calculation from power
+spectra and the right panel from correlation functions. The right panel
+is used to transfer {wγg, wγg|S } to {wGg, wIg} later in Fig. 6.
+dition to what has been addressed in Yao et al. (2020b). We use
+the galaxy positions (ra, dec), the two noiseless shear compo-
+nents (γ1, γ2), and BPZ-measured photo-z zB to calculate the
+SC correlations as in Eq. (11) and (12). We test the signal drop
+Qs of Eq. (13) and (14) with our photo-z PDF model and with
+true-z from simulation (van den Busch et al. 2020). We compare
+the results using MICE2 gold samples (which highly mimic the
+KiDS-1000 shear catalog galaxies) with magnification (Eq. 9)
+and without magnification. For the MICE2 galaxies with mag-
+nification, we tested how it will bias the IA measurement, and
+proved that when the magnification effect is also included in the
+model, IA can be measured in an unbiased way. The validations
+will be shown later in our results with some details in Appendix
+A.
+4. Measurements
+We show the estimation of the signal-drops for lensing and IA
+due to the SC selection (as in Eqs. 13 and 14), i.e. the lensing-
+drop QGg and the IA-drop QIg in Fig. 4. They are responsible for
+the lensing-IA separation later in Fig. 6, following Eq. (15) and
+(16). We follow the processes in Yao et al. (2020a,b) and adopt
+a bi-Gaussian photo-z probability distribution function (PDF)
+model with a secondary peak representing the photo-z outlier
+problem. We require the PDF model to have the same mean-z as
+in Fig. 3, while closest describing the projection from nP(zP) to
+n(z). We will also show for the first time how the assumed photo-
+z PDF model can affect the results in the next section, with more
+details shown in Appendix A.
+We calculate the SC correlation function estimator,
+wγg(θ) = B(θ)
+�
+ED wjγ+
+j
+(1 + ¯m) �
+ED wj
+−
+�
+ER wjγ+
+j
+(1 + ¯m) �
+ER wj
+,
+(24)
+to obtain the measurements of wγg and wγg|S from the tangential
+shear of each galaxy γ+
+j . Here we sum over the ellipticity-density
+pairs (�
+ED) and the ellipticity-random pairs (�
+ER) in an annulus
+centered on θ, where the shear weight wj of the j-th galaxy and
+the average multiplicative bias ¯m are accounted for. The estima-
+tor is binned in angular θ space, with 9 logarithmic bins from 0.5
+Article number, page 6 of 15
+
+Yao et al 2022: KiDS shear × Planck lensing and IA removal
+100
+101
+102
+θ [arcmin]
+1.0
+1.1
+1.2
+1.3
+1.4
+B
+Boost
+BoostS
+Fig. 5. The boost factors for wγgL and wγgL|S are shown in blue and
+orange, respectively. The overlapping lines suggest the two signals are
+affected by the boost factor in almost the same way. We show the boost
+factor is significant at small scales for the SC observables.
+to 300 arcmin. We use the averaged multiplicative bias ¯m from
+averaging over the three z-bins, weighted by the effective galaxy
+number density. This gives ¯m = −0.0036.
+We account for the impact of the boost factor (Mandelbaum
+et al. 2005; Singh et al. 2017b; Joachimi et al. 2021), which is B
+in Eq. (24). It is defined as
+B(θ) =
+�
+ED wj
+�
+RD wj
+,
+(25)
+which is used to quantify the small-scale bias due to the clus-
+tering of lens galaxies and source galaxies (Bernardeau 1998;
+Hamana et al. 2002; Yu et al. 2015). We show the measurements
+of the boost factor for wγgL and wγgL|S as in Eq. (11) and (12)
+in Fig. 5. The fact that the boost factors for wγgL and wγgL|S are
+identical suggests this bias can be absorbed by the galaxy bias
+bg,eff parameter if magnification is absent (gmag = 0), leading to
+an unbiased AIA and Alens. The impact from the boost factor can
+potentially break the linear galaxy bias assumption, but later in
+Fig.6 we show the linear assumption is fine. The impact of the
+boost factor and magnification existing together will be shown
+later.
+In Fig. 6 we show the SC measurements. In the left panel, the
+measured shape-galaxy correlations wγgL are shown in blue: (1)
+the boost factor ignored case (B = 1) is shown as blue crosses,
+while (2) the boost factor corrected case is shown as blue up-
+triangles. With the SC selection Eq. (8), requiring zP
+γ < zP
+g for
+each galaxy pair, the lensing component will drop to QGg ∼ 0.3
+and the IA component will drop to QIg ∼ 0.85 (for more details
+on QGg and QIg, see Fig. 4 and Appendix A). Therefore, the se-
+lected correlations wγg|S will drop to the orange down-triangles.
+Similarly, the boost factor ignored case is shown as crosses.
+The separated lensing-galaxy signal wGg and IA-galaxy sig-
+nal wIg (which is contaminated by magnification-shear signal
+gmagwGκ) are shown in the right panel of Fig. 6. The blue and or-
+ange curves are the theoretical predictions with the best-fit {AIA,
+bg,eff, gmag}. For the fitting, we cut off the shaded regions at both
+large scales and small scales. The small scale cut at θ = 1 ar-
+cmin is based on the linear galaxy bias assumption, as including
+the θ < 1 arcmin data will make the fitting significantly worse
+(increasing the fitting χ2 from 7.5 to 50, with degree-of-freedom
+changed from 8 to 10). We note this scale cut could include the
+impacts from the 3D non-linear galaxy bias (Fong & Han 2021)
+and other small-scale effects such as massive neutrinos or baryon
+feedback in the matter power spectrum (Hildebrandt et al. 2017;
+Asgari et al. 2021). We emphasize that these systematics will be
+absorbed by the effective galaxy bias parameter bg,eff —- with-
+out breaking the scale-independent bias assumption —- so that
+the IA amplitude will not be affected. As discussed previously
+in Yao et al. (2020a,b), the SC method requires significant sep-
+aration between wγgL and wγgL|S to accurately get wGg and wIg.
+Therefore, we introduce a large-scale cut at θ = 20 arcmin due
+to insufficient separation for the left panel of Fig. 6.
+Similarly, we measure the ⟨γκ⟩ correlation with the estimator
+wγκ(θ) =
+�
+i j wjγ+
+j κi
+(1 + ¯m) �
+i j wj
+,
+(26)
+where κi is the CMB lensing convergence in the i-th pixel of
+the pixelized map, taking the pixel center for its (ra, dec) co-
+ordinates, with nside = 2048 in Healpy. The measured wγκ are
+shown in Fig. 7. The tangential shear is shown as blue dots. We
+also show the measurements with randomly shuffling galaxy po-
+sitions and the shear in red crosses as a null test. We test the
+45 deg rotated cross shear for both the above cases and they are
+consistent with zero. The theoretical prediction with the best-fit
+Alens and AIA are shown as the green curve. If one assumes there
+is no IA in the measurements and uses AIA = 0, the theoretical
+values for the pure lensing signal are shown in orange.
+Note in Fig. 7, because we use the Wiener-filtered κ map
+from Planck, both the wγκ measurements and the theoretical pre-
+dictions are suppressed at small scales. The Wiener filter can
+significantly reduce the impact of the noise of the Planck lens-
+ing map and improve the S/N of the measurements.
+Together with the measurements in Figs. 6 and 7, we obtain
+observables of this work, which are the LHS terms of Eqs. (17),
+(18) and (19). We use Jackknife resampling to obtain the co-
+variance. 200 Jackknife regions are used, which is much larger
+than the length of the data vector (12), based on the analy-
+sis of Mandelbaum et al. (2006); Hartlap et al. (2007). The
+Jackknife regions are separated using the K-means algorithm
+kmeans_radec5. The normalized covariance matrix is shown in
+Fig. 8. We find strong anti-correlation between wGg and wIg as
+expected (Yao et al. 2020b). Note here in Fig. 8, wIg means the
+separated signal in the RHS of Eq. (16), including both the IA
+part and the contamination from magnification. There is no sig-
+nificant correlation between wγκ and the other two observables.
+This covariance will be used in the Monte Carlo Markov Chain
+(MCMC) to find the best-fit parameters of {AIA, bg,eff, gmag,
+Alens}, while all the other cosmological parameters are fixed to
+Planck as in Table 1.
+5. Results
+5.1. Validation with MICE2
+In this subsection, we apply the IA self-calibration to the MICE2
+mock catalog to test the impact of the systematics and validate
+the recovery of the IA signal. The processes of the mock data are
+identical to the descriptions in Sec. 4, but only focusing on the
+self-calibration part. The measurements are similar to Fig. 6 so
+5 https://github.com/esheldon/kmeansradec
+Article number, page 7 of 15
+
+A&A proofs: manuscript no. aanda
+1
+3
+10
+30
+θ [arcmin]
+-1
+0
+1
+2
+3
+4
+w(θ) × 104
+wγgL
+wγgL|S
+1
+3
+10
+30
+θ [arcmin]
+-1
+0
+1
+2
+3
+4
+wIg
+gmagwGκgal
+tot
+wGg
+wIg (+gmagwGκgal)
+Fig. 6. The measurements of SC. The left panel shows the measurement of the two introduced observables wγgL and the one with the SC selection
+wγgL|S, while the corresponding 45-deg rotation test is consistent with 0 for both measurements. The significant separation of the two signals shows
+that SC is applicable. The right panel shows the separated lensing signal wGg and wIg, where the latter is contaminated by the magnification signal
+as shown in Eq. (16). The up- and down-triangles are the results that take the boost factor (Fig. 5) into consideration, while the crosses are the
+results that ignore this correction, setting B = 1. The curves are the theoretical value with the best-fit {AIA, bg,eff, gmag} of this work. The blue curve
+represents the separated lensing signal as in Eq. (17). The orange curve represents the total contribution of IA and magnification as in Eq. (18).
+3
+10
+30
+100
+300
+θ [arcmin]
+-1
+0
+1
+2
+3
+wκγ(θ) × 106
+wGκ lensing
+w(G+I)κ
+best−fit
+⟨κγt⟩
+⟨κγshuffle⟩
+Fig. 7. The measurement of the cross-correlation between Planck con-
+vergence κ and KiDS-1000 shear γ, based on Eq. (19). The blue dots are
+the measurements using tangential shear, with the green curve showing
+the best-fit considering both lensing and IA, while the orange curve
+shows only the lensing-lensing component. The red crosses show the
+null test by randomly shuffling the shear galaxies. The 45-deg rotation
+tests for both the blue dots and the red dots are consistent with 0. The
+differently shaded regions correspond to our angular scale cuts at 2, 20
+(default), and 40 arcmin.
+we choose to skip them. We perform the MCMC calculation us-
+ing emcee (Foreman-Mackey et al. 2013). We consider flat priors
+in −5 < AIA < 5, 0 < bg,eff < 2 and −3 < gmag < 3.
+5.1.1. Impact from magnification
+We show how the magnification signal affects the original SC
+method (Zhang 2010b; Yao et al. 2020a,b) and the correction
+introduced in this work, focusing on the gmag − AIA space.
+wGg
+wIg
+wγκ
+wGg
+wIg
+wγκ
+correlation coefficient
+−0.75
+−0.50
+−0.25
+0.00
+0.25
+0.50
+0.75
+1.00
+Fig. 8. The normalized covariance matrix (i.e. the correlation coeffi-
+cient) used in this work. There exists a strong anti-correlation between
+the lensing-galaxy correlation wGg and the IA-galaxy correlation wIg
+(including the contamination from wGκgal) as we found in previous work.
+The covariance of the 12 data points is calculated from Jackknife re-
+sampling with 200 regions. We note the IA information is passed from
+1 < θ < 20 [arcmin] for wIg to 20 < θ < 300 [arcmin] for wγκ with the
+scale-independent AIA assumption.
+In Fig. 9, we show that if magnification is not included in the
+modeling, gmag is therefore not constrained. The existing mag-
+nification signal will be treated as the IA signal, leading to a
+non-vanishing AIA ∼ 0.3, which significantly deviates from the
+MICE2 input AIA = 0. When the magnification model is in-
+cluded in the analysis, AIA is then consistent with 0. This demon-
+strates the importance of including the magnification model in
+the SC analysis with high-z data. The results are also summa-
+Article number, page 8 of 15
+
+Yao et al 2022: KiDS shear × Planck lensing and IA removal
+MICE IA
+MICE IA+mag
+−0.30
+−0.15
+0.00
+0.15
+0.30
+AIA
+−0.45
+−0.30
+−0.15
+0.00
+gmag
+Fig. 9. The impact of the magnification signal on the IA measurement
+in MICE2. The green and blue contours are with and without magni-
+fication models, respectively. If the magnification model is used in the
+fitting, as in green, the IA amplitude AIA is consistent with 0, which is
+the MICE2 input.
+rized later in the comparisons in Fig. 11 for MICE2, and in
+Fig. 14 for KiDS data.
+We note that in the green case of Fig. 9 that considered both
+IA and magnification, gmag and AIA strongly degenerate. There-
+fore the constraining power in AIA has a significant loss com-
+pared with the blue case, which ignores magnification. This de-
+generacy can be broken in the future with higher S/N in the ob-
+servables. This is because the shape of wIg and wGκ are different
+at small scales for correlation functions as in Fig. 6, and on large
+scales for power spectra as in Fig. 2. The IA-model-dependency
+will be discussed later with other results. Based on the above
+analysis, we conclude it is important to include magnification
+modeling for SC when using high-z data.
+5.1.2. Impact from modeling p(z|zP)
+Since the SC selection Eq. (8) plays an important role in the
+lensing-IA separation process, it is crucial to understand how the
+following aspects affect SC: (1) the quality of the photo-z zP, (2)
+the true redshift distribution n(z), and (3) the link between them
+p(z|zP). The quality of photo-z and the reconstruction of n(z) has
+been studied thoroughly for KiDS data (Kuijken et al. 2019; van
+den Busch et al. 2022; Hildebrandt et al. 2021; van den Busch
+et al. 2020), we, therefore, trust these results and leave the al-
+ternative studies for SC to future works. The uncalibrated PDF
+that projects zP → z, on the other hand, has some known prob-
+lems, for example when Probability Integral Transform (PIT) is
+applied (Newman & Gruen 2022; Hasan et al. 2022).
+In this work, we use a bi-Gaussian PDF model to project the
+photo-z distribution nP(zP) to the SOM redshift distribution n(z),
+which are previously shown in Fig. 3. This modeling ignores the
+potential differences for galaxies in the same z-bin (Peng et al.
+2022; Xu et al. 2023). However, this is an alternative process,
+MICE Qsim+mag
+MICE Qmodel
+MICE Qmodel+mag
+−0.4
+−0.2
+0.0
+0.2
+AIA
+−0.4
+0.0
+0.4
+gmag
+Fig. 10. The impact from photo-z PDF model bias. The blue case uses
+photo-z from the BPZ algorithm and true-z for each galaxy to calcu-
+late Eq. A.7 and the resulting QGg and QIg, which are the “sim” cases
+in Fig. A.1. This AIA is consistent with 0, which is the MICE2 input.
+The green case uses the bi-Gaussian photo-z model for the calculation,
+which are the “model” cases in Fig. A.1, while ignoring the magnifica-
+tion contribution. This lead to unconstrained gmag and biased AIA. In the
+red case, which also uses the photo-z model, but includes the magnifi-
+cation model, the resulting AIA is still consistent with 0, with the bias
+from photo-z model error absorbed by gmag.
+considering the PDF problem for a single galaxy. This analytical
+approach is also much faster in calculation than using different
+PDFs for different galaxies.
+We use Fig. 10 to demonstrate how large this photo-z PDF
+modeling bias is with different approaches. We use MICE2 sim-
+ulation with galaxy number density affected by magnification.
+When the SC calculation uses true-z to calculate the signal drops
+QGg and QIg, and the magnification model is also considered, we
+find the resulting AIA is consistent with 0, which is the MICE2
+input. The scatter on AIA is ∼ 0.1, thanks to the noiseless shapes
+in MICE2. If the Qs are calculated with the assumed photo-z
+PDF model, without including the magnification model, then
+AIA will be biased towards the negative direction. We proved
+with our fiducial analysis that, even if there exists a bias in QGg
+due to the assumed photo-z model, as long as the magnification
+model is used, this bias will be absorbed by the gmag parameter,
+so that the IA amplitude AIA is unbiased (consistent with 0 in
+the MICE2 case). The results are also shown later in the com-
+parisons in Fig. 11 for MICE2, and in Fig. 14 for KiDS data.
+We note that the bias due to photo-z modeling is not an es-
+sential problem for SC. In the future, if the photo-z outlier prob-
+lem (or the redshift-color degeneracy problem) can be under-
+stood better, then a more reliable photo-z model can be used for
+our SC study. Alternatively, if the photo-z algorithms can give
+unbiased PDFs for each galaxy, this problem can also be directly
+solved.
+Article number, page 9 of 15
+
+A&A proofs: manuscript no. aanda
+-0.4
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+AIA
+MICE(mag), Q(sim), w/o mag
+MICE(mag), Q(sim), w/ mag
+MICE(mag), Q(model), w/o mag
+MICE(mag), Q(model), w/ mag
+MICE(nomag), Q(model), w/ mag
+Fig. 11. We validate our SC method with MICE2 simulation, which
+does not have IA implemented; therefore, AIA = 0 is expected. The re-
+sults are shown in green, with “MICE(mag)” meaning magnification is
+included in the MICE simulation, while “MICE(nomag)” means mag-
+nification is not included, “Q(sim)” and “Q(model)” mean if the signal
+drops Q values are calculated from true-z from simulation or photo-z
+PDF model, and “w/o mag” and “w/ mag” show if the case includes
+magnification model in the fitting process. The upper two data are the
+results from Fig. 9, showing the impact of the modeling magnification.
+The 2nd to the 4th data are the results from Fig. 10, showing the impact
+of Q calculation using different PDFs. The 4th data correspond to our
+fiducial analysis later for KiDS data, with potential bias ∆AIA < 0.1.
+The bottom data is a reference case assuming no magnification effects
+in the data, corresponding to our previous work Yao et al. (2020b,a).
+5.2. Inference on real data
+With the above demonstration that our treatments for magnifica-
+tion and photo-z PDF are appropriate, and the resulting bias in
+AIA is very small (∆AIA < 0.1 and < 1σ as shown in Fig. 11),
+we move on to apply SC to KiDS data and its cross-correlation
+with Planck lensing. We show the analysis of the following three
+situations:
+(1) The case “ignore IA”. We only use the observed wγκ, while
+only including Alens in the fit and ignoring the contamination by
+IA (by setting AIA = 0).
+(2) The case “IA w/o SC”. We only use the observed wγκ, but
+consider both Alens and AIA following Eq. (19).
+(3) The case “with SC”. We use both wγκ in Fig. 7 and the SC
+correlations in Fig. 6. Both the CMB lensing amplitude Alens and
+the nuisance parameters {AIA, bg,eff, gmag} will be used in the
+analysis, following Eqs. (17), (18) and (19).
+The results are shown in Fig. 12. We use flat priors in 0 <
+Alens < 2, −5 < AIA < 5, and for the IA self-calibration nuisance
+parameters we use 0 < bg,eff < 4, −5 < gmag < 5.
+For case (1) “ignore IA”, shown in blue, AIA is unconstrained in
+the fitting, giving the best-fit Alens = 0.74+0.18
+−0.17.
+For case (2) “IA w/o SC”, when we consider the existence of IA
+and apply the IA model as in Eq. (7), but do not use the mea-
+surements from SC (Fig. 6 and Eq. 17, 18), there will be a strong
+degeneracy between Alens and AIA, as shown in orange. There is
+a significant loss of constraining power in the lensing amplitude,
+with the best-fit Alens = 0.79+0.43
+−0.46 and AIA = 0.47+3.11
+−3.47.
+For case (3) “with SC”, the introduced measurements of wGg
+and wIg can not only break the degeneracy between Alens and AIA
+(see Eq. 17, 18 and 19), but also bring more constraining power
+to AIA, so that the best-fit of Alens will not only be unbiased
+(according to the validation using simulation) but also has sig-
+−2
+−1
+0
+1
+2
+3
+4
+AIA
+0.2
+0.6
+1.0
+1.4
+1.8
+Alens
+0.2
+0.6
+1.0
+1.4
+1.8
+Alens
+ignore IA
+IA w/o SC
+with SC
+Fig. 12. The constraints on lensing amplitude Alens and the IA ampli-
+tude AIA, with three different methods: assume there is no IA in the
+measured wκγ (blue), consider the impact of IA with conventional IA
+model but do not use SC (orange), use SC to subtract IA information
+and constrain together with the CMB lensing cross-correlation (green).
+When IA is ignored, AIA is unconstrained. The similar height and width
+of Alens PDFs between blue and green prove that by including SC, the
+AIA − Alens degeneracy can be efficiently broken so that the constraining
+power loss in Alens is very small.
+nificantly improved constraining power. The best-fit values are
+Alens = 0.84+0.22
+−0.22, AIA = 0.60+1.03
+−1.03, bg,eff = 0.88+0.06
+−0.06, and gmag =
+−0.30+1.60
+−1.62. In Fig. 12 we only show AIA and Alens, which are the
+focus of this work, while bg,eff and gmag are only related with
+the SC observables but not CMB lensing. Also as discussed in
+Yao et al. (2020b), the existence of the effective galaxy bias bg,eff
+can also absorb some systematics (so it could be a biased bias),
+leaving the constraint on AIA unbiased (as shown in Fig. 11). For
+example, we tested if magnification is absent, the effect of boost
+factor will be purely absorbed by bg,eff, giving unbiased AIA and
+Alens. The effective galaxy bias could also absorb the differences
+in the assumed fiducial cosmology, with bg,eff ∼ 1.24 with KiDS
+COSEBI cosmology, for example. The redshift distribution n(z)
+can differ slightly with/without accounting for the lensing weight
+(considering the lensing/clustering part in the galaxy-shape cor-
+relation), with a ∼ 0.024 difference in the mean-z, which can lead
+to ∼ 8% difference in the theoretical lensing signal and ∼ 2% dif-
+ference in the theoretical IA signal. Other unaddressed sources
+of systematics such as baryonic feedback and massive neutrinos
+could have similar effects. We can also see from the validation
+using MICE data that although the resulting bg,eff is lower than
+the expectation, the AIA result is unbiased. The gmag result also
+resides in a reasonable range, considering the KiDS i-band mag-
+nitude (Kuijken et al. 2019) and comparing it with Duncan et al.
+(2014). The above three cases of IA treatments are also summa-
+rized later in Fig. 13 and 14 together with more tests and other
+works.
+The corresponding best-fit curves are shown in Fig. 2 and 6
+with AIA = 0.60+1.03
+−1.03, bg,eff = 0.88+0.06
+−0.06, and gmag = −0.30+1.60
+−1.62
+. Even though the impact of magnification is comparable to the
+IA signal, we can see in both the angular power spectrum and
+correlation function that the shapes of IA and magnification are
+different. For example, as shown in Fig. 6, the tidal alignment
+model wIg and magnification gmagwGκ are comparable at large
+Article number, page 10 of 15
+
+Yao et al 2022: KiDS shear × Planck lensing and IA removal
+0.5
+1
+1.5
+2
+Alens
+with SC (Planck)
+ignore IA
+IA w/o SC
+wγκCMB scale > 40 arcmin
+wγκCMB scale > 2 arcmin
+SC ignore mag
+SC ignore boost
+with SC (KiDS COSEBI)
+Hand+ 2015 Planck
+Hand+ 2015 WMAP
+Liu+ 2015
+Kirk+ 2016 SPT
+Kirk+ 2016 SPT fix-IA
+Kirk+ 2016 Planck
+Harnois-Deraps+ 2016, CFHT
+Harnois-Deraps+ 2016, RCSLenS
+Singh+ 2017
+Harnois-Deraps+ 2017, KiDS
+Harnois-Deraps+ 2017, Planck
+Omori+ 2018 fix-IA
+Namikawa+ 2019
+Marques+ 2020
+Robertson+ 2021, Planck
+Robertson+ 2021, KiDS
+baseline
+comparisons
+previous w/o IA
+previous IA prior
+Fig. 13. The comparisons of the constraints on Alens with previous mea-
+surements. Our baseline analysis “with SC” is consistent with 1. We
+also show some cases where IA is ignored in the analysis and if IA is
+considered but the AIA − Alens degeneracy is not broken with SC. These
+main results in blue are similar to Fig. 12. We show tests with differ-
+ent scale cuts and different treatments to magnification, boost factor,
+and different (KiDS) fiducial cosmology in red. We compare with other
+works, separated into ignoring IA (orange) and assuming a strong prior
+of IA (green). We note that for different work, the different fiducial cos-
+mology (the “Planck”, “WMAP”, “KiDS” labels on the y-axis) can lead
+to ∼ 10% difference in Alens.
+scale, while different at small scale. Therefore, in principle, the
+degeneracy between IA and magnification can be broken for fu-
+ture data with higher S/N so that the shape/slope information of
+the observables can be used. The current degeneracy is due to
+the low S/N so that the amplitudes of AIA and gmag degenerate.
+Furthermore, if a more complicated IA model is used, for ex-
+ample, as in Blazek et al. (2019); Abbott et al. (2022), the small-
+scale IA will be different. Based on the study of Shi et al. (2021),
+for a wide range of stellar mass, the small-scale IA should have
+a higher amplitude (either a direct raise in the amplitude or a
+“drop-raise” pattern as we go to smaller scales) than the current
+model so that the IA-magnification degeneracy can be broken
+further. The appropriate IA model will require studies in many
+aspects, and with higher S/N in the measurements, thus we leave
+this topic for future work.
+We investigate how different choices can change our results.
+We first compare the different scale cuts for wκγ. Besides the
+baseline analysis of Alens = 0.84+0.22
+−0.22 with θ > 20 arcmin, two
+more tests are made with a larger scale cut of θ > 40 arcmin and
+a smaller scale cut of θ > 2 arcmin, as shown in Fig. 7, which
+give us Alens = 0.97+0.25
+−0.25 and Alens = 0.77+0.21
+−0.22, respectively. The
+comparisons are shown in Fig. 13. The large-scale lensing am-
+plitude is higher than the small-scale one, which agrees with the
+finding in Planck Collaboration et al. (2020c) and other cross-
+correlation work (Sun et al. 2022). In this work, we only re-
+port this large-scale v.s. small scale difference. However, the cur-
+rent S/N of CMB convergence - galaxy shear correlation and the
+model assumptions do not allow us to investigate further on this
+topic.
+-3
+-2
+-1
+0
+1
+2
+3
+4
+AIA
+with SC (Planck)
+IA w/o SC
+SC ignore mag
+SC ignore boost
+with SC (KiDS COSEBI)
+Robertson+ 2021 prior
+Asgari+ 2021 C(ℓ)
+Asgari+ 2021 COSEBI
+Asgari+ 2021 ξ±
+DES Y3 Secco+
+HSC Y1 ξ± Hamana+
+HSC Y1 C(ℓ) Hikage+
+this work
+KiDS
+others
+Fig. 14. The comparisons of the constraints on AIA. We show the results
+of this work in blue, which contains our fiducial analysis with SC ap-
+plied, and the comparisons of (1) without SC, (2) with SC but ignoring
+magnification, (3) with SC but ignoring boost factor, and (4) switching
+to KiDS fiducial cosmology. We show comparisons with other works
+using KiDS-1000 data in orange, and some works using DES or HSC
+data in green.
+We then compare the different choices in the SC method. We
+find that if the magnification model is ignored in the analysis,
+the existing magnification signal in the data will be treated as
+an IA signal, leading to an over-estimated AIA = 0.81+0.36
+−0.41 and
+an over-estimated Alens = 0.87+0.18
+−0.18. On the other hand, we pre-
+viously argued that, when magnification is absent, the impact
+from the boost factor will be purely absorbed by the effective
+galaxy bias bg,eff, leaving AIA and Alens unbiased. Unfortunately,
+this does not hold anymore when magnification is present: if the
+boost factor is not corrected, all the parameters will be biased
+as follows AIA = 1.86+1.01
+−1.05, bg,eff = 0.67+0.06
+−0.06, Alens = 1.00+0.23
+−0.23
+and gmag = 1.55+1.28
+−1.31. We include the comparisons of Alens and
+AIA for the above-described cases in Fig. 13 and 14 and empha-
+sis the importance of taking magnification and boost factor into
+consideration. We also show the impact of the assumed fiducial
+cosmology: if the fiducial cosmology is switched from Planck
+to KiDS-1000 COSEBI as in Table 1, both Alens and AIA will
+change as shown in Fig. 13 (bottom-red) and 14 (bottom-blue).
+With the above results in simulation and data, summarized
+in Fig. 11, 13 and 14, we show that our measurements on AIA
+and Alens are unbiased from magnification, boost factor, and the
+assumed photo-z PDF model. These are the new developments
+considering the existence of magnification at high redshift z ∼ 1,
+beyond the study of Yao et al. (2020b).
+Additionally, we compare our analysis with previous works.
+The comparisons of Alens are shown in Fig. 13. We find that most
+of the previous works ignored the IA contamination (Hand et al.
+2015; Liu & Hill 2015; Kirk et al. 2016; Harnois-Déraps et al.
+2016; Singh et al. 2017a; Harnois-Déraps et al. 2017; Namikawa
+et al. 2019; Marques et al. 2020). For the ones that considered IA,
+they either fixed the IA amplitude (Kirk et al. 2016; Omori et al.
+2019) or used a strong prior (Robertson et al. 2021) to break the
+degeneracy between Alens and AIA, which will otherwise cause
+a strong loss in constraining power as we show in Fig. 12. We
+are the first to directly achieve the IA amplitude measurement
+within the same data and break the lensing-IA degeneracy. Our
+Article number, page 11 of 15
+
+A&A proofs: manuscript no. aanda
+-0.5
+0
+0.5
+1
+1.5
+AIA
+Asgari+ 2021 C(ℓ)
+SC, C(ℓ) cosmo
+Asgari+ 2021 COSEBI
+SC, COSEBI cosmo
+Asgari+ 2021 ξ±
+SC, ξ± cosmo
+SC
+cosmic shear
+Fig. 15. The comparisons of AIA between SC-subtracted results (blue)
+and cosmic shear tomography subtracted results (orange) with cosmolo-
+gies from different 2-point statistics. The cosmologies are shown in Ta-
+ble 1.
+baseline analysis is consistent with most of the previous results,
+showing the contamination from IA is not significant, mainly due
+to the total S/N of CMB lensing - galaxy shear cross-correlation
+is only at 3 ∼ 5 σ level at the current stage. However, the correct
+treatment for IA will be more and more important in the future
+with stage IV cosmic shear surveys and CMB observations.
+The comparisons of the AIA constraint with other results us-
+ing KiDS-1000 data are shown in Fig. 14, including the prior
+assumed in Robertson et al. (2021) and the cosmic shear tomog-
+raphy constraint in Asgari et al. (2021). Although the redshift
+range is slightly different, the above works have consistent re-
+sults on AIA. These comparisons will become more interesting
+for the next-stage observations.
+As an extended study, we investigate how the choice of fidu-
+cial cosmology affects the SC results, namely AIA. In Fig. 14
+we show the results with the fiducial Planck cosmology and the
+KiDS-1000 two-point correlation function ξ± best-fit cosmology.
+We further compare the results with the KiDS-1000 band power
+C(ℓ) cosmology and the COSEBIs cosmology in Fig. 15. The re-
+sults from Asgari et al. (2021) (shown in orange) are arranged in
+increasing order from bottom to top. We find that when assuming
+the same cosmology, the SC results (shown in blue) also follow
+the same (weak) trend, meanwhile, they agree very well with the
+cosmic shear results. We note the SC results will provide extra
+information in constraining IA in cosmic shear in the future.
+6. Summary
+In this work, we achieved the first application of the self-
+calibration (SC) method of intrinsic alignment (IA) of galax-
+ies to its cosmological application. We proved that with SC, the
+lensing-IA degeneracy could be efficiently broken, i.e., in this
+CMB lensing × galaxy shear cross-correlation work, it means
+breaking the degeneracy between the lensing amplitude Alens and
+the IA amplitude AIA. We showed that for previous treatments,
+IA are either ignored or being considered with a strong assumed
+prior on AIA. We demonstrated in Fig. 12, 13 and 14 that with
+SC to break the degeneracy, the constraining power in both Alens
+and AIA is preserved.
+We demonstrated that the proper angular scale cuts on wκγ
+are important. Our baseline analysis using information from
+θ > 20 arcmin gives Alens = 0.84+0.22
+−0.22. If we use informa-
+tion only at larger scales with θ > 40 arcmin, the constraint is
+Alens = 0.97+0.25
+−0.25. If we include information at much smaller
+scales with θ > 2 arcmin, the constraint is Alens = 0.77+0.21
+−0.22.
+At the current stage, they do not differ significantly from each
+other (even considering they are strongly correlated), as shown
+in Fig. 13. However, we note that these differences at differ-
+ent scales also exist in other works Planck Collaboration et al.
+(2020c) and Sun et al. (2022). We, therefore, emphasize the im-
+portance of understanding the possible systematics at different
+scales for future studies with higher S/N.
+Comparing our CMB lensing amplitude Alens with other
+works in Fig. 13, we found consistent results with different treat-
+ments of IA throughout almost all the works. We conclude that
+IA is not a significant source of systematics for the current stage.
+However, it will soon become more important with the stage IV
+observations. Nevertheless, we emphasize that the correct treat-
+ment to break the lensing-IA degeneracy is very important to
+maintain the cosmological constraining power. Our constraint
+on the IA amplitude AIA in Fig. 14 is also consistent with the
+existing analysis on IA with KiDS-1000 data. We note that the
+SC-subtracted IA information can be used as extra constraining
+power for any of these analyses.
+On the technique side, we further developed the SC method
+considering more sources of systematics beyond Yao et al.
+(2020b). We showed at z ∼ 1, the impact of galaxy shear × cos-
+mic magnification component wGκgal contaminates the separated
+IA × galaxy number density signal wIg, and is non-negligible as
+shown in Fig. 2 and 6. We use Eq. (16) and (18) to show how the
+magnification term enters our observable and how we include
+it in the theory as a correction. We show in Fig. 13 and 14 that
+the correction of magnification is important when applying SC
+to higher redshift data, in order to get the correct constraint on
+IA. We also discussed that, with the contamination from mag-
+nification, boost factor can no longer be absorbed by the effec-
+tive galaxy bias bg,eff, and need to be accounted for correctly, as
+shown in Eq. (24), (25) and Fig. 6, 13, 14.
+We also validated our analysis with MICE2 simulation, fo-
+cusing on two aspects: (1) how good can the magnification
+model mitigate the contamination from the magnification-shear
+signal; and (2) will the assumed photo-z PDF model (which is
+used to calculate the signal drop QGg and QIg) bias the IA mea-
+surement. With the strong constraining power from MICE2 with
+no shape noise, we can show in Fig. 11 that, when the magnifi-
+cation model is included in the analysis, the IA amplitude can be
+obtained correctly (consistent within 1σ range of 0, which is the
+input of MICE2). Additionally, the bias from the assumed photo-
+z model is negligible when the magnification model is used, as
+the effective magnification prefactor gmag will absorb the intro-
+duced error. We, therefore, emphasize the importance of includ-
+ing the magnification model in the SC analysis, especially for fu-
+ture high-z surveys like LSST, Euclid, WFIRST, and CSST. We
+further notice the contamination from magnification will make
+SC no longer an IA-model-independent method, therefore, SC
+is more suitable for low-z data when considering alternative IA
+models.
+Comparing with our first measurements with KV-450 data
+(Yao et al. 2020a), a lot of improvements have been added in the
+SC method, including:
+(1) the covariance, the galaxy bias, the scale-dependency for the
+lensing-drop QGg, the IA-drop QIg, and appropriate scale-cuts,
+Article number, page 12 of 15
+
+Yao et al 2022: KiDS shear × Planck lensing and IA removal
+which have been introduced in Yao et al. (2020b);
+(2) the boost factor, the cosmic magnification, and the photo-z
+PDF modeling, which are introduced in this work;
+(3) its first validation using simulation, and its first application
+to cosmology in order to break the lensing-IA degeneracy, intro-
+duced in this work.
+With these improvements, we manage to achieve consistent IA
+results between SC and cosmic shear, as shown in Fig. 15, while
+previously we got AIA = 2.31+0.42
+−0.42 with the old version of SC
+(Yao et al. 2020a) and AIA = 0.981+0.694
+−0.678 for cosmic shear (Hilde-
+brandt et al. 2020) with KV-450 data.
+Despite SC-obtained AIA is consistent with the MICE input
+IA, and when applying to data it is consistent with the KiDS cos-
+mic shear results Asgari et al. (2021) and the other CMB lensing
+work Robertson et al. (2021), as well as gmag is in reasonable
+agreement with (Duncan et al. 2014), our results still suffer from
+an unrealisticly low effective galaxy bias bg,eff = 0.88, which
+is different from our previous work (Yao et al. 2020b). We dis-
+cussed this value may absorb the contribution from (1) fiducial
+cosmology, (2) lensing weight in n(z), (3) insufficient modeling
+in non-linear galaxy bias, baryonic effects, and massive neutri-
+nos, (4) incorrect photo-z v.s. true-z connection as discussed in
+Appendix A and (5) possible other sources of systematics. We
+emphasize the complication and leave this point for future stud-
+ies.
+We note that there could still exist other systematics other
+than the galaxy bias, such as beyond Limber approximation
+(Fang et al. 2020), non-flat ΛCDM (Yu et al. 2021), selection
+bias on shear measurement (Li et al. 2021). But they have either
+much smaller impacts compared with IA or are strongly reduced
+due to our scale cuts. Therefore, they are beyond the scope of
+this paper.
+Acknowledgements. The authors thank Yu Yu, Hai Yu, Jiaxin Wang for useful
+discussions.
+This work is supported by National Key R&D Program of China No.
+2022YFF0503403. JY acknowledges the support of the National Science
+Foundation of China (12203084), the China Postdoctoral Science Foundation
+(2021T140451), and the Shanghai Post-doctoral Excellence Program (2021419).
+HYS acknowledges the support from CMS-CSST-2021-A01 and CMS-CSST-
+2021-B01, NSFC of China under grant 11973070, the Shanghai Committee of
+Science and Technology grant No.19ZR1466600 and Key Research Program
+of Frontier Sciences, CAS, Grant No. ZDBS-LY-7013. PZ acknowledges the
+support of the National Science Foundation of China (11621303, 11433001).
+XL acknowledges the support of NSFC of China under Grant No. 11803028,
+YNU Grant No. C176220100008, and a grant from the CAS Interdisciplinary
+Innovation Team. BJ acknowledges support by STFC Consolidated Grant
+ST/V000780/1. MB is supported by the Polish National Science Center through
+grants no. 2020/38/E/ST9/00395, 2018/30/E/ST9/00698, 2018/31/G/ST9/03388
+and 2020/39/B/ST9/03494, and by the Polish Ministry of Science and Higher
+Education through grant DIR/WK/2018/12. HH is supported by a Heisenberg
+grant of the Deutsche Forschungsgemeinschaft (Hi 1495/5-1) as well as
+an ERC Consolidator Grant (No. 770935). TT acknowledges support from
+the Leverhulme Trust. AW is supported by an European Research Council
+Consolidator Grant (No. 770935). ZY acknowledges support from the Max
+Planck Society and the Alexander von Humboldt Foundation in the framework
+of the Max Planck-Humboldt Research Award endowed by the Federal Ministry
+of Education and Research (Germany). The computations in this paper were run
+on the π 2.0 cluster supported by the Center for High Performance Computing
+at Shanghai Jiao Tong University.
+The codes JY produced for this paper were written in Python. JY thanks all its
+developers and especially the people behind the following packages: SCIPY
+(Jones et al. 2001–), NUMPY (van der Walt et al. 2011), ASTROPY (Astropy
+Collaboration et al. 2013) and MATPLOTLIB (Hunter 2007), TreeCorr (Jarvis
+et al. 2004), CCL (Chisari et al. 2019), CAMB (Lewis et al. 2000), Healpy
+(Górski et al. 2005; Zonca et al. 2019), emcee (Foreman-Mackey et al. 2013),
+fitsio6, kmeans_radec7, corner (Foreman-Mackey 2016), ChainConsumer8. The
+6 https://github.com/esheldon/fitsio
+7 https://github.com/esheldon/kmeansradec
+8 https://github.com/Samreay/ChainConsumer
+KiDS-1000 results in this paper are based on data products from observations
+made with ESO Telescopes at the La Silla Paranal Observatory under pro-
+gramme IDs 177.A-3016, 177.A-3017 and 177.A-3018, and on data products
+produced by Target/OmegaCEN, INAF-OACN, INAF-OAPD, and the KiDS
+production team, on behalf of the KiDS consortium.
