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+A Multi-Objective Planning and Scheduling
+Framework for Community Energy Storage Systems
+in Low Voltage Distribution Networks
+K.B.J. Anuradha
+The Australian National University
+Canberra, Australia
+[email protected]
+Chathurika P. Mediwaththe
+The Australian National University & CSIRO
+Canberra, Australia
+[email protected]
+Masoume Mahmoodi
+The Australian National University
+Canberra, Australia
+[email protected]
+Abstract—This paper presents a methodology for optimizing
+the planning and scheduling aspects of a community energy
+storage (CES) system in the presence of solar photovoltaic (SPV)
+power in low voltage (LV) distribution networks. To this end, we
+develop a multi-objective optimization framework that minimizes
+the real power loss, the energy trading cost of LV customers and
+the CES provider with the grid, and the investment cost for the
+CES. Distribution network limits including the voltage constraint
+are also taken into account by combining the optimization
+problem with a linearized power flow model. Simulations for the
+proposed optimization framework with real power consumption
+and SPV generation data of the customers, highlight both real
+power loss and energy trading cost with the grid are reduced
+compared with the case without a CES by nearly 29% and 16%,
+respectively. Moreover, a case study justifies our methodology
+is competent in attaining the three objectives better than the
+optimization models which optimize only the CES scheduling.
+Keywords—Community energy storage, distribution networks,
+multi-objective optimization, planning and scheduling, power
+flow
+NOMENCLATURE
+Sets and Indices
+V, i, j
+Set of nodes, node indices
+E
+Set of lines in the network
+Wj
+Set of downstream nodes of node j includ-
+ing itself
+Cj, c
+Set of customers at node j, customer index
+T , t
+Set of time intervals, time index
+X, x
+Feasible set, decision variable vector
+Model Parameters
+rij, xij
+Resistance and reactance of line (i, j) - (Ω)
+Umin, Umax
+Minimum and maximum squared voltage
+magnitude limits - (V 2)
+ηch, ηdis
+Charging and discharging efficiencies of the
+CES
+λmin, λmax
+Percentage coefficients of the CES capacity
+aj
+Binary variable to find the optimal CES
+location
+pRate
+j
+Optimal CES rated power- (kW)
+Ecap
+j
+Optimal CES capacity - (kWh)
+pRate
+min , pRate
+max
+Minimum and maximum allowable CES
+rated power- (kW)
+ECES
+min , ECES
+max
+Minimum and maximum allowable CES
+capacity - (kWh)
+λp(t)
+Grid energy price at time t - (AUD/kWh)
+γCES
+Fixed part of the CES investment cost -
+(AUD)
+δCES
+Cost of the CES for a unit capacity -
+(AUD/kWh)
+∆t
+Time difference between two adjacent time
+instances - (h)
+wi
+Weight coefficient of the ith objective func-
+tion
+Power Flows and Injections
+pL
+cj(t), qL
+cj(t)
+Real and reactive power consumption of
+the customer c at node j at time t -
+(kW, kV AR)
+pP V
+cj (t)
+SPV generation of the customer c at node
+j at time t - (kW)
+Pij(t), Qij(t)
+Real and reactive power flow from i to j
+node at time t - (kW, kV AR)
+pj(t), qj(t)
+Real and reactive power absorption at node
+j at time t - (kW, kV AR)
+pG
+cj(t))
+Real power exchange with the grid by the
+customer c at node j at time t - (kW)
+pCES
+cj
+(t))
+Real power exchange with the CES by the
+arXiv:2301.02372v1  [eess.SY]  6 Jan 2023
+
+customer c at node j at time t - (kW)
+pG
+CES(t))
+Real power exchange with the grid by the
+CES at time t- (kW)
+pCES,ch
+j
+(t)
+Charging power of the CES at node j at
+time t - (kW)
+pCES,dis
+j
+(t)
+Discharging power of the CES at node j at
+time t - (kW)
+Other Notations
+ECES
+j
+(t)
+Energy level of the CES at node j at time
+t - (kWh)
+Vj(t)
+Voltage magnitude of node j at time t - (V )
+Uj(t)
+Squared voltage magnitude of node j at
+time t - (V 2)
+Iij(t)
+Current flow from node i to j at time t -
+(A)
+I. INTRODUCTION
+In the recent past, there has been a notable interest among
+the power systems research community and the industry for
+the uptake of community energy storage (CES) in low voltage
+(LV) power systems. This trend is driven by the benefits
+gained from a CES such as providing the opportunity to
+increase the hosting capacity of the network, enhancing the
+solar energy self consumption of the customers, and increasing
+the community access to renewable energy [1]. Additionally,
+CES devices can be deployed to gain technical merits such as
+real power loss minimization and economic benefits including
+the curtailment of energy purchase cost of the customers [2].
+As discussed in literature, a CES may be used in energy
+management problems to earn technical and monetary benefits
+together [3], [4]. Those merits can be fully exploited if the
+CES planning aspects including its location, the rated power
+and the capacity are optimized simultaneously with the CES
+scheduling aspects namely, its charging and discharging.
+The existing literature on CES utilization in LV distribution
+networks can be divided into two categories as; (i) optimiza-
+tion of CES scheduling only, (ii) optimization of both CES
+planning and scheduling. In the first category, the authors
+have presented optimization frameworks for CES scheduling
+without accounting for its planning aspects. For instance, a
+method built up on game theory concepts to maximize the
+revenue for the CES provider and minimize energy costs for
+the customers is discussed in [3]. A multi-objective framework
+to minimize the real energy loss and energy costs of the
+customers and the CES provider for trading energy with the
+grid is discussed in [4]. A method based on model predictive
+control to optimize the CES scheduling is presented in [5].
+In addition to the papers which have presented methods for
+optimizing only the CES scheduling, there are research work
+which have proposed methods for optimizing both planning
+and scheduling of CES simultaneously. For instance, a method
+for maximizing the hosting capacity in a distribution network
+in the presence of a CES is proposed in [6]. Analytical meth-
+ods to minimize the real energy loss of a network by finding
+the optimal CES location and its capacity are discussed in [7],
+[8]. A common feature of these methods is that the optimal
+CES planning aspects are determined based on analytical
+(such as graphical or numerical methods) and sensitivity based
+approaches (methods which decide the optimal values based
+on a calculated sensitivity parameter). These approaches can
+be computationally exhaustive as the optimal CES location and
+the capacity are found upon computing a sensitivity parameter
+for a large number of location-capacity combinations. Also,
+even after an exhaustive search, it is not always guaranteed to
+reach an optimal solution [7]. Thus, a robust formulation to
+optimize the capacity, the rated power and the location of a
+CES while generating the techno-economic benefits associated
+with such storage devices would be an effective alternative to
+overcome the challenges in the literature.
+In this paper, we study the extent to which the location,
+the capacity and the power rating of a CES in addition to its
+scheduling, affect network and economic benefits achievable
+from it. For this, we develop an optimization framework
+that optimizes both planning and scheduling of a CES. The
+optimized planning and scheduling aspects are then leveraged
+to minimize the network power loss, cost incurred by the
+customers and the CES provider for trading energy with the
+grid and the investment cost of the CES simultaneously. To the
+best of our knowledge, this problem has not been addressed in
+the literature. The contributions of this paper are as follows.
+• A linearized power flow model is exploited with the
+CES operational constraints to develop a multi-objective
+optimization framework. It is then solved as a mixed
+integer quadratic program according to the optimization
+algorithms in [9]. The analytic hierarchy process (AHP)
+is used for fairly weighting the objective functions [10].
+• The performance of the proposed optimization framework
+is evaluated on a real LV distribution network. Here,
+we do a comparison between our proposed optimization
+framework, and the models that arbitrarily choose the
+CES planning aspects such as its location, to assess the
+impact of it on the objectives. Finally, a comprehensive
+analysis of the results is also presented.
+The rest of the paper is structured as follows. Section II
+presents the mathematical models used in our problem. The
+proposed CES planning and scheduling optimization frame-
+work is illustrated in Section III. Section IV is about the
+numerical and graphical results along with their discussion.
+Eventually, the conclusion of the work and possible future
+developments are given in Section V.
+II. SYSTEM MATHEMATICAL MODELLING
+In this paper, the positive power absorption convention is
+considered for all nodes. Also, it is considered that there are
+multiple customers at each node. All the real and reactive
+power quantities are measured in kW and kVAR, respectively.
+It is assumed that power consumption (both real and reactive)
+and SPV generation of each customer are known ahead from
+their forecasts. A summary of the notations used in this paper,
+together with their definitions are given in the Nomenclature.
+
+Fig. 1: Possible power exchanges between a customer, the CES
+and the grid
+A. Power Flow Model
+The typical mutual power exchanges that can ensue between
+different entities (i.e. customers, CES and grid) in the presence
+of a CES is shown in Fig. 1. pCES
+cj
+(t) > 0 suggests a power
+import by the customer c at node j at time t from the CES,
+and pCES
+cj
+(t) < 0 occurs when that customer exports power
+to the CES. The same sign convention used for pCES
+cj
+(t) is
+valid for pG
+cj(t) and pG
+CES(t). The mathematical relationships
+between the power flows shown in Fig. 1 are described later.
+The relationship between the line power flows and node
+absorptions which follows the LinDistflow model are given
+by (1), (2) [11].
+Pij(t) = pj(t) +
+�
+k:j→k
+Pjk(t)
+(1)
+Qij(t) = qj(t) +
+�
+k:j→k
+Qjk(t)
+(2)
+The nodal real and reactive power absorptions are illustrated
+by the equations (3) and (4). The equation (3a) governs the
+real power absorption for the CES connected node and for the
+rest of the nodes (except the slack node), it is the equation
+(3b). Additionally, we assume all the SPV units and the CES
+operate at unity power factor.
+pj(t) =
+�
+c∈Cj
+pL
+cj(t) −
+�
+c∈Cj
+pP V
+cj (t) + pCES,ch
+j
+(t)
+−pCES,dis
+j
+(t)
+∀j ∈ V\ {0} , t∈ T
+(3a)
+pj(t) =
+�
+c∈Cj
+pL
+cj(t) −
+�
+c∈Cj
+pP V
+cj (t)
+∀j ∈ V\ {0} , t∈ T
+(3b)
+qj(t) =
+�
+c∈Cj
+qL
+cj(t)
+∀j ∈ V\ {0} , t∈ T
+(4)
+The equations (5) and (6) demonstrate how a customer
+exchanges power with the CES and the grid, when that
+customer encounters a mismatch of its real power consumption
+and SPV generation. When a customer experiences a deficit of
+its SPV generation to supply its real power consumption, that
+deficit can be fulfilled in share by the CES and the grid. On
+the other hand, if a customer has a surplus SPV generation,
+that customer exports the excess to both CES and the grid.
+The mathematical relationship between pG
+CES(t), pCES
+cj
+(t),
+pCES,ch
+j
+(t) and pCES,dis
+j
+(t) can be written as (7).
+If pL
+cj(t) ≥ pP V
+cj (t):
+0 ≤ pG
+cj(t) + pCES
+cj
+(t) = pL
+cj(t) − pP V
+cj (t)
+(5a)
+0 ≤ pG
+cj(t) ≤ pL
+cj(t) − pP V
+cj (t)
+∀j ∈ V\ {0} , c ∈ Cj, t∈ T
+(5b)
+Otherwise:
+pG
+cj(t) + pCES
+cj
+(t) = pL
+cj(t) − pP V
+cj (t) ≤ 0
+(6a)
+pL
+cj(t) − pP V
+cj (t) ≤ pG
+cj(t) ≤ 0
+∀j ∈ V\ {0} , c ∈ Cj, t∈ T
+(6b)
+pG
+CES(t) =
+N
+�
+j=1
+�
+�
+�
+�
+c∈Cj
+pCES
+cj
+(t) + pCES,ch
+j
+(t) − pCES,dis
+j
+(t)
+�
+�
+�
+(7)
+The Lindistflow equations given in (1)-(4) can be written in
+matrix format as ((8) [11]
+U = U01 − 2˜Rp − 2˜Xq
+∀t∈ T
+(8)
+where U = |V(t)|2 is the vector of squared voltage magni-
+tudes of nodes, 1 is a vector of all ones and U0 = |V0|2 is
+the squared voltage magnitude of the slack node. Also, p and
+q are the vectors of nodal real and reactive power absorption.
+The matrices ˜R and ˜X ∈ RN×N have the elements Rij =
+� (a.b) ∈ Li ∩ Ljrab and Xij = � (a.b) ∈ Li ∩ Ljxab, re-
+spectively where Li is the set of lines on the path connecting
+node 0 and “i” [3], [11].
+The squared voltage magnitudes at each node needs to be
+maintained within its allowable voltage magnitude limits. This
+is guaranteed by the inequality given in (9). Here, Umin =
+Umin1 and Umax = Umax1.
+Umin ≤ U ≤ Umax
+∀t∈ T
+(9)
+B. Community Energy Storage Model
+In this section we present the mathematical modelling of
+the CES. We consider the CES is owned by a third party, and
+the owner is designated as the CES provider.
+The set of constraints listed from (10) to (17) model the
+CES. The equations (10) and (11) imply that the CES charging
+and discharging power should not exceed the rated power
+pRate
+j
+of the CES. The temporal variation of the energy level
+of the CES is expressed by (12). Also, the CES energy level
+
+External Grid
+Grid
+pCEs(t) < 0
+pCes(t) > 0
+0 > ()号d
+pgj(t) > 0
+田田
+pCES(t) < 0
+cth Customer
+CES Device
+at node jat any time should exist within its upper and lower state of
+charge (SoC) limits. This is handled by (13). The continuity
+of the CES operation over the next day is guaranteed by the
+inequality given in (14) which is bounded by a small positive
+number ε [3], [4]. Note that td in (14) represents the day num-
+ber of the year. Here td ∈ TD, where TD = {1, 2, ...., NT /24}
+and NT is the cardinality of set T .
+0 ≤ pCES,ch
+j
+(t) ≤ pRate
+j
+∀j ∈ V\ {0} , t∈ T
+(10)
+0 ≤ pCES,dis
+j
+(t) ≤ pRate
+j
+∀j ∈ V\ {0} , t∈ T
+(11)
+ECES
+j
+(t) = ECES
+j
+(t − 1) + (ηchpCES,ch
+j
+(t)
+−
+1
+ηdis pCES,dis
+j
+(t))∆t
+∀j ∈ V\ {0} , t∈ T
+(12)
+λminEcap
+j
+≤ ECES
+j
+(t) ≤ λmaxEcap
+j
+∀j ∈ V\ {0} , t∈ T
+(13)
+��ECES
+j
+(24td) − ECES
+j
+(0)
+�� ≤ ε
+∀j ∈ V\ {0} , td∈ T D
+(14)
+The equation (15) is used to find the optimal CES location.
+Also, (15) ensures that only one CES is installed in the
+network. If aj = 0, it implies that there is no CES at node j.
+If aj = 1, then the CES is connected to node j. To determine
+the optimal CES capacity Ecap
+j
+, the inequality given in (16)
+is utilized. For a case aj = 0, (16) makes Ecap
+j
+also to be
+zero. When Ecap
+j
+= 0, the values ECES
+j
+(t), pCES,ch
+j
+(t) and
+pCES,dis
+j
+(t) in (12) and (13) also turn out to be zero. The
+inequality in (17) guarantees the rated power of the CES is
+bounded by its minimum and maximum allowable values.
+N
+�
+j=1
+aj = 1
+∀j ∈ V\ {0} , aj ∈ {0, 1}
+(15)
+ajEcap
+min ≤ Ecap
+j
+≤ ajEcap
+max
+∀j ∈ V\ {0} , aj ∈ {0, 1}
+(16)
+ajpRate
+min ≤ pRate
+j
+≤ ajpRate
+max
+∀j ∈ V\ {0} , aj ∈ {0, 1}
+(17)
+III. OPTIMIZATION FRAMEWORK & PROBLEM
+FORMULATION
+In our paper, it is expected to minimize the real power
+loss of the network, energy trading costs of the customers
+and the CES provider with the grid, and to minimize the
+CES investment cost. Therefore, a multi-objective function
+is obtained by combining those objectives functions, and its
+formulation is given as follows.
+A. Objective Functions
+1) Minimizing the Real Power Loss of the Network: The
+real power loss in a network can be written as (19), in terms
+of (18), and by taking Ui(t) ≈ U0(t) ∀i ∈ V\ {0} [4], [11].
+|Iij(t)|2 = Pij(t)2 + Q2
+ij(t)
+Ui(t)
+∀(i, j)∈ E, t∈ T
+(18)
+fP loss =
+�
+t∈T
+�
+(i,j)∈E
+rij |Iij(t)|2
+(19)
+2) Minimizing the Energy Trading Cost of the Customers
+and the CES Provider with the Grid: The first term of the
+objective function given in (20) relates to the energy trading
+cost with the grid by customers, and latter for the CES
+provider.
+fEn,cost =
+�
+t∈T
+λp(t)
+�
+�
+�
+N
+�
+j=1
+�
+c∈Cj
+pG
+cj(t) + pG
+CES(t)
+�
+�
+� ∆t (20)
+Here, it is considered a one-for-one non-dispatchable energy
+buyback scheme such that the same energy price for both
+imports and exports of energy from the grid by the customers
+and the CES is used [12]. This kind of an energy pricing
+scheme can effectively value the SPV power as being same as
+the power imported from the grid, which is usually generated
+by a conventional generation method.
+3) Minimizing the Investment Cost of the CES: The third
+objective is to minimize the investment cost of the CES device
+which is given by (21) [2].
+fInv,cost = γCES + δCESEcap
+j
+(21)
+B. Problem Formulation
+The three objective functions are normalized and weighted
+to form the multi-objective function in (22), according to the
+techniques described in [13]. The normalization guarantees
+the objective functions are converted into a form which can
+be added together (since fP loss is measured in kW, and
+fEn,cost, fInv,cost are measured in AUD ).
+x∗ = argmin
+x∈X
+w1
+� fP loss−f utopia
+P loss
+f Nadir
+P loss −f utopia
+P loss
+�
++ w2
+�
+fEn,cost−f utopia
+En,cost
+f Nadir
+En,cost−f utopia
+En,cost
+�
++w3
+�
+fInv,cost−f utopia
+Inv,cost
+f Nadir
+Inv,cost−f utopia
+Inv,cost
+�
+(22)
+where X is the feasible set which is constrained by (1)-(17).
+The utopia values, individual minimum point values and nadir
+values of the multi-objective function are found by (23), (24)
+and (25), respectively. Besides, the decision variable vector
+can be explicitly expressed as (26).
+f utopia
+P loss = fP loss(x∗
+Ploss)
+(23a)
+
+Fig. 2: 7-Node LV radial distribution network
+f utopia
+En,cost = fEn,cost(x∗
+En,cost)
+(23b)
+f utopia
+Inv,cost = fInv,cost(x∗
+Inv,cost)
+(23c)
+x∗
+Ploss = argmin
+x∈X
+fP loss
+(24a)
+x∗
+En,cost = argmin
+x∈X
+fEn,cost
+(24b)
+x∗
+Inv,cost = argmin
+x∈X
+fInv,cost
+(24c)
+f Nadir
+P loss = Max
+�
+fP loss(x∗
+Ploss), fP loss(x∗
+En,cost),
+fP loss(x∗
+Inv,cost)
+�
+(25a)
+f Nadir
+En,cost = Max
+�
+fEn,cost(x∗
+Ploss), fEn,cost(x∗
+En,cost),
+fEn,cost(x∗
+Inv,cost)
+�
+(25b)
+f Nadir
+Inv,cost = Max
+�
+fInv,cost(x∗
+Ploss), fInv,cost(x∗
+En,cost),
+fInv,cost(x∗
+Inv,cost)
+�
+(25c)
+x = (aj, pRate
+j
+, Ecap
+j
+, pCES,ch
+j
+, pCES,dis
+j
+, pG
+CES, pG
+cj) (26)
+In summary, the optimization framework can be written as
+(22), subject to a set of constraints (1)-(17). Also, as (22) being
+a quadratically-constrained convex multi-objective function, it
+is solved as a mixed-integer quadratic program.
+IV. NUMERICAL AND SIMULATION RESULTS
+In the simulations, a 7-node LV radial distribution network
+given in Fig. 2 is used and its line data can be found
+in [14]. Also, real power consumption and SPV generation
+data of 30 customers in an Australian residential community
+were used for simulations [15]. To be more practical, we
+randomly allocated multiple customers for each node. Hence,
+�N
+j=1 |Cj| = 30, and the number of customers at each node
+are marked in Fig. 2. Here, all the customers generate SPV
+power in addition to their real power consumption. However,
+reactive power consumption of the customers is not considered
+due to the lack of sufficient real data. As the optimization
+Fig. 3: Variation of grid energy price for 24 hours
+involves not only a scheduling problem but also a planning
+problem, the optimization is performed over a long time
+period. Thus, we consider one year time period split in one
+hour time intervals (i.e.|T | = 8760 ) for the simulations.
+The voltage and power base are taken as 400V and
+100 kVA, respectively. In addition to that, V0
+= 1p.u.,
+Umin = 0.9025p.u., Umax = 1.1025p.u., λmin = 0.05,
+λmax = 1, ηch = 0.98, ηdis = 1.02, Ecap
+min = 200kWh,
+Ecap
+max = 2000kWh, pRate
+min
+= 20kW, pRate
+max
+= 200kW,
+ε = 0.0001kWh and ∆t = 1h are used as the model
+parameters. The values of γCES and δCES are taken as 24000
+AUD and 300 AUD/kWh as specified in [2]. Additionally, the
+weighting factors w1, w2 and w3 were calculated according
+to the principles of AHP specified in [10]. We considered
+a moderate plus importance for both fEn,cost and fInv,cost
+compared to fP loss, and an equal importance for fEn,cost and
+fInv,cost. Hence, based on the AHP method, the values of
+w1, w2 and w3 were calculated as 1/9, 4/9 and 4/9, respec-
+tively. Fig. 3 depicts how the grid energy price varies with the
+time of the day following a time of use (ToU) tariff scheme.
+As seen in Fig. 3, the grid energy price is 0.24871 AUD/kWh
+during T1(from 12am-7am) & T5(from 10pm-12am), 0.31207
+AUD/kWh during T2(from 7am-3pm) & T4 (from 9pm-10pm)
+and 0.52602 AUD/kWh during T3(from 3pm-9pm) [16].
+A. Case Study - Proposed Optimization Framework Vs Opti-
+mization Models With Arbitrary CES Locations
+We did a case study to compare the results of our model with
+four different cases by arbitrarily changing the CES location.
+For this, we considered our optimization framework as Case I,
+while the rest as Case II-V. The same optimization framework
+(except the constraint that finds the optimal CES location),
+and the model parameters as for Case I were used for Case II-
+V. A synopsis of the results for the five cases are tabulated in
+Table I. The Case I lists the planning results and the minimized
+objective function values for our proposed model. The Cases
+II and III suggest the same optimal CES capacity and the rated
+power. Nevertheless, due to their difference in CES location,
+Case II provides less real energy loss and energy trading cost
+compared with the Case III. When the CES is at node 6, the
+optimization suggests the same optimal capacity as in Case I.
+However, as node 6 is not the optimal location for CES, the
+real energy loss and energy trading cost for Case IV are higher
+
+2
+6
+External
+Transformer
+IC2l = 4
+IC6l = 4
+Grid
+22/0.4 kV
+0
+1
+3
+4
+IC1l = 3
+IC3l = 5
+C4 = 6
+ICzl = 5
+5
+ICsl = 30.55
+I (AUD/kWh)
+0.50
+0.45
+ Signal (
+0.40
+0.35
+Price
+Energy I
+0.30
+0.25
+T4 T5
+0.20
+0
+5
+10
+15
+20
+25
+Time Duration (24 HoursTABLE I: SUMMARY OF THE RESULTS FOR CASE STUDIES
+CES
+Location
+(Node)
+Optimal CES
+Capacity (kWh)
+Optimal CES
+Power Rating (kW)
+Real Energy
+Loss1 (kWh)
+Energy Trading
+Cost With
+Grid1 (AUD)
+CES Investment
+Cost (AUD)
+Base Case (Without CES)
+Not applicable
+Not applicable
+Not applicable
+110116.68
+45585
+Not applicable
+Case I (Proposed Model)
+4 (optimal)
+482.15
+200
+78200.88 (71.02%)
+38520 (84.50%)
+168645
+Case II
+3 (chosen)
+601.32
+200
+80250.48 (72.88%)
+43362 (95.12%)
+204396
+Case III
+5 (chosen)
+601.32
+200
+81961.28 (74.43%)
+43840 (96.17%)
+204396
+Case IV
+6 (chosen)
+482.15
+200
+80761.16 (73.34%)
+44154 (96.86%)
+168645
+Case V
+7 (chosen)
+547.69
+200
+86082.52 (78.17%)
+43625 (95.70%)
+188307
+1 Percentage values are calculated with respect to their corresponding values without a CES
+Fig. 4: Total power exchange with the grid by the customers
+Fig. 5: Total power exchange with the CES by the customers
+than in Case I. Also, our model has produced the highest cost
+reduction percentages for real energy loss (28.98%) and the
+energy trading cost with the grid (15.5%), compared to all the
+other cases. Hence, it is clear that Case I yields the minimum
+values for all the three objective functions, and this justifies
+the effectiveness of our optimization framework compared to
+the models that optimize only the CES scheduling.
+B. Analysis of the Results-Mutual Power Exchanges Between
+the customers, the CES and the grid
+In order to understand the CES scheduling and power
+exchanges between different entities, we select a single day (24
+hours) for our discussion. Fig. 4 shows the variation of total
+Fig. 6: Power exchange with the grid by the CES
+Fig. 7: CES charging and discharging power pattern
+power exchange that occurs with the grid by the customers.
+Since �N
+j=1
+�
+c∈Cj pG
+cj(t) being a positive value approxi-
+mately during T1, T3, T4, T5 time intervals, it implies that the
+customers tend to import certain amount of power from the
+grid for satisfying their real power consumption during those
+time periods. On the other hand, during T2 (time period of the
+day usually the SPV generation is high), the customers have a
+tendency to export a portion of their surplus SPV generation
+to the grid. This is evident as �N
+j=1
+�
+c∈Cj pG
+cj(t) < 0 during
+T2. This behavior guarantees a cost benefit for the customers
+for their exported power according to equation (20).
+The Fig. 5 depicts how the customers exchange power
+
+) (kw)
+50
+()53
+0
+-50
+-100
+0
+5
+10
+15
+20
+2550
+(kw)
+0
+-50
+-100
+-150
+-200
+0
+5
+10
+15
+20
+25100
+(kW)
+50
+0
+CG
+-50
+-100
+0
+5
+10
+15
+20
+25
+Time
+e Duration(24 Hours100
+(kW)
+50
+0
+and
+-50
+-100
+p
+0
+5
+10
+15
+20
+25Fig. 8: Temporal variation of the CES energy level
+with the CES. During T2, the customers export a part of
+their surplus SPV generation to the CES. On the contrary,
+during rest of the time periods, the customers import a certain
+amount of power from the CES for satisfying their real power
+consumption. This action results in reducing the cost for the
+customers as the amount of power imported from the grid is
+minimized.
+The Fig. 6 illustrates how the CES exchanges power with
+the grid. As the grid energy price during T1 being the lowest,
+the CES tends to import power from the grid (i.e. pCES
+G
+(t) >
+0) during T1. This guarantees that the CES is charged with
+low priced energy from the grid. However, during T2, T3 and
+T4, it is seen that the CES exports its power back to the grid
+(i.e. pCES
+G
+(t) < 0). This happens as the CES provider can
+maximize its revenue by exporting power back to grid.
+In Fig. 7 and 8, it is observed that during T1, the CES
+charges (from the low priced grid energy) and partially dis-
+charges by the end of T1. During T2, the CES continues
+to charge and by the end of this time period, it reaches its
+maximum energy level. The stored energy in the CES is fully
+utilized during T3 and T4 for partially supplying the real
+power consumption of the customers. This facilitates monetary
+benefits for both the customers as the amount of expensive
+power imported from the grid is lowered. Additionally, when
+observing the temporal variation of the CES energy level, it
+is visualized that it is the peak value of the CES energy level
+which was obtained as the optimal CES capacity (i.e. 482.15
+kWh).
+V. CONCLUSION & FUTURE WORK
+In this work, we have explored how the optimization of
+the planning and scheduling aspects of a community energy
+storage (CES) can benefit both the network and the customers.
+To this end, we developed a multi-objective mixed-integer
+quadratic optimization framework to minimize three objec-
+tives: (i) network real power loss, (ii) energy trading cost of
+the customers and the CES provider with the grid, and (iii)
+the CES investment cost. The simulation results highlighted
+our optimization framework is competent in acquiring the
+expected merits compared with the case without a CES, and
+optimization models that optimize only the scheduling of CES.
+As future work, we expect to develop the work considering
+a stochastic model taking into account the uncertainties of
+real power consumption and SPV generation of the customers.
+Moreover, we look forward to extend the work by considering
+the unbalanced nature of LV distribution networks, and reac-
+tive power control capabilities of solar photovoltaic (SPV) and
+CES inverters.
+REFERENCES
+[1] M. Shaw, B. Sturmberg, C.P. Mediwaththe, H. Ransan-Cooper, D. Taylor
+and L. Blackhall “Community batteries: a cost/benefit analysis,” Tech-
+nical Report, Australian National University, 2020.
+[2] Y.
+Zheng, Y. Song, A. Huang, and D.J. Hill, “Hierarchical Optimal
+Allocation of Battery Energy Storage Systems for Multiple Services in
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+1911–1921, 2020.
+[3] C.P. Mediwaththe, and L. Blackhall, “Network-Aware Demand-Side
+Management Framework With A Community Energy Storage System
+Considering Voltage Constraints,” IEEE Trans.Power Syst., vol. 36,
+no. 2, pp. 1229–1238, 2021.
+[4] C.P. Mediwaththe, and L. Blackhall, “Community Energy Storage-based
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+in Proc. Int. Conf. Smart Grids and Energy Syst., 2020, pp. 516–521.
+[5] R. Zafar, J. Ravishankar, J.E. Fletcher and H.R. Pota, “Multi-Timescale
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+vol. 33, no. 6, pp. 7152–7161, 2018.
+[6] P. Hasanpor Divshali , and L. S¨oder, “Improving Hosting Capacity of
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+bridge U.K.: Cambridge Univ. Press, 2004.
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+[11] W. Lin, and E. Bitar, “Decentralized Stochastic Control of Distributed
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+2018.
+[12] J. Martin“1-to-1 solar buyback vs solar feed-in tariffs: The eco-
+nomics,”2012. [Online]. Available: https://www.solarchoice.net.au/blog/
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+[13] O. Grodzevich , and O. Romanko, “Normalization and other topics in
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+[15] “Solar Home Electricity Data,” [Online]. Available: https://www.ausgrid.
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+data/.”
+[16] “Origin, “VIC residential energy price fact sheet,” 2018.” [Online].
+Available: shorturl.at/gkmV5”
+
+500
+400
+(kWh)
+300
+200
+100
+0
+5
+10
+15
+20
+25
+Time Duration (24 Hours)

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+page_content='A Multi-Objective Planning and Scheduling  Framework for Community Energy Storage Systems  in Low Voltage Distribution Networks  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Anuradha  The Australian National University  Canberra, Australia  Jayaminda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='KariyawasamBovithanthri@anu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='au  Chathurika P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Mediwaththe  The Australian National University & CSIRO  Canberra, Australia  chathurika.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='mediwaththe@csiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='au  Masoume Mahmoodi  The Australian National University  Canberra, Australia  masoume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='mahmoodi@anu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='au  Abstract—This paper presents a methodology for optimizing  the planning and scheduling aspects of a community energy  storage (CES) system in the presence of solar photovoltaic (SPV)  power in low voltage (LV) distribution networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' To this end, we  develop a multi-objective optimization framework that minimizes  the real power loss, the energy trading cost of LV customers and  the CES provider with the grid, and the investment cost for the  CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Distribution network limits including the voltage constraint  are also taken into account by combining the optimization  problem with a linearized power flow model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Simulations for the  proposed optimization framework with real power consumption  and SPV generation data of the customers, highlight both real  power loss and energy trading cost with the grid are reduced  compared with the case without a CES by nearly 29% and 16%,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Moreover, a case study justifies our methodology  is competent in attaining the three objectives better than the  optimization models which optimize only the CES scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  Keywords—Community energy storage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' distribution networks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  multi-objective optimization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' planning and scheduling,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' power  flow  NOMENCLATURE  Sets and Indices  V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' j  Set of nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' node indices  E  Set of lines in the network  Wj  Set of downstream nodes of node j includ-  ing itself  Cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' c  Set of customers at node j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' customer index  T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' t  Set of time intervals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' time index  X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' x  Feasible set,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' decision variable vector  Model Parameters  rij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' xij  Resistance and reactance of line (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' j) - (Ω)  Umin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Umax  Minimum and maximum squared voltage  magnitude limits - (V 2)  ηch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' ηdis  Charging and discharging efficiencies of the  CES  λmin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' λmax  Percentage coefficients of the CES capacity  aj  Binary variable to find the optimal CES  location  pRate  j  Optimal CES rated power- (kW)  Ecap  j  Optimal CES capacity - (kWh)  pRate  min ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' pRate  max  Minimum and maximum allowable CES  rated power- (kW)  ECES  min ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' ECES  max  Minimum and maximum allowable CES  capacity - (kWh)  λp(t)  Grid energy price at time t - (AUD/kWh)  γCES  Fixed part of the CES investment cost -  (AUD)  δCES  Cost of the CES for a unit capacity -  (AUD/kWh)  ∆t  Time difference between two adjacent time  instances - (h)  wi  Weight coefficient of the ith objective func-  tion  Power Flows and Injections  pL  cj(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' qL  cj(t)  Real and reactive power consumption of  the customer c at node j at time t -  (kW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' kV AR)  pP V  cj (t)  SPV generation of the customer c at node  j at time t - (kW)  Pij(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Qij(t)  Real and reactive power flow from i to j  node at time t - (kW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' kV AR)  pj(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' qj(t)  Real and reactive power absorption at node  j at time t - (kW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' kV AR)  pG  cj(t))  Real power exchange with the grid by the  customer c at node j at time t - (kW)  pCES  cj  (t))  Real power exchange with the CES by the  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='02372v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='SY]  6 Jan 2023  customer c at node j at time t - (kW)  pG  CES(t))  Real power exchange with the grid by the  CES at time t- (kW)  pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='ch  j  (t)  Charging power of the CES at node j at  time t - (kW)  pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='dis  j  (t)  Discharging power of the CES at node j at  time t - (kW)  Other Notations  ECES  j  (t)  Energy level of the CES at node j at time  t - (kWh)  Vj(t)  Voltage magnitude of node j at time t - (V )  Uj(t)  Squared voltage magnitude of node j at  time t - (V 2)  Iij(t)  Current flow from node i to j at time t -  (A)  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' INTRODUCTION  In the recent past, there has been a notable interest among  the power systems research community and the industry for  the uptake of community energy storage (CES) in low voltage  (LV) power systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' This trend is driven by the benefits  gained from a CES such as providing the opportunity to  increase the hosting capacity of the network, enhancing the  solar energy self consumption of the customers, and increasing  the community access to renewable energy [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Additionally,  CES devices can be deployed to gain technical merits such as  real power loss minimization and economic benefits including  the curtailment of energy purchase cost of the customers [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  As discussed in literature, a CES may be used in energy  management problems to earn technical and monetary benefits  together [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Those merits can be fully exploited if the  CES planning aspects including its location, the rated power  and the capacity are optimized simultaneously with the CES  scheduling aspects namely, its charging and discharging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  The existing literature on CES utilization in LV distribution  networks can be divided into two categories as;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' (i) optimiza-  tion of CES scheduling only, (ii) optimization of both CES  planning and scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' In the first category, the authors  have presented optimization frameworks for CES scheduling  without accounting for its planning aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' For instance, a  method built up on game theory concepts to maximize the  revenue for the CES provider and minimize energy costs for  the customers is discussed in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' A multi-objective framework  to minimize the real energy loss and energy costs of the  customers and the CES provider for trading energy with the  grid is discussed in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' A method based on model predictive  control to optimize the CES scheduling is presented in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  In addition to the papers which have presented methods for  optimizing only the CES scheduling, there are research work  which have proposed methods for optimizing both planning  and scheduling of CES simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' For instance, a method  for maximizing the hosting capacity in a distribution network  in the presence of a CES is proposed in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Analytical meth-  ods to minimize the real energy loss of a network by finding  the optimal CES location and its capacity are discussed in [7],  [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' A common feature of these methods is that the optimal  CES planning aspects are determined based on analytical  (such as graphical or numerical methods) and sensitivity based  approaches (methods which decide the optimal values based  on a calculated sensitivity parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' These approaches can  be computationally exhaustive as the optimal CES location and  the capacity are found upon computing a sensitivity parameter  for a large number of location-capacity combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Also,  even after an exhaustive search, it is not always guaranteed to  reach an optimal solution [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Thus, a robust formulation to  optimize the capacity, the rated power and the location of a  CES while generating the techno-economic benefits associated  with such storage devices would be an effective alternative to  overcome the challenges in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  In this paper, we study the extent to which the location,  the capacity and the power rating of a CES in addition to its  scheduling, affect network and economic benefits achievable  from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' For this, we develop an optimization framework  that optimizes both planning and scheduling of a CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' The  optimized planning and scheduling aspects are then leveraged  to minimize the network power loss, cost incurred by the  customers and the CES provider for trading energy with the  grid and the investment cost of the CES simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' To the  best of our knowledge, this problem has not been addressed in  the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' The contributions of this paper are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  A linearized power flow model is exploited with the  CES operational constraints to develop a multi-objective  optimization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' It is then solved as a mixed  integer quadratic program according to the optimization  algorithms in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' The analytic hierarchy process (AHP)  is used for fairly weighting the objective functions [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  The performance of the proposed optimization framework  is evaluated on a real LV distribution network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Here,  we do a comparison between our proposed optimization  framework, and the models that arbitrarily choose the  CES planning aspects such as its location, to assess the  impact of it on the objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Finally, a comprehensive  analysis of the results is also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  The rest of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Section II  presents the mathematical models used in our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' The  proposed CES planning and scheduling optimization frame-  work is illustrated in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Section IV is about the  numerical and graphical results along with their discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  Eventually, the conclusion of the work and possible future  developments are given in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' SYSTEM MATHEMATICAL MODELLING  In this paper, the positive power absorption convention is  considered for all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Also, it is considered that there are  multiple customers at each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' All the real and reactive  power quantities are measured in kW and kVAR, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  It is assumed that power consumption (both real and reactive)  and SPV generation of each customer are known ahead from  their forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' A summary of the notations used in this paper,  together with their definitions are given in the Nomenclature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 1: Possible power exchanges between a customer, the CES  and the grid  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Power Flow Model  The typical mutual power exchanges that can ensue between  different entities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' customers, CES and grid) in the presence  of a CES is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' pCES  cj  (t) > 0 suggests a power  import by the customer c at node j at time t from the CES,  and pCES  cj  (t) < 0 occurs when that customer exports power  to the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' The same sign convention used for pCES  cj  (t) is  valid for pG  cj(t) and pG  CES(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' The mathematical relationships  between the power flows shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 1 are described later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  The relationship between the line power flows and node  absorptions which follows the LinDistflow model are given  by (1), (2) [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  Pij(t) = pj(t) +  �  k:j→k  Pjk(t)  (1)  Qij(t) = qj(t) +  �  k:j→k  Qjk(t)  (2)  The nodal real and reactive power absorptions are illustrated  by the equations (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' The equation (3a) governs the  real power absorption for the CES connected node and for the  rest of the nodes (except the slack node), it is the equation  (3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Additionally, we assume all the SPV units and the CES  operate at unity power factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  pj(t) =  �  c∈Cj  pL  cj(t) −  �  c∈Cj  pP V  cj (t) + pCES,ch  j  (t)  −pCES,dis  j  (t)  ∀j ∈ V\\ {0} , t∈ T  (3a)  pj(t) =  �  c∈Cj  pL  cj(t) −  �  c∈Cj  pP V  cj (t)  ∀j ∈ V\\ {0} , t∈ T  (3b)  qj(t) =  �  c∈Cj  qL  cj(t)  ∀j ∈ V\\ {0} , t∈ T  (4)  The equations (5) and (6) demonstrate how a customer  exchanges power with the CES and the grid, when that  customer encounters a mismatch of its real power consumption  and SPV generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' When a customer experiences a deficit of  its SPV generation to supply its real power consumption, that  deficit can be fulfilled in share by the CES and the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' On  the other hand, if a customer has a surplus SPV generation,  that customer exports the excess to both CES and the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  The mathematical relationship between pG  CES(t), pCES  cj  (t),  pCES,ch  j  (t) and pCES,dis  j  (t) can be written as (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  If pL  cj(t) ≥ pP V  cj (t):  0 ≤ pG  cj(t) + pCES  cj  (t) = pL  cj(t) − pP V  cj (t)  (5a)  0 ≤ pG  cj(t) ≤ pL  cj(t) − pP V  cj (t)  ∀j ∈ V\\ {0} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' c ∈ Cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' t∈ T  (5b)  Otherwise:  pG  cj(t) + pCES  cj  (t) = pL  cj(t) − pP V  cj (t) ≤ 0  (6a)  pL  cj(t) − pP V  cj (t) ≤ pG  cj(t) ≤ 0  ∀j ∈ V\\ {0} ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' c ∈ Cj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' t∈ T  (6b)  pG  CES(t) =  N  �  j=1  �  �  �  �  c∈Cj  pCES  cj  (t) + pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='ch  j  (t) − pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='dis  j  (t)  �  �  �  (7)  The Lindistflow equations given in (1)-(4) can be written in  matrix format as ((8) [11]  U = U01 − 2˜Rp − 2˜Xq  ∀t∈ T  (8)  where U = |V(t)|2 is the vector of squared voltage magni-  tudes of nodes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 1 is a vector of all ones and U0 = |V0|2 is  the squared voltage magnitude of the slack node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Also, p and  q are the vectors of nodal real and reactive power absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  The matrices ˜R and ˜X ∈ RN×N have the elements Rij =  � (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='b) ∈ Li ∩ Ljrab and Xij = � (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='b) ∈ Li ∩ Ljxab, re-  spectively where Li is the set of lines on the path connecting  node 0 and “i” [3], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  The squared voltage magnitudes at each node needs to be  maintained within its allowable voltage magnitude limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' This  is guaranteed by the inequality given in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Here, Umin =  Umin1 and Umax = Umax1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  Umin ≤ U ≤ Umax  ∀t∈ T  (9)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Community Energy Storage Model  In this section we present the mathematical modelling of  the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' We consider the CES is owned by a third party, and  the owner is designated as the CES provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  The set of constraints listed from (10) to (17) model the  CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' The equations (10) and (11) imply that the CES charging  and discharging power should not exceed the rated power  pRate  j  of the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' The temporal variation of the energy level  of the CES is expressed by (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Also, the CES energy level  External Grid  Grid  pCEs(t) < 0  pCes(t) > 0  0 > ()号d  pgj(t) > 0  田田  pCES(t) < 0  cth Customer  CES Device  at node jat any time should exist within its upper and lower state of  charge (SoC) limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' This is handled by (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' The continuity  of the CES operation over the next day is guaranteed by the  inequality given in (14) which is bounded by a small positive  number ε [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Note that td in (14) represents the day num-  ber of the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Here td ∈ TD, where TD = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='., NT /24}  and NT is the cardinality of set T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  0 ≤ pCES,ch  j  (t) ≤ pRate  j  ∀j ∈ V\\ {0} , t∈ T  (10)  0 ≤ pCES,dis  j  (t) ≤ pRate  j  ∀j ∈ V\\ {0} , t∈ T  (11)  ECES  j  (t) = ECES  j  (t − 1) + (ηchpCES,ch  j  (t)  −  1  ηdis pCES,dis  j  (t))∆t  ∀j ∈ V\\ {0} , t∈ T  (12)  λminEcap  j  ≤ ECES  j  (t) ≤ λmaxEcap  j  ∀j ∈ V\\ {0} , t∈ T  (13)  ��ECES  j  (24td) − ECES  j  (0)  �� ≤ ε  ∀j ∈ V\\ {0} , td∈ T D  (14)  The equation (15) is used to find the optimal CES location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  Also, (15) ensures that only one CES is installed in the  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' If aj = 0, it implies that there is no CES at node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  If aj = 1, then the CES is connected to node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' To determine  the optimal CES capacity Ecap  j  , the inequality given in (16)  is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' For a case aj = 0, (16) makes Ecap  j  also to be  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' When Ecap  j  = 0, the values ECES  j  (t), pCES,ch  j  (t) and  pCES,dis  j  (t) in (12) and (13) also turn out to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' The  inequality in (17) guarantees the rated power of the CES is  bounded by its minimum and maximum allowable values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  N  �  j=1  aj = 1  ∀j ∈ V\\ {0} , aj ∈ {0, 1}  (15)  ajEcap  min ≤ Ecap  j  ≤ ajEcap  max  ∀j ∈ V\\ {0} , aj ∈ {0, 1}  (16)  ajpRate  min ≤ pRate  j  ≤ ajpRate  max  ∀j ∈ V\\ {0} , aj ∈ {0, 1}  (17)  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' OPTIMIZATION FRAMEWORK & PROBLEM  FORMULATION  In our paper, it is expected to minimize the real power  loss of the network, energy trading costs of the customers  and the CES provider with the grid, and to minimize the  CES investment cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Therefore, a multi-objective function  is obtained by combining those objectives functions, and its  formulation is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Objective Functions  1) Minimizing the Real Power Loss of the Network: The  real power loss in a network can be written as (19), in terms  of (18), and by taking Ui(t) ≈ U0(t) ∀i ∈ V\\ {0} [4], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  |Iij(t)|2 = Pij(t)2 + Q2  ij(t)  Ui(t)  ∀(i, j)∈ E, t∈ T  (18)  fP loss =  �  t∈T  �  (i,j)∈E  rij |Iij(t)|2  (19)  2) Minimizing the Energy Trading Cost of the Customers  and the CES Provider with the Grid: The first term of the  objective function given in (20) relates to the energy trading  cost with the grid by customers, and latter for the CES  provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  fEn,cost =  �  t∈T  λp(t)  �  �  �  N  �  j=1  �  c∈Cj  pG  cj(t) + pG  CES(t)  �  �  � ∆t (20)  Here, it is considered a one-for-one non-dispatchable energy  buyback scheme such that the same energy price for both  imports and exports of energy from the grid by the customers  and the CES is used [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' This kind of an energy pricing  scheme can effectively value the SPV power as being same as  the power imported from the grid, which is usually generated  by a conventional generation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  3) Minimizing the Investment Cost of the CES: The third  objective is to minimize the investment cost of the CES device  which is given by (21) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  fInv,cost = γCES + δCESEcap  j  (21)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Problem Formulation  The three objective functions are normalized and weighted  to form the multi-objective function in (22), according to the  techniques described in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' The normalization guarantees  the objective functions are converted into a form which can  be added together (since fP loss is measured in kW, and  fEn,cost, fInv,cost are measured in AUD ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  x∗ = argmin  x∈X  w1  � fP loss−f utopia  P loss  f Nadir  P loss −f utopia  P loss  �  + w2  �  fEn,cost−f utopia  En,cost  f Nadir  En,cost−f utopia  En,cost  �  +w3  �  fInv,cost−f utopia  Inv,cost  f Nadir  Inv,cost−f utopia  Inv,cost  �  (22)  where X is the feasible set which is constrained by (1)-(17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  The utopia values, individual minimum point values and nadir  values of the multi-objective function are found by (23), (24)  and (25), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Besides, the decision variable vector  can be explicitly expressed as (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  f utopia  P loss = fP loss(x∗  Ploss)  (23a)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 2: 7-Node LV radial distribution network  f utopia  En,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost = fEn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost(x∗  En,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost)  (23b)  f utopia  Inv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost = fInv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost(x∗  Inv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost)  (23c)  x∗  Ploss = argmin  x∈X  fP loss  (24a)  x∗  En,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost = argmin  x∈X  fEn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost  (24b)  x∗  Inv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost = argmin  x∈X  fInv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost  (24c)  f Nadir  P loss = Max  �  fP loss(x∗  Ploss),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' fP loss(x∗  En,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  fP loss(x∗  Inv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost)  �  (25a)  f Nadir  En,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost = Max  �  fEn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost(x∗  Ploss),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' fEn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost(x∗  En,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  fEn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost(x∗  Inv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost)  �  (25b)  f Nadir  Inv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost = Max  �  fInv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost(x∗  Ploss),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' fInv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost(x∗  En,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  fInv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost(x∗  Inv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='cost)  �  (25c)  x = (aj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' pRate  j  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Ecap  j  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='ch  j  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' pCES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='dis  j  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' pG  CES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' pG  cj) (26)  In summary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' the optimization framework can be written as  (22),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' subject to a set of constraints (1)-(17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Also, as (22) being  a quadratically-constrained convex multi-objective function, it  is solved as a mixed-integer quadratic program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' NUMERICAL AND SIMULATION RESULTS  In the simulations, a 7-node LV radial distribution network  given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 2 is used and its line data can be found  in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Also, real power consumption and SPV generation  data of 30 customers in an Australian residential community  were used for simulations [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' To be more practical, we  randomly allocated multiple customers for each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Hence,  �N  j=1 |Cj| = 30, and the number of customers at each node  are marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Here, all the customers generate SPV  power in addition to their real power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' However,  reactive power consumption of the customers is not considered  due to the lack of sufficient real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' As the optimization  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 3: Variation of grid energy price for 24 hours  involves not only a scheduling problem but also a planning  problem, the optimization is performed over a long time  period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Thus, we consider one year time period split in one  hour time intervals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='|T | = 8760 ) for the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  The voltage and power base are taken as 400V and  100 kVA, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' In addition to that, V0  = 1p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=',  Umin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='9025p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=', Umax = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='1025p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=', λmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='05,  λmax = 1, ηch = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='98, ηdis = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='02, Ecap  min = 200kWh,  Ecap  max = 2000kWh, pRate  min  = 20kW, pRate  max  = 200kW,  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='0001kWh and ∆t = 1h are used as the model  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' The values of γCES and δCES are taken as 24000  AUD and 300 AUD/kWh as specified in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Additionally, the  weighting factors w1, w2 and w3 were calculated according  to the principles of AHP specified in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' We considered  a moderate plus importance for both fEn,cost and fInv,cost  compared to fP loss, and an equal importance for fEn,cost and  fInv,cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Hence, based on the AHP method, the values of  w1, w2 and w3 were calculated as 1/9, 4/9 and 4/9, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 3 depicts how the grid energy price varies with the  time of the day following a time of use (ToU) tariff scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 3, the grid energy price is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='24871 AUD/kWh  during T1(from 12am-7am) & T5(from 10pm-12am), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='31207  AUD/kWh during T2(from 7am-3pm) & T4 (from 9pm-10pm)  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='52602 AUD/kWh during T3(from 3pm-9pm) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Case Study - Proposed Optimization Framework Vs Opti-  mization Models With Arbitrary CES Locations  We did a case study to compare the results of our model with  four different cases by arbitrarily changing the CES location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  For this, we considered our optimization framework as Case I,  while the rest as Case II-V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' The same optimization framework  (except the constraint that finds the optimal CES location),  and the model parameters as for Case I were used for Case II-  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' A synopsis of the results for the five cases are tabulated in  Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' The Case I lists the planning results and the minimized  objective function values for our proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' The Cases  II and III suggest the same optimal CES capacity and the rated  power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Nevertheless, due to their difference in CES location,  Case II provides less real energy loss and energy trading cost  compared with the Case III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' When the CES is at node 6, the  optimization suggests the same optimal capacity as in Case I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  However, as node 6 is not the optimal location for CES, the  real energy loss and energy trading cost for Case IV are higher  2  6  External  Transformer  IC2l = 4  IC6l = 4  Grid  22/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='4 kV  0  1  3  4  IC1l = 3  IC3l = 5  C4 = 6  ICzl = 5  5  ICsl = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='55  I (AUD/kWh)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='45  Signal (  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='35  Price  Energy I  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='25  T4 T5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='20  0  5  10  15  20  25  Time Duration (24 HoursTABLE I: SUMMARY OF THE RESULTS FOR CASE STUDIES  CES  Location  (Node)  Optimal CES  Capacity (kWh)  Optimal CES  Power Rating (kW)  Real Energy  Loss1 (kWh)  Energy Trading  Cost With  Grid1 (AUD)  CES Investment  Cost (AUD)  Base Case (Without CES)  Not applicable  Not applicable  Not applicable  110116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='68  45585  Not applicable  Case I (Proposed Model)  4 (optimal)  482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='15  200  78200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='88 (71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='02%)  38520 (84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='50%)  168645  Case II  3 (chosen)  601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='32  200  80250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='48 (72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='88%)  43362 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='12%)  204396  Case III  5 (chosen)  601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='32  200  81961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='28 (74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='43%)  43840 (96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='17%)  204396  Case IV  6 (chosen)  482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='15  200  80761.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='16 (73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='34%)  44154 (96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='86%)  168645  Case V  7 (chosen)  547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='69  200  86082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='52 (78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='17%)  43625 (95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='70%)  188307  1 Percentage values are calculated with respect to their corresponding values without a CES  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 4: Total power exchange with the grid by the customers  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 5: Total power exchange with the CES by the customers  than in Case I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Also, our model has produced the highest cost  reduction percentages for real energy loss (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='98%) and the  energy trading cost with the grid (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='5%), compared to all the  other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Hence, it is clear that Case I yields the minimum  values for all the three objective functions, and this justifies  the effectiveness of our optimization framework compared to  the models that optimize only the CES scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Analysis of the Results-Mutual Power Exchanges Between  the customers, the CES and the grid  In order to understand the CES scheduling and power  exchanges between different entities, we select a single day (24  hours) for our discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 4 shows the variation of total  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 6: Power exchange with the grid by the CES  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 7: CES charging and discharging power pattern  power exchange that occurs with the grid by the customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  Since �N  j=1  �  c∈Cj pG  cj(t) being a positive value approxi-  mately during T1, T3, T4, T5 time intervals, it implies that the  customers tend to import certain amount of power from the  grid for satisfying their real power consumption during those  time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' On the other hand, during T2 (time period of the  day usually the SPV generation is high), the customers have a  tendency to export a portion of their surplus SPV generation  to the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' This is evident as �N  j=1  �  c∈Cj pG  cj(t) < 0 during  T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' This behavior guarantees a cost benefit for the customers  for their exported power according to equation (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 5 depicts how the customers exchange power  ) (kw)  50  ()53  0  50  100  0  5  10  15  20  2550  (kw)  0  50  100  150  200  0  5  10  15  20  25100  (kW)  50  0  CG  50  100  0  5  10  15  20  25  Time  e Duration(24 Hours100  (kW)  50  0  and  50  100  p  0  5  10  15  20  25Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 8: Temporal variation of the CES energy level  with the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' During T2, the customers export a part of  their surplus SPV generation to the CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' On the contrary,  during rest of the time periods, the customers import a certain  amount of power from the CES for satisfying their real power  consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' This action results in reducing the cost for the  customers as the amount of power imported from the grid is  minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  The Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 6 illustrates how the CES exchanges power with  the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' As the grid energy price during T1 being the lowest,  the CES tends to import power from the grid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' pCES  G  (t) >  0) during T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' This guarantees that the CES is charged with  low priced energy from the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' However, during T2, T3 and  T4, it is seen that the CES exports its power back to the grid  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' pCES  G  (t) < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' This happens as the CES provider can  maximize its revenue by exporting power back to grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 7 and 8, it is observed that during T1, the CES  charges (from the low priced grid energy) and partially dis-  charges by the end of T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' During T2, the CES continues  to charge and by the end of this time period, it reaches its  maximum energy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' The stored energy in the CES is fully  utilized during T3 and T4 for partially supplying the real  power consumption of the customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' This facilitates monetary  benefits for both the customers as the amount of expensive  power imported from the grid is lowered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' Additionally, when  observing the temporal variation of the CES energy level, it  is visualized that it is the peak value of the CES energy level  which was obtained as the optimal CES capacity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' 482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='15  kWh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' CONCLUSION & FUTURE WORK  In this work, we have explored how the optimization of  the planning and scheduling aspects of a community energy  storage (CES) can benefit both the network and the customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  To this end, we developed a multi-objective mixed-integer  quadratic optimization framework to minimize three objec-  tives: (i) network real power loss, (ii) energy trading cost of  the customers and the CES provider with the grid, and (iii)  the CES investment cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content=' The simulation results highlighted  our optimization framework is competent in acquiring the  expected merits compared with the case without a CES, and  optimization models that optimize only the scheduling of CES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  As future work, we expect to develop the work considering  a stochastic model taking into account the uncertainties of  real power consumption and SPV generation of the customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
+page_content='  Moreover, we look forward to extend the work by considering  the unbalanced nature of LV distribution networks, and reac-  tive power control capabilities of solar photovoltaic (SPV) and  CES inverters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
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+page_content='ausgrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE0T4oBgHgl3EQfdADd/content/2301.02372v1.pdf'}
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+Parallel Reasoning Network for Human-Object Interaction Detection
+Huan Peng1,2, Fenggang Liu2, Yangguang Li2, Bin Huang2, Jing Shao2, Nong Sang1, Changxin Gao1
+1Huazhong University of Science and Technology
+2SenseTime Group
+{nsang,cgao}@hust.edu.cn; [email protected];
+{penghuan,liufenggang,huangbin1,shaojing}@senseauto.com
+Abstract
+Human-Object Interaction (HOI) detection aims to learn
+how human interacts with surrounding objects. Previous
+HOI detection frameworks simultaneously detect human,
+objects and their corresponding interactions by using a
+predictor. Using only one shared predictor cannot differ-
+entiate the attentive field of instance-level prediction and
+relation-level prediction. To solve this problem, we pro-
+pose a new transformer-based method named Parallel Rea-
+soning Network(PR-Net), which constructs two indepen-
+dent predictors for instance-level localization and relation-
+level understanding. The former predictor concentrates on
+instance-level localization by perceiving instances’ extrem-
+ity regions. The latter broadens the scope of relation region
+to reach a better relation-level semantic understanding. Ex-
+tensive experiments and analysis on HICO-DET benchmark
+exhibit that our PR-Net effectively alleviated this problem.