+Author contributions: All authors contributed to the development and writing
+of this paper. The authorship list is given in three groups: the lead authors (JY,
+HS, PZ, XL) followed by two alphabetical groups. The first alphabetical group
+includes those who are key contributors to both the scientific analysis and the
+data products. The second group covers those who have either made a significant
+contribution to the data products, or to the scientific analysis.
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+
+Yao et al 2022: KiDS shear × Planck lensing and IA removal
+Appendix A: The signal drop Q
+We keep the main text of this paper focused on the physics while
+keeping the details of the SC method, more specifically the cal-
+culation for the lensing-drop QGg and the IA-drop QIg in this ap-
+pendix. The Qs are calculated through Eq. (13) and (14), while
+the correlation functions being used are just the Hankel trans-
+form (similar to Eq. (4)) of the angular power spectrum CGg and
+CIg. The associated CGg and CGg|S are calculated by:
+CGg
+ii (ℓ) =
+� ∞
+0
+qi(χ)ni(χ)
+χ2
+bg,effPδ
+�
+k = ℓ
+χ; χ
+�
+dχ,
+(A.1)
+CGg
+ii |S(ℓ) =
+� ∞
+0
+qi(χ)ni(χ)
+χ2
+bg,effPδ
+�
+k = ℓ
+χ; χ
+�
+ηGg
+i (z)dχ.
+(A.2)
+Similarly, the CIg and CIg|S are given by:
+CIg
+ii (ℓ) =
+� ∞
+0
+ni(χ)ni(χ)
+χ2
+bg,effPδ,γI
+�
+k = ℓ
+χ; χ
+�
+dχ,
+(A.3)
+CIg
+ii |S(ℓ) =
+� ∞
+0
+ni(χ)ni(χ)
+χ2
+bg,effPδ,γI
+�
+k = ℓ
+χ; χ
+�
+ηIg
+i (z)dχ.
+(A.4)
+Here ηGg
+i (z) = ηGg
+i (zL = zg = z) is the function that account
+for the effect of the SC selection Eq. (8) in the Limber integral,
+similarly for ηIg. They are expressed
+ηGg
+i (zL, zg) =
+2
+�
+dzP
+G
+�
+dzP
+g
+� ∞
+0 dzGWL(zL, zG)S (zP
+G, zP
+g)K
+�
+dzP
+G
+�
+dzPg
+� ∞
+0 dzGWL(zL, zG)K
+, (A.5)
+ηIg
+i (zL, zg) =
+2
+�
+dzP
+G
+�
+dzP
+g
+� ∞
+0 dzGS (zP
+G, zP
+g)K
+�
+dzP
+G
+�
+dzPg
+� ∞
+0 dzGK
+,
+(A.6)
+as in Yao et al. (2020b), where K is the galaxy-pair redshift dis-
+tribution kernel
+K(zG, zg, zP
+G, zP
+g) = p(zG|zP
+G)p(zg|zP
+g)nP
+i (zP
+G)nP
+i (zP
+g),
+(A.7)
+and S is the SC selection function
+S (zP
+G, zP
+g) =
+�1
+for zP
+G < zP
+g,
+0
+otherwise ,
+(A.8)
+which correspond to Eq. 8 in the main text, and the lensing kernel
+is
+WL(zL, zS ) =
+�������
+3
+2Ωm
+H2
+0
+c2 (1 + zL)χL(1 − χL
+χS )
+for zL < zS
+0
+otherwise
+.
+(A.9)
+Here zx is the true-z where x can be “G” the source, “L” the
+lens, or “g” the galaxy number density. The galaxy photo-z dis-
+tribution is nP(zP), and the redshift PDF (probability distribution
+function) is p(z|zP).
+As shown above, when the galaxy photo-z distribution and
+the corresponding true-z distribution are given, as shown in
+Fig. 3 in this work, we can follow the above procedure to calcu-
+late the lensing-drop QGg and QIg. The results of QGg and QIg for
+this work are shown in Fig. 4 for your interest. Generally, given
+the tomographic bin width, the better photo-z is, the smaller QGg
+will be (it reaches ∼ 0 for perfect photo-z). On the other hand,
+non-symmetric photo-z distribution and non-symmetric true-z
+distribution will make GIg deviate from 1. For more details on
+the Q calculation and its properties, see discussions in Yao et al.
+(2020a,b).
+101
+102
+103
+104
+ℓ
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Q
+gG model
+gI model
+gG sim
+gI sim
+100
+101
+102
+θ [arcmin]
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+gG model
+gI model
+gG sim
+gI sim
+Fig. A.1. The effect of photo-z modeling with MICE2. By applying the
+SC selection Eq. (8) or (A.8), the lensing-drop GGg from photo-z model
+(green) is slightly biased compared with the results from true-z (blue),
+while the IA-drop GIg from photo-z model (red) is immune to such bias
+and agrees with the true-z result (orange).
+We note that for the SC calculation, the redshift PDF p(z|zP)
+for each galaxy is required. Due to the fact that the PDFs from
+photo-z algorithm can be biased due to the color-redshift degen-
+eracy in the photometric surveys, calibration is needed (Hilde-
+brandt et al. 2017, 2021; Abbott et al. 2022). However, we can
+only statistically calibrate the overall redshift distribution n(z)
+but not the PDF p(z|zP) for each galaxy. This means in order to
+calculate Eq. A.7 we need to assume a photo-z PDF model. We
+choose to use a bi-Gaussian model Yao et al. (2020a)
+p2G(z|zP) = (1 − fout)pmain(z|zP; ∆1, σ1) + foutpoutlier(z|zP; ∆2, σ2),
+(A.10)
+with a main Gaussian peak and a Gaussian outlier peak with
+different bias ∆i and scatter σi, and an outlier rate fout.
+We fit the bi-Gaussian model Eq. (A.10), requiring it to have
+same mean redshift ⟨z⟩ with the SOM calibrated n(z) (Asgari
+et al. 2021), and minimize the difference between the resulting
+model z-distribution
+�
+nP(zP)p(z|zP)dzP and the SOM n(z). The
+best-fit will then be a good description of the photo-z quality and
+can be used in Eq. (A.7). The resulting signal drops are shown in
+Fig. 4 in the main text.
+We validate the bi-Gaussian photo-z model for SC with
+MICE2 simulation. We compare with the results that use the
+photo-z distribution and true-z distribution in the calculation of
+Eq. (A.7). We show in Fig. A.1 that the bi-Gaussian model can
+produce the IA-drop QIg measurement very consistent with the
+ones with true-z from simulation. However, we find the lensing-
+drop QGg from the photo-z model is slightly higher than the true
+values from the simulation. This error will be propagated to the
+separated lensing signal wGg and the IA+magnification signal
+wIg + gmagwGκ according to Eq. (15) and (16). Its impact in AIA
+is shown in Fig. 10 and 11.
+Article number, page 15 of 15
+

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+arXiv:2301.02180v1  [math.DS]  5 Jan 2023
+Existence of robust non-uniformly hyperbolic
+endomorphism in homotopy classes
+Victor Janeiro
+[email protected], ICEx-UFMG, Belo Horizonte-MG, Brazil.
+Abstract
+We extend the results of [1] by showing that any homothety in 핋2 is homo-
+topic to a non-uniformly hyperbolic ergodic area preserving map, provided that
+its degree is at least 52. We also address other small topological degree cases not
+considered in the previous article. This proves the existence of a 1 open set
+of non-uniformly hyperbolic systems, that intersects essentially every homotopy
+classes in 핋2, where the Lyapunov exponents vary continuously.
+1
+Introduction
+We study conservative maps of the two-torus 핋2 from the point of view of smooth
+ergodic theory. We are interested in the Lyapunov exponents of these systems, in
+particular, in extending the results obtained in [1] to the homothety case and some
+cases with lower topological degree, which were not included in the previous results.
+With this in mind, some familiarity with the results of [1] is desirable.
+For a differentiable covering map 푓 ∶ 핋2 → 핋2 and a pair (푥, 푣) ∈ 푇핋2, the number
+̃(푥, 푣) = lim sup
+푛→∞
+log ‖퐷푥푓 푛(푣)‖
+푛
+is the Lyapunov exponent of 푓 at (푥, 푣). See [2] for background in Smooth Ergodic
+Theory. Due to Oseledet’s Theorem [3] , there is a full area set 푀0 on 핋2 where the
+previous limit exists for every 푣, and there exists a measurable bundle 퐸− defined on
+푀0 such that for 푥 ∈ 푀0, 푣 ≠ 0 ∈ 퐸−(푥):
+(푥, 푣) ∶= lim
+푛→∞
+log ‖퐷푥푓 푛(푣)‖
+푛
+= lim
+푛→∞
+log 푚(퐷푥푓 푛)
+푛
+∶= −(푥),
+while for 푣 ∈ ℝ2 ⧵ 퐸−(푥):
+(푥, 푣) = lim
+푛→∞
+log ‖퐷푥푓 푛‖
+푛
+∶= +(푥),
+Moreover, if 휇 denotes the Lebesgue (Haar) measure on 핋2, then:
+∫ (+(푥) + −(푥))푑휇(푥) = ∫ log | det 퐷푥푓 |푑휇(푥) > 0,
+(1)
+1
+
+so +(푥) > 0 almost everywhere. At last, we say that 푓 is non-uniformly hyperbolic
+(NUH) if −(푥) < 0 < +(푥) almost everywhere.
+Non uniformly hyperbolic systems provide a generalization of the classical Anosov
+surface maps [4]. Here, we will only be concerned with the non-invertible case in an
+attempt to aid the understanding of their statistical properties, which is still under
+development. For the general ergodic theory of endomorphisms, the reader is directed
+to [5].
+Any map 푓 ∶ 핋2 → 핋2 is homotopic to a linear endomorphism 퐸 ∶ 핋2 → 핋2,
+induced by an integer matrix that we denote by the same letter. In [1], it is established
+the existence of a 1 open set of non-uniformly hyperbolic systems that intersects
+every homotopy class that does not contain a homothety, provided that the degree is
+not too small. The authors then conjecture that the same is true for homotheties. In
+this article, we prove this conjecture, provided that the degree is at least 52. There are
+other low topological degree cases not covered by Andersson, Carrasco and Saghin,
+which we also address here.
+Let End푟
+휇(핋2) be the set of 푟 local diffeomorphisms of 핋2 preserving the Lebesgue
+measure 휇, that are not invertible. For 푓 ∈ End푟
+휇(핋2), (푥, 푣) ∈ 푇 1핋2 define:
+퐼(푥, 푣; 푓 푛) =
+∑
+푦∈푓 −푛(푥)
+log ‖(퐷푦푓 푛)−1푣‖
+det(퐷푦푓 푛)
+,
+and
+퐶(푓 ) = ∑
+푛∈ℕ
+1
+푛
+inf
+(푥,푣)∈푇 1핋2 퐼(푥, 푣; 푓 푛).
+Define the set
+ ∶= {푓 ∈ End푟
+휇(핋2) ∶ 퐶(푓 ) > 0},
+which is open in the 1-topology. On Subsection 2.3 of the main reference [1], it is
+proved:
+Theorem 1. If 푓 ∈  , then 푓 is non-uniformly hyperbolic.
+Our main results are:
+Theorem A. For 퐸 = 푘 ⋅퐼푑 ∈ 푀2×2(ℤ), with |푘| ≥ 5, the intersection [퐸]∩ is non-empty
+and in fact contains maps that are real analytically homotopic to E.
+Theorem B. If 퐸 ∈ 푀2×2(ℤ) is not a homothety and 푑푒푡(퐸) > 4, the intersection [퐸] ∩ 
+is non-empty and in fact contains maps that are real analytically homotopic to E.
+Our Theorem B is equivalent to the Theorem A of [1] but includes three cases
+which are not proved there. The main difficulty for our results is that, in the case of
+2
+
+a homothety, the induced projective action is trivial; non-triviality of this projective
+action is a central piece in the method of Andersson et al.
+Finally, by inspection on the proofs of Theorems B and C of [1], we can see that it
+works for all cases included here. Hence, defining:
+퐶det(푓 ) ∶= sup
+푛∈ℕ
+1
+푛 inf
+푥∈핋2 log(det(퐷푥푓 푛)) > 0,
+and the open set:
+1 ∶=
+{
+푓 ∈ End푟
+휇(핋2) ∶ 퐶(푓 ) > −1
+2퐶det(푓 )
+}
+,
+we have from Theorems A and B that if a linear endomorphism 퐸 satisfies the condi-
+tions of either of the Theorems, then [퐸] ∩ 1 ≠ ∅. Therefore, by Theorem 퐵 of [1], we
+have conituity of the maps 1 ∋ 푓 ↦ ∫핋2 ±(푓 )푑휇 in the 1 topology.
+From Theorem C of [1], we conclude that for any linear endomorphism E as in
+Theorem A or B. If ±1 is not an eigenvalue of 퐸, then [퐸] ∩  contains stably ergodic
+endomorphisms. In fact, it contains stably Bernoulli endomorphisms and, in particular,
+maps that are mixing of all orders.
+Acknowledgements
+The results presented here were conjectured by Martin Andersson, Pablo D. Carrasco
+and Radu Saghin in [1], I thank Pablo D. Carrasco, who is also my MSc advisor, for the
+suggestion of the problem and for the hours of conversations on the subject that were
+crucial to this article.
+This work has been supported by the Brazillian research agencies CAPES and
+CNPq.
+2
+Preliminary
+In order to prove Theorems A and B, we require a result on the computation of the
+numbers 퐼(푥, 푣; 푓 푛) which the proof can be found in [1]:
+Proposition 2.1. For any 푛 ∈ ℕ, it holds:
+퐼(푥, 푣; 푓 푛) =
+푛−1
+∑
+푖=0
+∑
+푦∈푓 −푖(푥)
+퐼(푦, 퐹 −푖
+푦 푣; 푓 )
+det(퐷푦푓 푖) ,
+(2)
+where 퐹 −푖
+푦 푣 =
+(퐷푦푓 푖)−1푣
+‖(퐷푦푓 푖)−푖푣‖.
+3
+
+2.1
+Shears
+For fixed points 푧1, 푧2, 푧3, 푧4 ∈ 핋1, in this order, take the closed intervals 퐼1 = [푧1, 푧2],
+퐼3 = [푧3, 푧4], and the open intervals 퐼2 = (푧2, 푧3) and 퐼4 = (푧4, 푧1).
+Definition 2.1. We define the horizontal and vertical critical regions in 핋2 as ℎ = (퐼1 ∪
+퐼3) × 핋1, 푣 = 핋1 × (퐼1 ∪ 퐼3) and its complements ℎ = 핋2 ⧵ ℎ , 푣 = 핋2 ⧵ 푣 are respectively
+the horizontal and vertical good region.
+We then divide the good regions into +
+ℎ = 퐼4 × 핋1, −
+ℎ = 퐼2 × 핋1, +
+푣 = 핋1 × 퐼4 and
+−
+푣 = 핋1 × 퐼2.
+For fixed numbers 0 < 푎 < 푏, we take 푠 ∶ 핋1 → ℝ as an analytic map satisfying
+the following conditions:
+1. If 푧 ∈ 퐼4, then 푎 < 푠′(푧) < 푏;
+2. If 푧 ∈ 퐼2, then −푏 < 푠′(푧) < −푎;
+3. If 푧 ∈ 퐼1 ∪ 퐼3, then |푠′(푧)| < 푏.
+Consider the two families of conservative diffeomorphisms of the torus given by:
+ℎ푡(푥1, 푥2) = (푥1, 푥2 + 푡푠(푥1)), 푣푟(푥1, 푥2) = (푥1 + 푟푠(푥2), 푥2),
+푡, 푟 ∈ ℝ.
+Note that:
+퐷(푥1,푥2)ℎ푡 = (
+1
+0
+푡푠′(푥1)
+1) ,
+퐷(푥1,푥2)푣푟 = (
+1
+푟푠′(푥2)
+0
+1
+) .
+In order to simplify the computations we will consider the maximum norm on 푇핋2
+as ‖(푢1, 푢2)‖ = max{|푢1|, |푢2|}, and all the computations from now on are performed
+using this norm. This way, we get, for every 푥 ∈ 핋2:
+‖퐷푥ℎ푡‖ < 푏푡 + 1, and ‖퐷푥푣푟‖ < 푏푡 + 1.
+Definition 2.2. Given 훼 > 0, the corresponding horizontal cone is Δℎ
+훼 = {(푢1, 푢2) ∈ ℝ2 ∶
+|푢2| ≤ 훼|푢1|}, while the corresponding vertical cone is its complement Δ푣
+훼 = ℝ2 ⧵ Δℎ
+훼,
+Lemma 2.1. For 훼 > 1, let Δℎ
+훼 and Δ푣
+훼 be the corresponding horizontal and vertical cones.
+Then, for every 푡, 푟 > 2훼
+푎 , and, for every unit vector 푢 ∈ 푇푥핋2, the following holds:
+1. If 푢 ∈ Δ푣
+훼, and:
+(a) 푥 ∈ 푣, then
+• (퐷푥푣푟)−1푢 ∈ Δℎ
+훼 (퐷푥푣−1
+푟 Δ푣
+훼 ⊂ Δℎ
+훼);
+• ‖(퐷푥푣푟)−1푢‖ > 푎푟−훼
+훼
+= 푟
+푎− 훼
+푟
+훼 ;
+4
+
+(b) 푥 ∈ 푣, then ‖(퐷푥푣푟)−1푢‖ > 1
+훼 .
+2. If 푢 = ±(1, 푢2) ∈ Δℎ
+훼, then:
+(a) either for every 푥 ∈ +
+푣 ( if 푢2 ≤ 0) or for every 푥 ∈ −
+푣 (if 푢2 ≥ 0) it holds:
+• (퐷푥푣푟)−1푢 ∈ Δℎ
+훼;
+• ‖(퐷푥푣푟)−1푢‖ > 1;
+(b) for all other 푥, we have ‖(퐷푥푣푟)−1푢‖ >
+1
+푏푟+1.
+3. If 푢 ∈ Δℎ
+훼, and:
+(a) 푥 ∈ ℎ, then
+• (퐷푥ℎ푡)−1푢 ∈ Δ푣
+훼 (퐷푥ℎ−1
+푡 Δℎ
+훼 ⊂ Δ푣
+훼);
+• ‖(퐷푥ℎ푡)−1푢‖ > 푎푡−훼
+훼
+= 푡
+푎− 훼
+푡
+훼 ;
+(b) 푥 ∈ ℎ, then ‖(퐷푥ℎ푡)−1푢‖ > 1
+훼 .
+4. If 푢 = ±(푢1, 1) ∈ Δ푣
+훼, then:
+(a) either for every 푥 ∈ +
+ℎ ( if 푢1 ≤ 0) or for every 푥 ∈ −
+ℎ (if 푢1 ≥ 0) it holds:
+• (퐷푥ℎ푡)−1푢 ∈ Δ푣
+훼;
+• ‖(퐷푥ℎ푡)−1푢‖ > 1;
+(b) for all other 푥, we have ‖(퐷푥ℎ푡)−1푢‖ >
+1
+푏푡+1.
+Proof. We prove items 1 and 2, the case for ℎ푡 is analogous. Let 푥 = (푥1, 푥2) ∈ 푣, and
+푢± = (1, ±훼) then:
+(퐷푥푣푟)−1푢± = (
+1
+−푟푠′(푥2)
+0
+1
+) (
+1
+±훼) = (
+1 ∓ 푟푠′(푥2)훼
+±훼
+) ,
+also since 푥 ∈ 푣, 푎 < |푠′(푥2)| < 푏, we also have 훼 > 1 and 푟 > 2훼
+푎 , hence:
+|1 ∓ 푟푠′(푥2)훼| ≥ 푟훼푎 − 1 > 2훼2 − 1 > 훼 > 1,
+which shows that (퐷푥푣푟)−1Δ푣
+훼 ⊂ Δℎ
+훼. Also, ‖(퐷푥푣푟)−1푢‖ = |1 ∓ 푟푠′(푥2)훼| > 푟푎훼 − 1. Now,
+noticing that the minimal expansion of vectors in Δ푣
+훼 occurs on either of (1, ±훼), we
+have for every unit vector 푢 ∈ Δ푣
+훼:
+‖(퐷푥푣푟)−1푢‖ ≥ ‖(퐷푥푣푟)−1(1, ±훼)‖
+‖(1, ±훼)‖
+> 푟훼 − 1
+훼
+.
+For part 2 (a), we have for x ∈ +
+푣 푠′(푥2) > 푎 > 0, and for 푥 ∈ −
+푣, 푠′(푥2) < −푎 < 0,
+thus, by simple calculations analogous to the last one, we get the results. Finally, for
+(b) we just use 푚((퐷푥푣푟)−1) =
+1
+‖퐷푥푣푟‖ >
+1
+푏푟+1 for every 푥 ∈ 핋2.
+5
+
+3
+Endomorphisms and Shears: Proof of Theorem A
+Fix 퐸 = 푘 ⋅ 퐼푑, for some 푘 ∈ ℕ (we shall make the entire argument on 푘 ∈ ℕ for the
+sake of simplicity of notation, we emphasize that the entire argument works for 푘 ∈ ℤ
+by replacing 푘 for |푘| when necessary). Fix a 훿 <
+1
+4푘 and define the critical and good
+regions as in Def. 2.1 for points 푧1, 푧2, 푧3, 푧4 ∈ 핋1 such that:
+• 퐼1 = [푧1, 푧2] and 퐼3 = [푧3, 푧4] have size 2훿;
+• The translation of 퐼1 by a multiple of 1
+푘 does not intersect 퐼3.
+• 퐼2 = (푧2, 푧3) and 퐼4 = (푧4, 푧1) have size strictly larger than 1
+푘 [
+푘−1
+2 ], where [푝]
+denotes the floor of 푝.
+It is obtained directly from the definitions that:
+Proposition 3.1. For every 푥 = (푥1, 푥2) ∈ 핋2, 퐸−1(푥) has 푘2 points given by:
+퐸−1(푥1, 푥2) =
+{
+(
+푥1 + 푖
+푘
+, 푥2 + 푗
+푘
+) ∶ 푖, 푗 = 0, ⋯ , 푘 − 1
+}
+.
+At least 푘 [
+푘−1
+2 ] are inside each of +
+푣, −
+푣, +
+ℎ and −
+ℎ, and at most 푘 of them are inside
+each of 푣, ℎ.
+From now on, in this section, we fix any 훼 > 1 and the corresponding cones as in
+Def. 2.2. We consider the analytic maps:
+푓(푡,푟) = 퐸◦푣푟◦ℎ푡,
+which we shall denote only by 푓 = 푓(푡,푟). Clearly 푓 is an area preserving endomorphism
+isotopic to E. We observe that, given 푥 ∈ 핋2 and 푦 ∈ 푓 −1(푥), we have:
+(퐷푦푓 )−1 = (퐷푦ℎ푡)−1(퐷ℎ푡(푦)푣푟)−1퐸−1.
+The goal is for (퐷ℎ푡(푦)푣푟)−1 to take vectors in the vertical cone and expand them
+in the horizontal direction and then (퐷푦ℎ푡)−1 takes its images and expands them in
+the vertical direction, resulting in (퐷푦푓 )−1 expanding in the vertical direction for most
+points in 푓 −1(푥). Thus, in order to keep track of this derivative, we must localize the
+points 푦 ∈ 푓 −1(푥) in regard to which of ℎ or ℎ they belong, and {ℎ푡(푦) ∶ 푦 ∈ 푓 −1(푥)} =
+(퐸◦푣푟)−1(푥) regarding which of 푣 or 푣 they belong.
+Lemma 3.1. For every 푥 ∈ 핋2, we have:
+1. (푣푟◦퐸)−1(푥) has 푘2 points of which at least 푘 [
+푘−1
+2 ] of them are in each one of +
+푣 and
+−
+푣 and at most 푘 of them are in 푣;
+6
+
+2. 푓 −1(푥) has 푘2 points of which at least 푘 [
+푘−1
+2 ] of them are in each one of +
+ℎ and −
+ℎ
+and at most 푘 of them are in ℎ.
+Proof.
+1. It is a direct consequence of Prop. 3.1 along with the fact that the regions
++
+푣, −
+푣 and 푣 are invariant under 푣푟.
+2. Notice that in each row of pre-images by E of a point 푥 = (푥1, 푥2) given by
+{
+(
+푥1+푖
+푘 , 푥2+푗0
+푘 ) ∶ 푖 = 0, ⋯ , 푘 − 1
+}
+for a fixed 푗0 ∈ {0, ⋯ , 푘 − 1}, 푣−1
+푟
+is a rotation
+by −푟푠 (
+푥2+푗0
+푘 ) in the circle 핋1 ×
+{ 푥2+푗0
+푘
+}
+. Hence, at least [
+푘−1
+2 ] of the 푘 points of
+this row are inside each one of +
+ℎ and −
+ℎ, and at most 1 is in ℎ.
+As this is also true for all the 푘 rows of pre-images by E, we get at least 푘 [
+푘−1
+2 ]
+pre-images by 퐸◦푣푟 are inside each one of +
+ℎ and −
+ℎ, and at most 푘 pre-images
+by 퐸◦푣푟 are inside ℎ. Finally, since these sets are invariant under ℎ푡, we get the
+desired result.
+Remark 3.1. Even knowing which regions is a point 푦 ∈ (퐸◦푣푟)−1(푥), we cannot de-
+termine the region which ℎ−1
+푡 (푦) is inside, as 푡 is varying. That is, there may be points
+푦 ∈ 푓 −1(푥) that are in ℎ such that ℎ푡(푦) ∈ 푣 and vice-versa.
+Definition 3.1. In order to keep track of the vectors, define:
+• For 푢 = (푢1, 푢2) ∈ ℝ2 with 푢2 ≠ 0:
+∗ (푢) =
+{
+−sgn (
+푢1
+푢2) , if 푢1 ≠ 0,
+−sgn(푢2),
+if 푢1 = 0.
+Notice that ∗ (푢) = ∗ (퐸−1푢), for every 푢 ∈ ℝ2.
+• For 푥 ∈ 핋2, 푦 ∈ 푓 −1(푥) and 푢 ∈ ℝ2, let (푤1, 푤2) = (퐷ℎ푡(푦)푣푟)−1퐸−1푢:
+∗푦 (푢) =
+⎧⎪⎪
+⎨⎪⎪⎩
+−sgn (
+푤1
+푤2) , if 푤1, 푤2 ≠ 0,
+−sgn(푤2),
+if 푤2 ≠ 0, 푤1 = 0,
+−sgn(푤1),
+if 푤1 ≠ 0, 푤2 = 0.
+In view of item 4 of Lemma 2.1, even though (퐷ℎ푡(푦)푣푟)−1 may not send a vector
+푢 ∈ Δ푣
+훼 to the horizontal cone if ℎ푡(푦) ∈ 푣, we can still end up having expansion in the
+vertical direction, depending on whether 푦 ∈ 
+∗푦(푢)
+ℎ
+or not. In this regard, from Lemma
+3.1, there are 푘 points 푦 ∈ 푓 −1(푥) such that ℎ푡(푦) are in 푣, and these points (ℎ푡(푦)) are
+all in the same circle 핋1 ×
+{ 푥2+푗0
+푘
+}
+, hence the derivative (퐷ℎ푡(푦)푣푟)−1 is the same for those
+points. We get:
+7
+
+Proposition 3.2. For every 푢 ∈ ℝ2, 푥 ∈ 핋2, then the sign ∗푦 (푢) = sg (
+푤1
+푤2) is the same for
+all points 푦 ∈ 푓 −1(푥) such that ℎ푡(푦) ∈ 푣, where ∗푦 (푢) is as in Definition 3.1.
+Definition 3.2. For a fixed 푥 ∈ 핋2 and:
+• 푢 ∈ Δ푣
+훼, define:
+⎧⎪⎪⎪⎪
+⎨⎪⎪⎪⎪⎩
+퐴 = {푦 ∈ 푓 −1(푥) ∶ 푦 ∈ ℎ, ℎ푡(푦) ∈ 푣}.
+퐵 = {푦 ∈ 푓 −1(푥) ∶ 푦 ∈ 
+∗푦(푢)
+ℎ
+, ℎ푡(푦) ∈ 푣},
+푣 = 퐴 ∪ 퐵,
+ℎ = 푓 −1(푥) ⧵ 푣.
+• 푢 ∈ Δℎ
+훼, define:
+⎧⎪⎪⎪⎪
+⎨⎪⎪⎪⎪⎩
+퐶 = {푦 ∈ 푓 −1(푥) ∶ 푦 ∈ ℎ, ℎ푡(푦) ∈ ∗(푢)
+푣 }.
+퐷 = {푦 ∈ 푓 −1(푥) ∶ 푦 ∈ 
+∗푦(푢)
+ℎ
+, ℎ푡(푦) ∈ 푣 ∪ −∗(푢)
+푣
+},
+푣 = 퐶 ∪ 퐷,
+ℎ = 푓 −1(푥) ⧵ 푣.
+A direct consequence of Lemma 3.1 and Prop. 3.2, having Remark. 3.1 in mind, is
+the following:
+Lemma 3.2. For a fixed (푥, 푢) ∈ 푇핋2, 푓 −1(푥) has 푘2 points, of which:
+1. For 푢 ∈ Δ푣
+훼, at most 2푘 − 1 − [
+푘−1
+2 ] of them are in ℎ and at least (푘 − 1)2 + [
+푘−1
+2 ]
+are inside 푣, because:
+• At least (푘 − 1)2 are in A and,
+• at least [
+푘−1
+2 ] are in B.
+2. For 푢 ∈ Δℎ
+훼, at most 푘2 − [
+푘−1
+2 ] (푘 + [
+푘−1
+2 ]) are in ℎ and at least [
+푘−1
+2 ] (푘 + [
+푘−1
+2 ])
+are in 푣, because:
+• At least (푘 − 1) [
+푘−1
+2 ] are in C and,
+• at least [
+푘−1
+2 ] (1 + [
+푘−1
+2 ]) are in D.
+Knowing that for every unit vector 푢 ∈ ℝ2 we have ‖퐸−1푢‖ = 1
+푘 (maximum norm),
+from Lemma 2.1 we get:
+Lemma 3.3. For 푡, 푟 > 2훼
+푎 and for fixed 푥 ∈ 핋2, it holds:
+1. If 푢 ∈ Δ푣
+훼, then for all 푦 ∈ 푣 we have (퐷푦푓 )−1푢 ∈ Δ푣
+훼;
+2. If 푢 ∈ Δ푣
+훼 is a unit vector, then:
+‖(퐷푦푓 )−1푢‖ >
+⎧⎪⎪⎪
+⎨⎪⎪⎪⎩
+(
+푎− 훼
+푡
+훼 ) (
+푎− 훼
+푟
+훼 )
+푡푟
+푘 , 푦 ∈ 퐴,
+1
+훼푘,
+푦 ∈ 퐵,
+1
+(푏푡+1)훼푘,
+푦 ∈ ℎ;
+8
+
+3. If 푢 ∈ Δℎ
+훼, then for all 푦 ∈ 푣 we have (퐷푦푓 )−1푢 ∈ Δ푣
+훼;
+4. If 푢 ∈ Δℎ
+훼 is a unit vector, then:
+‖(퐷푦푓 )−1푢‖ >
+⎧⎪⎪⎪
+⎨⎪⎪⎪⎩
+(
+푎− 훼
+푡
+훼 )
+푡
+푘,
+푦 ∈ 퐶,
+1
+(푏푟+1)푘,
+푦 ∈ 퐷,
+1
+(푏푡+1)(푏푟+1)푘, 푦 ∈ ℎ.
+3.1
+Non-uniform hyperbolicity
+For (푥, 푢) ∈ 푇핋2 with 푢 ≠ 0 and for 푛 ∈ ℕ denote by
+퐷푓 −푛(푥, 푢) = {(푦, 푤) ∈ 푇핋2 ∶ 푓 푛(푦) = 푥, 퐷푦푓 푛푤 = 푢}.
+For any non-zero tangent vector (푥, 푢) and 푛 ≥ 0, define:
+푛 = {(푧, 푤) ∈ 퐷푓 −푛(푥, 푢) ∶ 푤 ∈ Δ푣
+훼},
+푛 = 퐷푓 −푛(푥, 푢) ⧵ 푛,
+푔푛 = #푛,
+푏푛 = #푛 = 푘2푛 − 푔푛.
+From Lemmas 3.2, 3.3 one deduces:
+Lemma 3.4. Let (푥, 푢) ∈ 푇핋2.
+1. If 푢 ∈ Δ푣
+훼, then at least (푘 − 1)2 + [
+푘−1
+2 ] of its pre-images under 퐷푓 are also in Δ푣
+훼;
+2. If 푢 ∈ Δℎ
+훼, then at least [
+푘−1
+2 ] (푘 + [
+푘−1
+2 ]) of its pre-images under 퐷푓 are in Δ푣
+훼.
+By the lemma above, we get:
+푔푛+1 ≥ ((푘 − 1)2 + [
+푘 − 1
+2
+]) 푔푛 + [
+푘 − 1
+2
+] (푘 + [
+푘 − 1
+2
+]) 푏푛
+= ((푘 − 1)2 − [
+푘 − 1
+2
+] (푘 − 1 + [
+푘 − 1
+2
+])) 푔푛 + [
+푘 − 1
+2
+] (푘 + [
+푘 − 1
+2
+]) 푘2푛,
+hence:
+푔푛+1
+푘2(푛+1) ≥ 1
+푘2 ((푘 − 1)2 − [
+푘 − 1
+2
+] (푘 − 1 + [
+푘 − 1
+2
+]))
+푔푛
+푘2푛
++ 1
+푘2 [
+푘 − 1
+2
+] (푘 + [
+푘 − 1
+2
+]) .
+9
+
+Denoting by 푎푛 = 푔푛
+푘2푛 and
+푐 = 1
+푘2 ((푘 − 1)2 − [
+푘 − 1
+2
+] (푘 − 1 + [
+푘 − 1
+2
+])) ,
+푒 = 1
+푘2 [
+푘 − 1
+2
+] (푘 + [
+푘 − 1
+2
+]) ,
+the inequality above becomes:
+푎푛+1 ≥ 푐 ⋅ 푎푛 + 푒.
+Lemma 3.5. For every (푥, 푢) ∈ 푇핋2, 푢 ≠ 0, and 푛 ≥ 0 it holds:
+푎푛 ≥
+푒
+1 − 푐 (1 − 푐푛)
+=
+[
+푘−1
+2 ] (푘 + [
+푘−1
+2 ])
+2푘 − 1 + [
+푘−1
+2 ] (푘 − 1 + [
+푘−1
+2 ])
+(1 − 푐푛)
+In particular,
+lim inf 푎푛 ≥
+[
+푘−1
+2 ] (푘 + [
+푘−1
+2 ])
+2푘 − 1 + [
+푘−1
+2 ] (푘 − 1 + [
+푘−1
+2 ])
+∶= 퐿(푘),
+uniformly in (푥, 푢) ∈ 핋2.
+From now on we shall denote by 퐿(푘) =
+[ 푘−1
+2 ](푘+[ 푘−1
+2 ])
+2푘−1+[ 푘−1
+2 ](푘−1+[ 푘−1
+2 ]). As another direct con-
+sequence of Lemmas 3.2 and 3.3 we have the following:
+Lemma 3.6. If 푟, 푡 > 2훼
+푎 , then for all (푥, 푢) ∈ 푇핋2 we have:
+1. If 푢 ∈ Δ푣
+훼, then:
+퐼(푥, 푢; 푓) ≥(푘 − 1)2
+푘2
+log 푟 + (
+푘2 − 4푘 + 2 + [
+푘−1
+2 ]
+푘2
+) log 푡
++ log (
+1
+훼푘 ((푎 − 훼
+푡 ) (푎 − 훼
+푟 ))
+(푘−1)2
+푘2
+(푏 + 1
+푡 )
+− 1
+푘2(2푘−1−[ 푘−1
+2 ])
+) .
+2. If 푢 ∈ Δℎ
+훼, then:
+퐼(푥, 푢; 푓) ≥ − (
+푘2 − (푘 − 1) [
+푘−1
+2 ]
+푘2
+) log 푟 − (
+푘2 − [
+푘−1
+2 ] (2푘 − 1 + [
+푘−1
+2 ])
+푘2
+) log 푡
++ log (
+1
+푘 (
+1
+훼 (푎 − 훼
+푡 ))
+푘−1
+푘2 [ 푘−1
+2 ]−1
+(푏 + 1
+푡 )
+1
+푘2[ 푘−1
+2 ](푘+[ 푘−1
+2 ])−1
+) .
+10
+
+Now, to calculate (푓 ), we use Prop. 2.1 to compute:
+퐼(푥, 푢; 푓 푛) =
+푛−1
+∑
+푖=0
+∑
+푦∈푓 −푖(푥)
+퐼(푦, (퐷푦푓 푖)−1푢; 푓)
+푘2푖
+∶=
+푛−1
+∑
+푖=0
+퐽푖,
+and, if 푡, 푟 > 2훼
+푎 , for each 푖 we obtain:
+퐽푖 = 1
+푘2푖
+∑
+푦∈푓 −1(푥)
+퐼(푦, (퐷푦푓 푖)−1푢; 푓 ) = 1
+푘2푖
+∑
+(푦,푤)∈푖
+퐼(푦, 푤; 푓) + 1
+푘2푖
+∑
+(푦,푤)∈푖
+퐼(푦, 푤; 푓)
+≥ 푎푖푉(푡, 푟, 푘) + (1 − 푎푖)퐻(푡, 푟, 푘),
+where V and H are the right side of the inequalities obtained in Lemma 3.6 for 푢 ∈ Δ푣
+훼
+and 푢 ∈ Δℎ
+훼 respectively. It follows from Lemma 3.5, with 퐿(푘) as above and 푐푘 = [
+푘−1
+2 ],
+to simplify the notation, that:
+lim
+푖→∞ 퐽푖 ≥ 퐿(푘)푉(푡, 푟, 푘) + (1 − 퐿(푘))퐻(푡, 푟, 푘)
+= 퐶(푡, 푟, 푘) + 1
+푘2 (퐿(푘) ((푘 − 1) (2푘 − 푐푘) + 1) − (푘2 − (푘 − 1)푐푘)) log 푟 +
+1
+푘2 (퐿(푘) (2(푘 − 1)2 − 푐푘 (2(푘 − 1) + 푐푘)) − (푘2 − 푐푘 (2푘 − 1 + 푐푘))) log 푡
+,
+where
+퐶(푡, 푟, 푘) = 퐿(푘)퐶1(푡, 푟, 푘) + (1 − 퐿(푘))퐶2(푡, 푟, 푘),
+with
+퐶1(푡, 푟, 푘) = log (
+1
+훼푘 ((푎 − 훼
+푡 ) (푎 − 훼
+푟 ))
+(푘−1)2
+푘2
+(푏 + 1
+푡 )
+− 1
+푘2(2푘−1−[ 푘−1
+2 ])
+)
+퐶2(푡, 푟, 푘) = log (
+1
+푘 (
+1
+훼 (푎 − 훼
+푡 ))
+푘−1
+푘2 [ 푘−1
+2 ]−1
+(푏 + 1
+푡 )
+1
+푘2[ 푘−1
+2 ](푘+[ 푘−1
+2 ])−1
+) ,
+as in Lemma 3.6. From this, we get that for any 푘, 퐶(푡, 푟, 푘) is growing as 푡 and 푟 grow,
+then for 푡, 푟 > 2훼
+푎 , 퐶(푡, 푟, 푘) > 퐶 is uniformly bounded from below by some constant 퐶.
+Now, in order to get lim
+푖→∞ 퐽푖 > 0, we can either make 푡 or 푟 large, depending on
+whether the constant (which depends on 푘) multiplying log 푡 or log 푟 is positive or
+negative. However, for both of them, we only get positivity of the constant if 푘 ≥ 5.
+Thus, for 푘 ≥ 5, since all the bounds above are uniform for all non-zero tangent
+vectors (푥, 푢), we obtain that for 푡 (or 푟) sufficiently large, for all 푖 greater than some 푖0,
+and for all nonzero tangent vectors (푥, 푢), 퐽푖(푥, 푢) > 푁 > 0 for some constant 푁. Hence,
+there exists some 푛0 such that
+1
+푛0
+퐼(푥, 푢; 푓 푛0) = 1
+푛0
+푛0−1
+∑
+푖=0
+퐽푖(푥, 푢) > 푁
+2 > 0,
+11
+
+for all nonzero tangent vectors (푥, 푢). Therefore, (푓 ) > 0 which by Theorem 1 con-
+cludes the proof of Theorem A.
+We finish this section by including some examples for a better visualization that
+for a fixed 푘 ∈ ℕ, the bounds obtained in this section are quite simple. For that, we fix
+푘 = 5, we get 퐿(5) = 2
+3, the limitations of our last calculations become:
+lim
+푖→∞ 퐽푖 ≥ 퐶(푡, 푟, 5) + 5 log 푟 + 5 log 푡,
+with
+퐶(푡, 푟, 5) = log (
+1
+5
+훼
+17
+25
+푎2/3 (푎 − 훼
+푡 )
+1
+5
+(푎 − 훼
+푟 )
+32
+75
+(푏 + 1
+푡 )
+− 18
+25
+)
+Thus, taking the map 푠 ∶ 핋1 → ℝ as 푠(푢) = sin(2휋푢), 훿 = 1
+20, 푎 = 2휋 sin( 휋
+10), 푏 = 2휋,
+and 훼 = 1.1, we get that for every 푡, 푟 ⪆ 2푎
+훼 ≈ 1.77 the number 퐶(푡, 푟, 5)+5 log 푟 +5 log 푡
+is positive. Thus, the maps 푓(푡, 푟) = 퐸◦푣푟◦ℎ푡 satisfy the results of Theorem A.
+4
+Proof of Theorem B
+For 푘 ⋅ 퐼푑 ≠ 퐸 ∈ 푀2×2(ℤ), let 휏1(퐸) be the greatest common divisor of the entries of
+E, 휏2(퐸) = det(퐸)/휏1(퐸), so that 푑 = 휏1 ⋅ 휏2 coincides with the topological degree of the
+induced endomorphism 퐸 ∶ 핋2 → 핋2.
+We want to make a slight change in the argument used in [1] so that for every
+푥 ∈ 핋2, 푓 −1(푥) has at most one point in the critical zone. This solves the cases where
+the pair (휏1, 휏2) is (2, 4), (3, 3) or (4, 4). For the remaining four cases (1, 2), (1, 3), (1, 4)
+and (2, 2), even with this improvement in the argument, the proportion we obtain for
+vectors in the good region (which in these cases is the optimum one for the argument
+presented here) is still insufficient to obtain expansion in the vertical direction, given
+the small amount of pre-images.
+The numbers 휏1, 휏2 are the elementary divisors of E and, as in Section 2.4 of [1],
+there exists 푃 ∈ 퐺퐿2(ℤ) such that the matrix 퐺 = 푃−1 ⋅ 퐸 ⋅ 푃 satisfies:
+퐺−1(ℤ) =
+{
+(
+푖
+휏2푗
+휏1) ∶ 푖, 푗 ∈ ℤ
+}
+Moreover, as E is not a homothety, by another change of coordinates if necessary
+we may assume that E does not have (0, 1) as an eigenvector.
+With this in mind, we assume that ℙ퐸 does not fix [(0, 1)] and that 퐸−1ℤ2 = 1
+휏2ℤ× 1
+휏1ℤ.
+So there exists an 훼 > 휏2 > 1 such that if Δℎ
+훼 and Δ푣
+훼 are the corresponding horizontal
+and vertical cones as in Def. 2.2, then 퐸−1Δ푣훼 ⊂ 퐼푛푡(Δℎ
+훼). From now on, we fix such
+훼 > 휏2.
+Let 퐿 < max
+{
+1
+4휏2, 휏−1
+2 −훼−1
+2
+}
+, choose points 푧1, 푧2, 푧2, 푧4 ∈ 핋1, in this order, such that:
+12
+
+• 퐼1 = [푧1, 푧2] and 퐼3 = [푧3, 푧4] have size 퐿;
+• the translation of 퐼1 by a multiple of 1/휏2 does not intersect 퐼3;
+• 퐼2 = (푧2, 푧3) and 퐼4 = (푧4, 푧1) have size strictly larger than 1
+휏2 [
+휏2−1
+2 ],
+and define the critical and good regions ℎ, ℎ and ±
+ℎ as in Def. 2.1. As an immediate
+consequence of the definition we get:
+Proposition 4.1. For every 푥 ∈ 핋2, 퐸−1(푥) has 푑 points of which at least 1
+휏2 [
+휏2−1
+2 ] are
+inside each of +
+ℎ and −
+ℎ, and at most 휏1 of them are inside of ℎ.
+In order to have at most one pre-image of each point in the critical zone of the shear
+ℎ푡(푥1, 푥2) = (푥1, 푥2+푡푠(푥1) defined as before, we define the conservative diffeomorphism
+of the torus 푣(푥1, 푥2) = (푥1 + ̃푠(푥2), 푥2), with ̃푠 ∶ 핋1 → ℝ an analytic map which we shall
+impose restrictions later. We then study the family:
+푓푡 = 퐸◦푣◦ℎ푡,
+of area preserving endomorphism of the torus isotopic to E. We shall denote 푓 = 푓푡 to
+simplify the notation.
+Given 푥 ∈ 핋2, the set 푓 −1(푥) = ℎ−1
+푡 ◦푣−1◦퐸−1(푥) is composed by d points, and given
+푦 ∈ 푓 −1(푥), we have (퐷푦푓 )−1 = (퐷푦ℎ푡)−1◦(퐷ℎ푡(푦)푣)−1◦퐸−1.
+In order to define 푣 in a way that only one pre-image of 푥 by 푓 remains in the
+critical zone, we notice that 퐸−1(푥) is composed by 푑 points which, by the change of
+coordinates made initially, are aligned in a lattice of height 휏1 and length 휏2. We also
+notice that the map ℎ−1
+푡 keeps the vertical lines invariant. Therefore, the map 푣−1 needs
+to act in a way that it moves points on a vertical line enough so that only one remains
+in the critical zone, and, also, it cannot move them so much that we have new points
+entering the critical zone.
+In this way, we took the analytic map ̃푠 ∶ 핋1 → ℝ satisfying:
+1. If 퐿 is the size of the intervals 퐼1, 퐼3 then |̃푠(푢)| < 1
+휏2 − 퐿, for all 푢 ∈ 핋1.
+2. For all 푢 ∈ 핋1, we have that |||̃푠 (푢 + 푗
+휏1)||| > 퐿 for all 푗 ∈ {0, 1, ⋯ , 휏1 − 1} except at
+most one index.
+3. |̃푠′(푢)| < (2훼)−1, for all 푢 ∈ 핋1, where 훼 is the size of the cones fixed in the previous
+subsection.
+Notice that conditions 2 and 3 are not mutually exclusives thanks to the conditions
+for 훼 and 퐿 imposed in the previous subsection. Now, conditions 1 and 2 give us:
+13
+
+Lemma 4.1. For all 푥 ∈ 핋2, 푓 −1(푥) is composed by 푑 points of which at most one is inside
+ℎ. At least 푑 − 1 of the pre-images are inside  of which at least 휏1 [
+휏2−1
+2 ] are inside each
+of +
+ℎ and −
+ℎ.
+Proof. In the case where 퐸−1(푥) has no points in the critical zone, due to condition 1
+together with the fact that ℎ푡 preserves vertical lines, the map ℎ−1
+푡 ◦푣−1 does not take
+any of those points to the critical zone.
+In the case where 퐸−1(푥) has a point in the critical zone, it implies that we have
+exactly 휏1 points there. Due to condition 2, only one of those points is able to remain
+there, and due to condition 1, none of the other points is getting inside.
+For the minimum amount of points in each of +
+ℎ and −
+ℎ, we notice that, by Prop.
+4.1, 퐸−1(푥) already has at least 휏1 [
+휏2−1
+2 ] points inside each one, and, due to condition 1,
+those points must remain there.
+At last, condition 3 gives us the next lemma, required for the whole construction
+to work:
+Lemma 4.2. There exists 훽 > 훼 such that for all 푦 ∈ 핋2, (퐷푦푣)−1◦퐸−1Δ푣
+훽 ⊂ Δℎ
+훽, where Δ푣
+훽
+and Δℎ
+훽 are the corresponding vertical and horizontal cones of size 훽 as in Def. 2.2.
+Proof. For 푦 = (푦1, 푦2), 퐷푦푣 = (
+1
+̃푠′(푦2)
+0
+1
+). Then, due to condition 3, for all 휆 ∈ ℝ,
+퐷푦푣 ⋅ 휆푒2 = 휆(̃푠′(푦2), 1) ∈ Δ푣
+2훼. Since, by the definition of 훼, we have 퐸−1 ⋅ 휆푒2 ∈ 푖푛푡(Δℎ
+훼),
+we conclude that for all 푦 ∈ 핋2, ℙ((퐷푦푣)−1◦퐸−1)⋅[푒2] is uniformly away from [푒2], hence
+there exists such 훽 as we wanted.
+Remark 4.1. Items 3 and 4 of Lemma 2.1 also works in this cases for Δ푣
+훽 and Δℎ
+훽.
+We give the correspondent to Lemma 3.3 for this case, as a consequence of items
+3 and 4 of Lemma 2.1, Remark 4.1 and Lemma 4.2 . From now on, we fix 훽 > 훼 as in
+Lemma 4.2 and let:
+푒푣 = inf
+{
+‖(퐷푥푣)−1◦퐸−1푢‖ ∶ (푥, 푢) ∈ 푇 1핋2, 푢 ∈ Δ푣
+훽
+}
+,
+푒ℎ = inf
+{
+‖(퐷푥푣)−1◦퐸−1푢‖ ∶ (푥, 푢) ∈ 푇 1핋2, 푢 ∈ Δℎ
+훽
+}
+.
+Lemma 4.3. For 푡 > 2훽
+푎 it holds:
+1. if 푦 ∈ ℎ then (퐷푦푓 )−1Δ푣
+훽 ⊂ Δ푣
+훽, it is strictly invariant.
+2. if 푢 ∈ Δ푣
+훽 is a unit vector, then
+‖(퐷푦푓 )−1푢‖ >
+{
+푒푣(푎−훽/푡))
+훽
+푡, 푦 ∈ ℎ,
+푒푣
+훽 ,
+푦 ∈ ℎ.
+14
+
+3. if 푢 ∈ Δℎ
+훽, and (퐷ℎ푡(푦)푣)−1◦퐸−1 ⋅ 푢 = (푤1, 푤2) let ∗푦 (푢) be as in Def. 3.1. Then if
+푦 ∈ 
+∗푦(푢)
+ℎ
+we have (퐷푦푓 )−1(푢) ∈ Δ푣
+훽.
+4. if 푢 ∈ Δℎ
+훽 is a unit vector, then
+‖(퐷푦푓 )−1푢‖ >
+{
+푒ℎ,
+푦 ∈ 
+∗푦(푢)
+ℎ
+,
+푒ℎ
+푏+ 1
+푡 푡−1, 푦 ∉ 
+∗푦(푢)
+ℎ
+.
+We notice that, analogously to the homothety case, we have the problem that ∗푦 (푢)
+depends on 푦 ∈ 푓 −1(푥), therefore even though we have at least 휏1 [
+휏2−1
+2 ] points in each
+of ±
+ℎ, there could be a vector 푢 ∈ ℝ2 such that for all 푦 ∈ +
+ℎ, ∗푦 (푢) = − and vice-versa.
+However, we can see that this is not the case:
+Proposition 4.2. For every 푥 ∈ 핋2, 푢 ∈ ℝ2, there are at least 휏2 [
+휏2−1
+2 ] points 푦 ∈ 푓 −1(푥)
+such that 푦 ∈ 
+∗푦(푢)
+ℎ
+, where ∗푦 (푢) is as in Def. 3.1 changing 푣푟 for 푣.
+Proof. By the same argument used in Prop. 3.2, we can see that ∗푦 (푢) is constant for
+points 푦 ∈ 푓 −1(푥) such that ℎ푡(푦) lies in the same horizontal line. There are exactly
+휏2 pre-images 푦′ such that ℎ푡(푦) and ℎ푡(푦′) are in the same horizontal line, hence at
+least [
+휏2−1
+2 ] of these lies in 
+∗푦(푢)
+ℎ
+. As 푣−1◦퐸−1(푥) has 휏1 different vertical lines, we get the
+result.
+4.1
+Non-uniform hyperbolicity
+We end up having calculations completely mirrored in those made in Subsection 3.1,
+and for that reason we will skip the details. For (푥, 푢) ∈ 푇핋2 with 푢 ≠ 0 and for 푛 ∈ ℕ,
+we define the sets 퐷푓 −푛(푥, 푢), 푛, 푛, and the numbers 푔푛, 푏푛 = 푑푛 − 푔푛 as before. From
+Lemmas 4.1, 4.3 and Prop. 4.2 we deduce:
+Lemma 4.4. Let (푥, 푢) ∈ 푇핋2.
+1. If 푢 ∈ Δ푣
+훽, then at least 푑 − 1 of its pre-images under 퐷푓 are also in Δ푣
+훽.
+2. If 푢 ∈ Δℎ
+훽, then at least 휏1 [
+휏2−1
+2 ] of its pre-images under 퐷푓 are in Δℎ
+훽.
+For that, we get for all 푛 ∈ ℕ:
+푔푛+1 ≥ (푑 − 1 − 휏1 [
+휏2 − 1
+2
+]) 푔푛 + 휏1 [
+휏2 − 1
+2
+] 푑푛,
+hence, putting 푎푛 = 푔푛
+푑푛 :
+푎푛+1 ≥ (
+푑 − 1
+푑
+− 1
+휏2 [
+휏2 − 1
+2
+]) 푎푛 + 1
+휏2 [
+휏2 − 1
+2
+] .
+Thus, we get:
+15
+
+Lemma 4.5. For every (푥, 푢) ∈ 푇핋2, 푢 ≠ 0, and 푛 ≥ 0, it holds:
+lim inf 푎푛 ≥ 1
+휏2 [
+휏2 − 1
+2
+]
+푑
+1 + 휏1 [
+휏2−1
+2 ]
+∶= 퐿(휏1, 휏2).
+Remark 4.2. This is where we are able to verify that this argument will work for the cases
+(휏1, 휏2) as (2, 4), (3, 3) and (4, 4), where we have 퐿(휏1, 휏2) as 2/3, 3/4 and 4/5, respectively.
+And it won’t work for the other cases (1, 2), (1, 3), (1, 4) and (2, 2) where we will get 퐿(휏1, 휏2)
+as 0, 1/2, 1/2 and 0, respectively. As we will see, for the rest of the argument to work, we
+need this lower bound strictly greater than 1/2.
+As another consequence of Lemmas 4.1, 4.3 and Prop. 4.2, we get:
+Lemma 4.6. If 푡 > 2훽
+푎 , then for all (푥, 푢) ∈ 푇핋2, it holds:
+1. If 푢 ∈ Δ푣
+훽, then:
+퐼(푥, 푢; 푓) ≥ 푑 − 1
+푑
+log 푡 + log (
+푒푣
+훽 (푎 − 훽
+푡 )
+푑−1
+푑
+) .
+2. If 푢 ∈ Δℎ
+훽, then:
+퐼(푥, 푢; 푓) ≥ − (1 − 1
+휏2 [
+휏2 − 1
+2
+]) log 푡 + log (푒ℎ (푏 + 1
+푡 )
+−(1− 1
+휏2[
+휏2−1
+2 ])
+) .
+Again, by Prop. 2.1, we have:
+퐼(푥, 푢; 푓 푛) =
+푛−1
+∑
+푖=0
+∑
+푦∈푓 −푖(푥)
+퐼(푦, (퐷푦푓 푖)−1푢; 푓)
+푘2푖
+∶=
+푛−1
+∑
+푖=0
+퐽푖,
+we compute, for 푡 > 2훽
+푎 , for all 푖 ≥ 0:
+퐽푖 = 1
+푑
+∑
+(푦,푤)∈푖
+퐼(푦, 푤; 푓) + 1
+푑
+∑
+(푦,푤)∈푖
+퐼(푦, 푤; 푓)
+≥ 푎푖푉(푡, 휏1, 휏2) + (1 − 푎푖)퐻(푡, 휏1, 휏2),
+where 푎푖 is as in Lemma 4.5, 푉 and 퐻 are the right side of the inequalities obtained in
+Lemma 4.6 for 푢 ∈ Δ푣
+훽 and 푢 ∈ Δℎ
+훽 respectively. It follows:
+lim
+푖→∞ 퐽푖 ≥ 퐿(휏1, 휏2)푉(푡, 휏1, 휏2) + (1 − 퐿(휏1, 휏2))퐻(푡, 휏1, 휏2)
+= (휏1 − 2
+휏2) [
+휏2−1
+2 ] − 1
+1 + 휏1 [
+휏2−1
+2 ]
+log 푡 + 퐶(푡, 휏1, 휏2),
+16
+
+where:
+퐶(푡, 휏1, 휏2) =퐿(휏1, 휏2) log (
+푒푣
+훽 (푎 − 훽
+푡 )
+푑−1
+푑
+)
++ (1 − 퐿(휏1, 휏2)) log (푒ℎ (푏 + 1
+푡 )
+−(1− 1
+휏2[
+휏2−1
+2 ])
+) > 퐶,
+for all 푡 > 2훽
+푎 , that is, 퐶(푡, 휏1, 휏2) is uniformly bounded from below by some constant C.
+Since 푑 = 휏1 ⋅ 휏2 > 4, the constant multiplying log 푡 is positive. Therefore, since all
+the bounds above are uniform for all non-zero tangent vectors (푥, 푢), as in the homo-
+thety case we obtain that for 푡 sufficiently large, for all 푛 greater than some 푛0, and for
+all nonzero tangent vectors (푥, 푢):
+1
+푛퐼(푥, 푢; 푓 푛) = 1
+푛
+푛−1
+∑
+푖=0
+퐽푖(푥, 푢) > 0,
+hence, (푓 ) > 0 which by Theorem 1 concludes the proof of Theorem B.
+References
+[1] M. Andersson, P. D. Carrasco, and R. Saghin, “Non-uniformly hyperbolic endo-
+morphisms,” 2022.
+[2] L. Barreira and Y. Pesin, Introduction to Smooth Ergodic Theory. Graduate Studies
+in Mathematics, American Mathematical Society, 2013.
+[3] V. I. Oseledets, “A multiplicative ergodic theorem. characteristic ljapunov, expo-
+nents of dynamical systems,” Trudy Moskovskogo Matematicheskogo Obshchestva,
+vol. 19, pp. 179–210, 1968.
+[4] D. V. Anosov, “Geodesic flows on closed riemannian manifolds of negative curva-
+ture,” Trudy Mat. Inst. Steklov, vol. 90, pp. 3–210, 1967.
+[5] M. Qian, J.-S. Xie, and S. Zhu, Smooth Ergodic Theory for Endomorphisms, vol. 1978
+of Lecture Notes in Mathematics. 01 2009.
+17
+

AdE0T4oBgHgl3EQfPgCp/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf,len=428
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='02180v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='DS]  5 Jan 2023  Existence of robust non-uniformly hyperbolic  endomorphism in homotopy classes  Victor Janeiro  victorgjaneiro@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='com, ICEx-UFMG, Belo Horizonte-MG, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Abstract  We extend the results of [1] by showing that any homothety in 핋2 is homo-  topic to a non-uniformly hyperbolic ergodic area preserving map, provided that  its degree is at least 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' We also address other small topological degree cases not  considered in the previous article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' This proves the existence of a \ue22f1 open set  of non-uniformly hyperbolic systems, that intersects essentially every homotopy  classes in 핋2, where the Lyapunov exponents vary continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  1  Introduction  We study conservative maps of the two-torus 핋2 from the point of view of smooth  ergodic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' We are interested in the Lyapunov exponents of these systems, in  particular, in extending the results obtained in [1] to the homothety case and some  cases with lower topological degree, which were not included in the previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  With this in mind, some familiarity with the results of [1] is desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  For a differentiable covering map 푓 ∶ 핋2 → 핋2 and a pair (푥, 푣) ∈ 푇핋2, the number  ̃\ue244(푥, 푣) = lim sup  푛→∞  log ‖퐷푥푓 푛(푣)‖  푛  is the Lyapunov exponent of 푓 at (푥, 푣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' See [2] for background in Smooth Ergodic  Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Due to Oseledet’s Theorem [3] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' there is a full area set 푀0 on 핋2 where the  previous limit exists for every 푣,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' and there exists a measurable bundle 퐸− defined on  푀0 such that for 푥 ∈ 푀0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푣 ≠ 0 ∈ 퐸−(푥):  \ue244(푥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푣) ∶= lim  푛→∞  log ‖퐷푥푓 푛(푣)‖  푛  = lim  푛→∞  log 푚(퐷푥푓 푛)  푛  ∶= \ue244−(푥),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  while for 푣 ∈ ℝ2 ⧵ 퐸−(푥):  \ue244(푥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푣) = lim  푛→∞  log ‖퐷푥푓 푛‖  푛  ∶= \ue244+(푥),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' if 휇 denotes the Lebesgue (Haar) measure on 핋2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' then:  ∫ (\ue244+(푥) + \ue244−(푥))푑휇(푥) = ∫ log | det 퐷푥푓 |푑휇(푥) > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  (1)  1  so \ue244+(푥) > 0 almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' At last, we say that 푓 is non-uniformly hyperbolic  (NUH) if \ue244−(푥) < 0 < \ue244+(푥) almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Non uniformly hyperbolic systems provide a generalization of the classical Anosov  surface maps [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Here, we will only be concerned with the non-invertible case in an  attempt to aid the understanding of their statistical properties, which is still under  development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For the general ergodic theory of endomorphisms, the reader is directed  to [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Any map 푓 ∶ 핋2 → 핋2 is homotopic to a linear endomorphism 퐸 ∶ 핋2 → 핋2,  induced by an integer matrix that we denote by the same letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' In [1], it is established  the existence of a \ue22f1 open set of non-uniformly hyperbolic systems that intersects  every homotopy class that does not contain a homothety, provided that the degree is  not too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' The authors then conjecture that the same is true for homotheties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' In  this article, we prove this conjecture, provided that the degree is at least 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' There are  other low topological degree cases not covered by Andersson, Carrasco and Saghin,  which we also address here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Let End푟  휇(핋2) be the set of \ue22f푟 local diffeomorphisms of 핋2 preserving the Lebesgue  measure 휇, that are not invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For 푓 ∈ End푟  휇(핋2), (푥, 푣) ∈ 푇 1핋2 define:  퐼(푥, 푣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓 푛) =  ∑  푦∈푓 −푛(푥)  log ‖(퐷푦푓 푛)−1푣‖  det(퐷푦푓 푛)  ,  and  퐶\ue244(푓 ) = ∑  푛∈ℕ  1  푛  inf  (푥,푣)∈푇 1핋2 퐼(푥, 푣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓 푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Define the set  \ue241 ∶= {푓 ∈ End푟  휇(핋2) ∶ 퐶\ue244(푓 ) > 0},  which is open in the \ue22f1-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' On Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='3 of the main reference [1], it is  proved:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푓 ∈ \ue241 , then 푓 is non-uniformly hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Our main results are:  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For 퐸 = 푘 ⋅퐼푑 ∈ 푀2×2(ℤ), with |푘| ≥ 5, the intersection [퐸]∩\ue241 is non-empty  and in fact contains maps that are real analytically homotopic to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 퐸 ∈ 푀2×2(ℤ) is not a homothety and 푑푒푡(퐸) > 4, the intersection [퐸] ∩ \ue241  is non-empty and in fact contains maps that are real analytically homotopic to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Our Theorem B is equivalent to the Theorem A of [1] but includes three cases  which are not proved there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' The main difficulty for our results is that, in the case of  2  a homothety, the induced projective action is trivial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' non-triviality of this projective  action is a central piece in the method of Andersson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Finally, by inspection on the proofs of Theorems B and C of [1], we can see that it  works for all cases included here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Hence, defining:  퐶det(푓 ) ∶= sup  푛∈ℕ  1  푛 inf  푥∈핋2 log(det(퐷푥푓 푛)) > 0,  and the open set:  \ue2411 ∶=  {  푓 ∈ End푟  휇(핋2) ∶ 퐶\ue244(푓 ) > −1  2퐶det(푓 )  }  ,  we have from Theorems A and B that if a linear endomorphism 퐸 satisfies the condi-  tions of either of the Theorems, then [퐸] ∩ \ue2411 ≠ ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Therefore, by Theorem 퐵 of [1], we  have conituity of the maps \ue2411 ∋ 푓 ↦ ∫핋2 \ue244±(푓 )푑휇 in the \ue22f1 topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  From Theorem C of [1], we conclude that for any linear endomorphism E as in  Theorem A or B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If ±1 is not an eigenvalue of 퐸, then [퐸] ∩ \ue241 contains stably ergodic  endomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' In fact, it contains stably Bernoulli endomorphisms and, in particular,  maps that are mixing of all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Acknowledgements  The results presented here were conjectured by Martin Andersson, Pablo D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Carrasco  and Radu Saghin in [1], I thank Pablo D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Carrasco, who is also my MSc advisor, for the  suggestion of the problem and for the hours of conversations on the subject that were  crucial to this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  This work has been supported by the Brazillian research agencies CAPES and  CNPq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  2  Preliminary  In order to prove Theorems A and B, we require a result on the computation of the  numbers 퐼(푥, 푣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓 푛) which the proof can be found in [1]:  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For any 푛 ∈ ℕ, it holds:  퐼(푥, 푣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓 푛) =  푛−1  ∑  푖=0  ∑  푦∈푓 −푖(푥)  퐼(푦, 퐹 −푖  푦 푣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓 )  det(퐷푦푓 푖) ,  (2)  where 퐹 −푖  푦 푣 =  (퐷푦푓 푖)−1푣  ‖(퐷푦푓 푖)−푖푣‖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1  Shears  For fixed points 푧1, 푧2, 푧3, 푧4 ∈ 핋1, in this order, take the closed intervals 퐼1 = [푧1, 푧2],  퐼3 = [푧3, 푧4], and the open intervals 퐼2 = (푧2, 푧3) and 퐼4 = (푧4, 푧1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' We define the horizontal and vertical critical regions in 핋2 as \ue22fℎ = (퐼1 ∪  퐼3) × 핋1, \ue22f푣 = 핋1 × (퐼1 ∪ 퐼3) and its complements \ue233ℎ = 핋2 ⧵ \ue22fℎ , \ue233푣 = 핋2 ⧵ \ue22f푣 are respectively  the horizontal and vertical good region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  We then divide the good regions into \ue233+  ℎ = 퐼4 × 핋1, \ue233−  ℎ = 퐼2 × 핋1, \ue233+  푣 = 핋1 × 퐼4 and  \ue233−  푣 = 핋1 × 퐼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  For fixed numbers 0 < 푎 < 푏, we take 푠 ∶ 핋1 → ℝ as an analytic map satisfying  the following conditions:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푧 ∈ 퐼4, then 푎 < 푠′(푧) < 푏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푧 ∈ 퐼2, then −푏 < 푠′(푧) < −푎;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푧 ∈ 퐼1 ∪ 퐼3, then |푠′(푧)| < 푏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Consider the two families of conservative diffeomorphisms of the torus given by:  ℎ푡(푥1, 푥2) = (푥1, 푥2 + 푡푠(푥1)), 푣푟(푥1, 푥2) = (푥1 + 푟푠(푥2), 푥2),  푡, 푟 ∈ ℝ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Note that:  퐷(푥1,푥2)ℎ푡 = (  1  0  푡푠′(푥1)  1) ,  퐷(푥1,푥2)푣푟 = (  1  푟푠′(푥2)  0  1  ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  In order to simplify the computations we will consider the maximum norm on 푇핋2  as ‖(푢1, 푢2)‖ = max{|푢1|, |푢2|}, and all the computations from now on are performed  using this norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' This way, we get, for every 푥 ∈ 핋2:  ‖퐷푥ℎ푡‖ < 푏푡 + 1, and ‖퐷푥푣푟‖ < 푏푡 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Given 훼 > 0, the corresponding horizontal cone is Δℎ  훼 = {(푢1, 푢2) ∈ ℝ2 ∶  |푢2| ≤ 훼|푢1|}, while the corresponding vertical cone is its complement Δ푣  훼 = ℝ2 ⧵ Δℎ  훼,  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For 훼 > 1, let Δℎ  훼 and Δ푣  훼 be the corresponding horizontal and vertical cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Then, for every 푡, 푟 > 2훼  푎 , and, for every unit vector 푢 ∈ 푇푥핋2, the following holds:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푢 ∈ Δ푣  훼, and:  (a) 푥 ∈ \ue233푣, then  (퐷푥푣푟)−1푢 ∈ Δℎ  훼 (퐷푥푣−1  푟 Δ푣  훼 ⊂ Δℎ  훼);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  ‖(퐷푥푣푟)−1푢‖ > 푎푟−훼  훼  = 푟  푎− 훼  푟  훼 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  4  (b) 푥 ∈ \ue22f푣, then ‖(퐷푥푣푟)−1푢‖ > 1  훼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푢 = ±(1, 푢2) ∈ Δℎ  훼, then:  (a) either for every 푥 ∈ \ue233+  푣 ( if 푢2 ≤ 0) or for every 푥 ∈ \ue233−  푣 (if 푢2 ≥ 0) it holds:  (퐷푥푣푟)−1푢 ∈ Δℎ  훼;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  ‖(퐷푥푣푟)−1푢‖ > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  (b) for all other 푥, we have ‖(퐷푥푣푟)−1푢‖ >  1  푏푟+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푢 ∈ Δℎ  훼, and:  (a) 푥 ∈ \ue233ℎ, then  (퐷푥ℎ푡)−1푢 ∈ Δ푣  훼 (퐷푥ℎ−1  푡 Δℎ  훼 ⊂ Δ푣  훼);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  ‖(퐷푥ℎ푡)−1푢‖ > 푎푡−훼  훼  = 푡  푎− 훼  푡  훼 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  (b) 푥 ∈ \ue22fℎ, then ‖(퐷푥ℎ푡)−1푢‖ > 1  훼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푢 = ±(푢1, 1) ∈ Δ푣  훼, then:  (a) either for every 푥 ∈ \ue233+  ℎ ( if 푢1 ≤ 0) or for every 푥 ∈ \ue233−  ℎ (if 푢1 ≥ 0) it holds:  (퐷푥ℎ푡)−1푢 ∈ Δ푣  훼;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  ‖(퐷푥ℎ푡)−1푢‖ > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  (b) for all other 푥, we have ‖(퐷푥ℎ푡)−1푢‖ >  1  푏푡+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' We prove items 1 and 2, the case for ℎ푡 is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Let 푥 = (푥1, 푥2) ∈ \ue233푣, and  푢± = (1, ±훼) then:  (퐷푥푣푟)−1푢± = (  1  −푟푠′(푥2)  0  1  ) (  1  ±훼) = (  1 ∓ 푟푠′(푥2)훼  ±훼  ) ,  also since 푥 ∈ \ue233푣, 푎 < |푠′(푥2)| < 푏, we also have 훼 > 1 and 푟 > 2훼  푎 , hence:  |1 ∓ 푟푠′(푥2)훼| ≥ 푟훼푎 − 1 > 2훼2 − 1 > 훼 > 1,  which shows that (퐷푥푣푟)−1Δ푣  훼 ⊂ Δℎ  훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Also, ‖(퐷푥푣푟)−1푢‖ = |1 ∓ 푟푠′(푥2)훼| > 푟푎훼 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Now,  noticing that the minimal expansion of vectors in Δ푣  훼 occurs on either of (1, ±훼), we  have for every unit vector 푢 ∈ Δ푣  훼:  ‖(퐷푥푣푟)−1푢‖ ≥ ‖(퐷푥푣푟)−1(1, ±훼)‖  ‖(1, ±훼)‖  > 푟훼 − 1  훼  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  For part 2 (a), we have for x ∈ \ue233+  푣 푠′(푥2) > 푎 > 0, and for 푥 ∈ \ue233−  푣, 푠′(푥2) < −푎 < 0,  thus, by simple calculations analogous to the last one, we get the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Finally, for  (b) we just use 푚((퐷푥푣푟)−1) =  1  ‖퐷푥푣푟‖ >  1  푏푟+1 for every 푥 ∈ 핋2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  5  3  Endomorphisms and Shears: Proof of Theorem A  Fix 퐸 = 푘 ⋅ 퐼푑, for some 푘 ∈ ℕ (we shall make the entire argument on 푘 ∈ ℕ for the  sake of simplicity of notation, we emphasize that the entire argument works for 푘 ∈ ℤ  by replacing 푘 for |푘| when necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Fix a 훿 <  1  4푘 and define the critical and good  regions as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1 for points 푧1, 푧2, 푧3, 푧4 ∈ 핋1 such that:  퐼1 = [푧1, 푧2] and 퐼3 = [푧3, 푧4] have size 2훿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  The translation of 퐼1 by a multiple of 1  푘 does not intersect 퐼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  퐼2 = (푧2, 푧3) and 퐼4 = (푧4, 푧1) have size strictly larger than 1  푘 [  푘−1  2 ], where [푝]  denotes the floor of 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  It is obtained directly from the definitions that:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For every 푥 = (푥1, 푥2) ∈ 핋2, 퐸−1(푥) has 푘2 points given by:  퐸−1(푥1, 푥2) =  {  (  푥1 + 푖  푘  , 푥2 + 푗  푘  ) ∶ 푖, 푗 = 0, ⋯ , 푘 − 1  }  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  At least 푘 [  푘−1  2 ] are inside each of \ue233+  푣, \ue233−  푣, \ue233+  ℎ and \ue233−  ℎ, and at most 푘 of them are inside  each of \ue22f푣, \ue22fℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  From now on, in this section, we fix any 훼 > 1 and the corresponding cones as in  Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' We consider the analytic maps:  푓(푡,푟) = 퐸◦푣푟◦ℎ푡,  which we shall denote only by 푓 = 푓(푡,푟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Clearly 푓 is an area preserving endomorphism  isotopic to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' We observe that, given 푥 ∈ 핋2 and 푦 ∈ 푓 −1(푥), we have:  (퐷푦푓 )−1 = (퐷푦ℎ푡)−1(퐷ℎ푡(푦)푣푟)−1퐸−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  The goal is for (퐷ℎ푡(푦)푣푟)−1 to take vectors in the vertical cone and expand them  in the horizontal direction and then (퐷푦ℎ푡)−1 takes its images and expands them in  the vertical direction, resulting in (퐷푦푓 )−1 expanding in the vertical direction for most  points in 푓 −1(푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Thus, in order to keep track of this derivative, we must localize the  points 푦 ∈ 푓 −1(푥) in regard to which of \ue233ℎ or \ue22fℎ they belong, and {ℎ푡(푦) ∶ 푦 ∈ 푓 −1(푥)} =  (퐸◦푣푟)−1(푥) regarding which of \ue233푣 or \ue22f푣 they belong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For every 푥 ∈ 핋2, we have:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' (푣푟◦퐸)−1(푥) has 푘2 points of which at least 푘 [  푘−1  2 ] of them are in each one of \ue233+  푣 and  \ue233−  푣 and at most 푘 of them are in \ue22f푣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓 −1(푥) has 푘2 points of which at least 푘 [  푘−1  2 ] of them are in each one of \ue233+  ℎ and \ue233−  ℎ  and at most 푘 of them are in \ue22fℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' It is a direct consequence of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1 along with the fact that the regions  \ue233+  푣, \ue233−  푣 and \ue22f푣 are invariant under 푣푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Notice that in each row of pre-images by E of a point 푥 = (푥1, 푥2) given by  {  (  푥1+푖  푘 , 푥2+푗0  푘 ) ∶ 푖 = 0, ⋯ , 푘 − 1  }  for a fixed 푗0 ∈ {0, ⋯ , 푘 − 1}, 푣−1  푟  is a rotation  by −푟푠 (  푥2+푗0  푘 ) in the circle 핋1 ×  { 푥2+푗0  푘  }  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Hence, at least [  푘−1  2 ] of the 푘 points of  this row are inside each one of \ue233+  ℎ and \ue233−  ℎ, and at most 1 is in \ue22fℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  As this is also true for all the 푘 rows of pre-images by E, we get at least 푘 [  푘−1  2 ]  pre-images by 퐸◦푣푟 are inside each one of \ue233+  ℎ and \ue233−  ℎ, and at most 푘 pre-images  by 퐸◦푣푟 are inside \ue22fℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Finally, since these sets are invariant under ℎ푡, we get the  desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Even knowing which regions is a point 푦 ∈ (퐸◦푣푟)−1(푥), we cannot de-  termine the region which ℎ−1  푡 (푦) is inside, as 푡 is varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' That is, there may be points  푦 ∈ 푓 −1(푥) that are in \ue233ℎ such that ℎ푡(푦) ∈ \ue22f푣 and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' In order to keep track of the vectors, define:  For 푢 = (푢1, 푢2) ∈ ℝ2 with 푢2 ≠ 0:  ∗ (푢) =  {  −sgn (  푢1  푢2) , if 푢1 ≠ 0,  −sgn(푢2),  if 푢1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Notice that ∗ (푢) = ∗ (퐸−1푢), for every 푢 ∈ ℝ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  For 푥 ∈ 핋2, 푦 ∈ 푓 −1(푥) and 푢 ∈ ℝ2, let (푤1, 푤2) = (퐷ℎ푡(푦)푣푟)−1퐸−1푢:  ∗푦 (푢) =  ⎧⎪⎪  ⎨⎪⎪⎩  −sgn (  푤1  푤2) , if 푤1, 푤2 ≠ 0,  −sgn(푤2),  if 푤2 ≠ 0, 푤1 = 0,  −sgn(푤1),  if 푤1 ≠ 0, 푤2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  In view of item 4 of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1, even though (퐷ℎ푡(푦)푣푟)−1 may not send a vector  푢 ∈ Δ푣  훼 to the horizontal cone if ℎ푡(푦) ∈ \ue22f푣, we can still end up having expansion in the  vertical direction, depending on whether 푦 ∈ \ue233  ∗푦(푢)  ℎ  or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' In this regard, from Lemma  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1, there are 푘 points 푦 ∈ 푓 −1(푥) such that ℎ푡(푦) are in \ue22f푣, and these points (ℎ푡(푦)) are  all in the same circle 핋1 ×  { 푥2+푗0  푘  }  , hence the derivative (퐷ℎ푡(푦)푣푟)−1 is the same for those  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' We get:  7  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For every 푢 ∈ ℝ2, 푥 ∈ 핋2, then the sign ∗푦 (푢) = sg (  푤1  푤2) is the same for  all points 푦 ∈ 푓 −1(푥) such that ℎ푡(푦) ∈ \ue22f푣, where ∗푦 (푢) is as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For a fixed 푥 ∈ 핋2 and:  푢 ∈ Δ푣  훼, define:  ⎧⎪⎪⎪⎪  ⎨⎪⎪⎪⎪⎩  퐴 = {푦 ∈ 푓 −1(푥) ∶ 푦 ∈ \ue233ℎ, ℎ푡(푦) ∈ \ue233푣}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  퐵 = {푦 ∈ 푓 −1(푥) ∶ 푦 ∈ \ue233  ∗푦(푢)  ℎ  , ℎ푡(푦) ∈ \ue22f푣},  \ue242푣 = 퐴 ∪ 퐵,  \ue242ℎ = 푓 −1(푥) ⧵ \ue242푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  푢 ∈ Δℎ  훼, define:  ⎧⎪⎪⎪⎪  ⎨⎪⎪⎪⎪⎩  퐶 = {푦 ∈ 푓 −1(푥) ∶ 푦 ∈ \ue233ℎ, ℎ푡(푦) ∈ \ue233∗(푢)  푣 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  퐷 = {푦 ∈ 푓 −1(푥) ∶ 푦 ∈ \ue233  ∗푦(푢)  ℎ  , ℎ푡(푦) ∈ \ue22f푣 ∪ \ue233−∗(푢)  푣  },  \ue234푣 = 퐶 ∪ 퐷,  \ue234ℎ = 푓 −1(푥) ⧵ \ue234푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  A direct consequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1 and Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='2, having Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1 in mind, is  the following:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For a fixed (푥, 푢) ∈ 푇핋2, 푓 −1(푥) has 푘2 points, of which:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For 푢 ∈ Δ푣  훼, at most 2푘 − 1 − [  푘−1  2 ] of them are in \ue242ℎ and at least (푘 − 1)2 + [  푘−1  2 ]  are inside \ue242푣, because:  At least (푘 − 1)2 are in A and,  at least [  푘−1  2 ] are in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For 푢 ∈ Δℎ  훼, at most 푘2 − [  푘−1  2 ] (푘 + [  푘−1  2 ]) are in \ue234ℎ and at least [  푘−1  2 ] (푘 + [  푘−1  2 ])  are in \ue234푣, because:  At least (푘 − 1) [  푘−1  2 ] are in C and,  at least [  푘−1  2 ] (1 + [  푘−1  2 ]) are in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Knowing that for every unit vector 푢 ∈ ℝ2 we have ‖퐸−1푢‖ = 1  푘 (maximum norm),  from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1 we get:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For 푡, 푟 > 2훼  푎 and for fixed 푥 ∈ 핋2, it holds:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푢 ∈ Δ푣  훼, then for all 푦 ∈ \ue242푣 we have (퐷푦푓 )−1푢 ∈ Δ푣  훼;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푢 ∈ Δ푣  훼 is a unit vector, then:  ‖(퐷푦푓 )−1푢‖ >  ⎧⎪⎪⎪  ⎨⎪⎪⎪⎩  (  푎− 훼  푡  훼 ) (  푎− 훼  푟  훼 )  푡푟  푘 , 푦 ∈ 퐴,  1  훼푘,  푦 ∈ 퐵,  1  (푏푡+1)훼푘,  푦 ∈ \ue242ℎ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푢 ∈ Δℎ  훼, then for all 푦 ∈ \ue234푣 we have (퐷푦푓 )−1푢 ∈ Δ푣  훼;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푢 ∈ Δℎ  훼 is a unit vector, then:  ‖(퐷푦푓 )−1푢‖ >  ⎧⎪⎪⎪  ⎨⎪⎪⎪⎩  (  푎− 훼  푡  훼 )  푡  푘,  푦 ∈ 퐶,  1  (푏푟+1)푘,  푦 ∈ 퐷,  1  (푏푡+1)(푏푟+1)푘, 푦 ∈ \ue234ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1  Non-uniform hyperbolicity  For (푥, 푢) ∈ 푇핋2 with 푢 ≠ 0 and for 푛 ∈ ℕ denote by  퐷푓 −푛(푥, 푢) = {(푦, 푤) ∈ 푇핋2 ∶ 푓 푛(푦) = 푥, 퐷푦푓 푛푤 = 푢}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  For any non-zero tangent vector (푥, 푢) and 푛 ≥ 0, define:  \ue233푛 = {(푧, 푤) ∈ 퐷푓 −푛(푥, 푢) ∶ 푤 ∈ Δ푣  훼},  \ue22e푛 = 퐷푓 −푛(푥, 푢) ⧵ \ue233푛,  푔푛 = #\ue233푛,  푏푛 = #\ue22e푛 = 푘2푛 − 푔푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  From Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='3 one deduces:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Let (푥, 푢) ∈ 푇핋2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푢 ∈ Δ푣  훼, then at least (푘 − 1)2 + [  푘−1  2 ] of its pre-images under 퐷푓 are also in Δ푣  훼;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푢 ∈ Δℎ  훼, then at least [  푘−1  2 ] (푘 + [  푘−1  2 ]) of its pre-images under 퐷푓 are in Δ푣  훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  By the lemma above, we get:  푔푛+1 ≥ ((푘 − 1)2 + [  푘 − 1  2  ]) 푔푛 + [  푘 − 1  2  ] (푘 + [  푘 − 1  2  ]) 푏푛  = ((푘 − 1)2 − [  푘 − 1  2  ] (푘 − 1 + [  푘 − 1  2  ])) 푔푛 + [  푘 − 1  2  ] (푘 + [  푘 − 1  2  ]) 푘2푛,  hence:  푔푛+1  푘2(푛+1) ≥ 1  푘2 ((푘 − 1)2 − [  푘 − 1  2  ] (푘 − 1 + [  푘 − 1  2  ]))  푔푛  푘2푛  + 1  푘2 [  푘 − 1  2  ] (푘 + [  푘 − 1  2  ]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  9  Denoting by 푎푛 = 푔푛  푘2푛 and  푐 = 1  푘2 ((푘 − 1)2 − [  푘 − 1  2  ] (푘 − 1 + [  푘 − 1  2  ])) ,  푒 = 1  푘2 [  푘 − 1  2  ] (푘 + [  푘 − 1  2  ]) ,  the inequality above becomes:  푎푛+1 ≥ 푐 ⋅ 푎푛 + 푒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For every (푥, 푢) ∈ 푇핋2, 푢 ≠ 0, and 푛 ≥ 0 it holds:  푎푛 ≥  푒  1 − 푐 (1 − 푐푛)  =  [  푘−1  2 ] (푘 + [  푘−1  2 ])  2푘 − 1 + [  푘−1  2 ] (푘 − 1 + [  푘−1  2 ])  (1 − 푐푛)  In particular,  lim inf 푎푛 ≥  [  푘−1  2 ] (푘 + [  푘−1  2 ])  2푘 − 1 + [  푘−1  2 ] (푘 − 1 + [  푘−1  2 ])  ∶= 퐿(푘),  uniformly in (푥, 푢) ∈ 핋2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  From now on we shall denote by 퐿(푘) =  [ 푘−1  2 ](푘+[ 푘−1  2 ])  2푘−1+[ 푘−1  2 ](푘−1+[ 푘−1  2 ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' As another direct con-  sequence of Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='3 we have the following:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푟, 푡 > 2훼  푎 , then for all (푥, 푢) ∈ 푇핋2 we have:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푢 ∈ Δ푣  훼, then:  퐼(푥, 푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓) ≥(푘 − 1)2  푘2  log 푟 + (  푘2 − 4푘 + 2 + [  푘−1  2 ]  푘2  ) log 푡  + log (  1  훼푘 ((푎 − 훼  푡 ) (푎 − 훼  푟 ))  (푘−1)2  푘2  (푏 + 1  푡 )  − 1  푘2(2푘−1−[ 푘−1  2 ])  ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푢 ∈ Δℎ  훼, then:  퐼(푥, 푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓) ≥ − (  푘2 − (푘 − 1) [  푘−1  2 ]  푘2  ) log 푟 − (  푘2 − [  푘−1  2 ] (2푘 − 1 + [  푘−1  2 ])  푘2  ) log 푡  + log (  1  푘 (  1  훼 (푎 − 훼  푡 ))  푘−1  푘2 [ 푘−1  2 ]−1  (푏 + 1  푡 )  1  푘2[ 푘−1  2 ](푘+[ 푘−1  2 ])−1  ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  10  Now, to calculate \ue22f\ue244(푓 ), we use Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1 to compute:  퐼(푥, 푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓 푛) =  푛−1  ∑  푖=0  ∑  푦∈푓 −푖(푥)  퐼(푦, (퐷푦푓 푖)−1푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓)  푘2푖  ∶=  푛−1  ∑  푖=0  퐽푖,  and, if 푡, 푟 > 2훼  푎 , for each 푖 we obtain:  퐽푖 = 1  푘2푖  ∑  푦∈푓 −1(푥)  퐼(푦, (퐷푦푓 푖)−1푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓 ) = 1  푘2푖  ∑  (푦,푤)∈\ue233푖  퐼(푦, 푤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓) + 1  푘2푖  ∑  (푦,푤)∈\ue22e푖  퐼(푦, 푤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓)  ≥ 푎푖푉(푡, 푟, 푘) + (1 − 푎푖)퐻(푡, 푟, 푘),  where V and H are the right side of the inequalities obtained in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='6 for 푢 ∈ Δ푣  훼  and 푢 ∈ Δℎ  훼 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' It follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' with 퐿(푘) as above and 푐푘 = [  푘−1  2 ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  to simplify the notation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' that:  lim  푖→∞ 퐽푖 ≥ 퐿(푘)푉(푡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푘) + (1 − 퐿(푘))퐻(푡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푘)  = 퐶(푡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푘) + 1  푘2 (퐿(푘) ((푘 − 1) (2푘 − 푐푘) + 1) − (푘2 − (푘 − 1)푐푘)) log 푟 +  1  푘2 (퐿(푘) (2(푘 − 1)2 − 푐푘 (2(푘 − 1) + 푐푘)) − (푘2 − 푐푘 (2푘 − 1 + 푐푘))) log 푡  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  where  퐶(푡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푘) = 퐿(푘)퐶1(푡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푘) + (1 − 퐿(푘))퐶2(푡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푘),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  with  퐶1(푡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푘) = log (  1  훼푘 ((푎 − 훼  푡 ) (푎 − 훼  푟 ))  (푘−1)2  푘2  (푏 + 1  푡 )  − 1  푘2(2푘−1−[ 푘−1  2 ])  )  퐶2(푡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푟,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푘) = log (  1  푘 (  1  훼 (푎 − 훼  푡 ))  푘−1  푘2 [ 푘−1  2 ]−1  (푏 + 1  푡 )  1  푘2[ 푘−1  2 ](푘+[ 푘−1  2 ])−1  ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' From this, we get that for any 푘, 퐶(푡, 푟, 푘) is growing as 푡 and 푟 grow,  then for 푡, 푟 > 2훼  푎 , 퐶(푡, 푟, 푘) > 퐶 is uniformly bounded from below by some constant 퐶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Now, in order to get lim  푖→∞ 퐽푖 > 0, we can either make 푡 or 푟 large, depending on  whether the constant (which depends on 푘) multiplying log 푡 or log 푟 is positive or  negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' However, for both of them, we only get positivity of the constant if 푘 ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Thus, for 푘 ≥ 5, since all the bounds above are uniform for all non-zero tangent  vectors (푥, 푢), we obtain that for 푡 (or 푟) sufficiently large, for all 푖 greater than some 푖0,  and for all nonzero tangent vectors (푥, 푢), 퐽푖(푥, 푢) > 푁 > 0 for some constant 푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Hence,  there exists some 푛0 such that  1  푛0  퐼(푥, 푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓 푛0) = 1  푛0  푛0−1  ∑  푖=0  퐽푖(푥, 푢) > 푁  2 > 0,  11  for all nonzero tangent vectors (푥, 푢).