+Our PR-Net has achieved competitive results on HICO-DET
+and V-COCO benchmarks.
+1. Introduction
+The real world contains large amounts of complex
+human-centric activities, which are mainly composed of
+various human-object interactions (HOIs). In order for ma-
+chines to better understand these complex activities, we
+need to detect all these HOIs accurately. To be specific,
+HOI detection can be defined as detecting the human-object
+pair and their corresponding interactions in an image. It
+can be divided into two sub-tasks, instance detection, and
+interaction understanding. Only if these two sub-tasks are
+completed can we build a good HOI detector.
+Previously, different methods were taken to process
+these two sub-tasks. Traditional methods like [4,11,23,28]
+first locates all instances and then extracts their correspond-
+ing features with an off-the-shelf object detector like [12,
+29]. After that, instance matching and feature fusing ap-
+proaches are used to construct human-object pairs which
+Figure 1. The attention fields for two different level predictors
+in our PR-Net. The first column shows these input images. The
+second column exhibits the attention fields of instance-level pre-
+dictor, in which the model concentrates on the extremity region of
+human and object. The third column exhibits the attention fields
+of interaction-level predictor, in which the model spreads its scope
+of attention to the relation-level region.
+are more likely to have interactive relations. These pairs are
+then sent into the intention parsing network as inputs, and
+HOI is classified and outpus, so as to obtain the humain-
+object position and corresponding interactive relation cate-
+gory. In summary, these traditional two-stage approaches
+suffer from the isolated training process of instance local-
+ization and interaction understanding, so they cannot lo-
+calize interactive human-object pairs and understand those
+complex HOI instances.
+To alleviate the above problems, multitask learning man-
+ners [5, 17, 18, 24, 30, 35, 40, 42] are proposed to com-
+plete these two sub-tasks simultaneously. Among these ap-
+proaches, they [5,18,24,35,40] process these two sub-tasks
+concurrently.
+Whereas they need an additional complex
+group composition procedure to match the predictions of
+these two sub-tasks, which reduces the computation effi-
+ciency. In addition, other one-stage methods [30, 42] pre-
+dict human-object pairs and corresponding interactions us-
+ing one shared prediction head, without needing matching
+or gathering processes. However, they accomplish instance
+1
+arXiv:2301.03510v1  [cs.CV]  9 Jan 2023
+
+localization and interaction understanding in a mixed and
+tied manner. This naive mixed prediction manner can cause
+inconsistent focus in attentive fields between the instance-
+level and the relation-level prediction. This inconsistent fo-
+cus has caused limited interaction understanding for those
+hard-negative HOIs, which leads to dissatisfactory HOI de-
+tection performance.
+To sum up, we propose a new transformer-based ap-
+proach named Parallel Reasoning Network (PR-Net) to alle-
+viate inconsistent focus of attentive fields for different level
+prediction. Specificly, two parallel predictos, instance-level
+predictor and relation-level predictor,are concluded in PR-
+Net. The former focuses on instance-level localization, and
+the latter keeps a watchful eye on relation-level semantic
+understanding. As can be seen from the two examples in
+the second columns of Figure 1, PR-Net’s attention to in-
+stances is focused on the endpoints of human skeleton and
+the particular edge regions of objects, indicating that the
+instance-level predictor can accurately locate the localiza-
+tion of human and objects by focusing on these critical ex-
+tremity regions of instances. From the two examples in the
+third column of Figure 1, it can be seen that PR-Net’s at-
+tention to relational areas is focused on the interaction con-
+tact areas between human and objects and some contextual
+areas containing helpful understanding of the interaction,
+which indicates that the relational level predictor spreads
+its vision to relation areas to better understand the subtle
+relationships between human and objects. In addition, the
+instance-level queries of our instance-level predictor strictly
+correspond to the relation-level queries of our relationship-
+level predictors. So there is no need for any instance-level
+queries between them, which greatly reduces the computa-
+tional cost [30].
+Our contribution can be concluded in the following three
+aspects:
+• We propose PR-Net, which leverages a parallel reason-
+ing architecture to effectively alleviate the problem of
+inconsistent focus in attention fields between instance-
+level and relation-level prediction. PR-Net achieves a
+better trade-off between two contradictory sub-tasks of
+HOI detection. The former needs more local informa-
+tion from the extremity region of instances, the latter is
+eager for more context information from the relation-
+level area.
+• With a decoupled prediction manner, PR-Net can de-
+tect various HOIs simultaneously without any match-
+ing or recomposition process to link the instance-level
+prediction and relation-level prediction.
+• Equipped with additional techniques, including Con-
+sistency Loss for better training and Trident-NMS for
+better post-processing, PR-Net achieves competitive
+results on both HICO-DET and V-COCO benchmark
+datasets in HOI detection.
+2. Related Works
+2.1. Two-stage Approaches in HOI Detection
+Most two-stage HOI detectors firstly detect all the hu-
+man and object instances with a modern object detection
+framework such as Faster R-CNN, Mask R-CNN [12, 29].
+After instance-level feature extraction and contextual infor-
+mation collection, these approaches pair the human and ob-
+ject instances for interaction recognition. In the process of
+interaction recognition, various contextual features are ag-
+gregated to acquire a better relation-level semantic repre-
+sentation. InteractNet [9] introduces an additional branch
+for interaction prediction, iCAN [8] captures contextual in-
+formation using attention mechanisms for interaction pre-
+diction. TIN [23] further extends HOI detection models
+with a transferable knowledge learner. In-GraphNet [37]
+presents a novel graph-based interactive reasoning model to
+infer HOIs. VSGNet [31] utilizes relative spatial reasoning
+and structual connections to analyze HOIs. IDN [22] repre-
+sents the implicit interaction in the transformation function
+space to learn a better HOI semantic. Hou proposes fabri-
+cating object representations in feature space for few-shot
+learning [16] and learning to transfer object affordance for
+HOI detection [15]. Zhang [38] proposes to merge multi-
+modal features using a graphical model to generate a more
+discriminative feature.
+2.2. One-stage Approaches in HOI Detection
+One-stage approaches directly detect Human-Object In-
+teractions without complicated coarse-to-fine bounding box
+regression [5, 17, 18, 24, 30, 35, 40, 42]. Among these ap-
+proaches, [24, 36] introduced a keypoint-style interaction
+detection method which performs inference at each interac-
+tion key point. [17] introduced a real-time method to pre-
+dict the interactions for each human-object union box. Re-
+cently, transformer-based detection approach was proposed
+to handle HOI detection as a sparse set prediction prob-
+lem [5, 30, 42]. Specifically, [30] designed a transformer
+encoder-decoder architecture to predict Human-Object In-
+teractions in an end-to-end manner directly and introduced
+additional cost terms for interaction prediction. On the other
+hand, Kim et al. [19] and Chen et al. [6] propose an in-
+teraction decoder to be used alongside the DETR instance
+decoder. It is equally important for predicting interactions
+and matching related human-object pairs. These aforemen-
+tioned one-stage approaches have enormously boosted the
+performance of Human-Object Interaction Detectors.
+2
+
+Pairwise
+Instance
+Decoder
+Instance-level 
+Queries
+Instance-level
+Feature
+Relation
+Decoder
+Relation-Level Predictor 
+Instance-level Predictor
+Relation-level 
+Queries
+Relation-level 
+Feature
+Convolutional
+Neural
+Network
+……
+Transformer
+Encoder
+…
+…
+Positional Encoding
+Input Feature
+Visual Memory
+Image Feature Extractor
+Classification Loss
+Regression Loss
+Consistency Loss
+Training
+Trident-NMS
+Testing
+Object Class
+Human Box
+Object Box
+Relation Box
+Relation class
+……
+……
+……
+……
+Figure 2. The framework of our PR-Net. It is comprised of four components:Image Feature Extractor, Pairwise Instance Predictor,
+Relation-level Predictor, Training and Post-processing Techniques.
+3. Proposed Method
+In this section, we present our Parallel Reasoning
+Network(PR-Net) for HOI detection, which is illustrated in
+the Figure 2. We can know that our PR-Net includes an
+Image Feature Extractor(CNN backbone and transformer
+encoder) and two parallel predictors (i.e., Instance-level
+Predictor and Relation-level Predictor). The two parallel
+predictors are designed to decode instance information(i.e.
+human-box, object-box, object-class) and relation informa-
+tion(i.e. relation-box, relation-class) respectively. Next, we
+introduce the proposed instance-level and relation-level loss
+functions to learn the location of instances and the interac-
+tions within each human-object pair. At last, we introduce
+the proposed Trident-NMS which is utilized to filter those
+duplicated HOI predictions effectively.
+3.1. Image Feature Extractor
+The overall Image Feature Extractor architecture con-
+sists of a standard CNN backbone fc and transformer en-
+coder fe. The conventional CNN backbone is used to pro-
+cess the input image xϵR3×H×W to a global context feature
+map zϵRc×H′×W ′, in which typically images are down-
+sampled to (H′, W ′) spatial shape with a dimension of c.
+Then, the global context feature map is serialized as to-
+kens, in which the spatial dimensions of the feature map
+are collapsed into one dimension, resulting in H′ × W ′
+tokens.
+Then, the tokens are linearly mapped to T =
+{ti|tiϵRc′}Nq
+i=1, where Nq = H′ × W ′. Afterward, these
+tokens are shaped as a sequence to feed into the transformer
+encoder.
+For the transformer encoder, each encoder layer fol-
+lows standard architecture of transformer, which con-
+sists of a multi-head self-attention module and a feed
+forward network (FFN). Additional position embedding
+qeϵRc′×H′×W ′ is also added to the serialized token to
+supplement the positional information.
+With the mech-
+anism of self-attention, the encoder can map the former
+global context feature map from CNN to richer contex-
+tual information. Finally, the set of encoded image fea-
+tures {di|diϵRc′}Nq
+i=1 can be formulated as visual memory
+E = fe(T, qe). The visual memory E contains richer con-
+textual information.
+3.2. Instance-level Predictor
+The Instance-level Predictor includes a standard trans-
+former decoder fip with just three layers.
+The decoder
+response for above visual memory E, according to a set
+of learnable instance query vectors Qp = {qi|qiϵRc′}Nq
+i=1
+which is added with position embedding plϵRc′×H′×W ′.
+The instance-level queries vectors are trained to learn a
+more precise location of instances, which focuses more
+on those local information about location of instances.
+The independent predictors are composed of three feed-
+forward networks (FFNs), including human-bounding-box
+FFN φhb, object-bounding-box FFN φob, and object-class
+FFN φoc, each of which response for decoding instance fea-
+ture to human-box ˆbh, object-box ˆbo and object-class ˆco re-
+spectively. The formulation can be denoted as:
+ˆbh = φhb(fip(Qp, pl, E)),
+ˆbo = φob(fip(Qp, pl, E)),
+ˆco = φoc(fip(Qp, pl, E)).
+(1)
+3.3. Relation-level Predictor
+We decouple the relation problems from HOI and use a
+Relation-level Predictor to reason relationships from larger-
+scale semantics. We propose a relation box to guide the
+predictor to percept the human-object relationship in the
+3
+
+human blow cakeRelation-level Predictor.
+The Relation-level Predictor consists of a standard trans-
+former decoder frd and two independent predictors(FFNs).
+Another relation-level queries Qr and position embedding
+pr are randomly initialed and fed into the Relation-level
+Predictor. One of the predictors φub predicts relation boxes
+ˆbu, the other predictor φac decodes the relation class in-
+formation ˆca. The relation boxes ˆbu and the relation class
+information ˆca can be formulated as Eq. 2.
+ˆbu = φub(fdr(Qr, pr, E)),
+ˆca = φac(fdr(Qr, pr, E)).
+(2)
+Attributed to the relation boxes, the decoder of Interaction-
+level Predictor is guided to enlarge the receptive field (as
+shown in Figure 1). The relation queries Qr can pay at-
+tention to the entire area where human and object interact.
+Thus, the predictor φac can predict a more accurate relation
+class.
+In addiction, to match the relation class information ˆca
+with the aforementioned human-box ˆbh, object-box ˆbo and
+object-class ˆco from the Instance-level Predictor, we ditch
+the complex matching method like HO pointer in HOTR.
+Instead, we just match the relation class information ˆca
+and the instances information ˆbh etc. one by one in order.
+Specifically, for a pair of instances {ˆbh
+i ,ˆbo
+, ˆco
+i , iϵNq}, ˆca
+i is
+the corresponding relation class. In this way, the instance-
+level query vectors Qp and the relation-level query vectors
+Qr represent the same human-object interaction, but have
+the ability to focus on different receptive field.
+3.4. Loss Functions
+The overall loss functions consist of the instance-level
+loss and relation-level loss, applied to Instance-level Predic-
+tor and Relation-level Predictor, respectively. The instance-
+level loss supervises the Instance-level Predictor to pre-
+dict instance-level target, i.e., human-box, object-box, and
+object-class. The relation-level loss assists the Relation-
+level Predictor to predict relation-class and relation-box
+from the larger receptive field.
+3.4.1
+The Instance-level loss function
+LIL supervises the instance information, including human-
+box ˆbh, object-box ˆbo and object-class ˆco. The instance-
+level loss function consists of human-box regression Lhr,
+object-box regression Lor and object-class classification
+Loc. Lhr and Lor are standard bounding-box regression
+loss, i.e. L1 loss, to locate the position of human and ob-
+ject. Loc is a classification loss to classify the categories of
+the object. The loss functions can be defined as Eq. 5.
+Lhr = 1
+N
+N
+�
+i
+||ˆbh
+i − bh
+i ||,
+Lor = 1
+N
+N
+�
+i
+||ˆbo
+i − bo
+i ||,
+Loc = 1
+N
+N
+�
+i
+CE(ˆco
+i , co
+i ),
+(3)
+where CE is cross entropy loss, co
+i is the ground truth of
+object class.
+The instance-level loss function LIL can be defined as:
+LIL = Whr ∗ Lhr + Wor ∗ Lor + Woc ∗ Loc.
+(4)
+3.4.2
+The Relation-level loss function
+LRL supervises the relationship information, i.e., the rela-
+tion class ˆca, primarily. In addition, auxiliary relation boxes
+are also supervised to pay attention to the entire area where
+the interaction happens. Thus, the Relation-level loss func-
+tion consists of relation-box regression Lur, relation-box
+consistency loss Luc and relation-class loss Lac. The Lac is
+a classification loss to classify the categories of the interac-
+tion. The relation-box regression loss function Lur is a L1
+loss to resemble the predicted relation boxes and its ground-
+ing truth. The grounding truth of relation boxes is the outer
+bounding box of human and object boxes. The relation-box
+regression loss function helps the Relation-level Predictor
+to be aware of the relation feature of human and object. The
+consistency loss Luc are used to keep the consistency of ˆbh,
+ˆbo and ˆbu. Specifically, a pseudo relation box ˆbho is gener-
+ated by taking the outer bounding box of ˆbh and ˆbo. Then,
+an L1 loss resemble ˆbu and ˆbho. With the relation box, the
+relation-class loss can supervise better relation semantics.
+Lur = 1
+N
+N
+�
+i
+||ˆbu
+i − bu
+i ||,
+Luc = 1
+N
+N
+�
+i
+||ˆbu
+i − ˆbho
+i ||,
+Lac = 1
+N
+N
+�
+i
+SigmoidCE(ˆca
+i , ca
+i ).
+(5)
+The relation-level loss function LRL can be defined as:
+LRL = Wur ∗ Lur + Wuc ∗ Luc + Wac ∗ Lac.
+(6)
+In all, the overall loss fucntion L can be denoted as:
+L = LIL + LRL.
+(7)
+4
+
+3.5. Inference for HOI Detection
+The inference process of our PR-Net can be divided into
+two parts: the calculation of the HOI predictions and the
+Trident-NMS post-processing technique.
+HOI Prediction To acquire the final HOI detection results,
+we need to predict human bounding box, object bounding
+box, and object class using both instance-level predictions
+and relation class and relation box using relation-level pre-
+diction. Based on the above predictions, we can calculate
+the final HOI prediction score as below:
+shoi
+i
+= {maxkso
+i (k)} ∗ srel
+i
+(8)
+Where maxkso
+i (k) means the most probable class score of
+the i-th output object from instance-level predictor; srel
+i
+means the multi-class scores of the i-th output interaction
+from relation-level predictor. Note that each human-object
+pair can only have one object with certain class, but there
+maybe exist multiple human-object interactions within one
+pair.
+Trident-NMS For each predicted HOI class in one image,
+we choose to filter its duplicated predictions according to
+the above calculated HOI prediction scores with our pro-
+posed Trident Non Maximal Suppression(Trident-NMS). In
+detail, if the TriIoU(i, j) between the i-th and the j-th HOI
+prediction is higher than the threshold Thresnms, we will
+filter the prediction which has a lower HOI score. And the
+calculation of TriIoU(i, j) is as below:
+TriIoU(i, j) =IoU(bh
+i , bh
+j )Wh
+× IoU(bo
+i , bo
+j)Wo
+× IoU(brel
+i , brel
+j )Wrel
+(9)
+Where IoU(bh
+i , bh
+j ), IoU(bo
+i , bo
+j), IoU(brel
+i , brel
+j ) repre-
+sent the Interaction over Union between the i-th and the j-
+th human boxes, object boxes and relation boxes; Wh, Wo,
+Wrel represent the weights of Human IoU, Object IoU and
+Relation IoU.
+4. Experiment
+4.1. Datasets and Evaluation Metrics
+We evaluate our method on two large-scale benchmarks,
+including V-COCO [10] and HICO-DET [3] datasets. V-
+COCO includes 10,346 images, which contains 16,199 hu-
+man instances in total and provides 26 common verb cate-
+gories. HICO-DET contains 47,776 images, where 80 ob-
+ject categories and 117 verb categories compose of 600 HOI
+categories. There are three different HOI category sets in
+HICO-DET, which are: (a) all 600 HOI categories (Full),
+(b) 138 HOI categories with less than 10 training instances
+(Rare), and (c) 462 HOI categories with 10 or more training
+instances (Non-Rare). Following the standard protocols, we
+use mean average precision (mAP) in HICO-DET [4] and
+role average precision (AProle) in V-COCO [10] to report
+evaluation results.
+4.2. Implementation Details
+We use ResNet-50 and ResNet-101 [13] as a backbone
+feature extractor.
+The transformer encoder consist of 6
+transformer layers with multi-head attention of 8 heads.
+The number of transformer layers in Instance-level Predic-
+tor and Interaction-level Predictor is both set to be 3. The
+reduced dimension size of visual memory is set to 256. The
+number of instance-level and relation-level queries is set
+to 100 for both HICO-Det and V-COCO benchmark. The
+human, object and relation box FFNs both have 3 linear
+layers with ReLU, while the object and relation category
+FFNs have one linear layer. During training, we initial-
+ize the network with the parameters of DETR [2] trained
+on the MS-COCO dataset. We set the weight coefficients
+of bounding box regression, Generalized IoU, object class,
+relation class and consistency loss to 2.5, 1, 1, 1 and 0.5,
+respectively, which follows QPIC [30]. We optimize the
+network by AdamW [26] with the weight decay 10−4. We
+train the model for 150 epochs with a learning rate of 10−5
+for the backbone and 10−4 for the other parts decreased by
+10 times at the 100th and the 130th epoch respectively. All
+experiments are conducted on the 8 Tesla A100 GPUs and
+CUDA11.2, with a batch size of 16.
+We select 100 detection results with the highest scores
+for validation and then adopt Trident-NMS to filter results
+further.
+4.3. Overall Performance
+We summarize the performance comparisons in this sub-
+section.
+Performance on HICO-DET. Table 1 shows the per-
+formance comparison on HICO-DET. Firstly, the detection
+results of our PR-Net are the best among all approaches un-
+der the Full and Non-Rare settings, demonstrating that our
+method is more competitive than the others in detecting the
+most common HOIs. It is noted that PR-Net is also pre-
+eminent in detecting rare HOIs (HOI categories with less
+than 10 training instances), because our parallel reasoning
+network can migrate the non-rare knowledge into a rare do-
+main. Besides, our PR-Net obtains 32.86 mAP on HICO-
+DET (Default Full), which achieves a relative gain of 9.8%
+compared with the baseline. These results quantitatively
+show the efficacy of our method.
+Performance on V-COCO. Comparison results on V-
+COCO in terms of mAProle are shown in Table 2. It can
+be seen that our proposed PR-Net has a mAP(%) of 62.4,
+obtaining the best performance among all approaches. Al-
+though we do not adopt previous region-based feature learn-
+ing (e.g., RPNN [41], Contextual Att [34]), or employ ad-
+5
+
+Table 1. Results on HICO-DET [4]. “COCO” is the COCO pre-
+trained detector, “HICO-DET” means that the detector is further
+fine-tuned on HICO-DET.
+Default Full
+Method
+Detector
+Backbone
+Full
+Rare Non-Rare
+CNN-based
+VCL [14]
+COCO
+ResNet-50
+19.43 16.55
+20.29
+VSGNet [31]
+COCO
+ResNet-152
+19.80 16.05
+20.91
+DJ-RN [21]
+COCO
+ResNet-50
+21.34 18.53
+22.18
+PPDM [24]
+HICO-DET Hourglass-104 21.73 13.78
+24.10
+Bansal et al. [1]
+HICO-DET
+ResNet-101
+21.96 16.43
+23.62
+TIN [23]DRG
+HICO-DET
+ResNet-50
+23.17 15.02
+25.61
+VCL [14]
+HICO-DET
+ResNet-50
+23.63 17.21
+25.55
+GG-Net [40]
+HICO-DET Hourglass-104 23.47 16.48
+25.60
+IDNDRG [22]
+HICO-DET
+ResNet-50
+26.29 22.61
+27.39
+Transformer-based
+HOI-Trans [42]
+HICO-DET
+ResNet-50
+23.46 16.91
+25.41
+HOTR [18]
+HICO-DET
+ResNet-50
+25.10 17.34
+27.42
+AS-Net [5]
+HICO-DET
+ResNet-50
+28.87 24.25
+30.25
+QPIC [30]
+HICO-DET
+ResNet-50
+29.07 21.85
+31.23
+PR-Net (Ours)
+HICO-DET
+ResNet-50
+31.17 25.66
+32.82
+PR-Net (Ours)
+HICO-DET
+ResNet-101
+32.86 28.03
+34.30
+ditional human pose (e.g., PMFNet [32], TIN [23]), our
+method outperforms these approaches with sizable gains.
+Besides, our method achieves an absolute gain of 3.6 points,
+a relative improvement of 6.1% compared with the baseline,
+validating its efficacy in the HOI detection task.
+Table 2. Performance comparison on V-COCO dataset.
+Method
+Backbone Network
+APS1
+role
+APS2
+role
+CNN-based
+VSGNet [31]
+ResNet-152
+51.8
+57.0
+PMFNet [32]
+ResNet-50-FPN
+52.0
+-
+PD-Net [39]
+ResNet-152-FPN
+52.6
+-
+CHGNet [33]
+ResNet-50-FPN
+52.7
+-
+FCMNet [25]
+ResNet-50
+53.1
+-
+ACP [20]
+ResNet-152
+53.23
+-
+IDN [22]
+ResNet-50
+53.3
+60.3
+GG-Net [40]
+Hourglass-104
+54.7
+-
+DIRV [7]
+EfficientDet-d3
+56.1
+-
+Transformer-based
+HOI-Trans [42]
+ResNet-101
+52.9
+-
+AS-Net [5]
+ResNet-50
+53.9
+-
+HOTR [18]
+ResNet-50
+55.2
+64.4
+QPIC [30]
+ResNet-50
+58.8
+61.0
+PR-Net (Ours)
+ResNet-50
+61.4
+62.5
+PR-Net (Ours)
+ResNet-101
+62.9
+64.2
+4.4. Ablation Analysis
+To evaluate the contribution of different components
+in our PR-Net, we first conduct a comprehensive ablation
+analysis on the HICO-DET dataset. Next, we analyze the
+impact of the number of different-level predictors. At last,
+we analyze the effects of different post-processing manners.
+Contribution of different components.
+Compared
+Table 3. Ablation analysis of the proposed PR-Net with the back-
+bone of ResNet-101 on HICO-DET test set. Parallel Predictor
+means we parallelly predict instance-level locations and relation-
+level semantics. Consistency Loss means we constrain the union
+box of the human-object pair and the relation box to be consis-
+tent. Trident-NMS means duplicate filtering through human, ob-
+ject, and relation bounding boxes.
+Parallel Predictor
+Consistency Loss
+Trident-NMS
+HICO-DET
+Full
+Rare
+NonRare
+-
+-
+-
+29.90
+23.92
+31.69
+✓
+-
+-
+31.62
+25.43
+33.47
+✓
+✓
+-
+31.87
+27.59
+33.14
+✓
+✓
+✓
+32.86
+28.03
+34.30
+with our baseline [30], the performance improvements of
+our PR-Net are from three components: Parallel Predictor,
+Consistency Loss, and Trident-NMS. From Table 3, we can
+know the contribution of different components.
+Among
+these components, Parallel Predictor is our core approach.
+With that, we can observe a noticeable gain of mAP in
+HICO-DET by 1.72. It proves that the parallel reasoning
+structure can significantly improve instance localization
+and interaction understanding for an HOI detection model.
+Additionally, we design a consistency loss between the
+union box of the human-object pair and the relation box,
+which can contribute about 0.25 mAP gain in the HICO-
+DET test set. It shows that it is meaningful and helpful
+to constrain the union region of instance-level predictions
+and the relation region of relation-level predictions.
+At
+last, we design a more effective post-processing technique
+named Trident-NMS, which brings about 1.0 mAP gain in
+the HICO-DET test set. It reveals that the set-prediction
+method can also benefit from duplicate filtering technique
+and post-processing technique like NMS is essential for
+HOI detection.
+Impacts of different numbers of parallel predictors.
+In our PR-Net, two parallel predictors are significant for
+HOI detection, and we detailedly analyze the impact of
+different numbers of parallel predictors. From Table 4, we
+can know that equipped with three layers of instance-level
+predictor and relation-level predictor, our PR-Net can
+acquire the best mAP performance in the HICO-DET test
+set. It reveals that our PR-Net can significantly outperform
+the baseline QPIC [30] without additional computational
+cost.
+Interestingly, we can also observe that even with
+only one layer of parallel predictors, our PR-Net can also
+outperform the baseline equipped with a six-layer predictor.
+Effects of different implements of Trident-NMS. In
+Table 5, we analyze the effects of different implements
+of Trident-NMS. We find that Product-based Trident-
+NMS performs better than Sum-based Trident-NMS.
+6
+
+Table 4. Ablation analysis of the number of instance-level predic-
+tor Ndec and the number of relation-level predictor Nreldec.
+approaches
+Backbone Ndec Nreldec Full Rare Non-Rare
+QPIC(Baseline) [30] ResNet50
+6
+-
+29.07 21.85
+31.23
+PR-Net(Ours)
+ResNet50
+1
+1
+29.64 24.18
+31.27
+PR-Net(Ours)
+ResNet50
+3
+3
+31.17 25.66
+32.82
+PR-Net(Ours)
+ResNet50
+6
+6
+31.04 24.87
+32.89
+QPIC(Baseline) [30] ResNet101
+6
+-
+29.90 23.92
+31.69
+PR-Net(Ours)
+ResNet101
+1
+1
+30.26 23.27
+32.34
+PR-Net(Ours)
+ResNet101
+3
+3
+32.86 28.03
+34.30
+PR-Net(Ours)
+ResNet101
+6
+6
+32.52 27.04
+34.16
+Additionally, we can also observe that when the weight
+of Human-IoU in TriIoU increases, the HOI detection
+performance will be better. This reveals that human box
+duplication is more frequent than that of object box or
+relation box. In summary, with either the Product-based
+or Sum-based TriIoU calculation, we should pay more
+attention to the non-maximal suppression of the human box.
+Table 5. Ablation analysis of the Trident-NMS module on HICO-
+DET test set. Product means we calculate TriIoU by multiply-
+ing these weighted Human-IoU, Object-IoU, and Relation-IoU.
+Sum means we calculate TriIoU by adding all these weighted
+Human-IoU, Object-IoU, and Relation-IoU. Wh, Wo, Wrel rep-
+resent the weights of Human-IoU, Object-IoU and Relation-IoU
+respectively. Thresnms means the threshold of non-maximum
+suppression.
+Product
+Sum
+Wh
+Wo
+Wrel
+Thresnms
+HICO-DET
+Full
+Rare
+NonRare
+-
+-
+-
+-
+-
+-
+31.87
+27.59
+33.14
+-
+✓
+0.33
+0.33
+0.33
+0.5
+30.61
+27.00
+31.69
+-
+✓
+0.33
+0.33
+0.33
+0.7
+32.53
+27.88
+33.91
+-
+✓
+0.4
+0.4
+0.2
+0.7
+32.63
+27.96
+34.02
+-
+✓
+0.5
+0.4
+0.1
+0.7
+32.66
+27.91
+34.00
+-
+✓
+0.6
+0.3
+0.1
+0.7
+32.56
+27.70
+34.01
+✓
+-
+1.0
+1.0
+1.0
+0.5
+32.77
+27.98
+34.20
+✓
+-
+1.0
+1.0
+0.5
+0.5
+32.81
+28.02
+34.25
+✓
+-
+0.5
+0.5
+0.5
+0.5
+32.61
+27.65
+34.08
+✓
+-
+0.5
+1.0
+0.5
+0.5
+32.61
+27.67
+34.09
+✓
+-
+1.0
+0.5
+0.5
+0.5
+32.86
+28.03
+34.30
+4.5. Visualization of features
+Using the t-SNE visualization technique [27], we visual-
+ize 20000 samples of output feature. These object and inter-
+action features are extracted from the last layer of Instance-
+level Predictor and Relation-level Predictor in our PR-Net,
+respectively. From the Figure 3, we can observe that our
+PR-Net can obviously distinguish different class of objects
+and interactions. Interestingly, from this visualization of
+features, our PR-Net can even learn better the complex
+interaction representations then the object representations
+which benefits from our advantageous parallel reasoning ar-
+chitecture.
+Figure 3. Visualization of object features and relation features on
+HICO-DET dataset via t-SNE technique. Left is object features
+and right is relation features.
+4.6. Qualitative Examples
+From Figure 4, we can observe that our PR-Net can accu-
+rately detect both human box, object box, and relation box
+as well as their corresponding interactions. From the first
+row and second column of Figure 4, we can know that our
+PR-Net can precisely distinguish which man is riding the
+horse in the image. From the second row and third column
+of Figure 4, our PR-Net can precisely detect those subtle
+and indiscernible HOIs. In summary, our PR-Net can cor-
+rectly detect those complex and hard HOIs.
+Figure 4. Visualization of some HOI detection examples (Top 1
+result) detected by the proposed Parallel Reasoning Network on
+the HICO-DET test set.
+5. Conclusion
+In this paper, we propose a new Human-Object Inter-
+action Detector named Parallel Reasoning Network(PR-
+Net), which consists of an instance-level predictor and
+a relation-level predictor, to alleviate the problem of in-
+consistent focus in attentive fields between instance-level
+and interaction-level predictions.
+In addition, our PR-
+Net achieves a better trade-off between instance localiza-
+tion and interaction understanding. Furthermore, equipped
+with Consistency Loss and Trident-NMS, our PR-Net has
+achieved competitive results on two main HOI benchmarks,
+validating its efficacy in detecting Human-Object Interac-
+tions.
+7
+
+100
+100
+75
+75
+50
+50
+25
+25
+0
+0
+-25
+-25
+-50
+-50
+-75
+-75
+-100
+-100
+-100
+-75
+-50
+-25
+0
+25
+50
+75
+100
+-100
+-75
+-50
+-25
+0
+25
+50
+75
+100uman
+human lie_on chair
+ridehorse
+LOG
+numan sit on benchReferences
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+

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+UDK 539.125.17; 539.126.17 
+Ya. D. Krivenko-Emetov* 
+Institute for Nuclear Research, National Academy of Sciences of Ukraine, Kyiv, Ukraine 
+National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute», Kyiv, Ukraine 
+*Corresponding author: [email protected]; [email protected] 
+MULTICOMPONENT VAN DER WAALS MODEL OF A NUCLEAR FIREBALL IN THE 
+FREEZE-OUT STAGE 
+ 
+Abstract. A two-component van der Waals gas model is proposed to describe the hadronic stages of the 
+evolution of a nuclear fireball in the cooling stage. At the first stage of hadronization, when mesons 
+dominate, a two-component meson model (
+0
+ - and 
+
+ -mesons) with an effective two-particle interaction 
+potential of a rectangular well is proposed. At the late-stage hadronization, when almost all mesons have 
+decayed, a two-component nucleon model of protons and neutrons is proposed with the corresponding 
+effective rectangular nucleon potential. The saddle point method has been applied for analytical calculations 
+of the partition function. This made it possible to uniformly obtain analytical expressions for both the 
+pressure and density, taking into account the finite dimensions of the system, and the analytical expressions 
+for chemical potentials. It is assumed that the proposed models and derived formulas can be used to analyze 
+experimental data connected to the quantitative characteristics of the particle yields of different types in the 
+final state from the hadronic stages of the evolution of a nuclear fireball, as well as to determine the critical 
+parameters of the system in high-energy nucleus-nucleus collisions. 
+Keywords: fireball, freeze-out, van der Waals equation, effective nuclear capability, Grand Canonical 
+Ensemble, pressure fluctuation, quark-gluon plasma, experimental data 
+ 
+Introduction 
+ 
+Experimental observations of an elliptical flow in non-central collisions of heavy nuclei at high 
+energies provide much evidence that in these collisions of nuclei a state of quark-gluon plasma 
+appears and thermalization occurs, which is associated with the fact that particles collide with each 
+other more than once. For this state, one can introduce the concept of temperature, viscosity, 
+density, and other thermodynamic quantities that characterize the substance. In these terms, one can 
+describe and study the phenomena that occur during the cooling of a hadron gas formed after a 
+phase transition from the state of a quark-gluon plasma. It is believed that at a critical temperature 
+(
+150
+
+T
+ MeV, the so-called Hagedorn temperature), hadrons "melt" and a phase transition of the 
+hadron gas (hadron matter) into the quark-gluon phase occurs. Therefore, in recent decades, 
+statistical models of hadron gas have been actively used to describe the data of the Large Hadron 
+Collider (LHC), the Relativistic Heavy Ion Collider (RHIC), and even earlier to describe the data of 
+
+Alternating Gradient Synchrotron (AGS) and Super proton synchrotron (SPS), on the particle yields 
+in a relativistic nuclear-nuclear (
+A
+A 
+) collision at high energies [1, 2]. The van der Waals (vdW) 
+model, taking into account hadron-hadron interactions at short distances, is especially useful in this 
+description [3-10]. This is due to the fact that taking into account the effect of repulsion (off 
+volumes) leads to the prevention of an undesirably high density at high temperatures, which appears 
+in ideal gas models [11]. In addition, collisions of heavy high-energy ions in the LHC produce a 
+large number of different types of particles. The number of these particles is not fixed. Therefore, 
+the formalism of the Grand Canonical Ensemble (GCE) is one of the adequate mathematical 
+formalisms for these phenomena. In this case, the thermodynamic quantities do not depend on the 
+number of particles, but on the chemical potentials. For many years, researchers have proposed and 
+applied different versions of vdW models. These models have been mainly used to describe 
+experimental data on the number of particles at high energies, when tens or even hundreds of 
+hadrons of different types can be generated. Naturally, this generation process is limited only by the 
+energy of collisions. 
+Among these models, the model proposed in [11] should be noted. In this model, the 
+phenomenological parameters of the radii of the hard-core 
+ii
+R  and 
+ij
+R  are introduced, which 
+significantly changes the number of the yield particles with different types 
+i
+N  (i  is the particle 
+sort) and is mainly confirmed by experimental data. In order to describe more subtle effects in the 
+dependence of the hadronic gas pressure on density, various authors (e.g. [12, 13]) proposed the 
+development of this model [11]. Here, the effects of attraction between hadrons at a large distance 
+have been taken into account, which leads to the appearance of a corresponding contribution to the 
+pressure as 
+2
+an
+Pattr
+
+~
+ ( n  is the density). For a multicomponent gas, the parameter a , 
+corresponding to attraction, transforms into parameters 
+ij
+a , and the repulsive parameter b  
+transforms into parameters 
+ij
+b . At the same time, the parameters of the effective potential 
+corresponding to attraction and repulsion depend on the effective radii of repulsion 
+0
+iR  and 
+attraction 
+iR  as follows: 
+
+
+ij
+ij
+ij
+ij
+b
+c
+u
+a
+
+0
+~
+ , 
+
+
+3
+0
+0
+3
+2
+j
+i
+ij
+R
+R
+b
+
+
+
+, 
+
+3
+3
+2
+j
+i
+ij
+R
+R
+c
+
+
+
+, 
+ij
+u0  is the depth 
+of the effective potential well [12]. 
+However, even this vdW model cannot be developed properly when considering a finite nuclear 
+system. So, in the case of nuclear collisions, a nuclear fireball with dimensions 
+10
+7 
+
+
+~
+r
+ Fm is 
+observed. In a fairly general case, this problem (without taking into account the effects of reflection 
+from the system wall) has been solved for a two-component system. In this case, the GCE 
+formalism leads to the use of a double sum, which, in turn, can be transformed into a 
+
+multidimensional integral, which can be integrated by the saddle point method in the vicinity of a 
+saddle point with coordinates 
+
+
+2
+1 N
+N ,
+ [12]. 
+Of course, it would be good to apply this model to the analysis of experimental data obtained for 
+collisions of heavy nuclei at CERN. One of the variants of such a model was presented in [14] in a 
+concise form. It was believed that the collision energies were not high enough, and one could limit 
+oneself to only two varieties: protons and neutrons. It was assumed that the characteristic 
+temperatures do not exceed the temperatures at which new particles can be generated. The 
+temperatures of the nuclear fireball are of the order of 
+140
+135 
+~
+T
+ MeV in units 
+1
+
+B
+k
+ (freeze-
+out phase, see Fig. 1) at the stage after the transition to the hadronic gas phase. Therefore, the model 
+itself should have a transparent nonrelativistic limit, taking into account the law of conservation of 
+the total number of nucleons without the generation of new particles. 
+The successive stages of the evolution of a nuclear fireball are schematically shown in Fig. 1 
+[15]. From left to right: two touching ultrarelativistic nuclei; the state of a hot and superdense 
+nuclear system; quark-gluon phase; hadronization and chemical freeze-out; kinetic freeze-out. 
+ 
+ 
+Fig. 1. The successive stages of the evolution of a nuclear fireball 
+ 
+A more detailed and consistent description of the mathematical apparatus of the model [14] is 
+proposed in the article. Some more subtle effects are estimated (additional corrections for pressure, 
+density, and root-mean-square (RMS) fluctuations). A new two-component meson model [16] has 
+been proposed for the case of temperatures above the production threshold (
+135
+
+T
+ MeV ), when 
+the number of mesons is not conserved. 
+ 
+1 One-component vdW gas 
+ 
+According to various estimates, the lifetime of a nuclear fireball is much longer than the 
+characteristic nuclear interaction time 
+23
+22
+10
+10
+
+
+
+~
+'t
+ c. (see Fig. 1). It is of the order of the 
+relaxation time 
+23
+21
+10
+10
+
+
+
+ ~
+ c for sufficiently small local fireball volumes (subsystem). 
+
+0
+10.01
+1
+10
+1100
+t (fm/c)Therefore, we will assume that at each moment of time exceeding the relaxation time, a local 
+statistical equilibrium has time to be established in the subsystem. That is, such a local fireball 
+region is quasi-stationary, and therefore the methods of statistical physics can be applied to it. Since 
+all thermodynamic potentials, as well as entropy and volume, are additive (extensive) quantities, 
+therefore, the corresponding potentials (values) of the entire system (fireball) can be defined as the 
+sum of the corresponding thermodynamic potentials of quasi-closed subsystems [17]. Then, at each 
+moment of time, one can give a standard representation of the partition function of a rarefied  
+quasi-ideal van der Waals gas in the canonical ensemble (CE) for such quasi-closed subsystems. In 
+the approximation of pair interaction and condition  
+1
+
+V
+N
+T
+B
+, this quantity has the form [17]: 
+
+
+
+
+ 
+
+N
+N
+N
+T
+B
+V
+m
+T
+N
+N
+T
+V
+Z
+
+
+
+,
+!
+,
+,
+1
+, 
+ 
+ 
+ 
+(1) 
+where, respectively, N  and m  are the number and mass of particles, V  and T  are the volume and 
+temperature of the gas. 
+Formula (1) uses the notation [11]: 
+
+
+
+
+T
+m
+K
+T
+m
+dp
+T
+p
+m
+p
+m
+T
+2
+2
+2
+0
+2
+2
+2
+2
+2
+ 
+exp
+2
+1
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+,
+, 
+ 
+(2) 
+where 
+ z
+K2
+ is the modified Bessel function, and the second virial coefficient in (1) has the form: 
+ 
+
+
+
+ dV
+T
+U
+T
+B
+
+
+
+
+
+0
+exp
+1
+2
+1
+  
+ 
+ 
+ 
+(3) 
+and includes pairwise interaction of particles, 
+ 
+
+j
+i
+ij
+U
+U
+. 
+In relativistic limit 
+T
+m 
+ one can easy obtain, given the asymptotes of the Bessel function: 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+T
+m
+mT
+m
+T
+exp
+2
+2
+3
+~
+,
+. 
+This formula further leads to the effect of exponential suppression of the particle yields with 
+large mass, which is important in the study of quark-gluon plasma. 
+The pressure in the system is easy to find from the partition function: 
+
+
+
+
+
+
+ N
+T
+B
+V
+TN
+N
+T
+V
+Z
+V
+T
+N
+T
+V
+
+
+
+
+
+,
+,
+,
+,
+ln
+P
+.                                     (4) 
+Note that if the Stirling formula is used in the partition function for the factorial: 
+
+N
+e
+N
+N
+N
+
+
+2
+!
+, then the final pressure formula (4) will not change. 
+*THE MODEL. According to the above, all calculations for subsystems will be carried out by 
+methods of statistical physics. This implies, in addition to the local statistical equilibrium, the 
+
+fulfillment of the condition of the statistical (thermodynamic) boundary: 
+A
+N
+N 
+, where 
+A
+N  is the 
+Avogadro constant.  
+In this case, the final formulas can be applied to the nuclear fireball due to the indicated 
+additivity of thermodynamic potentials and volume. Since the number of generated particles in a 
+fireball is about 4-6 thousand during high-energy nucleus-nucleus interactions, this assumption is 
+more or less justified at the first stages of its evolution.  
+Of course, at later stages of evolution, since the number of nucleons at the nonrelativistic 
+boundary is limited by the baryon number conservation law and equal to ( heavy element nuclei 
+collide with mass number ), this assumption is, in general, somewhat doubtful. 
+Of course, at later stages of evolution, this assumption is, in general, somewhat doubtful, since 
+the number of nucleons n the nonrelativistic limit is confined by the baryon number conservation 
+law and is equal to 
+300
+200 
+
+N
+ (the nuclei of heavy elements collide with the mass number 
+200
+~
+A
+). However, the practical application of the van der Waals equation quite often goes 
+beyond the conditions under which the virial approximation has been obtained as experience 
+shows. Considering this fact, as well as the fact that we can always restrict ourselves to the first 
+stage (see Section 3), we believe that our approximation is sufficiently justified. 
+Although calculations by the saddle point method are made when 
+ 
+0
+
+T
+B
+, however, for the 
+reasons stated above, the final formulas are extended to region where the second virial coefficient 
+ 
+T
+B
+ is not necessarily negative.  
+From the partition function 
+
+
+N
+T
+V
+Z
+,
+,
+ one can also get: free energy 
+
+ 
+N
+T
+V
+F
+,
+,
+ 
+
+
+
+
+N
+T
+V
+Z
+T
+,
+,
+ln
+
+
+, chemical potential 
+
+
+
+
+
+
+
+
+ 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+V
+N
+T
+B
+T,m
+V
+N
+T
+N
+N
+T
+V
+F
+2
+ln
+ln
+,
+,
+ 
+ 
+(5) 
+and the derivative of the chemical potential which in the statistical limit has the form: 
+
+
+
+
+
+ 
+ 
+V
+T
+T
+B
+V
+T
+T
+B
+N
+T
+N
+V
+V
+P
+N
+2
+2
+lim
+A
+N
+N
+2
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+. 