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Therefore, \ue22f\ue244(푓 ) > 0 which by Theorem 1 con-  cludes the proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  We finish this section by including some examples for a better visualization that  for a fixed 푘 ∈ ℕ, the bounds obtained in this section are quite simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For that, we fix  푘 = 5, we get 퐿(5) = 2  3, the limitations of our last calculations become:  lim  푖→∞ 퐽푖 ≥ 퐶(푡, 푟, 5) + 5 log 푟 + 5 log 푡,  with  퐶(푡, 푟, 5) = log (  1  5  훼  17  25  푎2/3 (푎 − 훼  푡 )  1  5  (푎 − 훼  푟 )  32  75  (푏 + 1  푡 )  − 18  25  )  Thus, taking the map 푠 ∶ 핋1 → ℝ as 푠(푢) = sin(2휋푢), 훿 = 1  20, 푎 = 2휋 sin( 휋  10), 푏 = 2휋,  and 훼 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1, we get that for every 푡, 푟 ⪆ 2푎  훼 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='77 the number 퐶(푡, 푟, 5)+5 log 푟 +5 log 푡  is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Thus, the maps 푓(푡, 푟) = 퐸◦푣푟◦ℎ푡 satisfy the results of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  4  Proof of Theorem B  For 푘 ⋅ 퐼푑 ≠ 퐸 ∈ 푀2×2(ℤ), let 휏1(퐸) be the greatest common divisor of the entries of  E, 휏2(퐸) = det(퐸)/휏1(퐸), so that 푑 = 휏1 ⋅ 휏2 coincides with the topological degree of the  induced endomorphism 퐸 ∶ 핋2 → 핋2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  We want to make a slight change in the argument used in [1] so that for every  푥 ∈ 핋2, 푓 −1(푥) has at most one point in the critical zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' This solves the cases where  the pair (휏1, 휏2) is (2, 4), (3, 3) or (4, 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For the remaining four cases (1, 2), (1, 3), (1, 4)  and (2, 2), even with this improvement in the argument, the proportion we obtain for  vectors in the good region (which in these cases is the optimum one for the argument  presented here) is still insufficient to obtain expansion in the vertical direction, given  the small amount of pre-images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  The numbers 휏1, 휏2 are the elementary divisors of E and, as in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='4 of [1],  there exists 푃 ∈ 퐺퐿2(ℤ) such that the matrix 퐺 = 푃−1 ⋅ 퐸 ⋅ 푃 satisfies:  퐺−1(ℤ) =  {  (  푖  휏2푗  휏1) ∶ 푖, 푗 ∈ ℤ  }  Moreover, as E is not a homothety, by another change of coordinates if necessary  we may assume that E does not have (0, 1) as an eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  With this in mind, we assume that ℙ퐸 does not fix [(0, 1)] and that 퐸−1ℤ2 = 1  휏2ℤ× 1  휏1ℤ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  So there exists an 훼 > 휏2 > 1 such that if Δℎ  훼 and Δ푣  훼 are the corresponding horizontal  and vertical cones as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='2, then 퐸−1Δ푣훼 ⊂ 퐼푛푡(Δℎ  훼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' From now on, we fix such  훼 > 휏2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Let 퐿 < max  {  1  4휏2, 휏−1  2 −훼−1  2  }  , choose points 푧1, 푧2, 푧2, 푧4 ∈ 핋1, in this order, such that:  12  퐼1 = [푧1, 푧2] and 퐼3 = [푧3, 푧4] have size 퐿;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  the translation of 퐼1 by a multiple of 1/휏2 does not intersect 퐼3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  퐼2 = (푧2, 푧3) and 퐼4 = (푧4, 푧1) have size strictly larger than 1  휏2 [  휏2−1  2 ],  and define the critical and good regions \ue22fℎ, \ue233ℎ and \ue233±  ℎ as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' As an immediate  consequence of the definition we get:  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For every 푥 ∈ 핋2, 퐸−1(푥) has 푑 points of which at least 1  휏2 [  휏2−1  2 ] are  inside each of \ue233+  ℎ and \ue233−  ℎ, and at most 휏1 of them are inside of \ue22fℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  In order to have at most one pre-image of each point in the critical zone of the shear  ℎ푡(푥1, 푥2) = (푥1, 푥2+푡푠(푥1) defined as before, we define the conservative diffeomorphism  of the torus 푣(푥1, 푥2) = (푥1 + ̃푠(푥2), 푥2), with ̃푠 ∶ 핋1 → ℝ an analytic map which we shall  impose restrictions later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' We then study the family:  푓푡 = 퐸◦푣◦ℎ푡,  of area preserving endomorphism of the torus isotopic to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' We shall denote 푓 = 푓푡 to  simplify the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Given 푥 ∈ 핋2, the set 푓 −1(푥) = ℎ−1  푡 ◦푣−1◦퐸−1(푥) is composed by d points, and given  푦 ∈ 푓 −1(푥), we have (퐷푦푓 )−1 = (퐷푦ℎ푡)−1◦(퐷ℎ푡(푦)푣)−1◦퐸−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  In order to define 푣 in a way that only one pre-image of 푥 by 푓 remains in the  critical zone, we notice that 퐸−1(푥) is composed by 푑 points which, by the change of  coordinates made initially, are aligned in a lattice of height 휏1 and length 휏2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' We also  notice that the map ℎ−1  푡 keeps the vertical lines invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Therefore, the map 푣−1 needs  to act in a way that it moves points on a vertical line enough so that only one remains  in the critical zone, and, also, it cannot move them so much that we have new points  entering the critical zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  In this way, we took the analytic map ̃푠 ∶ 핋1 → ℝ satisfying:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 퐿 is the size of the intervals 퐼1, 퐼3 then |̃푠(푢)| < 1  휏2 − 퐿, for all 푢 ∈ 핋1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For all 푢 ∈ 핋1, we have that |||̃푠 (푢 + 푗  휏1)||| > 퐿 for all 푗 ∈ {0, 1, ⋯ , 휏1 − 1} except at  most one index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' |̃푠′(푢)| < (2훼)−1, for all 푢 ∈ 핋1, where 훼 is the size of the cones fixed in the previous  subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Notice that conditions 2 and 3 are not mutually exclusives thanks to the conditions  for 훼 and 퐿 imposed in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Now, conditions 1 and 2 give us:  13  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For all 푥 ∈ 핋2, 푓 −1(푥) is composed by 푑 points of which at most one is inside  \ue22fℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' At least 푑 − 1 of the pre-images are inside \ue233 of which at least 휏1 [  휏2−1  2 ] are inside each  of \ue233+  ℎ and \ue233−  ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' In the case where 퐸−1(푥) has no points in the critical zone, due to condition 1  together with the fact that ℎ푡 preserves vertical lines, the map ℎ−1  푡 ◦푣−1 does not take  any of those points to the critical zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  In the case where 퐸−1(푥) has a point in the critical zone, it implies that we have  exactly 휏1 points there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Due to condition 2, only one of those points is able to remain  there, and due to condition 1, none of the other points is getting inside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  For the minimum amount of points in each of \ue233+  ℎ and \ue233−  ℎ, we notice that, by Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1, 퐸−1(푥) already has at least 휏1 [  휏2−1  2 ] points inside each one, and, due to condition 1,  those points must remain there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  At last, condition 3 gives us the next lemma, required for the whole construction  to work:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' There exists 훽 > 훼 such that for all 푦 ∈ 핋2, (퐷푦푣)−1◦퐸−1Δ푣  훽 ⊂ Δℎ  훽, where Δ푣  훽  and Δℎ  훽 are the corresponding vertical and horizontal cones of size 훽 as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For 푦 = (푦1, 푦2), 퐷푦푣 = (  1  ̃푠′(푦2)  0  1  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Then, due to condition 3, for all 휆 ∈ ℝ,  퐷푦푣 ⋅ 휆푒2 = 휆(̃푠′(푦2), 1) ∈ Δ푣  2훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Since, by the definition of 훼, we have 퐸−1 ⋅ 휆푒2 ∈ 푖푛푡(Δℎ  훼),  we conclude that for all 푦 ∈ 핋2, ℙ((퐷푦푣)−1◦퐸−1)⋅[푒2] is uniformly away from [푒2], hence  there exists such 훽 as we wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Items 3 and 4 of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1 also works in this cases for Δ푣  훽 and Δℎ  훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  We give the correspondent to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='3 for this case, as a consequence of items  3 and 4 of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1, Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' From now on, we fix 훽 > 훼 as in  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='2 and let:  푒푣 = inf  {  ‖(퐷푥푣)−1◦퐸−1푢‖ ∶ (푥, 푢) ∈ 푇 1핋2, 푢 ∈ Δ푣  훽  }  ,  푒ℎ = inf  {  ‖(퐷푥푣)−1◦퐸−1푢‖ ∶ (푥, 푢) ∈ 푇 1핋2, 푢 ∈ Δℎ  훽  }  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For 푡 > 2훽  푎 it holds:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' if 푦 ∈ \ue233ℎ then (퐷푦푓 )−1Δ푣  훽 ⊂ Δ푣  훽, it is strictly invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' if 푢 ∈ Δ푣  훽 is a unit vector, then  ‖(퐷푦푓 )−1푢‖ >  {  푒푣(푎−훽/푡))  훽  푡, 푦 ∈ \ue233ℎ,  푒푣  훽 ,  푦 ∈ \ue22fℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  14  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' if 푢 ∈ Δℎ  훽, and (퐷ℎ푡(푦)푣)−1◦퐸−1 ⋅ 푢 = (푤1, 푤2) let ∗푦 (푢) be as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Then if  푦 ∈ \ue233  ∗푦(푢)  ℎ  we have (퐷푦푓 )−1(푢) ∈ Δ푣  훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' if 푢 ∈ Δℎ  훽 is a unit vector, then  ‖(퐷푦푓 )−1푢‖ >  {  푒ℎ,  푦 ∈ \ue233  ∗푦(푢)  ℎ  ,  푒ℎ  푏+ 1  푡 푡−1, 푦 ∉ \ue233  ∗푦(푢)  ℎ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  We notice that, analogously to the homothety case, we have the problem that ∗푦 (푢)  depends on 푦 ∈ 푓 −1(푥), therefore even though we have at least 휏1 [  휏2−1  2 ] points in each  of \ue233±  ℎ, there could be a vector 푢 ∈ ℝ2 such that for all 푦 ∈ \ue233+  ℎ, ∗푦 (푢) = − and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  However, we can see that this is not the case:  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For every 푥 ∈ 핋2, 푢 ∈ ℝ2, there are at least 휏2 [  휏2−1  2 ] points 푦 ∈ 푓 −1(푥)  such that 푦 ∈ \ue233  ∗푦(푢)  ℎ  , where ∗푦 (푢) is as in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1 changing 푣푟 for 푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' By the same argument used in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='2, we can see that ∗푦 (푢) is constant for  points 푦 ∈ 푓 −1(푥) such that ℎ푡(푦) lies in the same horizontal line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' There are exactly  휏2 pre-images 푦′ such that ℎ푡(푦) and ℎ푡(푦′) are in the same horizontal line, hence at  least [  휏2−1  2 ] of these lies in \ue233  ∗푦(푢)  ℎ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' As 푣−1◦퐸−1(푥) has 휏1 different vertical lines, we get the  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1  Non-uniform hyperbolicity  We end up having calculations completely mirrored in those made in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1,  and for that reason we will skip the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For (푥, 푢) ∈ 푇핋2 with 푢 ≠ 0 and for 푛 ∈ ℕ,  we define the sets 퐷푓 −푛(푥, 푢), \ue233푛, \ue22e푛, and the numbers 푔푛, 푏푛 = 푑푛 − 푔푛 as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' From  Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='3 and Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='2 we deduce:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Let (푥, 푢) ∈ 푇핋2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푢 ∈ Δ푣  훽, then at least 푑 − 1 of its pre-images under 퐷푓 are also in Δ푣  훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푢 ∈ Δℎ  훽, then at least 휏1 [  휏2−1  2 ] of its pre-images under 퐷푓 are in Δℎ  훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  For that, we get for all 푛 ∈ ℕ:  푔푛+1 ≥ (푑 − 1 − 휏1 [  휏2 − 1  2  ]) 푔푛 + 휏1 [  휏2 − 1  2  ] 푑푛,  hence, putting 푎푛 = 푔푛  푑푛 :  푎푛+1 ≥ (  푑 − 1  푑  − 1  휏2 [  휏2 − 1  2  ]) 푎푛 + 1  휏2 [  휏2 − 1  2  ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Thus, we get:  15  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' For every (푥, 푢) ∈ 푇핋2, 푢 ≠ 0, and 푛 ≥ 0, it holds:  lim inf 푎푛 ≥ 1  휏2 [  휏2 − 1  2  ]  푑  1 + 휏1 [  휏2−1  2 ]  ∶= 퐿(휏1, 휏2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' This is where we are able to verify that this argument will work for the cases  (휏1, 휏2) as (2, 4), (3, 3) and (4, 4), where we have 퐿(휏1, 휏2) as 2/3, 3/4 and 4/5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  And it won’t work for the other cases (1, 2), (1, 3), (1, 4) and (2, 2) where we will get 퐿(휏1, 휏2)  as 0, 1/2, 1/2 and 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' As we will see, for the rest of the argument to work, we  need this lower bound strictly greater than 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  As another consequence of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='3 and Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='2, we get:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푡 > 2훽  푎 , then for all (푥, 푢) ∈ 푇핋2, it holds:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푢 ∈ Δ푣  훽, then:  퐼(푥, 푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓) ≥ 푑 − 1  푑  log 푡 + log (  푒푣  훽 (푎 − 훽  푡 )  푑−1  푑  ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' If 푢 ∈ Δℎ  훽, then:  퐼(푥, 푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓) ≥ − (1 − 1  휏2 [  휏2 − 1  2  ]) log 푡 + log (푒ℎ (푏 + 1  푡 )  −(1− 1  휏2[  휏2−1  2 ])  ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Again, by Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='1, we have:  퐼(푥, 푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓 푛) =  푛−1  ∑  푖=0  ∑  푦∈푓 −푖(푥)  퐼(푦, (퐷푦푓 푖)−1푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓)  푘2푖  ∶=  푛−1  ∑  푖=0  퐽푖,  we compute, for 푡 > 2훽  푎 , for all 푖 ≥ 0:  퐽푖 = 1  푑  ∑  (푦,푤)∈\ue233푖  퐼(푦, 푤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓) + 1  푑  ∑  (푦,푤)∈\ue22e푖  퐼(푦, 푤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓)  ≥ 푎푖푉(푡, 휏1, 휏2) + (1 − 푎푖)퐻(푡, 휏1, 휏2),  where 푎푖 is as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='5, 푉 and 퐻 are the right side of the inequalities obtained in  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='6 for 푢 ∈ Δ푣  훽 and 푢 ∈ Δℎ  훽 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' It follows:  lim  푖→∞ 퐽푖 ≥ 퐿(휏1, 휏2)푉(푡, 휏1, 휏2) + (1 − 퐿(휏1, 휏2))퐻(푡, 휏1, 휏2)  = (휏1 − 2  휏2) [  휏2−1  2 ] − 1  1 + 휏1 [  휏2−1  2 ]  log 푡 + 퐶(푡, 휏1, 휏2),  16  where:  퐶(푡, 휏1, 휏2) =퐿(휏1, 휏2) log (  푒푣  훽 (푎 − 훽  푡 )  푑−1  푑  )  + (1 − 퐿(휏1, 휏2)) log (푒ℎ (푏 + 1  푡 )  −(1− 1  휏2[  휏2−1  2 ])  ) > 퐶,  for all 푡 > 2훽  푎 , that is, 퐶(푡, 휏1, 휏2) is uniformly bounded from below by some constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  Since 푑 = 휏1 ⋅ 휏2 > 4, the constant multiplying log 푡 is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Therefore, since all  the bounds above are uniform for all non-zero tangent vectors (푥, 푢), as in the homo-  thety case we obtain that for 푡 sufficiently large, for all 푛 greater than some 푛0, and for  all nonzero tangent vectors (푥, 푢):  1  푛퐼(푥, 푢;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 푓 푛) = 1  푛  푛−1  ∑  푖=0  퐽푖(푥, 푢) > 0,  hence, \ue22f\ue244(푓 ) > 0 which by Theorem 1 concludes the proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  References  [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Andersson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Carrasco, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Saghin, “Non-uniformly hyperbolic endo-  morphisms,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Barreira and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Pesin, Introduction to Smooth Ergodic Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Graduate Studies  in Mathematics, American Mathematical Society, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  [3] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Oseledets, “A multiplicative ergodic theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' characteristic ljapunov, expo-  nents of dynamical systems,” Trudy Moskovskogo Matematicheskogo Obshchestva,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 19, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 179–210, 1968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Anosov, “Geodesic flows on closed riemannian manifolds of negative curva-  ture,” Trudy Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Steklov, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 90, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 3–210, 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Qian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Xie, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' Zhu, Smooth Ergodic Theory for Endomorphisms, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 1978  of Lecture Notes in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content=' 01 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}
+page_content='  17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE0T4oBgHgl3EQfPgCp/content/2301.02180v1.pdf'}

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+First Principles Assessment of CdTe as a Tunnel Barrier at the α-Sn/InSb Interface
+Malcolm J. A. Jardine,1, ∗ Derek Dardzinski,2, ∗ Maituo Yu,2 Amrita Purkayastha,1
+A.-H. Chen,3 Yu-Hao Chang,4 Aaron Engel,4 Vladimir N. Strocov,5 Mo¨ıra
+Hocevar,3 Chris J. Palmstrøm,4, 6 Sergey M. Frolov,1 and Noa Marom2, 7, 8, †
+1Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA, 15260, USA
+2Department of Materials Science and Engineering,
+Carnegie Mellon University, Pittsburgh, PA 15213, USA
+3Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut N´eel, 38000 Grenoble, France
+4Materials Department, University of California-Santa Barbara, Santa Barbara, CA, USA
+5Paul Scherrer Institut, Swiss Light Source, CH-5232 Villigen PSI, Switzerland
+6Department of Electrical and Computer Engineering,
+University of California-Santa Barbara, Santa Barbara, CA, USA
+7Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA
+8Department of Chemistry, Carnegie Mellon University, Pittsburgh, PA 15213, USA
+Majorana zero modes, with prospective applications in topological quantum computing, are ex-
+pected to arise in superconductor/semiconductor interfaces, such as β-Sn and InSb. However, prox-
+imity to the superconductor may also adversely affect the semiconductor’s local properties. A tunnel
+barrier inserted at the interface could resolve this issue. We assess the wide band gap semiconduc-
+tor, CdTe, as a candidate material to mediate the coupling at the lattice-matched interface between
+α-Sn and InSb. To this end, we use density functional theory (DFT) with Hubbard U corrections,
+whose values are machine-learned via Bayesian optimization (BO) [npj Computational Materials 6,
+180 (2020)]. The results of DFT+U(BO) are validated against angle resolved photoemission spec-
+troscopy (ARPES) experiments for α-Sn and CdTe. For CdTe, the z-unfolding method [Advanced
+Quantum Technologies, 5, 2100033 (2022)] is used to resolve the contributions of different kz values
+to the ARPES. We then study the band offsets and the penetration depth of metal-induced gap
+states (MIGS) in bilayer interfaces of InSb/α-Sn, InSb/CdTe, and CdTe/α-Sn, as well as in tri-layer
+interfaces of InSb/CdTe/α-Sn with increasing thickness of CdTe. We find that 16 atomic layers
+(3.5 nm) of CdTe can serve as a tunnel barrier, effectively shielding the InSb from MIGS from the
+α-Sn. This may guide the choice of dimensions of the CdTe barrier to mediate the coupling in
+semiconductor-superconductor devices in future Majorana zero modes experiments.
+I.
+INTRODUCTION
+A promising route toward the realization of fault-
+tolerant quantum computing schemes is developing ma-
+terials systems that can host topologically protected Ma-
+jorana zero modes (MZMs) [1, 2].
+MZMs may ap-
+pear in one-dimensional topological superconductors [3–
+5], which can be effectively realized by proximity cou-
+pling a conventional superconductor and a semiconduc-
+tor nanowire that possesses strong spin-orbit coupling
+(SOC). Adding in a magnetic field enables this system to
+behave as an effective spinless p-wave topological super-
+conductor, which allows for MZM states [6]. Recently,
+there have been new developments in material choices
+and experimental methods to identify MZMs in semicon-
+ductor nanowire-superconductor systems [7], designed to
+overcome challenges identified during the first wave of
+experiments [8–10]. These include trying new combina-
+tions of semiconductors and epitaxial superconductors,
+e.g. Pb, Sn, Nb, to maximize the electron mobility and
+utilize larger superconducting gaps and higher critical
+magnetic fields [11–16]. Additionally, new proposed ar-
+∗ These authors contributed equally to this work
+† Corresponding author: [email protected]
+chitectures include creating nanowire networks and in-
+ducing the field via micromagnets [17, 18].
+One of the challenges presented by the superconduc-
+tor/semiconductor nanowire construct, is that excessive
+coupling between the superconducting metal and semi-
+conductor may “metallize” the semiconductor, thus ren-
+dering the topological phase out of reach. Theoretical
+studies that treated the semiconducting and supercon-
+ducting properties via the Poisson-Schr¨odinger equation,
+have shown that excessive coupling between the mate-
+rials may lead to the semiconductor’s requisite proper-
+ties, such as the Lande´e g-factor and spin-orbit-coupling
+(SOC), being renormalized to a value closer to the
+metal’s. In addition, large unwanted band shifts may be
+induced [12, 19–22]. Having a tunnel barrier could modu-
+late the superconductor-semiconductor coupling strength
+and thus the induced proximity effect, which is critical
+for controlling experiments. It is currently unknown what
+the required width range of a tunnel barrier is. Another
+potential benefit of a CdTe layer is InSb surface passiva-
+tion.
+InSb and Sn are among the materials used to fabricate
+devices for Majorana search [23]. InSb is the backbone of
+such systems because it has the highest electron mobility,
+strongest spin-orbit coupling (SOC) and a large Land´e
+g-factor in the conduction band compared to other III-
+V semiconductors. β−Sn has a bulk critical field of 30
+arXiv:2301.02879v1  [cond-mat.mtrl-sci]  7 Jan 2023
+
+2
+mT and a superconducting critical temperature of 3.7 K,
+higher than the 10 mT and 1 K, respectively, of Al. Re-
+cently, β-Sn shells have been grown on InSb nanowires,
+inducing a hard superconducting gap [12].
+The large
+band gap semiconductor CdTe is a promising candidate
+to serve as a tunnel barrier. Thanks to its relative in-
+ertness, it may simultaneously act as a passivation layer
+protecting the InSb from environmental effects and po-
+tentially minimizing disorder [24, 25]. Advantageously,
+CdTe is lattice matched to InSb [26].
+Sn has two al-
+lotropes. The β form, with a BCT crystal structure, is
+of direct relevance to MZM experiments thanks to its su-
+perconducting properties. However, the semi-metallic α
+form has a diamond structure, which is lattice matched
+to InSb and CdTe, making it an ideal model system for
+investigating, both theoretically and experimentally, the
+electronic structure of Sn/InSb heterostructures.
+Much experimental work, such as growth and ARPES
+studies, has been undertaken on α-Sn. Previously, α-Sn
+has been found to possess a topologically trivial band in-
+version, with SOC inducing a second band inversion and
+a topological surface state (TSS) [27, 28]. The effect of
+strain on the topological properties of α-Sn has also been
+studied [21, 29–38]. In-plane compressive strain has been
+reported to make α-Sn a topological Dirac-semi-metal
+and induce a second TSS to appear [27].
+Conversely,
+tensile strain has been reported to induce a transition
+to a topological insulator. CdTe [25] and α-Sn [12, 28]
+have been epitaxially grown on InSb. Depositing Sn on
+InSb often leads to growth of epitaxially matched α-Sn,
+although β-Sn may appear under some conditions [39].
+In addition, α-Sn can transition to β-Sn if the Sn layer is
+above a critical thickness or if heat is applied during fab-
+rication processes [40, 41]. Studying the interface with
+the lattice matched α-Sn may provide insight, which is
+also pertinent to β-Sn as both could be present in hy-
+brid systems. Therefore, these are promising materials
+to investigate for future device construction.
+MZM experiments rely on finely tuned proximity cou-
+pling between a superconducting metal and a semicon-
+ductor. By adding a tunnel barrier at the interface be-
+tween the two materials and varying its width, one could
+potentially mediate the proximity coupling strength to
+achieve precise control over the interface transparency.
+To the best of our knowledge, this idea has not yet been
+tested in experiments and it is presently unknown which
+material(s) would be the best choice for a barrier and
+what would be the optimal thickness.
+Simulations of
+a tri-layer system with a tunnel barrier are therefore
+needed to inform MZM experiments. Here, we use den-
+sity functional theory (DFT) to study a tri-layer system,
+in which InSb is separated from α-Sn by a CdTe tunnel
+barrier. Despite recent progress towards treating super-
+conductivity within the framework of DFT [42, 43] the
+description of proximity-induced superconductivity at an
+interface with a semiconductor is still outside the reach of
+present-day methods. However, DFT can provide useful
+information on properties, such as the band alignment at
+the interface. Conduction band offsets are of particular
+importance because the proximity effect in most experi-
+ments on InSb primarily concerns the conduction band.
+In addition, DFT can provide information on the pene-
+tration depth of metal induced gap states (MIGS) into
+the semiconductor [20, 25, 44, 45], which is important
+for determining the appropriate thickness of the tunnel
+barrier.
+Within DFT, computationally efficient (semi-)local
+exchange-correlation functionals severely underestimate
+the band gap of semiconductors to the extent that
+some narrow-gap semiconductors, such as InSb, are er-
+roneously predicted to be metallic [46–49]. This is at-
+tributed to the self-interaction error (SIE), a spurious
+repulsion of an electron from its own charge density [50–
+52]. Hybrid functionals, which include a fraction of exact
+(Fock) exchange, mitigate the SIE and yield band gaps in
+better agreement with experiment. However, their com-
+putational cost is too high for simulations of large in-
+terface systems, such as the α-Sn/CdTe/InSb tri-layer
+system studied here. The DFT+U approach, whereby a
+Hubbard U correction is added to certain atomic orbitals,
+provides a good balance between accuracy and computa-
+tional cost[46, 53, 54]. Recently, some of us have pro-
+posed a method of machine learning the U parameter for
+a given material by Bayesian optimization (BO) [55]. The
+DFT+U(BO) method has been employed successfully for
+InSb and CdTe [56].
+It has been shown that (semi-)local functionals fail
+to describe the bulk band structure of α-Sn correctly,
+specifically the band ordering and the orbital compo-
+sition of the valence bands at the Γ point.
+DFT+U,
+hybrid functionals, or many-body perturbation theory
+within the GW approximation are necessary to obtain a
+correct description of the band structure [29, 35, 57–59].
+DFT+U simulations have required slab models of more
+than 30 monolayers of Sn to converge towards a bulk
+regime, where quantum confinement is no longer domi-
+nant. With a small number of layers α-Sn may exhibit
+topological properties [26, 60, 61].
+Some DFT studies
+have considered slab models of bi-axially strained α-Sn.
+DFT simulations of strained α-Sn on InSb have been con-
+ducted with a small number of layers of both materials
+[26, 62]. The DFT+U approach has reproduced the ef-
+fects of strain and compared well with experimental data
+[28, 60, 62].
+Here,
+we perform first principles calculations us-
+ing DFT+U(BO) for a (110) tri-layer semiconduc-
+tor/tunnel barrier/metal interface composed of the ma-
+terials InSb/CdTe/α-Sn, owing to their relevance to cur-
+rent Majorana search experiments [12, 25]. To date, DFT
+studies of large interface slab models with a vacuum re-
+gion have not been conducted for these interfaces. Pre-
+viously, the results of DFT+U(BO) for InSb(110) have
+been shown to be in good agreement with angle-resolved
+photoemission spectroscopy (ARPES) experiments [63].
+Here, we also compare the results of DFT+U(BO) to
+ARPES for α-Sn (Section III A) and CdTe (Section
+
+3
+III B). Excellent agreement with experiment is obtained.
+In particular, for CdTe the z-unfolding scheme (Section
+II A) helps resolve the contributions of different kz values
+and modelling the 2 × 2 surface reconstruction repro-
+duces the spectral signatures of surface states. We then
+proceed to study the bi-layer interfaces of InSb/CdTe,
+CdTe/α-Sn, and InSb/α-Sn (Section III C). Finally, to
+assess the effectiveness of the tunnel barrier, we study
+tri-layer interfaces with 2 to 16 monolayers (0.5 nm to 3.5
+nm) of CdTe inserted between the InSb substrate and the
+α-Sn (Section III D). This thickness is within the thick-
+ness range of CdTe shells grown on InSb nanowires. For
+all interfaces, our simulations provide information on the
+band alignment and the presence of MIGS. We find that
+16 layers of CdTe (about 3.5 nm) form an effective tunnel
+barrier, insulating the InSb from the α-Sn. However, this
+may be detrimental for transport at the interface. Based
+on this, we estimate that the relevant thickness regime
+for tuning the coupling between InSb and Sn may be in
+the range of 6-10 layers of CdTe.
+II.
+METHODS
+A.
+Z-Unfolding
+Simulations of large supercell models produce complex
+band structures with a large number of bands, as shown
+in Figure 1a,b for a CdTe(111) slab with 25 atomic layers,
+whose band structure was calculated using PBE+U(BO),
+as described in Section II B. Band structure unfolding
+is a method of projecting the band structure of a su-
+percell model onto the appropriate smaller cell ([63–68].
+This can help resolve the contributions of states emerg-
+ing from of e.g., defects and surface reconstructions vs.
+the bulk bands of the material. In addition, it can fa-
+cilitate the comparison to angle-resolved photoemission
+spectroscopy (ARPES) experiments. The “bulk band un-
+folding” scheme [63] projects the supercell band struc-
+ture onto the primitive unit cell, illustrated in Figure
+1c. The resulting band structure, shown in Figure 1d,
+appears bulk-like. Bulk-unfolded band structures have
+been shown to compare well with ARPES experiments
+using high photon energies, which are not surface sensi-
+tive owing to the large penetration depth.
+The “z-unfolding” scheme [63] projects the band struc-
+ture of a slab model with a finite thickness onto the Bril-
+louin zone (BZ) of a single layer of the slab supercell
+with the same orientation, illustrated in Figure 1e. The
+resulting band structure, shown in Figure 1f, contains ex-
+tra bands that are not present in the bulk-unfolded band
+structure. The extra bands originate from different kz
+values in the 3D primitive Brillouin zone projecting onto
+the surface Brillouin zone (SBZ), creating overlapping
+paths. For example, panel Figure 1g shows cross sections
+through the BZ at values of kz = 0 and kz = 0.5. The
+bulk-paths of Γ − L, Γ − K and Γ − X all overlap with
+the surface k-path Γ − M, possibly with contributions
+from additional paths, such as X − U. The plane cuts at
+different kz values are derived from the tessellated bulk
+BZ structure, shown in Figure 1h. When z-unfolding is
+performed, the value of kz may be treated as a free pa-
+rameter. The dependence on kz manifests as a smooth
+change in the spectral function over the possible range
+of kz which varies the mixture of different constituent
+bulk-paths that overlap the SBZ-path, as shown in Fig-
+ure 1i for Γ − M. The BZ for z-unfolding is a surface BZ
+with a finite thickness, shown in red in Figure 1j. The
+simulation cell for the DFT calculations is set up to be
+the corresponding real-space unit cell. The z-unfolded k-
+paths are parallel to the (111) surface at a constant value
+of kz.
+In ARPES experiments, the relation of the experimen-
+tal spectra to kz may be less straightforward. First, the
+dependence of the inelastic mean free path of the elec-
+trons on their kinetic energy is given by the universal
+curve [69, 70].
+Using photon energies that correspond
+to a small mean free path is advantageous for probing
+surface states.
+However, it can produce prominent kz
+broadening due to the Heisenberg uncertainty principle
+[71–75] that implies integration of the ARPES signal over
+kz through the broadening interval. Second, deviations of
+the photoemission final states from the free electron ap-
+proximation can cause contributions from different values
+of kz to appear in the ARPES spectra. The photoelec-
+trons are often treated as free electrons, based on the as-
+sumption that the photoelectron kinetic energy is much
+larger than the modulations of the crystal potential. In
+this case, kz for a given photoelectron kinetic energy, Ek,
+and the in-plane momentum, K//, is one single value,
+which is determined by:
+kz =
+√2m0
+ℏ
+�
+Ek − ℏ2
+2m0
+K2
+// − V0
+(1)
+where m0 is the free-electron mass and V0 the inner po-
+tential in the crystal. However, a considerable body of
+evidence has accumulated that the final states even in
+metals [76, 77] and to a greater extent in complex ma-
+terials such as transition metal dichalcogenides [78, 79]
+can significantly deviate from the free electron approx-
+imation.
+Such deviations can appear, first, as non-
+parabolic dispersions of the final states and, second, as
+their multiband composition. The latter means that for
+given Ek and K// the final-state wavefunction Φf incor-
+porates a few Bloch waves φkz with different kz values,
+Φf = �
+kz Akzφkz, which give comparable contributions
+to the total photocurrent determined by the Akz ampli-
+tudes [76]. A detailed theoretical description of the multi-
+band final states, treated as the time-reversed low-energy
+electron diffraction (LEED) states [73] within the wave-
+function matching approach, as well as further examples
+for various materials can be found in Refs. [78, 79] and
+the references therein. An insightful analysis of the multi-
+band final states extending into the soft-X-ray photon
+energies can be found in Ref. [77]. A rigorous analysis of
+
+4
+FIG. 1.
+(a) Side view of the CdTe(111) slab (b) Folded band structure of CdTe(111) 25 monolayer slab. (c) Primitive unit cell
+of CdTe (d) bulk-unfolded band structure (e) unit cell of CdTe(111) slab used in z-unfolding. (f) Z-unfolded band structure
+along k-path M − Γ − M for kz = 0.5, and (g) as a function of kz. (h) FCC bulk BZ (grey), (111) unit-cell BZ (red) and
+(111) surface BZ (blue). (i) Intersecting planes slice through the bulk BZ for kz = 0 (green) and kz = 0.5 (red) with the SBZ
+indicated. (j) tessellated bulk BZs showing (111) orientated intersecting planes for given kz values.
+final state effects in ARPES is beyond the scope of this
+work. Here, we will only mention that all these effects
+trace back to hybridization of free-electron plane waves
+through the higher Fourier components of the crystal po-
+tential. In cases where significant kz broadening and/or
+final states effects are present, z-unfolding, rather than
+bulk unfolding, should be used in order to resolve the
+contributions of different kz values to the measured spec-
+trum. This is demonstrated for CdTe in Section III B,
+where the final states appear to incorporate two Bloch
+waves with kz = 0 and kz = 0.5.
+B.
+Computational Details
+DFT calculations were conducted using the Vienna Ab
+Initio Simulation Package (VASP) [80] with the projector
+augmented wave method (PAW) [81, 82]. The general-
+ized gradient approximation (GGA) of Perdew, Burke,
+and Ernzerhof (PBE) [83] was employed to describe the
+exchange-correlation interactions among electrons with a
+Hubbard U correction [84]. The U values were machine
+learned using Bayesian optimization (BO) [55]. Briefly,
+the BO objective function is formulated to reproduce as
+closely as possible the band structure obtained from the
+Heyd-Scuseria-Ernzerhof (HSE) [85] hybrid functional.
+The reference HSE calculations were conducted for bulk
+CdTe with a lattice parameter of 6.482 ˚A and α-Sn with
+a lattice parameter of 6.489 ˚A and compared to the re-
+sults with the lattice constant of InSb, 6.479 ˚A, which
+was used for interface models. It was verified that using
+the lattice constant of InSb does not have an appreciable
+effect on the electronic properties of CdTe and α-Sn, as
+shown in the SI.
+The hyperparameters of our BO implementation are
+the coefficients α1 and α2, which assign different weights
+to the band gap vs. the band structure in the objective
+function, the number of valence and conduction bands
+used for the calculation of the objective function, Nb, and
+the parameter κ that controls the balance between explo-
+ration and exploitation in the upper confidence bound ac-
+quisition function. For InSb the values of U In,p
+eff
+= −0.2
+and U Sb,p
+eff
+= −6.1 were used, following Ref. [55, 63].
+It has been shown that PBE+U(BO) produces a band
+structure in good agreement with ARPES for InSb [63].
+Because α-Sn is a semi-metal, only the band shape
+was considered in the optimization, i.e. α1 was set to
+0 and α2 = 1 [59].The other BO hyperparameters used
+for Sn were κ = 7.5 and Nb = (5, 5). This resulted in a
+value of U Sn,p
+eff
+= −3.04 eV, slightly different than in Refs.
+[29, 35, 61], which used empirical methods to choose a
+U value that yields a correct band ordering. As shown
+in Ref. [59], PBE+U(BO) reproduces the correct band
+ordering of α-Sn with the band inversion at the Γ point,
+
+a) full slab side view
+c) CdTe primitive cell
+e) CdTe(111) unit cell
+h)
+↑[111]
+g)
+k²= 0.0
+k,= 0.5
+SBZ
+k,= 0.5
+M-
+0
+k_= 0.0
+CdO In
+ z unfolded - kz
+0.00
+0.25
+0.50
+b)
+d)
+f)
+folded
+)
+bulk unfolded
+z unfolded - k_ = 0.5
+3
+3
+D
+3
+[111]
+2 
+2
+2
+2
+1
+1
+M
+「k
+M
+1
+02
+6
+0
+(eV)
+0 
+0
+0
+出
+-1
+-1
+.1
+1
+-2
+2
+-3
+-3 
+-3
+-4 
+-4 
+-5
+5
+M
+L
+L
+M
+M
+L
+M5
+in agreement with other studies using DFT+U [27, 28].
+For CdTe, we applied a U correction to both the Cd-d
+orbitals and Te-p orbitals, unlike earlier studies [56, 86].
+The hyperparameters used for CdTe were κ = 7.5, Nb =
+(5, 5), α1 = 0.5 and α2 = 0.5. The latter two parameters
+were chosen to assign equal weights to the band gap and
+the band shape. This led to U values of U Cd,d
+eff
+= 7.381
+and U T e,p
+eff
+= −7.912. The Cd-d U value obtained here
+is similar to the 7 eV used in Ref. [86] and somewhat
+lower than U Cd,d
+eff
+= 8.3 eV in Ref. [56]. The gap of 1.21
+eV, obtained here by applying the Hubbard U correction
+to both the Te-p states and the Cd-d states is closer to
+experimental values of around 1.5 eV [87, 88] and the
+HSE value of 1.31 eV than previous calculations [56].
+Spin-orbit coupling (SOC) was used in all calculations
+and dipole corrections were applied to slab models [89].
+The tags used for convergence of calculations were BMIX
+= 3, AMIN = 0.01, ALGO = Fast, and EDIFF = 1·10−5.
+The kinetic energy cutoff was set to 400 eV for all bulk
+calculations and 350 eV for surface and interface slab
+models.
+A 9 × 9 × 9 k-point mesh was used for bulk
+calculations and a k-point mesh of 7 × 7 × 1 was used for
+surface and interface calculations. All interface density
+of states (DOS) calculations used a k-point mesh of 13 ×
+13 × 1.
+All band structure and density of states plots were gen-
+erated using the open-source Python package, VaspVis
+[59], which is freely available from The Python Package
+Index (PyPI) via the command: pip install vaspvis, or on
+GitHub at:
+https://github.com/DerekDardzinski/
+vaspvis
+C.
+Slab Construction
+All slab models were constructed using the experimen-
+tal InSb lattice constant value of 6.479 ˚A [90], assuming
+that the epitaxial films of CdTe and α-Sn would conform
+to the substrate. The length of two monolayers of a (110)
+slab was 4.5815 ˚Ain the z-direction. A vacuum region of
+around 40 ˚A was added to each slab model in the z-
+direction to avoid spurious interactions between periodic
+replicas. The surfaces of all slab models were passivated
+by pseudo-hydrogen atoms such that there were no sur-
+face states from dangling bonds [91]. Despite α-Sn being
+a semi-metal passivation is required to remove spurious
+surface states, as shown in the supplemental information
+(SI). The pseudo-hydrogen fractional charges utilized to
+passivate each atom were 1.25 for In and 0.75 for Sb
+in InSb, 1.5 for Cd and 0.5 for Te in CdTe, and 1 for
+Sn. Structural relaxation of the pseudo-hydrogen atoms
+was performed until the maximal force was below 0.001
+eV/˚A. The InSb/CdTe interface structure has In-Te and
+Sb-Cd bonds with each In interface atom connected to 3
+Sb and 1 Te. The configuration with In-Cd and Sb-Te
+bonds was also considered but this was found to be less
+stable by 1.33 eV. Ideal interfaces were considered with
+no intermixing and no relaxation of the interface atoms
+was performed.
+When constructing such slab models, it is necessary
+to converge the number of layers to avoid quantum size
+effects and approach the bulk properties [92]. For InSb
+it has previously been shown that 42 monolayers are suf-
+ficiently converged [63]. Plots of the band gap vs. the
+number of atomic layers for CdTe(110) and α-Sn (110)
+slabs are provided in the SI. CdTe was deemed converged
+with 42 monolayers with a gap value of 1.23 eV, which
+is only slightly larger than the bulk PBE+U(BO) value.
+The z-unfolded band structures of CdTe(111) were cal-
+culated for a 40 monolayer slab. A 26 monolayer slab
+model was used to simulate the 2 × 2 reconstruction,
+due to the higher computational cost of the 2 × 2 su-
+percell. Structural relaxation was performed for the top
+two monolayers of the 2 × 2 reconstruction. For the slab
+of unstrained (110) α-Sn, 70 monolayers were needed to
+close the gap at the zero-gap point of the semi-metal,
+which corresponds to around 16 nm. The tri-layer slab
+models comprised 42 layers of InSb, 70 layers of α-Sn and
+between 0 and 16 layers of CdTe in two-layer increments,
+amounting to a total slab thickness of around 300 nm
+(not including vacuum). The (110) bi-layer slab models
+comprised 42 layers of CdTe and InSb, and 70 layers of
+α-Sn as these were deemed converged.
+D.
+ARPES Experimental details
+The α-Sn samples were grown by molecular beam epi-
+taxy on an In-terminated c(8 × 2) InSb(001) surface pre-
+pared by atomic hydrogen cleaning. 51 monolayers (16.5
+nm) of α-Sn were deposited as calibrated via Rutherford
+backscattering spectrometry. Growth was performed at a
+substrate temperature of -20 ◦C and a base pressure bet-
+ter than 1·10−10 Torr. The ARPES measurements were
+taken at Beamline 10.0.1.2 at the Advanced Light Source
+in Berkeley. The base pressure was better than 5·10−11
+Torr while the sample temperature was held at 68 K.
+The sample was illuminated with 63 eV p-polarized light
+and spectra were collected using a Scienta R4000 detector
+with energy resolution better than 40 meV and angular
+resolution better than 0.1◦. The sample was transferred
+via vacuum suitcase with a base pressure better than
+·10−11 Torr between the growth chamber and beamline.
+A photon energy of 63 eV corresponds to a kz approxi-
+mately 0.15 ˚A−1 above the Γ002 point.
+III.
+RESULTS AND DISCUSSION
+A.
+α-Sn
+Figure 2a shows the bulk unfolded PBE+U(BO) band
+structure for a 51 monolayer thick α-Sn (001) slab, com-
+pared to ARPES data for a sample of the same thickness
+taken at a photon energy of 63 eV . The point M is at
+
+6
+FIG. 2. Electronic structure of α-Sn: (a) Bulk-unfolded band
+structure of an α-Sn (001) slab with 51 atomic layers (light
+blue) compared with ARPES data for a sample of the same
+thickness. The point M is at 0.9298 ˚A−1. The ARPES data
+is cutoff at 0.9 ˚A−1 due to experimental artifacts at the edges.
+Spin-polarized band structures projected onto (b) the top sur-
+face atoms and (c) the bottom surface atoms, indicated by the
+green boxes on the slab structure illustrated in (d).
+0.9298 ˚A−1. The ARPES data is cutoff at 0.9 ˚A−1 due to
+experimental artifacts at the edges. The PBE+U (BO)
+band structure is in excellent agreement with ARPES.
+The top of the valence band in the ARPES and the sim-
+ulated band structure lines up and the bulk bands are
+reproduced well. The bandwidth of the heavy hole band,
+Γ8, is slightly underestimated, consistent with Ref. [63].
+This is corrected by the HSE functional, as shown in the
+SI for a bulk unit cell of α-Sn with a (001) orientation.
+However, it is not feasible to use HSE for the large inter-
+face models studied here, owing to its high computational
+cost.
+The previously reported topological properties of α-
+Sn slabs are also observed here [27–31, 35, 36, 62]. The
+spin-polarized topological surface state (TSS) is shown
+in panels (b) and (c) of Fig. 2 for a (001) 51 monolayer
+slab along the X − Γ − X k-path. As expected, the TSS
+is characterized by a linear dispersion with the top and
+bottom surfaces having opposite spin polarization. The
+associated Rashba-like surface states are also observed
+along the K − Γ − K k-path, as shown in the SI. This
+linear surface state is also observed in the (110) slabs used
+to construct the bilayer and tri-layer models. Notably
+there is an energy gap between the top and bottom TSSs,
+which closes at 70 layers, the same thickness at which
+the band gap closes.
+This gap is possibly induced by
+the hybridization of the top and bottom surface states in
+under-converged slabs. We note that the effect of strain
+on the electronic structure of α-Sn is not studied here.
+B.
+CdTe
+Fig. 3 shows a comparison of band structures obtained
+using PBE+U(BO) to the ARPES experiments of Ren et
+al. [93] for CdTe(111). Ren et al. collected ARPES data
+at photon energies of 19, 25 and 30 eV . Here, we com-
+pare our results with the second-derivative maps of the
+ARPES data taken at 25 eV along the k-paths Γ − M
+(panels (a) and (b)) and Γ − K − M (panels (c) and
+(d)). The original data has been converted to gray scale
+and reflected around kx = 0.
+To facilitate the quali-
+tative comparison of the DFT band structure features
+with the ARPES experiment, we apply a Fermi energy
+shift of 0.25 eV to line up the VBM and a stretch factor
+of 1.22 to compensate for the bandwidth underestima-
+tion of PBE+U(BO), particularly for bands deep below
+the Fermi energy [94]. Bandwidth underestimation by
+PBE+U(BO) compared with HSE and ARPES has also
+been reported for InAs and InSb in [63, 95]. The original
+computed band structure without the shift and stretch is
+provided in the SI.
+Owing to the low mean free path at this photon energy,
+the spectrum appears integrated over a certain kz inter-
+val and surface contributions are readily visible in the
+ARPES [69, 70]. To account for the different kz contribu-
+tions, the z-unfolding method was employed, as described
+in Section II A. Panels (a) and (c) show the z-unfolded
+band structures as a function of kz for slab models with-
+out a surface reconstruction (figures with single values of
+kz are provided in the SI). This is used determine which
+kz values are likely present in the experiment. A mixture
+of kz = 0 and k = 0.5 provides the best agreement with
+the ARPES data. This combination of kz values is used
+for the DFT data shown in cyan in panels (b) and (d).
+This is consistent with the kz broadening with contribu-
+tions centered around kz = 0 and k = 0.5 often present
+in ARPES data taken at low mean field path energies in
+gapped materials [71? , 72].
+To account for the presence of surface states, we mod-
+eled the CdTe(111)A-(2 × 2) surface reconstruction [96],
+illustrated in panel (e). The atom-projected band struc-
+tures of the bottom layer (indicated by pink dashed box)
+are plotted in pink in panels (b) and (d).
+The addi-
+tional bands arising from the surface reconstruction are
+in close agreement with the bands in the ARPES labeled
+as surface states by Ren et al., indicated by red arrows.
+These surface states are unaffected by the choice of kz.
+By accounting for the contributions of different kz values
+and for the presence of surface states excellent agreement
+with experiment is achieved, as the DFT band structures
+reproduce all the features of the ARPES.
+C.
+Bilayer Interfaces
+We begin by probing the local electronic structure at
+the the InSb/α-Sn bi-layer interface. Fig. 4a shows the
+
+d)
+0
+b)
+UP
+a)
+↓ DOWN
+(Λa)
+0.0
+-1
+00
+1
+E-0.5
+00
+-2
+E(eV)
+x
+-3
+00
+UP
+C)
+(a)
+DOWN
+00
+0.0
+00
+-4
+1
+E -0.5
+0
+0.4
+-0.8-0.4
+0.0
+0.8
+T
+↑M
+M→
+X
+k, (A-1)7
+FIG. 3.
+Electronic structure of CdTe: Z-unfolded band structures of CdTe(111) compared with second-derivative map of
+ARPES data (black and white), adapted with permission from “Spectroscopic studies of CdTe(111) bulk and surface electronic
+structure” by J. Ren et al., Phys. Rev. B, 91, 235303 (2015); Copyright (2015) by the American Physical Society [93]. Z-
+unfolded band structures compared to ARPES data along (a), (b) Γ − M and (c), (d) Γ − K − M. (a), (c) Dependence of the
+band structure on kz. (b), (d) Mixture of kz = 0.0 and kz = 0.5 (cyan) for a model with a 2 × 2 surface reconstruction with
+the contributions of the surface atoms shown in pink. DFT has shift of -0.25 eV and stretch factor of 1.22 for comparison. (e)
+Illustration of the 2 × 2 surface reconstruction with the Cd atom removed indicated by a blue circle. The atoms used for the
+surface projection are indicated by a pink dashed box
+DOS as a function of position across the interface, in-
+dicated by the atomic layer number. Fig. 4b shows the
+local DOS at select positions. The Fermi level is posi-
+tioned at the semi-metal point of the α-Sn and in the
+gap of the InSb. We note that the α-Sn appears as if it
+has a small gap due to an artifact of the 10−4 cutoff ap-
+plied in the log plot in panels (a) and (d). The local DOS
+plots shown in panels (b) and (e) and the band structure
+plots shown in panels (c) and (f) clearly show the semi-
+metal point. No significant band bending is found for
+InSb, as expected from branching point theory [97, 98].
+Based on the element-projected band structure, shown
+in panel (c), the InSb conduction band minimum (CBM)
+lies 0.09 eV above the α-Sn semi-metal point and the
+InSb valence band maximum (VBM) lies 0.16 eV below
+it. A linear TSS is present in the α-Sn. Based on an
+atom projected band structure, shown in the SI, the ori-
+gin of this state is the top surface of α-Sn, adjacent to
+the vacuum region. A TSS is no longer present in the
+α-Sn layers at the interface with InSb, possibly owing to
+hybridization between the α-Sn and InSb [62]. Metal-
+induced gap states (MIGS) are an inherent property of
+a metal/semiconductor interface, produced by the pen-
+etration of exponentially decaying metallic Bloch states
+into the gap of the semiconductor [99–102]. The pres-
+ence of MIGS manifests in Figure Fig. 4a as a gradually
+decaying non-zero DOS in the band gap of the InSb in
+the vicinity of the interface. Figure 4b shows that the
+MIGS are prominent in the first few atomic layers and
+become negligible beyond 8 layers from the interface.
+Fig. 4d shows the DOS as a function of position across
+the CdTe/α-Sn interface, indicated by the atomic layer
+number. Fig. 4e shows the local DOS at select positions.
+The Fermi level is positioned at the semi-metal point of
+the α-Sn and in the gap of the CdTe.
+Based on the
+projected band structure, shown in panel (f), the CdTe
+CBM is positioned 0.18 eV above the Fermi level and
+the CdTe VBM is located 1.03 eV below the Fermi level.
+This agrees with previous reports that interfacing with
+Sn brings the conduction band of the CdTe closer to the
+Fermi energy, with downward band-bending of 0.25 eV
+[103] and 0.1 eV [104].
+We find a valence band offset
+of around 1 eV, similar to the (110) and (111) interface
+reported in [33, 104–108]. Close to the interface there is
+a significant density of MIGS, which decay within about
+10 layers (3-4 nm) into the CdTe. This suggests that this
+number of CdTe layers may be required for an effective
+tunnel barrier.
+Fig. 4g shows the DOS as a function of position across
+the InSb/CdTe interface, indicated by the atomic layer
+number. Fig. 4h shows the local DOS at select positions.
+The band alignment is type-I with the CdTe band gap
+straddling the InSb band-edges. The Fermi level is close
+to the InSb VBM and around the middle of the gap of
+the CdTe. No band bending is found in either material.
+Based on the projected band structure, shown in panel
+(i), the CdTe CBM lies 0.28 eV above the InSb CBM
+and the CdTe VBM lies 0.75 eV below the InSb VBM.
+These values are similar to the band offsets reported in
+references [25, 88, 109]. Because the band gap of InSb
+is significantly smaller than that of CdTe, states from
+the InSb penetrate into the gap of the CdTe, similar to
+MIGS. These states decay gradually and vanish at a dis-
+tance greater than 12 layers from the interface.
+D.
+Tri-layer Interfaces
+Fig. 5 shows the DOS as a function of position across
+InSb/CdTe/α-Sn tri-layer interfaces with varying thick-
+
+Kz
+0.2s
+OCdOIn
+0.25
+0.00
+0.50
+0.00
+0.50
+a)
+b)
+d)
+c)
+e)
+.
+0:
+0
+0
+-1
+-1
+-2
+-2
+(na)
+-3
+山
+4
+-4
+-4 
+-5 
+-5 
+-5
+-5 
+surface
+states
+surface
+-6 
+-6
+19-
+-6.
+states
+12 
+8
+0
+4
+12
+8
+4
+8
+4
+4
+12 
+8
+12
+4
+0
+4
+8
+0
+8
+8
+8
+M
+M
+K
+IF
+K
+M
+M
+M
+M
+T
+M
+K
+K
+M
+k (A-1)
+k (A-1)
+k (A-1)
+k (A-1)8
+FIG. 4. Electronic structure of bilayer interfaces: Density of states in the (a) InSb/α-Sn, (d) CdTe/α-Sn and (g) InSb/CdTe
+interfaces as a function of position. The atomic layers are numbered based on distance from the interface, which is located at
+zero. The structure of each interface is illustrated on top. (b Local density of states for selected layers in the (b) InSb/α-Sn, (e)
+CdTe/α-Sn and (h) InSb/CdTe interfaces, indicated by dashed lines in the same colors in panels (a), (d), and (g), respectively.
+Element projected band structures of the (c) InSb/α-Sn, (f) CdTe/α-Sn and (i) InSb/CdTe interfaces, with bands originating
+from α-Sn colored in red, bands originating from InSb colored in light blue, and bands originating from CdTe colored in purple.
+ness of the CdTe tunnel barrier. The position is indicated
+by the atomic layer number, with the layer of InSb clos-
+est to the CdTe considered as zero. Panels (a) and (b)
+show that with 6 atomic layers of CdTe, the MIGS from
+the α-Sn penetrate through the tunnel barrier into the
+first 12 layers of the InSb.
+For a thin layer of CdTe,
+the band gap is expected to be significantly larger than
+the bulk value because of the quantum size effect (see
+the gap convergence plot in the SI). However, owing to
+the presence of MIGS, the gap of the CdTe remains con-
+siderably smaller than its bulk value. With 10 layers of
+CdTe, shown in panels (c) and (d), there is still a sig-
+nificant presence of MIGS throughout the CdTe, which
+decay by 6 layers into the InSb. Panels (e) and (f) show
+that with 16 layers of CdTe the InSb is completely insu-
+lated from MIGS coming from the α-Sn. The gap of the
+CdTe reaches a maximum of around 0.3 eV at a distance
+of 5 layers from the InSb. This is because MIGS from
+the α-Sn penetrate into the CdTe from one side, whereas
+states from the InSb penetrate from the other side, such
+that the band gap of the CdTe never reaches its expected
+value.
+Figure 6 summarizes the band alignment at the bilayer
+and tri-layer interfaces studied here. For the tri-layer in-
+terfaces, the band alignment between the InSb and the
+α-Sn is not significantly affected by the presence of CdTe,
+as shown in the element-projected band structures in the
+SI. The α-Sn semi-metal point remains pinned at the
+Fermi level, as in the bilayer InSb/α-Sn (see also Fig-
+ure 4c). The InSb VBM remains at 0.17 eV below the
+Fermi level, similar to its position in the bilayer interface,
+regardless of the CdTe thickness. The InSb CBM posi-
+tion decreases slightly with the thickness of the CdTe
+from 0.09 eV above the Fermi level without CdTe, to
+
+8.8.8.8181818818.818
+.8.818.8181818.818
+.:1:
+:
+:1:
+:1:
+:
+a)(
+d)
+g)
+:
+:
+I
+0.2
+CdTe/Sn
+100
+InSb/Sn
+InSb/CdTe
+0.2
+0.4
+(arb. units)
+0.0
+0.2
+EF (eV)
+0.1
+-0.2
+0.0
+log(DOS)(
+0.0
+-0.4 1
+-0.2
+E
+-0.6-
+10-
+-3
+-0.1
+0.4
+-0.8
+-0.6
+-0.2
+8 -15
+3
+3
+18
+5 -12
+-9
+-6-3
+-18
+3 -15 -12 
+9-　6
+m-
+-3 
+0
+3
+6
+12
+0
+0
+9
+一
+b)
+Layers
+e)
+Layers
+h)
+Layers
+5 J
+5
+-17
+Sn
+-12
+41
+41
+0
+4
+-8 
+-17
+4
+(e-OL) 
+3 1
+-8 
+3 1
+6
+-6
+Sb
+DOS 
+-4
+8
+2 1
+2 1
+2
+-4
+0
+12
+Cd
+一
+0
+4
+17
+一
+11
+1
+11
+4
+Te
+01
+0,
+0.2
+0.2
+-0.2
+-0.1
+0.0
+0.1
+-1.0 -0.8 -0.6 -0.4 -0.2
+0.2
+-0.2
+0.4
+-1.2
+0.0
+-0.6 
+-0.4
+0.0
+0.6
+E-E (eV)
+E-Ef (eV)
+(
+f)
+E-Eε (eV)
+i)
+0.2
+0.18.
+.0.38.
+0.2
+0.4 
+0.09
+Sn
+0.1
+TSS
+0.0
+0.2
+0.1.
+0.0
+InSb
+-0.2 
+0.0
+(eV)
+-0.1
+-0.1.
+CdTe
+-0.16
+-0.4
+-0.2
+-0.2
+TSS
+-0.6
+-0.4
+E -0.3
+-0.8
+-0.6
+0.4
+-1.03
+-1.0
+-0.83
+-0.5
+-0.8
+-1.2
+-0.6
+1.0
+x
+1X
+1X
+x9
+FIG. 5.
+Electronic structure of InSb/CdTe/α-Sn tri-layer interfaces: Density of states as a function of distance from the
+interface for (a) 6, (c) 10 and (e) 16 CdTe barrier layers. The atomic layers are numbered based on distance from the interface,
+which is located at zero. Interface structures are illustrated on top. (b), (d), (f) Local density of states for selected layers,
+indicated by dashed lines in the same colors in panels (a), (c), and (e), respectively.
+FIG. 6. Valence and conduction band edge positions for InSb
+and CdTe in the bilayer and tri-layer interfaces. The Fermi
+level is at the semi-metal point of the α-Sn.
+0.054 eV with 6 layers of CdTe, 0.04 eV with 10 layers,
+and 0.037 eV with 16 layers. This may be attributed to
+the quantum size effect, which causes a slight narrowing
+of the InSb gap because of the increase in the overall
+size of the system. Based on the element-projected band
+structures provided in the SI, the band edge positions
+of the CdTe are dominated by the interface with the α-
+Sn, rather than the interface with the InSb. The CdTe
+CBM remains at 0.18 eV above the Fermi level, as in
+the bilayer CdTe/α-Sn interface (see also Figure 4f), re-
+gardless of the number of layers. As the band gap of the
+CdTe narrows with increasing thickness, the CdTe VBM
+shifts from 1.24 eV below the Fermi level with 6 layers
+to 1.105 eV with 10 layers, and 1.05 eV with 16 layers,
+approaching the bilayer VBM position of 1.03 eV below
+the Fermi level with 42 layers. Although the band gap of
+the CdTe is significantly reduced due to MIGS, a type I
+band alignment with the InSb is maintained, similar to
+the bilayer InSb/CdTe interface (Figure 4g,i), as shown
+in Fig. 5 panels (a), (c), and (e).
+Figure 7 show the LDOS in the second layer of InSb
+from the interface as a function of the number of CdTe
+layers. Without CdTe and with two layers of CdTe, there
+is no band gap in the InSb close to the interface, owing
+to the significant density of MIGS. With 6 layers of CdTe
+the gap of the InSb close to the interface is still consid-
+erably narrower than its bulk value. The band gap in
+the second layer of InSb from the interface approaches
+its bulk value with 10 layers of CdTe and finally reaches
+it with 16 layers of CdTe. This suggests that 16 CdTe
+layers provide an effective barrier to electronically insu-
+late the InSb from the α-Sn. It is reasonable to assume
+that a barrier of this thickness or higher would all but
+eliminate transport through the interface into the InSb.
+Therefore, we estimate that the relevant barrier thickness
+regime to modulate the coupling at an interface with a
+
+::*:18::+::+1:++:+::+::+:: (0
+a)
+C)
+8
+8
+：
+:i:
+:::
+1100
+0.2
+0.2
+0.2
+(arb. units)
+10-1
+(eV)
+0.0
+0.0
+0.0
+10-2
+log(DOS) (
+-3
+-0.2
+-0.2
+-0.2
+-0.4
+-0.4
+-0.4
+10-4
+-12-9
+-12 -8 -4０　4　8　121620
+-12
+-9
+-6-30
+3
+6
+9
+-6-3０36
+９1215
+Layers
+Layers
+Layers
+b)
+d)
+f)
+5
+5
+5
+-12
+-12
+-12
+Sn
+-6
+-6
+-6
+4 1
+4
+4
+0
+0
+0
+In
+(t-OL)
+2
+31
+2
+2
+3
+3
+3
+5
+Sb
+DOS
+6
+9
+8
+2 1
+21
+2
+15
+Cd
+11
+1
+Te
+0 -
+0
+0
+-0.15-0.10-0.05 0.00
+0.05
+0.100.15
+-0.15-0.10-0.05 0.00 0.05
+0.10
+0.15
+-0.15-0.10-0.05 0.00 0.05
+0.100.15
+E- EF (eV)
+E- EF (eV)
+E- EF (eV)0.5
+0.0
+(eV)
+InSb
+CdTe
+-0.5
+-1.0
+CdTe/α-Sn -
+InSb/α-Sn
+InSb/(CdTe)6/α-Sn
+InSb/(CdTe)10/α-Sn
+InSb/(CdTe)16/α-Sn
+InSb/CdTe
+Interface10
+FIG. 7. Density of states in the second InSb layer from the
+interface (layer -2 in Figure 5) as a function of the number of
+CdTe barrier layers.
+superconductor and tune the proximity effect would be
+in the range of 6-10 layers, where MIGS still exist. We
+note, however, that the interface with β-Sn may have
+somewhat different characteristics in terms of the band
+alignment and the penetration depth of MIGS.
+IV.
+CONCLUSION
+In summary, we have used DFT with a Hubbard
+U correction machine-learned by Bayesian optimization
+to study CdTe as a prospective tunnel barrier at the
+InSb/α-Sn interface. The results of PBE+U(BO) were
+validated by comparing the band structures of slab mod-
+els of α-Sn(001) and CdTe(111) with ARPES experi-
+ments (the PBE+U(BO) band structure of InSb(110)
+had been compared to ARPES experiments previously
+[63]). Excellent agreement with experiment is obtained
+for both materials. In particular, for the low-mean-free-
+path ARPES of CdTe, the z-unfolding scheme success-
+fully reproduces the contributions of different kz values
+and modelling the 2 × 2 surface reconstruction success-
+fully reproduces the contributions of surface states.
+We then proceeded to use PBE+U(BO) to calculate
+the electronic structure of bilayer InSb/α-Sn, CdTe/α-
+Sn, and InSb/CdTe, as well as tri-layer InSb/CdTe/α-Sn
+interfaces with varying thickness of CdTe. Simulations
+of these very large interface models were possible thanks
+to the balance between accuracy and computational cost
+provided by PBE+U(BO). We find that the most stable
+configuration of the InSb/CdTe interface is with In-Te
+and Sb-Cd bonding. MIGS penetrate from the α-Sn into
+the InSn and CdTe. Similarly, states from the band edges
+of InSb penetrate into the larger gap of the CdTe. No
+interface states are found in any of the interfaces studied
+here, in contrast to the EuS/InAs interface, for example,
+in which a quantum well interface state emerges [110].
+For all interfaces comprising α-Sn, the semi-metal
+point is pinned at the Fermi level. For the tri-layer inter-
+face, the band alignment between the InSb and the α-Sn
+remains the same as in the bilayer interface regardless of
+the thickness of the CdTe barrier, with the Fermi level
+closer to the conduction band edge of the InSb. The band
+edge positions of the CdTe are dominated by the inter-
+face with the α-Sn rather than the interface with InSb,
+with the conduction band edge being closer to the Fermi
+level. A type-I band alignment is maintained between
+CdTe and InSb with the gap of the former straddling
+the latter. The CBM of the CdTe is pinned whereas the
+VBM shifts upwards towards the Fermi level as the gap
+narrows with the increase in thickness.
+We find that 16 layers of CdTe (about 3.5 nm) serve as
+an effective barrier, preventing the penetration of MIGS
+from the α-Sn into the InSb. However, in the context of
+Majorana experiments, it is possible that a barrier thick
+enough to completely insulate the semiconductor from
+the superconductor would also all but eliminate trans-
+port.
+Therefore, we estimate that the relevant regime
+for tuning the coupling at the interface would be in the
+thickness range where some MIGS are still present, while
+thicker CdTe layers could be used to passivate exposed
+InSb surfaces. We note, however, that the interface with
+the superconducting β-Sn, which is not lattice matched
+to InSb and CdTe, may have different characteristics than
+the interface with α-Sn. In practice, careful experimen-
+tation with varying barrier thickness would be needed to
+determine the optimal configuration for MZM devices.
+We have thus demonstrated that DFT simulations
+can provide useful insight into the electronic properties
+of semiconductor/tunnel barrier/metal interfaces. This
+includes the interface bonding configuration, the band
+alignment, and the presence of MIGS (and, possibly, of
+interface states). Such simulations may be conducted for
+additional interfaces to explore other prospective mate-
+rial combinations. This may inform the choice of inter-
+face systems and the design of future Majorana experi-
+ments. More broadly, similar DFT simulations of inter-
+faces may be performed to evaluate prospective tunnel
+barriers e.g., for semiconductor devices.
+V.
+ACKNOWLEDGEMENTS
+We thank Guang Bian from the University of Mis-
+souri, Li Fu from Northwestern Polytechnical University,
+China, and Tai C. Chiang from the University of Illinois
+at Urbana-Champaign for sharing their ARPES data for
+CdTe. Work at the University of Pittsburgh was sup-
+ported by the Department of Energy through grant DE-
+SC-0019274. Work at CMU and UCSB was funded by
+the National Science Foundation (NSF) through grant
+OISE-1743717. Work in Grenoble is supported by the
+ANR-NSF PIRE:HYBRID, Transatlantic Research Part-
+nership and IRP-CNRS HYNATOQ. This research used
+computing resources of the University of Pittsburgh Cen-
+ter for Research Computing, which is supported by NIH
+award number S10OD028483 and of the National Energy
+
+10
+0
+ CdTe
+Density of States (10-4)
+2 
+CdTe
+8
+6 CdTe
+10 CdTe
+6
+16 CdTe
+4
+2
+0
+-0.1
+0.0
+0.1
+0.2
+E- Er (eV)11
+Research Scientific Computing Center (NERSC), a U.S.
+Department of Energy Office of Science User Facility op-
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+

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+arXiv:2301.08662v1  [math.PR]  20 Jan 2023
+On the construction and identification of Boltzmann
+processes
+S. Albeverio∗, B. R¨udiger†and P. Sundar ‡
+January 23, 2023
+Abstract
+Given the existence of a solution {f(t, x, v)}t≥0 of the Boltzmann equation
+for hard spheres, we introduce a stochastic differential equation driven by
+a Poisson random measure that depends on f(t, x, v). The marginal distri-
+butions of its solution solves a linearized Boltzmann equation in the weak
+form. Further, if the distributions admit a probability density, we establish,
+under suitable conditions, that the density at each t coincides with f(t, x, v).
+The stochastic process is therefore called the Boltzmann process.
+AMS Subject Classification: 35Q20; 60H20; 60H30
+Keywords:
+Boltzmann equation; Poisson random measures; stochastic
+differential equations; relative entropy
+1
+Introduction
+The Boltzmann equation describes the time evolution of the density of molecules
+in dilute (or rarified) gas for a given initial distribution.
+Each molecule (or
+particle) moves in a straight line without any external forces acting on it until it
+collides with another particle and gets deflected. The Boltzmann equation forms
+the basis for the kinetic theory of gases [6].
+∗Institute of Applied Mathematics and HCM, BiBoS, IZKS, University of Bonn, Germany.
+Email: [email protected]
+†Bergische Universit¨at Wuppertal Fakult¨at 4 -Fachgruppe Mathematik-Informatik, Gauss
+str. 20, 42097 Wuppertal, Germany. Email: [email protected]
+‡Department of Mathematics, Louisiana State University, Baton Rouge, La 70803, USA.
+Email: [email protected]
+1
+
+The Boltzmann equation has the general form
+∂f
+∂t (t, x, z) + z · ∇xf(t, x, z) = Q(f, f)(t, x, z),
+(1.1)
+where f is a probability density function that depends on time t ≥ 0, the space
+(location) variable x ∈ R3, and velocity, z ∈ R3. The function Q is a certain
+quadratic form in f, called collision operator (or integral).
+Set Ξ := (0, π] × [0, 2π). Then Q can be written in the general form
+Q(f, f)(t, x, z) =
+�
+R3×Ξ
+{f(t, x, z⋆)f(t, x, v⋆) − f(t, x, z)f(t, x, v)}B(z, dv, dθ)dφ.
+(1.2)
+The dynamics of collisions are encoded in B(z, dv, dθ).
+Each v ∈ R3 in (1.2)
+denotes the velocity of an incoming particle which may hit, at the fixed loca-
+tion x ∈ R3, particles whose velocity is z. Let z⋆ ∈ R3 and v⋆ ∈ R3 denote
+the resulting outgoing (post-collision) velocities corresponding to the incoming
+(pre-collision) velocities z and v respectively. The angle θ ∈ (0, π] denotes the az-
+imuthal or colatitude angle of the deflected velocity, v⋆, and φ ∈ [0, 2π) measures
+the longitude of v∗.
+In the Boltzmann equation, the collisions are assumed to be elastic and hence,
+conservation of momentum and kinetic energy hold, i.e. considering particles of
+mass m = 1, the following equalities hold:
+�
+z⋆ + v⋆ = z + v
+|z⋆|2 + |v⋆|2 = |z|2 + |v|2
+(1.3)
+In fact,
+�
+z⋆ = z + (n, v − z)n
+v⋆ = v − (n, v − z)n
+(1.4)
+where
+n = z⋆ − z
+|z⋆ − z|
+(1.5)
+where (·, ·) denotes the scalar product, and | · |, the Euclidean norm in R3.
+Remark 1.1. The Jacobian of the transformation (1.4) is 1 in magnitude, and
+(z⋆)⋆ = z since the collision dynamics are reversible.
+The outgoing velocity z∗ is then uniquely determined in terms of the colatitude
+angle θ ∈ (0, π] measured from the center, and longitude angle φ ∈ [0, 2π) of
+the deflection vector n in a sphere with north-pole z and south-pole v centered
+2
+
+at z+v
+2
+(and with radius determined by the conserved kinetic energy) which are
+used in equation (1.1) and (1.2) (see e.g. the article by H. Tanaka [19] for futher
+details). It follows
+(v − z, n) = |v − z| cos(π
+2 − θ
+2) = |v − z| sin(θ
+2),
+(1.6)
+where θ is the angle between v − z and v⋆ − z⋆, and π
+2 − θ
+2 is the angle between
+n and v − z. In polar coordinates we obtain
+(v − z, n)dn = |v − z| sin(θ
+2) cos(θ
+2)dθdφ
+(1.7)
+The collision measure B(z, dv, dθ) is a σ-additive positive measure defined on the
+Borel σ-field B(R3) × B((0, π]), for each z, and measurable in z for each fixed set
+in the above Borel σ-field. The form of B depends on the version of Boltzmann
+equation one has in mind.
+There are several models of the Boltzmann equation (see e.g. [20]) In general,
+one sets
+B(z, dv, dθ) = σ(|v − z|)dvQ(dθ)
+(1.8)
+where σ, known as velocity cross-section, is a positive function on R+, and Q is
+a σ-finite measure on B((0, π]). If Q is a finite measure, it is called the cut-off case.