+ 
+(6) 
+Then, we obtain the Grand partition function (GPF) 
+
+
+
+,
+,T
+V
+Z
+ from the partition function 
+
+
+N
+T
+V
+Z
+,
+,
+ taking into account the above physical considerations (see, e.g., [18, 19]): 
+
+
+
+
+N
+T
+V
+Z
+T
+N
+T
+V
+N
+,
+,
+,
+,
+
+
+
+
+
+ 
+
+
+exp
+Z
+. 
+ 
+ 
+ 
+ 
+(7) 
+At high temperatures (which, for example, are realized during collisions of heavy ions in the 
+GCE, and 
+'
+dN
+T
+N
+
+
+) one can turn from the sum to the integral using the Euler-Maclaurin 
+
+formula. In this case, the first integral term remains and the logarithm of the statistical sum is 
+introduced into the exponent. Let's denote this indicator by 
+
+'
+N
+
+: 
+
+
+ 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+0
+0
+exp
+ln
+exp
+'
+'
+'
+,
+'
+'
+,
+,
+N
+dN
+T
+T
+N
+V
+Z
+N
+dN
+T
+T
+V
+Z
+.  
+ 
+(8) 
+Further integration is performed by the saddle point method [16], since at high temperatures the 
+integrand has a strongly pronounced maximum. We obtain the following expression for finding the 
+maximum point (
+
+N ) for the integrand from the extremum condition imposed on the saddle point: 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+N
+N
+N
+N
+N
+N
+N
+T
+V
+Z
+N
+T
+N
+,
+,
+ln
+,  
+ 
+ 
+ 
+(9) 
+
+
+
+
+
+
+
+
+
+
+m
+T
+V
+N
+T
+N
+,
+
+
+
+
+
+
+
+ln
+ln
+, 
+ 
+ 
+ 
+ 
+(10) 
+where 
+
+  is the chemical potential at the saddle point. 
+As a result, we obtain: 
+
+
+
+
+
+
+ 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+N
+V
+N
+T
+B
+T
+V
+e
+N
+N
+m
+T
+N
+N
+T
+V
+N
+N
+N
+N
+N
+exp
+2
+2
+2
+2
+,
+,
+,
+Z
+,         (11) 
+where the second derivative of the exponent   at the saddle point is defined as follows: 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+N
+N
+N
+N
+N
+N
+N
+N
+N
+N
+T
+V
+Z
+N
+T
+N
+T
+N
+N
+2
+2
+2
+2
+2
+2
+ln
+2
+,
+,
+ 
+ 
+ 
+0
+2
+2
+1
+1
+1
+2
+2
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+V
+T
+B
+V
+T
+B
+N
+N
+N
+T
+N
+T
+N
+N
+N
+N
+N
+, 
+because                
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+N
+N
+T
+N
+N
+N
+1
+2
+2
+,  
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+N
+N
+N
+N
+N
+N
+T
+V
+Z
+N
+T
+2
+2ln
+1
+,
+,
+. 
+The pressure in the GCE is defined as follows in terms of the temperature and the logarithm of 
+the GPF (see, e.g., [18]): 
+
+
+
+
+V
+T
+V
+T
+T
+P
+
+
+
+,
+,
+,
+Z
+ln
+. 
+ 
+ 
+ 
+ 
+(12) 
+It's easy to show that pressure (12), taking into account (5) and (11), can be rewritten as follows: 
+
+
+ 
+
+
+ 
+ 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+V
+T
+B
+T
+B
+T
+V
+N
+T
+B
+T
+T
+P
+N
+N
+2
+ln
+1
+2
+ln
+1
+2
+2
+,
+,                (13) 
+where the saddle point 
+
+ V
+T
+V
+N
+
+
+
+
+
+,
+,
+ is defined according to (5) and (10) as 
+ 
+
+
+ 
+
+
+T
+N
+T
+m
+N
+
+
+
+
+
+exp
+,
+. The parameter   can be eliminated from equation (13) using the 
+definition of density, which in the thermodynamic limit turns into the well-known formula [1]: 
+
+
+
+ 
+
+
+ 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+T
+B
+V
+T
+B
+T
+P
+n
+2
+1
+2
+1
+2
+1
+,
+. 
+ 
+ 
+(14) 
+In the thermodynamic limit (
+A
+N
+N 
+, 
+
+
+V
+) the chemical potential of the saddle point 
+
+  
+from (10) when 
+ 
+
+
+V
+N
+T
+B
+N
+N
+2
+1
+
+
+ turns into the chemical potential  (
+
+
+
+), which is 
+determined by the well-known thermodynamic equation (5).  
+Both equations (13) and (14) in parametric form (the saddle point   acts as a parameter) 
+determine the relationship between pressure P , temperature T , and density n . We obtain the state 
+equation in GCE by excluding explicitly this parameter from the system of equations (13) and (14): 
+
+
+ 
+
+
+dP
+n
+T
+B
+Tn
+n
+T
+P
+
+
+
+
+1
+,
+,
+.      
+ 
+ 
+ 
+        (15) 
+Of course, the resulting state equation is implicitly a parametric equation, since the saddle point 
+  (and, hence, and n ) determines the chemical potential  according to (5) and (10). 
+It is important that the resulting formula takes into account the contribution to the pressure of the 
+finite volume of the system, 
+s
+V . This contribution naturally vanishes in the thermodynamic limit, 
+where there is no difference between CE and GCE. 
+Only the last of the three terms remains for large but finite volumes, as in the nuclear fireball 
+model discussed in Sec. 4: 
+ 
+ 
+
+
+
+
+ 
+
+
+s
+V
+V
+V
+n
+T
+B
+T
+n
+T
+B
+n
+T
+B
+V
+T
+dP
+s
+2
+ln
+ln
+1
+2
+
+
+
+
+
+
+lim
+.                              (16) 
+RMS fluctuations of pressure and density calculated by known formulas (see, e.g., [20]) give 
+estimates of the found corrections to the corresponding quantities: 
+
+
+
+
+ 
+
+
+n
+T
+B
+V
+n
+T
+n
+P
+V
+Tn
+P
+s
+2
+1
+2
+2
+
+
+
+
+
+
+~
+,     
+ 
+ 
+    (17) 
+
+
+
+
+ 
+
+
+n
+T
+B
+V
+N
+V
+N
+T
+n
+N
+N
+2
+1
+1
+2
+2
+2
+
+
+
+
+
+
+
+
+
+
+~
+. 
+ 
+ 
+ 
+ (18) 
+ 
+2 Two-component vdW gas 
+ 
+Let us consider the procedure for taking into account the excluded volume and attraction in the 
+vdW model for the case of a two-component hadron gas of two types of particles “i ” and “ j ”. 
+1
+N  
+and 
+2
+N  are the number of particles of the first and second sorts. In this case, the partition function 
+has the form: 
+
+ 
+2
+1 N
+N
+T
+V
+Z
+,
+,
+,
+ 
+
+ 
+
+
+
+
+ 
+
+
+
+
+ 
+ 
+
+
+
+
+T
+U
+r
+d
+r
+d
+T
+m
+T
+m
+N
+N
+N
+k
+k
+N
+l
+l
+N
+N
+12
+1
+2
+3
+1
+1
+3
+2
+1
+2
+1
+exp
+!
+!
+1
+2
+1
+2
+1
+
+
+
+
+
+
+
+
+
+
+
+,
+,
+,            (19) 
+This expression for the pair-interactions approximation (
+
+
+
+
+12
+123
+U
+U
+
+) and a weakly ideal gas 
+(
+1
+2
+
+V
+NB
+) can be rewritten as follows [11]: 
+
+
+ 
+
+
+ 
+
+
+
+
+
+2
+1
+2
+1
+2
+1
+2
+1
+!
+!
+1
+N
+N
+T
+m
+T
+m
+N
+N
+N
+N
+T
+V
+Z
+,
+,
+~
+,
+,
+,
+ 
+
+
+
+
+2
+1
+1
+12
+2
+22
+2
+21
+1
+11
+N
+N
+N
+B
+N
+B
+V
+N
+B
+N
+B
+V
+~
+~
+
+
+
+
+
+
+. 
+ 
+ 
+ (20) 
+Here we have introduced the notation: 
+jj
+ii
+ij
+ii
+ij
+B
+B
+B
+B
+B
+
+ 2
+~
+. 
+The two-particle partition function 
+
+
+2
+1 
+ ,
+,
+,T
+V
+Z
+ in GCE is expressed in terms of the  
+two-particle partition function 
+
+2
+1 N
+N
+T
+V
+Z
+,
+,
+,
+ in CE [12, 17], as: 
+
+
+
+ 
+
+
+
+
+
+
+
+
+
+
+2
+1
+2
+0
+1
+0
+2
+2
+1
+2
+2
+1
+1
+e
+d
+d
+'
+,
+'
+,
+,
+'
+'
+,
+,
+,
+'
+'
+N
+N
+T
+V
+Z
+N
+N
+T
+T
+V
+N
+N
+Z
+ 
+
+
+
+
+2
+1
+2
+0
+1
+0
+2
+exp
+d
+d
+'
+,
+'
+'
+'
+N
+N
+N
+N
+T
+
+
+
+
+
+
+.          (21) 
+Integration of (21) by the saddle point method [21] leads us to the following result: 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+2
+1
+2
+2
+1
+1
+2
+1
+exp
+2
+N
+N
+T
+V
+Z
+T
+N
+N
+N
+N
+,
+,
+,
+,
+~
+Z
+, where the coordinates of the saddle 
+point 
+
+i
+N  (
+2
+1,
+
+i
+) are found from the extremum conditions: 
+
+
+0
+
+
+
+
+
+
+
+i
+j
+i
+N
+N
+N ,
+, 
+
+
+22
+21
+12
+11
+2
+1
+c
+c
+c
+c
+N
+N
+det
+,
+
+
+
+
+
+
+, 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+N
+N
+j
+i
+j
+i
+ij
+N
+N
+N
+N
+c
+,
+2
+. 
+Substituting the value of the partition function into the definition of pressure in the GCE [18], we 
+obtain the following expression [12]: 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+V
+C
+B
+B
+B
+B
+T
+V
+T
+V
+T
+T
+P
+2
+ln
+ln
+2
+1
+21
+12
+22
+2
+2
+11
+2
+1
+2
+1
+2
+1
+2
+1
+~
+~
+~
+,
+,
+,
+,
+,
+Z
+, (22) 
+where 
+21
+2
+12
+1
+22
+2
+11
+1
+B
+B
+B
+B
+C
+~
+~ 
+
+
+
+
+
+. 
+Using such a mathematical apparatus, one can organically introduce the law of conservation of 
+chemical potentials. The latter are related to the condition imposed on the integrand when finding 
+the saddle point. In the thermodynamic limit the chemical potential determined by the extremum 
+condition coincides with the definition of the chemical potential itself: 
+
+
+i
+j
+i
+i
+i
+N
+N
+N
+T
+V
+F
+
+
+
+
+
+
+,
+,
+,
+, 
+
+where 
+
+
+
+
+
+2
+1
+2
+1
+ln
+N
+N
+T
+V
+Z
+T
+N
+N
+T
+V
+F
+,
+,
+,
+,
+,
+,
+
+
+ is the definition of free energy (10). 
+We get from the definition of density  
+
+
+
+
+
+
+
+
+ji
+ij
+j
+ii
+i
+i
+i
+j
+i
+i
+B
+B
+B
+T
+P
+n
+~
+~
+~
+,
+,
+
+
+
+
+
+
+
+
+
+
+
+
+2
+1
+.                              (23) 
+The virial expansion (22) can be rewritten, taking into account (23), as a two-component vdW 
+equation in the approximation 
+1
+
+V
+N
+b
+i
+ij
+ and 
+
+
+1
+
+ij
+ij
+Tb
+a
+): 
+
+ 
+
+
+2
+1
+2
+1
+n
+n
+T
+P
+,
+,
+,
+,
+ 
+
+
+
+
+dP
+n
+a
+n
+a
+n
+n
+a
+n
+a
+n
+n
+b
+n
+b
+Tn
+n
+b
+n
+b
+Tn
+
+
+
+
+
+
+
+
+
+
+
+1
+12
+2
+22
+2
+2
+21
+1
+11
+1
+1
+12
+2
+22
+2
+2
+21
+1
+11
+1
+1
+1
+~
+~
+~
+~
+,    (24) 
+where dP , according to (22), takes into account the finite size of the fireball. When formula (24) 
+was derived, the expression 
+T
+a
+b
+B
+ij
+ij
+ij
+~
+~
+~
+
+
+ was used (see, e.g., [12]), and for each type of particles 
+the corresponding parameters of attraction and repulsion were introduced [11]: 
+
+
+jj
+ii
+ii
+ij
+ij
+a
+a
+a
+a
+a
+
+
+
+
+2
+~
+, 
+
+
+jj
+ii
+ij
+ii
+ij
+b
+b
+b
+b
+b
+
+
+2
+~
+,   is a phenomenological parameter reflecting the 
+complexity of the problem. 
+ 
+3 The asymmetric two-component freeze-out model with non-conservation of the number 
+of particles 
+ 
+The considering nucleus-nucleus collisions 
+
+A
+A
+ have very high energies, more than 1 GeV 
+per nucleon. At the same time, mesons of different sorts dominate in the initial freeze-out stages. 
+Therefore, to describe the nucleus-nucleus interactions at this stage of the freeze-out above the 
+production threshold of new particles (
+135
+
+T
+ MeV), we propose a generalization of the vdW 
+model to a medium-sized nuclear fireball [16]: 
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+A
+r
+b
+a
+V
+V
+V
+f
+f
+f
+3
+0
+2
+4
+3
+4
+3
+2
+~
+~
+~
+max
+min
+. 
+Here 
+2
+1
+1
+1
+0
+.
+. 
+
+r
+ Fm, 
+
+ a
+, 
+
+ b
+ are the mean semiaxes of the ellipsoid, and 
+
+ A
+ is the 
+mass number of nuclei left in the fireball after the collision. In our considerations we assume that 
+the fireball consists, mainly, of mesons, given that the number of nucleons is much less than the 
+number of mesons (
+
+ 300
+200
+~
+pn
+N
+5000
+4000 
+ ~
+N
+). We neglect the contribution of other 
+particles. Therefore, we introduce the following additional natural assumptions. 
+1. The average internucleon energies do not exceed the production threshold of the heavy 
+mesons. Therefore, we restrict ourselves to two sorts of particles ("0" is the 
+0
+ -meson, "+" is the 
+
+ -meson). 
+
+2. Since 
+
+ -meson production reactions are twice as likely as 
+0
+ -meson production reactions, 
+we assume that 
+n
+kn
+n
+
+
+
+0
+, where, 
+1
+
+k
+, 
+0
+n  is the 
+0
+ -meson density, and 
+
+n  is the 
+
+ -meson 
+density. This corresponds to a more probable production of the 
+
+ -mesons in reactions 
+
+
+
+
+
+
+n
+d
+d
+p
+, 
+0
+
+
+
+
+
+p
+d
+d
+p
+ than production of the 
+0
+ -mesons. 
+3. We introduce the effective potential of the meson interaction 
+
+j
+i
+U
+, where 
+ 
+
+0,
+,
+
+
+j
+i
+. That 
+is, "(0+)" is the interaction of 
+0
+ -mesons with 
+
+ -mesons, "(++)" is the interaction of 
+
+ -mesons  
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+r
+R
+R
+if
+R
+R
+r
+R
+R
+if
+u
+R
+R
+r
+if
+U
+j
+i
+j
+i
+j
+i
+j
+i
+j
+i
+j
+i
+0
+0
+0
+0
+0
+0
+,
+.                                        (25) 
+Since the effective rectangular potential a well leads to approximately the same values of 
+pressure and density as the real potential (see Fig. 2, where 
+
+
+U
+U
+
+0
+, 
+
+
+
+
+0
+0
+u
+). Therefore, the 
+real meson-meson potential (a) can be replaced by a similar effective rectangular potential (b). 
+ 
+a                                           b 
+Fig. 2. Meson-meson potential 
+ 
+4. We accept that the 
+0
+ -meson hard-core radius is much smaller than the 
+
+ -meson hard-core 
+radius: 
+0
+0
+0
+
+ R
+R
+. The radius of the hard-core of the 
+
+ -meson is assumed to be known. 
+Average pressure and density fluctuations are easily found within the framework of the proposed 
+model, similarly to formulas (18) and (19): 
+
+
+
+
+
+
+Tn
+B
+V
+n
+T
+P
+f
+
+
+
+
+
+
+1
+~
+, 
+ 
+ 
+ 
+ 
+ 
+(26) 
+
+
+
+
+
+
+Tn
+B
+V
+n
+V
+n
+f
+f
+
+
+
+
+
+
+
+
+1
+1
+~
+.  
+ 
+ 
+ 
+ 
+(27) 
+The following results are obtained (Fig. 3, Fig. 4). Such data have been used (Fig. 3): 
+3
+142,
+
+T
+ MeV, the effective radius of the 
+
+ -meson, 
+45
+0
+0
+,
+
+
+R
+ Fm, and 
+0
+ -meson, 
+01
+0
+0
+0
+,
+
+R
+ Fm, the average value of the volume of the meson fireball is taken as the value 
+600
+~
+
+
+f
+V
+ Fm 3 , 
+5
+0.
+
+k
+, the parameter of the potential depth, 
+
+
+100
+80
+0
+0
+
+
+~
+,
+u
+ MeV. One can 
+
+U来
+U*clearly see (Fig. 4) an increase in the correction 
+
+ P
+dP
+ at low densities, which is typical in the 
+final stages of the freeze-out. 
+ 
+ 
+Fig. 3. Dependence of the meson pressure P  (24) on the meson density 
+n
+kn
+n
+
+
+
+0
+ for the  
+two-component asymmetric vdW model with correction (upper isotherm) and without correction (lower 
+isotherm) 
+ 
+ 
+Fig. 4. Ratio of the correction to pressure dP  from the size of the meson fireball to the value of the RMS 
+pressure fluctuation 
+
+ P
+ (26) as a function of the meson density 
+n
+kn
+n
+
+
+
+0
+ 
+ 
+4 Two-component model of a nucleon fireball at the final stage of the freeze-out 
+ 
+The average lifetime of mesons dominating in the initial stages of the freeze-out is relatively 
+short (
+16
+8
+10
+10
+
+ 
+ ~
+ c). That's why they decay pretty quickly. Accordingly, baryons, namely 
+protons and neutrons, begin to dominate at the final stage of freezing. In addition, as shown above, 
+the effects of the finite volume size become noticeable at sufficiently low density values. This 
+formally corresponds to just such final stages of the fireball evolution. Therefore, despite a certain 
+doubt about the existence of a fireball at such late stages, when the boundary between the gas and 
+the aggregate of individual nucleons gradually disappears, to describe the nucleus-nucleus 
+
+P(T=142.3, n), MeV/Fm-3
+10
+n, Fm-3
+0.05
+0.10
+0.20
+0.25
+0.30
+-10
+-20
+-30
+-40
+-50 FdP/<P>
+2.0
+1.5
+1.0
+n, Fm-3
+0.05
+0.10
+0.15
+9.20
+0.25
+0.30interactions at the last stage of the freeze-out, which is below the production threshold of new 
+particles (
+135
+
+T
+ MeV), in [14] the following generalization of the vdW model to the nucleon 
+fireball was proposed. We accept the following simplifications by analogy with the previous 
+section. 
+1. The average energies of internucleon collisions do not exceed the production threshold of 
+other hadrons. Therefore, we restrict ourselves to two varieties (“ p ” is the proton, “ n ” is the 
+neutron). 
+2. We take the relation between the density of protons and neutrons in the form 
+n
+n
+n
+n
+p
+
+
+ 
+following from the law of conservation of the baryon number, 
+A
+N
+Z
+
+
+. 
+3. We assume that the nucleon composition of colliding nuclei is known as such 
+n
+p
+kn
+n 
+, where 
+1
+
+k
+, since heavy nuclei have an excess of neutrons. 
+4. The effective potential of the proton-neutron, proton-proton and neutron-neutron interactions, 
+which leads to approximately the same values of pressure and density as the real potential (Fig. 3), 
+can be represented by analogy to (25) as 
+
+j
+i
+U
+,  where 
+
+
+n
+p
+j
+i
+,
+,
+
+. 
+5. The hard-core radius of the proton is assumed to be known, 
+5
+0
+0
+,
+
+p
+R
+ Fm. We accept that the 
+radius of the neutron is much less than the radius of the proton: 
+0
+0
+p
+n
+R
+R 
+. 
+We get from equation (27): 
+
+
+
+
+ 
+ 
+dP
+n
+a
+n
+n
+n
+k
+Tn
+T
+P
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+2
+2
+2
+1
+1
+1
+,
+,
+,                                          (28) 
+where   
+
+
+k
+n
+n
+
+
+
+1
+, k  is a dimensionless quantity, 
+k
+b
+k
+b
+b
+k
+b
+22
+2
+21
+12
+11
+
+
+
+
+
+~
+~
+, 
+22
+21
+12
+11
+b
+k
+b
+b
+k
+b
+
+
+
+
+
+~
+~
+, 
+k
+b
+b
+b
+b
+b
+b
+k
+b
+kb
+21
+12
+21
+22
+12
+11
+2
+22
+11
+~
+~
+~
+~
+
+
+
+
+
+, 
+
+
+22
+21
+12
+2
+11
+a
+k
+a
+a
+k
+a
+a
+
+
+
+
+
+~
+~
+. 
+It follows from the condition 
+0
+1
+0
+2
+R
+R 
+ that 
+11
+22
+b
+b
+
+, 
+
+
+
+. By analogy with Eqs. (18) and 
+(19), we find the corresponding average fluctuations of pressure and density. 
+Functional dependences for pressure, obtained by Eq. (28), and the ratio of dP  to RMS pressure 
+fluctuations are shown in Figs. 5 and 6. 
+ 
+
+ 
+Fig. 5. Dependence of nucleon pressure P  (28) on nucleon density, 
+n
+kn
+n
+n
+p
+
+
+, in the two-component 
+asymmetric vdW model with correction (upper isotherm) and without correction (lower isotherm) 
+ 
+ 
+ 
+Fig. 6. The ratio of correction from the size of the nucleon fireball to pressure dP  to the value of the 
+RMS pressure fluctuation 
+
+ P
+ depending on the density of nucleons, 
+n
+kn
+n
+n
+p
+
+
+ 
+ 
+It can be seen that the correction dP  makes a nonzero contribution to the total pressure also in 
+this case. On the other hand, it is negligibly small almost everywhere in comparison with the 
+contribution from fluctuations. The correction makes a contribution comparable to fluctuations only 
+in the region near zero density that is nonphysical for a nuclear fireball. But it can be neglected in 
+this region, as can be seen from Fig. 6. 
+ 
+Summary 
+ 
+The effect of taking into account the excluded volume and attraction is analyzed in the case of a 
+two-component gas: (i) 
+0
+ - and 
+
+ -mesons; (ii) protons and neutrons. The calculations have been 
+performed in the Canonical and Grand Canonical ensembles by the saddle point method for a two-
+component system. The particles interact with the potentials of the hard-core at short distances and 
+with relatively high potentials at large distances (effective attraction radii). For effective 
+
+P(T=142.3. n), MeV/Fm-3
+10
+n, Fm-3
+0.05
+0.10
+N15
+0.20
+0.25
+0.30
+-10
+-20
+-30
+-40
+-50 FdP/<P>
+1.0
+0.9
+0.8
+F
+0.7
+n, Fm-3
+0.05
+0.10
+0.15
+0.20
+0.23
+0.30interparticle interactions of this type, an equation of state has been obtained with corrections that 
+take into account the finite dimensions of the nuclear fireball, as well as RMS fluctuations of 
+pressure and density. 
+The pressure correction disappears in the thermodynamic limit, when, according to statistical 
+physics, there is no difference between various statistical ensembles. The formulas for pressure and 
+density obtained by the saddle point method can be used to analyze experimental data concerning 
+the relative number of the yield particleshe of various sorts and critical parameters in high-energy 
+nuclear-nucleus collisions. A generalization of the presented vdW model to the asymmetric case of 
+a two-component model (
+0
+ - and 
+
+ -mesons) with realistic parameters of the hard-core and 
+attraction has been proposed as an example of such a use. The ratio of the pressure correction to the 
+RMS value of pressure fluctuation is estimated for the case of an asymmetric two-component 
+meson fireball model. An increase in the correction has been found at low density values 
+corresponding to the final stages of freezing. 
+It is found that the contribution to pressure and relative fluctuations, taking into account different 
+radii and the finiteness of the nuclear fireball, is noticeable in the case of the meson model with 
+nonconservation of the number of particles. However, this correction can be neglected for the final 
+stages of the freeze-out, when nucleons begin to dominate (the model of Sec. 4). Therefore, the 
+developed model is applicable in the analysis of experimental data on the study of the initial meson 
+phase of a nuclear fireball (the model of Sec. 3), which occurs, in particular, in experiments on the 
+study of quark-gluon plasma. 
+The research was carried out within the framework of the initiative scientific topic 0122U200549 
+(“National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, 
+Ukraine is the customer). 
+ 
+REFERENCES 
+ 
+1. J. Stachel, U. Heidelberg. Tests of thermalization in relativistic nucleus nucleus collisions. Nucl. Phys. A 
+610 (1996) 509C.  
+2. P. Braun-Munzinger, J. Stachel. Dynamics of ultra-relativistic nuclear collisions with heavy beams: An 
+experimental overview. Nucl. Phys. A 638 (1998) 3C.  
+3. J. Cleymans, H. Satz. Thermal Hadron Production in High Energy Heavy Ion Collisions. Z. Phys. C 57 
+(1993) 135. 
+4. J. Cleymans et al. The hadronisation of a quark-gluon plasma. Z. Phys. C 58 (1993) 347. 
+5. K. Redlich et al. Hadronisation of quark-gluon plasma. Nucl. Phys. A 566 (1994) 391. 
+
+6. P. Braun-Munzinger et al. Thermal equilibration and expansion in nucleus-nucleus collisions at the AGS. 
+Phys. Lett. B 344 (1995) 43. 
+7. P. Braun-Munzinger et al. Thermal and hadrochemical equilibration in nucleus-nucleus collisions at the 
+SPS. Phys. Lett. B 365 (1996) 1. 
+8. R.A. Ritchie, M.I. Gorenstein, H.G. Miller. The excluded volume hadron gas model and pion production 
+at the SPS. Z. Phys. C 75 (1997) 535. 
+9. G.D. Yen et al. Excluded volume hadron gas model for particle number ratios in 
+A
+A 
+ collisions. Phys. 
+Rev. C 56 (1997) 2210. 
+10. G.D. Yen at al. Chemical freezeout in relativistic 
+A
+A 
+ collisions: is it close to the quark-gluon plasma? 
+J. Phys. G 24 (1998) 1777. 
+11. M.I. Gorenstein, A.P. Kostyuk, Ya.D Krivenko. Van der Waals excluded-volume model of 
+multicomponent hadron gas. J. Phys. G 25 (1999) 75. 
+12. Ya.D. Krivenko-Emetov. Attractive inter-particle force in van der Waals model of multicomponent 
+hadron gas in the grand canonical ensemble. 2019 arXiv:1909.08441v1 [hep-ph]; Ya.D. Krivenko-
+Emetov. Interparticle attractive forces account of the multicomponent hadron gas in the grand canonical 
+ensenble. Book of abstract of the 24th Annual Scientific Conf. of Inst. for Nucl. Research, Kyiv, Ukraine, 
+April 10-13, 2017 (Kyiv, 2017) p. 36. 
+13. V. Vovchenko at al. Multicomponent van der Waals equation of state: Applications in nuclear and 
+hadronic physics. Phys. Rev. C 96 (2017) 045202. 
+14. Ya.D. Krivenko-Emetov. Finite volume effects in the two-component van der Waals model in 
+relativistic nucleus-nucleus collisions of heavy ions. Book of abstract of the 28th Annual Scientific 
+Conf. of Inst. for Nucl. Research, Kyiv, Ukraine, Sept. 27 – Oct. 01, 2021 (Kyiv, 2021) p. 27. 
+15. Quark-Gluon 
+Plasma 
+(QGP) 
+Physics 
+with 
+ALICE 
+at 
+the 
+CERN 
+LHC. 
+URL: 
+https://indico.cern.ch/event/1013634/contributions/4255256/attachments/2227069/3772748/IoP-
+April2021.pdf. 
+16. Krivenko-Emetov, Ya.D. Pressure corrections for one-component and two-component van der Waals 
+nuclear fireball models at the freezeout stage. Book of abstract of 29th Annual Scientific Conf. of Inst. for 
+Nucl. Research, Kyiv, Sept. 26 – 30, 2022, p.21-22. (Ukr). D. Sokolyuk, Ya. Krivenko-Emetov. Two-
+component van der Waals model of a nuclear fireball in the cooling stage (freezeout). Mat. of XX All-
+Ukrainian science and practice conf. students, postgraduates and young scientists “Theoretical and 
+applied problems of physics, mathematics and informatics”, Kyiv, June 15, 2022 (Igor Sikorsky Kyiv 
+Polytechnic Institute, 2022) p. 88. (Ukr). 
+17. L.D. Landau, E.M. Lifshitz. Statistical Physics Vol. 5 of Course of Theoretical Physics. (2 ed. Addison 
+Wesley, 1969) 484 p.  
+18. R. Kubo. Statistical mechanics (Moskva: Mir, 1967) 452 p. (Rus). 
+19. R.P. Feynman. Statistical Mechanics: a set of lectures. Advanced Book Classics (2 ed. Perseus Books, 
+Reading, Mass., 1998) 354 p. 
+
+20. A.M. Fedorchenko. Theoretical physics. T.2. Quantum mechanics, thermodynamics and statistical 
+physics (Kyiv: Vyshcha shkola, 1993) 415 p. (Ukr). 
+21. M.V. Fedoruk. Saddle point method (Moskva, 1977) 368 p. (Rus). 
+ 
+Я. Д. Кривенко-Еметов* 
+Інститут ядерних досліджень НАН України, Київ, Україна 
+Національний технічний університет України «Київський політехнічний інститут  
+імені Ігоря Сікорського», Київ, Україна 
+*Відповідальний автор: [email protected]; [email protected] 
+БАГАТОКОМПОНЕНТНА МОДЕЛЬ ВАН ДЕР ��АЛЬСА  
+ЯДЕРНОГО ФАЄРБОЛУ НА СТАДІЇ ФРІЗАУТУ 
+ 
+Двокомпонентна газова модель Ван-дер-Ваальсу запропонована для опису адронних етапів 
+еволюції ядерного фаєрболу у стадії охолодження. Для першого етапу адронізації, коли домінують 
+мезони, запропонована двокомпонентна мезонна модель(
+0
+ - та 
+
+ -мезонів) з ефективним 
+двочастинковим потенціалом взаємодії прямокутної ями. Для останнього етапу, коли майже усі 
+мезони розпались, запропонована двокомпонентна нуклонна модель протонів та нейтронів з 
+відповідним ефективним прямокутним нуклонним потенціалом. При аналітичних розрахунках 
+статистичної суми використовувався методу перевалу, що дозволило єдиним чином отримати 
+аналітичні вирази, як для тиску та щільності з урахуванням скінченних розмірів системи, так і вирази 
+для хімічних потенціалів. Очікується, що запропоновані моделі й отримані формули можуть бути 
+використані для аналізу експериментальних даних щодо кількісних характеристик виходу частинок 
+різних сортів у кінцевому стані від адронних стадій еволюції ядерного фаєрболу, а також для 
+визначення критичних параметрів системи у ядро-ядерних зіткненнях за високих енергій. 
+Ключові слова: фаєрбол, фрізаут, рівняння Ван-дер Ваальса, ефективний ядерний потенціал, 
+Великий канонічний ансамбль, флуктуація тиску, кварк-глюонна плазма, експериментальні дані. 
+

3dAzT4oBgHgl3EQffPzi/content/tmp_files/2301.01451v1.pdf.txt

@@ -0,0 +1,1824 @@
+arXiv:2301.01451v1  [quant-ph]  4 Jan 2023
+Reduced dynamics with Poincar´e symmetry in open quantum system
+Akira Matsumura∗
+Department of Physics, Kyushu University, Fukuoka, 819-0395, Japan
+Abstract
+We consider how the reduced dynamics of an open quantum system coupled to an environment admits
+the Poincar´e symmetry. The reduced dynamics is described by a dynamical map, which is given by tracing
+out the environment from the total dynamics. Introducing the notion of covariant map, we investigate the
+dynamical map which is symmetric under the Poincar´e group. Based on the representation theory of the
+Poincar´e group, we develop a systematic way to give the dynamical map with the Poincar´e symmetry. Using
+this way, we derive the dynamical map for a massive particle with a finite spin and a massless particle with
+a finite spin and a nonzero momentum. We show that the derived map gives the unitary evolution of a
+particle when its energy is conserved. We also find that the dynamical map for a particle does not have the
+Poincar´e symmetry when the superposition state of the particle decoheres into a mixed state.
+∗Electronic address: [email protected]
+1
+
+Contents
+I. Introduction
+2
+II. Quantum dynamical map and its symmetry
+4
+III. Dynamical map with Poincar´e symmetry
+5
+IV. A model of the dynamical map for a single particle
+10
+V. Conclusion
+13
+Acknowledgments
+14
+A. Derivation of Eqs.(54),(55),(56),(57),(58) and (59)
+14
+B. Analysis of a massive particle
+15
+C. Analysis on a massless particle
+21
+References
+27
+I.
+INTRODUCTION
+It is difficult to isolate a quantum system perfectly, which is affected by the inevitable influence
+of a surrounding environment. Such a quantum system is called an open quantum system. Since
+we encounter open quantum systems in a wide range of fields such as quantum information science
+[1, 2], condensed matter physics [3, 4] and high energy physics [5], it is important to understand
+their dynamics.
+In general, the dynamics of an open quantum system, the so-called reduced
+dynamics, is very complicated. This is because the environment may have infinitely many degrees
+of freedom and they are uncontrollable. One needs the effective theory with relevant degrees of
+freedom to describe the reduced dynamics of an open quantum system [2].
+As is well-known, symmetry gives a powerful tool for capturing relevant degrees of freedom in
+the dynamics of interest. For example, let us focus on the symmetry in the Minkowski spacetime,
+which is called the Poincar´e symmetry. Imposing the Poincar´e symmetry on a quantum theory,
+one finds that quantum dynamics in the theory is described by the fundamental degrees of freedom
+such as a massive particle and a massless particle [6]. The approach based on symmetries provides
+2
+
+a way to get the effective theory of open quantum systems.
+In this paper, we discuss the consequences of the Poincar´e symmetry on the reduced dynamics
+of an open quantum system. This may give the understanding of the relativistic theories of open
+quantum systems (for example, [7–14]).
+The present paper is also motivated by the theory of
+quantum gravity. Since the unification of quantum mechanics and gravity has not been completed
+yet, we do not exactly know how gravity is incorporated in quantum mechanics. This situation
+has prompted to propose many models on the gravity of quantum systems. In the previous work
+[15], the model with a classical gravitational interaction between quantum systems was proposed,
+which is called the Kafri-Taylor-Milburn model. In addition, the Diosi-Penrose model [16–18] and
+the Tilloy-Diosi model [19] were advocated, for which the gravity of a quantum system intrinsically
+induces decoherence.
+They are formulated by the theory of open quantum systems in a non-
+relativistic regime.
+One may concern how the models are consistent with a relativistic theory.
+Our analysis on reduced dynamics with the Poincar´e symmetry would help to obtain a relativistic
+extension of the above proposed models.
+For our analysis, we assume that the reduced dynamics of an open quantum system is described
+by a dynamical map. The dynamical map is obtained by tracing out the environment from the total
+unitary evolution with an initial product state. It is known that the dynamical map is represented
+by using the Kraus operators [2, 20–22]. The notion of covariant map is adopted for incorporating
+a symmetry into a dynamical map. We derive the condition of a dynamical map with the Poincar´e
+symmetry in terms of the Kraus operators. With the help of the representation theory of the
+Poincar´e group, we obtain a systematic way to deduce those Kraus operators.
+Applying the way, we exemplify the dynamical map with the Poincar´e symmetry.
+To get
+the concrete Kraus operators, we focus on the dynamics of a single particle, which is possible to
+decay into the vacuum state. Assuming that the particle is a massive particle with a finite spin or a
+massless particle with a finite spin and a nonzero momentum, we get a model of the dynamical map
+with the Poincar´e symmetry. In the model, we find the following consequences: (i) if the particle
+is stable or the energy of particle is conserved, the obtained map turns out to be the unitary map
+given by the Hamiltonian of particle. (ii) If the superposition state of a particle decoheres into
+a mixed state, the dynamical map for the particle does not have the Poincar´e symmetry. These
+consequences imply that the Poincar´e symmetry can strongly constraint the reduced dynamics of
+an open quantum system.
+The structure of this paper is as follows. In Sec.II, we discuss a dynamical map describing the
+reduced dynamics of an open quantum system and consider how symmetries are introduced in the
+3
+
+dynamical map. In Sec.III, we derive the condition that the dynamical map is symmetric under the
+Poincar´e group. In Sec.IV, focusing on the dynamics of a single particle, we present a model of the
+dynamical map with the Poincar´e symmetry. We then investigate the properties of the dynamical
+map in details. Sec.V is devoted as the conclusion. We use the unit ℏ = c = 1 in this paper.
+II.
+QUANTUM DYNAMICAL MAP AND ITS SYMMETRY
+In this section, we consider the reduced dynamics of an open quantum system and discuss the
+symmetry of the dynamics. The reduced dynamics is given as the time evolution of the density
+operator of the system. The time evolution from a time τ = s to τ = t is assumed to be given by
+ρ(t) = Φt,s[ρ(s)] = TrE[ ˆU(t, s)ρ(s) ⊗ ρE ˆU †(t, s)],
+(1)
+where ρ(τ) is the system density operator, ρE is the density operator of an environment and ˆU(t, s)
+is the unitary evolution operator of the total system.
+In this paper, the map Φt,s is called a
+dynamical map, which has the property called completely positive and trace-preserving (CPTP)
+[2, 20–22]. The dynamical map Φt,s is rewritten in the operator-sum representation,
+Φt,s[ρ(s)] =
+�
+λ
+ˆF t,s
+λ ρ(s) ˆF t,s †
+λ
+,
+(2)
+where ˆF t,s
+λ
+called the Kraus operators satisfy the completeness condition,
+�
+λ
+ˆF t,s †
+λ
+ˆF t,s
+λ
+= ˆI.
+(3)
+In this notation, λ takes discrete values. When the label λ is continuous, we should replace the
+summation �
+λ with the integration
+�
+dµ(λ) with an appropriate measure µ(λ). It is known that
+two dynamical maps Φ and Φ′ with
+Φ[ρ] =
+�
+λ
+ˆFλ ρ ˆF †
+λ,
+Φ′[ρ] =
+�
+λ
+ˆF ′
+λ ρ ˆF
+′†
+λ ,
+(4)
+are equivalent to each other (i.e. Φ[ρ] = Φ′[ρ] for any density operator ρ) if and only if there is a
+unitary matrix Uλλ′ satisfying �
+λ Uλ1λU∗
+λ2λ = δλ1λ2 = �
+λ Uλλ1U∗
+λλ2 and
+ˆF
+′
+λ =
+�
+λ′
+Uλλ′ ˆFλ′.
+(5)
+This is the uniqueness of a dynamical map [2, 20–22].
+We introduce the notion of covariant map [22–24] to impose symmetry on dynamical maps. A
+dynamical map Φt,s is covariant under a group G if
+Φt,s[ ˆUs(g) ρ(s) ˆU †
+s (g)] = ˆUt(g)Φt,s[ρ(s)] ˆU †
+t (g),
+(6)
+4
+
+where ˆUs(g) and ˆUt(g) with g ∈ G are the unitary representations of G. In this paper, the dynamical
+map Φt,s satisfying (6) is called symmetric under the group G. In the next section, we will discuss
+the dynamical map which is symmetric under the Poincar´e group.
+III.
+DYNAMICAL MAP WITH POINCAR´E SYMMETRY
+In this section, we consider a quantum theory with the Poincar´e symmetry and discuss the
+general conditions on a dynamical map with the Poincare symmetry. The generators of the unitary
+representation of the Poincar´e group in the Schr¨odinger picture [6] are given by
+ˆPµ =
+�
+d3x ˆT 0
+µ,
+ˆJµν =
+�
+d3x ˆ
+Mµν0,
+(7)
+where ˆTµν is the energy-momentum tensor satisfying
+∂µ ˆT µ
+ν = 0,
+ˆTµν = ˆTνµ
+(8)
+and ˆ
+Mµνρ with
+ˆ
+Mµνρ = xµ ˆT ρ
+ν − xν ˆT ρ
+µ
+(9)
+is the Noether current associated with the Lorentz transformations. From Eq.(8), we can show
+that ∂ρ ˆ
+Mµνρ = 0. Focusing on each component of the generators, we have
+ˆH = ˆP 0 =
+�
+d3x ˆT 00,
+ˆP i =
+�
+d3x ˆT 0i,
+(10)
+ˆJk = 1
+2ǫijk ˆJij =
+�
+d3x ǫijkxi ˆT 0
+j ,
+ˆKk =
+�
+d3(xk ˆT 00 − t ˆT 0k),
+(11)
+where note that the boost generator ˆKk explicitly depends on a time t. These operators satisfy
+the commutation relations,
+[ ˆPi, ˆPj] = 0,
+(12)
+[ ˆPi, ˆH] = 0,
+(13)
+[ ˆJi, ˆH] = 0,
+(14)
+[ ˆJi, ˆJj] = iǫijk ˆJk,
+(15)
+[ ˆJi, ˆPj] = iǫijk ˆP k,
+(16)
+[ ˆJi, ˆKj] = iǫijk ˆKk,
+(17)
+[ ˆKi, ˆPj] = iδij ˆH,
+(18)
+[ ˆKi, ˆH] = i ˆPi,
+(19)
+[ ˆKi, ˆKj] = −iǫijk ˆJk,
+(20)
+5
+
+which correspond to the Poincar´e algebra.
+We consider a dynamical map Φt,s from ρ(s) to ρ(t) = Φt,s[ρ(s)]. The Poincar´e symmetry of
+the dynamical map requires that
+ˆUt(Λ, a)Φt,s[ρ(s)] ˆU †
+t (Λ, a) = Φt,s[ ˆUs(Λ, a)ρ(s) ˆU †
+s (Λ, a)],
+(21)
+where the unitary operator ˆUt(Λ, a) depends on the proper (detΛ = 1) orthochronous (Λ00 ≥ 1)
+Lorentz transformation matrix Λµν and the real parameters aµ for the spacetime translations. The
+unitary operator ˆUt(Λ, a) generated by ˆH, ˆPi, ˆJi and ˆKi has the group multiplication rule
+ˆUt(Λ′, a′) ˆUt(Λ, a) = ˆUt(Λ′Λ, a′ + Λ′a),
+(22)
+where we used the fact that we can always adopt the non-projective unitary representation of the
+Poincar´e group [6]. The explicit time dependence of ˆUt comes from the boost generator ˆKi. Using
+the operator-sum representation, we have
+ˆUt(Λ, a)
+�
+λ
+ˆF t,s
+λ ρ(s) ˆF t,s†
+λ
+ˆU †
+t (Λ, a) =
+�
+λ
+ˆF t,s
+λ
+ˆUs(Λ, a)ρ(s) ˆU †
+s (Λ, a) ˆF t,s†
+λ
+.
+From the uniqueness of the Kraus operators ˆF t,s
+λ
+(see Eq.(5)), we obtain
+ˆU †
+t (Λ, a) ˆF t,s
+λ ˆUs(Λ, a) =
+�
+λ′
+Uλλ′(Λ, a) ˆF t,s
+λ′ .
+(23)
+We can always choose ˆF t,s
+λ
+so that { ˆF t,s
+λ }λ is the set of linearly independent operators. This linear
+independence and the group multiplication rule of ˆUt(Λ, a) given in (22) lead to the fact that the
+unitary matrix Uλλ′(Λ, a) satisfies the group multiplication rule
+�
+λ′
+Uλλ′(Λ′, a′)Uλ′λ′′(Λ, a) = Uλλ′′(Λ′Λ, Λa + a′).
+(24)
+Hence, the unitary matrix Uλλ′(Λ, a) is a representation of the Poincar´e group.
+Before discussing the condition of symmetry, Eq.(23), we present the useful relation
+ˆKi = e−i ˆ
+Ht ˆKi
+0 ei ˆ
+Ht,
+(25)
+where
+ˆKi
+0 =
+�
+d3x xi ˆT 00.
+(26)
+According to the Poincar´e algebra, we have
+ˆUt(Λ, a) = e−i ˆ
+Ht ˆU0(Λ, a)ei ˆ
+Ht,
+(27)
+6
+
+where ˆU0(Λ, a) is the unitary representation of the Poincar´e group with the genrators ˆH, ˆP i, ˆKi
+0 and
+ˆJi. In the scattering theory, Eq. (27) is consistent with the Poincar´e invariance of the S-operator
+ˆS(∞, −∞), where ˆS(tf, ti) = ei ˆ
+H0tfe−i ˆ
+H(tf−ti)e−i ˆ
+H0ti and ˆH = ˆH0 + ˆV . This is because
+ˆU I†
+tf (Λ, a) ˆS(tf, ti) ˆU I
+ti(Λ, a) = ei ˆ
+H0tf ˆU †
+tf(Λ, a)e−i ˆ
+H(tf−ti) ˆUti(Λ, a)e−i ˆ
+H0ti
+= ei ˆ
+H0tfe−i ˆ
+Htf ˆU †
+0(Λ, a) ˆU0(Λ, a)ei ˆ
+Htie−i ˆ
+H0ti
+= ˆS(tf, ti),
+(28)
+where ˆU I
+t(Λ, a) = ei ˆ
+H0t ˆUt(Λ, a)e−i ˆ
+H0t. Eq.(27) also implies that the unitary evolution generated by
+ˆH is symmetric under the Poincar´e group. Indeed, we can show that the unitary map
+Ut,s[ρ(s)] = e−i ˆ
+H(t−s) ρ(s) ei ˆ
+H(t−s)
+(29)
+satisfies the condition of symmetry (21) as
+Ut,s[ ˆUs(Λ, a)ρ(s) ˆU †
+s (Λ, a)] = e−i ˆ
+H(t−s) ˆUs(Λ, a) ρ(s) ˆU †
+s (Λ, a)ei ˆ
+H(t−s)
+= e−i ˆ
+H(t−s) ˆUs(Λ, a)ei ˆ
+H(t−s) Ut,s[ρ(s)] e−i ˆ
+H(t−s) ˆU †
+s(Λ, a)ei ˆ
+H(t−s)
+= ˆUt(Λ, a) Ut,s[ρ(s)] ˆU †
+t (Λ, a).
+Eq.
+(27) helps us to simplify the condition of symmetry, Eq.(23), on the Kraus operators.
+Defining the Kraus operators ˆEt,s
+λ
+as
+ˆEt,s
+λ = ei ˆ
+Ht ˆF t,s
+λ e−i ˆ
+Hs
+(30)
+which have the completeness condition,
+�
+λ
+ˆEt,s†
+λ
+ˆEt,s
+λ = ˆI,
+(31)
+we can rewrite Eq.(23) as
+ˆU †
+0(Λ, a) ˆE ˆU0(Λ, a) = U(Λ, a) ˆE.
+(32)
+Here, we introduced the vector ˆE with the λ component given by ˆEt,s
+λ
+and the matrix U(Λ, a) with
+the (λ, λ′) component given by Uλλ′(Λ, a). We define the dynamical map Et,s as
+Et,s[ρ] =
+�
+λ
+ˆEt,s
+λ ρ ˆEt,s†
+λ
+.
+(33)
+The condition (32) gives the fact that the map Et,s is symmetric under the Poincar`e group in the
+sense that
+ˆU0(Λ, a)Et,s[ρ] ˆU †
+0(Λ, a) = Et,s[ ˆU0(Λ, a) ρ ˆU †
+0(Λ, a)].
+(34)
+7
+
+The dynamical map Φt,s is written by the unitary map Ut,s and the dynamical map Et,s as
+Φt,s[ρ] =
+�
+λ
+ˆF t,s
+λ ρ ˆF t,s†
+λ
+= e−i ˆ
+Ht �
+λ
+ˆEt,s
+λ ei ˆ
+Hsρe−i ˆ
+Hs ˆEt,s†
+λ
+ei ˆ
+Ht
+= e−i ˆ
+HtEt,s[ei ˆ
+Hsρe−i ˆ
+Hs]ei ˆ
+Ht
+= e−i ˆ
+H(t−s)Et,s[ρ]ei ˆ
+H(t−s)
+= Ut,s ◦ Et,s[ρ],
+(35)
+where in the fourth equality we used the symmetric condition (34) noticing that ei ˆ
+Hs is the unitary
+transformation of the time translation.
+Our task is to determine ˆE satisfying Eq.(32) (or Et,s
+satisfying Eq.(34)). Since Eq. (32) is decomposed into equations for each irreducible representation
+subspace, the irreducible unitary representations of the Poincar´e group is useful for our analysis.
+Let us present how to classify the unitary representations of the Poincar´e group [6]. We consider
+the standard momentum ℓµ and the Lorentz transformation matrix (Sq)µν with
+qµ = (Sq)µνℓν.
+(36)
+The unitary matrix U(Λ, a) is written as
+U(Λ, a) = U(I, a)U(Λ, 0) = T (a)V(Λ),
+(37)
+where I is the identity matrix, U(I, a) = T (a) = e−iPµaµ and U(Λ, 0) = V(Λ). We define the vector
+vq,ξ as
+vq,ξ = NqV(Sq)vℓ,ξ,
+(38)
+where Pµvℓ,ξ = ℓµvℓ,ξ, Nq is the normalization and the label ξ describes the degrees of freedom
+other than them determined by ℓµ. We obtain the following transformation rules for the vector
+vq,ξ:
+T (a)vq,ξ = Nqe−iP µaµV(Sq)vℓ,ξ
+= NqV(Sq)e−i(Sq)µνP νaµvℓ,ξ
+= NqV(Sq)e−i(Sq)µνℓνaµvℓ,ξ
+= NqV(Sq)e−iqµaµvℓ,ξ
+= e−iqµaµvq,ξ
+(39)
+8
+
+and
+V(Λ)vq,ξ = NqV(Λ)V(Sq)vℓ,ξ
+= NqV(ΛSq)vℓ,ξ
+= NqV(SΛq)V(S−1
+Λq ΛSq)vℓ,ξ
+= NqV(SΛq)
+�
+ξ′
+Dξ′ξ(Q(Λ, q))vℓ,s′
+= Nq
+NΛq
+�
+ξ′
+Dξ′ξ(Q(Λ, q))vΛq,s′,
+(40)
+where Q(Λ, q) = S−1
+Λq ΛSq is an element of the little group, which satisfies Qµνℓν = ℓµ, and Dξ′ξ(Q)
+is the unitary representation of the little group. The irreducible unitary representations of the
+Poincar´e group are classified by the standard momentum ℓµ and the irreducible unitary represen-
+tations of the little group which does not change ℓµ.
+standard momentum ℓµ
+little group composed of Qµν with Qµνℓν = ℓµ
+(a)
+ℓµ = [M, 0, 0, 0], M > 0
+SO(3)
+(b)
+ℓµ = [−M, 0, 0, 0], M > 0
+SO(3)
+(c)
+ℓµ = [κ, 0, 0, κ], κ > 0
+ISO(2)
+(d)
+ℓµ = [−κ, 0, 0, κ], κ > 0
+ISO(2)
+(e)
+ℓµ = [0, 0, 0, N], N 2 > 0
+SO(2,1)
+(f)
+ℓµ = [0, 0, 0, 0]
+SO(3,1)
+TABLE I: Classification of the standard momentum ℓµ and the little group associated with ℓµ.