+To write (1.1), (1.2) in its weak form (in the functional analytic sense), we need
+a result of Tanaka [18]:
+Proposition 1.1. Let ψ(u, v, y, z) ∈ C0(R12), as a function of u, v, y, z ∈ R3. For
+each θ ∈ (0, π] fixed
+�
+R6×[0,2π)
+ψ(u, v, u⋆, v⋆)dφdudv =
+�
+R6×[0,2π)
+ψ(u⋆, v⋆, u, v)dφdudv
+(1.9)
+Consider the Boltzmann equation (1.1) with collision operator (1.2). Multiply
+(1.1) by a function ψ (of (x, z) ∈ R6) belonging to C2
+0(R6), and integrate with
+respect to x and z, using integration by parts, we arrive at the weak formulation
+of the Boltzmann equation:
+�
+R6 ψ(x, z)∂f
+∂t (t, x, z)dxdz −
+�
+R6 f(t, x, z)(z, ∇xψ(x, z))dxdz
+=
+�
+R6 f(t, x, z)Lfψ(x, z)dxdz
+(1.10)
+3
+
+for all t ∈ R+ with
+Lfψ(x, z) =
+�
+R3×(0,π]×[0,2π)
+{ψ(x, z⋆) − ψ(x, z)}f(t, x, v)B(z, dv, dθ)dφ,
+where B is as in (1.8).
+In its weak form, the Boltzmann equation cannot be treated as the Kolmogorov
+equation that corresponds to a Markov process with jumps. In fact, to over-
+come this obstacle we proposed and studied the Boltzmann-Enskog equation [2]
+which corresponds to the dynamics of moderately dense gases. For hard and soft
+potentials, solvability and uniqueness of the Boltzmann-Enskog equation were
+carried out in subsequent works [11] and [12]. Indeed, there is a vast literature
+on various aspects of the Boltzmann equation and we refer the reader to [7], [9],
+[10], [15] [18, 19], [20], and the references therein. Standing on the shoulders of
+these giants, our aim is to build a stochastic analytic treatment of the (spatially
+non-homogeneous) Boltzmann equation in the non-cutoff case for hard spheres
+with σ(|z−v|)= |z−v|. An ingenious suggestion given to us by Professor Presutti
+provided the impetus to this work, and made the problem tractable.
+Definition 1.1. A collection of densities {f(t, x, z}t∈[0,T], with x, z ∈ R3, is
+a strong (resp.
+weak) solution of the Boltzmann equation in [0, T] if for any
+t ∈ [0, T] it solves (1.1) (resp. (1.10)).
+We denote by D := D([0, T], R3) the space of all right continuous functions with
+left limits on [0, T] taking values in R3, and equipped with the topology induced
+by the Skorohod metric.
+Definition 1.2. A stochastic process (Xs, Zs)s∈[0,T] with values on D × D, and
+having time t marginals with density denoted by f(t, x, z) which solve (1.10) for
+all t ∈ [0, T], is called a ”Boltzmann process”.
+We remark that the infinitesimal generator of a Boltzmann process is given by
+(z, ∇x) + Lf. Costantini and Marra [8] analyzed hydrodynamical limits of a pro-
+cess given by a the drift term involving (z, ∇x) and Lf in addition to a martingale.
+We use the following notation R0
++ := {t ∈ R : t ≥ 0}. In this article we assume
+the following hypothesis.
+Hypotheses A:
+A1. The measure Q on [0, π) is finite outside any neighbourhood of 0, and for
+all ǫ > 0, it satisfies
+� ǫ
+0
+θQ(dθ) < ∞.
+4
+
+A2. The function σ : R0
++ → R0
++ (entering (1.8)) is given by σ(z) := czγ, with
+c > 0, γ ∈ (−1, 1] fixed.
+There are many useful consequences of A1. Let us set
+α(z, v, θ, φ) := (n, (v − z))n
+(1.11)
+Define ˆα(z, v, θ, φ) := α(z, v, θ, φ)σ(|z − v|).
+Condition A1 implies that there
+exists a constant C such that the following estimates hold.
+�
+Ξ
+|ˆα(z, v, θ, φ)|Q(dθ)dφ ≤ C|z − v|1+γ,
+(1.12)
+and hence
+�
+Ξ
+|ˆα(z, v, θ, φ)|Q(dθ)dφ ≤ C(|z|1+γ + |v|1+γ).
+(1.13)
+Moreover, the following parameter transformation was introduced for each z ̸= v
+in [18] (see also [9], Section 3, or [10]).
+α(z, v, θ, φ) = 1 − cos(θ)
+2
+(v − z) + sin(θ)
+2
+Γ(v − z, φ)
+= sin2(θ
+2)(v − z) + sin(θ)
+2
+Γ(v − z, φ)
+(1.14)
+for all φ ∈ [0, 2π), where
+Γ(v − z, φ) = I(v − z) cos(φ) + J(v − z) sin(φ)
+and
+v−z
+|v−z|, I(v−z)
+|v−z| , J(v−z)
+|v−z| form an orthogonal basis. It follows in particular that
+� 2π
+0
+Γ(v − z, φ)dφ = 0.
+(1.15)
+In order to study solutions to the Boltzmann equation it is feasible to study
+continuity properties of (u − v, n)n in u, v for fixed θ, φ. However, it was already
+pointed out by Tanaka that (u, v) �−→ (u−v, n)n cannot be smooth. To overcome
+this problem Tanaka introduced in Lemma 3.1 of [18] another transformation of
+parameters, which describes a rotation around the longitude angle, is bijective and
+has Jacobian 1. As a consequence of this transformation φ0 he proved following
+Lemma 1.1 (see also Lemma 2.6 in [16]).
+Lemma 1.1. [18][10] There exists a measurable function φ0 : R12 → (0, 2π] such
+that
+|Γ(v − z, φ) − Γ(v′ − z′, φ + φ0(z, v, z′, v′))| ≤ 3|z − v − (z′ − v′)|
+(1.16)
+5
+
+and hence
+|α(z, v, θ, φ) − α(z′, v′, θ, φ + φ0(z, v, z′, v′))| ≤ 2θ(|z − z′| + |v − v′|)
+(1.17)
+and
+|α(z, v, θ, φ)| ≤ 2θ(|z| + |v|)
+(1.18)
+Moreover by his transformation [18] (see also [10]), Section 3) and using (1.14)
+Tanaka proved the following inequality:
+� π
+0
+���
+� 2π
+0
+α(z, v, θ, φ) − α(z′, v′, θ, φ)dφ
+��� Q(dθ)
+≤ C(|z − z′| + |v − v′|).
+(1.19)
+where by an abuse of notation we use the same symbol C > 0 in (1.13) and (1.19),
+even though the constants are different.
+Let {f(t, x, z)}t∈R0
++ be a collection of densities on (R3 × R3, B(R3 × R3)). Let
+us introduce the operator Qt(f, f)(·) defined through the right side of equation
+(1.10)
+Qt(f, f)(ψ) :=
+�
+R6 f(t, x, z)Lfψ(x, z)dxdz
+(1.20)
+It is easy to verify that
+Qt(f, f)(ψ) = 0
+for
+ψ(x, z) = a + (b, z) + c|z|2
+(1.21)
+∀ a, c ∈ R , b ∈ R3. (For a rigorous proof see Chapter II.7 [5] or [7], [4].)
+The integral form of equation (1.10) corresponds to
+�
+R6 ψ(x, z)f(t, x, z)dxdz =
+�
+R6 ψ(x, z)f(0, x, z)dxdz
++
+� t
+0
+�
+R6 f(t, x, z){(z, ∇xψ(x, z)) + Lfψ(x, z)}dxdzds,
+(1.22)
+It is worthwhile to note that if a second collection of densities {g(t, x, z)}t∈R0
++ on
+(R3 × R3, B(R3 × R3)) is given, then
+QS
+t (f, g)(ψ) = 0
+for
+ψ(x, z) = a + (b, z) + c|z|2
+(1.23)
+∀ a, c ∈ R , b ∈ R3 with the operator QS
+t defined through
+QS
+t (f, g)(·) := Qt(f, g)(·) + Qt(g, f)(·)
+(1.24)
+6
+
+with
+Qt(f, g)(ψ) :=
+�
+R6 g(t, x, z)Lf ψ(x, z)dxdz.
+(1.25)
+The following Povzner type inequality is essentially contained in [15, Lemma 3.6].
+(See also [7], Theorem 6.2.1 and Appendix B of Chapter 6 for p ≥ 2 and references
+there.)
+Lemma 1.2. For all θ ∈ (0, π], p ≥ 2 and γ ∈ (0, 1],
+� 2π
+0
+�
+⟨z + α(z, v, θ, φ)⟩2p + ⟨v − α(z, v, θ, φ)⟩2p − ⟨z⟩2p − ⟨v⟩2p�
+dφ
+≤ −sin2(θ)
+2
+�
+⟨v⟩2p + ⟨z⟩2p�
++ Cp sin2(θ)
+⌊ p+1
+2 ⌋
+�
+k=1
+�
+⟨v⟩2k⟨z⟩2p−2k + ⟨v⟩2p−2k⟨z⟩2k�
+,
+where ⟨v⟩ := (1 + |v|2)
+1
+2 , ⌊x⌋ ∈ Z is defined by ⌊x⌋ ≤ x < ⌊x⌋ + 1 and Cp > 0 is
+some constant.
+Using conservation laws and Lemma 1.2, it can be proven that if {f(t, x, u)}t∈[0,T]
+is a weak solution of the Boltzmann equation in [0, T], with initial finite second
+moment, i.e.
+�
+R6 |z|2f(0, x, z)dxdz < ∞, then for all p ≥ 1
+�
+R6 |z|pf(t, x, z)dxdz < ∞
+∀t ∈ [0, T].
+(1.26)
+For a proof we refer the reader to [15, Theorem 3.6].
+Let {µt(dx, dz)}t∈R0
++ be a collection of probabilities on (R3 × R3, B(R3 × R3)).
+Let us define the operator
+Qt(f, µ)(ψ) :=
+�
+R6 Lfψ(x, z)µt(dx, dz).
+(1.27)
+acting on all ψ for which the integral on the right side is finite.
+Lemma 1.3.
+Qt(f, µ)(|z|2)
+=
+�
+R9×Ξ
+(|v|2 − |z|2)σ(|z − v|) sin2(θ
+2)Q(dθ)dφf(t, x, v)dvµt(dx, dz)
+(1.28)
+Proof.
+Qt(f, µ)(|z|2) =
+�
+R6 Lf|z|2µt(dx, dz)
+(1.29)
+=
+�
+R9×Ξ
+(|z⋆|2 − |z|2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz).
+7
+
+Moreover,
+Qt(f, µ)(|z|2) =
+�
+R9×Ξ
+(|z⋆|2 + |v⋆|2 − |z|2 − |v|2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz)
+−
+�
+R9×Ξ
+(|v⋆|2 − |v|2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz)
+= −
+�
+R9×Ξ
+(|v⋆|2 − |v|2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz).
+(1.30)
+where in the last equality we used that the kinetic energy is conserved during the
+elastic collision, see (1.3).
+Combining equation (1.29) and (1.30), we obtain
+Qt(f, µ)(|z|2) =
+1
+2
+�
+R9×Ξ
+(|z⋆|2 − |z|2 − (|v⋆|2 − |v|2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz)
+= 1
+2
+�
+R9×Ξ
+(|z|2 + 2(z, α) + |α|2 − |z|2) − (|v|2 − 2(v, α) + |α|2 − |v|2)
+× σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz)
+=
+�
+R9×Ξ
+(z + v, α)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz),
+(1.31)
+Using the parametrization (1.14) for α = α(z, v, θ, φ) and (1.15) we obtain
+Qt(f, µ)(|z|2) =
++
+�
+R9×Ξ
+(z + v, v − z) sin2(θ
+2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz)
+= −
+�
+R9×Ξ
+(|z|2 − |v|2) sin2(θ
+2)σ(|z − v|)Q(dθ)dφf(t, x, v)dvµt(dx, dz)
+2
+The Boltzmann process
+We use the following notation throughout the paper. U0 = D × [0, π) × (0, 2π].
+Let {f(t, x, v)}t∈R0
++ be a collection of densities on (R3 × R3, B(R3 × R3)). Then
+m(t, v) denotes the marginal density of velocity v at time t, i.e.
+m(t, v) :=
+�
+R3 f(t, x, v)dx so that f(t, x|v)m(t, v) := f(t, x, v), upon disintagration of mea-
+sures.
+8
+
+Hypotheses B: We assume that t → f(t, x, v) is differentiable for each x, v
+∈ R3 fixed, and satisfies
+B0. |∂f
+∂t | is bounded on any compact subset of R0
++ × R6.
+B1.
+∂f
+∂t (t, ·) ∈ L1(R6),
+∀t ∈ R0
++,
+B2. supx∈R3
+�
+R3 |u|1+γf(s, x, u)du ∈ C([0, T])
+∀ T > 0.
+B3. supx∈R3
+�
+R3 |u|1+γ ∂
+∂tf(t, x, u)du ∈ L1([0, T])
+∀ T > 0.
+Theorem 2.1. Let {f(t, x, v)}t∈R0
++ be a collection of densities which satisfies
+hypothesis B. Suppose hypothesis A hold.
+Let X0 and Z0 be R3- valued ran-
+dom variables. Suppose that for any fixed T > 0 there exists a stochastic basis
+(Ω, F, (Ft)t∈[0,T], P), an adapted process (Xt, Zt)t∈[0,T] with values on D × D,
+which has time marginals with density f(t, x, u), and such that it satisfies a.s.
+the following stochastic equation for t ∈ [0, T]:
+
+
+
+
+
+
+
+
+
+Xt = X0 +
+� t
+0
+Zsds
+Zt = Z0 +
+� t
+0
+�
+U0×R0
++
+α(Zs, vs, θ, φ)1[0, σ(|Zs−vs|)f(s,Xs|vs)](r)dN,
+(2.1)
+where in the above equation, dN := N(dv, dθ, dφ, dr, ds) is a Poisson random
+measure with compensator m(s, v)dvQ(dθ)dφdsdr. Then (Xt, Zt)t∈[0,T] is a Boltz-
+mann process.
+Proof. From (1.13) it follows for each T > 0
+� T
+0
+E[
+�
+U0×R0
++
+|α(Zs, vs, θ, φ)|1[0, σ(|Zs−vs|)f(s,Xs|vs)](r)m(s, v)dvQ(dθ)dφdr]ds
+=
+� T
+0
+E[
+�
+U0
+|ˆα(Zs, vs, θ, φ)|f(s, Xs, v)dvQ(dθ)dφ]ds
+≤ C
+� T
+0
+�
+R9(|z|1+γ + |v|1+γ)f(s, x, z)f(s, x, v)dxdzdvds,
+≤ 2C
+� T
+0
+sup
+x∈R3
+�
+R6
+�
+|z|1+γf(s, x, z)dz
+�
+f(s, x, v)dvdxds < ∞.
+for some constant C > 0. In the above estimates we have used that the function
+f(t) is the probability density of the process (Xt, Zt), as well the assumption A2
+and B2. It follows that we can apply the Itˆo formula to (Xs, Zs)s∈R+ [17]. In
+9
+
+fact let t, ∆t > 0, ψ ∈ C2
+0(R3 × R3), then
+ψ(Xt+∆t, Zt+∆t)
+= ψ(Xt, Zt) +
+� t+∆t
+t
+(Zs, ∇xψ(Xs, Zs))ds
++
+� t+∆t
+t
+�
+U0×R+
+0
+{ψ(Xs, Zs + α(Zs, vs, θ, φ)1[0, σ(|Zs−vs|)f(s,Xs|vs)](r)) − ψ(Xs, Zs)}dN
+It follows
+E[ψ(Xt+∆t, Zt+∆t) − ψ(Xt, Zt)] =
+E
+�� t+∆t
+t
+(Zs, ∇xψ(Xs, Zs))ds
+�
++
+E
+
+
+� t+∆t
+t
+�
+U0
+{ψ(Xs, Zs+ α(Zs, vs, θ, φ))−ψ(Xs, Zs)}σ(|Zs−vs|)f(s, Xs,vs)dvQ(dθ)dφds
+
+
+Upon dividing by ∆t on both sides, we obtain
+lim
+∆t↓0
+1
+∆t
+�
+R6 ψ(x, u){f(t + ∆t, x, u) − f(t, x, u)}dxdu
+= lim
+∆t↓0
+1
+∆t
+� t+∆t
+t
+�
+R6(u, ∇xψ(x, u))f(s, x, u)dxduds +
+lim
+∆t↓0
+1
+∆t
+� t+∆t
+t
+�
+R6×R3×[0,π)×(0,2π]
+{ψ(x, u + α(u, v, θ, φ)) − ψ(x, u)}
+× σ(|u − v|)f(s, x, v)f(s, x, u)dvQ(dθ)dφdxduds
+(2.2)
+Letting ∆t → 0 in every term of (2.2) we obtain (1.10). Indeed, for e.g., the
+second term on the right side of (2.2) and prove the continuity of the function
+g(s) :=
+�
+R6×R3×[0,π)×(0,2π]{ψ(x, u + α(u, v, θ, φ)) − ψ(x, u)}
+×σ(|u − v|)f(s, x, v)f(s, x, u)dvQ(dθ)dφdxdu
+Since
+{ψ(x, u + α(u, v, θ, φ)) − ψ(x, u)} ≃ ∇uψ(x, z)α(u, v, θ, φ)
+with (x.z) ∈ K compact set, and
+|α(u, v, θ, φ)|σ(|u − v|) ≤ |u − v|1+γ| sin(θ
+2)|,
+by denoting with F a compact set in R3 which includes all projections x of
+(x.z) ∈ K, it follows that
+|g(s) − g(s0)| ≤ C
+�
+F ×R6 |f(s, x, u)f(s, x, v) − f(s0, x, u)f(s0, x, v)|
+×(|u|γ+1 + |v|γ+1)dxdudv,
+(2.3)
+10
+
+with
+C := ∥∇uψ∥∞2π
+� π
+0
+θQ(dθ)
+We split the integral on the right side of (2.3) into two terms, one with |u|γ+1
+(resp. |u|γ+1), and get
+�
+F ×R6 |f(s, x, u)f(s, x, v) − f(s0, x, u)f(s0, x, v)||u|γ+1dxdudv
+=
+�
+F ×R6 |f(s, x, u)f(s, x, v) − f(s, x, u)f(s0, x, v)||u|γ+1dxdudv
++
+�
+F ×R6 |f(s, x, u)f(s0, x, v) − f(s0, x, u)f(s0, x, v)||u|γ+1dxdudv
+= J1(s) + J2(s)
+(2.4)
+where J1(s) (resp. J2(s)) is the first (resp. second) term on the right side of
+(2.4).
+J1(s) =
+�
+F ×R6 |u|γ+1f(s, x, u)|
+� s
+s0
+∂f
+∂r (r, x, v)dr|dxdudv
+≤
+�
+supx∈R3
+�
+R3 |u|γ+1f(s, x, u)du
+� � s
+s0
+�
+R6 |∂f
+∂r (r, x, v)|dxdvdr
+By B1 and B2 lims→s0 J1(s) = 0.
+Let us consider J2(s).
+J2(s) =
+�
+F ×R6 |u|γ+1f(s0, x, v)|
+� s
+s0
+∂f
+∂r (r, x, u)dr|dxdudv
+≤
+� s
+s0
+�
+F ×R3
+��
+R3 |u|γ+1|∂f
+∂r (r, x, u)|du
+�
+f(s0, x, v)dxdvdr
+Since supx∈R3
+�
+R3 |u|γ+1|∂f
+∂r (r, x, u)|du is integrable in [s0, s] by B3, we obtain
+lims→s0 J2(s) = 0.
+Likewise, and without any changes in the arguments it follows
+lim
+s→s0 C
+�
+F ×R6 |f(s, x, u)f(s, x, v) − f(s0, x, u)f(s0, x, v)||v|γ+1dxdudv = 0
+Hence lims→s0 g(s) = g(s0), so that g is a continuous function and
+lim
+∆t↓0
+1
+∆t
+� t+∆t
+t
+g(s)ds = g(t).
+Note that in the above arguments we have taken s > s0 for simplicity. One may
+also take s0 > s.
+Theorem 2.1 motivates the following Definition.
+Definition 2.1. Let {f(t, x, v)}t∈R0
++ be a collection of densities satisfying Hy-
+pothesis B. Suppose that for any fixed T > 0, there exists a stochastic basis
+(Ω, F, (Ft)t∈[0,T], P) and an adapted process (Xt, Zt)t∈[0,T] with values on D × D
+such that
+11
+
+i) (Xt, Zt)t∈[0,T] has time marginals with density f(t, x, u), for t ∈ [0, T],
+ii) (Xt, Zt)t∈[0,T] is a solution of the McKean -Vlasov SDE (2.1).
+Then we say that “the McKean -Vlasov equation (2.1) with density functions
+{f(t, x, v)}t∈R0
++ is associated to the the Boltzmann equation (1.1)”.
+If the above property holds for T ∈ [0, S] with S > 0, then the McKean -Vlasov
+SDE (2.1) with density functions {f(t, x, v)}t∈[0,S] is associated to the Boltzmann
+equation (1.1) up to time S.
+Remark 2.1. Let us assume hypothesis A. From Theorem 2.1 it follows that
+any stochastic process (Xt, Zt)t∈[0,T] solving a McKean -Vlasov equation (2.1)
+associated to the Boltzmann equation (1.10) is (according to Definition 1.2) a
+Boltzmann process.
+The Boltzmann equation (1.10) is hence the Kolmogorov
+equation associated to the McKean -Vlasov equation (2.1).
+3
+Existence of the Boltzmann process
+In Theorem 2.1 we proved that any process (Xt, Zt)t∈[0,T] solving the McKean
+-Vlasov equation (2.1) associated to (1.10) in [0, T] is a Boltzmann process. In
+this section we analyze the following: given a strong solution {f(t, x, z)}t∈[0,T] of
+the Boltzmann equation (1.1), we find sufficient conditions for the existence of
+a solution of the McKean -Vlasov equation (2.1) with density {f(t, x, z)}t∈[0,T].
+The solution process (Xt, Zt)t∈[0,T] is then a Boltzmann process.
+We present an overview on the construction of Boltzmann processes. We briefly
+outline the construction of the process (Xt, Zt)t∈[0,T] under suitable conditions
+before stating the main result on Boltzmann processes. The proofs of the ensuing
+results on the existence of a solution to a certain linearized stochastic system will
+appear in a separate paper [1].
+3.1
+Construction of a solution of a SDE defined through a col-
+lection of densities solving (1.1)
+In this paragraph, we assume that {f(t, x, z)}t∈[0,T] is a collection of densities
+which solves the Boltzmann equation (1.1) and satisfies the following conditions:
+B4. sups∈[0,T],x∈R3
+�
+R3 f(s, x, v)dv ≤ CT < ∞.
+12
+
+B5. There exists for every K > 0 a constant CK
+T > 0 such that
+sup
+s∈[0,T],|x|≤K
+�
+R3 max(1, |v|1+γ)|∇xf(s, x, v)|dv ≤ CK
+T < ∞.
+B6. sups∈[0,T],x∈R3
+�
+R3 |v|γ+2f(s, x, v)dv ≤ cT < ∞.
+On any fixed filtered probability space (Ω, F, (Ft)t∈[0,T], P) satisfying the usual
+conditions, let ST := S1
+T (Rd) denote the linear space of all adapted c`adl`ag pro-
+cesses (Xt)t∈[0,T] with values on Rd equipped with norm
+∥X∥S1
+T := E[ sup
+s∈[0,T]
+|Xs|].
+(3.1)
+Under hypotheses B4 - B6, and adopting the notation f(s, x, v) = f(s, x | v)m(s, v)
+upon disintegration of measures, we first prove the existence of a weak solution
+to the stochastic system
+Xt = X0 +
+� t
+0
+Zsds,
+∀t ∈ [0, T]
+(3.2)
+Zt = Z0 +
+� t
+0
+�
+U0×R+
+0
+α(Zs, vs, θ, φ)1[0, σ(|Zs−vs|)f(s,Xs|vs)](r)dN
+∀t ∈ [0, T]
+(3.3)
+for t ∈ [0, T] where dN := N(dv, dθ, dφ, dr, ds) with its compensator given by
+m(s, v)dvQ(dθ)dφdsdr with values in S1
+T := S1
+T (R3 × R3).
+Here we do not assume that (3.3) is of McKean -Vlasov type.
+First, we recall the definition of weak solutions in the context of stochastic analysis
+[14].
+Definition 3.1. A ”weak solution” of equation ((3.3), (3.2)) in the time inter-
+val [0, T] is a triplet ((Ω, F, (Ft)t∈[0,T], P), N(dv, dθ, dφ, dr, ds), (Xt, Zt)t∈[0,T])
+for which the following properties hold:
+• (Ω, F, (Ft)t∈[0,T], P) is a stochastic basis;
+• N(dv, dθ, dφ, dr, ds) is an adapted Poisson random measure with compen-
+sator m(s, v)dvQ(dθ)dφdsdr;
+• (X·, Z·) := (Xt, Zt)t∈[0,T]) is an adapted c`adl`ag stochastic process with val-
+ued in Rd × Rd which satisfies ((3.3), (3.2)) P -a.s.
+The existence of solutions to the stochastic system (3.3),(3.2) is stated in the
+following theorem, proven in [1].
+13
+
+Theorem 3.1. Let γ = 1 and Hypothesis A be satisfied.
+Let T > 0 and
+{f(t, x, v)}t∈[0,T] be a collection of densities which satisfy f(t, x, u) ∈ C([0, T] ×
+R6) and Hypotheses B. Let the initial distribution of (X0, Z0) admit finite second
+moment. There exists a weak solution
+((Ω, F, (Ft)t∈[0,T], P), N(dv, dθ, dφ, dr, ds), (Xt, Zt)t∈[0,T])
+of (3.2), (3.3) such that (X·, Z·) ∈ S1
+T . Moreover,
+sup
+t∈[0,T]
+E[|Xt|2] + sup
+t∈[0,T]
+E[|Zt|2] < ∞
+(3.4)
+We remark that the estimate (3.4) is proven by symmetry arguments similar to
+those appearing in the proof of Lemma 1.2. The form of the stochastic system
+(3.2), (3.3) with the process taking values in R6 at each t ∈ [0, T], one obtains
+that the solution lies in D × D.
+3.2
+Construction of Boltzmann processes with densities satisfy-
+ing (1.1)
+We recall the concept of relative entropy which plays a key role in the proof of
+the following theorem. Recall that for any two probability measures µ, ν on a
+common measurable space (X, X), the relative entropy of ν with respect to µ,
+denoted R(ν || µ), is defined by
+R(ν || µ) =
+�
+X
+�
+log dν
+dµ
+�
+dν
+if ν is absolutely continuous with respect to µ. Otherwise, we set R(ν || µ) = ∞.
+The following Lemma is well known.
+Lemma 3.2. Let µ, ν be two probability measures on a measurable space (X, X).
+Then R(ν || µ) ≥ 0 and R(ν || µ) = 0 if and only if µ = ν.
+We assume that {f(t, x, z)}t∈[0,T] is a collection of densities which solves the
+Boltzmann equation (1.1) and satisfies hypotheses B as well as the following
+condition:
+C1. The densities f(t, x, z) and g(t, x, z) are in C1,2([0, T] × R6) and are strictly
+positive-valued functions with g log g, g log f ∈ L1(R6) for each t ∈ [0, T] and
+lim|x|→∞ g(t, x, z) = 0 and g(0, x, z) = f(0, x, z) a.s.
+Theorem 3.3. Let (Xt, Zt)t∈[0,T] be a stochastic process that solves the stochastic
+system (3.2), (3.3). Suppose that (Xt, Zt)t∈[0,T] has time marginals with density
+g(t, x, z), for each t ∈ [0, T]. Suppose that {f(t, x, z)}t∈[0,T] and {g(t, x, z)}t∈[0,T]
+satisfy hypotheses B0 − B6 and C1. Then g(t, x, z) = f(t, x, z)
+a.e. for all
+t ∈ [0, T].
+14
+
+Proof. We will write R(g || f) for the relative entropy of the measure with prob-
+ability density g with respect to the measure with probability density f. The
+theorem is proved by establishing the following equality.
+Rt(g|f) :=
+�
+R6 log
+� g(t.x.z)
+f(t, x, z)
+�
+g(t, x, z)dxdz = 0
+∀t ∈ [0, T]
+(3.5)
+We first apply the Itˆo formula [13] to log(g(t, Xt, Zt)), where (Xt, Zt)t∈[0,T] solves
+(3.3), (3.2) and take expectation to obtain
+�
+R6 log (g(t, x, z))g(t, x, z)dxdz −
+�
+R6 log (g(0, x, z))g(0, x, z)dxdz
+=
+� t
+0
+�
+R6×R3×Ξ
+{log (g(s, x, z⋆)) − log (g(s, x, z))}
+× σ(|z − v|)f(s, x, v)g(s, x, z)Q(dθ)dφdvdxdzds.
+(3.6)
+Indeed, in arriving at (3.6), we have used the following two calculations:
+(i)
+� t
+0
+�
+R6
+∂
+∂s log (g(s, x, z))g(s, x, z)dxdzds =
+� t
+0
+�
+R6
+∂
+∂sg(s, x, z)dxdzds
+=
+�
+R6(g(t, x, z) − g(0, x, z))dxdz = 0
+since g is a probability density.
+(ii)
+�
+R6 z · ∇x log (g(s, x, z))g(s, x, z)dxdzds =
+�
+R6 z · ∇xg(s, x, z)dxdzds = 0
+where the last equality is obtained by integrating and using the condition that
+lim
+|x|→∞g(t, x, z) = 0. Likewise, one obtains upon taking expectation and recall-
+ing that {f(t, x, z)}t∈[0,T] is a collection of densities which solves the Boltzmann
+equation (1.1),
+�
+R6 log (f(t, x, z))g(t, x, z)dxdz −
+�
+R6 log (f(0, x, z))g(0, x, z)dxdz
+=
+� t
+0
+�
+R6
+Q(f, f)(s, x, z)
+f(s, x, z)
+g(s, x, z)dxdzds
++
+� t
+0
+�
+R6×R3×Ξ
+{log (f(s, x, z⋆)) − log (f(s, x, z))}
+× σ(|z − v|)f(s, x, v)g(s, x, z)Q(dθ)dφdvdxdzds
+=
+� t
+0
+�
+R9×Ξ
+{g(s, x, z⋆)
+f(s, x, z⋆) − g(s, x, z)
+f(s, x, z)}
+× σ(|z − v|)f(s, x, v)f(s, x, z)Q(dθ)dφdvdxdzds ,
++
+� t
+0
+�
+R6×R3×Ξ
+{log (f(s, x, z⋆)) − log (f(s, x, z))}
+× σ(|z − v|)f(s, x, v)g(s, x, z)Q(dθ)dφdvdxdzds.
+(3.7)
+15
+
+It is worthwhile to note that the last equality in the above display results upon
+rewriting
+� t
+0
+�
+R6
+Q(f, f)(s, x, z)
+f(s, x, z)
+g(s, x, z)dxdzds
+=
+� t
+0
+�
+R9×Ξ
+{f(s, x, z⋆)f(s, x, v⋆) − f(s, x, z)f(s, x, v)}
+× σ(|z − v|) g(s, x, z)
+f(s, x, z)Q(dθ)dφdvdxdzds
+=
+� t
+0
+�
+R9×Ξ
+{g(s, x, z⋆)
+f(s, x, z⋆) − g(s, x, z)
+f(s, x, z)}
+× σ(|z − v|)f(s, x, v)f(s, x, z)Q(dθ)dφdvdxdzds ,
+by using Proposition 1.1.
+Combining equations (3.6) with (3.7) we obtain that
+Rt(g|f) =
+�
+R6 log (g(t, x, z))g(t, x, z)dxdz −
+�
+R6 log (f(t, x, z))g(t, x, z)dxdz
+=
+� t
+0
+�
+R9×Ξ
+� g(s, x, z)
+f(s, x, z){1 + log
+�g(s, x, z⋆)/f(s, x, z⋆)
+g(s, x, z)/f(s, x, z)
+�
+} − g(s, x, z⋆)
+f(s, x, z⋆)
+�
+× σ(|z − v|)f(s, x, v)f(s, x, z)Q(dθ)dφdvdxdzds
+(3.8)
+Transforming (3.8) in the equivalent equation below, and recalling that for x ≥ 0
+we have 1 + log (x) − x ≤ 0, we easily see that
+Rt(g|f) =
+� t
+0
+�
+R9×Ξ
+�
+1 + log
+�g(s, x, z⋆)/f(s, x, z⋆)
+g(s, x, z)/f(s, x, z)
+�
+− g(s, x, z⋆)/f(s, x, z⋆)
+g(s, x, z)/f(s, x, z)
+�
+× g(s, x, z)
+f(s, x, z)σ(|z − v|)f(s, x, v)f(s, x, z)Q(dθ)dφdvdxdzds
+≤ 0.
+However, by Lemma 3.2, Rt(g|f) ≥ 0, and hence, Rt(g|f) = 0.
+From Theorem 3.3 it follows that (Xt, Zt)t∈[0,T] in Theorem 3.3 solves the McKean-
+Vlasov equation associated to the Boltzmann equation (1.1) and is a Boltzmann
+process with densities {f(t, x, z)}t∈[0,T] up to time T.
+Based on Theorem 3.1, the main result of this work is given below:
+Theorem 3.4. Let Hypotheses A be satisfied and γ = 1. Assume that {f(t, x, u)}t∈[0,T]
+is a collection of densities which solves the Boltzmann equation (1.1), and satis-
+fies the hypotheses B0 − B6. Let the random vector (X0, Z0) have finite second
+16
+
+moment. Suppose that the weak solution of the stochastic system (3.3), (3.2) has
+its distribution that admits a probability density at each time t ∈ [0, T] given by
+g(t, x, u). If condition C1 is satisfied by {f(t, x, u)}t∈[0,T] and {g(t, x, u)}t∈[0,T],
+then the McKean-Vlasov equation (2.1) (that involves {f(t, x, u)}t∈[0,T]) has a
+weak solution in [0, T] with values in D × D, and its Kolmogorov equation solves
+equation (1.1).
+Proof. The result follows from Theorem 3.1 and Theorem 3.3.
+Acknowledgments: The authors are very thankful to Professor Errico Presutti
+for suggesting that a given solution of the Boltzmann equation be used in order to
+construct a Boltzmann process. The second author considers herself blessed for
+having had the opportunity to write her Doctoral Thesis under the supervision
+of Errico Presutti.
+References
+[1] S. Albeverio, B. R¨udiger, P. Sundar, Boltzmann processes and their construc-
+tion. In preparation (2023).
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+167(1), 90-122 (2017).
+[3] Boltzmann, L.: Vorlesungen ¨uber Gastheorie. (1896) J. A. Barth, Leipzig,
+Part I; Part II. (1898) transl. by S. B. Brush, Lectures on Gas Theory. Univ.
+Calif. Press, Berkeley (1964).
+[4] Bressan, A. : Notes on the Boltzmann equation. Lecture Notes for a Summer
+Course given at S.I.S.S. A. 2005.
+[5] Cercignani, C.: Theory and application of the Boltzmann Equation and its
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+[6] Cercignani C.: The Boltzmann Equation and its Applications. Springer Ver-
+lag, New York (1988).
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+(42 pages) (2019).
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+Amsterdam 2002.
+18
+

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+Four-body Semileptonic Charm Decays D → P1P2ℓ+νℓ Based on
+SU(3) Flavor Analysis
+Ru-Min Wang1,†,
+Yi Qiao1,
+Yi-Jie Zhang1,
+Xiao-Dong Cheng2,§,
+Yuan-Guo Xu1,♯
+1College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang, Jiangxi 330022, China
+2College of Physics and Electronic Engineering, Xinyang Normal University, Xinyang, Henan 464000, China
+†[email protected]
+§[email protected]
+♯[email protected]
+Motivated by the significant experimental progress in probing semileptonic decays D
+→
+P1P2ℓ+νℓ (ℓ = µ, e), we analyze the branching ratios of the D → P1P2ℓ+νℓ decays with the non-
+resonant, the light scalar meson resonant and the vector meson resonant contributions in this work.
+We obtain the hadronic amplitude relations between different decay modes by the SU(3) flavor
+analysis, and then predict relevant branching ratios of the D → P1P2ℓ+νℓ decays by the present ex-
+perimental data with 2σ errors. Most of our predicted branching ratios are consistent with present
+experimental data within 2σ error bars, and others are consistent with the data within 3σ error
+bars. We find that the branching ratios of the non-resonant decays D0 → π−K
+0ℓ+νℓ, π0K−ℓ+νℓ,
+D+ → π+K−ℓ+νℓ, π0K
+0ℓ+νℓ, π+π−ℓ+νℓ, π0π0ℓ+νℓ, and D+
+s
+→ K+K−ℓ+νℓ, K0K
+0ℓ+νℓ are on
+the order of O(10−3 − 10−4).
+The vector meson resonant contributions are dominant in the
+D0 → π−K
+0ℓ+νℓ, π0K−ℓ+νℓ, π0π−ℓ+νℓ, D+ → π+K−ℓ+νℓ, π0K
+0ℓ+νℓ, π+π−ℓ+νℓ, and D+
+s
+→
+K+K−ℓ+νℓ, K0K
+0ℓ+νℓ, K+π−ℓ+νℓ, K0π0ℓ+νℓ decays. The non-resonant, the vector meson reso-
+nant and the scalar resonant contributions are all important in the D0 → ηπ−ℓ+νℓ decays. The
+D0 → K−K0ℓ+νℓ, η′π−ℓ+νℓ and D+ → K
+0K0ℓ+νℓ, π0π0ℓ+νℓ, ηπ0ℓ+νℓ, η′π0ℓ+νℓ decays only receive
+both the non-resonant and the scalar resonant contributions, and both contributions are important
+in their branching ratios. According to our predictions, many decay modes could be observed in
+the experiments like BESIII, LHCb and BelleII, and some decay modes might be measured in these
+experiments in near future.
+arXiv:2301.00090v1  [hep-ph]  31 Dec 2022
+
+2
+I.
+INTRODUCTION
+Semileptonic heavy meson decays dominated by tree-level exchange of W-bosons in the SM are very important
+processes in testing the stand model and in searching for the new physics beyond the stand model, for example, the
+extraction of the Cabbibo-Kobayashi-Maskawa (CKM) matrix elements. Four-body semileptonic exclusive decays
+D → P1P2ℓ+νℓ are generated by the c → s/dℓ+νℓ transitions, and they can receive contributions from the non-
+resonant, the light scalar meson resonant and the vector meson resonant contributions, etc. Therefore, these decays
+are also a good laboratory for probing the internal structure of light hadrons [1–3]. Some non-resonant D → P1P2ℓ+νℓ
+decays, the light scalar meson resonant decays D → S(S → P1P2)ℓ+νℓ and the vector meson resonant decays D →
+S(S → P1P2)ℓ+νℓ have been observed by BESIII, BABAR, CLEO and MARKIII, etc [4–11]. Present experimental
+measurements give us an opportunity to additionally test theoretical approaches.
+Experimental backgrounds of the semileptonic decays are cleaner than ones of the hadronic decays, and theoretical
+description of the semileptonic exclusive decays are relatively simple. Since leptons do not participate in the strong
+interaction, the weak and strong dynamics can be separated in these processes.
+All the strong dynamics in the
+initial and final hadrons is included in the hadronic transition form factors, which are important for testing the
+theoretical calculations of the involved strong interaction. The form factors can be calculated, for examples, by the
+chiral perturbation theory [12], the unitarized chiral perturbation theory [13, 14], the light-cone sum rules [15–17] and
+the QCD factorization [18]. Nevertheless, due to our poor understanding of hadronic interactions, the evaluations
+of the form factors are difficult and often plugged with large uncertainties. One needs to find ways to minimize the
+uncertainties to extract useful information.
+In the lack of reliable calculations, symmetries provide very important information for particle physics. SU(3)
+flavor symmetry is a symmetry in QCD for strong interaction. From the perspective of the SU(3) flavor symmetry,
+the leptonic part of the D → P1P2ℓ+νℓ decay is SU(3) flavor singlet, which makes no difference between different
+decay modes with certain lepton (e or µ). The different hadronic parts (the hadronic amplitudes or the hadronic form
+factors) of the D → P1P2ℓ+νℓ decays could be related by the SU(3) flavor symmetry without the detailed dynamics.
+Nevertheless, the size of the hadronic amplitudes or the form factors can not be determined by itself in the SU(3)
+flavor symmetry approach. However, if experimental data are enough, one may use the data to extract the hadronic
+amplitudes or the form factors, which can be viewed as predictions based on symmetry, has a smaller dependency
+on estimated form factors. Although the SU(3) flavor symmetry is only an approximate symmetry because up, down
+and strange quarks have different masses, it still provides some very useful information about the decays. The SU(3)
+flavor symmetry has been widely used to study hadron decays, for instance, b-hadron decays [19–32], c-hadron decays
+[31–46] and light hadron decays [31, 47–52].
+Although the SU(3) flavor symmetry works well in heavy hadron decays, the calculations of SU(3) flavor breaking
+effects would play a key role in the precise theoretical predictions of the observables and a precise test of the the
+unitarity of the CKM matrix. If up and down quark masses are neglected, a non-zero strange quark mass breaks
+the SU(3) flavor symmetry down to the isospin symmetry. When up and down quark mass difference is kept, isospin
+symmetry is also broken. Applications of the SU(3) flavor breaking approach on hadron decays can be found in Refs.
+
+3
+[53–60]. The SU(3) flavor breaking effects due to the fact of ms ≫ mu,d will be considered in our analysis of the
+non-resonant D → P1P2ℓ+νℓ decays.
+Four body semileptonic decay D → P1P2ℓ+νℓ have been studied, for instance, in Refs. [13, 61–66]. In this work, we
+will study the D → P1P2ℓ+νℓ decays with the SU(3) flavor symmetry/breaking. In three cases of the non-resonant
+decays, the light scalar meson resonant decays and the vector meson resonant decays, we will firstly construct the
+hadronic amplitude relations between different decay modes, use the available data to extract the hadronic amplitudes,
+then predict the not-yet-measured modes for further tests in experiments, and finally analyze the contributions with
+the non-resonance, the light scalar meson resonances and the vector meson resonances in the branching ratios.
+This paper is organized as follows. In Sec. II, the expressions of the branching ratios are given. In Sec. III, we will
+give our numerical results of the D → P1P2ℓ+ν decays with the non-resonant, the light scalar meson resonant and
+the vector meson resonant contributions. Our conclusions are given in Sec. IV.
+II.
+Theoretical frame
+A.
+Decay branching ratios
+The effective Hamiltonian for c → qiℓ+νℓ transition can be written as
+Heff(c → qiℓ+νℓ) = GF
+√
+2 Vcqi ¯qiγµ(1 − γ5)c ¯νℓγµ(1 − γ5)ℓ,
+(1)
+where GF is the Fermi constant, Vcqi is the CKM matrix element, and qi = d, s for i = 2, 3. The decay amplitude of
+the D(p) → P1(k1)P2(k2)ℓ+(q1)νℓ(q2) decay can be divided into leptonic and hadronic parts
+A(D → P1P2ℓ+νℓ) = ⟨P1(k1)P2(k2)ℓ+(q1)νℓ(q2)|Heff(c → qiℓ+νℓ)|D(p)⟩
+(2)
+= GF
+√
+2 VcqiLµHµ,
+(3)
+where Lµ = ¯νℓγµ(1 − γ5)ℓ is leptonic charged current, and Hµ = ⟨P1(k1)P2(k2)|¯s/ ¯dγµ(1 − γ5)c|D(p)⟩ is hadronic
+matrix element. The leptonic part Lµ is calculable using the perturbation theory, while the hadronic part Hµ are
+encoded into the transition form factors. Following Refs. [18, 67], the D → P1P2 form factors are given as
+⟨P1(k1)P2(k2)|¯s/ ¯dγµc|D(p)⟩ = iF⊥
+1
+√
+k2 qµ
+⊥,
+(4)
+−⟨P1(k1)P2(k2)|¯s/ ¯dγµγ5c|D(p)⟩ = Ft
+qµ
+�
+q2 + F0
+2
+�
+q2
+√
+λ
+kµ
+0 + F∥
+1
+√
+k2
+¯kµ
+∥ ,
+(5)
+with
+kµ
+0 = kµ − k · q
+q2 qµ,
+(6)
+¯kµ
+∥ = ¯kµ − 4(k · q)(q · ¯k)
+λ
+kµ + 4k2(q · ¯k)
+λ
+qµ,
+(7)
+qµ
+⊥ = 2ϵµαβγ qαkβ¯kγ
+√
+λ
+,
+(8)
+
+4
+where k ≡ k1+k2, q ≡ q1+q2, ¯k ≡ k1−k2, ¯q ≡ q2−q1, and λ = λ(m2
+D, q2, k2) with λ(a, b, c) = a2+b2+c2−2ab−2bc−2ac.
+In terms of the form factors, the differential branching ratio of the non-resonant D → P1P2ℓ+νℓ decays can be
+written as [18]
+dB(D → P1P2ℓ+ν)N
+dq2 dk2
+= 1
+2τD|N|2βℓ(3 − βℓ)|FA|2,
+(9)
+with
+|N|2 = G2
+F |Vcq|2 βℓq2�
+λ(m2
+D, q2, k2)
+3 · 210π5m3
+D
+with
+βℓ = 1 − m2
+ℓ
+q2 ,
+|FA|2 = |F0|2 + 2
+3(|F∥|2 + |F⊥|2) +
+3m2
+ℓ
+q2(3 − βℓ)|Ft|2,
+(10)
+where τM(mM) is lifetime(mass) of M particle. In this work, we ignore the small contributions of |Ft|2 term, which
+is proportional to m2
+ℓ. The corresponding limits of integration are given by (mP1 + mP2)2 ≤ k2 ≤ (mDq − mℓ)2
+and m2
+ℓ ≤ q2 ≤ (mDq −
+√
+k2)2. The calculations of the form factors F0, F∥, F⊥ and Ft are quite complicated, and
+their specific expressions in the QCD factorization limit can be found in Ref. [18]. Nevertheless, we will not use
+the specific expressions in this work, and we will relate the different hadronic decay amplitudes or the different form
+factors between different decay modes by the SU(3) flavor symmetry/breaking, which are discussed in later Sec. II C.
+Except for the non-resonant D → P1P2ℓ+νℓ decays, the resonant D → R(R → P1P2)ℓ+νℓ decays with the
+scalar(R = S) resonance and the vector(R = V ) resonance are also studied in this work. In the case of the de-
+cay widths of the resonances are very narrow, the resonant decay branching ratios respect a simple factorization
+relation
+B(D → Rℓ+νℓ, R → P1P2) = B(D → Rℓ+νℓ) × B(R → P1P2),
+(11)
+and this result is also a good approximation for wider resonances. Above Eq. (11) will be used in our analysis for the
+scalar resonant D → S(S → P1P2)ℓ+νℓ decays and the vector resonant D → V (V → P1P2)ℓ+νℓ decays in Sec. III B
+and III C, respectively. Relevant B(D → Rℓ+νℓ) and B(R → P1P2) are also obtained by the SU(3) flavor symmetry
+in our later analysis.
+B.
+Meson multiplets
+Before giving the hadronic amplitudes based on the SU(3) flavor analysis, we will collect the representations for the
+multiplets of the SU(3) flavor group first in this subsection.
+Charmed mesons containing one heavy c quark are flavor SU(3) anti-triplets
+Di =
+�
+D0(c¯u), D+(c ¯d), D+
+s (c¯s)
+�
+.
+(12)
+Light pseudoscalar meson (P) and vector meson (V ) octets and singlets under the SU(3) flavor symmetry of light
+u, d, s quarks are [68]
+P =
+�
+�
+�
+�
+�
+π0
+√
+2 + η8
+√
+6 + η1
+√
+3
+π+
+K+
+π−
+− π0
+√
+2 + η8
+√
+6 + η1
+√
+3
+K0
+K−
+K
+0
+− 2η8
+√
+6 + η1
+√
+3
+�
+�
+�
+�
+� ,
+(13)
+
+5
+V
+=
+�
+�
+�
+�
+�
+ρ0
+√
+2 +
+ω
+√
+2
+ρ+
+K∗+
+ρ−
+− ρ0
+√
+2 +
+ω
+√
+2
+K∗0
+K∗−
+K
+∗0
+φ
+�
+�
+�
+�
+� ,
+(14)
+where the η and η′ are mixtures of η1 = u¯u+d ¯d+s¯s
+√
+3
+and η8 = u¯u+d ¯d−2s¯s
+√
+6
+with the mixing angle θP
+�
+� η
+η′
+�
+� =
+�
+� cosθP −sinθP
+sinθP
+cosθP
+�
+�
+�
+� η8
+η1
+�
+� .
+(15)
+And θP = [−20◦, −10◦] from Particle Data Group (PDG) [11] will be used in our numerical analysis.
+The structures of the light scalar mesons are not fully understood yet. Many suggestions are discussed, such as
+ordinary two quark state, four quark state, meson-meson bound state, molecular state, glueball state or hybrid state,
+for examples, in Refs. [69–77]. In this work, we will consider the two quark and the four quark scenarios for the scalar
+mesons below or near 1 GeV . In the two quark picture, the light scalar mesons can be written as [78]
+S =
+�
+�
+�
+�
+�
+a0
+0
+√
+2 +
+σ
+√
+2
+a+
+0
+K+
+0
+a−
+0
+− a0
+0
+√
+2 +
+σ
+√
+2
+K0
+0
+K−
+0
+K
+0
+0
+f0
+�
+�
+�
+�
+� .
+(16)
+The two isoscalars f0(980) and f0(500) are obtained by the mixing of σ = u¯u+d ¯d
+√
+2
+and f0 = s¯s,
+�
+� f0(980)
+f0(500)
+�
+� =
+�
+� cosθS
+sinθS
+−sinθS cosθS
+�
+�
+�
+� f0
+σ
+�
+� ,
+(17)
+where the three possible ranges of the mixing angle θS [69, 79], 25◦ < θS < 40◦, 140◦ < θS < 165◦ and −30◦ < θS <
+30◦ will be analyzed in our numerical results. In the four quark picture, the light scalar mesons are given as [11, 80]
+σ = u¯ud ¯d,
+f0 = (u¯u + d ¯d)s¯s/
+√
+2,
+a0
+0 = (u¯u − d ¯d)s¯s/
+√
+2,
+a+
+0 = u ¯ds¯s,
+a−
+0 = d¯us¯s,
+K+
+0 = u¯sd ¯d,
+K0
+0 = d¯su¯u,
+K
+0
+0 = s ¯du¯u,
+K+
+0 = s¯ud ¯d,
+(18)
+and the two isoscalars are expressed as
+�
+� f0(980)
+f0(500)
+�
+� =
+�
+� cosφS
+sinφS
+−sinφS cosφS
+�
+�
+�
+� f0
+σ
+�
+� ,
+(19)
+where the constrained mixing angle φS = (174.6+3.4
+−3.2)◦ [70].
+C.
+Non-resonant hadronic amplitudes
+Since the hadronic amplitudes of the semileptonic D → V/Sℓ+νℓ decays based on the SU(3) flavor symme-
+try/breaking have been discussed in previous Ref. [81], we will focus on the hadronic amplitudes of the non-resonant
+D → P1P2ℓ+νℓ decays in this subsection.
+
+6
+In terms of the SU(3) flavor symmetry, the quark current ¯qiγµ(1−γ5)c can be expressed as a SU(3) flavor anti-triplet
+(¯3), and the effective Hamiltonian in Eq. (1) is transformed as [41]
+Heff(c → qiℓ+νℓ) = GF
+√
+2 H(¯3) ¯νℓγµ(1 − γ5)ℓ,
+(20)
+with H(¯3) = (0, Vcd, Vcs). The decay amplitude of the non-resonant D → P1P2ℓ+νℓ decay can be written as
+A(D → P1P2ℓ+νℓ)N = GF
+√
+2 H(D → P1P2)N ¯νℓγµ(1 − γ5)ℓ,
+(21)
+and the hadronic amplitude H(D → P1P2)N can be parameterized as
+H(D → P1P2)N = c10DiP i
+jP j
+kH(¯3)k + c20DiP i
+jH(¯3)jP k
+k + c30DiH(¯3)iP j
+kP k
+j + c40DiH(¯3)iP k
+k P j
+j ,
+(22)
+where ci0(i = 1, 2, 3, 4) are the nonperturbative coefficients under the SU(3) flavor symmetry. Feynman diagrams for
+the non-resonant D → P1P2ℓ+νℓ decays are displayed in Fig. 1.
+SU(3) flavor breaking effects come from different masses of u, d and s quarks, and they will become useful once we
+have measurements of several D → P1P2ℓ+νℓ decays that are precise enough to see deviations from the SU(3) flavor
+νℓ
+ℓ+
+c
+H(3)k
+qk
+¯qj
+qj
+¯qi
+¯qi
+( a )
+νℓ
+ℓ+
+c
+H(3)j
+qj
+¯qi
+¯qi
+¯qk
+qk
+( b )
+νℓ
+ℓ+
+c
+H(3)i
+¯qi
+¯qk
+qk
+qj
+¯qj
+( c )
+νℓ
+ℓ+
+c
+H(3)i
+¯qi
+¯qk
+qk
+¯qj
+qj
+( d )
+FIG. 1: Diagrams of the non-resonant D → P1P2ℓ+νℓ decays.
+
+7
+symmetry. The diagonalized mass matrix can be expressed as [59, 60]
+�
+�
+�
+�
+�
+mu
+0
+0
+0
+md
+0
+0
+0
+ms
+�
+�
+�
+�
+� = 1
+3(mu + md + ms)I + 1
+2(mu − md)X + 1
+6(mu + md − 2ms)W,
+(23)
+with
+X =
+�
+�
+�
+�
+�
+1
+0
+0
+0 −1 0
+0
+0
+0
+�
+�
+�
+�
+� ,
+W =
+�
+�
+�
+�
+�
+1 0
+0
+0 1
+0
+0 0 −2
+�
+�
+�
+�
+� .
+(24)
+Compared with s quark mass, the u and d quark masses are much smaller which can be ignored. The SU(3) flavor
+breaking effects due to a non-zero s quark mass dominate the SU(3) breaking effects. When u and d quark mass
+difference is ignored, the residual SU(3) flavor symmetry becomes the isospin symmetry and the term proportional to
+X can be dropped. The identity I part contributes to the D → P1P2ℓ+νℓ decay amplitudes in a similar way as that
+given in Eq. (21) which can be absorbed into the coefficients ci0. Only W part will contribute to the SU(3) breaking
+effects. The SU(3) breaking contributions to the hadronic amplitudes due to the fact of ms ≫ mu,d are
+∆H(D → P1P2)N = c11DiW i
+aP a
+j P j
+kH(¯3)k + c12DiP i
+jW j
+aP a
+k H(¯3)k + c13DiP i
+jP j
+kW k
+a H(¯3)a
++ c21DiW i
+aP a
+j H(¯3)jP k
+k + c22DiP i
+jW j
+aH(¯3)aP k
+k
++ c31DiW i
+aH(¯3)aP j
+kP k
+j + c32DiH(¯3)iP j
+kW k
+a P a
+j
++ c41DiW i
+aH(¯3)aP k
+k P j
+j ,
+(25)
+where cij (i, j = 1, 2, 3, 4) are the nonperturbative SU(3) flavor breaking coefficients.
+Full hadronic amplitudes of the different non-resonant D → P1P2ℓ+ν decays and their relations under the SU(3)
+flavor symmetry/breaking are given in later Sec. III A.
+III.
+Numerical results of the D → P1P2ℓ+ν decays
+The branching ratios with the non-resonant contributions, the light scalar meson resonant contributions and the
+vector meson resonant contributions will be analyzed in this section. If not special specified, the theoretical input
+parameters, such as the lifetimes and the masses, and the experimental data within the 2σ error bars from PDG [11]
+will be used in our numerical analysis.
+A.
+Non-resonant D → P1P2ℓ+ν decays
+The hadronic amplitudes of the non-resonant D → P1P2ℓ+νℓ decays including both the SU(3) flavor symmetry and
+the SU(3) flavor breaking terms are summarized in the second column of Tab. I, in which we can see the relations
+of different hadronic amplitudes. The following relations are hold in both the SU(3) flavor symmetry and the SU(3)
+
+8
+flavor breaking due to a strange quark mass.
+H(D0 → π−K
+0ℓ+νℓ)N = H(D+ → π+K−ℓ+νℓ)N =
+√
+2H(D0 → π0K−ℓ+νℓ)N = −
+√
+2H(D+ → π0K
+0ℓ+νℓ)N,
+H(D0 → η8K−ℓ+νℓ)N = H(D+ → η8K
+0ℓ+νℓ)N,
+H(D0 → η1K−ℓ+νℓ)N = H(D+ → η1K
+0ℓ+νℓ)N,
+H(D+
+s → K+K−ℓ+νℓ)N = H(D+
+s → K0K
+0ℓ+νℓ)N,
+H(D0 → K−K0ℓ+νℓ)N = H(D+ → K
+0K0ℓ+νℓ)N − H(D+ → K+K−ℓ+νℓ)N,
+H(D+
+s → K+π−ℓ+νℓ)N = −
+√
+2H(D+
+s → K0π0ℓ+νℓ)N.
+(26)
+If assuming the SU(3) flavor breaking effects are small and can be ignored, more amplitude relations will be obtained.
+Moreover, as shown in Fig. 1, the SU(3) flavor symmetry contributions of Fig. 1 (b-d) are suppressed by the Okubo-
+Zweig-Iizuka (OZI) rule [82–84]. If ignoring both the OZI suppressed SU(3) flavor symmetry contributions and the
+SU(3) flavor breaking contributions, almost all hadronic amplitudes of the non-resonant D → P1P2ℓ+νℓ decays can
+be related by the coefficient c10.
+Since the leptonic charged current ¯νℓγµ(1 − γ5)ℓ is the SU(3) flavor singlet, and it is completely generic between
+different decay modes with certain ℓ = e or µ. The same relations as the hadronic amplitudes listed in Tab. I are
+valid in the decay amplitudes of the D → P1P2ℓ+νℓ decays and the form factors of the D → P1P2 transitions. For the
+non-resonant D → P1P2ℓ+νℓ decays, only B(D+ → π+K−µ+νµ)N has been measured, and B(D+ → π+K−e+νe)N
+has been upper limited. Because the non-resonant D → P1P2ℓ+νℓ decays have not been measured enough to reveal
+the OZI suppressed SU(3) flavor symmetry contributions and the SU(3) symmetry breaking effects, we ignore both
+of them in our analysis, and then almost all hadronic amplitudes, form factors or decay amplitudes can be related
+by the SU(3) flavor symmetry coefficient c10. The simple relations associated by the coefficient c10 for FA given in
+Eq. (10) will be used to obtain our numerical results. Noted that, for consistency, only the SU(3) flavor symmetry
+contributions will be considered in the light scalar meson resonant D → S(S → P1P2)ℓ+νℓ decays and the vector
+meson resonant D → V (V → P1P2)ℓ+νℓ decays in later Sec. III B and Sec. III C, respectively.
+The experimental data of B(D+ → π+K−µ+νµ)N within 2σ errors and the upper limit of B(D+ → ��+K−e+νe)N at
+90% confidence level from PDG [11] are listed in the second column of Tab. II, which will be used to determine c10 in
+the non-resonant D+ → π+K−ℓ+νℓ decays, and then many other branching ratios of the non-resonant D → P1P2ℓ+νℓ
+decays can be predicted by using the constrained c10 from the data of B(D+ → π+K−ℓ+νℓ)N listed in the second
+column of Tab. II. Our predictions are listed in the third column of Tab. II for the c → sℓ+νℓ transitions and in the
+second column of Tab. III for the c → dℓ+νℓ transitions.
+From Tabs.
+II-III, one can see that many branching ratios of the non-resonant D → P1P2ℓ+νℓ decays, such
+as B(D0 → π−K
+0ℓ+νℓ)N, B(D0 → π0K−ℓ+νℓ)N, B(D+ → π+K−ℓ+νℓ)N, B(D+ → π0K
+0ℓ+νℓ)N, B(D+
+s
+→
+K+K−ℓ+νℓ)N, B(D+
+s
+→ K0K
+0ℓ+νℓ)N, B(D+ → π+π−ℓ+νℓ)N and B(D+ → π0π0ℓ+νℓ)N, are on the orders of
+O(10−3 −10−4), which could be measured by the BESIII, LHCb and BelleII experiments. Nevertheless, other decays,
+for examples, the non-resonant D → ηPℓ+νℓ decays, are strongly suppressed by the narrow phase spaces, the mixing
+angle θP or the CKM matrix element Vcd, their branching ratios are on the orders of O(10−5 − 10−7), and many of
+them might be observed by the BESIII and BelleII experiments in the near future.
+
+9
+TABLE I: The hadronic amplitudes for the D → P1P2ℓ+νℓ decays.
+C1 ≡ c10 + c11 + c12 − 2c13, C2 ≡ c20 + c21 − 2c22,
+C3 ≡ c30 − 2c31, C4 ≡ c40 − 2c41, and [C
+′,′′]R denotes the contributions come from the decays with R resonances.
+Decay modes
+Non-resonant hadronic amplitudes
+Scalar resonant ones
+Vector resonant ones
+c → sℓ+νℓ:
+D0 → π−K
+0ℓ+νℓ
+C1
+�
+C′
+1
+�
+K−
+0
+�
+C′′
+1
+�
+K∗−
+D0 → π0K−ℓ+νℓ
+1
+√
+2 C1
+� 1
+√
+2 C′
+1
+�
+K−
+0
+� 1
+√
+2 C′′
+1
+�
+K∗−
+D0 → η8K−ℓ+νℓ
+− 1
+√
+6 C1 +
+√
+6c12
+· · ·
+· · ·
+D0 → η1K−ℓ+νℓ
+2
+√
+3
+�
+C1 + 3
+2 C2
+�
+−
+√
+3c12
+· · ·
+· · ·
+D+ → π+K−ℓ+νℓ
+C1
+�
+C′
+1
+�
+K0
+0
+�
+C′′
+1
+�
+K∗0
+D+ → π0K
+0ℓ+νℓ
+− 1
+√
+2 C1
+� 1
+√
+2 C′
+1
+�
+K0
+0
+� 1
+√
+2 C′′
+1
+�
+K∗0
+D+ → η8K
+0ℓ+νℓ
+− 1
+√
+6 C1 +
+√
+6c12
+· · ·
+· · ·
+D+ → η1K
+0ℓ+νℓ
+2
+√
+3
+�
+C1 + 3
+2 C2
+�
+−
+√
+3c12
+· · ·
+· · ·
+D+
+s → K+K−ℓ+νℓ
+C1 + 2C3 −3c11
+�
+cos2θS C′
+1
+�
+f0(980)
+�
+C′′
+1
+�
+φ
+D+
+s → K0K
+0ℓ+νℓ
+C1 + 2C3 −3c11
+�
+cos2θS C′
+1
+�
+f0(980)
+�
+C′′
+1
+�
+φ
+D+
+s → π0π0ℓ+νℓ
+√
+2C3 +
+√
+2c32
+�
+sinθScosθSC′
+1
+�
+f0(980)
+�
+−sinθScosθSC′
+1
+�
+f0(500)
+· · ·
+D+
+s → π+π−ℓ+νℓ
+2C3
+�√
+2sinθScosθSC′
+1
+�
+f0(980)
+�
+−
+√
+2sinθScosθSC′
+1
+�
+f0(500)
+· · ·
+D+
+s → η8η8ℓ+νℓ
+2
+√
+2
+3
+�
+C1 + 3
+2 C3
+�
+−
+√
+2�
+2c11 + 2c12 + c32
+�
+· · ·
+· · ·
+D+
+s → η1η1ℓ+νℓ
+√
+2
+3 (C1 + 3C2 + 3C3 + 9C4) −
+√
+2(c11 + c12 + 3c21)
+· · ·
+· · ·
+D+
+s → η8η1ℓ+νℓ
+− 2
+√
+2
+3
+�
+C1 + 3
+2 C2
+�
++2
+√
+2(c11 + c12 + 3
+2 c21 + c32)
+· · ·
+· · ·
+c → dℓ+νℓ:
+D0 → K−K0ℓ+νℓ
+C1 −3(c12 − c13)
+�
+C′
+1
+�
+a0(980)
+· · ·
+D0 → π0π−ℓ+νℓ
+· · ·
+· · ·
+� 1
+√
+2 C′′
+1
+�
+ρ−
+D0 → η8π−ℓ+νℓ
+� 2
+3 C1 +
+√
+6c13
+�� 2
+3 C′
+1
+�
+a0(980)
+� 1
+√
+6 C′′
+1
+�
+ρ−
+D0 → η1π−ℓ+νℓ
+2
+√
+3
+�
+C1 + 3
+2 C2
+�
++
+√
+3(2c13 + 3c22)
+� 2
+√
+3 C′
+1
+�
+a0(980)
+� 1
+√
+3 C′′
+1
+�
+ρ−
+D+ → K
+0K0ℓ+νℓ
+C1 + 2C3 −3(c12 − c13 − 2c31)
+�
+1
+2 C′
+1
+�
+a0(980)0
+�
+1
+√
+2 sinθScosθSC′
+1
+�
+f0(980)
+· · ·
+D+ → K+K−ℓ+νℓ
+2C3 +6c31
+�
+− 1
+2 C′
+1
+�
+a0(980)0
+�
+1
+√
+2 sinθScosθSC′
+1
+�
+f0(980)
+· · ·
+D+ → π+π−ℓ+νℓ
+C1 + 2C3 +3c13 + 6c31
+�
+sin2θSC′
+1
+�
+f0(980)
+�
+cos2θSC′
+1
+�
+f0(500)
+� 1
+2 C′′
+1
+�
+ρ0,ω
+D+ → π0π0ℓ+νℓ
+1
+√
+2 (C1 + 2C3) + 1
+√
+2 (3c13 + 6c31 + 2c32)
+�
+1
+√
+2 sin2θSC′
+1
+�
+f0(980)
+�
+1
+√
+2 cos2θSC′
+1
+�
+f0(500)
+· · ·
+D+ → η8π0ℓ+νℓ
+− 1
+√
+3
+�
+C1 + C2
+�
+−
+√
+3�
+c13 + c22
+�
+�
+−
+1
+√
+6 C′
+1
+�
+a0(980)
+· · ·
+D+ → η1π0ℓ+νℓ
+−
+� 2
+3
+�
+C1 + C2
+�
+− 1
+√
+6
+�
+6c13 + 9c22
+�
+�
+−
+1
+√
+3 C′
+1
+�
+a0(980)
+· · ·
+D+ → η8η8ℓ+νℓ
+√
+2
+6
+�
+C1 + 6C3
+�
++ 1
+√
+2 (c13 + 6c31 − 2c32)
+· · ·
+· · ·
+D+ → η1η1ℓ+νℓ
+√
+2
+3 (C1 + 3C2 + 3C3 + 9C4) +
+√
+2(c13 + 3c22 + 3c31 + 9c41)
+· · ·
+· · ·
+D+ → η8η1ℓ+νℓ
+√
+2
+3
+�
+C1 + 3
+2 C2
+�
++
+√
+2�
+c13 + 3
+2 c22 + 2c32
+�
+· · ·
+· · ·
+D+
+s → K+π−ℓ+νℓ
+C1 −3c11 + 3c13
+�
+C′
+1
+�
+K0
+0
+�
+C′′
+1
+�
+K∗0
+D+
+s → K0π0ℓ+νℓ
+− 1
+√
+2 C1 − 1
+√
+2 (−3c11 + 3c13)
+�
+−
+1
+√
+2 C′
+1
+�
+K0
+0
+� 1
+√
+2 C′′
+1
+�
+K∗0
+D+
+s → η8K0ℓ+νℓ
+− 1
+√
+6 C1 + 1
+√
+6
+�
+3c11 + 6c12 − 3c13
+�
+· · ·
+· · ·
+D+
+s → η1K0ℓ+νℓ
+2
+√
+3
+�
+C1 + 3
+2 C2
+�
+−
+√
+3�
+2c11 + c12 − 2c13 + 3c21 − 3c22
+�
+· · ·
+· · ·
+
+10
+TABLE II: The experimental data and the SU(3) flavor symmetry predictions of the non-resonant branching ratios and the
+total branching ratios of the D → P1P2ℓ+νℓ decays with the c → sℓ+νℓ transitions within the 2σ errors. The experimental
+data are taken from PDG [11], ‘N’ denotes the non-resonant contributions, and ‘T’ denotes the total contributions including
+the non-resonance, the light scalar meson resonances as well as the vector meson resonances. The same below.
+Branching ratios
+Exp. data with N
+Ones with N
+Exp. data with T
+Ones with T
+B(D0 → π−K
+0e+νe)(×10−2)
+· · ·
+0.076 ± 0.041
+1.44 ± 0.08
+1.57 ± 0.14
+B(D0 → π0K−e+νe)(×10−2)
+· · ·
+0.039 ± 0.021
+1.6+2.6
+−1.0
+0.80 ± 0.07
+B(D0 → ηK−e+νe)(×10−6)
+· · ·
+3.51 ± 3.51
+· · ·
+3.51 ± 3.51
+B(D0 → η′K−e+νe)(×10−6)
+· · ·
+4.03 ± 2.17
+· · ·
+4.03 ± 2.17
+B(D+ → π+K−e+νe)(×10−2)
+< 0.7
+0.20 ± 0.10
+4.02 ± 0.36
+4.06 ± 0.30
+B(D+ → π0K
+0e+νe)(×10−2)
+· · ·
+0.100 ± 0.052
+· · ·
+2.01 ± 0.15
+B(D+ → ηK
+0e+νe)(×10−5)
+· · ·
+0.89 ± 0.89
+· · ·
+0.89 ± 0.89
+B(D+ → η′K
+0e+νe)(×10−5)
+· · ·
+1.03 ± 0.55
+· · ·
+1.03 ± 0.55
+B(D+
+s → K+K−e+νe)(×10−2)
+· · ·
+0.034 ± 0.018
+· · ·
+1.27 ± 0.13
+B(D+
+s → K0K
+0e+νe)(×10−3)
+· · ·
+0.33 ± 0.18
+· · ·
+8.58 ± 0.95
+B(D+
+s → π+π−e+νe)(×10−3)
+· · ·
+· · ·
+· · ·
+1.47 ± 0.79
+B(D+
+s → π0π0e+νe)(×10−4)
+· · ·
+· · ·
+· · ·
+8.58 ± 3.50
+B(D+
+s → ηηe+νe)(×10−4)
+· · ·
+0.56 ± 0.49
+· · ·
+0.56 ± 0.49
+B(D+
+s → ηη′e+νe)(×10−6)
+· · ·
+5.38 ± 3.19
+· · ·
+5.38 ± 3.19
+B(D0 → π−K
+0µ+νµ)(×10−2)
+· · ·
+0.073 ± 0.039
+· · ·
+1.47 ± 0.13
+B(D0 → π0K−µ+νµ)(×10−2)
+· · ·
+0.038 ± 0.020
+· · ·
+0.75 ± 0.07
+B(D0 → ηK−µ+νµ)(×10−6)
+· · ·
+3.18 ± 3.18
+· · ·
+3.18 ± 3.18
+B(D0 → η′K−µ+νµ)(×10−6)
+· · ·
+2.76 ± 1.49
+· · ·
+2.76 ± 1.49
+B(D+ → π+K−µ+νµ)(×10−2)
+0.19 ± 0.10
+0.19 ± 0.10
+3.65 ± 0.68
+3.80 ± 0.27
+B(D+ → π0K
+0µ+νµ)(×10−2)
+· · ·
+0.095 ± 0.050
+· · ·
+1.89 ± 0.13
+B(D+ → ηK
+0µ+νµ)(×10−5)
+· · ·
+0.81 ± 0.81
+· · ·
+0.81 ± 0.81
+B(D+ → η′K
+0µ+νµ)(×10−5)
+· · ·
+0.71 ± 0.38
+· · ·
+0.71 ± 0.38
+B(D+
+s → K+K−µ+νµ)(×10−2)
+· · ·
+0.032 ± 0.017
+· · ·
+1.19 ± 0.12
+B(D+
+s → K0K
+0µ+νµ)(×10−3)
+· · ·
+0.30 ± 0.16
+· · ·
+8.02 ± 0.88
+B(D+
+s → π+π−µ+νµ)(×10−3)
+· · ·
+· · ·
+· · ·
+1.25 ± 0.69
+B(D+
+s → π0π0µ+νµ)(×10−4)
+· · ·
+· · ·
+· · ·
+7.34 ± 3.09
+B(D+
+s → ηηµ+νµ)(×10−4)
+· · ·
+0.51 ± 0.45
+· · ·
+0.51 ± 0.45
+B(D+
+s → ηη′µ+νµ)(×10−6)
+· · ·
+3.98 ± 2.36
+· · ·
+3.98 ± 2.36
+
+11
+TABLE III: The experimental data and the SU(3) flavor symmetry predictions of the non-resonant branching ratios and the
+total branching ratios of the D → P1P2ℓ+νℓ decays with the c → dℓ+νℓ transitions within the 2σ errors.
+Branching ratios
+Ones with N
+Exp. data with T
+Ones with T
+B(D0 → K−K0e+νe)(×10−5)
+0.83 ± 0.45
+· · ·
+1.25 ± 0.64
+B(D0 → π0π−e+νe)(×10−3)
+0
+1.45 ± 0.14
+1.85 ± 0.11
+B(D0 → ηπ−e+νe)(×10−5)
+4.34 ± 2.68
+· · ·
+16.38 ± 5.10
+B(D0 → η′π−e+νe)(×10−5)
+0.39 ± 0.26
+· · ·
+0.57 ± 0.35
+B(D+ → K
+0K0e+νe)(×10−5)
+2.11 ± 1.13
+· · ·
+3.31 ± 1.69
+B(D+ → K+K−e+νe)(×10−5)
+· · ·
+· · ·
+1.31 ± 0.63
+B(D+ → π+π−e+νe)(×10−3)
+0.26 ± 0.14
+2.45 ± 0.20
+3.08 ± 0.51
+B(D+ → π0π0e+νe)(×10−4)
+1.33 ± 0.71
+· · ·
+2.88 ± 1.75
+B(D+ → ηπ0e+νe)(×10−5)
+5.68 ± 3.50
+· · ·
+9.68 ± 4.49
+B(D+ → η′π0e+νe)(×10−6)
+5.21 ± 3.46
+· · ·
+8.28 ± 5.00
+B(D+ → ηηe+νe)(×10−6)
+3.16 ± 2.26
+· · ·
+3.16 ± 2.26
+B(D+ → ηη′e+νe)(×10−8)
+3.96 ± 2.37
+· · ·
+3.96 ± 2.37
+B(D+
+s → K+π−e+νe)(×10−3)
+0.075 ± 0.041
+· · ·
+1.66 ± 0.17
+B(D+
+s → K0π0e+νe)(×10−4)
+0.38 ± 0.21
+· · ·
+8.24 ± 0.85
+B(D+
+s → ηK0e+νe)(×10−5)
+1.70 ± 1.06
+· · ·
+1.70 ± 1.06
+B(D+
+s → η′K0e+νe)(×10−7)
+5.21 ± 3.47
+· · ·
+5.21 ± 3.47
+B(D0 → K−K0µ+νµ)(×10−5)
+0.76 ± 0.43
+· · ·
+1.11 ± 0.57
+B(D0 → π0π−µ+νµ)(×10−3)
+0
+· · ·
+1.76 ± 0.10
+B(D0 → ηπ−µ+νµ)(×10−5)
+4.13 ± 2.55
+· · ·
+15.04 ± 4.76
+B(D0 → η′π−µ+νµ)(×10−5)
+0.34 ± 0.23
+· · ·
+0.50 ± 0.31
+B(D+ → K
+0K0µ+νµ)(×10−5)
+1.93 ± 1.04
+· · ·
+2.94 ± 1.50
+B(D+ → K+K−µ+νµ)(×10−5)
+· · ·
+· · ·
+1.09 ± 0.53
+B(D+ → π+π−µ+νµ)(×10−3)
+0.25 ± 0.14
+· · ·
+2.92 ± 0.48
+B(D+ → π0π0µ+νµ)(×10−4)
+1.29 ± 0.69
+· · ·
+2.68 ± 1.65
+B(D+ → ηπ0µ+νµ)(×10−5)
+5.40 ± 3.33
+· · ·
+8.71 ± 4.16
+B(D+ → η′π0µ+νµ)(×10−6)
+4.67 ± 3.10
+· · ·
+7.23 ± 4.37
+B(D+ → ηηµ+νµ)(×10−6)
+2.83 ± 2.02
+· · ·
+2.83 ± 2.02
+B(D+ → ηη′µ+νµ)(×10−8)
+2.43 ± 1.46
+· · ·
+2.43 ± 1.46
+B(D+
+s → K+π−µ+νµ)(×10−3)
+0.072 ± 0.039
+· · ·
+1.58 ± 0.16
+B(D+
+s → K0π0µ+νµ)(×10−4)
+0.36 ± 0.20
+· · ·
+7.81 ± 0.80
+B(D+
+s → ηK0µ+νµ)(×10−5)
+1.57 ± 0.98
+· · ·
+1.57 ± 0.98
+B(D+
+s → η′K0µ+νµ)(×10−7)
+4.08 ± 2.72
+· · ·
+4.08 ± 2.72
+
+12
+B.
+D → S(S → P1P2)ℓ+νℓ decays
+We will analyze the D → P1P2ℓ+νℓ decays with the light scalar resonances in this subsection. As given in Eq.
+(11), their branching ratios can be obtained by using B(D → Sℓ+νℓ) and B(S → P1P2). The detail analysis of
+B(D → Sℓ+νℓ) by the SU(3) flavor symmetry can be found in Ref. [81].
+1.
+Branching ratios of the S → P1P2 decays
+As for the S → P1P2 decays, the partial decay widths can be written as [85]
+Γ(S → P1P2) =
+pc
+8πm2
+S
+g2
+S→P1P2,
+(27)
+where the center of mass momentum pc ≡
+�
+λ(m2
+S,m2
+P1,m2
+P2)
+2mS
+, and gS→P1P2 is the strong coupling constant. With the
+SU(3) flavor symmetry, the strong coupling constant can be parameterized as
+g2q
+S→P1P2 = g2Si
+jP k
+i P j
+k
+(28)
+for the two quark scalar states, and
+g4q
+S→P1P2 = g4Sim
+jn P j
+i P n
+m + g′
+4Sim
+jmP n
+i P j
+n
+(29)
+for the four quark scalar states, where g2, g4 and g′
+4 are the nonperturbative parameters. The strong coupling constants
+of these decays are listed in the second and third columns of Tab. IV for the two quark scalar states and the four
+quark scalar states, respectively.
+Since the width determination is very model dependent, there are not accurate values about the decay widths
+of a0(980), f0(980) and f0(500) mesons in Ref. [11]. Therefore, it is difficult to obtain accurate B(S → P1P2) in
+terms of Γ(S → P1P2)/ΓS, where ΓS is the decay width of scalar meson. We assume the light scalar mesons decay
+dominantly into pairs of pseudoscalar mesons and all other decay channels are negligible, and then one can obtain
+B(S → P1P2) without the decay width values of the light scalar mesons, for an example, B(f0(500) → π+π−) ≈
+Γ(f0(500)→π+π−)
+Γ(f0(500)→π+π−)+Γ(f0(500)→π0π0).
+In the two quark picture, the parameter g2 is canceled in the branching ratios. Therefore, B(K0 → πK, a0(980) →
+KK, f0(500) → ππ) only depend on the masses of relevant mesons, B(a0(980) → η′π, η′π) depend on the meson masses
+and the mixing angle θP , and B(f0(980) → ππ, KK) depend on the meson masses and the mixing angle θS. The
+numerical results of B(S → P1P2) in the two quark picture are listed in the second column of Tab. V. One can see that
+the branching ratios of the K0, a0(980), f0(500) decays are accurately predicted, nevertheless, B(f0(980) → ππ, KK)
+are predicted with large error due to the indeterminate mixing angle θS. The three possible ranges for the mixing
+angle θS, 25◦ < θS < 40◦, 140◦ < θS < 165◦ and −30◦ < θS < 30◦ [69, 79], have been considered, and the predictions
+of B(f0(980) → ππ, KK) are quite dependent on the mixing angle θS.
+In the third column of Tab.
+V, we also give the predictions with two quark picture of B(S → P1P2) further
+constrained from the relevant experimental data of B(D → Sℓ+νℓ, S → P1P2) listed in later Tabs. VI-VII. The
+
+13
+TABLE IV: The strong coupling constants of the S → P1P2 decays by the SU(3) flavor symmetry.