+For simplicity, ξ is regarded as the label of basis vectors of the irreducible representation sub-
+spaces of the little group. Other degeneracies not represented by q and ξ will be reintroduced in
+the form of the dynamical map Φt,s, which we will see in the next section. We investigate Eq.(32)
+restricted on each irreducible representation. For convenience, we separately focus on the Lorentz
+transformation and the spacetime translation in Eq.(32). The unitary operator ˆU0(Λ, a) is written
+as
+ˆU0(Λ, a) = ˆU0(I, a) ˆU0(Λ, 0) = ˆT(a) ˆV (Λ),
+(41)
+where ˆU0(I, a) = ˆT(a) = e−i ˆPµaµ with ˆP µ = [ ˆH, ˆP 1, ˆP 2, ˆP 3] and ˆU0(Λ, 0) = ˆV (Λ) with the genera-
+tors ˆJi and ˆKi
+0. From Eq.(32) for Λ = I, we have
+ˆT †(a) ˆE ˆT(a) = T (a) ˆE.
+(42)
+9
+
+Eq.(32) for aµ = 0 gives
+ˆV †(Λ) ˆE ˆV (Λ) = V(Λ) ˆE.
+(43)
+Introducing ˆEq,ξ = v†
+q,ξ ˆE, we obtain the following equations from Eqs.(42) and (43):
+ˆT †(a) ˆEq,ξ ˆT(a) = e−iqµaµ ˆEq,ξ
+(44)
+and
+ˆV †(Λ) ˆEq,ξ ˆV (Λ) =
+N ∗
+q
+N ∗
+Λ−1q
+�
+ξ′
+D∗
+ξ′ξ(Q(Λ−1, q)) ˆEΛ−1q,ξ′,
+(45)
+where we used Eqs.(39) and (40), and Q(Λ, q) = S−1
+Λq ΛSq. The label ξ can take discrete or contin-
+uous values. For the continous case, the summation �
+ξ is replaced with the integration
+�
+dµ(ξ)
+with a measure µ(ξ). Focusing on Eq.(45) for Λ = Sq, we get
+ˆV †(Sq) ˆEq,ξ ˆV (Sq) = N ∗
+q ˆEℓ,ξ,
+(46)
+where note that Nℓ = 1 and Q(S−1
+q , q) = S−1
+S−1
+q
+qS−1
+q Sq = S−1
+ℓ
+= I hold by the definition of vq,ξ.
+Eq.(46) tells us that the Kraus operators ˆEq,ξ is determined from the Kraus operators ˆEℓ,ξ with
+the standard momentum ℓµ. All we have to do is to give the form of the Kraus operators ˆEℓ,ξ.
+To this end, we present the following equations given by Eq.(44) for qµ = ℓµ and by Eq.(45) for
+qµ = ℓµ and Λ = W with W µνℓν = ℓµ, respectively:
+ˆT †(a) ˆEℓ,ξ ˆT(a) = e−iℓµaµ ˆEℓ,ξ,
+(47)
+ˆV †(W) ˆEℓ,ξ ˆV (W) =
+�
+ξ′
+D∗
+ξ′ξ(W −1) ˆEℓ,ξ′,
+(48)
+where Q(Λ−1, q) = Q(W −1, ℓ) = S−1
+W −1ℓW −1Sℓ = W −1. In the next section, we construct a model
+of the dynamical map with the Poincar´e symmetry to describe the reduced dynamics of a single
+particle.
+IV.
+A MODEL OF THE DYNAMICAL MAP FOR A SINGLE PARTICLE
+In this section, based on Eqs.(47) and (48), we give a model of the dynamical map with the
+Poincar´e symmetry. To simplify the analysis, we consider the Hilbert space H0 ⊗ H1, where H0 is
+the one-dimensional Hilbert space with a vacuum state |0⟩ and H1 is the irreducible subspace with
+one-particle states. Any state vector |Ψ⟩ in H1 ( |Ψ⟩ ∈ H1 ) is given by
+|Ψ⟩ =
+�
+d3q
+�
+σ
+Ψ(p, σ) ˆa†(p, σ)|0⟩,
+(49)
+10
+
+where |0⟩ is the vacuum state satisfying ˆa(p, σ)|0⟩ = 0, Ψ(p, σ) with the momentum p and the spin
+σ is the wave function, ˆa(p, σ) and ˆa†(p, σ) are the annihilation and creation operators satisfying
+[ˆa(p, σ), ˆa(p′, σ′)]± = 0 = [ˆa†(p, σ), ˆa†(p′, σ′)]±,
+[ˆa(p, σ), ˆa†(p′, σ′)]± = δ3(p − p′)δσσ′.
+(50)
+In the above notation, [ ˆA, ˆB]± is defined as [ ˆA, ˆB]± = ˆA ˆB ± ˆB ˆA, in which the signs − and + apply
+bosons and fermions, respectively. In Ref.[6, 26], the transformation rules of ˆa†(p, σ) are given by
+ˆT(a)ˆa†(p, σ) ˆT †(a) = e−ipµaµˆa†(p, σ),
+(51)
+ˆV (Λ)ˆa†(p, σ) ˆV †(Λ) =
+�
+EpΛ
+Ep
+�
+σ′
+Dσ′σ(Q(Λ, p))ˆa†(pΛ, σ′),
+(52)
+where Ep = p0, EpΛ = (Λp)0 and pΛ is the vector with the component (pΛ)i = (Λp)i. The matrix
+Q(Λ, p) = S−1
+Λp ΛSp is the element of the little group which satisfies Q(Λ, p)µνkν = kµ, where kµ
+is the standard momentum for a massive particle (kµ = [m, 0, 0, 0], m > 0) or a massless particle
+(kµ = [k, 0, 0, k], k > 0). The momentum pµ and the standard momentum kµ are connected with
+(Sp)µνkν = pµ, and Dσ′σ(Q(Λ, p)) is the irreducible unitary representation of the little group.
+We consider the Kraus operators ˆEℓ,ξ acting on the Hilbert space H0
+� H1, that is, ˆEℓ,ξ :
+H0
+� H1 → H0
+� H1, which have the following form
+ˆEℓ,ξ = Aℓ,ξˆI +
+�
+d3p
+�
+σ
+Bℓ,ξ(p, σ)ˆa(p, σ) +
+�
+d3p′d3p
+�
+σ′,σ
+Cℓ,ξ(p′, σ′, p, σ)ˆa†(p′, σ′)ˆa(p, σ).
+(53)
+The dynamical map given by these operators describes the reduced dynamics of a single particle,
+which can possibly decay into the vacuum state. Substituting the above operators into Eq.(47)
+and Eq.(48), we obtain the equations
+Aℓ,ξ = e−iℓµaµAℓ,ξ,
+(54)
+Bℓ,ξ(p, σ)e−ipµaµ = Bℓ,ξ(p, σ)e−iℓµaµ,
+(55)
+Cℓ,ξ(p′, σ′, p, σ)ei(p′µ−pµ)aµ = Cℓ,ξ(p′, σ′, p, σ)e−iℓµaµ,
+(56)
+and
+Aℓ,ξ =
+�
+ξ′
+D∗
+ξ′ξ(W −1)Aℓ,ξ′,
+(57)
+�
+EpW
+Ep
+�
+σ
+Bℓ,ξ(pW , σ)D∗
+σ′σ(Q) =
+�
+ξ′
+D∗
+ξ′ξ(W −1)Bℓ,ξ′(p, σ′),
+(58)
+�
+Ep′
+W EpW
+Ep′Ep
+�
+σ′,σ
+Cℓ,ξ(p′
+W, σ′, pW , σ)D¯σ′σ′(Q′)D∗
+¯σσ(Q) =
+�
+ξ′
+D∗
+ξ′ξ(W −1)Cℓ,ξ′(p′, ¯σ′, p, ¯σ),
+(59)
+11
+
+where Q = Q(W −1, Wp) and Q′ = Q(W −1, Wp′). The derivation of these equations is devoted in
+Appendix A.
+We can analyze the explicit form of Aℓ,ξ, Bℓ,ξ(p, σ) and Cℓ,ξ(p′, σ′, p, σ) for a massive particle
+and a massless particle, respectively. For the analysis, we assume that the massive particle has
+a finite spin and the massless particle has a finite spin and a nonzero momentum. Through the
+long computations presented in Appendices B and C, we get the following dynamical map with
+the Poincar´e symmetry,
+Φt,s[ρ(s)] = Ut,s◦Et,s[ρ(s)],
+Et,s[ρ] =
+�
+j
+�
+β(j)
+t,s
+�
+d3p
+�
+σ
+ˆa(p, σ)ρˆa†(p, σ)+α(j)
+t,s
+�ˆI+γ(j)
+t,s ˆN
+�
+ρ
+�ˆI+γ(j)∗
+t,s
+ˆN
+��
+,
+(60)
+where α(j)
+t,s , β(j)
+t,s and γ(j)
+t,s are the parameters depending on time, ˆN is the number operator defined
+as
+ˆN =
+�
+d3p
+�
+σ
+ˆa†(p, σ)ˆa(p, σ),
+(61)
+and Ut,s is the unitary map given in (29). In the form of the dynamical map Φt,s, we recovered
+the labels j which represent the degeneracies other than the labels q and ξ appearing in the Kraus
+operators ˆEq,ξ defined around (44). The parameters α(j)
+t,s, β(j)
+t,s and γ(j)
+t,s in Eq.(60) satisfy
+0 ≤ α(j)
+t,s ≤ 1,
+�
+j
+α(j)
+t,s = 1,
+0 ≤ β(j)
+t,s ,
+0 ≤
+�
+j
+β(j)
+t,s ≤ 1
+�
+j
+�
+β(j)
+t,s + α(j)
+t,s
+�
+γ(j)
+t,s + γ(j)∗
+t,s
++ |γ(j)
+t,s |2��
+= 0.
+(62)
+These conditions come from the completeness condition of the Kraus operators (3). For the com-
+putation of the completeness condition, note that the number operator ˆN satisfies ˆN 2 = ˆN on the
+Hilbert space H0 ⊗ H1, since we assume that the dynamical map describes the reduced dynamics
+of a single particle.
+From the transformation rules of the creation and the annihilation operators, Eqs.(51) and (52),
+it is easy to check that the map Et,s satisfies the condition of symmetry given in (34). Since the
+unitary map Ut,s is symmetric under the Poincar´e group, which is checked around Eq.(29), we can
+confirm that Φt,s is also symmetric.
+Let us consider the case where there is no decay under the dynamical map Φt,s and focus on the
+dynamics of one-particle states. In this case, the parameter �
+j β(j)
+t,s vanishes. Since the density
+operator ρ given by one-particle states satisfies ˆNρ = ρ = ρ ˆN, we have
+Φt,s[ρ(s)] = Ut,s ◦ Et,s[ρ(s)] =
+�
+j
+α(j)
+t,s |1 + γ(j)
+t,s |2Ut,s[ρ(s)] = Ut,s[ρ(s)],
+(63)
+12
+
+where we used the condition (62) with �
+j β(j)
+t,s = 0 in the third equality. This means that the
+dynamical map with the Poincar´e symmetry for a non-decaying particle is the unitary map. The
+result corresponds to a non-perturbative extension of the analysis in [25], which gives an implication
+on the particle dynamics. For example, if the superposition state of a particle decoheres under
+a non-unitary evolution, the Poincar´e symmetry breaks in the particle dynamics described by a
+dynamical map.
+We discuss the energy conservation. The expectation value of ˆHn at a time t, where n is a
+natural number, is computed as
+Tr[ ˆHnρ(t)] =
+�
+j
+�
+β(j)
+t,s Tr[ ˆHn
+�
+d3p
+�
+σ
+ˆa(p, σ)ρ(s)ˆa†(p, σ)] + α(j)
+t,s Tr[ ˆHn(ˆI + γ(j)
+t,s ˆN)ρ(s)(ˆI + γ(j)∗
+t,s
+ˆN)]
+�
+= (1 −
+�
+j
+β(j)
+t,s )Tr[ ˆHnρ(s)].
+(64)
+In the reduced dynamics by the dynamical map Φt,s, the energy of a single particle is not conserved
+unless �
+j β(j)
+t,s is a constant, even when the map is symmetric under the Poincar´e group. Such
+a deviation between symmetry and conservation law was discussed in, for example, Refs [23] and
+[24]. If the parameter �
+j β(j)
+t,s is a constant, then �
+j β(j)
+t,s = �
+j β(j)
+s,s = 0 and the dynamical map
+Φt,s is reduced to the unitary map Ut,s as discussed above.
+V.
+CONCLUSION
+We discussed what a dynamical map describing the reduced dynamics of an open quantum
+system is realized under the Poincar´e symmetry. The unitary representation theory of the Poincar´e
+group refines the condition for the dynamical map with the Poincar´e symmetry. For a massive
+particle and a massless particle, we derived a concrete model of the dynamical map. In the model,
+the particle can decay into the vacuum state. If there is no decay process, the dynamical map
+describes the unitary evolution generated by the Hamiltonian of the particle. This means that
+the non-decaying single particle does not decohere as long as the dynamical map for the particle
+has the Poincar´e symmetry. In this way, it was exemplified that the Poincar´e symmetry strongly
+constrains the possible dynamics of an open quantum system.
+In this paper, we assumed an open system with a single particle. Our analysis is possible to
+be extended to the case with many particles.
+Considering interactions among many particles,
+we can understand more general effective theories in terms of the Poincar´e symmetry. For the
+particles interacting via gravity, the models which induce intrinsic gravitational decoherence have
+13
+
+been proposed [15–19]. These models are written in the theory of open quantum systems. In the
+weak field regime of gravity, the Poincar´e symmetry may provide a guidance for establishing the
+theory of an open quantum system with gravitating particles.
+This paper has a potential to develop the theory of open quantum systems. To describe the
+reduced dynamics of an open quantum system, a quantum master equation is often adopted. It
+has been discussed how the quantum Markov dynamics given by the equation is consistent with
+a relativistic theory [27, 28]. Applying the present approach, it will be possible to discuss the
+quantum Markov dynamics with the Poincar´e symmetry.
+It is hoped that this paper paves the way to understand a relativistic theory of open quantum
+systems and to study the interplay between quantum theory and gravity.
+Acknowledgments
+We thank Y. Kuramochi for useful discussions and comments related to this paper. A.M. was
+supported by 2022 Research Start Program 202203.
+Appendix A: Derivation of Eqs.(54),(55),(56),(57),(58) and (59)
+We present the transformation rules of Aℓ,ξ, Bℓ,ξ and Cℓ,ξ given in Eqs.(54),(55),(56),(57),(58)
+and (59). Using the assumed form of the Kraus operators ˆEℓ,ξ defined by (53), we can compute
+the right hand side of Eq.(47) as
+ˆT †(a) ˆEℓ,ξ ˆT(a) = Aℓ,ξˆI +
+�
+d3p
+�
+σ
+Bℓ,ξ(p, σ)e−ipµaµˆa(p, σ)
++
+�
+d3p′d3p
+�
+σ′,σ
+Cℓ,ξ(p′, σ′, p, σ)ei(p′µ−pµ)aµˆa†(p′, σ′)ˆa(p, σ).
+From Eq.(47), we have
+Aℓ,ξ = e−iℓµaµAℓ,ξ,
+(A1)
+Bℓ,ξ(p, σ)e−ipµaµ = Bℓ,ξ(p, σ)e−iℓµaµ
+(A2)
+Cℓ,ξ(p′, σ′, p, σ)ei(p′µ−pµ)aµ = Cℓ,ξ(p′, σ′, p, σ)e−iℓµaµ.
+(A3)
+14
+
+The right hand side of Eq.(48) is evaluated as
+ˆV †(W) ˆEℓ,ξ ˆV (W)
+= Aℓ,ξˆI +
+�
+d3p
+�
+σ
+Bℓ,ξ(p, σ)
+�
+EpW −1
+Ep
+�
+σ′
+D∗
+σ′σ(Q(W −1, p))ˆa(pW −1, σ′)
++
+�
+d3p′d3p
+�
+σ′,σ
+Cℓ,ξ(p′, σ′, p, σ)
+×
+�
+Ep′
+W −1
+Ep′
+�
+EpW −1
+Ep
+�
+¯σ,¯σ′
+D¯σ′σ′(Q(W −1, p′))D∗
+¯σσ(Q(W −1, p))ˆa†(p′
+W −1, ¯σ′)ˆa(pW −1, ¯σ)
+= Aℓ,ξˆI +
+�
+d3p
+�
+σ
+Bℓ,ξ(pW, σ)
+�
+EpW
+Ep
+�
+σ′
+D∗
+σ′σ(Q(W −1, Wp))ˆa(p, σ′)
++
+�
+d3p′d3p
+�
+σ′,σ
+Cℓ,ξ(p′
+W, σ′, pW , σ)
+×
+�
+Ep′
+W
+Ep′
+�
+EpW
+Ep
+�
+¯σ,¯σ′
+D¯σ′σ′(Q(W −1, Wp′))D∗
+¯σσ(Q(W −1, Wp))ˆa†(p′, ¯σ′)ˆa(p, ¯σ),
+where note that the Lorentz invariant measure is d3p/Ep and hence f(p)d3p = Epf(p)d3p/Ep =
+EpΛf(pΛ)d3p/Ep. From Eq.(48), we have
+Aℓ,ξ =
+�
+ξ′
+D∗
+ξ′ξ(W −1)Aℓ,ξ′,
+(A4)
+�
+EpW
+Ep
+�
+σ
+Bℓ,ξ(pW, σ)D∗
+σ′σ(Q) =
+�
+ξ′
+D∗
+ξ′ξ(W −1)Bℓ,ξ′(p, σ′)
+(A5)
+�
+Ep′
+W EpW
+Ep′Ep
+�
+σ′,σ
+Cℓ,ξ(p′
+W , σ′, pW, σ)D¯σ′σ′(Q′)D∗
+¯σσ(Q) =
+�
+ξ′
+D∗
+ξ′ξ(W −1)Cℓ,ξ′(p′, ¯σ′, p, ¯σ),
+(A6)
+where Q = Q(W −1, Wp) and Q′ = Q(W −1, Wp′).
+Appendix B: Analysis of a massive particle
+We assume that the spectrum of ˆP µ on any state |Ψ⟩ in the Hilbert space of one-particle states,
+H1, satisfies
+ˆP µ ˆPµ|Ψ⟩ = −m2|Ψ⟩,
+⟨Ψ| ˆP 0|Ψ⟩ > 0.
+(B1)
+The above equations are equivalent to the fact that the Hamiltonian ˆH = ˆP 0 has the form ˆH =
+�
+ˆPk ˆP k + m2, which implies that |Ψ⟩ is the state of a massive particle. In this appendix, we derive
+the form of the dynamical map Et,s for a massive particle.
+15
+
+Case (a) ℓµ = [M, 0, 0, 0], M > 0 or (b) ℓµ = [−M, 0, 0, 0], M > 0 : We focus on the spectrum
+ℓµ = [±M, 0, 0, 0], M > 0. From Eq.(A1) for all aµ = [a, 0, 0, 0], we have
+Aℓ,ξ = e±iMaAℓ,ξ
+∴
+Aℓ,ξ = 0.
+Eq.(A2) for all aµ = [0, a] leads to
+Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ)
+∴
+Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p).
+From Eq.(A2) for all aµ = [a, 0, 0, 0], we get
+Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ)e±iMa,
+and combined with Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p), we obtain
+Bℓ,ξ(σ)eima = Bℓ,ξ(σ)e±iMa.
+Since the mass m is positive, to get a nontrivial result, we should choose +M with M = m. Using
+Eq.(A5) for Q = R ∈ SO(3) and adopting the result Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p), we find
+�
+σ
+Bℓ,ξ(σ)D∗
+σ′σ(R−1) =
+�
+ξ′
+D∗
+ξ′ξ(R−1)Bℓ,ξ′(σ′),
+where note that Q = Q(W −1, Wp) = Q(R−1, Rℓ) = S−1
+ℓ R−1SRℓ = R−1 for ℓµ = [m, 0, 0, 0]. Since
+the representations Dσ′σ and Dξ′ξ are irreducible and unitary, by Schur’s lemma we have
+Bℓ,ξ(σ) = Bℓ uξσ,
+where uξσ is a unitary matrix. From Eq.(A3) for all aµ = [0, a], we deduce
+Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ)
+∴
+Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p).
+Eq.(A3) for all aµ = [a, 0, 0, 0] leads to
+Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ)e±iMa,
+and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p) into the above equation, we have
+Cℓ,ξ(p, σ′, σ) = Cℓ,ξ(p, σ′, σ)e±iMa
+∴
+Cℓ,ξ(p, σ′, σ) = 0.
+The above results imply that the Kraus operator ˆEℓ,ξ with ℓµ = [m, 0, 0, 0] has the following form,
+ˆEℓ,ξ = Bℓ
+�
+σ
+uξσˆa(0, σ).
+16
+
+Eq.(46) tells us that
+ˆEq,ξ = N ∗
+q ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = N ∗
+q Bℓ
+�
+Eq
+m
+�
+σ
+uξσˆa(q, σ),
+where Eq = (Sq ℓ)0 and qi = (Sq ℓ)i. To determine the normalization Nq, the inner product v†
+q,ξvq,ξ
+is assumed to be
+v†
+q′,s′vq,ξ = δ3(q′ − q)δξ′ξ,
+which leads to Nq =
+�
+m/Eq up to a phase factor. For this normalization, the following complete-
+ness condition is given as
+�
+d3q
+�
+s
+vq,ξv†
+q,ξ = I.
+Under the completeness condition, we derive a part of the dynamical map Et,s as
+Et,s[ρ(s)] ⊃ |Bℓ|2
+�
+d3q
+�
+σ
+ˆa(q, σ)ρ(s)ˆa†(q, σ),
+(B2)
+where we used the fact that uξσ is the unitary matrix.
+Case (c) ℓµ = [κ, 0, 0, κ], κ > 0 or (d) ℓµ = [−κ, 0, 0, κ], κ > 0 : We consider the spectrum
+ℓµ = [±κ, 0, 0, κ], κ > 0. From Eq.(A1) for all aµ = [a, 0, 0, 0], we have
+Aℓ,ξ = e±iκaAℓ,ξ
+∴
+Aℓ,ξ = 0.
+From Eq.(A2) for all aµ = [0, a], we get
+Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ)e−iℓ·a
+∴
+Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ),
+where ℓ = [0, 0, κ]T. Eq.(A2) for all aµ = [a, 0, 0, 0] leads to
+Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ)e±iκa,
+and from the equation Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), we obtain
+Bℓ,ξ(σ)ei
+√
+κ2+m2a = Bℓ,ξ(σ)e±iκa
+∴
+Bℓ,ξ(σ) = 0,
+where Eℓ =
+√
+ℓ2 + m2 =
+√
+κ2 + m2. Eq.(A3) for all aµ = [0, a] gives
+Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ)e−iℓ·a
+∴
+Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′−p+ℓ).
+Using Eq.(A3) for all aµ = [a, 0, 0, 0], we get
+Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ)e±iκa,
+17
+
+and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p + ℓ) into the above equation, we have
+Cℓ,ξ(p, σ′, σ)e−i(Ep−ℓ−Ep)a = Cℓ,ξ(p, σ′, σ)e±iκa.
+Noticing the fact that Ep−ℓ − Ep ± κ ̸= 0, we get the result Cℓ,ξ(p, σ′, σ) = 0. Combined with the
+above analysis, the Kraus operator ˆEℓ,ξ vanishes:
+ˆEℓ,ξ = 0
+∴
+ˆEq,ξ = Nq ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = 0.
+(B3)
+Case (e) ℓµ = [0, 0, 0, N], N 2 > 0 : We focus on the spectrum ℓµ = [0, 0, 0, N], N 2 > 0. From
+Eq.(A1) for all aµ = [0, a], we have
+Aℓ,ξ = e−iℓ·aAℓ,ξ
+∴
+Aℓ,ξ = 0.
+Eq.(A2) for all aµ = [a, 0, 0, 0] leads to
+Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ)
+∴
+Bℓ,ξ(p, σ) = 0,
+where note that Eq =
+�
+q2 + m2 ̸= 0. From Eq.(A3) for all aµ = [0, a], we deduce
+Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ)e−iℓ·a
+∴
+Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′−p+ℓ),
+where ℓ = [0, 0, N]T. From Eq.(A3) for all aµ = [a, 0, 0, 0], we get
+Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ),
+and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p + ℓ) into the above equation, we have
+Cℓ,ξ(p, σ′, σ)e−i(Ep−ℓ−Ep)a = Cℓ,ξ(p, σ′, σ)
+∴
+Cℓ,ξ(p, σ′, σ) = Cℓ,ξ(σ′, σ)δ3(p − ℓ/2).
+Combined with the above analysis, the function Cℓ,ξ(p′, σ′, p, σ) is
+Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(σ′, σ)δ3(p′ + ℓ/2)δ3(p − ℓ/2),
+and the Kraus operator ˆEℓ,ξ is written as
+ˆEℓ,ξ =
+�
+σ′,σ
+Cℓ,ξ(σ′, σ)ˆa†(−ℓ/2, σ′)ˆa(ℓ/2, σ).
+By the completeness condition of the Kraus operators, Eq.(31), the above Kraus operator ˆEℓ,ξ
+should satisfy ˆE†
+ℓ,ξ ˆEℓ,ξ ≤ ˆI. Concretely, ˆE†
+ℓ,ξ ˆEℓ,ξ is evaluated as
+ˆE†
+ℓ,ξ ˆEℓ,ξ =
+�
+¯σ′,¯σ
+C∗
+ℓ,ξ(¯σ′, ¯σ)ˆa†(ℓ/2, ¯σ)ˆa(−ℓ/2, ¯σ′)
+�
+σ′,σ
+Cℓ,ξ(σ′, σ)ˆa†(−ℓ/2, σ′)ˆa(ℓ/2, σ)
+= δ3(0)
+�
+σ′
+� �
+¯σ
+Cℓ,ξ(σ′, ¯σ)ˆa(ℓ/2, ¯σ)
+�† �
+σ
+Cℓ,ξ(σ′, σ)ˆa(ℓ/2, σ),
+18
+
+where the term given by the linear combination of ˆa†ˆa†ˆaˆa vanishes on H0
+� H1.
+To satisfy
+ˆE†
+ℓ,ξ ˆEℓ,ξ ≤ ˆI, we find that
+�
+σ′
+� �
+¯σ
+Cℓ,ξ(σ′, ¯σ)ˆa(ℓ/2, ¯σ)
+�† �
+σ
+Cℓ,ξ(σ′, σ)ˆa(ℓ/2, σ) = 0
+∴
+Cℓ,ξ(σ′, σ) = 0
+The consequence of Cℓ,ξ(σ′, σ) = 0 is that the Kraus operator ˆEℓ,ξ vanishes as ⟨Φ| ˆEℓ,ξ|Ψ⟩ = 0 for
+all |Ψ⟩, |Φ⟩ ∈ H0
+� H1, and hence
+ˆEq,ξ = N ∗
+q ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = 0,
+(B4)
+on the Hilbert space H0
+� H1.
+Case (f) ℓµ = [0, 0, 0, 0] : We consider the case where ℓµ = [0, 0, 0, 0]. In the following, we drop
+the label ℓ. Eq.(A1) is identical for all aµ. Since the little group associated with ℓµ is SO(3, 1),
+Eq.(A4) for W = Λ ∈ SO(3, 1) is given as
+Aξ =
+�
+ξ′
+D∗
+ξ′ξ(Λ−1)Aξ′.
+(B5)
+Eq.(A2) for all aµ = [a, 0, 0, 0] gives the condition
+Bξ(p, σ)eiEpa = Bξ(p, σ)
+∴
+Bξ(p, σ) = 0,
+where note that Eq =
+�
+q2 + m2 ̸= 0. From Eq.(A3) for all aµ, we obtain
+Cξ(p′, σ′, p, σ)ei(p′µ−pµ)aµ = Cξ(p′, σ′, p, σ)
+∴
+Cξ(p′, σ′, p, σ) = Cξ(p, σ′, σ)δ3(p′ − p).
+Eq. (A6) for W = Λ ∈ SO(3, 1) is written as
+�
+Ep′
+ΛEpΛ
+Ep′Ep
+�
+σ′,σ
+Cξ(p′
+Λ, σ′, pΛ, σ)D¯σ′σ′(Q′)D∗
+¯σσ(Q) =
+�
+ξ′
+D∗
+ξ′ξ(Λ−1)Cξ′(p′, ¯σ′, p, ¯σ),
+where Q
+=
+Q(Λ−1, Λp) and Q′
+=
+Q(Λ−1, Λp′).
+From the equation Cξ(p′, σ′, p, σ)
+=
+Cξ(p, σ′, σ)δ3(p′ − p) and noticing the fact that the invariant delta function is Epδ3(p − p′), we
+get the condition
+�
+σ′,σ
+Cξ(pΛ, σ′, σ)D¯σ′σ′(Q)D∗
+¯σσ(Q) =
+�
+ξ′
+D∗
+ξ′ξ(Λ−1)Cξ′(p, ¯σ′, ¯σ),
+(B6)
+where Q′ = Q(Λ−1, Λp′) turns out to be Q = Q(Λ−1, Λp) by the presence of the delta function
+δ3(p − p′). It is known that the dimension of irreducible unitary representations Dξ′ξ of SO(3,1)
+19
+
+is one or infinite [29]. For the one-dimensional representation, dropping the label ξ, we find that
+Eq.(B5) trivially holds and that Eq.(B6) is reduced to
+�
+σ′,σ
+C(pΛ, σ′, σ)D¯σ′σ′(Q)D∗
+¯σσ(Q) = C(p, ¯σ′, ¯σ).
+For p = 0 and Λ = R ∈ SO(3), we get
+�
+σ′,σ
+C(0, σ′, σ)D¯σ′σ′(R−1)D∗
+¯σσ(R−1) = C(0, ¯σ′, ¯σ)
+∴
+C(0, σ′, σ) = Cδσ′σ,
+where this holds by the Schur’s lemma. Choosing p = 0 and Λ = Sp with (Sp)µνkν = pµ for
+kµ = [m, 0, 0, 0], we have
+C(p, σ′, σ) = C(0, σ′, σ),
+where we used Q = Q(S−1
+p , p) = S−1
+k S−1
+p Sp = I and Dσ′σ(I) = δσ′σ. Hence C(p, σ′, σ) = Cδσ′σ.
+For the infinite dimensional representation, Eq.(B5) leads to Aℓ,ξ = 0, and Eq.(B6) for p = 0 and
+Λ = R ∈ SO(3) gives
+�
+σ′,σ
+Cξ(0, σ′, σ)D¯σ′σ′(R−1)D∗
+¯σσ(R−1) =
+�
+ξ′
+D∗
+ξ′ξ(R−1)Cξ′(0, ¯σ′, ¯σ).
+Assuming that the massive particle has a finite spin and using the Schur’s lemma, we get
+Cξ′(0, ¯σ′, ¯σ) = 0. Eq.(B6) for p = 0 and Λ = Sp with (Sp)µνkν = pµ for kµ = [m, 0, 0, 0] pro-
+vides
+Cξ(p, σ′, σ) =
+�
+ξ′
+D∗
+ξ′ξ(S−1
+p )Cξ′(0, σ′, σ) = 0.
+The above analysis on ℓµ = [0, 0, 0, 0] tells us the following Kraus operator
+ˆE = AˆI + C ˆN,
+where ˆN is the number operator defined in (61). A part of the dynamical map Et,s is given as
+Et,s[ρ(s)] ⊃
+�
+AˆI + C ˆN
+�
+ρ(s)
+�
+AˆI + C ˆN
+�†
+.
+(B7)
+The above results given in Eqs.(B2), (B3), (B4) and (B7) provide the following form of Et,s:
+Et,s[ρ(s)] = |Bℓ|2
+�
+d3q
+�
+σ
+ˆa(q, σ)ρ(s)ˆa†(q, σ) +
+�
+AˆI + C ˆN
+�
+ρ(s)
+�
+AˆI + C ˆN
+�†
+.
+(B8)
+Recovering other degeneracies labeled by j differently from q and ξ, introducing �
+j and redefining
+the parameters as |A|2 = α(j)
+t,s , C/A = γ(j)
+t,s and |Bℓ|2 = β(j)
+t,s , we get the form of the dynamical map
+Et,s given in (60).
+20
+
+Appendix C: Analysis on a massless particle
+We assume that the spectrum of ˆP µ on any state |Ψ⟩ in the Hilbert space of one-particle states,
+H1, satisfies
+ˆP µ ˆPµ|Ψ⟩ = 0,
+⟨Ψ| ˆP 0|Ψ⟩ > 0.
+(C1)
+The above equations leads to the fact that the Hamiltonian ˆH = ˆP 0 has the form ˆH =
+�
+ˆPk ˆP k,
+which means that |Ψ⟩ is the state of a massless particle. In this appendix, we derive the form of
+the dynamical map Et,s for a massless particle with nonzero momentum.
+Case (a) ℓµ = [M, 0, 0, 0], M > 0 or (b) ℓµ = [−M, 0, 0, 0], M > 0 : We focus on the spectrum
+ℓµ = [±M, 0, 0, 0], M > 0. Eq.(A1) for all aµ = [a, 0, 0, 0] gives
+Aℓ,ξ = e±iMaAℓ,ξ
+∴
+Aℓ,ξ = 0.
+Eq.(A2) for all aµ = [0, a] leads to
+Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ)
+∴
+Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p).
+From Eq.(A2) for all aµ = [a, 0, 0, 0], we get
+Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ)e±iMa,
+and combined with Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p), we obtain
+Bℓ,ξ(σ) = Bℓ,ξ(σ)e±iMa
+∴
+Bℓ,ξ(σ) = 0.
+Using Eq.(A3) for all aµ = [0, a], we deduce
+Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ)
+∴
+Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p).
+Eq.(A3) for all aµ = [a, 0, 0, 0] leads to
+Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ)e±iMa,
+and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p) into the above equation, we have
+Cℓ,ξ(p, σ′, σ) = Cℓ,ξ(p, σ′, σ)e±iMa
+∴
+Cℓ,ξ(p, σ′, σ) = 0.
+The above results imply that the Kraus operator ˆEℓ,ξ vanishes and has the following form,
+ˆEq,ξ = N ∗
+q ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = 0.
+(C2)
+21
+
+Case (c) ℓµ = [κ, 0, 0, κ], κ > 0 or (d) ℓµ = [−κ, 0, 0, κ], κ > 0 : We consider the spectrum
+ℓµ = [±κ, 0, 0, κ], κ > 0. From Eq.(A1) for all aµ = [a, 0, 0, 0], we have
+Aℓ,ξ = e±iκaAℓ,ξ
+∴
+Aℓ,ξ = 0.
+Eq.(A2) for all aµ = [0, a] gives
+Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ)e−iℓ·a
+∴
+Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ),
+where ℓ = [0, 0, κ]T. Eq.(A2) for all aµ = [a, 0, 0, 0] leads to
+Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ)e±iκa,
+and from the equation Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), we have
+Bℓ,ξ(σ)eiκa = Bℓ,ξ(σ)e±iκa,
+where Eℓ =
+√
+ℓ2 = κ.
+To get a nontrivial result, we should choose +κ.
+Using Eq.(A5) for
+Q = L ∈ ISO(2) and adopting the result Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), we find
+�
+σ
+Bℓ,ξ(σ)D∗
+σ′σ(L−1) =
+�
+ξ′
+D∗
+ξ′ξ(L−1)Bℓ,ξ′(σ′),
+where note that Q = Q(W −1, Wp) = Q(L−1, Lℓ) = S−1
+ℓ L−1SLℓ = L−1 for ℓµ = [κ, 0, 0, κ]. Since
+the representations Dσ′σ and Dξ′ξ are irreducible and unitary, by Schur’s lemma we get
+Bℓ,ξ(σ) = Bℓ uξσ,
+where uξσ is a unitary matrix. Using Eq.(A3) for all aµ = [0, a], we deduce
+Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ)e−iℓ·a
+∴
+Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′−p+ℓ).
+Eq.(A3) for all aµ = [a, 0, 0, 0] leads to
+Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ)e±iκa,
+and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p + ℓ) into the above equation, we have
+Cℓ,ξ(p, σ′, σ)e−i(Ep−ℓ−Ep)a = Cℓ,ξ(p, σ′, σ)e±iκa.
+The condition of Ep−ℓ − Ep + κ = 0 is written by p⊥ = [p1, p2] = 0 and p3 ≥ κ, and the condition
+of Ep−ℓ − Ep − κ = 0 is given by p⊥ = 0 and p3 ≤ 0. Hence, the form of Cℓ,ξ(p, σ′, σ) is
+Cℓ,ξ(p, σ′, σ) =
+�
+C+
+ℓ,ξ(p3, σ′, σ)θ(p3 − κ) + C−
+ℓ,ξ(p3, σ′, σ)θ(−p3)
+�
+δ2(p⊥)
+22
+
+Combined with the above analysis, the Kraus operator ˆE+
+ℓ,ξ for +κ is
+ˆE+
+ℓ,ξ = Bℓ
+�
+σ
+uξσˆa(ℓ, σ) +
+�
+dp3
+�
+σ,σ′
+C+
+ℓ,ξ(p3, σ′, σ)θ(p3 − κ)ˆa†(0, p3 − κ, σ′)ˆa(0, p3, σ),
+and the Kraus operator ˆE−
+ℓ,ξ for −κ is
+ˆE−
+ℓ,ξ =
+�
+dp3
+�
+σ,σ′
+C−
+ℓ,ξ(p3, σ′, σ)θ(−p3)ˆa†(0, p3 − κ, σ′)ˆa(0, p3, σ).
+By the completeness condition of the Kraus operators, Eq.(31), the above Kraus operator ˆE±
+ℓ,ξ
+should satisfy ˆE±†
+ℓ,ξ ˆE±
+ℓ,ξ ≤ ˆI. Concretely, ˆE+†
+ℓ,ξ ˆE+
+ℓ,ξ is evaluated as
+ˆE+†
+ℓ,ξ ˆE+
+ℓ,ξ =
+�
+Bℓ
+�
+σ
+uξσˆa(ℓ, σ) +
+�
+dp3
+�
+σ,σ′
+C+
+ℓ,ξ(p3, σ′, σ)θ(p3 − κ)ˆa†(0, p3 − κ, σ′)ˆa(0, p3, σ)
+�†
+×
+�
+Bℓ
+�
+σ
+uξσˆa(ℓ, σ) +
+�
+dp3
+�
+σ,σ′
+C+
+ℓ,ξ(p3, σ′, σ)θ(p3 − κ)ˆa†(0, p3 − κ, σ′)ˆa(0, p3, σ)
+�
+= |Bℓ|2 �
+σ
+u∗
+sσˆa†(ℓ, σ)
+�
+σ′
+usσ′ˆa(ℓ, σ′)
++ δ2(0)
+�
+dp3
+�
+σ′
+�
+σ,¯σ
+C+∗
+ℓ,ξ (p3, σ′, σ)C+
+ℓ,ξ(p3, σ′, ¯σ)θ(p3 − κ)ˆa†(0, p3, σ)ˆa(0, p3, ¯σ),
+where the term given by the linear combination of ˆa†ˆaˆa, ˆa†ˆa†ˆa, and ˆa†ˆa†ˆaˆa vanishes on H0
+� H1.
+To satisfy ˆE+†
+ℓ,ξ ˆE+
+ℓ,ξ ≤ ˆI, we find
+�
+dp3
+�
+σ′
+�
+σ,¯σ
+C+∗
+ℓ,ξ (p3, σ′, σ)C+
+ℓ,ξ(p3, σ′, ¯σ)θ(p3−κ)ˆa†(0, p3, σ)ˆa(0, p3, ¯σ) = 0
+∴
+C+
+ℓ,ξ(p3, σ′, ¯σ) = 0
+In the same manner, we have
+ˆE−†
+ℓ,ξ ˆE−
+ℓ,ξ = δ2(0)
+�
+dp3
+�
+σ′
+�
+σ,¯σ
+C−∗
+ℓ,ξ (p3, σ′, σ)C−
+ℓ,ξ(p3, σ′, ¯σ)θ(−p3)ˆa†(0, p3, σ)ˆa(0, p3, ¯σ),
+and to satisfy ˆE−†
+ℓ,ξ ˆE−
+ℓ,ξ ≤ ˆI, we obtain
+�
+dp3
+�
+σ′
+�
+σ,¯σ
+C−∗
+ℓ,ξ (p3, σ′, σ)C−
+ℓ,ξ(p3, σ′, ¯σ)θ(−p3)ˆa†(0, p3, σ)ˆa(0, p3, ¯σ) = 0
+∴
+C−
+ℓ,ξ(p3, σ′, ¯σ) = 0.
+These analyses give the following form of the Kraus operators,
+ˆE+
+ℓ,ξ = Bℓ
+�
+σ
+uξσˆa(ℓ, σ),
+ˆE−
+ℓ,ξ = 0,
+on the Hilbert space H0
+� H1. By Eq. (46), we get
+ˆE+
+q,ξ = N ∗
+q ˆV (Sq) ˆE+
+ℓ,ξ ˆV †(Sq) = N ∗
+q Bℓ
+�
+Eq
+κ
+�
+σ
+uξσˆa(q, σ),
+ˆE−
+q,ξ = N ∗
+q ˆV (Sq) ˆE−
+ℓ,ξ ˆV †(Sq) = 0.
+23
+
+Setting that the inner product v†
+q,ξvq,ξ is
+v†
+q′,s′vq,ξ = δ3(q′ − q)δξ′ξ,
+the normalization Nq is given as Nq =
+�
+κ/Eq up to a phase factor. For this normalization, we
+get the following completeness condition as
+�
+d3q
+�
+s
+vq,ξv†
+q,ξ = I.
+Taking account for the completeness, we can derive a part of the dynamical map Et,sas
+Et,s[ρ(s)] ⊃ |Bℓ|2
+�
+d3q
+�
+σ
+ˆa(q, σ)ρ(s)ˆa†(q, σ),
+(C3)
+where we used the fact that uξσ is the unitary matrix.
+Case (e) ℓµ = [0, 0, 0, N], N 2 > 0 : We focus on the spectrum ℓµ = [0, 0, 0, N], N 2 > 0. From
+Eq.(A1) for all aµ = [0, a], we have
+Aℓ,ξ = e−iℓ·aAℓ,ξ
+∴
+Aℓ,ξ = 0.
+Eq.(A2) for all aµ = [0, a] leads to
+Bℓ,ξ(p, σ)e−ip·a = Bℓ,ξ(p, σ)e−iℓ·a
+∴
+Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ).
+Eq.(55) for all aµ = [a, 0, 0, 0] gives
+Bℓ,ξ(p, σ)eiEpa = Bℓ,ξ(p, σ),
+and then combined with Bℓ,ξ(p, σ) = Bℓ,ξ(σ)δ3(p − ℓ), we get
+Bℓ,ξ(σ)eiκa = Bℓ,ξ(σ)
+∴
+Bℓ,ξ(σ) = 0
+where we used Eℓ =
+√
+ℓ2 = κ > 0. Adopting Eq.(A3) for all aµ = [0, a], we deduce
+Cℓ,ξ(p′, σ′, p, σ)ei(p′−p)·a = Cℓ,ξ(p′, σ′, p, σ)e−iℓ·a
+∴
+Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′−p+ℓ),
+where ℓ = [0, 0, N]T. From Eq.(A3) for all aµ = [a, 0, 0, 0], we get
+Cℓ,ξ(p′, σ′, p, σ)e−i(Ep′−Ep)a = Cℓ,ξ(p′, σ′, p, σ),
+and substituting Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(p, σ′, σ)δ3(p′ − p + ℓ) into the above equation, we have
+Cℓ,ξ(p, σ′, σ)e−i(Ep−ℓ−Ep)a = Cℓ,ξ(p, σ′, σ)
+∴
+Cℓ,ξ(p, σ′, σ) = Cℓ,ξ(σ′, σ)δ3(p − ℓ/2).
+24
+
+Combined with the above analysis, the function Cℓ,ξ(p′, σ′, p, σ) is
+Cℓ,ξ(p′, σ′, p, σ) = Cℓ,ξ(σ′, σ)δ3(p′ + ℓ/2)δ3(p − ℓ/2),
+and the Kraus operator ˆEℓ,ξ is written as
+ˆEℓ,ξ =
+�
+σ′,σ
+Cℓ,ξ(σ′, σ)ˆa†(−ℓ/2, σ′)ˆa(ℓ/2, σ).
+By the completeness condition of the Kraus operators, Eq.(31), the above Kraus operator ˆEℓ,ξ
+should satisfy ˆE†
+ℓ,ξ ˆEℓ,ξ ≤ ˆI. Explicitly, ˆE†
+ℓ,ξ ˆEℓ,ξ is evaluated as
+ˆE†
+ℓ,ξ ˆEℓ,ξ =
+�
+¯σ′,¯σ
+C∗
+ℓ,ξ(¯σ′, ¯σ)ˆa†(ℓ/2, ¯σ)ˆa(−ℓ/2, ¯σ′)
+�
+σ′,σ
+Cℓ,ξ(σ′, σ)ˆa†(−ℓ/2, σ′)ˆa(ℓ/2, σ)
+= δ3(0)
+�
+σ′
+� �
+¯σ
+Cℓ,ξ(σ′, ¯σ)ˆa(ℓ/2, ¯σ)
+�† �
+σ
+Cℓ,ξ(σ′, σ)ˆa(ℓ/2, σ),
+where the term associated with the linear combination of ˆa†ˆa†ˆaˆa vanishes on H0
+� H1. To satisfy
+ˆE†
+ℓ,ξ ˆEℓ,ξ ≤ ˆI, we find that
+�
+σ′
+� �
+¯σ
+Cℓ,ξ(σ′, ¯σ)ˆa(ℓ/2, ¯σ)
+�† �
+σ
+Cℓ,ξ(σ′, σ)ˆa(ℓ/2, σ) = 0
+∴
+Cℓ,ξ(σ′, σ) = 0
+Hence, the Kraus operator ˆEℓ,ξ vanishes, and we have that
+ˆEq,ξ = N ∗
+q ˆV (Sq) ˆEℓ,ξ ˆV †(Sq) = 0,
+(C4)
+on the Hilbert space H0
+� H1.
+Case (f) ℓµ = [0, 0, 0, 0] : We focus on the spectrum ℓµ = [0, 0, 0, 0]. In the following, we do
+not write the label ℓ. Eq.(A1) is identical for all aµ. Since the little group associated with ℓµ is
+SO(3, 1), Eq.(A4) for W = Λ ∈ SO(3, 1) is given as
+Aξ =
+�
+ξ′
+D∗
+ξ′ξ(Λ−1)Aξ′.
+(C5)
+Eq.(A2) for all aµ = [0, a] gives the condition
+Bξ(p, σ)e−ip·a = Bξ(p, σ)
+∴
+Bξ(p, σ) = Bξ(σ)δ3(p).
+This equation makes Eq.(A2) for all aµ = [a, 0, 0, 0] and Eq. (A5) for W = Λ ∈ SO(3, 1) trivial.
+This form Bξ(p, σ) = Bξ(σ)δ3(p) leads to ˆEℓ,ξ ⊃ �
+σ Bξ(σ)ˆa(0, σ). However, this operator vanishes
+on the Hilbert space of massless particles since we assumed that there are no states with zero
+momentum. Eq.(A3) for all aµ gives us the condition
+Cξ(p′, σ′, p, σ)ei(p′µ−pµ)aµ = Cξ(p′, σ′, p, σ)
+∴
+Cξ(p′, σ′, p, σ) = Cξ(p, σ′, σ)δ3(p′ − p).
+25
+
+Eq. (A6) for W = Λ ∈ SO(3, 1) is written as
+�
+Ep′
+ΛEpΛ
+Ep′Ep
+�
+σ′,σ
+Cξ(p′
+Λ, σ′, pΛ, σ)D¯σ′σ′(Q′)D∗
+¯σσ(Q) =
+�
+ξ′
+D∗
+ξ′ξ(Λ−1)Cξ′(p′, ¯σ′, p, ¯σ),
+where Q
+=
+Q(Λ−1, Λp) and Q′
+=
+Q(Λ−1, Λp′).
+From the equation Cξ(p′, σ′, p, σ)
+=
+Cξ(p, σ′, σ)δ3(p′ − p) and noticing the fact that the invariant delta function is Epδ3(p − p′), we
+get the condition
+�
+σ′,σ
+Cξ(pΛ, σ′, σ)D¯σ′σ′(Q)D∗
+¯σσ(Q) =
+�
+ξ′
+D∗
+ξ′ξ(Λ−1)Cξ′(p, ¯σ′, ¯σ),
+(C6)
+where note that the delta function δ3(p − p′) leads to Q′ = Q(Λ−1, Λp′) = Q(Λ−1, Λp) = Q. The
+(proper orthochronous) Lorentz group SO(3, 1) has one and infinite dimensional unitary irreducible
+representations [29]. Choosing the one-dimensional representation of Dξ′,ξ and dropping the label
+ξ, we find that Eq.(C5) trivially holds and that Eq.(C6) is reduced to
+�
+σ′,σ
+C(pΛ, σ′, σ)D¯σ′σ′(Q)D∗
+¯σσ(Q) = C(p, ¯σ′, ¯σ).
+For p = ℓ = [0, 0, κ] and Λ = L ∈ ISO(2), we get
+�
+σ′,σ
+C(ℓ, σ′, σ)D¯σ′σ′(L−1)D∗
+¯σσ(L−1) = C(ℓ, ¯σ′, ¯σ)
+∴
+C(ℓ, σ′, σ) = Cδσ′σ,
+where we used the Schur’s lemma. Choosing p = ℓ and Λ = Sp with (Sp)µνℓν = pµ for ℓµ =
+[κ, 0, 0, κ], we have
+C(p, σ′, σ) = C(ℓ, σ′, σ),
+where we used Q = Q(S−1
+p , p) = S−1
+k S−1
+p Sp = I and Dσ′σ(I) = δσ′σ. Hence C(p, σ′, σ) = Cδσ′σ.
+If we adopt the infinite dimensional representation of Dξ′ξ, Eq.(C5) leads to Aℓ,ξ = 0 and Eq.(C6)
+for p = ℓ and Λ = L ∈ ISO(2) gives
+�
+σ′,σ
+Cξ(ℓ, σ′, σ)D¯σ′σ′(L−1)D∗
+¯σσ(L−1) =
+�
+ξ′
+D∗
+ξ′ξ(L−1)Cξ′(ℓ, ¯σ′, ¯σ).
+Assuming that the massless particle has a finite spin and using the Schur’s lemma, we get
+Cξ′(ℓ, ¯σ′, ¯σ) = 0. Eq.(C6) for p = ℓ and Λ = Sp with (Sp)µνℓν = pµ for ℓµ = [κ, 0, 0, κ] pro-
+vides
+Cξ(p, σ′, σ) =
+�
+ξ′
+D∗
+ξ′ξ(S−1
+p )Cξ′(ℓ, σ′, σ) = 0.
+26
+
+The above analysis tells us that the Kraus operator has the following form
+ˆE = AˆI + C ˆN,
+where ˆN is the number operator defined in (61).
+A part of the dynamical map Et,s with the
+Poincar´e symmetry is given as
+Et,s[ρ(s)] ⊃
+�
+AˆI + C ˆN
+�
+ρ(s)
+�
+AˆI + C ˆN
+�†
+.
+(C7)
+Gathering the above results (C2), (C3), (C4) and (C7), we have the following form of Et,s:
+Et,s[ρ(s)] = |Bℓ|2
+�
+d3q
+�
+σ
+ˆa(q, σ)ρ(s)ˆa†(q, σ) +
+�
+AˆI + C ˆN
+�
+ρ(s)
+�
+AˆI + C ˆN
+�†
+.
+(C8)
+In the same manner performed around (B8), we obtain the form of the dynamical map Et,s given
+in (60).
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+(Springer-Verlag, Berlin, Heidelberg, 2001).
+[22] M. Keyl, “Fundamentals of quantum information theory”, Phys. Rep. 369, 431 (2002).
+[23] C. Cˆırstoiu, K. Korzekwa, and D. Jennings, “Robustness of Noether’s Principle: Maximal Disconnects
+between Conservation Laws and Symmetries in Quantum Theory”, Phys. Rev. X 10, 041035 (2020).
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+28
+

3tE3T4oBgHgl3EQfPwn9/content/tmp_files/2301.04407v1.pdf.txt

@@ -0,0 +1,1588 @@
+Beyond-Standard-Model
+Physics Associated with the
+Top Quark
+Roberto Franceschini
+Università degli Studi and INFN Roma Tre, Via della Vasca Navale
+84, I-00146, Roma
+email: [email protected]
+xxxxxx 0000. 00:1–25
+Copyright © 0000 by Annual Reviews.
+All rights reserved
+Keywords
+top quark, beyond the standard model, hierarchy problem, flavor,
+dark matter, new physics
+Abstract
+We review scenarios of physics beyond the Standard Model in which the
+top quark plays a special role. Models that aim at the stabilization of
+the weak scale are presented together with the specific phenomenology
+of partner states that are characteristic of this type of model.
+Fur-
+ther, we present models of flavor in which the top quark is singled out
+as a special flavor among the SM ones. The flavor and collider phe-
+nomenology of these models is broadly presented. Finally, we discuss
+the possibility that dark matter interacts preferably with the top quark
+flavor and broadly present the dark matter phenomenology of these
+scenarios, as well as collider and flavor signals.
+1
+arXiv:2301.04407v1  [hep-ph]  11 Jan 2023
+
+Contents
+1. Introduction .................................................................................................
+2
+2. Top quark and BSM related to the Higgs boson and the origin of the weak scale .........................
+3
+2.1. Supersymmetry..........................................................................................
+3
+2.2. Composite and pNGB/Little Higgs .....................................................................
+8
+3. EFT at current and future colliders..........................................................................
+11
+4. Top quark and BSM related to Flavor Dynamics or Dark Matter (or both)................................
+12
+4.1. Top quark and BSM related to flavor ..................................................................
+12
+4.2. Flavored dark matter models ...........................................................................
+15
+5. Conclusions ...................................................................................................
+18
+1. Introduction
+The top quark is a singular object amidst the fermions of the Standard Model as it is the
+heaviest among them. This entails several peculiar properties: in the domain of QCD it
+stands out as it is the only quark never to be observed into a hadron, as its decay is much
+faster than the hadron formation time; after the discovery of the Higgs boson, and for
+all the time before in which the Higgs mechanism has dominated the landscape of model
+building for the electroweak sector of the SM, the top quark stands out as the only one
+with a “normal” size coupling with the Higgs boson. This latter property has made the top
+quark very interesting both for the question about the origin of the structure of flavor in
+the SM and for the origin of the electroweak scale itself. The special interest about top
+flavor has to do with its strong preference to decay into bottom quarks, i.e not involving
+other flavor families, which in the CKM picture results in Vtb = 1 up to small corrections,
+and its large mass, which can possibly act as a magnifier of the effects of physics beyond
+the Higgs boson as origin of flavor. For electroweak physics the top quark plays a crucial
+role in that it affects the properties of the Higgs boson, and by the Higgs mechanism for
+weak bosons mass generation, also in the physics of weak gauge bosons: its effect can be
+seen in their masses and decay rates, which are sensitive to the strength of the top quark
+gauge and Yukawa couplings and to its mass. Deviations of these properties from the SM
+predictions can be signs of new physics related to the top quark. While the importance of
+the top quark can be appreciated already from these general facts, the detailed role played
+by the top quark can be better understood going closer to explicit new physics models,
+which will pace the exposition of the greatest part of the following material.
+In sections 2.1 and 2.2 we discuss models in which the top quark plays a special role for
+the origin of the electroweak symmetry. The discussion is further extended in section 3 in a
+more model independent direction using a flavor-conserving effective field theory of the top
+quark sector, which also allow to discuss prospects for top quark physics at future colliders.
+In section 4.1 we attack a different problem, that of the origin of the flavors of the SM. In
+section 4.2 we extend the discussion to the possibility that SM flavor plays a part in the
+stabilization of the dark matter in a way that makes the dark matter interact preferably
+with the top quark flavor and discuss the phenomenology of dark matter in these scenarios.
+Finally in section 5 we offer some conclusions.
+Being the subjects list rather large, the discussion is necessarily kept free from some
+details, which are available in the provided references. This review is conceived so that it
+2
+
+can also be useful for younger graduate students seeking an high-level introduction to the
+subject(s) discussed. Hopefully the readers can start here their own exploration on topics
+that would otherwise require to go through a large stack of literature. References are kept
+to a minimum of key works as to encourage the reader to actually study these selected
+works.
+2. Top quark and BSM related to the Higgs boson and the origin of the weak
+scale
+2.1. Supersymmetry
+Supersymmetry has been proposed as a space-time symmetry involving fermionic genera-
+tors. Unlike in gauge symmetries, this makes possible to involve spin and momentum in the
+definition of the symmetry algebra, which, up to violations of the symmetry itself, would
+require interactions and masses of bosonic and fermionic particles to be tightly related.
+One such relation would require the electron to be accompanied by exactly mass degener-
+ate states of spin-0, pretty much the same as Lorentz symmetry of space-time built-in the
+Dirac equation implies the existence of exactly mass degenerate anti-particles of the elec-
+tron. The absence of any evidence in experiments for spin-0 electron-like state motivates
+to consider supersymmetry as an approximate symmetry, broken at some unknown scale so
+that all the supersymmetric partners of the SM states are pushed beyond the mass scale
+presently probed by experiments.
+The mechanism for supersymmetry breaking is a subject for model building, which is
+outside of the scope of this review. For our purpose it is key to recall that the supersym-
+metry breaking top quark sector has the rather model-independent tendency to determine
+the Higgs bosons mass and quartic coupling, thus leading to the identification of the su-
+persymmetric top scalar quark, most often called “stop squark”, as the main player setting
+the Higgs boson potential. In the Minimal Supersymmetric Standard Model (see (1) for an
+extensive review) this is represented by equations for the constraints on the minimization
+of the Higgs boson potential
+m2
+Z =
+��m2
+Hd − m2
+Hu
+��
+�
+1 − sin2(2β)
+− m2
+Hu − m2
+Hd − 2 |µ|2 ,
+sin(2β) =
+2b
+m2
+Hu + m2
+Hd + 2 |µ|2 ,
+coupled with the 1-loop effect of the top quark and top squark on the bilinear terms of the
+2 Higgs doublets Hu and Hd . In particular, for the Higgs doublet Hu that interacts with
+up-type quarks, hence feels the top quark sector, the RGE equations is
+d
+d log Qm2
+Hu = 3Xt − 6g2 |M2|2 − 6
+5g2
+1 |M1|2 + 3
+5g2
+1S ,
+1.
+where Xt = 2 |yt|2 �
+m2
+Hu + m2
+Q3 + m2
+¯u3
+�
++ 2 |at|2, M1,2 are the U(1) and SU(2) gaugino
+mass terms, and S = Tr[Yjm2
+φj].
+These equations naturally lead to possibility that the supersymmetry breaking stop
+masses m2
+Q3 and m2
+¯u3 or a large A-term |at| might induce a large Xt, which in turn drives
+m2
+Hu < 0 as log Q diminishes from some high-scale down to the weak scale. This possibility
+has made the role of stop squarks a very central one in supersymmetric models. In essence,
+www.annualreviews.org •
+3
+
+the supersymmetric partner of the top quark is responsible for breaking the electroweak
+symmetry, by making m2
+Hu < 0 hence making the Higgs boson potential unstable at the
+origin of the Hd, Hu fields space, and setting the value of the masses that set the weak scale,
+e.g. mZ from the above equation or the mass of the Higgs boson that receives the above
+mentioned large radiative corrections from the stop squark.
+As a matter of fact, once the Higgs boson was discovered and its mass was known,
+a number of works tried to determine the impact of this measurement on the properties
+of the stop squark (e.g. see Ref. (2) for the MSSM and some extensions). In turn, the
+necessity for peculiar supersymmetry breaking to accommodate the Higgs mass has spurred
+investigations on the possible supersymmetry breaking models that can lead to such peculiar
+stop squarks (see e.g. (3–8) for some examples of supersymmetry breaking models emerged
+or re-emerged to address the null searches of supersymmetry and the Higgs discovery).
+2.1.1. Phenomenology. The phenomenology of the supersymmetric partners of the top quark
+is largely dictated by one feature of the supersymmetric models: the existence of a conserved
+quantum number that distinguishes SM states from their supersymmetric partners. The
+standard choice for such quantity is called R-parity, a Z2 symmetry under which all SM
+states are even and all partners states are odd. The conservation of this symmetry implies
+that partners states can appear in interaction vertexes only in even number, e.g.
+one
+SM states can interact with two supersymmetric states and it is not possible for a single
+supersymmetric state to interact with a pair of SM states. For particle colliders this implies
+that the lowest order process to produce supersymmetric states in collisions is
+SM SM → SUSY SUSY,
+and the decay of supersymmetric particles to any number of SM states is forbidden unless
+there is at least one supersymmetric particle (or an odd number of them), e.g.
+SUSY → SUSY SM .
+When R-parity is exact a most copious production mechanism for stop squarks at the LHC
+is
+gg → ˜ti˜t∗
+j,
+2.
+where we denoted ˜tk for k = 1, 2 the two stop squarks mass eigenstates 1. Other production
+mechanisms are possible, e.g. in decays of supersymmetric partners heavier than the stops
+or via production of stops in association with other supersymmetric states.
+Once produced, the stop squark can decay in a number of possible channels, depending
+on which supersymmetric states are lighter than the state ˜tk at hand. Most studied 2-body
+decay modes are
+˜t → tχ0, ˜t → bχ+ ,
+3.
+which feature fermions χ that are mixtures of supersymmetric partners of gauge bosons of
+the electroweak interactions and of the Higgs bosons of the model. The motivation for the
+1The definition of mass eigenstate as “stops” assumes that flavor labels we give in the SM are
+the same for the partners states. It must be stressed that the fate of flavor in the supersymmetric
+partners sector is largely model dependent and it is possible to use flavor mixing to change the
+phenomenology of stop squarks, see e.g.
+(9).
+See (1) for more details on the gauge and flavor
+structure of the squark sector.
+4
+
+prevalence of these decay modes is that, by the rules of unbroken supersymmetry, these
+decays are mediated by couplings given by gauge and Yukawa couplings of the SM, hence
+they are pretty much impossible to switch off unless m˜t − mχ < 0. As a matter of fact
+the quantity m˜t − mχ plays a major role in determining the stop phenomenology. When
+m˜t − mχ → 0 it becomes necessary to consider multi-body processes are also possible and
+may be phenomenologically relevant, e.g.
+˜t → bW +χ0, ˜t → b ¯ff
+′χ0,
+4.
+as well as possible flavor violating decays that may be induced at loop level, such as
+˜t → cχ0 .
+5.
+In the above discussion the particle χ0 is considered as the lightest supersymmetric state
+(LSP), so that, by the conservation of R-parity, it is absolutely stable. As χ0 is not elec-
+trically charged and it is color neutral, pretty much like neutrinos it does not leave directly
+observables traces in detectors. For this reason the presence of χ0 can be detected only as
+momentum missing in the overall momentum conservation in each collision. As we cannot
+reliably measure the fractions of the longitudinal momentum of the colliding protons taken
+by the partons initiating the production of stops, e.g. the gluons entering in eq.(2), and
+the fraction taken by the rest of the partons, the longitudinal momentum conservation is
+usually not exploited in hadron colliders, therefore the presence of χ0 is usually sought for
+as missing transverse momentum, most often (mis)named missing transverse energy mET.
+Being an electrically neutral stable particle charged only under supersymmetric Yukawa
+and electroweak gauge interactions, χ0 qualifies as perfect candidate for a WIMP Dark Mat-
+ter. The possibility to have a Dark Matter candidate stemming out of supersymmetry has
+given formidable motivation to pursue this scenario for the past decades. So much so, that
+missing transverse energy searches have becomes synonymous of searches for supersymme-
+try. It must be said, however, that the null searches of supersymmetric particles, as well as
+WIMP Dark Matter in the mass range suitable for χ0(10), has put this idea under great
+pressure lately (11,12).
+Given these experimental results, and the vast range of possible models for supersym-
+metry breaking, it must be recalled that in general it is possible to have other states than χ0
+as lightest supersymmetric particles. For instance the supersymmetric partner of a neutrino
+or even top sector squarks. The latter leads to peculiar phenomena due to the formation of
+hadrons containing supersymmetric states(13)(14), but these models typically suffer from
+quite stringent limits (15–17). Therefore the majority of the searches for supersymmetric
+states in the top quark sector are carried out in the χ0 LSP setting.
+Wholly alternative phenomenological scenarios for supersymmetric top quark partners
+are possible and are actively pursued in experimental searches. The main possible alter-
+native has to do with the non-conservation of R-parity (18).
+With broken R-parity all
+supersymmetric particles can in principle be produced singly and can decay into just SM
+states, e.g.
+SM SM → SUSY and SUSY → SM SM ,
+are now possible processes. In this situation there is no longer an absolutely stable weak
+scale particle to purse the idea of Dark Matter as a WIMP2 and the phenomenology of
+2Alternative DM candidates can be found in these models, see e.g. (19) for a possible gravitino
+dark matter scenarios and issues related to this possibility.
+www.annualreviews.org •
+5
+
+supersymmetric states linked to the top quark is now greatly different from the picture
+given above (20). For instance R-parity violating couplings, still respecting the full gauge
+symmetry of the SM, allow, among other possibilities, the decays
+˜t → bs or ˜t → ℓd .
+As the final states of stop decay can now be made entirely of SM particles it is possible
+to detect stop squarks as resonances, a very powerful signature, that is not possible to
+pursue when χ0 is forced to appear among the decay products. Furthermore these decays,
+being mediated by R-parity breaking couplings, that need to be small for a number of
+constraints (18), can lead to meta-stable supersymmetric states, which can live measurable
+lengths in experiments.
+2.1.2. Experimental searches. In a detailed model it is possible to derive very specific signals
+from top sector supersymmetric partners, including both signatures at collider experiments
+and as well as low energy precision ones. The latter, however, turn out to be usually very
+much dependent on the model considered for low energy precision experiments (21). A
+similar issue exists with early universe physics, on top of the signals being quite difficult
+to detect.
+For this reason collider searches are the prime way to search for top sector
+supersymmetric partners.
+Before listing relevant searches it is necessary to clarify a point on their scope. The
+above searches are sensitive in principle to any sign of new physics related to the top quark
+sector involving mET or some kind of pair produced resonances.
+Although the search
+is optimized for supersymmetric partners, it can indeed be used to set bounds on other
+models. The interested reader can refer for instance to Ref. (22) for an interpretation of the
+“supersymmetry searches” in the context of fermionic top partners to be discussed in later
+Section 2.2.2.
+The searches for top sector supersymmetric partners can be divided into two main
+categories:
+• searches in large momentum transfer signals, which feature detector objects (jets,
+leptons, photons, ...) with energy and transverse momentum greater than the typical
+SM events;
+• searches in low momentum transfer signal, in which the detector objects arising from
+top sector supersymmetric partners production are not very different from that of
+typical SM events.
+The large momentum transfer ones are “classic” searches for new physics, and were envisaged
+already at the time of design of the experiments (23,24). Currently these searches can probe
+supersymmetric top partners up to a mass around 1.2 TeV, although not in full generality.
+Indeed it is quite hard to probe in full generality even a model as “minimal” as one having the
+full freedom to vary the branching ratios of decays eqs.(3)-(5). For a complete assessment is
+then necessary to test very accurately a large number of searches at once, often relying on a
+“phenomenological” incarnation of a sufficiently general supersymmetric model, as studied
+for instance in Ref. (25).
+The interpretation of these results is quite difficult, as many
+constraints on the model are imposed at once, e.g. the top partners states are required to
+“fix” the mass of the SM Higgs boson to its measured value by the dynamics of radiative
+corrections embodied in eq.(1). This requirement, while being a sensible one in the context
+of the specific model, can significantly alter the conclusion of that study.
+Therefore it
+6
+
+remains difficult to answer questions as simple as finding the lightest not excluded values
+of the mass of stop-like top partners 3.
+Further difficulties can arise and make nearly impossible to probe experimentally su-
+persymmetric top partners, e.g when special kinematical configurations become the typical
+configuration of top partners decay products. In these cases the search in low momentum
+transfer signatures can help. Indeed, these searches have been developed to overcome the
+difficulty that arise in the limit m˜t − mχ → 0. The shortcomings of the large momentum
+transfer searches can be clearly seen in Figure 1, as the excluded stop mass for large m˜t−mχ
+is much larger than for small values of this mass difference. In addition, when the stop-LSP
+mass gap is small and the stop becomes lighter, its production and decay cannot be reliably
+distinguished from other SM processes, e.g. the SM top quark production. This observa-
+tion motivates a zoom inset in the figure to display how these peculiar cases are covered.
+The most useful strategies to attack these difficult signatures have turned out to be the
+studies of angular observables and fiducial rates of top-like final states (27–31). Especially
+in angular observables there are modest, but persistent disagreement between the measure-
+ments in the top quark sample (32) and theoretical predictions. These disagreement are
+also accompanied by other disagreements of small entity, but persisting from Run1 LHC
+through Run2, in the kinematics of the reconstructed top quarks e.g. in Refs. (33,34). The
+possibility to see effects of BSM related to the top quark and the precision in measurements
+afforded by the LHC and the HL-LHC has motivated the great improvement of predictions
+for top quark SM observables, e.g. (35) for a seamless description of fixed NLO and PS cal-
+culations of top quark resonant and non-resonant rates, (36,37) for specific NNLO and EW
+corrections to the BSM sensitive rates and more in general drawing attention on possibly
+BSM-sensitive high energy top quarks (see e.g. (38)) and other production modes which
+may be of interest for both SM studies and BSM searches (see e.g. (39,40)).
+The searches mentioned above, though motivated and sometimes optimized on super-
+symmetry searches, are rather general. Thus it is important to stress that the observation
+of an excess in one of these “supersymmetry searches” would not at all prove the supersym-
+metric nature of the discovered state. A reliable statement on the supersymmetric nature
+of the newly discovered object would require several measurements. For some proposal at
+the LHC the interested reader can look for instance at (41). In general it is believed that a
+machine cleaner than a hadron collider, e.g. an e+e− collider, capable of producing the new
+particle would be needed to truly confer it the status of “supersymmetric partner” state of
+some SM state.
+At the time of writing there are no statistically significant and convincing signs of new
+physics in searches for new physics, the searches for supersymmetric top partners being no
+exception. Despite the absence of signals for top sector supersymmetric partners these are
+still believed to one our best chances to find new physics. Looking at the glass as “half
+full” one could even argue that in the minimal model of supersymmetry the relatively large
+observed Higgs boson mass requires large loop level corrections from contributions of the
+kind of eq.(1). These large loop corrections point towards a stop squarks mass scale at the
+TeV or larger, thus perfectly compatible with the present limits and possibly awaiting us
+for a next discovery at one of the next updates of the searches as more data is collected at
+the LHC.
+3One possible answer in the context of (25) is offered in the supplementary material of that
+analysis(26).
+www.annualreviews.org •
+7
+
+Observed limits
+Expected limits
+  
+  -1
+ = 13 TeV, 139 fb
+s
+Data 15-18, 
+0
+1
+χ∼
+ bff' 
+→
+ 
+1t~
+monojet, 
+[2102.10874]
+0
+1
+χ∼
+ bff' 
+→
+ 
+1t~
+ / 
+0
+1
+χ∼
+ bW
+→
+ 
+1t~
+ / 
+0
+1
+χ∼
+ t
+→
+ 
+1t~
+0L, 
+[2004.14060]
+0
+1
+χ∼
+ bff' 
+→
+ 
+1t~
+ / 
+0
+1
+χ∼
+ bW
+→
+ 
+1t~
+ /  
+0
+1
+χ∼
+ t
+→
+ 
+1t~
+1L,  
+[2012.03799] 
+0
+1
+χ∼
+ bff' 
+→
+ 
+1t~
+ / 
+0
+1
+χ∼
+ bW
+→
+ 
+1t~
+ /  
+0
+1
+χ∼
+ t
+→
+ 
+1t~
+2L,  
+[2102.01444]
+   
+  -1
+ = 13 TeV, 36.1 fb
+s
+Data 15-16, 
+0
+1
+χ∼
+ bff' 
+→
+ 
+1t~
+ / 
+0
+1
+χ∼
+ bW
+→
+ 
+1t~
+ / 
+0
+1
+χ∼
+ t
+→
+ 
+1t~
+[1709.04183, 1711.11520, 
+ 1708.03247, 1711.03301]
+0
+1
+χ∼
+ t
+→
+ 
+1t~
+,  
+tt
+[1903.07570]
+   
+  -1
+ = 8 TeV, 20.3 fb
+s
+Data 12, 
+0
+1
+χ∼
+ bff' 
+→
+ 
+1t~
+ / 
+0
+1
+χ∼
+ bW
+→
+ 
+1t~
+ / 
+0
+1
+χ∼
+ t
+→
+ 
+1t~
+[1506.08616]
+200
+400
+600
+800
+1000 1200
+) [GeV]
+1 
+t~
+m( 
+100
+200
+300
+400
+500
+600
+700
+800
+900
+) [GeV]
+ 0
+ 1
+χ∼
+m(
+  -1
+ = 8,13 TeV, 20.3-139 fb
+s
+March 2021
+ATLAS Preliminary
+ production
+1t~
+1t~
+Limits at 95% CL
+180
+200
+220
+0
+10
+20
+30
+40
+50
+60
+70
+) = 0
+0
+1
+χ∼
+, 
+1t~
+ m( 
+∆
+W
+ + m
+b
+) = m
+0
+1
+χ∼
+, 
+1t~
+ m( 
+∆
+t
+) = m
+0
+1
+χ∼
+, 
+1t~
+ m( 
+∆
+Figure 1
+Searches for top sector supersymmetric partners in the Stop-LSP mass plane.
+As the mass scale of top quark supersymmetric partners is not entirely fixed it often
+considered that these particles may be too heavy for the LHC to discover them. Therefore
+the discovery reach for these particles is often considered in the evaluation of the physics case
+of future particle accelerators. Projections for a 100 TeV pp collider (42, 43) usually cover
+a mass range 5-8 times larger than what can be probed at the LHC, while the expectation
+for a high energy lepton collider, such as multi-TeV muon collider(44–48), is to probe the
+existence of top partners up to the kinematic limits at √s/2.
+2.2. Composite and pNGB/Little Higgs
+2.2.1. Models . New physics associated to the top quark sector has been motivated also from
+a series of model building activities aimed at explaining the origin of the electroweak scale
+through the Goldstone boson nature of the agent of its breaking, resulting in theories of
+the Higgs boson as a pseudo Nambu-Goldstone boson. From a low energy effective point of
+view these theories can be put in the language of a composite Higgs boson, whose lightness
+compared to its scale of compositeness is justified by its goldstonian nature. Models built
+in this family are reviewed in Refs. (49–52) and they all share the need to enlarge the
+symmetries of the SM by a new global symmetry, that is broken at some scale above the
+TeV to a smaller symmetry, with the associated Nambu-Goldstone bosons, which will host
+the yet smaller symmetry group of the SM at even lower energies. The minimal model of
+this type (53) that is able to pass bounds from electroweak precision tests including Zb¯b
+8
+
+couplings assumes an SO(5) global symmetry, broken to SO(4) ≃ SU(2) × SU(2) which
+contain the weak interactions gauged SU(2).
+The enlargement of the symmetry of the SM motivates appearance of matter repre-
+sentations in multiplets that are necessarily larger than the usual doublets and singlets of
+the SM. In particular, in order to obtain Yukawa interactions the constructions of pNGB
+and little Higgs model converges in the existence of “partner” states for the top quark, the
+bottom quark and in principle for all the fermions of the SM. The precise phenomenological
+manifestation of the “partner” states is highly model dependent, as it depends on the choice
+the new global symmetry group that one has in building this type of models, the repre-
+sentation of this symmetry group that one chooses for the new matter and the imagined
+mechanism to originate the SM fermion masses at the most microscopic level.
+One possible limitation to the model building choices may comes from the requirement of
+not introducing large deviations in well known couplings, e.g. the Zbb couplings (54), still a
+large set of possibilities exists. For this review we focus on a unifying feature of many models,
+that is the presence of “partner” states directly connected to the SM top quark sector via
+Yukawa and gauge interactions with relatively universal decay patterns (55–57), although
+other decay modes and more “exotic” partners may exist including possible couplings to
+scalar states accompanying the Higgs boson in some models (58–60).
+2.2.2. Phenomenology. At the core of the experimental tests of the idea of fermion top part-
+ners lies the assumption that the main interaction leading to the decay of these top partners
+into SM states is the Yukawa of the top quark, in which the Higgs boson or longitudinal
+components of the weak gauge bosons appear. For this reason the large majority of the
+searches are presented in terms of exclusions for branching fractions of the top partners
+states into the following pairs of SM states
+T → tZ, th, Wb,
+where T is a charge 2/3 top partner and
+B → bZ, bh, Wt,
+where B is a charge -1/3 partner of the bottom quark, whose existence is consequence of
+the SU(2) weak isospin symmetry that must hold in the theory that supersedes the SM at
+high energies. In models with a symmetry larger than SU(2), e.g. (54)(53), it is typical
+to have further partners states that appear as necessary to furnish full representations of
+the larger symmetry. A much studied case is the state of charge 5/3 that leads to a very
+characteristic decay
+X5/3 → W +t ,
+which in turn gives a characteristic same-sign di-lepton signal (61). For little Higgs models
+the appearance of this type of exotic partners requires the formulation of somewhat more
+involved models, but it is definitively a possibility(58,62).
+2.2.3. Experimental searches at colliders. Experimental searches for new states are carried
+out at the LHC exploiting the color charge of the top partners in processes such as
+gg → TT ,
+www.annualreviews.org •
+9
+
+that are analogous to previous processes for supersymmetric partners and depend only on
+the QCD charge of T. Unlike for supersymmetric partners, for which the conservation of
+R-parity plays a crucial role, the single production of top partners
+gq → q′Tb ,
+is possible in the most minimal models and can in principle lead to a deeper understanding of
+the BSM physics, as this process involves directly new physics couplings for the production
+of the top partners state (63). For instance the rate of single production of top partners
+states can be a discriminant with respect to so-called “vector-like” quarks, whose couplings
+are not dictated by Goldstone property of the Higgs (see Ref. (64) for a more in-depth
+discussion).
+A great difference in the search for the partners discussed in this section is that they
+can in principle give rise to resonant signals, e.g. in the invariant mass of an hadronic top
+and one hadronic Higgs boson in the decay T → th and other signals discussed for instance
+in the search of Ref.(65).
+Another consequence of the top partner decaying in purely SM final states is that even
+the “heavy” SM particles, such as t, Z, W, h, are produced with significant boost in the
+majority of the events. This motivates the use of special experimental techniques for the
+identification of those detector objects (66) as for instance in the search of Ref.(67).
+The search strategies mentioned above are combined by the experimental collaborations,
+that present results in a plane with axes spanning the possible values of two decays, e.g. if
+figure 2 an example is shown for T → Ht and T → Wb. The underlying assumption of this
+presentation of the results is that the top partner does not decay in any BSM states, hence
+the branching ratio of T → Zt is determined by the two branching rations displayed. The
+right panel of the same figure shows how the different searches have different sensitivity to
+each decay mode and can be patched together to better exclude top partners of a given
+mass. For more exotic signals from X5/3 searches are carried out as well, e.g. in Ref. (68).
+Results of searches at LHC collected in figure 2 and newer results (67, 69) on the kinds of
+top partners described so far put bounds on the top partners mass at around 1.2 TeV.
+As mentioned above it is possible to have larger groups and larger representations in the
+symmetry breaking pattern. For instance if the large global symmetry of which breaking
+the Higgs is a pNGB is chosen to be SO(6) broken to SO(5) and top quark partners states
+are chosen to furnish a 6-dimensional representation there is one extra top partners state
+compared to the case of top quark partners in the 5 of SO(5) considered for the minimal
+model of Ref. (54). If we call this new top partner state Ψ1, the name signals the fact
+that it is a singlet under the remnant SO(5) symmetry, we can have new signals from
+its production via QCD interactions and decay that do not fit into any of the previously
+considered categories e.g.
+Ψ1 → th, tZ, tη, Wb ,
+where η is an extra pNGB that arises due to the larger number of broken generators in the
+breaking SO(6) → SO(5) → SO(4) ≃ SU(2) × SU(2) .
+In general the extensions of pNGB models can include possible FCNC of top quarks
+with new physics states, e.g. Ref. (73) has considered decays of the SM top quark that
+violate flavor
+t → cη
+as a consequence of underlying flavor-changing dynamics in the top partners by a coupling
+Tcη which would also yield a new possible search channel for a top partner T → cη. Other
+10
+
+Figure 2
+Searches for top fermionic partners (70,71) in the plane BR(T → Ht) vs BR(T → Wb) with the
+constraint B(bW) + B(th) + B(tZ) = 1. For reference, some model-dependent choices of the
+branching ratios introduced in Ref.(72) are shown.
+exotic possibilities are covered in the literature, e.g. T → tg, tγ, X5/3 → tφ+ and more
+exotics ones are presented in Refs. (74–76) and can in principle lead to new signals for top
+quark partners.
+3. EFT at current and future colliders
+The previous sections dealt with explicit models of new physics giving rise to signals from
+direct production of particles beyond those of the Standard Model. As these searches have
+so far yield no evidence of new physics a growing interest and motivation have risen for
+the description of new physics in Effective Field Theories. The effective character of these
+theories is due to the fact that they arise by the removal of heavy states from a theory more
+microscopic than the SM and they lead to a set of BSM interactions, that is usually in overlap
+with the set generated by other microscopic theories. Therefore it has been done a great
+work in identifying the most general sets of interactions under given assumptions (77,78),
+so that new physics studies can be carried in a “model-independent” fashion, e.g. searching
+for very characteristic interactions involving four top quarks (79–82) or other four-fermion
+operators involving top quarks, or other kinds of contact interactions independently of their
+microscopic origin.
+The plus side of the EFT approach is that it is very comprehensive. The converse of
+this comprehensiveness is the possible loss of contact with the microscopic origin of physics
+beyond the Standard Model which gives rise to specific patterns and organization principles
+for the size of each contact interaction. Thus it is necessary to strike a balance between a
+fully general EFT and a “physically efficacious” effective theory. Where this balance lies is
+very much dependent on the amount of data that one can use in constraining the couplings
+of the effective interactions, as well as the theoretical prejudice on what effects are worth
+being considered, e.g. pure top sector effects (78,83–87), or effects involving EW and Higgs
+physics as well (88,89) or exploring flavor changing effects (90–95).
+As the effect of BSM contact interactions from the EFT affects precision measurements
+of SM processes, this enhanced attention towards signals of BSM associated to top quarks
+www.annualreviews.org •
+11
+
+(↑H
+1
+m, = 800 Gev
+m, = 900 GevV 
+ATLAS
+0.8
+ATLAS Preliminary
+1420
+Vs = 13 TeV, 36.1 fb-1
+BR(T 
+0.6
+0.9
+个
+..Exp.exclusion Obs.exclusion
+Vs = 13 TeV, 36.1 fb'
+ limit [
+0.
+W(lv)b+X [arxiv:1707.03347]
+1400
+0.8
+VLQ combination
+1400
+0.2
+H(bb)+X [arxiv:1803.09678]
+R
+Z(vv)t+X [ariv:1705.10751]
+B
+0.7
+Observed limit
+nass 
+m = 950 Gev 
+m = 1000 Gev 
+1375
+0.8
+Trilep./same-sign [CERN-EP-2018-171]
+0.6
+Z(I)/b+X [arxiv:1806.10555]
+0.6
+1380
+m
+0.4
+All-had [CERN-EP-2018-176]
+★ SU(2) doublet
+★ sU(2) doublet ● sU(2) singlet 
+O sU(2) singlet
+1360
+95%
+1320
+m = 1050 Gev 
+m = 1100 GeV
+m = 1150 Gev 
+0.4
+0.8F
+0.6
+0.3
+1340
+0.4
+0.2
+0.2
+1320
+m = 1200 GeV ±
+m = 1300 GeV
+m, = 1400 GeV
+0.8
+0.1
+★
+1300
+0.4
+0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
+0.2
+.
+BR(T → Wb)
+0.2 0.4 0.6 0.8
+00.2 0.4 0.6 0.8
+0.2 0.4 0.6 0.8
+0
+BR(T →> Wb)Figure 3
+Summary from Ref. (101,102) of the constraints on contact interactions involving the top quark.
+The left panel shows the effect of HL-LHC compared to present constraints. The right panel
+shows the effect of future e+e− machines. The taller (and lighted) bars for each case represent the
+looser bounds that are obtained when the coupling of interest is bound while the others are
+allowed to float, see (101,102) and references therein for details.
+has produced activity on the improvement of the description of several processes that are
+either backgrounds or serve as SM reference on top of which search for signs of BSM,
+e.g. see recent Ref. (96) for four-top production, recent ttV results discussed in Refs. (97–
+99), tth results in Ref. (100) and references therein. For an up to date snapshot of the
+characterization of the top quark electroweak interactions and possible BSM in deviations
+from the SM we refer the reader to Refs. (83–87). The upshot of the work is that present
+measurements, also thanks to the availability of differential measurements and trustable
+computations in the same phase-space regions, can put bounds on generic new physics in
+the top quark sector in the TeV ballpark.
+The possibility to identify indirect signs of new physics in signatures related to the top
+quark has become a commonly used benchmark in the evaluation of performances of future
+colliders, especially clean e+e− machines, whose best chance to see new physics in the top
+sector is through indirect effects. Works such as (101–104) have studied the outcome of
+analyses to be carried out at future colliders and the interplay between present and future
+colliders probes of new physics in top quark effective field theory. The results are summa-
+rized in Figure 3, which shows the significant improvement that will be attained by the
+HL-LHC, especially on single-couplings effects. The figure also shows the strong tightening
+of the bounds with the addition of data from future e+e− data at the Zh threshold, the t¯t
+threshold and above, which will make the global EFT constraints particularly robust by the
+removal of possible flat directions in couplings-space and providing new data in channels
+that can be probed best at clean e+e− machines.
+4. Top quark and BSM related to Flavor Dynamics or Dark Matter (or both)
+4.1. Top quark and BSM related to flavor
+The top quark flavor remains a special one in the SM. Indeed the top quark is so heavy that
+one can easily single out the third generation of quarks as a peculiars source of breaking of
+12
+
+102
+FIT
+I LHC Run 2 + Tevatron + LEP
+ +HL-LHC S2
+HL-LHC +CEPC
+HL-LHC + FCCee
+HL-LHC +ILC
+ HL-LHC + CLIC
+FIT
+HEPfit
+101
+HEPfit
+101
+val (TeV-2)
+100
+100
+95% Interv
+10-1
+10-2
+10-2
+c
+10-3
+Ctw
+Cot
+CoQ
+Ctz
+Cb
+Ctp
+Ctw
+Cot
+Cpb
+Ceb
+CeQ
+Clb
+Cet
+Cit
+Cio
+Operator Coefficients
+Operator Coefficientsthe flavor symmetry
+GF = U(3)qL × U(3)uR × U(3)dR
+that the SM would enjoy if all quark masses were zero. A hierarchy of breaking dominated by
+the third generation can be accommodated easily, thanks to the freedom about the possible
+symmetry breaking patterns and possible mechanisms for breaking the flavor symmetry of
+the SM that one can consider. In addition, this way of organizing the breaking of flavor
+symmetry is most compatible with experimental bounds. In fact, bounds on first and second
+generation flavor changing processes are the most tight, whereas there is a relative lack of
+constraints on the third generation. If the sole breaking of the symmetry GF arises from
+the Yukawa couplings of the SM, or new sources are aligned with the Yukawa matrices,
+the breaking is said to comply with “Minimal Flavor Violation” (MFV) (105–107). In this
+setting the bounds from flavor observables are most easily accommodated, but it is not the
+only possibility to comply with observations. The fact that the top quark Yukawa coupling
+is a possible large source of flavor symmetry breaking motivates to consider BSM related
+to the top flavor, but this conclusion holds also in other settings.
+A classification of possible states that can couple to quark bilinears charged under the
+flavor symmetry, e.g. a new scalar coupled as φtutu, has proven useful in the past to assess
+the possibility of flavorful signs of new physics. For a recent listing of the possible states
+one can read tables on Ref. (108). From a phenomenological point of view these models
+give rise to transitions in four-quark scatterings that do not conserve the flavor charge. For
+instance the scattering
+uu → tt
+can arise via a t-channel exchange of a flavored boson. This can alter the kinematic of
+top quark production as well as the net charge of the top quark sample at hadron colliders.
+Indeed new flavorful boson of this kind were advocated in response to TeVatron experiments
+claiming disagreements between the SM predictions and measured top quark properties,
+such as the forward-backward asymmetry in the production of top quarks (109–111). In
+addition these new flavored states coupled to the top quark can give rise to transitions
+f ¯f → tφtjuj ,
+that can be observed quite easily at e+e− colliders in multi-jet final states, the detailed
+final state depending on the model-dependent decay of the flavored state φ.
+The possibility that a flavored state connected to the top quark might be among the
+lightest new states from the new physics sector has appeared also in models of gauged
+flavor symmetries.
+In these models the flavor symmetry GF is gauged, as to not have
+to deal with unobserved massless Goldstone bosons.
+For instance Refs. (112, 113) have
+proposed a new set of states that would have the notable property to make the GF gauging
+free from triangular anomalies by the addition of vector-like new quarks. In this kind of
+models the new quarks are charged under the SM flavor symmetry and can be arranged as
+to have top-flavor new states to be the lightest ones. Indeed in these models the masses of
+the the SM quarks would be explained by a see-saw-like mechanism in which the lightest
+SM fermions are mixed with a very heavy new state, whereas the heaviest SM states are
+mixed with the lightest of the new physics states. In this case the SM top quark would be
+the state coupled to the lightest of the new physics states, named t′, possibly accompanied
+by a partner state for the bottom quark, named b′. Remarkably this type of model gives
+www.annualreviews.org •
+13
+
+Λ ��
+(���)
+�������� �����
+������
+��� ������
+������ �����
+��� ������
+������ �����
+��������
+��
+ϵ�
+�� → μ+μ-
+������� ���
+�������� ���
+μ → � γ
+��
+��
+��
+���
+�
+�
+��
+��
+Figure 4: Lower bounds on ⇤IR on the various flavor scenarios. The first set of bounds corresponds
+to our scenario with multiple flavor scales, the second and third sets assume partial compositeness
+at ⇤IR for the whole third and second family respectively, while the last set gives the bounds for the
+anarchic flavor scenario. To derive the numerical values we have taken g⇤ ' 3, xt ' xc ' 0.5, and
+set all free ↵L,R parameters to one.
+where
+gij ⌘ Ytxt(V †
+CKM)i3(VCKM)3j ,
+(7.2)
+and dLi denotes the left-handed down-type quark component in the i-th family. A remarkable
+feature of these corrections is the fact that they automatically follow a MFV structure. The first
+operator contributes to �F = 2 transitions and generates correlated e↵ects in the ✏K, �MBd and
+�MBs observables, which are of the order of the present experimental sensitivity if we take ⇤IR ⇠
+TeV and we allow for a slight reduction of the left-handed top compositeness, xt < 1. The second
+operator of Eq. (7.1) gives flavor-changing Z-couplings. At present it only pushes the ⇤IR scale in
+the few TeV range. In the future it can be seen either in deviations in the decays K ! µµ or
+B ! (X)``. This contribution can however be significantly smaller if the strong sector is invariant
+under a custodial PLR symmetry, which protects the down-type quark couplings to the Z boson [30].
+Additional contributions to �F = 2 operators can also be generated at the scales ⇤c,s,d at
+which the second and first family quarks get their masses. These corrections however only give
+a sizable e↵ect on ✏K, that pushes the ⇤IR scale in the multi-TeV range (⇤IR & 6 TeV), which
+is still a milder bound with respect to the anarchic one. It must however be stressed that these
+bounds depend on the coe�cients of the e↵ective operators which are a↵ected by some degree
+of uncertainty. These contributions to ✏K severely constrain the maximal dimension of the OH
+operator, requiring dH . 2.
+We also considered possible variations of the framework described above. For example, a more
+economical scenario has been proposed in which each family is associated to a single flavor scale
+at which the bilinear mass operators are generated. A few additional new-physics flavor e↵ects
+20
+Figure 4
+Lower bound on the scale of new physics related to the SM fermion mass generation in a composite Higgs scenario (117)
+under different assumptions on the compositeness of SM fermions.
+phenomenological signatures very similar to those of top partners states of composite and
+little Higgs, e.g. the partner states can be produced by strong interactions and decay as
+b′ → bh, bZ, tW
+and
+t′ → th, tZ, bW .
+These ideas also lend themselves to be paired with supersymmetry. Although super-
+symmetry is not necessary for the idea of gauged flavor symmetries in general, these models
+can provide a setup to originate R-parity breaking with an underlying structure for the
+flavor structure of the RPV couplings (114,115), that for instance would motivate
+˜t → bs
+as the main channel to search RPV stops (116).
+A solution with a hierarchy of flavored new physics scales inverted with respect to that of
+the SM quarks has been proposed also for composite Higgs models (117–120), which would
+otherwise suffer from severe bounds from high-pT and flavor observables (see e.g. (121–123)),
+even in presence of some degree of model building (124–126) aimed at keeping all the new
+physics at a common low-scale and still survive flavor tests thanks to a friendly, possibly
+MFV-like structure, of the flavor origin in the microscopic completion of the composite
+Higgs model. As it can be appreciated in Fig. 4 the top quark sector emerges still as a less
+constrained one and further motivates to consider BSM physics related to the top quark,
+and possibly exclusively to the top quark or to the third generation of SM fermions.
+14
+
+Observables of interests include indirect probes such as electric dipoles moments (see
+e.g. (127)), meson oscillations and decays, and in principle rare Z and Higgs bosons flavor-
+violating decays which usually receive important contributions from the top quark sec-
+tor (117). In addition, it is possible to have phenomena more directly related to the top
+quark such as
+t → cV,
+where V = γ, Z, g(128,129) and deviations from Vtb = 1 in the CKM matrix (64,130–132).
+4.2. Flavored dark matter models
+Given the strength of the bounds from direct searches of dark matter scattering on heavy
+nuclei it has become interesting to consider dark matter models in which the flavor of SM
+quarks and leptons plays a role, as the strongest bounds hinge on effective couplings of the
+dark matter to first and, to a slightly lesser extent, to second generation quarks and gluons.
+Rather interestingly the flavor puzzle of the SM comes equipped with a symmetry,
+which, though not exact, can be used to stabilize the dark matter if it is broken according to
+Minimal Flavor Violation (133,134) and even with more general patterns of flavor symmetry
+and its breaking (135). As a dark matter coupling sensitive to flavor could mediate flavor
+changing transitions the option of the MFV structure, or slight departures from it, has been
+so far been a main route in model building aimed at removing possible tensions with flavor
+observables.
+Among the possible flavor structures that the Dark Matter and the SM can fields can
+be cast in, for our work here we focus on the possibility that the top quark flavor has a
+special role. Explicit models have appeared in the context of possible explanations of the
+CDF AF B anomaly (109–111), e.g. see the model built in Ref. (136), but the idea stands
+out on itself even without anomalies in top quark physics. Indeed if one considers that the
+complexity of the SM may be replicated in the sector of dark matter it is natural to consider
+multiple species of dark matter, that are “flavors” of dark matter (137–139). These flavors
+can be separated from our own SM flavors or can be related to our species of fermions. In
+case some relation exists between flavors of the SM and of the dark sector the possibility
+that the top quark flavored dark matter is the lightest state is at least as probable as any
+other flavor assumption. For example, when Minimal Flavor Violation is advocated one can
+explicitly write a mass term for the dark-flavor fermion multiplet χ which in general has
+the form
+¯χ (m0 + Υ(Y Y )) χ ,
+where Υ is a function of combinations of the Yukawa matrices of the SM that form singlets
+under the flavor group that is dominated by the piece proportional to Y †
+u Yu, hence the top
+quark flavor tends to be special just from the principle of MFV itself. In a concrete case
+we can have interactions of SM fermions u(i)
+R
+and mass terms for the dark matter flavor
+multiplet χ
+φ¯χ
+�
+g0 + g1Y †
+u Yu
+�
+u(i)
+R + h.c. + ¯χ
+�
+m0 + m1Y †
+u Yu + ...
+�
+χ ,
+6.
+where φ is a suitable representation of GSM ⊗ GF .
+In Ref. (136) for instance φ ∼
+(3, 1, 2/3)SM ⊗ (1, 1, 1)F , χ ∼ (1, 1, 0)SM ⊗ (1, 3, 1)F and the Yukawa matrices, as in
+general in MFV, transform as spurions Yu ∼ (3, ¯3, 1)F and Yd ∼ (3, 1, ¯3)F . We see that
+it is possible to pick m1 as to partly cancel the flavor universal m0 term, making χt the
+www.annualreviews.org •
+15
+
+lightest particle of the χ multiplet while retaining full freedom to pick the combinations of
+g0 and g1 that corresponds to the couplings of the mass eigenstates χi.
+In absence of a field φ one can imagine contact operators to couple the Dark Matter
+and the SM flavors i and j, e.g. operators of the type
+(¯χΓSχ)
+�
+¯ψ(i)ΓSψ(j)�
+7.
+for some Lorentz structure ΓS have been considered as low energy remnants of flavored
+gauge bosons (137) or other heavy scalar and fermion states charged under a MFV-broken
+flavor symmetry or in a horizontal symmetry model (138). Operators involving the SM
+Higgs boson, e.g.
+� ¯Qχ
+�
+(χ∗Hu)
+have also been considered in (133) for a scalar χ ∼ (1, 1, 0)SM ⊗ (3, 1, 1)F . A variation of
+the model of Ref. (137) could lead to top quark flavor being singled out, the other referred
+works already consider the third generation, hence the top quark and/or the bottom quark,
+as special due to either the MFV structure or as a result of the horizontal symmetry.
+The phenomenology of top flavored dark matter is very rich as it comprises both possible
+signals in dark matter searches and in precision flavor observables as well as in high energy
+collider searches. Flavor observables put in general stringent bounds on flavored dark matter
+models, the case of top-flavored dark matter being significantly less constrained due to
+majority of data belonging to u, d, s, c, b quark systems. Dark matter direct detection is also
+in general suppressed because nucleons involved in dark matter scattering do not contain
+top flavor, hence the interactions are usually originated at loop level or via breaking of the
+flavor alignments, i.e. the dark matter interacts almost exclusively with top quark flavor,
+but it may have a small, though not completely negligible coupling to light flavors. The
+existence of such coupling depends on the model. A specific analysis for a case in which only
+top quark flavor interacts with the DM in the model eq.(6) is presented in Ref. (140) for both
+dark matter direct detection and collider prospects in a MFV scenario. The annihilation
+rate for the thermal freeze-out is set by the scattering
+χχ → tt
+8.
+mediated by a mediator φ (other scatterings are discussed in detail for instance in (141)).
+In this specific case the direct detection scattering on nucleons
+χN → χN
+is mediated by a loop induced couplings of Z, γ to χ from a bubble loop of t and φ from
+eq.(6).
+Despite the smallness of these couplings the reach of current and future large
+exposure experiments, e.g.
+see (142), could probe such low level of scattering rates for
+exposure around 1 ton year, that means the model can be tested with presently available
+data (10).
+A more recent analysis (143) considered flavor, direct dark matter detection and collider
+searches for a model featuring a top-flavored dark matter χ and a new state φ. In this work
+a “Dark Minimal Flavor Violation” flavor structure that extends MFV, but can recover it as
+a limit, is considered and allows for a more generic structure in flavor space for the vertex
+λij ¯u(i)
+R φχ + h.c. .
+16
+
+In this context it is possible to delay the observation of χ in direct detection experiments,
+as new contributions to the direct detection rate appear compared to the MFV case and
+it is possible to arrange for cancellations among scattering amplitudes. It remains an open
+questions if it is going to be possible to claim an observation in spite of the so-called
+“neutrino fog” that future Xenon experiments (142) face when probing rates so small that
+neutrinos from the Sun, supernovae and other natural sources are expected to contribute
+an event rate comparable or larger than that of the dark matter.
+In principle it is possible to have mχ < mt so that the thermal freeze-out is controlled by
+other processes than the simple tree-level exchange of eq.(8). Reference (143) experimented
+with this possibility in Dark Minimal Flavor Violations, but it appears in tension with the
+direct detection experiments. This conclusion concurs with what can be extrapolated from
+the earlier MFV analysis of (136).
+The search for models with mediators, that are colored in all models considered so far,
+can be carried out very effectively at hadron colliders searching for signals
+pp → φφ → tχtχ ,
+that very much resemble the search for supersymmetric top partners. Depending on the
+model there can be more general combinations of flavors of quarks
+pp → φφ → qjχqiχ .
+Therefore it is in general useful to consider the whole list of squark searches to put bounds on
+this type of models. References (143,144) reports bounds in the TeV ballpark which inherit
+the strengths and weaknesses discussed for the search of supersymmetric quark partners.
+Other possible signals at hadron collider are the
+pp → tχχ
+scattering, which can arise from interactions such as eq.(7), studied in (138), or associated
+production φχ, followed by φ → tχ studied for instance in (144).
+It is also possible to consider models that go beyond what we have considered here
+starting from the notable feature that MFV and some extensions may render the DM
+stable. In a model of such “top-philic” dark matter model on can have (145) scalars that
+couple to tχ as well as to light quark bilinears, e.g. from RPV supersymmetry, so that they
+mediate scatterings of the type
+qi¯qj → Sij → tχ .
+Other potentially interesting signals possible flavored gauge bosons with couplings ρijqiqj
+can appear, replacing Sij with ρij in the above process. Further signals in this type of
+models arise, e.g.
+qig → tρti
+possibly followed by ρ → χt, and similarly for S. A model with a flavored gauge boson
+has been studied in (146) with the goal of pinning down the flavor of light quark that
+interacts with the top quark and the dark matter leveraging charm-tagging and lepton
+charge asymmetry at the LHC.
+Though many general issues follow the same path for scalar and fermionic dark matter
+it is worth mentioning that references (147, 148) contain a full study of the case in which
+the partner and the dark matter are a fermion and a scalar, respectively, at the converse of
+most of what we discussed above. Further studies of top and dark matter related matters
+can be found in the context of simplified models building (141,149,150).