+strong couplings
+ones for two quark state
+ones for four quark state
+gK−
+0 →π0K−
+1
+√
+2 g2
+− 1
+√
+2g4
+gK−
+0 →π−K0
+g2
+g4
+gK0
+0→π+K−
+g2
+g4
+gK0
+0→π0K0
+− 1
+√
+2 g2
+1
+√
+2g4
+ga0(980)−→ηπ−
+2 g2
+� 1
+√
+6cosθP −
+1
+√
+3sinθP
+�
+2 g′
+4
+� 1
+√
+6cosθP −
+1
+√
+3sinθP
+�
+ga0(980)−→η′π−
+2 g2
+� 1
+√
+6sinθP +
+1
+√
+3cosθP
+�
+2 g′
+4
+� 1
+√
+6sinθP +
+1
+√
+3cosθP
+�
+ga0(980)−→K0K−
+g2
+g4
+ga0(980)0→ηπ0
+g2
+� 1
+√
+3cosθP −
+� 2
+3sinθP
+�
+g′
+4
+� 1
+√
+6cosθP −
+1
+√
+3sinθP
+�
+ga0(980)0→η′π0
+g2
+� 1
+√
+3sinθP +
+� 2
+3cosθP
+�
+g′
+4
+� 1
+√
+6sinθP +
+1
+√
+3cosθP
+�
+ga0(980)0→K+K−
+1
+√
+2 g2
+1
+√
+2 g4
+ga0(980)0→K0K0
+− 1
+√
+2 g2
+− 1
+√
+2 g4
+gf0(980)→π+π−
+√
+2 g2 sinθS
+√
+2 g′
+4 cosφS + g4sinφS
+gf0(980)→π0π0
+g2 sinθS
+g′
+4 cosφS −
+1
+√
+2g4sinφS
+gf0(980)→K+K−
+g2 cosθS
+1
+��
+2g4cosφS
+gf0(980)→K0K0
+g2 cosθS
+1
+√
+2g4cosφS
+gf0(500)→π+π−
+√
+2 g2 cosθS
+−
+√
+2 g′
+4 sinφS + g4cosφS
+gf0(500)→π0π0
+g2 cosθS
+−g′
+4 sinφS −
+1
+√
+2g4cosφS
+predictions of B(f0(980) → P1P2) are quite accurate when θS is further constrained from [25◦, 40◦] to [25◦, 36◦],
+from [140◦, 165◦] to [144◦, 151◦] and from |φS| ≤ 30◦ to 22◦ ≤ |φS| ≤ 30◦ by the relevant experimental data of
+B(D → Sℓ+νℓ, S → P1P2) with 2σ errors.
+Since θS in the two quark picture has been further constrained by
+B(D → Sℓ+νℓ, S → P1P2), the predictions of B(f(980) → ππ, KK) are more accurate as listed in the third column
+of Tab. V. Other B(S → P1P2) are not further constrained from the data of B(D → Sℓ+νℓ, S → P1P2), so we do not
+list them in the third column of Tab. V.
+In the four quark picture, the two nonperturbative parameters g4 and g′
+4 in the a0(980), f0(980), f0(500) decays,
+and |g′
+4/g4| = 0.61 ± 0.13 are obtained by the data Γ(a0(980) → K ¯K)/Γ(a0(980) → ηπ) = 0.177 ± 0.048 from PDG
+[11]. In this work, we treat g4 and g′
+4 as real number, then two possible cases (g′
+4/g4 > 0 and g′
+4/g4 < 0) are analyzed.
+The numerical results with the four quark picture are listed in the last column of Tab. V. As for B(f0(980) → ππ) and
+B(f0(500) → ππ), very large errors come from the mixing angles φS, and they are obviously different in the g′
+4/g4 > 0
+and g′
+4/g4 < 0 cases. In general, there is a relative strong phase between g′
+4 and g4, therefore, the common relevant
+branching ratios are between ones in the g′
+4/g4 > 0 case and ones in the g′
+4/g4 < 0 case. In addition, B(K0 → P1P2)
+are same in both the two quark and four quark pictures.
+
+14
+2.
+Branching ratios of the D → S(S → P1P2)ℓ+νℓ decays
+Then B(D → Sℓ+νℓ, S → P1P2) can be obtained in terms of B(S → P1P2) listed in Tab. V and the expressions
+of B(D → Sℓ+νℓ) given in Ref. [81]. Using the experimental data of B(D+
+s → f0(980)e+νe) = (2.3 ± 0.8) × 10−3
+[11] as well as B(D → Sℓ+νℓ, S → P1P2) listed in the second columns of Tabs. VI-VII. The numerical results of
+B(D → Sℓ+νℓ, S → P1P2) with 2σ errors for the two quark and four quark pictures are given in Tab. VI and Tab.
+VII for the c → sℓ+νℓ and c → dℓ+νℓ transitions, respectively. Our comments on the results are as follows.
+• The
+experimental
+lower
+limits
+of
+B(D0
+→
+a0(980)−e+νe,
+a0(980)−
+→
+ηπ−)
+and
+B(D+
+→
+f0(500)e+νe, f0(500) → π+π−) have not been used to constrain the predictions of B(D → Sℓ+νℓ, S → P1P2),
+since the two lower limits of the SU(3) flavor symmetry predictions are slightly lower than their experimental
+data in both the two quark and four quark pictures. For B(D0 → a0(980)−e+νe, a0(980)− → ηπ−), one can see
+that the prediction in the two quark picture agrees with experimental data within 2σ error bars, nevertheless, the
+prediction in the four quark picture is smaller, which only agrees with experimental data within 3σ error bars.
+As for B(D+ → f0(500)e+νe, f0(500) → π+π−), the prediction in the two quark picture is much smaller than
+its experimental lower limit with 2σ error, nevertheless, the prediction with g′
+4
+g4 > 0 ( g′
+4
+g4 < 0 ) in the four quark
+picture agrees with its data within 2σ (3σ) error bars. Therefore, in the later analysis of total contributions to
+B(D → P1P2ℓ+νℓ), the predictions of B(D → Sℓ+νℓ, S → P1P2) with g′
+4
+g4 > 0 in the four quark picture will be
+used.
+• In the two quark picture, though the mixing angle θS only appears in the D → P1P2ℓ+νℓ decays with f0(980)
+and f0(500) resonances, all other predictions of the branching ratios are slightly affected by the experimental
+constraints. So we list all predictions in the three possible ranges of the mixing angle θS in the 3rd-5th columns
+of Tabs.
+VI-VII. One can see the all predictions included the decays with f0(980) and f0(500) resonances
+are similar in the three possible ranges of the mixing angle θS. As mentioned before, θS is constrained from
+[25◦, 40◦] to [25◦, 36◦], from [140◦, 165◦] to [144◦, 151◦] and from |φS| ≤ 30◦ to 22◦ ≤ |φS| ≤ 30◦ by the relevant
+experimental data with 2σ errors.
+• A lot of the branching ratio predictions are quite different between the two quark picture and the four quark
+picture. Present datum of B(D+ → f0(500)e+νe, f0(500) → π+π−) favors the four quark picture of scalar
+mesons. B(D → Sℓ+νℓ, S → P1P2) with the c → sℓ+νℓ transitions are predicted on the order of O(10−3 −10−4).
+Due to the CKM matrix element Vcd suppressed, B(D → Sℓ+νℓ, S → P1P2) with the c → dℓ+νℓ transitions are
+predicted on the order of O(10−4 − 10−6).
+• Some branching ratios of the D → S(S → P1P2)ℓ+νℓ decays have been obtained in Refs. [13, 61]. B(D+ →
+Se+νe, S → π+π���) = (6.99 ± 2.46) × 10−4 [13], B(D+ → Sµ+νµ, S → π+π−) = (7.20 ± 2.52) × 10−4 [13],
+B(D0 → a0(980)−ℓ+νℓ, a0(980)− → ηπ−) = (1.36 ± 0.21) × 10−4 [61]. Our predictions in the four quark picture
+of B(D+ → Sℓ+νℓ, S → π+π−) are consistent with ones in Ref. [13], our predictions in the two quark picture
+of B(D0 → a0(980)−ℓ+νℓ, a0(980)− → ηπ−) are consistent with ones in Ref. [61], nevertheless, our predictions
+in the four quark picture are smaller than ones in Ref. [61].
+
+15
+TABLE V: Branching ratios of the S → P1P2 decays within 2σ errors. The results are obtained by the SU(3) flavor symmetry
+relations and Γ(a0(980) → K ¯K)/Γ(a0(980) → ηπ) = 0.177 ± 0.048 [11].
+†denotes the results with
+g′
+4
+g4 > 0, and ♯denotes ones
+with
+g′
+4
+g4 < 0.
+Branching ratios
+ones with 2q state in S1 case
+ones with 2q state in S2 case
+ones with 4q state
+B(K−
+0 → π0K−)
+0.34 ± 0.00
+0.34 ± 0.00
+B(K−
+0 → π−K
+0)
+0.66 ± 0.00
+0.66 ± 0.00
+B(K
+0
+0 → π+K−)
+0.67 ± 0.00
+0.67 ± 0.00
+B(K
+0
+0 → π0K
+0)
+0.33 ± 0.00
+0.33 ± 0.00
+B(a0(980)− → ηπ−)
+0.64 ± 0.04
+0.86 ± 0.03
+B(a0(980)− → η′π−)
+0.03 ± 0.01
+0.04 ± 0.01
+B(a0(980)− → K0K−)
+0.33 ± 0.03
+0.10 ± 0.02
+B(a0(980)0 → ηπ0)
+0.60 ± 0.04
+0.67 ± 0.06
+B(a0(980)0 → η′π0)
+0.04 ± 0.01
+0.05 ± 0.02
+B(a0(980)0 → K+K−)
+0.19 ± 0.02
+0.15 ± 0.03
+B(a0(980)0 → K0 ¯K0)
+0.17 ± 0.01
+0.13 ± 0.03
+0.45 ± 0.09θS=[25◦,40◦]
+0.43 ± 0.07θS=[25◦,35◦]
+0.42 ± 0.16†
+B(f0(980) → π+π−)
+0.36 ± 0.17θS=[140◦,165◦]
+0.41 ± 0.09θS=[144◦,158◦]
+0.59 ± 0.13♯
+0.22 ± 0.22θS=[−30◦,30◦]
+0.38 ± 0.06[22◦≤|θS|≤30◦]
+0.22 ± 0.04θS=[25◦,40◦]
+0.21 ± 0.03θS=[25◦,35◦]
+0.34 ± 0.11†
+B(f0(980) → π0π0)
+0.18 ± 0.09θS=[140◦,165◦]
+0.21 ± 0.04θS=[144◦,158◦]
+0.20 ± 0.10♯
+0.11 ± 0.11θS=[−30◦,30◦]
+0.19 ± 0.03[22◦≤|θS|≤30◦]
+0.17 ± 0.07θS=[25◦,40◦]
+0.19 ± 0.05θS=[25◦,35◦]
+B(f0(980) → K+K−)
+0.24 ± 0.14θS=[140◦,165◦]
+0.20 ± 0.07θS=[144◦,158◦]
+0.12 ± 0.04
+0.35 ± 0.17θS=[−30◦,30◦]
+0.22 ± 0.04[22◦≤|θS|≤30◦]
+0.16 ± 0.06θS=[25◦,40◦]
+0.17 ± 0.05θS=[25◦,35◦]
+B(f0(980) → K0 ¯K0)
+0.22 ± 0.12θS=[140◦,165◦]
+0.18 ± 0.06θS=[144◦,158◦]
+0.11 ± 0.04
+0.32 ± 0.16θS=[−30◦,30◦]
+0.20 ± 0.04[22◦≤|θS|≤30◦]
+B(f0(500) → π+π−)
+0.66 ± 0.00
+0.73 ± 0.09†
+0.57 ± 0.12♯
+B(f0(500) → π0π0)
+0.34 ± 0.00
+0.27 ± 0.09†
+0.43 ± 0.12♯
+
+16
+TABLE VI: The experimental data and the SU(3) flavor symmetry predictions of the D → S(S → P1P2)ℓ+νℓ decays with the c → sℓ+νℓ transitions within 2σ errors.
+†denotes the results with
+g′
+4
+g4 > 0, and ♯ denotes ones with
+g′
+4
+g4 < 0.
+Branching ratios
+Exp. Data
+Ones in the 2-quark picture with
+Ones in the 4-quark picture
+θS = [25◦, 35◦]
+θS = [144◦, 158◦]
+22◦ ≤ |θS| ≤ 30◦
+B(D0 → K−
+0 e+νe, K−
+0 → π−K
+0)(×10−4)
+· · ·
+19.99 ± 7.34
+19.86 ± 7.26
+19.74 ± 6.97
+8.37 ± 3.01
+B(D0 → K−
+0 e+νe, K−
+0 → π0K−)(×10−4)
+· · ·
+10.18 ± 3.77
+10.12 ± 3.73
+10.05 ± 3.57
+4.19 ± 1.50
+B(D+ → K
+0
+0e+νe, K
+0
+0 → π+K−)(×10−3)
+· · ·
+5.17 ± 1.92
+5.19 ± 1.85
+5.12 ± 1.86
+2.24 ± 0.83
+B(D+ → K
+0
+0e+νe, K
+0
+0 → π0K
+0)(×10−3)
+· · ·
+2.57 ± 0.96
+2.59 ± 0.92
+2.55 ± 0.92
+1.12 ± 0.42
+B(D+
+s → f0(980)e+νe, f0(980) → π+π−)(×10−3)
+1.30 ± 0.63 [86]
+1.19 ± 0.18
+1.17 ± 0.17
+1.18 ± 0.17
+1.22 ± 0.55†,
+1.44 ± 0.49♯
+B(D+
+s → f0(980)e+νe, f0(980) → π0π0)(×10−4)
+7.9 ± 2.9 [4]
+5.95 ± 0.92
+5.89 ± 0.85
+5.90 ± 0.86
+7.91 ± 2.85†,
+7.13 ± 2.10♯
+B(D+
+s → f0(980)e+νe, f0(980) → K+K−)(×10−4)
+· · ·
+5.11 ± 2.34
+5.53 ± 2.78
+6.28 ± 2.07
+3.33 ± 1.53†,
+3.07 ± 1.34♯
+B(D+
+s → f0(980)e+νe, f0(980) → K0K
+0)(×10−4)
+· · ·
+4.62 ± 2.12
+5.01 ± 2.52
+5.68 ± 1.87
+3.01 ± 1.39†,
+2.78 ± 1.22♯
+B(D+
+s → f0(500)e+νe, f0(500) → π+π−)(×10−4)
+· · ·
+9.91 ± 2.83
+9.67 ± 3.07
+9.44 ± 3.30
+2.49 ± 2.49†,
+0.90 ± 0.90♯
+B(D+
+s → f0(500)e+νe, f0(500) → π0π0)(×10−5)
+< 64 [4]
+49.77 ± 14.23
+48.57 ± 15.43
+47.44 ± 16.56
+6.66 ± 6.66†,
+0.78 ± 0.78♯
+B(D0 → K−
+0 µ+νµ, K−
+0 → π−K0)(×10−4)
+· · ·
+17.27 ± 6.48
+17.16 ± 6.41
+17.04 ± 6.14
+7.19 ± 2.63
+B(D0 → K−
+0 µ+νµ, K−
+0 → π0K−)(×10−4)
+· · ·
+8.63 ± 3.24
+8.58 ± 3.20
+8.52 ± 3.07
+3.59 ± 1.32
+B(D+ → K
+0
+0µ+νµ, K
+0
+0 → π+K−)(×10−3)
+· · ·
+4.43 ± 1.68
+4.46 ± 1.62
+4.40 ± 1.62
+1.92 ± 0.73
+B(D+ → K
+0
+0µ+νµ, K
+0
+0 → π0K0)(×10−3)
+· · ·
+2.22 ± 0.84
+2.23 ± 0.81
+2.20 ± 0.81
+0.96 ± 0.36
+B(D+
+s → f0(980)µ+νµ, f0(980) → π+π−)(×10−3)
+· · ·
+1.01 ± 0.16
+1.00 ± 0.15
+1.00 ± 0.16
+1.02 ± 0.46†,
+1.23 ± 0.42♯
+B(D+
+s → f0(980)µ+νµ, f0(980) → π0π0)(×10−4)
+· · ·
+5.05 ± 0.83
+4.99 ± 0.77
+5.00 ± 0.78
+6.72 ± 2.48†,
+6.04 ± 1.82♯
+B(D+
+s → f0(980)µ+νµ, f0(980) → K+K−)(×10−4)
+· · ·
+4.31 ± 1.94
+4.70 ± 2.34
+5.34 ± 1.75
+2.79 ± 1.28†,
+2.59 ± 1.14♯
+B(D+
+s → f0(980)µ+νµ, f0(980) → K0K
+0)(×10−4)
+· · ·
+3.90 ± 1.76
+4.25 ± 2.12
+4.83 ± 1.58
+2.52 ± 1.16†,
+2.34 ± 1.03♯
+B(D+
+s → f0(500)µ+νµ, f0(500) → π+π−)(×10−4)
+· · ·
+8.88 ± 2.62
+8.70 ± 2.86
+8.49 ± 3.05
+2.30 ± 2.30†,
+0.83 ± 0.83♯
+B(D+
+s → f0(500)µ+νµ, f0(500) → π0π0)(×10−5)
+· · ·
+44.67 ± 13.23
+43.85 ± 14.53
+42.77 ± 15.49
+6.16 ± 6.16†,
+7.23 ± 7.23♯
+
+17
+TABLE VII: The experimental data and the SU(3) flavor symmetry predictions of the D → S(S → P1P2)ℓ+νℓ decays with the c → dℓ+νℓ transitions within 2σ errors.
+† denotes the results with
+g′
+4
+g4 > 0, ♯ denotes ones with
+g′
+4
+g4 < 0, and a denotes the experimental lower limits have not used to constrain the predictions.
+Branching ratios
+Exp. Data
+Ones in the 2-quark picture with
+Ones in the 4-quark picture
+θS = [25◦, 35◦]
+θS = [144◦, 158◦]
+22◦ ≤ |θS| ≤ 30◦
+B(D0 → a0(980)−e+νe, a0(980)− → ηπ−)(×10−5)
+13.3+6.8
+−6.0a
+5.99 ± 2.69
+5.86 ± 2.48
+6.05 ± 2.57
+3.81 ± 0.98
+B(D0 → a0(980)−e+νe, a0(980)− → η′π−)(×10−6)
+· · ·
+2.88 ± 1.71
+2.97 ± 1.77
+2.97 ± 1.73
+1.88 ± 0.98
+B(D0 → a0(980)−e+νe, a0(980)− → K0K−)(×10−6)
+· · ·
+29.99 ± 13.81
+30.73 ± 13.81
+30.57 ± 13.70
+4.22 ± 1.93
+B(D+ → a0(980)0e+νe, a0(980)0 → ηπ0)(×10−5)
+17+16
+−14
+7.35 ± 3.28
+7.25 ± 3.13
+7.32 ± 3.17
+4.00 ± 1.00
+B(D+ → a0(980)0e+νe, a0(980)0 → η′π0)(×10−6)
+· · ·
+5.53 ± 3.26
+5.69 ± 3.32
+5.65 ± 3.20
+3.08 ± 1.56
+B(D+ → a0(980)0e+νe, a0(980)0 → K+K−)(×10−5)
+· · ·
+2.28 ± 1.06
+2.30 ± 1.00
+2.29 ± 0.99
+0.88 ± 0.36
+B(D+ → a0(980)0e+νe, a0(980)0 → K0K
+0)(×10−5)
+· · ·
+1.99 ± 0.92
+2.01 ± 0.88
+2.00 ± 0.86
+0.77 ± 0.31
+B(D+ → f0(980)e+νe, f0(980) → π+π−)(×10−5)
+< 2.8 [5]
+1.15 ± 0.50
+1.10 ± 0.58
+0.96 ± 0.43
+1.65 ± 1.15†,
+2.14 ± 0.65♯
+B(D+ → f0(980)e+νe, f0(980) → π0π0)(×10−6)
+· · ·
+5.75 ± 2.53
+5.51 ± 2.92
+4.80 ± 2.18
+10.53 ± 3.67†,
+10.10 ± 5.37♯
+B(D+ → f0(980)e+νe, f0(980) → K+K−)(×10−6)
+· · ·
+5.07 ± 0.88
+5.06 ± 0.85
+5.01 ± 0.80
+4.35 ± 2.78†,
+4.60 ± 2.76♯
+B(D+ → f0(980)e+νe, f0(980) → K0K
+0)(×10−6)
+· · ·
+5.07 ± 0.88
+5.06 ± 0.85
+5.01 ± 0.80
+4.35 ± 2.78†,
+4.60 ± 2.76♯
+B(D+ → f0(500)e+νe, f0(500) → π+π−)(×10−4)
+6.3 ± 1.0a
+1.44 ± 0.64
+1.72 ± 0.92
+1.79 ± 0.85
+3.64 ± 2.57†,
+2.95 ± 1.87♯
+B(D+ → f0(500)e+νe, f0(500) → π0π0)(×10−4)
+· · ·
+0.72 ± 0.32
+0.87 ± 0.46
+0.91 ± 0.43
+1.45 ± 1.02†,
+2.08 ± 1.57♯
+B(D+
+s → K0
+0e+νe, K0
+0 → π−K+)(×10−5)
+· · ·
+22.34 ± 8.09
+22.13 ± 7.97
+22.34 ± 7.64
+9.54 ± 3.38
+B(D+
+s → K0
+0e+νe, K0
+0 → π0K0)(×10−5)
+· · ·
+11.17 ± 4.04
+11.07 ± 3.99
+11.17 ± 3.82
+4.77 ± 1.69
+B(D0 → a0(980)−µ+νµ, a0(980)− → ηπ−)(×10−5)
+· · ·
+4.95 ± 2.27
+4.84 ± 2.10
+5.00 ± 2.18
+3.14 ± 0.84
+B(D0 → a0(980)−µ+νµ, a0(980)− → η′π−)(×10−6)
+· · ·
+2.39 ± 1.44
+2.46 ± 1.48
+2.45 ± 1.45
+1.56 ± 0.82
+B(D0 → a0(980)−µ+νµ, a0(980)− → K0K−)(×10−6)
+· · ·
+24.78 ± 11.68
+25.37 ± 11.62
+25.20 ± 11.53
+3.51 ± 1.62
+B(D+ → a0(980)0µ+νµ, a0(980)0 → ηπ0)(×10−5)
+· · ·
+6.09 ± 2.78
+6.00 ± 2.65
+6.06 ± 2.69
+3.30 ± 0.86
+B(D+ → a0(980)0µ+νµ, a0(980)0 → η′π0)(×10−6)
+· · ·
+4.58 ± 2.74
+4.72 ± 2.79
+4.67 ± 2.69
+2.55 ± 1.31
+B(D+ → a0(980)0µ+νµ, a0(980)0 → K+K−)(×10−5)
+· · ·
+1.89 ± 0.89
+1.91 ± 0.85
+1.89 ± 0.83
+0.73 ± 0.30
+B(D+ → a0(980)0µ+νµ, a0(980)0 → K0K
+0)(×10−5)
+· · ·
+1.65 ± 0.78
+1.66 ± 0.74
+1.65 ± 0.73
+0.64 ± 0.27
+B(D+ → f0(980)µ+νµ, f0(980) → π+π−)(×10−5)
+· · ·
+0.94 ± 0.43
+0.91 ± 0.48
+0.79 ± 0.36
+1.37 ± 0.96†,
+1.76 ± 0.55♯
+B(D+ → f0(980)µ+νµ, f0(980) → π0π0)(×10−6)
+· · ·
+4.74 ± 2.14
+4.58 ± 2.43
+3.97 ± 1.82
+8.67 ± 3.13†,
+8.32 ± 4.47♯
+B(D+ → f0(980)µ+νµ, f0(980) → K+K−)(×10−6)
+· · ·
+4.21 ± 0.73
+4.19 ± 0.71
+4.15 ± 0.67
+3.55 ± 2.29†,
+3.76 ± 2.26♯
+B(D+ → f0(980)µ+νµ, f0(980) → K0K
+0)(×10−6)
+· · ·
+4.21 ± 0.73
+4.19 ± 0.71
+4.15 ± 0.67
+3.55 ± 2.29†,
+3.76 ± 2.26♯
+B(D+ → f0(500)µ+νµ, f0(980) → π+π−)(×10−4)
+· · ·
+1.28 ± 0.59
+1.54 ± 0.84
+1.61 ± 0.79
+3.30 ± 2.39†,
+2.68 ± 1.74♯
+B(D+ → f0(500)µ+νµ, f0(980) → π0π0)(×10−4)
+· · ·
+0.64 ± 0.30
+0.78 ± 0.43
+0.81 ± 0.40
+1.32 ± 0.95†,
+1.89 ± 1.46♯
+B(D+
+s → K0
+0µ+νµ, K0
+0 → π−K+)(×10−5)
+· · ·
+19.61 ± 7.20
+19.43 ± 7.10
+19.60 ± 6.80
+8.38 ± 3.01
+B(D+
+s → K0
+0µ+νµ, K0
+0 → π0K0)(×10−5)
+· · ·
+9.80 ± 3.60
+9.71 ± 3.55
+9.80 ± 3.40
+4.19 ± 1.50
+
+18
+C.
+D → V (V → P1P2)ℓ+νℓ decays
+We will analyze the D → P1P2ℓ+νℓ decays with the vector resonances in this subsection. Since the light vector
+mesons are understood well, the calculations of B(D → V ℓ+νℓ, V → P1P2) are much easier than ones of B(D →
+Sℓ+νℓ, S → P1P2). From Eq. (11), their branching ratios of D → V (V → P1P2)ℓ+νℓ can be obtained by using
+B(D → V ℓ+νℓ) and B(V → P1P2). The D → V ℓ+νℓ decays have been studied by the SU(3) flavor symmetry in Ref.
+[81]. Many B(D → V ℓ+νℓ) have been accurately measured and have been listed in the second column of Tab. V in
+Ref. [81]. The expressions of B(D → V ℓ+νℓ) within the C3 case in Ref. [81] will be taken for our analysis.
+Following Ref. [85], B(V → P1P2) can be written as
+B(V → P1P2) = τV p′3
+c
+6πm2
+V
+g2
+V →P1P2,
+(30)
+where p′
+c ≡
+�
+λ(m2
+V ,m2
+P1,m2
+P2)
+2mV
+and gV →P1P2 are the strong coupling constants. Similar to g2q
+S→P1P2 in Eq. (28), gV →P1P2
+can be parameterized by the SU(3) flavor symmetry
+gV →P1P2 = gV V i
+j P k
+i P j
+k,
+(31)
+where gV is the corresponding nonperturbative parameter.
+At present, many involved B(V → P1P2) have been well measured [11]
+B(K∗+ → πK) = (99.902 ± 0.018)%,
+B(K∗0 → πK) = (99.754 ± 0.042)%,
+B(ρ+ → π0π+) = 100%,
+B(ρ0 → π+π−) = 100%,
+B(φ → K+K−) = (49.1 ± 1.0)%,
+B(ω → π+π−) = (1.53+0.22
+−0.26)%.
+(32)
+Using the following relations from Eq. (31)
+√
+2gK∗−→π0K− = gK∗−→π−K0,
+√
+2gK∗0→π0K0 = gK∗0→π−K+,
+gρ−→π0π− =
+√
+3gρ−→η8π− =
+�
+3/2gρ−→η1π−,
+gφ→K+K− = gφ→K0K
+0,
+(33)
+following B(V → P1P2) can be obtained
+B(K∗0 → π0K0) = (33.02 ± 0.02)%,
+B(K∗0 → π−K+) = (66.74 ± 0.04)%,
+B(K∗+ → π0K+) = (33.62 ± 0.01)%,
+B(K∗+ → π−K0) = (66.28 ± 0.01)%,
+B(ρ+ → ηπ+) = (4.38 ± 0.66)%,
+B(φ → K0K0) = (32.42 ± 1.04)%.
+(34)
+For D → V (V → P1P2)ℓ+νℓ decays, the branching ratios of D+ → K
+∗0(K
+∗0 → π+K−)e+νe and D+ → K
+∗0(K
+∗0 →
+π+K−)µ+νµ have been measured, and the experimental data with 2σ errors are listed in the second column of Tab.
+VIII. Using the experimental data of B(D+ → K
+∗0ℓ+νℓ, K
+∗0 → π+K−), B(V → P1P2) and B(D → V ℓ+νℓ), we
+obtain the predictions of B(D → V ℓ+νℓ, V → P1P2) by the SU(3) flavor symmetry, which are given in the third
+column of Tab. VIII. We can see that B(D → V ℓ+νℓ, V → P1P2) with the c → sℓ+νℓ transitions are predicted on
+the order of O(10−2 − 10−3), and B(D → V ℓ+νℓ, V → P1P2) with the c → dℓ+νℓ transitions are predicted on the
+
+19
+TABLE VIII: The experimental data and the SU(3) flavor symmetry predictions of D → V (V → P1P2)ℓ+νℓ decays within 2σ
+errors.
+Branching ratios
+Exp. Data
+Our predictions
+Previous ones
+c → se+νe:
+B(D0 → K∗−e+νe, K∗− → π−K
+0)(×10−2)
+. . .
+1.42 ± 0.07
+. . .
+B(D0 → K∗−e+νe, K∗− → π0K−)(×10−3)
+. . .
+7.18 ± 0.37
+7.17 [62]
+B(D+ → K
+∗0e+νe, K
+∗0 → π+K−)(×10−2)
+3.77 ± 0.34
+3.64 ± 0.11
+3.51 [62]
+B(D+ → K
+∗0e+νe, K
+∗0 → π0K
+0)(×10−2)
+. . .
+1.80 ± 0.06
+. . .
+B(D+
+s → φe+νe, φ → K+K−)(×10−2)
+. . .
+1.20 ± 0.10
+. . .
+B(D+
+s → φe+νe, φ → K0K
+0)(×10−3)
+. . .
+7.94 ± 0.65
+. . .
+c → sµ+νµ:
+B(D0 → K∗−µ+νµ, K∗− → π−K
+0)(×10−2)
+. . .
+1.33 ± 0.07
+. . .
+B(D0 → K∗−µ+νµ, K∗− → π0K−)(×10−3)
+. . .
+6.76 ± 0.35
+7.17 [62]
+B(D+ → K
+∗0µ+νµ, K
+∗0 → π+K−)(×10−2)
+3.52 ± 0.20
+3.43 ± 0.11
+3.51 [62]
+B(D+ → K
+∗0µ+νµ, K
+∗0 → π0K
+0)(×10−2)
+. . .
+1.70 ± 0.05
+. . .
+B(D+
+s → φµ+νµ, φ → K+K−)(×10−2)
+. . .
+1.13 ± 0.09
+. . .
+B(D+
+s → φµ+νµ, φ → K0K
+0)(×10−3)
+. . .
+7.46 ± 0.62
+. . .
+c → de+νe:
+B(D0 → ρ−e+νe, ρ− → π0π−)(×10−3)
+. . .
+1.85 ± 0.11
+1.63 [62]
+B(D0 → ρ−e+νe, ρ− → ηπ−)(×10−5)
+. . .
+8.23 ± 1.59
+. . .
+B(D+ → ρ0e+νe, ρ0 → π+π−)(×10−3)
+. . .
+2.40 ± 0.12
+1.57 ± 0.07 [13],
+2.10 [62]
+B(D+ → ωe+νe, ω → π+π−)(×10−5)
+. . .
+3.55 ± 0.82
+. . .
+B(D+
+s → K∗0e+νe, K∗0 → π−K+)(×10−3)
+. . .
+1.49 ± 0.10
+. . .
+B(D+
+s → K∗0e+νe, K∗0 → π0K0)(×10−4)
+. . .
+7.39 ± 0.51
+. . .
+c → dµ+νµ:
+B(D0 → ρ−µ+νµ, ρ− → π0π−)(×10−3)
+. . .
+1.76 ± 0.10
+. . .
+B(D0 → ρ−µ+νµ, ρ− → ηπ−)(×10−5)
+. . .
+7.83 ± 1.51
+. . .
+B(D+ → ρ0µ+νµ, ρ0 → π+π−)(×10−3)
+. . .
+2.29 ± 0.11
+1.57 ± 0.07 [13]
+B(D+ → ωµ+νµ, ω → π+π−)(×10−5)
+. . .
+3.38 ± 0.78
+. . .
+B(D+
+s → K∗0µ+νµ, K∗0 → π−K+)(×10−3)
+. . .
+1.42 ± 0.10
+. . .
+B(D+
+s → K∗0µ+νµ, K∗0 → π0K0)(×10−4)
+. . .
+7.03 ± 0.48
+. . .
+
+20
+order of O(10−3 − 10−5). The predictions of B(D → V ℓ+νℓ, V → P1P2) are about one order larger than ones of the
+corresponding B(D → Sℓ+νℓ, S → P1P2).
+Previous predictions are also listed in the last column of Tab. VIII. Our predictions of B(D0 → K∗−ℓ+νℓ, K∗− →
+π0K−) and B(D+ → K
+∗0ℓ+νℓ, K
+∗0 → π+K−) are in good agreement with ones in Ref. [62]. And our predictions of
+B(D+ → ρ0ℓ+νℓ, ρ0 → π+π−) are slight larger than ones obtained by the light-front quark model and the light-cone
+sum rules in Ref. [13].
+D.
+Total branching ratios
+As analyzed in above, some four-body semileptonic decays of D mesons receive the contributions of the non-resonant
+states, the scalar resonant states and the vector resonant states, nevertheless, some decay modes only receive one or two
+kinds of them. For clearly showing the resonant contributions, we also list the scalar and vector resonant amplitudes in
+the third and last columns of Tab. I, respectively. The resonant amplitudes are obtained by multiplying the hadronic
+helicity amplitudes H(D → Rℓ+νℓ) given in Ref. [81] and the strong coupling constants gR→P1P2 obtained in this
+work. Noted that the resonant amplitudes listed in the last two columns of Tab. I only for clearly see the kinds of the
+resonant contributions, and we do not using them to obtain the numerical total branching ratios B(D → P1P2ℓ+νℓ)T .
+We have some comments for the contributions in Tab. I. For D(s) → ηKℓ+νℓ, η′Kℓ+νℓ, ηηℓ+νℓ, ηη′ℓ+νℓ decays,
+since the both final state mesons are quite heavy, they only receive the non-resonant contributions.
+The decays
+D+
+s → π0π0ℓ+νℓ, D+
+s → π+π−ℓ+νℓ, D0 → K−K0ℓ+νℓ, D+ → K
+0K0ℓ+νℓ, D+ → K+K−ℓ+νℓ, D+ → π0π0ℓ+νℓ and
+D+ → η(′)π0ℓ+νℓ receive both the non-resonant contributions and the scalar resonant contributions, moreover, the
+non-resonant contributions in the D+
+s → π0π0ℓ+νℓ, D+
+s → π+π−ℓ+νℓ and D+ → K+K−ℓ+νℓ decays are suppressed
+by the OZI rule, and the main contributions of these decay branching ratios come from the scalar resonant states. All
+other decay modes except the D0 → π0π−ℓ+νℓ decays receive all three kinds of the contributions, and their branching
+ratios are dominant by the vector resonant states. Due to the quantum number constraint, the D0 → π0π−ℓ+νℓ
+decays only receive the contributions of the vector resonant states.
+In the last columns of Tabs. II-III, total branching ratio predictions of the D → P1P2ℓ+ν decays including the
+possible non-resonant, scalar resonant and vector resonant contributions are listed. The present six experimental data
+with 2σ errors are also listed in the forth column of Tab. II and in third column of Tab. III for convenient comparison.
+One can see that, for B(D0 → π−K
+−e+νe), B(D0 → π0K−e+νe), B(D+ → π+K−e+νe), B(D+ → π+K−µ+νµ) and
+B(D+ → π+π−e+νe), our SU(3) flavor symmetry predictions are consistent with present data within 2σ error bars.
+Our prediction of B(D0 → π0π−e+νe) is slightly larger than its experimental datum, nevertheless, the prediction will
+be very close to the datum within 3σ error bars.
+For some Cabibbo suppressed decays due to c → dℓ+νℓ transitions, such as the D0 → K−K0ℓ+νℓ, D0 → η′π−ℓ+νℓ,
+D+ → K
+0K0ℓ+νℓ, D+ → π0π0ℓ+νℓ, D+ → ηπ0ℓ+νℓ and D+ → η′π0ℓ+νℓ decays, they only receive both the non-
+resonant contributions and the scalar resonant contributions, and we can see that both the non-resonant and the
+scalar resonant contributions are important. The non-resonant contributions in the D+ → K+K−ℓ+νℓ decays are
+suppressed by the OZI rule, and the scalar resonant contributions in the D+ → K+K−ℓ+νℓ decays are dominant.
+
+21
+IV.
+Summary
+Semileptonic decays of heavy mesons are quite interesting because of not only relatively simple theoretical description
+but also the clean experimental signals. Some semileptonic decays D → P1P2ℓ+νℓ have been measured by BESIII,
+CLEO and BABAR, etc. Using the present data of B(D → P1P2ℓ+νℓ) and the SU(3) flavor symmetry, we have
+presented a theoretical analysis of the D → P1P2ℓ+νℓ decays with the non-resonant, the light scalar meson resonant
+and the vector meson resonant contributions.
+• Non-resonant D → P1P2ℓ+νℓ decays: The amplitude relations included the SU(3) flavor breaking effects have
+been obtained. Almost all amplitudes can be related after ignoring the OZI suppressed and the SU(3) flavor
+breaking contributions. Via the experimental data of the non-resonant branching ratios B(D+ → π+K−ℓ+νℓ)N,
+we have predicted other non-resonant branching ratios. We have found that the branching ratios of the non-
+resonant decays D0 → π−K
+0ℓ+νℓ, π0K−ℓ+ν���, D+ → π+K−ℓ+νℓ, π0K
+0ℓ+νℓ, π+π−ℓ+νℓ, π0π0ℓ+νℓ, and D+
+s →
+K+K−ℓ+νℓ, K0K
+0ℓ+νℓ are on the order of O(10−3 − 10−4), which might be measured by the BESIII, LHCb
+and BelleII experiment, and some other decays might be measured at these experiments in near future.
+• Decays with the light scalar meson resonances: Using the SU(3) flavor symmetry and the present
+experimental data of B(D → Sℓ+νℓ), B(D → Sℓ+νℓ, S → P1P2) as well as B(S → P1P2), the not-
+measured B(D → Sℓ+νℓ, S → P1P2) have been obtained by the SU(3) flavor symmetry. We have found that
+B(D → Sℓ+νℓ, S → P1P2) with the c → sℓ+νℓ transitions are predicted on the order of O(10−3 − 10−4), and
+B(D → Sℓ+νℓ, S → P1P2) with the c → dℓ+νℓ transitions are predicted on the order of O(10−4−10−6). The two
+quark picture and the four quark picture for the scalar mesons have been analyzed in the D → S(S → P1P2)ℓ+νℓ
+decays. Present experimental data might favorite the four quark picture for the scalar mesons.
+• Decays with the vector meson resonances:
+Using the experimental data of B(D+ → K
+∗0e+νe, K
+∗0 →
+π+K−), B(D+ → K
+∗0µ+νµ, K
+∗0 → π+K−), many B(D → V ℓ+νℓ) and many B(V → P1P2), the not-measured
+B(D → V ℓ+νℓ, V → P1P2) have been predicted by the SU(3) flavor symmetry. We have found that B(D →
+V ℓ+νℓ, V → P1P2) with the c → sℓ+νℓ transitions are predicted on the order of O(10−2 − 10−3), and B(D →
+V ℓ+νℓ, V → P1P2) with the c → dℓ+νℓ transitions are predicted on the order of O(10−3 − 10−5).
+• Total branching ratios: Total branching ratio predictions including the possible non-resonant, light scalar
+meson resonant and vector meson resonant contributions have been obtained.
+The six total branching ra-
+tios have been measured, and we did not use them to further constrain the predictions.
+Our five predic-
+tions are consistent with present data within 2σ errors, and the prediction of B(D0 → π0π−e+νe) will be
+very close to the datum within 3σ error bars. We have found that the vector meson resonant contributions
+are dominant in the D0 → π−K
+0ℓ+νℓ, π0K−ℓ+νℓ, π0π−ℓ+νℓ, D+ → π+K−ℓ+νℓ, π0K
+0ℓ+νℓ, π+π−ℓ+νℓ, and
+D+
+s → K+K−ℓ+νℓ, K0K
+0ℓ+νℓ, K+π−ℓ+νℓ, K0π0ℓ+νℓ decays. All three kinds of contributions are important
+in D0 → ηπ−ℓ+νℓ decays.
+Both the non-resonant and the scalar resonant contributions are important in
+D0 → K−K0ℓ+νℓ, η′π−ℓ+νℓ and D+ → K
+0K0ℓ+νℓ, π0π0ℓ+νℓ, ηπ0ℓ+νℓ, η′π0ℓ+νℓ decays.
+
+22
+Although SU(3) flavor symmetry is approximate, it can still provide very useful information about these decays.
+According to our rough predictions, many decay modes could be observed at BESIII, LHCb and BelleII, and some
+decay modes might be measured in near future experiments. Therefore, the SU(3) flavor symmetry will be further
+tested by these semileptonic decays in future experiments.
+ACKNOWLEDGEMENTS
+The work was supported by the National Natural Science Foundation of China (12175088).
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