+www.annualreviews.org •
+17
+
+5. Conclusions
+The connection between new physics and the top quark sector is well established and has
+lead to a large amount of model building and phenomenological studies.
+Here we have
+presented supersymmetric top partners, motivated by supersymmetry as the symmetry that
+stabilizes the weak scale, and top partners states motivated by the possible compositeness
+and pseudo-Nambu-Goldstone boson nature of the Higgs boson.
+The phenomenological
+relevance of these incarnations of “BSM in the top quark sector” is tightly tied to the
+motivations of the models to which the top partners states belong.
+As the models in
+question are themselves in a “critical” phase at the moment, so is the situation for this
+type of new physics in the top quark sector. We say this in the sense that on one hand
+we have reached a point at which the expectation was to have already discovered signs
+of new physics, especially in the top quark sector in the mass range explored by current
+experiments, hence we should start to dismiss these ideas, while on the other hand we are
+still largely convinced of the validity of the arguments that lead to the formulation of these
+models. Furthermore no serious alternatives have appeared in the model building landscape
+and we still have plenty of evidence for the existence of physics beyond the Standard Model.
+Thus one can be lead to reconsider if the entire motivational construction for these models
+was somewhat wrong or at least biased towards a “close-by” and experimentally friendly
+solution.
+The way out of this crisis, in absence of experimental results changing the situation,
+is for everyone to decide. A possibility is to conclude that we need to update our beliefs
+about “where” (151) new physics can appear in the top quark sector and more in general
+in going beyond the SM. In this sense top partner searches are a gauge of our progress on
+testing well established ideas on new physics.
+It should be remarked that the top quark sector remains central also in the formulation
+of new physics models that try alternatives to the more well established ideas, see e.g.
+Refs. (152,153) on possible ways the top quark can lead the way to construct new physics
+models of a somewhat different kind that the two mainstream ideas discussed here.
+Given the absence of clear signs and directions in model building into which entrust our
+hopes for new physics we have discussed the power of general effective field theory analyses
+that can be used to search for new physics in precise SM measurements. These tools have
+become the weapon of choice in a post-LHC epoch for the so-called model-independent
+search of new physics. We have presented the power of current LHC and future HL-LHC
+analyses to see deviations from the SM due to top quark interactions. Overall the LHC
+has a chance to see deviation in some more friendly observables for a new physics scale in
+the TeV range. In order to secure this result and avoid possible blind-spots a new particle
+accelerator is needed, a most popular option being an e+e− capable of operating at or above
+the t¯t threshold with the luminosity to produce around 106 top quark pairs.
+Other great mysteries beyond the origin of the electroweak scale remain unsolved in
+the Standard Model. We have looked at possible solutions of the flavor puzzle in which
+the top quark flavor plays a special role. The phenomenology of models with lowest lying
+new physics states charged under top flavor has some similarity with that of top quark
+partners at colliders, but there is also the possibility to generate observable flavor violations
+as further distinctive experimental signatures.
+We have examined the possibility that the top quark may be a key to solve the mystery
+of dark matter of the Universe. We have presented scenarios in which the dark matter
+interacts predominantly or exclusively with the top quark flavor, possibly ascribing the
+18
+
+stability of the dark matter to the same flavor structure that makes the top quark flavor
+special among the SM flavors. Such possibility appears very well motivated as a way to
+reduce otherwise intolerably large couplings of dark matter with lighter generations and
+explain the stability of dark matter. The flavor dependence of the couplings has motivated
+efforts to build models for the realization of this idea in a coherent, though maybe still
+effective, theory of favor of which we have presented a few instances. We remarked how
+in these scenarios the dark matter phenomenology is quite different from other types of
+thermal dark matter and we have summarized dedicated analyses that have been carried
+out to identify the relevant bounds and constraints. The upshot is that idea can be broadly
+tested with current and future direct detection dark matter experiments. At the same time
+the new states associated with the dark matter may be observed on-shell at colliders, which
+can in principle also probe contact interactions that originate from off-shell states associated
+with the dark matter. Low energy flavor observables can also help to restrict the range of
+possible models of flavored dark matter leading to significant constraints both on MFV
+and non-MFV scenarios when a thermal relic abundance and a significant suppression of
+spin-dependent and spin-independent direct detection rates are required.
+Acknowledgments
+It is a pleasure to thank Kaustubh Agashe for discussions on top quark partners in composite
+Higgs models.
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+Title: Decreased serum vitamin D level as a prognostic marker in patients with 
+COVID-19 
+ 
+Ruyi Qu 1#, Qiuji Yang 1#, Yingying Bi 2#, Jiajing Cheng 2, Mengna He 3, Xin Wei 
+4, Yiqi Yuan 5, Yuxin Yang 6* and Jinlong Qin2* 
+ 
+1 Department of Geriatrics, Shanghai Fourth People's Hospital, School of Medicine, 
+Tongji University, Shanghai 200434, China 
+2 Department of Obstetrics and Gynecology, Shanghai Fourth People’s Hospital, 
+School of Medicine, Tongji University, Shanghai 200434, China 
+3 Information Department, Shanghai Fourth People's Hospital, School of Medicine, 
+Tongji University, Shanghai 200434, China 
+4 Department of Radiology, Shanghai Fourth People's Hospital, School of Medicine, 
+Tongji University, Shanghai 200434, China 
+5 Clinical Laboratory, Shanghai Fourth People's Hospital, School of Medicine, Tongji 
+University, Shanghai 200434, China 
+6 Department of Obstetrics and Gynecology, Shanghai Fourth People’s Hospital, 
+School of Life and Sciences and Technology, Tongji University, Shanghai 200434, 
+China 
+* Correspondence: [email protected] (Yuxin Yang); [email protected] 
+(Jinlong Qin) 
+# These authors contributed equally to this work. 
+ 
+ 
+ 
+
+Abstract 
+Background: The corona virus disease 2019 (COVID-19) pandemic, which is caused 
+by severe acute respiratory syndrome coronavirus 2, is still localized outbreak and has 
+resulted in a high rate of infection and severe disease in older patients with 
+comorbidities. The vitamin D status of the population has been found to be an important 
+factor that could influence outcome of COVID-19. However, whether vitamin D can 
+lessen the symptoms or severity of COVID-19 still remains controversial.  
+Methods: A total of 719 patients with confirmed COVID-19 were enrolled 
+retrospectively in this study from April 13 to June 6, 2022 at Shanghai Forth People’s 
+Hospital. The circulating levels of 25(OH)D3, inflammatory factors, and clinical 
+parameters were assayed. Time to viral RNA clearance (TVRC), classification and 
+prognosis of COVID-19 were used to evaluate the severity of COVID-19 infection.  
+Results: The median age was 76 years (interquartile range, IQR, 64.5-84.6), 44.1% of 
+patients were male, and the TVRC was 11 days (IQR, 7-16) in this population. The 
+median level of 25(OH)D3 was 27.15 (IQR, 19.31-38.89) nmol/L. Patients with lower 
+serum 25(OH)D3 had prolonged time to viral clearance, more obvious inflammatory 
+response, more severe respiratory symptoms and higher risks of impaired hepatic and 
+renal function. Multiple regression analyses revealed that serum 25(OH)D3 level was 
+negatively associated with TVRC independently. ROC curve showed the serum 
+vitamin D level could predict the severity classification and prognosis of COVID-19 
+significantly. 
+Conclusions: Serum 25(OH)D3 level is independently associated with the severity of 
+COVID-19 in elderly, and it could be used as a predictor of the severity of COVID-19. 
+In addition, supplementation with vitamin D might provide beneficial effects in old 
+patients with COVID-19.   
+ 
+Keywords: COVID-19; vitamin D; time to viral RNA clearance; inflammatory 
+response 
+ 
+ 
+
+Introduction 
+As of 30 November 2022, the corona virus disease 2019 (COVID-19) pandemic 
+has resulted in more than 639 million confirmed cases with more than 6.6 million 
+deaths[1]. China has also experienced several waves of COVID-19 pandemic and has 
+explored various strategies to protect the susceptible people from infection, especially 
+the elderly and patients with comorbidities. Therefore, it is of great significance to 
+analyze the risk factors of COVID-19 and investigate inventions to reduce the risks of 
+infection or serious illness for the prevention and treatment of COVID-19 in China. 
+In addition to the injury of alveolar epithelial cells mediated by angiotensin-
+converting enzyme 2 (ACE2), severe acute respiratory syndrome coronavirus 2 (SARS-
+CoV-2) could activate macrophages via ACE2 receptors [2, 3]. The interleukins 1 (IL-
+1), IL-6 and tumor necrosis factor (TNF-α) released by the activated macrophage could 
+further stimulate the neutrophils and T lymphocytes, followed by the release of a large 
+number of inflammatory factors. This inflammatory cascade and inflammation storm 
+are considered to be the main pathogenesis of acute respiratory distress syndrome in 
+COVID-19 [4, 5]. 
+In addition to regulating calcium and phosphorus metabolism, vitamin D is also 
+closely related to immune regulation, cardiovascular diseases, metabolic syndrome, 
+obesity, diabetes, hypertension, cancer, infection and other diseases [6-10]. The 
+multiple effects of vitamin D are related to the wide distribution of the vitamin D 
+receptor (VDR). After binding with VDR, the activated vitamin D acts on the cis-acting 
+elements in the promoter of target genes and thus regulating the transcription of the 
+target genes. Most immune cells, including dendritic cells, T lymphocytes, and B 
+lymphocytes, have high levels of VDR that could modulate the cellular response to 
+viruses as binding with vitamin D [11, 12]. In addition, there are expressions of VDR 
+in the lung tissue, which are related with the severity of lung injury. Previous studies 
+showed that mice with VDR knockout had more serious lung injury induced by LPS 
+compared with WT mice [13, 14]. The histological study showed increased alveolar 
+permeability, pulmonary vascular exudation, neutrophil infiltration and inflammatory 
+factors in the lungs of VDR knockout mice [13, 14]. 
+
+Although vitamin D has multiple beneficial effects, the nutritional status of 
+vitamin D in the population is unsatisfactory. An epidemiological study in more than 
+40 countries conducted by Lips P et al. found that vitamin D deficiency was present in 
+more than 50% of the population, especially among nursing home residents (mainly 
+elderly) [15]. In Europe, approximately 40% of the population is vitamin D deficient 
+(< 20 nmol/L), as well as in USA (24%) and Canada (37%) [16, 17]. 
+Previous study showed there was a close relationship between the risk of 
+respiratory tract infection and vitamin D [18]. Patients with daily or weekly vitamin D 
+supplementation, especially those with 25(OH)D3 < 10 nmol/L, were found to have a 
+reduced risk of respiratory tract infections [18]. Studies in patients with COVID-19 also 
+demonstrated that vitamin D levels were related with the infectious risk and severity of 
+COVID-19 [18-20]. A retrospective study by Angelidi et al. found that patients with 
+low serum 25(OH)D3 levels had increased mortality and risk of invasive mechanical 
+ventilation. The median 25(OH)D3 level in this population was 28 nmol/L. The 
+mortality in patients with 25(OH)D3 < 30 nmol/L was 25%, compared with the 9% 
+mortality in patients with > 30 nmol/L. The results were similar when the cutoff value 
+was adjusted as 20 nmol/L [19, 20]. Although treatment with vitamin D failed to 
+improve the survival in critically ill patients with COVID-19 [21], epidemiological 
+studies showed that patients with high vitamin D levels had low risks of infection with 
+COVID-19 [22], which demonstrated that vitamin D could prevent people from 
+COVID-19. Therefore, vitamin D levels have the potential to be a predictor of the risk 
+of infection and the severity of COVID-19. In addition, it is unclear whether some 
+subgroups of patients would benefit from treatment with vitamin D. 
+In this study, we aimed to assess vitamin D levels in patients with COVID-19 
+infection, and to investigate the relationship between vitamin D levels and time to 
+clearance of virus, the classification and progression of the disease, which might 
+provide evidence for identifying of high-risk patients for COVID-19, and protecting 
+these vulnerable patients from infection and critical illness of COVID-19. 
+ 
+Patients and Methods 
+
+Study Population 
+This is a retrospective cohort study of 719 patients aged 22 to 92 years with 
+confirmed COVID-19 pneumonia hospitalized at the Shanghai Fourth People's Hospital, 
+School of Medicine, Tongji University, Shanghai, China. All patients were diagnosed 
+with COVID-19 pneumonia according to World Health Organization interim guidance. 
+According to hospital data, patients were admitted from April 13 to June 6, 2022, the 
+final date of follow-up was June 20, 2022. The study was approved by the ethics 
+committee of Shanghai Fourth People’s Hospital (No. 2020012) and individual consent 
+for this retrospective analysis was waived. 
+Data Collection 
+The epidemiological, clinical evaluation and outcomes data of all participants 
+during hospitalization were collected from electronic medical records by a trained team 
+of physicians. The individual components of clinical outcomes were reviewed 
+independently and recorded into the computer data base by 2 authors (R.L. and Q.L). 
+The clinical outcomes (including the time to viral RNA clearance (TVRC), the 
+classification and progression of COVID-19) were monitored up to June 20, 2022, the 
+final date of follow-up. The Viral Nucleic Acid Kit (Health) was used to extract nucleic 
+acids from clinical throat swab samples obtained from all patients at admission. A 2019-
+nCoV detection kit (Bioperfectus) was used to detect the ORF1ab gene (nCovORF1ab) 
+and the Ngene (nCoV-NP) according to the manufacturer’s instructions using real-time 
+reverse transcriptase–polymerase chain reaction (qPCR). COVID-19 infection was 
+considered laboratory-confirmed if both the nCovORF1ab and nCoV-NP tests showed 
+positive results. Liver and kidney function, lipids and electrolytes were measured by 
+CH930, Atellica Solution, Siemens, Germany. Cytokines were measured by Deflex, 
+Beckman flow cytometry. Blood Routine and C-reactive protein (CRP) were measured 
+by BC7500, Mindray, China. Serum 25 hydroxyvitamin D was determined by cobas 
+8000, Roche. 
+Statistical Analysis 
+All statistical analyses were performed using IBM SPSS Statistics (Version 22.0, 
+147 SPSS, IBM Corp., Armonk, New York, USA), GraphPad Prism 8.0.2 (GraphPad 
+
+Software, Inc., San Diego CA, USA) and R (version 3.4.1, R Foundation for Statistical 
+Computing, Vienna, Austria). Continuous variables were presented as mean ± Standard 
+Deviation (SD) or median (quartile), and categorical variables were summarized as 
+counts (frequency percentages). χ2 or Fisher exact test (for small cell counts) was 
+applied to compare categorical variables. For continuous variables, normal distribution 
+was evaluated with Kolmogorov-Smirnov test. Then One-way ANOVA (if 
+homogeneity of variances was assumed) or Wilcoxon-Mann-Whitney U test (if 
+homogeneity of variances was not met) was used. Furthermore, receiver operating 
+characteristics (ROC) curves were performed to investigate the value of serum vitamin 
+D level in predicting the severity classification and prognosis of COVID-19 in the 
+population. 
+All reported values were two-sided and P < 0.05 was considered as statistical 
+significance.  
+ 
+ 
+Results 
+1. Clinical baseline characteristics of enrolled patients 
+A total of 719 patients with confirmed COVID-19 were enrolled retrospectively 
+in this study from April 13 to June 6, 2022 at Shanghai Forth People’s Hospital. In these 
+patients, the median age was 76 years (interquartile range, IQR, 64.5-84.6), 44.1% of 
+patients were male, and the TVRC was 11 days (IQR, 7-16). The median level of 
+25(OH)D3 was 27.15 (IQR, 19.31-38.89) nmol/L in these patients. The body mass 
+index (BMI) was 23.08 ± 2.59 Kg/m2 in these patients. There were slightly increased 
+levels of mean systolic blood pressure (SBP, 140.85 ± 20.76 mmHg) and respiratory 
+rate (RR, 19.53 ± 1.39 bpm), but normal levels of mean diastolic blood pressure (DBP, 
+79.85 ± 11.84 mmHg), heart rate (HR, 87.07 ± 15.11 bpm), temperature (Temp, 36.71 
+± 0.47℃) and oxygen saturation (SaO2, 97.29 ± 3.54%). The fraction of inspiration O2 
+(FiO2) was 29 (21- 33) %. The CRP levels were 9.45 (3.39 - 27.9) mg/L, but almost 
+normal levels of white blood cells (WBC, 6.37 ± 3.1×10^9/L), red blood cell (RBC, 
+4.05 ± 0.70×10^9/L) and hemoglobin (Hb, 121.61 ± 21.07g/L). In addition, the levels 
+
+of serum bilirubin (Bil), albumin (Alb), alanine aminotransferase (ALT)，fasting 
+glucose (FBG) and renal function were in normal ranges (Table 1). 
+ 
+Table 1 Clinical baseline characteristics of enrolled patients. 
+Parameter 
+Value 
+Age (years) 
+76.0 (64.5, 84.6) 
+Sex (M/F) 
+317/402 
+TVRC (days) 
+11 (7, 16) 
+25(OH)D3 (nmol/L) 
+27.15 (19.31, 38.89) 
+BMI (Kg/m2) 
+23.08 ± 2.59 
+CRP (mg/L) 
+9.45 (3.39, 27.9) 
+Temp (℃) 
+36.71 ± 0.47 
+SBP (mmHg) 
+140.85 ± 20.76 
+DBP (mmHg) 
+79.85 ± 11.84 
+HR (bpm) 
+87.07 ± 15.11 
+RR (bpm) 
+19.53 ± 1.39 
+SaO2 (%) 
+97.29 ± 3.54 
+FiO2 (%) 
+29 (21, 33) 
+WBC (10^9/L) 
+6.37 ± 3.1 
+RBC (10^12/L) 
+4.05 ± 0.70 
+FBG (mmol/L) 
+6.28 ± 2.57 
+Hb (g/L) 
+121.61 ± 21.07 
+T-Bil (mmol/L) 
+13.08 ± 7.89 
+ALT (U/L) 
+19.96 (13.77, 30.87) 
+T-Pro (g/L) 
+61.55 ± 6.11 
+Alb (g/L) 
+39.57 (35.81, 42.64) 
+Pre-Alb (g/L) 
+183.86 (136.10, 225.50) 
+BUN (mmol/L) 
+5.77 (4.57, 7.81) 
+Cr (umol/L) 
+57.9 (48.1, 73.7) 
+UA (umol/L) 
+288.16 (225.48, 363.44) 
+Cystatin C (mg/mL) 
+1.09 (0.91, 1.44) 
+Abbreviations: M: male; F: female; TVRC: time to viral RNA clearance; BMI: body mass index; 
+CRP: C-reaction protein; Temp: temperature; SBP: systolic blood pressure; DBP: diastolic blood 
+pressure; HR: heart rate; RR: respiration rate; SaO2: oxygen saturation; FiO2: fraction of inspiration 
+O2; WBC: white blood corpuscle; RBC: red blood corpuscle; FBG: fasting blood glucose; Hb: 
+hemoglobin; T-Bil: total bilirubin; ALT: alanine aminotransferase; T-Pro: total protein; Alb: 
+albumin; Pre-Alb: prealbumin; BUN: blood urea nitrogen; Cr: crea; UA: uric acid. 
+ 
+2. Comparison of clinical baseline characteristics and comorbidities among 
+patients with different levels of vitamin D 
+
+The levels of serum vitamin D were measured in 609 patients with COVID-19. 
+Then patients were divided into 4 groups according to the quartile values of serum 
+vitamin D levels: Q1 < 13.14 (9.59, 16.56) nmol/L, 13.14 (9.59, 16.56) nmol/L < Q2 < 
+23.1 (21.37, 25.13) nmol/L, 23.1 (21.37, 25.13) nmol/L < Q3 < 32.42 (29.9, 35.35) 
+nmol/L, and 32.42 (29.9, 35.35) nmol/L < Q4 < 49.29 (43.21, 63.29) nmol/L.  
+Table 2 showed the clinical baseline characteristics among the four groups. 
+Compared with patients with higher levels of serum vitamin D, patients in Q1 group 
+were older, and had more severe illness, which manifested as longer TVRC, lower 
+oxygen saturation, and FiO2. Patients in the Q1 group also had higher levels of 
+inflammation, which included higher levels of CRP and WBC. There was increased 
+percentage of neutrophil and decreased percentage of monocyte and lymphocyte. 
+Patients in Q1 group also had decreased levels of total protein (T-Pro), Alb, and 
+increased levels of lactate dehydrogenase (LDH), which indicated that patients with 
+low vitamin D levels had impaired hepatic synthetical function and nutritional state. 
+Although there was no significant difference in the blood urea nitrogen (BUN) and Crea 
+(Cr), patients in the Q1 group had increased levels of cystatin C, a biomarker of early 
+renal injury. In addition, there were decreased levels of serum magnesium in patients 
+with lower levels of vitamin D. However, there was no significant difference in male 
+proportion, BMI, basic vital signs, and other biochemical tests (Table 2). 
+The rates of comorbidities were high in this study. However, there was no 
+significant difference in comorbidities among patients with different levels of vitamin 
+D (Table 2). 
+ 
+Table 2 Comparison of clinical and biochemical characteristics and comorbidities 
+among patients with different levels of vitamin D 
+ 
+Q1 group 
+(n=162) 
+Q2 group  
+(n=164) 
+Q3 group  
+(n=164) 
+Q4 group  
+(n=165) 
+Age (years) 
+86.0 (70.0, 89.0) 
+75.6 (63.75, 87.0)a 
+72.5 (63.75, 81.25)ab 
+73.0 (64.0, 81.0)a 
+Sex (M/F) 
+63/99 
+71/93 
+78/86 
+82/83 
+TVRC (days) 
+14 (8, 19) 
+10 (6, 15)a 
+10 (7.25, 15)a 
+11 (7, 13)a 
+CVD, n (%) 
+44 (27.16) 
+37 (22.56) 
+13 (7.93) 
+28 (16.97) 
+HT, n (%) 
+98 (60.49) 
+84 (51.22) 
+93 (56.71) 
+89 (53.94) 
+T2DM, n (%) 
+36 (22.22) 
+36 (21.95) 
+48 (29.27) 
+50 (30.30) 
+
+Tumor, n (%) 
+17 (10.49) 
+15 (9.15) 
+15 (9.15) 
+14 (8.48) 
+BMI (Kg/m2) 
+22.41 ± 3.62 
+22.63 ± 3.57 
+23.68 ± 36.66ab 
+23.29 ± 3.60 
+CRP (mg/L) 
+15.46 (5.32, 42.08) 
+9.45 (3.87, 26.82) a 
+6.77 (2.68, 19.31) ab 
+6.29 (2.44, 18.33)a 
+Temp (℃) 
+36.67 ± 0.46 
+36.66 ± 0.45 
+36.74 ± 0.49 
+36.74 ± 0.49 
+SBP (mmHg) 
+139.61±22.03 
+140.87±22.43 
+141.07±19.76 
+142.95±19.03 
+DBP (mmHg) 
+76.68 ± 12.55 
+81.26 ± 12.43a 
+80.95 ± 10.80a 
+80.53 ± 10.90a 
+HR (bpm) 
+85.41 ± 15.33 
+86.81 ± 17.01 
+89.55 ± 14.5 
+87.24 ± 14.1 
+RR (bpm) 
+19.47 ± 1.63 
+19.66 ± 1.40 
+19.50 ± 1.11 
+19.45 ± 1.15 
+SaO2 (%) 
+96.69 ± 3.40 
+97.46 ± 1.78a 
+97.52 ± 1.65a 
+97.62 ± 1.13a 
+FiO2 (%) 
+29 (29, 33) 
+29 (21, 33) a 
+21 (21, 33) ab 
+21 (21, 29) ab 
+WBC (10^9/L)  7.14 ± 3.83 
+6.12 ± 2.49a 
+6.12 ± 2.69a 
+5.95 ± 3.30a 
+Monocyte % 
+7.45 ± 2.90 
+8.29 ± 2.70a 
+8.68 ± 3.31a 
+8.02 ± 2.96 
+Lymphocyte %  21.36 ± 12.13 
+25.60 ± 11.72a 
+26.58 ± 11.46a 
+28.76 ± 12.81ab 
+Neutrophil %  
+69.48 ± 13.30 
+63.84 ± 12.84a 
+62.37 ± 12.48a 
+61.10 ± 13.25a 
+PLT (10^9/L) 
+208.73 ± 90.71 
+211.61 ± 87.55 
+206.95 ± 72.97 
+189.82 ± 68.62a 
+RBC 
+(10^12/L) 
+3.76 ± 0.76 
+4.10 ± 0.63a 
+4.19 ± 0.68a 
+4.20 ± 0.61a 
+Hct (%) 
+34.19 ± 6.97 
+38.04 ± 5.87a 
+39.03 ± 5.62a 
+39.27 ± 5.29a 
+Hb (g/L) 
+116.36 ± 24.47 
+123.06 ± 19.16a 
+126.55 ± 17.76a 
+127.58 ± 17.93ab 
+T-Bil (umol/L) 
+12.72 ± 7.38 
+13.29 ± 6.60 
+13.38 ± 5.95 
+13.24 ± 10.94 
+ALT (U/L) 
+16.47 (12.77, 28.15) 
+22.34 (13.34, 33.24) 
+20.14 (13.95, 28.42) 
+20.00 (14.57, 32.89) 
+AST (U/L) 
+24.83 (19.60, 37.41) 
+24.38 (19.23, 32.94) 
+23.5 (18.97, 31.12) 
+24.59 (19.34, 34.13) 
+AKP (U/L) 
+83.41 (67.57, 102.71) 
+79.69 (67.88, 99.37) 
+78.94 (70.42, 100.54) 
+76.97 (61.96, 
+95.47)a 
+T-Pro (g/L) 
+58.17 ± 6.46 
+61.74 ± 5.68a 
+63.20 ± 5.40ab 
+63.18 ± 5.78ab 
+Alb (g/L) 
+35.71 ± 4.81 
+39.10 ± 4.33a 
+40.64 ± 4.16ab 
+41.02 ± 4.26ab 
+Pre-Alb (g/L) 
+149.02 (102.26, 
+203.19) 
+179.43 (136.11, 
+227.65) a 
+194.49 (162.13, 232.86) 
+ab 
+190.31 (161, 
+233.61)a 
+BUN (umol/L) 
+6.35 (4.68, 9.10) 
+5.66 (4.48, 7.16) a 
+5.77 (4.69, 7.82) 
+5.64 (4.57, 7.49) 
+Cr (umol/L) 
+56.50 (45.85, 82.90) 
+56.3 (48.7, 74.1)  
+60.85 (49.35, 72.23) 
+59.95 (49.3, 73.55) 
+UA (umol/L) 
+251.43 (193.66, 
+349.31) 
+278.97 (239.34, 
+373.91) 
+325.84 (235.18, 372.76) 
+a 
+295.65 (257.76, 
+349.96)a 
+Cystatin C 
+(mg/mL) 
+1.26 (0.98, 1.71) 
+1.09 (0.93, 1.38) a 
+1.05 (0.88, 1.37) a 
+1.03 (0.89, 1.33)a 
+Lactate 
+(mmol/L) 
+2.10 ± 0.93 
+1.95 ± 0.76 
+2.04 ± 0.80 
+2.23 ± 1.06 
+FBG (mmol/L)  6.63 ± 3.28 
+5.93 ± 2.66a 
+6.65 ± 3.24b 
+5.88 ± 2.24ac 
+LDH (U/L) 
+232.76 ± 84.93 
+209.88 ± 76.38a 
+203.97 ± 56.08a 
+201.54 ± 54.01a 
+K+ (mmol/L) 
+3.90 ± 0.69 
+3.76 ± 0.59a 
+3.82 ± 0.50 
+3.80 ± 0.50 
+Na+ (mmol/L) 
+141.61 ± 5.77 
+141.94 ± 5.23 
+141.92 ± 3.97 
+142.44 ± 3.55 
+Cl- (mmol/L) 
+104.11 ± 5.86 
+104.61± 4.83 
+104.36 ± 3.77 
+104.53 ± 3.61 
+Ca2+ (mmol/L) 
+1.77 ± 0.46 
+1.87 ± 0.48 
+1.99 ± 0.43 ab 
+2.05 ± 0.40 ab 
+Mg2+ 
+(mmol/L) 
+0.84 ± 0.11 
+0.87 ± 0.09a 
+0.88 ± 0.09a 
+0.87 ± 0.10a 
+
+Phosphate 
+(mmol/L) 
+1.08 ± 0.59 
+1.09 ± 0.37 
+1.14 ± 0.23 
+1.15 ± 0.28 
+Abbreviations: CVD: cardiovascular disease; HT: hormone therapy; T2DM: diabetes mellitus type 
+2; PLT: platelet count; Hct: red blood cell specific volume; AST: glutamic oxaloacetic transaminase; 
+AKP: alkaline phosphatase; LDH: lactate dehydrogenase. 
+ a, p < 0.05 compared with Q1 group; b, p<0.05 compared with Q2 group; c, p < 0.05 compared 
+with Q3 group. 
+ 
+3. Comparison of inflammatory factors among patients with different levels of 
+vitamin D 
+The increased levels of WBC and CRP in patients from Q1 group implicated that 
+patients with lower levels of serum vitamin D might had high inflammatory state. 
+Therefore, we measured the serum levels of inflammatory factors in patients from 
+different groups. However, there was no significant difference of inflammatory factors 
+among these groups except for lower levels of interferon-γ (IFN-γ) and TNF-α in 
+patients with lower levels vitamin D (Figure 1, Table S1). 
+ 
+ 
+Figure 1 Comparison of inflammatory factors among patients with different levels of vitamin D.**, 
+p < 0.01; ****, p < 0.0001. Abbreviations: IL: interleukins; IFN: interferon; TNF: tumor necrosis 
+factor. 
+ 
+4. Association between serum vitamin D level and the severity of COVID-19 
+
+To further assess the association between serum vitamin D level and the severity 
+of COVID-19, patients were first divided into 4 groups according to the quartile of 
+TVRC, which was used as an indicator for the severity of COVID-19. Patients in the 
+longest TVRC group (TVRC-Q4) had significantly lower serum vitamin D levels 
+(23.19 [IQR, 14.46-33.77] nmol/L) compared with patients in shorter TVRC groups 
+(26.74 [IQR, 20.76-38.97] nmol/L in TVRC-Q1, p = 0.0075; 31.02 [IQR, 22.87-41.03] 
+nmol/L in TVRC-Q2, p < 0.0001; 26.19 [IQR, 18.08, 41.44] nmol/L in TVRC-Q3, p = 
+0.0461) (Figure 2A). Patients were also grouped into mild, moderate, severe and 
+critical groups based on the severity classification of COVID-19 according to the 
+guideline for management of patients with COVID-19 (9th version). There were 
+significantly lower levels of serum vitamin D in patients with severe (19.53 [IQR, 
+12.71-27.01] nmol/L) and critical (15.54 [IQR, 8.51-20.68] nmol/L) groups compared 
+with patients in the mild (31.10 [IQR, 22.73-42.01] nmol/L) and moderate (26.31 [IQR, 
+17.98-36.51] nmol/L) groups (Figure 2B). Furthermore, patients were divided into 3 
+groups based on the prognosis of the disease according to the progression of the disease 
+changes of the severity classification of COVID-19 when the virus RNA was cleared, 
+and the relation between serum vitamin D levels and the prognosis was investigated. 
+Patients with good prognosis had significantly higher levels of serum vitamin D levels 
+(28.21 [IQR, 20.46-40.22] nmol/L) compared with patients with poor prognosis 
+(Prognosis-Q1, 19.53 [IQR, 12.11-27.44] nmol/L in Prognosis-Q2, p < 0.0001; 18.03 
+[IQR, 10.96-21.56] nmol/L in Prognosis-Q3, p = 0.016) (Figure 2C). 
+ 
+ 
+Figure 2 Association of vitamin D level with TVRC, classification and prognosis of COVID-19. (A) 
+Vitamin D levels in each group stratified by TVRC quartile 11 (IQR, 7-16). (B) Vitamin D levels 
+in each group divided by severity classification of COVID-19. (C) Vitamin D levels in patients with 
+different progression.  
+
+A
+B
+**
+C
+****
+200
+200-
+***
+**
+200-
+****
+150
+25(OH)D3 (nmol/L)
+150
+25(OH)D3 (
+100
+25(OH)D3 (
+100
+100
+50-
+50
+50
+TVRC-Q4
+Mild
+Moderate
+Severe
+Critica 
+ROC curve showed the serum vitamin D level could predict the severity 
+classification and prognosis of COVID-19 significantly (the area under the curve [AUC] 
+= 0.695, 95% CI [0.627-0.764], p < 0.001, for severe and critical of COVID-19, Figure 
+3A; AUC=0.728, 95% CI [0.585-0.872], p = 0.009, for the aggravation of COVID-19, 
+Figure 3B). 
+ 
+Figure 3 ROC curve to investigate the serum vitamin D level in predicting the severity classification 
+(A) and prognosis (B) of COVID-19. Abbreviations: AUC: the area under the curve; ROC: receiver 
+operating characteristics. 
+ 
+5. Association between serum vitamin D levels and clinical parameters 
+In univariate analyses, serum vitamin D level was negatively associated with 
+TVRC, age, FiO2, prognosis, IL-10, cystatin C, alkaline phosphatase (AKP), LDH, 
+direct bilirubin (D-Bil), and CRP. However, BMI, SaO2, DBP, Alb, IL-4, TNF-α, 
+serum calcium (Ca) levels, indirect bilirubin (I-Bil), serum magnesium (Mg) level, 
+serum sodium (Na) level, uric acid (UA), pre-albumin (pre-Alb), LDH, Hb, red blood 
+cell specific volume (Hct) and T-Pro were positively associated with serum vitamin D 
+level (Table 3).  
+ 
+Table 3 Correlation between serum vitamin D and other variables 
+Parameter 
+r 
+p-Value 
+Age (years) 
+-0.239 
+< 0.001 
+TVRC (days) 
+-0.135 
+0.001 
+BMI (Kg/m2) 
+0.091 
+0.048 
+CRP (mg/L) 
+-0.196 
+< 0.001 
+DBP (mmHg) 
+0.096 
+0.014 
+SaO2 (%) 
+0.095 
+0.016 
+
+A
+B
+1.0
+1.0
+0.8
+8'0
+Sensivity
+0.6
+Sensivity
+0.6
+AUC = 0.695
+AUC = 0.728
+p ≤ 0.001
+p = 0.009
+0.4 
+0.4 
+0.2 
+0.2 
+0.0
+0.0
+0'0
+0.2
+0.4 
+0.6
+0.8
+1.0
+0.0
+0.2 
+0.4 
+0.6
+0.8
+1.0
+1-Specifity
+1-SpecifityFiO2 (%) 
+-0.227 
+< 0.001 
+WBC (10^9/L) 
+-0.116 
+0.003 
+Monocyte % 
+0.078 
+0.047 
+Lymphocyte % 
+0.226 
+< 0.001 
+Neutrophil % 
+-0.23 
+< 0.001 
+Hct (%) 
+0.294 
+< 0.001 
+Hb (g/L) 
+0.298 
+< 0.001 
+D-Bil (mmol/L) 
+-0.112 
+0.009 
+I-Bil (umol/L) 
+0.102 
+0.017 
+AKP (U/L) 
+-0.104 
+0.035 
+T-Pro (g/L) 
+0.3 
+< 0.001 
+Alb (g/L) 
+0.405 
+< 0.001 
+Pre-Alb (g/L) 
+0.24 
+< 0.001 
+UA (umol/L) 
+0.144 
+0.001 
+Cystatin C (umol/L) 
+-0.191 
+< 0.001 
+LDH (U/L) 
+-0.151 
+< 0.001 
+Na+ (mmol/L) 
+0.087 
+0.027 
+Ca2+ (mmol/L) 
+0.343 
+< 0.001 
+Mg2+ (mmol/L) 
+0.106 
+0.015 
+Phosphate (mmol/L) 
+0.211 
+< 0.001 
+IL-10 
+-0.109 
+0.007 
+IL-4 
+0.067 
+0.01 
+TNF-α 
+0.202 
+< 0.001 
+DSS 
+-0.242 
+< 0.001 
+Prognosis 
+-0.194 
+< 0.001 
+Abbreviations: D-Bil: direct Bilirubin; I-Bil: indirect bilirubin; DSS: disease severity score. 
+ 
+6. Association between TVRC and clinical parameters 
+Spearman correlation coefficients were used to evaluate correlations between 
+TVRC and clinical parameters. The results showed that serum vitamin D level, BMI, 
+HR, ALB, TNF-α, serum calcium level, serum sodium level, serum phosphorus level, 
+serum chlorine level, uric acid, pre-Alb, T-Pro, Hb, hematocrit (Hct) and RBC were 
+negatively associated with TVRC. In addition, age, prognosis, IL-10, Il-12, IL-17, IL-
+2, WBC, CRP, Alb, LDH, ALT, glutamic oxaloacetic transaminase (AST), alkaline 
+phosphatase (AKP), BUN, cystatin C, creatinine, and serum potassium level were 
+positively associated with TVRC (Table 4). Multiple regression analyses revealed that 
+only serum vitamin D level was negatively associated with TVRC independently (Table 
+5). 
+
+ 
+Table 4. Correlation between TVRC and other variables 
+Variables 
+Beta coefficient 
+p-Value 
+Age (years) 
+0.253 
+< 0.001 
+25(OH)D3 (nmol/L) 
+-0.135 
+0.001 
+BMI (Kg/m2) 
+-0.164 
+< 0.001 
+CRP (mg/L) 
+0.157 
+< 0.001 
+HR (bpm) 
+-0.074 
+0.047 
+FiO2 (%) 
+0.242 
+< 0.001 
+RBC (10^12/L) 
+-0.185 
+< 0.001 
+WBC (10^9/L) 
+0.09 
+0.016 
+Lymphocyte % 
+-0.159 
+< 0.001 
+Neutrophil % 
+0.158 
+< 0.001 
+Hct (%) 
+-0.167 
+< 0.001 
+Hb (g/L) 
+-0.186 
+< 0.001 
+ALT (U/L) 
+0.076 
+0.048 
+AST (U/L) 
+0.081 
+0.03 
+AKP (U/L) 
+0.148 
+0.001 
+r-GT (U/L) 
+0.074 
+0.05 
+T-Pro (g/L) 
+-0.167 
+< 0.001 
+Alb (g/L) 
+-0.287 
+< 0.001 
+Pre-Alb (g/L) 
+-0.165 
+0.001 
+BUN (umol/L) 
+0.244 
+< 0.001 
+UA (umol/L) 
+-0.096 
+0.026 
+Cystatin C (mg/mL) 
+0.191 
+< 0.001 
+LDH (U/L) 
+0.127 
+0.002 
+Cr (umol/L) 
+0.116 
+0.002 
+K+ (mmol/L) 
+0.207 
+< 0.001 
+Na+ (mmol/L) 
+-0.204 
+< 0.001 
+Cl- (mmol/L) 
+-0.109 
+0.004 
+Ca2+ (mmol/L) 
+-0.119 
+0.003 
+Phosphate (mmol/L) 
+-0.229 
+< 0.001 
+IL-10 
+0.087 
+0.025 
+IL-12 
+0.08 
+0.04 
+IL-17 
+0.14 
+< 0.001 
+IL-1 
+0.076 
+0.05 
+IL-2 
+0.124 
+0.001 
+TNF-α 
+-0.095 
+0.014 
+DSS 
+0.235 
+< 0.001 
+Prognosis 
+0.178 
+< 0.001 
+Abbreviations: r-GT: γ-glutamyl transpeptidase. 
+ 
+Table 5. Multivariate regression analyses of predictors of TVRC in patients with 
+COVID-19 
+
+Variables 
+Beta coefficient  
+p-Value 
+95% CI 
+25(OH)D3 (nmol/L) 
+-0.230 
+0.016 
+-0.168 to -0.018 
+ 
+Discussion 
+As a kind of steroid hormone, vitamin D is tightly linked to a number of different 
+metabolic processes and immune regulation in the human body. Vitamin D activates 
+the innate immune system by binding with VDR in immune cells to defend the invasion 
+of foreign pathogenic microorganisms. For example, 1,25 dihydroxyvitamin D3 (1,25-
+(OH)2-D3) could induce the generation of antimicrobial peptides in monocytes to clean 
+the Mycobacterium tuberculosis[23, 24]. 1,25-(OH)2-D3 also could tune the cellular 
+and humoral immunity by regulating the differentiation and proliferation of T and B 
+lymphocytes and the secretion of Th1/Th2 cytokines. In addition, 1,25-(OH)2-D3 could 
+inhibit the exaggerated inflammatory response via inducing the differentiation of 
+regulatory T cells (Treg), and have protective effects in inflammatory responses and 
+autoimmune diseases. 
+Considering that 25(OH)D3 is the main form of vitamin D in the body and its 
+stable concentration in circulation, serum 25(OH)D3 was used as an indicator to 
+evaluate the nutritional status of vitamin D. Presently, vitamin D deficiency, 
+insufficiency, normal, and sufficiency are defined as <25, 25 to 50, 51 to 75, and > 
+75nmol/L, respectively[25]. Vitamin D deficiency was defined when the serum level 
+of 25(OH)D3 was less than 50nmol/L. An epidemiological study in East China showed 
+that the serum levels of 25(OH)D3 were 40.5 ± 12.5 nmol/L in the normal population, 
+and 80.3% of the population were vitamin D deficiency[26], which was significantly 
+higher than that in western countries[27, 28]. In addition, a study in elderly inpatients 
+showed that the vitamin D levels were 34.6 ± 16.2 nmol/L in the population, of which 
+17.5% were severely deficient, 73.0% were mildly deficient, 7.5% were insufficient, 
+and only 2.0% were sufficient[29]. These data suggest that vitamin D deficiency may 
+be common in the Han population, especially in the elderly and bedridden patients. 
+This study enrolled 719 patients with COVID-19 and assessed the levels of serum 
+vitamin D, cytokines and other clinical indicators to investigate the relationship 
+
+between vitamin D levels and TVRC, the classification and prognosis of the disease. 
+Higher levels of vitamin D were associated with the higher levels of T-Pro, Alb, pre-
+Alb, hemoglobin and BMI, which indicated that higher vitamin D levels were 
+associated with better protein synthesis ability of liver and better nutritional status of 
+patients. Conversely, the lower levels of vitamin D were associated with the longer 
+TVRC, higher levels of WBC and CRP, as well as worse oxygenation capacity of the 
+lung, suggesting that lower vitamin D was associated with severe conditions in these 
+patients. Meanwhile, lower levels of vitamin D were related with the biomarkers of 
+early hepatic and renal function impairments, such as lower levels of pre-Alb and higher 
+levels of cystatin C. In addition, there was positive relationship between vitamin D 
+levels and serum calcium and phosphorus concentrations. All these results 
+demonstrated that vitamin D had benefit effects on the clearance of the virus and 
+alleviating the condition in patients with COVID-19. Further investigation validated 
+that lower vitamin D levels were associated with longer TVRC, more severe disease 
+and worse prognosis. Therefore, serum vitamin D level is a predictor of the severity of 
+disease and prognosis in patients with COVID-19. 
+Previous studies have shown that the risks of severe infection and mortality were 
+increased in vulnerable groups (with comorbidities such as diabetes, hypertension, 
+coronary artery disease and tumors) in patients with COVID-19[29-32]. In this study, 
+we compared the comorbidities in patients with different vitamin D levels, and found 
+no significant difference in comorbidities among different groups. The results indicated 
+the association between vitamin D levels and the prognosis of the disease was less 
+affected by these chronic comorbidities. Further investigation showed that serum 
+vitamin D level was correlated with TVRC negatively, and serum vitamin D level was 
+an independent predictor of TVRC in patients with COVID-19, which further validated 
+the close relationship between vitamin D and the severity and prognosis of COVID-19. 
+Vitamin D deficiency is a common phenomenon in Chinese, especially in the 
+elderly. For its detrimental effects on the immune system, vitamin D deficiency would 
+impair the clearance of invasive pathogens. This concern is more obvious under the 
+current situation of the panic of COVID-19 and consistent virus variants. Therefore, it 
+
+is of great significance to investigate how to protect patients with high risk from 
+infection and improve the prognosis of these patients. Supplement with vitamin D 
+routinely in patients with COVID-19 is still in debate presently[33-35]. However, the 
+results of this study demonstrated that early supplement with vitamin D in patients with 
+COVID-19 and vitamin D deficiency could improve the ability of defensing the 
+infection of SARS-CoV-2, promoting the clearance of virus and improving the 
+prognosis in these high-risk patients. However, for the limitation of the observed study, 
+further prospective randomized controlled trails were needed to investigate the benefits 
+of supplement of vitamin D in these patients.  
+Limitations 
+There are several limitations of our study. Although our study implied that early 
+supplemented with vitamin D in patients with COVID-19 and vitamin D deficiency 
+might improve the prognosis of these patients. However, we did not give the therapy 
+with vitamin D in this population in this retrospective study. In addition, this 
+retrospective study has some disadvantages compared with prospective studies. 
+Therefore, further prospective studies are needed to validate the clinical value of serum 
+vitamin D levels in risk stratifications of patients with COVID-19. 
+Conclusion 
+This study demonstrated that serum 25(OH)D3 level was independently associated 
+with the severity of COVID-19 in elderly, and it could be used as a predictor of the 
+severity of COVID-19. In addition, supplementation with vitamin D might provide 
+beneficial effects in old patients with COVID-19. 
+Sources of Funding 
+This work was supported by Shanghai Committee of Science and Technology, 
+China (grant No. 22dz1202304 to Jiajing Cheng). 
+Author Contributions 
+Conceptualization, Ruyi Qu, Yuxin Yang and Jinlong Qin; Data curation, Ruyi 
+Qu, Qiuji Yang and Yingying Bi; Formal analysis, Ruyi Qu and Jinlong Qin; Funding 
+acquisition, Jiajing Cheng; Inves-tigation, Jiajing Cheng, Mengna He, Xin Wei and 
+Yiqi Yuan; Methodology, Ruyi Qu, Qiuji Yang, Yingying Bi, Jiajing Cheng, Yuxin 
+
+Yang and Jinlong Qin; Project administration, Jinlong Qin; Re-sources, Jiajing Cheng; 
+Software, Yingying Bi and Xin Wei; Supervision, Yuxin Yang and Jinlong Qin; 
+Validation, Qiuji Yang and Yingying Bi; Visualization, Yingying Bi, Mengna He and 
+Yiqi Yuan; Writing – original draft, Ruyi Qu, Qiuji Yang and Yuxin Yang; Writing – 
+review & editing, Yuxin Yang and Jinlong Qin. 
+Acknowledgments 
+The authors are grateful to all the participants in this study. 
+Conflicts of Interest 
+The authors declare no conflict of interest. 
+ 
+ 
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+12. 
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+16. 
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+Eur J Clin Nutr, 2020. 74(11): p. 1498-1513. 
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+19. 
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+Mechanical Ventilation: A Cohort of COVID-19 Hospitalized Patients. Mayo Clin Proc, 
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+hydroxyvitamin D levels. PLoS One, 2020. 15(9): p. e0239252. 
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+Rook, G.A., et al., Vitamin D3, gamma interferon, and control of proliferation of 
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+Zhen, C., et al., Serum vitamin D levels of the natural population in eastern China. Chinese 
+Journal of Endocrinology and Metabolism, 2017. 33(9): p. 4. 
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+seasonal subpopulations from NHANES III. Bone, 2002. 30(5): p. 7. 
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+COVID-19 and Pre-existing Type 2 Diabetes. Cell Metab, 2020. 31(6): p. 1068-1077 e3. 
+30. 
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+disease in COVID-19 pneumonia: A systematic review, meta-analysis and meta-regression. 
+J Renin Angiotensin Aldosterone Syst, 2020. 21(2): p. 1470320320926899. 
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+31. 
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+COVID-19 in Wuhan, China: a retrospective cohort study. The Lancet, 2020. 395(10229): 
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+The Lancet Oncology, 2020. 21(3): p. 335-337. 
+33. 
+Gibbons, J.B., et al., Association between vitamin D supplementation and COVID-19 
+infection and mortality. Sci Rep, 2022. 12(1): p. 19397. 
+34. 
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+risk of all cause acute respiratory tract infection and covid-19: phase 3 randomised 
+controlled trial (CORONAVIT). BMJ, 2022. 378: p. e071230. 
+35. 
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+inflammatory markers in COVID-19 patients admitted to the ICU. Sci Rep, 2022. 12(1): p. 
+18604. 
+ 
+Supplemental data 
+ 
+Table 1. Comparison of inflammatory factors among patients with different levels of 
+vitamin D 
+ 
+Q1 group 
+(n=150) 
+Q2 group  
+(n=153) 
+Q3 group  
+(n=157) 
+Q4 group  
+(n=149) 
+IL-1 
+0.80(0.16,1.69) 
+0.96(0.21,2.03) 
+0.89(0.37,2.07) 
+0.95(0.29,2.07) 
+IL-2 
+0.08(0.05,0.82) 
+0.08(0.04,0.88) 
+0.08(0.04,0.77) 
+0.08(0.05,0.88) 
+IL-4 
+0.44(0.07,1.90) 
+0.78(0.07,2.37) 
+0.98(0.08,5.47) 
+0.6(0.08,3.77) 
+IL-5 
+0.07(0.03,0.14) 
+0.07(0.04,0.14) 
+0.07(0.03,0.21) 
+0.07(0.04,0.1) 
+IL-6 
+80.44(21.22,204.04) 
+67.63(24.65,228.57) 
+107.85(26.4,312.4) 
+70.84(19.37,288.68) 
+IL-8 
+101.77(32.25,229.92) 
+108.2(23.96,255.95) 
+122.3(44.53,242.43) 
+108.06(32.42,244.3) 
+IL-10 
+4.88(2.79,9.32) 
+4.28(2.51,7.21) 
+4.26(2.53,8.01) 
+3.85(2.09,6.45)a 
+IL-12 
+0.07(0.04,0.18) 
+0.07(0.04,0.62) 
+0.07(0.04,0.5) 
+0.08(0.04,0.4) 
+IL-17 
+1.11(0.36,2.93) 
+0.91(0.26,2.37) 
+1.05(0.33,2.67) 
+1.05(0.32,2.31) 
+IFN-α 
+0.07(0.04,0.10) 
+0.07(0.03,0.09) 
+0.07(0.04,0.1) 
+0.06(0.04,0.1) 
+IFN-γ 
+1.27(0.38,3.79) 
+1.93(0.56,5)a 
+2.35(0.4,7.37)a 
+1.72(0.42,5.34) 
+TNF-α 
+4.32(1.82,9.12) 
+6.29(2.91,11.04)a 
+7.38(3.52,12.74)a 
+8.78(3.39,17.19)ab 
+ 
+ 
+Table 2. Comparison of clinical and biochemical characteristics among patients with 
+different severity rating 
+ 
+Q1 group 
+(n=330) 
+Q2 group  
+(n=309) 
+Q3 group  
+(n=71) 
+Q4 group  
+(n=9) 
+Age (years) 
+68(59,78) 
+81(71,88) a 
+86(77,90) a 
+87(79.5,91) a 
+Sex (M/F) 
+149/181 
+130/179 
+34/37 
+4/5 
+TVRC (days) 
+9(6,13) 
+12(8,17) a 
+13(9,20.25) a 
+12(3.25,14.75) 
+25(OH)D3 
+31.10(22.73,42.01) 
+26.31(17.98,36.51) a 
+19.53(12.71,27.01) ab 
+15.54(8.51,20.68) ab 
+BMI (Kg/m2) 
+23.42±3.59 
+22.74±3.61 
+21.73±3.18 
+NA 
+CRP (mg/L) 
+5.46(2.33,14.47)  
+10.27(4.38,27.91) a 
+36.65(19.13,76.31) ab 
+99.08(56.11,155.47) ab 
+
+Temp (℃) 
+36.74±0.48 
+36.68±0.46 
+36.70±0.50 
+37.00±0.41 
+SBP (mmHg) 
+141.06±19.57 
+140.53±21.84 
+141.38±21.72 
+140.00±21.47 
+DBP (mmHg) 
+81.42±11.06 
+78.75±12.02 
+77.87±13.31 
+76.11±15.60 
+HR (bpm) 
+88.33±15.10 
+86.01±14.55 
+85.58±17.10 
+89.22±17.14 
+RR (bpm) 
+19.41±1.24 
+19.58±1.23 
+19.69±2.14 
+20.78±2.99 
+SaO2 (%) 
+97.72±1.21 
+97.38±1.49 
+95.57±3.22 ab 
+91.38±11.88 abc 
+Fio2 (%) 
+21(21,29) 
+29(21,33) a  
+33(33,41) ab  
+61(29,141) ab  
+RBC 
+(10^12/L) 
+4.25±0.57 
+3.99±0.72 a 
+3.53±0.74 ab 
+3.20±0.71 abc 
+WBC (10^9/L)  5.75±2.58 
+6.45±3.03 a 
+8.41±3.93 ab 
+10.49±5.51 abc 
+Monocyte % 
+8.27±2.94 
+8.15±2.92 
+6.54±2.75 
+7.97±6.25 
+Lymphocyte 
+% 
+29.23±11.37 
+24.32±11.64 a 
+13.59±8.56 ab 
+8.78±5.00 abc 
+Neutrophil % 
+60.23±11.98 
+65.46±12.49 a 
+78.03±10.61 ab 
+82.88±9.49 abc 
+PLT (10^9/L) 
+196.96±65.99 
+210.42±86.97 
+221.68±94.84 
+252.33±159.25 
+Hct (%) 
+39.35±5.35 
+36.91±6.27 a 
+32.64±6.81 ab 
+29.43±6.56 abc 
+Hb (g/L) 
+127.60±18.05 
+118.93±20.43 a 
+107.37±26.52 ab 
+106.56±21.07 ab 
+T-Bil (umol/L) 
+12.73±5.94 
+12.89±6.22 
+14.98±16.83 
+17.20±9.87 
+ALT (U/L) 
+19.93(13.52,30.35) 
+20.20(13.91,31.77) 
+20.35(13.17,30.83) 
+19.02(14.79,30.27) 
+AST (U/L) 
+22.71(18.48,29.55) 
+24.98(19.32,34.96) a 
+30.38(21.24,42.76) a 
+45.24(34.48,50.38) a 
+AKP (U/L) 
+78.10(63.65,95.12) 
+83.77(69.13,99.65) 
+80.64(67.78,102.90) 
+84.03(67.08,112.51) 
+T-Pro (g/L) 
+62.91±5.54 
+61.30±5.92 
+57.54±6.56 ab 
+52.45±7.15 abc 
+Alb (g/L) 
+41.01±4.28 
+38.29±4.43 a 
+34.38±4.45 ab 
+31.75±4.24 abc 
+Pre-Alb (g/L) 
+196.58(160.86,238.85) 
+181.60(127.98,291.55) 
+a  
+98.48(80.37,148.75) ab 
+98.73(65.29,151.81) a 
+BUN (umol/L) 
+5.43(4.39,6.79) 
+6.13(4.80,8.32) a  
+6.33(4.92,11.15) a 
+13.85(7.56,20.02) a 
+Cr (umol/L) 
+57.50(48.80,71.28) 
+58.40(47.80,78.70)  
+54.20(37.70,80.60) 
+89.50(38.10,169.30) 
+UA (umol/L) 
+299.76(243.44,359.89) 
+291.43(224.63,376.58) 
+242.34(142.73,319.98) ab 
+252.97(148.49,377.04)
+a 
+Cystatin C 
+(mg/mL) 
+0.98(0.86,1.21) 
+1.16(0.95,1.56) a  
+1.39(1.06,1.88) ab  
+1.60(1.16,2.55) a 
+Lactate 
+(mmol/L) 
+1.94±0.71 
+2.00±0.78 
+2.21±1.04 
+2.77±1.23 
+FBG (mmol/L)  5.81±2.72 
+6.35±2.88 a 
+7.69±3.00 ab 
+8.07±1.25 abc 
+LDH (U/L) 
+194.28±48.28 
+215.18±70.80 a 
+244.00±90.07 ab 
+358.20±107.77 abc 
+K (mmol/L) 
+3.75±0.51 
+3.87±0.63 
+3.89±0.54 
+4.20±0.66 
+Na(mmol/L) 
+142.95±3.54 
+141.19±5.27 
+140.75±6.28 
+139.56±7.09 
+Cl (mmol/L) 
+105.00±3.52 
+103.94±5.25 
+103.75±6.22 
+102.33±7.35 ab 
+Ca (mmol/L) 
+2.05±0.37 
+1.84±0.49 a 
+1.53±0.49 ab 
+1.68±0.44 ab 
+Mg (mmol/L) 
+0.88±0.08 
+0.86±0.10 
+0.82±0.11 a 
+0.81±0.10 a 
+P (mmol/L) 
+1.19±0.34 
+1.12±0.43 
+0.88±0.36 ab 
+0.72±0.37 abc 
+ 
+ 
+
+Table 3. Comparison of clinical and biochemical characteristics among patients with 
+different prognosis 
+ 
+P1 group 
+(n=638) 
+P2 group  
+(n=70) 
+P3 group  
+(n=11) 
+Age (years) 
+74.5(64,86) 
+85.5(78.75,90.25) a 
+81(66,89) 
+Sex (M/F) 
+277/361 
+34/36 
+6/5 
+TVRC (days) 
+10(7,15) 
+14(9,23) a 
+18.5(14.75,22) a 
+BMI (Kg/m2) 
+23.17±3.60 
+21.79±2.70 
+19.92±4.98 
+25(OH)D3 
+28.21(20.46,40.22) 
+19.53(12.11,27.44) a 
+18.03(10.96,21.56) a 
+CRP (mg/L) 
+7.22(2.96,20.25) 
+45.98(22.03,93.56) a  
+69.08(17.99,105.47) a 
+Temp (℃) 
+36.71±0.47 
+36.71±0.49 
+36.78±0.41 
+SBP (mmHg) 
+140.69±20.61 
+142.04±22.37 
+142.55±20.86 
+DBP (mmHg) 
+79.84±11.47 
+80.24±14.89 
+77.91±12.76 
+HR (bpm) 
+87.47±14.69 
+83.21±17.82 
+88.45±18.97 
+RR (bpm) 
+19.48±1.22 
+19.99±2.36 a 
+19.64±1.86 
+SaO2 (%) 
+97.49±1.52 
+95.68±4.87 a 
+96.00±3.55 
+Fio2 (%) 
+29(21,33) 
+33(29,41) a  
+41(37,53) a  
+RBC (10^12/L) 
+4.12±0.65 
+3.47±0.80 a 
+3.72±0.64 
+WBC (10^9/L)  
+6.09±2.84 
+8.35±3.54 a 
+10.12±6.57 a 
+Monocyte % 
+8.21±2.94 
+6.95±2.83 a 
+5.48±5.14 a 
+Lymphocyte % 
+26.83±11.81 
+13.69±7.76 a 
+11.76±11.01 a 
+Neutrophil % 
+62.76±12.58 
+77.69±9.44 a 
+82.25±13.39 a 
+PLT (10^9/L) 
+205.06±78.08 
+211.16±93.12 
+219.82±133.25 
+Hct (%) 
+38.16±5.90 
+32.22±7.32 a 
+33.75±6.11 
+Hb (g/L) 
+123.41±19.53 
+104.79±23.48 a 
+125.00±39.87 b 
+T-Bil (umol/L) 
+12.80±6.08 
+15.83±17.02 a 
+11.66±6.80 
+ALT (U/L) 
+20.08(13.79,30.75) 
+18.68(13.30,32.71) a 
+21.80(10.16,32.22) a 
+AST (U/L) 
+23.58(18.86,32.43) 
+31.92(20.11,47.30) a 
+35.85(23.21,47.03) 
+AKP (U/L) 
+79.99(65.62,95.93) 
+88.07(69.46,113.97) 
+80.64(65.92,98.15) 
+T-Pro (g/L) 
+62.21±5.75 
+56.77±6.47 a 
+54.37±6.76 a 
+Alb (g/L) 
+39.72±4.53 
+34.15±4.47 a 
+32.74±4.76 a 
+Pre-Alb (g/L) 
+190.22(147.24,232.20) 
+100.75(81.27,146.45) a  
+129.31(82.19,197.08) 
+BUN (umol/L) 
+5.65(4.51,7.43) 
+7.32(5.28,12.94) a  
+10.41(6.20,18.20) a 
+Cr (umol/L) 
+57.80(48.20,72.48) 
+62.15(41.40,90.65)  
+67.20(32.10,129.40) 
+UA (umol/L) 
+291.73(230.62,365.03) 
+275.16(171.31,363.57) 
+148.57(79.31,240.75) ab 
+Cystatin C (mg/mL) 
+1.05(0.89,1.38) 
+1.47(1.23,2.15) a  
+1.20(1.01,2.35)  
+Lactate (mmol/L) 
+1.96±0.74 
+2.24±1.07 
+2.89±1.09 a 
+FBG (mmol/L)  
+6.00±2.67 
+8.07±3.41 a 
+9.64±3.43 a 
+LDH (U/L) 
+203.50±58.94 
+251.11±97.54 a 
+334.63±115.47 ab 
+K (mmol/L) 
+3.79±0.55 
+4.01±0.71 a 
+4.17±0.62 
+Na(mmol/L) 
+142.20±4.30 
+139.64±6.25 a 
+140.82±12.58 
+Cl (mmol/L) 
+104.46±4.39 
+103.93±5.77 
+102.91±11.60 
+Ca (mmol/L) 
+1.93±0.45 
+1.67±0.50 a 
+1.29±0.39 ab 
+Mg (mmol/L) 
+0.87±0.09 
+0.83±0.11 a 
+0.81±0.10 
+
+P (mmol/L) 
+1.15±0.39 
+0.95±0.37 a 
+0.63±0.19 ab 
+ 
+

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+Fabrication and characterization of iodine 
+photonic microcell for sub-Doppler 
+spectroscopy and laser stabilization 
+CLÉMENT GOÏCOECHÉA,1,2 THOMAS BILLOTTE,2 MATTHIEU CHAFER,1,2 
+MARTIN MAUREL,1,2 JENNY JOUIN,3 PHILIPPE THOMAS,3 DEVANG NAIK,2 
+FRÉDÉRIC GÉRÔME,1,2 BENOÎT DEBORD,1,2 AND FETAH BENABID1,2,* 
+1GLOphotonics SAS, 123 avenue Albert Thomas, 87060 Limoges, France 
+2GPPMM group, Xlim Research Institute, CNRS UMR 7252, University of Limoges, Limoges, France 
+3IRCER UMR CNRS 7315, Centre Européen de la Céramique, 12 rue Atlantis, 87068 Limoges, France 
+*[email protected] 
+Abstract: We report on the development of all-fiber stand-alone Iodine-filled Photonic 
+Microcells demonstrating record absorption contrast at room temperature. The microcell’s fiber 
+is made of inhibited coupling guiding hollow-core photonic crystal fibers. The fiber-core 
+loading with Iodine was undertaken at 10-1-10-2mbar vapor pressure using a novel gas-manifold 
+based on metallic vacuum parts with ceramic coated inner surfaces for corrosion resistance. 
+The fiber is then sealed on the tips and mounted on FC/APC connectors for better integration 
+with standard fiber components. The stand-alone microcells display Doppler lines with 
+contrasts up to 73% in the 633nm wavelength range, and an insertion loss between 3 to 4dB. 
+Sub-Doppler spectroscopy based on saturable absorption has been carried out to resolve the 
+hyperfine structure of the P(33)6-3 lines at room temperature with a full-width at half maximum 
+of 24MHz on the b4 component with the help of lock-in amplification. Also, we demonstrate 
+distinguishable hyperfine components on the R(39)6-3 line at room temperature without any 
+recourse to signal-to-noise ratio amplification techniques. 
+ 
+1. Introduction 
+In the last 20 years, efforts have been made towards the miniaturization of frequency standards 
+with the emergence of new technological devices, such as clocks based on 
+microelectromechanical systems (MEMS) [1], planar devices mounted on silicon chip, 
+using hollow-core anti-resonant reflecting optical waveguides (ARROW) [2], or compact 
+engineered circular multi-pass cells [3]. Amongst these devices, the hollow-core photonic 
+crystal fiber (HCPCF) technology appeared to be an excellent and promising alternative to 
+bulky cells for portable, small footprint applications by filling and sealing atomic or molecular 
+vapor inside its core [4], culminating in the form of the photonic microcell (PMC) an atom-
+photonics component that can be integrated with small insertion loss while using existing fiber 
+connectors into any optical set-up [5]. By confining the atoms/molecules alongside light over 
+modal areas of as small as a few μm2, whilst keeping them in interaction over length scales a 
+million times longer than the Rayleigh range, the resulting enhanced atom-laser interaction 
+efficiency leads to strong absorption despite low gas density and low light-level, resulting in 
+an increased signal to noise ratio compared to other technologies. The unique optical properties 
+of HCPCF offer a large range of core sizes and lengths coupled with low loss, alongside 
+reduced power consumption and micro-structured cladding providing versatile modes 
+composition allowing transverse light structuring [6]. 
+
+HCPCFs light guiding performances are continuously improving on a broad spectral range with 
+loss figures competing with standard solid-core fibers on telecom spectral ranges with loss 
+down to 0.174dB/km in the C-band [7]. Low-loss figures are also accessible in the visible range 
+down to 0.9dB/km at 558nm [8].  
+Since the advent of the PMC as a photonic component, a plethora of work has been carried out 
+on the types of enclosed gas and the sealing techniques. The evolution of the different 
+fabrication techniques of PMC has been mainly dictated by the type of fiber as gas-container 
+and by the nature of the gas. In fact, compatibility of the solid-core single mode fiber (SMF) 
+splicing technique with photonic bandgap (PBG) fibers [9,10] with its associated core diameter 
+range of 5-20µm is no longer viable with inhibited-coupling guiding HCPCF (IC-HCPCF) 
+because of their larger core sizes (typically between 20 and 100 µm), and thus implies a strong 
+mode mismatch induced loss. Therefore, mode-field adapters have been introduced to render 
+IC guiding fiber technology compatible with SMF, either by tapering HCPCF as reported by 
+Wheeler et al. [11] or by implementing graded-index fibers [12,13]. Another configuration has 
+been also reported based on glass cells glued on the tips of PBG HCPCF to manage gas filling 
+and proceed to gas-enclosing by collapsing a section of the glass cell [14]. All the different 
+techniques mentioned above suffer from drawbacks linked to the use of glue or exposition to 
+ambient air leading to gas contamination and degradation of PMC spectroscopic performances 
+on the long term. Recently, Billotte et al. introduced a novel PMC assembly process that no 
+longer requires helium buffer gas or gluing stage [5]. Based on a glass sleeve collapse, a 1.5dB 
+insertion loss microcell has been achieved with a 7m long IC-HCPCF filled with acetylene at 
+80µbar pressure. Doppler-free spectroscopy showed constant linewidth and contrast over more 
+than 3 months, thus highlighting the quality/purity of the sealed-gas medium and the impact of 
+this contaminant-free technique for the creation of next-generation PMCs. 
+The above work on PMC assembly was chiefly motivated for frequency standards, which thus 
+far was limited to molecular gases, such as Acetylene (C2H2, 1550nm region) [4,5,11] or 
+Carbon Dioxide (CO2) [15–17]. The actual state-of-the-art for a sealed PMC is set by Triches 
+et al. and Billotte et al., each exhibiting an instability around 2.10-11 @1000s [14,18]. Both are 
+working with acetylene around the telecom wavelength band. 
+Extending PMC technology to atomic or molecular vapors, such as alkali vapors or molecular 
+halogen gas (such as iodine (I2)), have been very challenging because of their physio-chemical 
+reactivity and the required vacuum environment. This in turn limited the impact of PMC based 
+optical frequency references to a restricted number of spectral ranges. For example, in the green 
+and red spectral range, I2 is known for displaying a very dense Doppler-lines spectrum useful 
+for the development of visible broadband frequency references [19,20]. This is illustrated by 
+the fact that several I2 ro-vibrational absorption lines are among Bureau International des Poids 
+et Mesures (BIPM) recommended frequency references for the realization of the meter [21]. 
+Also, I2 based frequency stabilization [22–26] proved to be valuable for several technological 
+applications [27–30]. Consequently, the advent of iodine filled PMC (I2-PMC) would be 
+extremely useful for applications like guiding star lasers [31], high resolution LIDAR [32], or 
+laser frequency stabilization [26], which requires these performances to be delivered in a 
+compact and mobile physical package. However, encapsulating iodine vapor into glass cells 
+carries specific challenges because of I2 physical and chemical properties. In particular, its high 
+
+reactivity and corrosion with metals [33,34] requires the use of complex glass-manifold filling 
+system. Consequently, the development of I2-PMC represents a technical challenge both in 
+vapor handling, loading and in-HCPCF sealing. Indeed, the use of common metallic vacuum 
+parts is inadequate for the I2-PMC assembly process, and the commonly used glass manifold 
+alternative for I2 cell manufacturing is too complex and expensive for scalable I2-PMC 
+fabrication. Within this context, we note the previous work on I2-filled HCPCF [24,35,36]. 
+Lurie et al. have shown that the hollow-core fiber for I2-microcell development and saturated 
+absorption spectroscopy applications demonstrated a strong efficiency thanks to strong overlap 
+of pump and probe beams [35], and laser stabilization with fractional frequency stability of 
+2.3×10-12 at 1s for a HCPCF mounted on a glass vacuum manifold was achieved [24]. Impact 
+of residual gases in the vacuum system alongside I2 pressure have been raised by the authors 
+as obstacles to reach transit limited sub-Doppler linewidths. This seminal work within the I2-
+filled HCPCF framework has been followed up in 2015 by the first demonstration of a 
+hermetically sealed I2 Kagome HCPCF [36]. However, despite a demonstration of laser 
+stabilization with fractional frequency stabilities of 3.10-11 at 100s, the sealed HCPCF has been 
+marred by a strong 21.5dB insertion loss and cannot be coined PMC because the gas sealing 
+was achieved by fusion-collapsing a section of the fiber, thus eliminating its optical guidance 
+features at the collapsed section.  
+We report in this work on an I2-PMC development based on a scalable, corrosion resistant and 
+contamination free fabrication process [5]. For example, a 2.5 and 4 meter long patch-cord like 
+iodine PMC based on tubular IC-HCPCF [37] and an overall transmission efficiency as high 
+as 40% have been fabricated. These PMCs exhibit, at room temperature, high performances in 
+term of absorption contrast reaching respectively 60% and 53% on the P(33)6-3 transition - at 
+632.991nm [38]. Furthermore, we demonstrate room temperature resolution of Iodine 
+hyperfine spectrum observation, over 10GHz spectral range around 633nm, of several sub-
+Doppler spectral transparencies from different broadened lines, thanks to saturable absorption. 
+The exceptional lifetime of these microcells is demonstrated through the unaffected P(33)6-3 
+absorption contrast and Doppler linewidth of 4 year-old I2-PMC. 
+2. Experimental set-up for I2-PMC fabrication 
+An IC-HCPCF based on tubular lattice cladding has been designed and fabricated for optimal 
+guidance on several hundred thousand hyperfine transitions of iodine in the green-to-red 
+spectral range (Fig. 1(a)) with loss below 30dB/km level between 530nm and 668nm. The fiber 
+(Fig. 1(a)) with an outer diameter of 200µm exhibits a core diameter of 30µm surrounded by 8 
+isolated tubes cladding. This fiber presents an excellent modal behavior with quasi single-mode 
+guidance as illustrated by the near-field intensity profile at 633nm (see Fig. 1(a)) measured at 
+the output of a 4m long fiber. 
+
+ 
+Fig. 1. (a) Measured loss spectrum of the experimental IC tubular fiber related to the 
+developed PMCs (see Fig. 2). On the right : micrograph picture of the fiber cross section and 
+near field intensity distribution at 633nm. (b) Overview of the experimental set-up for I2-filling 
+and fiber-sealing. The vacuum system is represented in gray with valves and the gauge 
+represented by yellow circles. The fiber is represented in dark blue. PBS: polarizing beam 
+splitter. Lock-in system is represented in blue color. AOM : acousto-optic modulator. (c) 
+Schematic of set-up for PMC characterization. 
+Two PMCs, PMC#1 and PMC#2, have been fabricated 3 years apart in 2017 and 2020 
+respectively from similar fibers using an in-house gas-vacuum manifold, represented 
+schematically in Fig. 1(b) and purposely designed for I2 loading into HCPCF and for I2-PMC 
+assembly. 
+The manifold is composed of three main compartments separated by vacuum valves. The 
+central part holds the HCPCF and acts as the fiber loading section. On the right side of the fiber 
+loading section, a turbo-molecular vacuum-pump is connected via a cryogenic trap to prevent 
+any contamination to the vacuum-pump during I2 releasing. The left side corresponds to the I2 
+dispenser. Here, iodine chips are placed in glass test-tube, which is hermetically connected to 
+the manifold via a metallic fitting. This section is under pressure and/or temperature regulation 
+for I2 sublimation and release into the fiber loading section. The fiber loading goes through the 
+following sequence. First, the fiber loading section is evacuated to a vacuum pressure of less 
+than 10-6mbar. Similarly, the iodine dispenser section is evacuated while ensuring the I2 
+remains solid by regulating the iodine chip temperature. The above ensures the leakproofness 
+of the manifold and high vacuum quality. Once this process is achieved, the I2 is sublimated by 
+increasing the temperature until a pressure of around 10-1 mbar is reached. Second, the I2 vapor 
+is released in the fiber loading section by opening the valve between the two sections and 
+
+W
+Pump
+WWclosing the valve to the vacuum pump section. During this loading process, we continuously 
+monitor the fiber transmission spectrum derived from a tunable external cavity diode laser 
+(TOPTICA DL-PRO, 631-635nm range) tuned to a frequency range corresponding to one of 
+the iodine rovibrational lines. A second beam from the same laser is sent to an Iodine 
+macroscopic gas cell and serves as a reference. The spectroscopic signature of the Iodine 
+absorption lines (spanning over a set of 3 lines around the P(33)6-3 transition) can be observed 
+after 10 minutes of loading. The loading is then kept on until the desired contrast is reached. 
+It is noteworthy, that the metallic parts of the whole manifold have been post-processed against 
+corrosion and chemical reaction with iodine by applying a ceramic coating on the inner metallic 
+surfaces. This allows an outstanding ease-of-use with several HCPCFs being loaded and 
+assembled over several years.   
+Before mounting and splicing the HCPCF (described in Fig. 1(a)), it was flushed with Helium 
+or Argon gas and heated for several hours in the oven at ~100°C to reduce any residual gas 
+inside the fiber. The fiber is then end-capped on one extremity by collapsing a borosilicate 
+capillary with an inner diameter fitting the outer diameter of the fiber, following the process 
+mentioned in [5]. The sealed and polished extremity is then mounted on a FC/APC optical 
+connector with a measured coupling loss in the range of 1 to 1.5dB at 633nm, 20dB lower than 
+the splicing loss obtained by collapsing the fiber on itself in [36]. 
+The second extremity of the HCPCF is connected by borosilicate sleeve fusion splicing to a 
+30cm piece of the same HCPCF. The second tip of 30cm long HCPCF is hermetically attached 
+to the loading compartment of the manifold via a home-made fiber-feedthrough (Fig. 1(a) of 
+[5]). The end-capped fiber is then evacuated by pumping the valve-controlled middle chamber 
+of the vacuum system down to the range of 10-6 mbar. Once the desired contrast is reached 
+(here around 60%), we hermetically seal the fiber by end-capping with a splicing machine 
+based on sleeve collapse around the tip of the HCPCF, as described in [5]. The tips of the 
+resulted PMC are then polished and mounted on FC/APC fiber connectors. Figure 2(a) shows 
+the photography of a typical FC/APC connectorized patch-cord like PMC in its final form. 
+Figure 2(b) shows the reconstructed near-field intensity profile of the transmitted light from 
+the developed I2-PMC. The measured transmission was in the range of 40-50%, corresponding 
+to an insertion loss of less than 4 dB, which is 17.5dB lower than the one measured in [36]. 
+Figure 2(d) represents the normalized transmission spectra at the output of PMC#1 (red curve), 
+PMC#2 (orange curve) and the commercial macroscopic cell (black curve) measured 
+consecutively with the same laser configurations and room temperature conditions. The bottom 
+axis and the top axis of the graph give the frequency and the relative frequency from that of the 
+R(39)6-3 transition, respectively. The two PMCs exhibit contrast between 53% to 60% for the 
+P(33)6-3 line and 61% to 73% for the R(39)6-3. This is respectfully 5.9 to 6.7 and 5.1 to 6.1 
+times larger than the ones obtained with the 10cm long commercial I2 macroscopic gas cell 
+(resp. 9% & 12%) corresponding to 1.3.10-1 – 1.6.101 mbar of I2 vapor pressure specifications 
+given by manufacturer. These contrasts are 2.2 to 2.9 times larger than the one obtained at room 
+temperature in previously reported work [36], which is to our knowledge the only reported 
+work on low-loss I2-loaded sealed HCPCF.  
+
+ 
+Fig. 2. (a) Photography of I2 PMC#1 mounted on FC/APC connectors. (b) Measured near field 
+intensity profile at 633nm at the output of PMC#1. (c) Contrast evolution of the PMC#2 over 2 
+years. (d) Normalized transmission spectra through the fabricated PMCs (PMC#1 in red color, 
+PMC#2 in orange) and through a macroscopic commercial gas cell (black). 
+Finally, comparison of the shown transmission spectra with those recorded at the time of PMC 
+sealing and more recently on October 2022 (see Fig.2(c)) shows comparable contrasts. In fact, 
+evolution of the contrast of P(33)6-3 line through PMC#2 has been studied along 2 years since 
+its encapsulation at t0. The different measurements are summarized on table from Fig. 2(c). The 
+measured contrast of the P(33)6-3 line of I2 was found to remain constant within a range of 
+8,5% around the extrema average value of 61%. Observed fluctuations have been attributed 
+to the different temperature conditions and laser diode ampereage setpoint, and corroborated 
+by an additional study using another PMC (based on the same fiber and fabrication process). 
+The result has shown that by considering the extreme experimental values of room temperature 
+(i.e. from 19 to 22.5°C) and laser diode ampereage, these two major contributions of contrast 
+change can lead to a variation of 9%.  
+The stability of the absorption contrast highlights the leak proofness of the PMCs, and the 
+reliability and repeatability of the developed process. 
+3. Sub-Doppler spectroscopy with I2 PMC   
+The fabricated PMCs have shown their potential for sub-Doppler spectroscopy through the 
+resolution of the hyperfine structure of the P(33)6-3 line. To do so, Saturated Absorption 
+
+PMC#2ContrastevolutionofP(33)6-3
+Deviation from
+Acquisitiontime
+Contrast
+average (%)
+to
+0.691
+10.6
+to+5months
+0.527
+15.7
+to+10 months
+0.658
+5.3Spectroscopy (SAS) measurements have been done following the set-up shown in Fig. 1(c). 
+The laser beam is separated into counter-propagating pump and probe beams with 4mW and 
+40µW output power, respectively. The pump beam is obtained from the first order diffracted 
+beam off an acousto-optic modulator (AOM) operating at 64MHz frequency. This was 
+motivated so to avoid interference between the probe and back-reflected pump during the 
+propagation in the fiber. The spectroscopic transparency signal is obtained by redirecting the 
+PMC-transmitted probe beam on to a photodiode with the help of a polarizing cube. Half and 
+quarter waveplates are used to improve both the optical PMC transmission and the intensity of 
+the redirected probe beam on the photodetector. 
+ 
+ 
+Fig. 3. (a) Example of Iodine ro-vibrational hyperfine energy levels. This structure is usually 
+not observable through a simple macroscopic cell at room temperature. (b) Measured R(39)6-
+3 Doppler line at room temperature through PMC#1. (c) Hyperfine components structure of 
+the P(33)6-3 Doppler line obtained with a lock-in amplifier detection scheme. Measured peaks 
+(red) are compared with tabulated values components for P(33)6-3 & R(127)11-15 
+(respectively in blue and green). Data have been fitted with lorentzian multi-peak fit function. 
+Figure 3(b) shows the probe signal when the laser frequency is tuned in the vicinity of R(39)6-
+3 line and recorded directly by the photodetector at room temperature. In addition of the 
+Doppler-broadened absorption line, the trace shows the 21 hyperfine b-lines [39]. To our 
+knowledge, this is the first time that such Iodine transparencies are observed on a cell without 
+any means of signal-to-noise ratio (SNR) post-acquisition amplification such as lock-in 
+
+[(b)
+0.24
+Signal input (a.u.)
+0.26
+0.28
+-0.30
+-25-20-15-10
+-5
+0
+5
+10
+5
+20
+25
+Acquisitiontime(ms)
+PMC#1 Lock-inamplifier output (a.u.)
+6
+PMC#1Lock-inamplifieroutput
+Lorentzianmulti-fitpeak
+P(33)6-3bcomponents(BIPMdatabase)
+CumulativeFitPeak
+R(127)11-15acomponents(BIPMdatabase)
+D
+4
+2
+-1000
+-900
+-800
+-700
+-600
+-500
+-400
+-300
+-200
+-100
+0
+100
+RelativefrequencyfromP(33)6-3b21component(MHz)detection. In order to improve the hyperfine structure resolution we used a lock-in amplification 
+detection scheme with a squared-modulated pump beam of 1MHz (amplitude modulation). The 
+spectrum obtained as output of the lock-in amplifier is shown in Fig. 3(c). The 21 hyperfine b-
+components of the P(33)6-3 line (“b” energy level scheme shown in Fig.3(a)) have been 
+identified and are in good agreement with the optical frequencies tabulated by the BIPM in blue 
+[38]. One can notice some shift and/or additional peak, such as between b11 and b18, that could 
+be explained by the overlap between P(33)6-3 and R(127)11-5 absorption lines of I2. Hence, 
+additional weaker peaks could come from the SAS of R(127)11-5 line.  
+Figure 4 shows the b4 hyperfine component of the P(33)6-3 line, previously displayed in Fig. 
+3(c), on which a Lorentzian fitting shows a FWHM of 21MHz for 4mW pump beam. 
+Contribution of broadening sources such as wall collisions and the natural linewidth can be 
+directly calculated [4] leading to 2.95MHz and 3.23MHz respectively (lifetime about 310ns 
+[40,41]). The laser linewidth provided by the manufacturer is about 0.20MHz. Considering the 
+pressure of I2 inside the PMC of around 10-2mbar, we can estimate a few 40kHz [42] 
+intermolecular collision broadening. Therefore, the minimum linewidth obtainable using this 
+setup is 6.42MHz. The larger measured linewidth of 21MHz is explained by power broadening 
+coming from the pump beam intensity of 10MW/m², 357 times bigger than saturation intensity 
+of 28kW/m² [43]. 
+ 
+Fig. 4. Zoom-in on b4 component of the hyperfine structure displayed in Fig. 3(c). Lock-in 
+signal is displayed in red line. A 21MHz FWHM Lorentzian curve has been fitted in black 
+line. 
+ 
+ 
+ 
+
+3.0
+PMC#1Lock-inamplifieroutput
+Lorentzianmulti-fitpeak
+PMC#1 Lock-in amplifier output (a.u.)
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+-710
+-700
+-690
+-680
+-670
+-660
+-650
+Relative frequency from P(33)6-3 b21 component (MHz)4. Conclusion 
+As a summary, we reported on the first fabrication of meter-long low-optical loss pure I2 PMCs 
+based on a new process for creating all-fibered stand-alone PMCs. Absorption contrasts up to 
+73% have been measured at room temperature with PMC insertion loss of 4dB, 5.4 times lower 
+than the state-of-the-art. The good sealing quality is demonstrated by a PMC being still 
+functional after 4 years and the stable absorption measured throughout both PMCs, as well as 
+the performance of Doppler-free signal measurements for the first PMC on the P(33)6-3 line 
+of I2 at room temperature. The b4 component of this line shows a FWHM of 24MHz at 4mW 
+output pump beam power. By calculating the different broadening sources, we identify the 
+dominant one as power broadening. An optimization of I2 pressure, PMC length and pump 
+power should allow us to reduce this FWHM while keeping the same SA contrast. These results 
+are very promising for many compact sensing applications and laser stabilization. 
+Funding: Région Nouvelle-Aquitaine. 
+Disclosures: The authors declare no conflicts of interest. 
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+ 
+

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+arXiv:2301.03132v1  [math.AC]  9 Jan 2023
+FREE DIVISORS, BLOWUP ALGEBRAS OF JACOBIAN IDEALS,
+AND MAXIMAL ANALYTIC SPREAD
+RICARDO BURITY, CLETO B. MIRANDA-NETO, AND ZAQUEU RAMOS
+Abstract. Free divisors form a celebrated class of hypersurfaces which has been extensively studied
+in the past fifteen years. Our main goal is to introduce four new families of homogeneous free divisors
+and investigate central aspects of the blowup algebras of their Jacobian ideals.
+For instance, for
+all families the Rees algebra and its special fiber are shown to be Cohen-Macaulay – a desirable
+feature in blowup algebra theory. Moreover, we raise the problem of when the analytic spread of the
+Jacobian ideal of a (not necessarily free) polynomial is maximal, and we characterize this property
+with tools ranging from cohomology to asymptotic depth. In addition, as an application, we give an
+ideal-theoretic homological criterion for homaloidal divisors, i.e., hypersurfaces whose polar maps are
+birational.
+Dedicated with gratitude to the memory of Professor Wolmer V. Vasconcelos,
+mentor of generations of commutative algebraists.
+Introduction
+The well-studied theory of free divisors – or free hypersurfaces – has its roots in the seminal work
+of K. Saito [35], and in subsequent papers of H. Terao [43, 44, 45] mostly concerned with the case
+of hyperplane arrangements. The original environment was the complex analytic setting, and the
+motivation was the computation of Gauss-Manin connections for the universal unfolding of an isolated
+singularity; for instance, it was proved that the discriminant in the parameter space of the universal
+unfolding is a free divisor. Over time, different approaches, viewpoints, and interests have emerged,
+including algebraic (and algebro-geometric) adaptations and even generalizations that have drawn
+the attention of an increasing number of researchers over the last fifteen years. The list of references
+is huge; see, e.g., Abe [1], Abe, Terao and Yoshinaga [2], Buchweitz and Conca [8], Buchweitz and
+Mond [9], Calder´on-Moreno and Narv´aez-Macarro [12], Damon [15], Dimca [17, 18], Dimca and
+Sticlaru [20], Miranda-Neto [31, 32], Schenck [37], Schenck, Terao and Yoshinaga [38], Schenck and
+Tohˇaneanu [39], Simis and Tohˇaneanu [41], Tohˇaneanu [47], and Yoshinaga [51]. In particular, nice
+references containing interesting open problems on the subject (including the celebrated Terao’s
+Conjecture) are Dimca’s book [17] and Schenck’s survey [37].
+In the present paper, the general goal is to present progress on the algebraic side of the theme,
+by means of various techniques.
+First, we explicitly describe four new families of homogeneous
+free divisors in standard graded polynomial rings over a field k with char k = 0.
+Second, and
+in the same graded setup, we turn our angle to investigating blowup algebras of Jacobian ideals of
+polynomials. More precisely, we prove that for our families the Rees algebra is Cohen-Macaulay – i.e.,
+from a geometric point of view, blowing-up their singular loci yields arithmetically Cohen-Macaulay
+schemes. Furthermore we characterize in various ways, via tools varying from (local) cohomology to
+asymptotic depth, the maximality of the dimension of the special fiber ring for polynomials which
+are no longer required to be free. The relevance of the latter lies in connections to the important
+2010 Mathematics Subject Classification. Primary: 14J70, 32S05, 13A30, 14E05, 14M05; Secondary: 13C15, 13H10,
+14E07, 32S22, 32S25.
+Key words and phrases. Free divisor, Jacobian ideal, blowup algebra, Rees algebra, analytic spread, homaloidal
+divisor.
+Corresponding author: Cleto B. Miranda-Neto ([email protected]).
+1
+
+2
+R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS
+theory of homaloidal divisors, i.e., homogeneous polynomials f ∈ k[x1, . . . , xn] (where, typically, the
+field k is assumed to be algebraically closed) for which the associated polar map
+Pf =
+� ∂f
+∂x1
+: · · · :
+∂f
+∂xn
+�
+: Pn−1 ��� Pn−1
+is birational – i.e., Pf is a Cremona transformation. In fact we provide, as an application, an ideal-
+theoretic (also homological) homaloidness criterion. It is worth mentioning that the modern theory
+about such polynomials began in Ein and Shepherd-Barron [23], where it was proved for instance
+that the relative invariant of a regular prehomogeneous complex vector space is homaloidal. Another
+classical reference is Dolgachev [21].
+In order to introduce the other main concepts of interest to this paper, let k denote a field of
+characteristic zero and, given n ≥ 3, let R = k[x1, . . . , xn] be a standard graded polynomial ring
+over k.
+Let R+ = (x1, . . . , xn) be the irrelevant ideal.
+Given a non-zero reduced homogeneous
+polynomial f ∈ R2
++ (whose partial derivatives will be assumed, to avoid pathologies, to be k-linearly
+independent), it is well-known that the property of f being a free divisor can be translated into saying
+that the corresponding Jacobian ideal Jf = (∂f/∂x1, . . . , ∂f/∂x1) ⊂ R is perfect of codimension 2
+– in particular, a free f must be highly singular in the sense that codim Sing V (f) = 2 regardless
+of n.
+Therefore, the intervention of Jf in the theory is (naturally) crucial, and as a bonus this
+allows for an interesting link to the study of blowup algebras, particularly the traditional problem
+of describing ideals for which such rings are Cohen-Macaulay. Here, we are especially interested in
+the Rees algebra
+R(Jf) = R
+� ∂f
+∂x1
+t, . . . , ∂f
+∂xn
+t
+�
+⊂ R[t]
+and its special fiber ring F(Jf) = R(Jf) ⊗R k, which, as is well-known from blowup theory, encode
+relevant geometric information. Recall that the analytic spread of Jf, denoted ℓ(Jf), is the Krull
+dimension of F(Jf), which is bounded above by n. Saying that Jf has maximal analytic spread
+means ℓ(Jf) = n.
+Next we briefly describe the contents of each section of the paper.
+Section 1 invokes the definitions that are central to this paper, such as the notions of free divisor
+and blowup algebras of ideals, as well as a few auxiliary facts which will be used in some parts of
+the paper. Also, some conventions are established.
+In Section 2 we present our first family of free divisors in R, with n ≥ 4. They are reducible and,
+in fact, linear in the sense that in addition the Jacobian ideal Jf is linearly presented, i.e., the entries
+of the corresponding Hilbert-Burch matrix are (possibly zero) linear forms. We also determine the
+defining equations of F(Jf) and compute the analytic spread as well as the reduction number of Jf.
+Moreover, we prove that R(Jf) and F(Jf) are Cohen-Macaulay.
+Section 3 describes our second family of free divisors, in an even number of at least 4 variables, and
+again reducible and linear in the above sense. We exhibit a well-structured minimal set of generators
+for the module of syzygies of Jf. In addition, as in the previous family – but via different methods
+– we show that R(Jf) and F(Jf) are Cohen-Macaulay (the latter is in fact shown to be a generic
+determinantal ring) and determine the analytic spread and the reduction number of Jf.
+In Section 4, our third family is presented as a two-parameter family of (no longer linear) free
+divisors f = fα,β in 3 variables and of degree αβ, where α, β ≥ 2. For the one-parameter family with
+β = 2 (also for (α, β) = (2, 3)), we show that R(Jf) is Cohen-Macaulay and derive that F(Jf) is
+isomorphic to a polynomial ring over k (so that Jf has reduction number zero). Also we prove that
+f is reducible if k = C, and in case k ⊆ R we verify that f is reducible if β is odd and irreducible
+otherwise. In addition, if β ≥ 3 is odd, we show how to derive yet another two-parameter family of
+free divisors g = gα,β (of degree αβ − α) from fα,β; for the one-parameter family with β = 3, we
+deduce that R(Jg) is Cohen-Macaulay.
+
+FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD
+3
+In Section 5 we introduce our fourth family of free divisors, in 3 variables. Such (reducible) divisors
+have the linearity property – as in the two first families – and are constructed as the determinant of
+the Jacobian matrix of a set of quadrics which we associate to 3 given suitable linear forms. This
+family is in fact partially new, because if k = C then its members are recovered by the well-known
+classification of linear free divisors in at most 4 variables, whereas on the other hand our (permanent)
+assumption on k is that char k = 0. Furthermore, we show that Jf is of linear type – i.e., the canonical
+epimorphism from the symmetric algebra of Jf onto R(Jf) is an isomorphism – and we derive that
+R(Jf) is a complete intersection ring.
+For f ∈ R belonging to any of the four (or five) families, we use a result from Miranda-Neto [31]
+to easily determine the Castelnuovo-Mumford regularity of the graded module Derk(R/(f)) formed
+by the k-derivations of the ring R/(f). Needless to say, the regularity is an important invariant
+which controls the complexity of a module (being related to bounds on the degrees of syzygies),
+whereas the derivation module is a classical object as it collects the tangent vector fields defined on
+the hypersurface V (f) ⊂ Pn−1
+k
+.
+We close the paper with Section 6, where we address the question as to when, for a (not necessarily
+free) polynomial f ∈ R, the ideal Jf has maximal analytic spread – the relevance of this task is the
+already mentioned connection to the theory of homaloidal divisors. We provide a number of charac-
+terizations of when such maximality holds, including cohomological conditions on a suitable auxiliary
+module as well as the asymptotic depth associated to both adic and integral closure filtrations of Jf.
+We also point out that the main problems we raise in this paper appear in this section. This includes
+a conjecture predicting that if f is linear free divisor satisfying ℓ(Jf) = n then Jf is of linear type
+(the case of interest is n ≥ 5), as well as the question of whether the reduced Hessian determinant of
+a homaloidal polynomial must necessarily be a (linear) free divisor. For all such problems we were
+motivated and guided by several examples, and the computations were performed with the aid of
+the program Macaulay of Bayer and Stillman [5].
+1. Preliminaries: Free divisors and blowup algebras
+We begin by invoking some definitions and auxiliary facts.
+First we establish the convention
+that, throughout the entire paper, k denotes a field of characteristic zero. A few other conventions
+(including notations) will be made in this section.
+Let R = k[x1, . . . , xn] be a standard graded
+polynomial ring in n ≥ 3 indeterminates x1, . . . , xn over k, and let R+ = (x1, . . . , xn) denote the
+homogeneous maximal ideal of R.
+1.1. Free divisors. Fix a non-zero homogeneous polynomial f ∈ R2
++. A logarithmic derivation of
+f is an operator θ = �n
+i=1 gi∂/∂xi, for homogeneous polynomials g1, . . . , gn ∈ R satisfying
+θ(f) =
+n
+�
+i=1
+gi
+∂f
+∂xi
+∈ (f).
+Geometrically, θ can be interpreted as a vector field defined on Pn−1
+k
+that is tangent along the
+(smooth part of the) hypersurface V (f). From now on we suppose f is reduced in the usual sense
+that fred = f, that is, f is (at most) a product of distinct irreducible factors. In addition, we assume
+throughout – with no further mention – that the partial derivatives of f are k-linearly independent
+so as to prevent f from being a cone (recall that a polynomial g ∈ R is a cone if, after some linear
+change of coordinates, g depends on at most n − 1 variables). Denote by TR/k(f) the R-module
+formed by the logarithmic derivations of f, which is also called tangential idealizer (or Saito-Terao
+module) of f, and commonly denoted Derlog(−V (f)). It is easy to see that TR/k(f) has (generic)
+rank n as an R-module.
+Definition 1.1. f is a free divisor if the R-module TR/k(f) is free.
+
+4
+R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS
+This concept, which originated in [35], has been shown to be of great significance to a variety of
+branches in mathematics. We recall yet another classical object.
+Definition 1.2. If fxi := ∂f/∂xi, i = 1, . . . , n, then the Jacobian ideal of f (also called gradient
+ideal of f) is given by
+Jf = (fx1, . . . , fxn) ⊂ R.
+Note that, because f is not a cone, the ideal Jf is minimally generated by the n partial derivatives
+of f. Also recall that the Euler derivation
+εn :=
+n
+�
+i=1
+xi
+∂
+∂xi
+is logarithmic for the homogeneous polynomial f by virtue of the well-known Euler’s identity
+�n
+i=1 xifxi = (deg f)f.
+Now we remark that, since TR/k(f) decomposes into the direct sum of
+the module of syzygies of Jf and the cyclic module Rεn (see [31, Lemma 2.2]), a free basis of TR/k(f)
+if f is a free divisor consists of the derivations corresponding to the columns of a minimal syzygy
+matrix of fx1, . . . , fxn together with εn.
+Next we recall a useful characterization which is even adopted as the definition of free divisor by
+some authors, and moreover highlights the central role that commutative algebra plays in the theory.
+Lemma 1.3. ([31, Lemma 4.1]) f ∈ R is a free divisor if and only if Jf is a codimension 2 perfect
+ideal (equivalently, the ideal Jf has projective dimension 1).
+In other words, f is a free divisor if and only if R/Jf is a Cohen-Macaulay ring and ht Jf = 2,
+where, here and in the entire paper, ht I stands for the height of an ideal I ⊂ R. It follows that the
+classical Hilbert-Burch theorem plays a major role in the algebraic side of free divisor theory. It is
+also worth mentioning that this fruitful interplay holds in a more general setting (see [32]).
+Below we invoke a well-known and very useful criterion of freeness detected by Saito himself in
+case k = C, but which is known to hold over any field of characteristic zero (see [8, Theorem 2.4]).
+Lemma 1.4. ([35, Theorem 1.8(ii)], also [17, Theorem 8.1]) f ∈ R is a free divisor if and only if
+there exist n vector fields θ1, . . . , θn ∈ TR/k(f) such that
+det [θj(xi)]i,j=1,...,n = λf
+for some non-zero λ ∈ k. In this case, the set {θ1, . . . , θn} is a free basis of TR/k(f).
+As already pointed out, up to elementary operations in the columns of an n × (n − 1) syzygy
+matrix ϕ of Jf, the derivations θj’s of the free basis above correspond to the columns of ϕ along with
+the Euler vector field εn.
+There is also the following important subclass introduced in [9].
+Definition 1.5. f is a linear free divisor if f is a free divisor and the ideal Jf is linearly presented.
+Stated differently, f is a linear free divisor if and only if Jf admits a minimal graded R-free
+resolution of the form
+0 −→ R(−n)n−1 −→ R(−n + 1)n −→ Jf −→ 0.
+In particular, the degree of a linear free divisor is necessarily equal to n (and thus has minimal
+degree, since any free divisor is seen to have degree at least n).
+Now let us provisionally consider a more general setup. Let S be any Noetherian commutative
+ring containing k. A k-derivation of S is defined as an additive map ϑ: S → S which vanishes on
+k and satisfies Leibniz’ rule: ϑ(uv) = uϑ(v) + vϑ(u), for all u, v ∈ S. Such objects are collected
+in an S-module, denoted Derk(S). In particular, if again R = k[x1, . . . , xn], we get the R-module
+Derk(R), which is free on the ∂/∂xi’s.
+Now if f ∈ R2
++ is as above, we can also consider the
+
+FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD
+5
+derivation module Derk(R/(f)), which can be graded as follows.
+First, assume that Derk(R) is
+given the grading inherited from the natural Z-grading of the Weyl algebra of R, so that each ∂/∂xi
+has degree −1. We endow TR/k(f) with the induced grading from Derk(R), that is, a logarithmic
+derivation �n
+i=1 gi∂/∂xi ∈ TR/k(f) has degree δ if g1, . . . , gn ∈ R have degree δ + 1. For example,
+εn ∈ [TR/k(f)]0. Finally recall that there is an identification (see, e.g., [31, Lemma 2.1])
+Derk(R/(f)) = TR/k(f)/fDerk(R).
+Then we let Derk(R/(f)) be graded with the grading induced from TR/k(f) by means of this quotient.
+The next auxiliary lemma is concerned with the graded module Derk(R/(f)). Let us first recall
+the concept of Castelnuovo-Mumford regularity of a finitely generated graded module E over the
+graded polynomial ring R. Let 0 → Fp → . . . → F0 → E → 0 be a minimal graded R-free resolution
+of E, where Fi := �bi
+j=1 R(−ai,j), i = 0, . . . , p. Note that p is the projective dimension of E.
+Definition 1.6. If mi := max{ai,j | 1 ≤ j ≤ bi}, i = 0, . . . , p, then the Castelnuovo-Mumford
+regularity of E is defined as reg E = max{mi − i | 0 ≤ i ≤ p}.
+This gives in some sense a numerical measure of the complexity of the module. There are more
+general definitions given in terms of sheaf and local cohomologies (which in turn are related), but
+the one given above suffices for our purposes in this paper. We refer, e.g., to [4] and [7, Chapter 15].
+Lemma 1.7. ([31, Corollary 2.5(i)]) If f ∈ R is a free divisor of degree d, then reg Derk(R/(f)) =
+d − 2. In particular, if f ∈ R is a linear free divisor then reg Derk(R/(f)) = n − 2.
+It is worth mentioning that some authors have investigated the Castelnuovo-Mumford regularity
+of other objects that are also “differentially related” to f, such as the Milnor algebra R/Jf (see [11])
+and the module TR/k(f) itself (see [16, Theorem 5.5] and [36, Section 3]).
+1.2. Blowup algebras. We close the section with a brief review on blowup algebras and a few
+closely related notions. We fix a homogeneous proper ideal I of R.
+Definition 1.8. The Rees algebra of I is the graded ring
+R(I) =
+�
+i≥0
+Iiti ⊂ R[t],
+where t is an indeterminate over R. This R-algebra defines the blowup along the subscheme corre-
+sponding to I. The special fiber ring of I, sometimes dubbed fiber cone of I, is the special fiber of
+R(I), i.e., the (standard) graded k-algebra
+F(I) = R(I) ⊗R k ∼=
+�
+i≥0
+Ii/R+Ii.
+The analytic spread of I is ℓ(I) = dim F(I). There are bounds ht I ≤ ℓ(I) ≤ n.
+Alternatively, R(I) can be realized as the quotient of the symmetric algebra SymRI (a basic
+construct in algebra) by its R-torsion submodule, which is in fact an ideal. Thus there is a natural
+R-algebra epimorphism SymRI → R(I). If this map is an isomorphism, I is said to be of linear type.
+Since R is in particular a domain, this is tantamount to saying that SymRI is a domain as well. For
+instance, any ideal generated by a regular sequence is of linear type.
+Next we provide a useful formula for the computation of the analytic spread by means of a
+Jacobian matrix (in characteristic zero, as we have permanently assumed). To this end we consider
+an even more concrete description of the Rees algebra (hence of its special fiber), to wit, if we fix
+generators I = (f1, . . . , fν) ⊂ R = k[x1, . . . , xn], then R(I) is just the R-subalgebra generated by
+f1t, . . . , fνt ∈ R[t]. In the particular case where the fi’s are all homogeneous of the same degree
+– e.g., the partial derivatives of a homogeneous polynomial – we can write the special fiber as
+F(I) ∼= k[f1, . . . , fν] as a k-subalgebra of R.
+
+6
+R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS
+Lemma 1.9. ([40, Proposition 1.1]) Write I = (f1, . . . , fν) and suppose all the fi’s are homogeneous
+of the same degree. Set Θ :=
+�
+∂fi
+∂xj
+�
+, 1 ≤ i ≤ ν, 1 ≤ j ≤ n. Then ℓ(I) = rank Θ.
+Finally recall that a subideal K ⊂ I is a reduction of I if the induced extension of Rees algebras
+R(K) ⊂ R(I) is integral; equivalently, there exists r ≥ 0 such that Ir+1 = KIr. The minimal such
+r is denoted rK(I). The reduction K is minimal if it is minimal with respect to inclusion. Now the
+reduction number of I is defined as
+r(I) = min{rK(I) | K is a minimal reduction of I}.
+For instance, it is a standard fact (as k is infinite) that r(I) = 0 if and only if I can be generated by
+ℓ(I) elements, which occurs if for example I is of linear type. More generally, the following basic result
+gives a way to compute this number in the presence of a suitable condition on the standard graded
+k-algebra F(I), which can be also regarded (for the purpose of reading the Castelnuovo-Mumford
+regularity off a minimal graded free resolution) as a cyclic graded module over a polynomial ring
+k[t1, . . . , tν] whenever I can be generated by ν forms in R.
+Lemma 1.10. ([26, Proposition 1.2]) If F(I) is Cohen-Macaulay, then r(I) = reg F(I).
+2. First family: linear free divisors in Pn−1
+Before presenting our first family of free divisors as well as properties of related blowup algebras,
+let us record a couple of basic calculations which will be used without further mention in the proof
+of Theorem 2.2 below.
+Remark 2.1. Let S = k[w, u] be a standard graded polynomial ring in 2 variables w, u, and consider
+the ideal n = (w, u). Given an integer r ≥ 2, the following facts are well-known and easy to see.
+(a) The ideal nr = (wr, wr−1u, . . . , wur−1, ur) is a perfect ideal of codimension 2, having the
+following (r + 1) × r syzygy matrix:
+(1)
+ϕr =
+
+
+−w
+0
+. . .
+0
+u
+−w
+. . .
+0
+0
+u
+. . .
+0
+...
+...
+...
+...
+0
+0
+. . .
+−w
+0
+0
+. . .
+u
+
+
+;
+(b) The presentation ideal of the Rees algebra R(nr), that is, the kernel of the surjective map of
+S-algebras
+S[y1, . . . , yr+1] ։ R(nr),
+yi �→ wr−i+1ui−1,
+is equal to Q = (I1(y · ϕr), I2(B)), where y =
+�
+y1
+· · ·
+yr+1
+�
+and B =
+�
+y1
+· · ·
+yr
+y2
+· · ·
+yr+1
+�
+.
+Theorem 2.2. Consider the standard graded polynomial ring R = k[x1, . . . , xn], where n ≥ 4.
+Denote xn−1 = w and xn = u. Let
+f = 2wn−1u +
+n−2
+�
+i=1
+xiwi−1un−i.
+Then f is a linear free divisor.
+Proof. First notice that
+(2)
+fxi = wi−1un−i
+for each
+1 ≤ i ≤ n − 2,
+
+FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD
+7
+fw = 2(n − 1)wn−2u +
+n−2
+�
+i=2
+(i − 1)xiwi−2un−i
+and
+fu = 2wn−1 +
+n−2
+�
+i=1
+(n − i)xiwi−1un−(i+1).
+In particular, the subideal (fx1, . . . , fxn−2) of Jf is equal to the ideal u2(w, u)n−3. Thus, if ϕn−3 is the
+(n − 2) × (n − 3) syzygy matrix of the ideal (w, u)n−3 (see (1)), then the columns of the n × (n − 3)
+matrix
+η =
+� ϕn−3
+0
+�
+are syzygies of the gradient ideal Jf. We also have the following equalities
+(3)
+ufw = 2(n − 1)wn−2u2 +
+n−2
+�
+i=2
+(i − 1)xiwi−2un−(i−1) = 2(n − 1)wfxn−2 +
+n−2
+�
+i=2
+(i − 1)xifxi−1
+and
+(4) (n − 1)ufu = 2(n − 1)wn−1u +
+n−2
+�
+i=1
+(n − 1)(n − i)xiwi−1un−i = wfw +
+n−1
+�
+i=1
+(n(n − i − 1) + 1)xifxi.
+Now note that (3) and (4) yield two new (linear) syzygies of Jf, to wit,
+δ1 =
+� α2x2
+α3x3
+· · ·
+αn−2xn−2
+2(n − 1)w
+−u
+0 �t
+and
+δ2 =
+�
+β1x1
+β2x2
+· · ·
+βn−2xn−2
+w
+−(n − 1)u
+�t
+where αi = i − 1 if 2 ≤ i ≤ n − 2, and βi = n(n − i − 1) + 1 whenever 1 ≤ i ≤ n − 2.
+Claim 1. The minimal graded free resolution of Jf is
+(5)
+0 → R(−n)n−1
+ψ
+−→ R(−n + 1)n → Jf → 0
+where ψ =
+�
+η
+δ1
+δ2
+�
+.
+From the discussion above, we already know that the sequence (5) is a complex. To prove that it
+is in fact a minimal graded free resolution of Jf, it suffices to verify that ht In−1(ψ) ≥ 2. Note we
+can write ψ in the form
+ψ =
+� ϕn−3
+∗
+0
+Φ
+�
+where Φ =
+�
+−u
+w
+0
+−(n − 1)u
+�
+. Thus, det Φ · In−3(ϕn−3) = u2 · (w, u)n−3 ⊂ In−1(ψ). In particular,
+un−1 ∈ In−1(ψ). On the other hand, if we specialize the entries of ψ via the k-algebra endomorphism
+of R that fixes the variables w, u and maps the remaining ones to 0, we obtain the matrix
+ψ =
+
+
+−w
+0
+. . .
+0
+0
+0
+u
+−w
+. . .
+0
+0
+0
+0
+u
+. . .
+0
+0
+0
+...
+...
+...
+...
+...
+...
+0
+0
+. . .
+−w
+0
+0
+0
+0
+. . .
+u
+2(n − 1)w
+0
+0
+0
+. . .
+0
+−u
+w
+0
+0
+. . .
+0
+0
+−(n − 1)u
+
+
+.
+The (n − 1)-minor of ψ obtained by omitting the last row is cwn−1 for a certain non-zero c ∈ k.
+Therefore, the (n − 1)-minor of ψ obtained by omitting the last row has the shape cwn−1 + G, for a
+suitable G ∈ (x1, . . . , xn−2). Hence, (un−1, cwn−1 + G) ⊂ In−1(ψ). Hence, ht In−1(ψ) ≥ 2 as desired.
+
+8
+R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS
+A computation shows that, in the first case covered by the theorem (i.e., n = 4), the Jacobian
+ideal Jf is of linear type; in particular, ℓ(Jf) = 4 and r(Jf) = 0. The case n ≥ 5 is treated in
+Proposition 2.3 below. As ingredients we consider the polynomial rings
+T := k[y1, . . . , yn−2, s, t]
+and
+T ′ := k[y1, . . . , yn−2]
+as well as the k-algebras
+A := k[fx1, . . . , fxn−2, fw, fu]
+and
+A′ := k[fx1, . . . , fxn−2].
+By factoring out u2 from the polynomials fx1, . . . , fxn−2 (see (2)), we see that
+A′ ∼= A′′ := k[wn−3, wn−4u, . . . , un−3].
+All these rings are related via the following commutative diagram of k-algebras:
+(6)
+T
+� � A
+T ′���
+�
+� � A′
+∼
+=
+�
+��
+�
+A′′
+Proposition 2.3. Maintain the above notations, and let f ∈ R be as in Theorem 2.2, with n ≥ 5.
+The following assertions hold:
+(i) F(Jf) ∼= T/I2
+�
+y1
+· · ·
+yn−3
+y2
+· · ·
+yn−2
+�
+as k-algebras. In particular, F(Jf) is Cohen-Macaulay;
+(ii) ℓ(Jf) = 4;
+(iii) r(Jf) = 1;
+(iv) reg Derk(R/(f)) = n − 2 (also for n = 4).
+Proof. (i) Since Jf is a homogeneous ideal generated in the same degree, there is an isomorphism
+of graded k-algebras F(Jf) ∼= A, so that F(Jf) ∼= T/Q where Q := ker (T ։ A). By the diagram
+(6), we get Q′T ⊂ Q, where Q′ := ker (T ′ ։ A′). From Remark 2.1(b) we have
+Q′T = I2
+�
+y1
+· · ·
+yn−3
+y2
+· · ·
+yn−2
+�
+.
+Hence, ht Q ≥ ht Q′T = n−4. Thus, in order to prove that Q = I2
+�
+y1
+· · ·
+yn−3
+y2
+· · ·
+yn−2
+�
+, we must show
+ht Q ≤ n − 4, or equivalently, dim A ≥ 4. Now, on the other hand, Lemma 1.9 gives dim A = rank Θ,
+where Θ is the Hessian matrix of f. Notice that the (Jacobian) matrix Θ can be written in blocks as
+Θ =
+�
+0
+Θw,u
+Θt
+w,u
+∗
+�
+where Θw,u is the (n − 2) × 2 Jacobian matrix of fx1, . . . , fxn−2 with respect to w, u.
+Clearly,
+I2(Θw,u) ̸= 0. In particular, I4(Θ) ̸= 0 because I2(Θw,u)2 ⊂ I4(Θ). Hence rank Θ ≥ 4, so that
+dim A = rank Θ ≥ 4, as needed. The Cohen-Macaulayness of F(Jf) will be confirmed below, in the
+proof of item (iii).
+(ii) Since ℓ(Jf) = dim F(Jf), the statement follows directly from the proof of (i).
+(iii) It is well-known that I2
+�
+y1
+· · ·
+yn−3
+y2
+· · ·
+yn−2
+�
+is a perfect ideal with linear resolution (in fact, this
+ideal is resolved by the Eagon-Northcott complex). In particular, the ring F(Jf) ∼= T/Q is Cohen-
+Macaulay and its Castelnuovo-Mumford regularity is 1. Thus, by Lemma 1.10, r(Jf) = reg F(Jf) =
+1.
+(iv) By Theorem 2.2, f is a linear free divisor. Now the assertion follows from Lemma 1.7.
+
+FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD
+9
+Our next goal is to prove that, for f as above, R(Jf) is Cohen-Macaulay.
+First, we need an
+auxiliary lemma.
+Lemma 2.4. Let k[z, v] = k[z1, . . . , zm, v1, . . . , vm] be a polynomial ring, with m ≥ 2. Consider
+F = a1v1z1 + · · · + amvmzm,
+where a1, . . . , an are non-zero elements of k. Let
+M =
+�
+z1
+z2
+. . .
+zm−1
+z2
+z3
+. . .
+zm
+�
+.
+Then, k[z, v]/(I2(M), F) is a Cohen-Macaulay domain of codimension m − 1.
+Proof. By [24, Section 4], we have that k[z, v]/I2(M) is a Cohen-Macaulay domain of codimension
+m−2. In particular, since F /∈ I2(M), the ring k[z, v]/(I2(M), F) is Cohen-Macaulay of codimension
+m − 1, and so it remains to show that it is a domain. Clearly, we can assume m ≥ 3.
+Claim. z1 is a (k[z, v]/(I2(M), F))-regular element.
+Suppose that z1 is not (k[z, v]/(I2(M), F))-regular. Then, z1 ∈ p for some associated prime p of
+k[z, v]/(I2(M), F), and hence in particular (z1, I2(M), F) ⊂ p. Now let M1 be the matrix obtained
+from M by deletion of its first column. Then it is easy to see that (z1, z2, I2(M1), F) ⊂ p. If M2 is the
+matrix obtained by deletion of the first column of M1, then (z1, z2, z3, I2(M2), F) ⊂ p. Proceeding
+in this way, we get
+(z1, z2, . . . , zm−1, F) ⊂ p.
+Since ht (z1, z2, . . . , zm−1, F) = m, it follows that ht p ≥ m. But, this is a contradiction because p
+is a associated prime of the Cohen-Macaulay (hence unmixed) ring k[z, v]/(I2(M), F), which has
+codimension m − 1. This proves the Claim.
+Finally, localizing in z1 we deduce the isomorphism
+k[z, v][z−1
+1 ]
+(I2(M), F)k[z, v][z−1
+1 ]
+∼=
+k[z, v2, . . . , vm][z−1
+1 ]
+I2(M)k[z, v2, . . . , vm][z−1
+1 ].
+But the ring on the right side of the isomorphism is a domain.
+By the claim, it follows that
+k[z, v]/(I2(M), F) is a domain.
+Theorem 2.5. Let f ∈ R be as in Theorem 2.2. Then, R(Jf) is Cohen-Macaulay.
+Proof. Consider the natural epimorphism
+C := k[x1, . . . , xn−2, w, u, y1, . . . , yn−2, s, t] ։ R(Jf),
+whose kernel we denote J . From the previous considerations (and notations), it follows that
+K := (I1(γ · ψ), Q) ⊂ J
+where γ =
+� y1
+· · ·
+yn−2
+s
+t �
+. Note we can rewrite the ideal K as
+K = I2
+�
+u
+y1
+· · ·
+yn−3
+w
+y2
+· · ·
+yn−2
+�
++ (G, H),
+G = γ · δ1 = −su + 2(n − 1)yn−2w +
+n−2
+�
+i=2
+αiyi−1xi
+and
+H = γ · δ2 = −(n − 1)tu + sw +
+n−2
+�
+i=1
+βiyixi.
+Claim 1. Let K1 := I2
+�
+u
+y1
+· · ·
+yn−3
+w
+y2
+· · ·
+yn−2
+�
++(H) ⊂ C. Then, C/K1 is a Cohen-Macaulay domain.
+Denote K0 := I2
+�
+u
+y1
+· · ·
+yn−3
+w
+y2
+· · ·
+yn−2
+�
+. It is well-known that C/K0 is a Cohen-Macaulay integral
+domain of dimension n + 3. Moreover, since H /∈ K0, this polynomial must be C/K0-regular. Hence,
+
+10
+R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS
+the ring C/K1 = C/(K0, H) is Cohen-Macaulay of dimension n + 2. In particular, C/K1 satisfies
+Serre’s condition S2 (see the definition in Subsection 6.1). We claim that, even more, the ring C/K1
+is normal, from which its integrality will follow. In order to show that C/K1 is normal, it remains
+to verify that it is locally regular in codimension 1. Note ht K1 = n − 2. By the classical Jacobian
+criterion, it suffices to prove that
+ht (K1, In−2(Θ)) ≥ n,
+where Θ denotes the Jacobian matrix of K1. Notice that the matrix Θ, after a reordering of its
+columns (which obviously does not affect ideals of minors), can be written in the format
+Θ =
+� Θ′
+0
+∗
+Θ′′
+�
+.
+Precisely, Θ′ is the Jacobian matrix of K0 with respect the variables w, u, y1, . . . , yn−2 and Θ′′ is the
+(row) Jacobian matrix of H with respect to the variables x1, . . . , xn−2, s, t. In particular,
+(K1, I1(Θ′′) · In−3(Θ′)) ⊂ (K1, In−2(Θ)).
+Now pick a minimal prime q of (K1, In−2(Θ)). In particular, I1(Θ′′) · In−3(Θ′) ⊂ q, which yields
+I1(Θ′′) ⊂ q or In−3(Θ′) ⊂ q. If I1(Θ′′) ⊂ q then ht q ≥ n because I1(Θ′′) = (y1, . . . , yn−2, w, u). On
+the other hand, if In−3(Θ′) ⊂ q then (K0, In−3(Θ′)) ⊂ q. But it is well-known that
+ht (K0, In−3(Θ′)) = n.
+Therefore, ht q ≥ n in any case, and we get ht (K1, In−2(Θ)) ≥ n, as desired.
+Claim 2. C/K is a Cohen-Macaulay domain of dimension n + 1.
+By Claim 1 and its proof, C/K1 is a Cohen-Macaulay domain of dimension n + 2. Thus, since
+G /∈ K1, the ring C/K = C/(K1, G) is Cohen-Macaulay of dimension n + 1. It remains to prove that
+C/K is a domain. First we claim that u is C/K-regular. Suppose otherwise. Then u ∈ p for some
+associated prime p of C/K, which gives
+(u, wy1, . . . , wyn−3, I2(N), G, H) ⊂ p,
+where
+N :=
+�
+y1
+· · ·
+yn−3
+y2
+· · ·
+yn−2
+�
+.
+In particular,
+(7)
+Q1 := (u, w, I2(N), G, H) ⊂ p
+or
+Q2 := (u, y1, . . . , yn−3, G, H) ⊂ p.
+We have
+C/(u, w, I2(N), H) ∼= (k[x1, . . . , xn−2, y1, . . . , yn−2]/(I2(N), β1y1x1 + · · · + βn−2yn−2xn−2))[s, t].
+From this isomorphism and Lemma 2.4, the ring C/(u, w, I2(N), H) is a Cohen-Macaulay domain
+of dimension (n − 1) + 2 = n + 1.
+Thus, since G /∈ (u, w, I2(N), H), we obtain that C/Q1 =
+C/(u, w, I2(N), G, H) is a Cohen-Macaulay ring of dimension n. In particular, ht Q1 = n. On the
+other hand,
+C/Q2 ∼= k[x1, . . . , xn−2, w, yn−2, s, t]/(yn−2w, sw + βn−2yn−2xn−2)
+is a Cohen-Macaulay ring of dimension n, which yields ht Q2 = n. It follows, by (7), that ht p ≥ n.
+This is a contradiction, because p is an associated prime of C/K, which is Cohen-Macaulay of codi-
+mension n−1. So, u is C/K-regular. Now, by localizing in u and setting D := k[x1, . . . , xn−2, w, u, y1, s, t],
+routine calculations give
+(C/K)[u−1] ∼= D[u−1]/(G, H)D[u−1] ∼= k[x1, . . . , xn−2, w, u, y1][u−1],
+which is a domain. Hence, C/K is a domain, which proves Claim 2.
+To conclude the proof of the theorem, we notice that since K ⊂ J are prime ideals of the same
+codimension, then necessarily K = J . In particular, R(Jf) ∼= C/J is Cohen-Macaulay.
+
+FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD
+11
+3. Second family: linear free divisors in P2n−1
+In order to describe our second family of free divisors, consider the standard graded polynomial
+ring R = k[x1, . . . , x2n−2, w, u] in 2n ≥ 4 indeterminates over k. Let
+f = wuq,
+q = (x1u − x2w)(x3u − x4w) · · · (x2(n−1)−1u − x2(n−1)w).
+For every 1 ≤ i ≤ n − 1, denote qi = q/(x2i−1u − x2iw) ∈ R. Then,
+(8)
+fx2i−1 = u2wqi
+and
+fx2i = −w2uqi
+(1 ≤ i ≤ n − 1),
+(9)
+fw = qu − wu
+n−1
+�
+i=1
+x2iqi
+and
+fu = qw + wu
+n−1
+�
+i=1
+x2i−1qi.
+Using (8) and (9) we easily deduce the following relations:
+(10)
+det
+� fx2i−1
+fx2j−1
+fx2i
+fx2j
+�
+= 0
+(1 ≤ i < j ≤ n − 1),
+(11)
+wfx2i−1 + ufx2i = 0
+(1 ≤ i ≤ n − 1),
+wfw + ufu = (n + 1)f,
+(12)
+x2i−1fx2i−1 + x2ifx2i = f
+(1 ≤ i ≤ n − 1),
+(13)
+(n + 1)x2i−1fx2i−1 + (n + 1)x2ifx2i − ufu − wfw = 0
+(1 ≤ i ≤ n − 1).
+Set α = a1a2, β = b1b2 and γ = a1b2 + a2b1. In addition to the equalities above, we have
+(n + 1)
+n−1
+�
+i=1
+x2ifx2i
+=
+−(n + 1)w2u
+n−1
+�
+i=1
+x2iqi
+=
+(n + 1)[w(fw − qu)]
+=
+nwfw − ufu + (ufu + wfw) − (n + 1)f
+=
+nwfw − ufu.
+(14)
+Now we are in a position to prove the first result of this section.
+Theorem 3.1. Maintain the above notations. The following assertions hold:
+(i) f is a linear free divisor;
+(ii) The 2n × (2n − 1) matrix
+(15)
+ψn =
+
+
+w (n + 1)x1
+. . .
+0
+0
+0
+u
+(n + 1)x2
+. . .
+0
+0
+(n + 1)x2
+...
+...
+...
+...
+...
+...
+0
+0
+· · ·
+w (n + 1)x2n−3
+0
+0
+0
+· · ·
+u
+(n + 1)x2n−2 (n + 1)x2n−2
+0
+−w
+· · ·
+0
+−w
+−nw
+0
+−u
+· · ·
+0
+−u
+u
+
+
+is a syzygy matrix of Jf. Thus a free basis of TR/k(f) is {θ1, . . . , θ2n−1, ε2n}, where the θi’s
+correspond to the columns of ψn;
+(iii) reg Derk(R/(f)) = 2(n − 1).
+
+12
+R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS
+Proof. (i) Consider the 2n × 2n matrix
+M =
+
+
+w x1
+. . .
+0
+0
+0
+x1
+u
+x2
+. . .
+0
+0
+(n + 1)x2
+x2
+...
+...
+...
+...
+...
+...
+...
+0
+0
+· · ·
+w x2n−3
+0
+x2n−3
+0
+0
+· · ·
+u
+x2n−2 (n + 1)x2n−2 x2n−2
+0
+0
+· · ·
+0
+0
+−nw
+w
+0
+0
+· · ·
+0
+0
+u
+u
+
+
+.
+Using (11), (12), (14), and the Euler relation, it is easy to see that
+∇f · M = [ 0
+f
+· · ·
+0
+f
+0
+2nf ],
+so that ∇f · M ≡ 0 mod f. Moreover, (n + 1)f = det M . Thus, by Lemma 1.4 (or, in this case, by
+the version of Saito’s criterion stated in [8, Theorem 2.4]), we conclude that f is a linear free divisor.
+(ii) For simplicity, write ψn = ψ. By (i) and Lemma 1.3, we know that Jf is a codimension 2
+perfect ideal, so it suffices to prove that ∇f ·ψ = 0 and that ψ has maximal rank. The former follows
+by (11), (13) and (14). Now denote by ∆ the (2n − 1)-minor of ψ obtained by omitting the 2n-th
+row of ψ. It is easy to see that ∆ modulo w is given by (n + 1)nx1x3 · · · x2n−3un. In particular, ∆ is
+non-zero as well. Hence, ψ has maximal rank.
+(iii) By part (i), f is a linear free divisor (in 2n variables). Now we apply Lemma 1.7.
+For the next results, we consider a set of 2n variables z1, . . . , z2n−2, s, t over R as well as the natural
+epimorphism
+S := k[x1, . . . , x2n−2, w, u, z1, . . . , z2n−2, s, t] ։ R(Jf)
+whose kernel we denote J . By the equalities (10) we have an inclusion
+I2
+�
+z1
+z3
+. . .
+z2n−3
+z2
+z4
+. . .
+z2n−2
+�
+⊂ J .
+Therefore,
+K :=
+�
+I1(γ · ψn), I2
+�
+z1
+z3
+. . .
+z2n−3
+z2
+z4
+. . .
+z2n−2
+��
+⊂ J
+where γ =
+�
+z1
+. . .
+z2n−2
+s
+t
+�
+. The generators of I1(γ · ψn) are of three types:
+(16)
+wz2i−1 + uz2i
+(1 ≤ i ≤ n − 1),
+Fi := (n + 1)(x2i−1z2i−1 + x2iz2i) − ws − ut
+(1 ≤ i ≤ n − 1),
+G := (n + 1)
+n−1
+�
+i=1
+x2iz2i − nws + ut.
+We can use the generators of type (16) as well as the ideal I2
+�
+z1
+z3
+. . .
+z2n−3
+z2
+z4
+. . .
+z2n−2
+�
+to rewrite K as
+K =
+
+
+
+
+I2
+�
+z1
+z3
+. . .
+z2n−3
+−u
+z2
+z4
+. . .
+z2n−2
+w
+�
+�
+��
+�
+=:K0
+, F1, . . . , Fn−1, G
+
+
+
+
+
+With this, we have
+S/K ∼= A[x1, . . . , x2n−2, s, t]/(F1, . . . , Fn−1, G)A[x1, . . . , x2n−2, s, t]
+
+FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD
+13
+where A := k[z1, . . . , z2n−2, w, u]/K0. Now, consider the 2n × n matrix
+ζ =
+
+
+(n + 1)z1
+0
+. . .
+0
+0
+(n + 1)z2
+0
+. . .
+0
+(n + 1)z2
+...
+...
+...
+...
+...
+0
+0
+. . . (n + 1)z2n−3
+0
+0
+0
+. . . (n + 1)z2n−2 (n + 1)z2n−2
+−w
+−w . . .
+−w
+−nw
+−u
+−u . . .
+−u
+u
+
+
+taken as a matrix with entries in the domain A. We denote by M the A-module defined as the
+cokernel of ζ.
+Proposition 3.2. Maintain the above notations. Then:
+(i) M is an A-module of projective dimension 1;
+(ii) The symmetric algebra SymAM is a Cohen-Macaulay domain of dimension 2n + 1.
+Proof. (i) Consider the complex
+(17)
+0 −→ An
+ζ
+−→ A2n −→ M −→ 0.
+By the well-known Buchsbaum-Eisenbud acyclicity criterion, in order to show that (17) is exact it
+suffices to confirm that rank ζ = n. To this end, consider the following n × n submatrix of ζ:
+η :=
+
+
+(n + 1)z1
+· · ·
+0
+0
+...
+...
+...
+...
+0
+. . .
+(n + 1)z2n−3
+0
+−w
+. . .
+−w
+−nw
+
+ .
+We have det η = −n(n + 1)n−1z1 · · · z2n−3w ̸= 0 (modK0). Hence, ζ has rank n.
+(ii) In addition to the property given in (i), recall A is a Cohen-Macaulay domain and ht K0 = n−1.
+Then, because of [29, Theorem 1.1], it suffices to show that
+ht(It(ζ) + K0) ≥ 2n − t + 1
+for every 1 ≤ t ≤ n. For this note first that, by suitably permuting the rows of ζ, we obtain a matrix
+N of the form
+N =
+
+
+∗z1 · · ·
+0
+. . .
+0
+0
+...
+...
+...
+. . .
+...
+...
+0
+0
+∗z2i−1 . . .
+0
+0
+...
+...
+...
+...
+...
+...
+0
+0
+0
+. . . ∗z2n−3
+0
+−u −u
+−u
+. . .
+−u
+u
+∗z2 · · ·
+0
+. . .
+0
+∗z2
+...
+...
+...
+· · ·
+...
+...
+0
+· · ·
+∗z2i
+. . .
+0
+∗z2i
+...
+...
+...
+...
+...
+...
+0
+0
+0
+. . . ∗z2n−2 ∗z2n−2
+−w −w
+−w
+. . .
+−w
+−nw
+
+
+where all coefficients ∗ are equal to n + 1. Let us denote the top and bottom blocks of N by Nodd
+and Neven, respectively. Our goal is to prove ht(It(N) + K0) ≥ 2n − t + 1 whenever 1 ≤ t ≤ n, where
+
+14
+R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS
+as before
+K0 = I2
+�
+z1
+z3
+. . .
+z2n−3
+−u
+z2
+z4
+. . .
+z2n−2
+w
+�
+.
+Let P be a prime ideal containing It(N) + K0 and having the same codimension. From Nodd it is
+easy to see that the ideal Ct generated by the t-products of the set {z1, z3, . . . , z2n−3, z2n−1 := u} is
+contained in P. But, by [48, Section 2], the minimal primes of Ct are of the form (z2j1−1, . . . , z2jn−t+1−1)
+for certain 1 ≤ j1 < · · · < jn−t+1 ≤ n. Hence, we can suppose that (z2j1−1, . . . , z2jn−t+1−1) ⊂ P with
+1 ≤ j1 < · · · < jn−t+1 ≤ n. Analogously, from Neven we can write (z2i1, . . . , z2in−t+1) ⊂ P for certain
+1 ≤ i1 < · · · < in−t+1 ≤ n (we put z2n := w).
+Let us assume that the following condition takes place:
+(†)
+There exists j ∈ {1, . . . , n} such that {z2j−1, z2j} ∩ P = {z2j−1} or {z2j}.
+Suppose {z2j−1, z2j} ∩ P = {z2j−1}. Then, by the relations in K0, all the odd variables belong to
+P. Therefore, we have
+(n − t + 1)
+�
+��
+�
+even variables
++
+n
+����
+odd variables
+= 2n − t + 1 variables in P,
+which gives ht(It(N) + K0) ≥ 2n − t + 1 for 1 ≤ t ≤ n. The argument for the case {z2j−1, z2j} ∩ P =
+{z2j} is similar.
+Now, suppose that (†) is not true. Without loss of generality, we may assume (j1, . . . , jn−t+1) =
+(1, . . . , n − t + 1). It follows that z1, z2, . . . , z2(n−t+1)−1, z2(n−t+1) ∈ P. Consider the following t × t
+submatrix of ζ:
+
+
+0
+∗z2(n−t+1)+1
+0
+. . .
+0
+0
+0
+0
+∗z2(n−t+2)+1 . . .
+0
+0
+...
+...
+...
+...
+...
+...
+0
+0
+0
+. . . ∗z2n−3
+0
+−u
+−u
+−u
+. . .
+−u
+u
+−w
+−w
+−w
+. . .
+−w
+−nw
+
+
+.
+The determinant of this matrix is cz2(n−t+1)+1 · · · z2n−3z2n−1z2n for some c ∈ k; in particular, this
+determinant lies in P and hence zs ∈ P for some s with 2(n − t + 1) + 1 ≤ s ≤ 2n. Therefore,
+since we are assuming that (†) does not hold, there exist two consecutive indices 2j − 1, 2j with
+2(n − t + 1) + 1 ≤ 2j − 1, 2j ≤ 2n satisfying z2j−1, z2j ∈ P. Now we can suppose, without loss of
+generality, that 2j − 1 = 2(n − t + 1) + 1. We have
+(z1, z2, . . . , z2(n−t+1)−1, z2(n−t+1), z2(n−t+1)+1, z2(n−t+1)+2) + K0 ⊂ P.
+Hence, considering the subideal
+�K0 := I2
+� z2(n−t+2)+1
+z2(n−t+3)+1
+. . .
+z2n−3
+−u
+z2(n−t+3)
+z2(n−t+4)
+. . .
+z2n−2
+w
+�
+⊂ K0,
+we observe that all the variables appearing in �K0 are different from the 2(n − t + 2) variables that
+already belong to P; since in addition ht �K0 = t − 3, we conclude
+ht(It(N) + K0) = ht P ≥ 2(n − t + 2) + (t − 3) = 2n − t + 1
+whenever 1 ≤ t ≤ n, as needed.
+So we have shown that SymAM is a Cohen-Macaulay domain. Note that M possesses a rank as
+an A-module (equal to n, by (17)). Now recall that the Rees algebra of the A-module M, denoted
+RA(M), can be defined as the quotient of SymAM by its A-torsion submodule (see [42] for the
+general theory). Consequently, since in this case A and SymAM are both domains, we can identify
+SymAM = RA(M); in particular, using [42, Proposition 2.2] (which gives a formula for the dimension
+
+FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD
+15
+of the Rees algebra of a module with rank) and noticing that dim A = 2n − (n − 1) = n + 1, we
+finally get
+dim SymAM = dim RA(M) = dim A + rankAM = (n + 1) + n = 2n + 1.
+Theorem 3.3. Maintain the above notations. Then:
+(i) K = J ;
+(ii) The Rees algebra R(Jf) is Cohen-Macaulay;
+(iii) Let T = k[z1, . . . , z2n−2, s, t], with n ≥ 3. Then,
+F(Jf) ∼= T/I2
+�
+z1
+z3
+. . .
+z2n−3
+z2
+z4
+. . .
+z2n−2
+�
+as k-algebras. In particular, F(Jf) is Cohen-Macaulay, ℓ(Jf) = n + 2, and r(Jf) = 1.
+Proof. We have a natural epimorphism
+SymAM ∼= S/K ։ S/J ∼= R(Jf).
+By Proposition 3.2, K is a prime ideal of S, and dim SymAM = 2n + 1 = dim R + 1 = dim R(Jf).
+So ht K = ht J , and then K = J .
+Using Proposition 3.2 once again, we obtain that R(Jf) is
+Cohen-Macaulay. This proves (i) and (ii).
+In order to prove (iii), let R+ be the homogeneous maximal ideal of R. We have
+F(Jf) ∼= R(Jf)/R+R(Jf) ∼= S/(R+S, K) ∼= T/I2
+�
+z1
+z3
+. . .
+z2n−3
+z2
+z4
+. . .
+z2n−2
+�
+,
+which, as is well-known (being a generic determinantal ring), is Cohen-Macaulay of dimension n + 2
+and moreover has regularity 1. The latter, by Lemma 1.10, gives r(Jf) = 1.
+Remark 3.4. (a) The ideal Jf is of linear type if and only if n = 2 (i.e., the case where R is a
+polynomial ring in 4 variables). Indeed, by Theorem 3.3(iii), if n ≥ 3 then r(Jf) = 1 ̸= 0, hence Jf
+cannot be of linear type. Conversely, let n = 2, so that R = k[x1, x2, w, u]. We can check that the
+ideals of minors of ψ2 (see (15)) satisfy
+ht Is(ψ2) ≥ 5 − s = (2n − 1) + 2 − s
+for
+s = 1, 2, 3.
+It follows by [29, Theorem 1.1] that Jf is of linear type (in particular, r(Jf) = 0). Note in addition
+that SymRJf is a complete intersection, i.e., the polynomials
+L1 = 3x1z1 + 3x2z2 − ws − ut, L2 = wz1 + uz2, L3 = x2z1 + x1z2 + us + wt
+form an R[z1, z2, s, t]-sequence.
+It is also worth mentioning that, in 3 variables, if g ∈ k[x, y, z] defines a rank 3 central hyperplane
+arrangement, then it has been recently shown that Jg is of linear type and moreover that the property
+of the symmetric algebra of Jg being a complete intersection characterizes the freeness of g (see [10,
+Proposition 2.14 and Corollary 2.15]).
+(b) Let n = 3, i.e., R = k[x1, x2, x3, x4, w, u]. In this case, a computation shows that the entries of the
+product
+�
+z1
+· · ·
+z4
+s
+t
+�
+· ψ3 form a regular sequence, i.e., SymRJf is a complete intersection
+once again.
+Question 3.5. For an arbitrary n, is SymRJf a complete intersection ring?
+
+16
+R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS
+4. Third family: non-linear free plane curves
+In this section we furnish our third family of free divisors and some of its properties. In fact, from
+such a family we will derive yet another one; see Remark 4.3. Similar examples (also in 3 variables)
+can be found, e.g., in [20] and [34].
+Theorem 4.1. Consider the two-parameter family of polynomials
+f = fα,β = (xα − yα−1z)β + yαβ ∈ R = k[x, y, z],
+for integers α, β ≥ 2. The following assertions hold:
+(i) f is a free divisor;
+(ii) R(Jf) is Cohen-Macaulay if (α, β) = (2, 3) or if α ≥ 2 and β = 2;
+(iii) Jf is not of linear type if α = 2 and β ≥ 3 or if α ≥ 3 and β = 2. In these cases, F(Jf) is
+a polynomial ring over k and then r(Jf) = 0;
+(iv) f is reducible over k = C. If k ⊆ R then f is reducible if β is odd and irreducible otherwise;
+(v) reg Derk(R/(f)) = αβ − 2.
+Proof. (i) We have
+fx = αβxα−1(xα−yα−1z)β−1,
+fy = αβyαβ−1−(α−1)βyα−2z(xα−yα−1z)β−1, fz = −βyα−1(xα−yα−1z)β−1
+Note that we can write fx, fy and fz as
+fx = αxα−1G,
+fy = xα−1P + yα−1Q,
+fz = −yα−1G
+for certain G, P, Q ∈ R. Thus,
+(18)
+Jf = I2
+
+
+yα−1
+−α−1P
+0
+G
+αxα−1
+Q
+
+ .
+In particular, since Jf has codimension two, it follows by the Hilbert-Burch theorem that Jf is a
+perfect ideal. By Lemma 1.3, f is a free divisor.
+(ii) In the specific cases (α, β) = (2, 2) and (α, β) = (2, 3), the Cohen-Macaulayness of R(Jf) can
+be confirmed by a routine computation. Therefore, we may suppose α ≥ 3 and β = 2. Determining
+G, P, Q in this situation, we obtain from (18) that a syzygy matrix of Jf is
+ϕ =
+
+
+yα−1
+(α − 1)xyα−2z
+0
+α(xα − yα−1z)
+αxα−1
+α2yα + α(α − 1)yα−2z2
+
+ .
+Let us denote by Q the (prime) ideal of k[x, y, z, s, t, u] = R[s, t, u] defining R(Jf). Notice that
+(19)
+I1
+�� s
+t
+u �
+· ϕ
+�
+= (syα−1 + αuxα−1, yα−2H + αxαt) ⊂ Q,
+where H := (α − 1)xzs + (α2y2 + α(α − 1)z2)u − αyzt. Clearly, we can rewrite I1([ s
+t
+u ] · ϕ) as
+I1
+�� yα−2
+xα−2 �
+·
+�
+sy
+H
+αux
+αx2t
+��
+.
+Now it follows from Cramer’s rule that
+(20)
+det
+�
+sy
+H
+αux
+αx2t
+�
+= αx2yst − αuxH = αx(xyst − uH) ∈ Q.
+From (19) and (20) we deduce an inclusion
+P := (syα−1 + αuxα−1, yα−2H + αxαt, xyst − uH) ⊂ Q.
+Claim 1. P is a perfect ideal of height 2.
+
+FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD
+17
+Clearly, ht P ≥ 2 and
+P = I2
+
+
+H
+−xt
+−ys
+u
+αxα−1
+yα−2
+
+
+Now, Claim 1 follows by the Hilbert-Burch theorem.
+Claim 2. P = Q.
+Since P ⊆ Q and ht P = ht Q = 2, it suffices to prove that the ideal P is prime. Note first that x is
+regular modulo P. To show this, suppose otherwise. Then we would have x ∈ p for some associated
+prime p of R[s, t, u]/P. In particular, (x, syα−1, yα−2H, uH) ⊂ p. Using the explicit format of H
+given above, it is easy to see that
+ht (x, syα−1, yα−2H, uH) ≥ 3.
+In particular, ht p ≥ 3. But this is a contradiction because, by Claim 1, P is perfect of height 2.
+Now, by inverting the element x we get
+D :=
+k[x, y, z, s, t, u][x−1]
+Pk[x, y, z, s, t, u][x−1]
+=
+k[x, y, z, s, t, u][x−1]
+(u + α−1x1−αyα−1s, xt + α−1x1−αyα−2H, xyst − uH)
+=
+k[x, y, z, s, t, u][x−1]
+(u + α−1x1−αyα−1s, xt + α−1x1−αyα−2H)
+∼=
+k[x, y, z, s, t][x−1]
+(at + bs)
+∼=
+k[x, y, z, x−1][s, t]
+(at + bs)
+where a := x(1 − x−αyα−1z)
+and
+b := x1−αyα−2[(α − 1)za + x1−αyα+1] are elements in the
+coefficient ring k[x, y, z, x−1]. Since a and b are easily seen to be relatively prime in this facto-
+rial domain, the element at + bs must be irreducible in k[x, y, z, x−1][s, t], so that the quotient
+k[x, y, z, x−1][s, t]/(at+bs) ∼= D is a domain. This means (as x is regular modulo P) that R[s, t, u]/P
+is a domain, as needed.
+Finally, by Claim 1 and Claim 2, we conclude that R(Jf) is Cohen-Macaulay.
+(iii) First, if α = 2, a simple inspection shows that the linear type property of Jf fails in case
+β ≥ 3 (and holds if β = 2), by analyzing the saturation of the ideal S of 2 linear forms defining
+SymRJf in R[s, t, u] by the ideal Jf. The resulting ideal – which thus defines R(Jf) (see, e.g., [31,
+Lemma 2.11]) – turns out to strictly contain S . In addition, it is contained in (x, y, z)R[s, t, u], so
+that F(Jf) ∼= k[s, t, u]. As to the case where α ≥ 3 and β = 2, we can use a previous calculation.
+Precisely, by the structure of the defining ideal Q = P ⊂ k[x, y, z, s, t, u] of R(Jf) as obtained in
+item (ii), we readily get that Jf is not of linear type. Moreover, by looking at the non-linear Rees
+equation
+xyst − uH ∈ (x, y, z)R[s, t, u]
+we conclude that, once again, F(Jf) ∼= k[s, t, u]. In either case, ℓ(Jf) = 3 and (by Lemma 1.10)
+r(Jf) = 0.
+(iv) Let k = C and assume first that β = 2m, m ≥ 1. We have f = (xα − yα−1z)2m + y2mα and
+hence, for i = √−1 and A := xα − yα−1z,
+(21)
+f = (iAm + ymα)(−iAm + ymα).
+Now, assume β ≥ 3 is odd. Write f = (xα − yα−1z + yα − yα)β + yαβ and set B := xα − yα−1z + yα.
+Thus we can rewrite
+f = (B − yα)β + yαβ = [Bg + (−1)βyαβ] + yαβ = Bg
+
+18
+R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS
+for a suitable g := gα,β ∈ R of degree α(β − 1). Of course, this also shows that if k ⊆ R and β is
+odd, then f is reducible over k.
+Finally, if k ⊆ R then the irreducibility of f over k for even β follows from the structure of the
+factors described in (21) over the unique factorization domain C[x, y, z].
+(v) According to item (i), f is a free divisor. Noticing that its degree is αβ, the assertion follows
+by Lemma 1.7.
+Remark 4.2. Computations strongly suggest that the cases described in item (ii) are precisely the
+ones where the Cohen-Macaulayness of R(Jf) takes place. Concerning the linear type property of
+Jf, computations also indicate that Jf is not of linear type if α, β ≥ 3 – cases not covered by part
+(iii). The (partially computer-assisted) conclusion is that Jf is of linear type if and only if α = β = 2.
+Remark 4.3. Here we want to point out that the form g = gα,β ∈ Rα(β−1) defined in the proof of
+item (iv) is also a free divisor provided that β ≥ 3 is odd. First, note that g is defined by means of
+Bg = (B − yα)β + yαβ, where B = xα − yα−1z + yα. Explicitly, from
+Bg = (B − yα)β + yαβ =
+β
+�
+j=0
+�β
+j
+�
+Bβ−j(−yα)j + yαβ = B ·
+
+
+β−1
+�
+j=0
+(−1)j
+�β
+j
+�
+Bβ−j−1yαj
+
+
+we get g =
+β−1
+�
+j=0
+(−1)j
+�β
+j
+�
+Bβ−j−1yαj. An elementary calculation shows that we can write gx, gy, gz
+as gx = αxα−1T, gy = xα−1U + yα−1V , gz = yα−1T, for certain T, U, V ∈ R. Thus,
+Jg = I2
+
+
+yα−1
+−α−1U
+0
+T
+αxα−1
+V
+
+ .
+Since ht Jg = 2, the ideal Jg must be perfect by the Hilbert-Burch theorem. Therefore, g is a free
+divisor whenever β ≥ 3 is odd. In particular, by Lemma 1.7 we have reg Derk(R/(g)) = αβ − α − 2.
+We also observe that, for every odd β ≥ 3, the form g is reducible over k = C. Indeed, let
+Φ =
+β−1
+�
+j=0
+(−1)j
+�β
+j
+�
+Zβ−j−1W j ∈ C[Z, W],
+which then factors as a product of linear forms in Z and W. Now let σ: C[Z, W] → C[x, y, z] be the
+homomorphism given by Z �→ B and W �→ yα. Then, the form σ(Φ) = g is reducible in C[x, y, z].
+Finally, let us study the algebra R(Jgα,β) in the cases β = 3 and β = 5.
+If β = 3 then first as a matter of illustration we explicitly have
+gα,3 = B2 − 3Byα + 3y2α
+=
+(B − yα)2 − yα(B − 2yα) = (B − yα)2 − yα(B − yα) + y2α
+=
+(xα − yα−1z + yα)(xα − yα−1z − 2yα) + 3y2α
+=
+x2α − 2xαyα−1z − xαyα + y2α−2z2 + y2α−1z + y2α.
+Computations show that, for all α ≥ 2, the ring R(Jgα,3) is Cohen-Macaulay, r(Jgα,3) = 0, but Jf is
+of linear type if and only if α = 2; more precisely, if L (resp. Q) defines SymRJgα,3 (resp. R(Jgα,3))
+in the polynomial ring R[s, t, u], then we have found the relation
+L : Q = (xα−1, yα−2)R[s, t, u].
+If β = 5, then in the cases α = 2 and α = 3 we have confirmed that depth R(Jgα,5) = 3, i.e.,
+the ring R(Jgα,5) is almost Cohen-Macaulay in the sense that its depth is 1 less than its dimension.
+Also, we have r(Jgα,5) = 0. We strongly believe such properties hold for α ≥ 4 as well.
+
+FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD
+19
+Question 4.4. Let β ≥ 7 be odd. Is R(Jgα,β) almost Cohen-Macaulay? Is it true that r(Jgα,β) = 0 ?
+If k ⊆ R and β ≥ 3 is odd, is gα,β irreducible?
+5. Fourth family: linear free plane curves
+In this section, we let R = k[x, y, z] and our objective is to exhibit our fourth family of free divisors,
+which as we shall prove have the linearity property as in two of the previous families. Although in
+[27, 6.4, p. 837] a classification of linear free divisors in at most 4 variables in the case k = C is
+given, the approach provided here describes concretely a recipe to detect some linear free divisors
+in 3 variables starting from a suitable 2 × 3 matrix L of linear forms (in fact, from only 3 linear
+forms, as we shall clarify), where, we recall, k is not required to be algebraically closed; in regard to
+this point, it should be mentioned that, even though most of the existing results in the literature are
+established over C, there has always been an interest in free divisor theory over arbitrary fields (see,
+e.g., [46] and [49]).
+We could start focusing on the case where the 6 entries of L are general linear forms, but there is in
+fact no need for this setting as we only suppose the forms in the first row to be linearly independent
+over k and, naturally, the rank of the matrix to be 2. Thus, after eventually a linear change of
+variables, we let
+L = L (L1, L2, L3) :=
+�
+x
+y
+z
+L1
+L2
+L3
+�
+,
+for linear forms
+L1 = a1x + a2y + a3z, L2 = a4x + a5y + a6z, L3 = a7x + a8y + a9z,
+where at least one of the 2 × 2 minors
+Q1 = xL2 − yL1, Q2 = xL3 − zL1, Q3 = yL3 − zL2
+does not vanish. We also consider the Jacobian matrix of Q = {Q1, Q2, Q3},
+Θ = Θ(Q) =
+
+
+Q1x
+Q2x
+Q3x
+Q1y
+Q2y
+Q3y
+Q1z
+Q2z
+Q3z
+
+ =
+
+
+L2 − a4x − a1y
+L3 + a7x − a1z
+a7y − a4z
+a5x − L1 − a2y
+a8x − a2z
+L3 − a8y − a5z
+a6x − a3y
+a9x − L1 − a3z
+a9y − L2 − a6z
+
+ .
+Our result in this section is as follows.
+Theorem 5.1. Maintain the above notations. If the cubic f := det Θ is non-zero and reduced, then
+f is a linear free divisor. Precisely, a free basis of TR/k(f) is {θ1, θ2, ε3}, where ε3 is the Euler
+derivation, θ2 = L1 ∂
+∂x + L2 ∂
+∂y + L3 ∂
+∂z, and
+θ1 = (a1L1 + a2L2 + a3L3) ∂
+∂x + (a4L1 + a5L2 + a6L3) ∂
+∂y + (a7L1 + a8L2 + a9L3) ∂
+∂z .
+Proof. Routine calculations show that η1 and η2 below are syzygies of Jf,
+η1 =
+
+
+(2a1 − a5 − a9)x + 3a2y + 3a3z
+3a4x + (2a5 − a1 − a9)y + 3a6z
+3a7x + 3a8y + (2a9 − a1 − a5)z
+
+ =
+
+
+3L1 − (a1 + a5 + a9)x
+3L2 − (a1 + a5 + a9)y
+3L3 − (a1 + a5 + a9)z
+
+
+3×1
+and η2 being the 3 × 1 column-matrix given by
+
+
+(−3a5 − 3a9)L1 + 6a2L2 + 6a3L3 + (−4a6a8 − (a5 − a9)2 − 4a3a7 − 4a2a4 + 3a1(a5 + a9))x
+6a4L1 + (−6a1 + 3a5 − 3a9)L2 + 6a6L3 + (−4a6a8 − (a5 − a9)2 − 4a3a7 − 4a2a4 + 3a1(a5 + a9))y
+6a7L1 + 6a8L2 + (−6a1 − 3a5 + 3a9)L3 + (−4a6a8 − (a5 − a9)2 − 4a3a7 − 4a2a4 + 3a1(a5 + a9))z
+
+ .
+Now let
+A := a1 + a5 + a9,
+B := −4a6a8 − (a5 − a9)2 − 4a3a7 − 4a2a4 + 3a1(a5 + a9),
+
+20
+R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS
+so that the following is a submatrix of the matrix of syzygies of Jf:
+
+
+(−3a5 − 3a9)L1 + 6a2L2 + 6a3L3 + Bx
+3L1 − Ax
+6a4L1 + (−6a1 + 3a5 − 3a9)L2 + 6a6L3 + By
+3L2 − Ay
+6a7L1 + 6a8L2 + (−6a1 − 3a5 + 3a9)L3 + Bz
+3L3 − Az
+
+
+3×2
+.
+Multiplying the second column by C := a1 + A = 2a1 + a5 + a9 and adding it to the first column,
+we obtain an equivalent matrix
+ϕ =
+
+
+6(a1L1 + a2L2 + a3L3) + (B − AC)x
+3L1 − Ax
+6(a4L1 + a5L2 + a6L3) + (B − AC)y
+3L2 − Ay
+6(a7L1 + a8L2 + a9L3) + (B − AC)z
+3L3 − Az
+
+
+3×2
+.
+Finally, attaching to ϕ a third column corresponding to the Euler derivation, the resulting 3 × 3
+matrix is easily seen to be equivalent to
+Φ =
+
+
+a1L1 + a2L2 + a3L3
+L1
+x
+a4L1 + a5L2 + a6L3
+L2
+y
+a7L1 + a8L2 + a9L3
+L3
+z
+
+
+3×3
+and satisfies det Φ = 1
+2f. Now the proposed assertions follow by Lemma 1.4.
+Numerous comments are in order.
+Remark 5.2. (a) Recall that, by definition, being reduced is a necessary condition for a polynomial
+to be free. Now we point out that, in general, it is possible for the cubic f := det Θ (with Θ as
+defined above) to be non-reduced. For instance, if we start with the matrix L (x−y, x+y +z, y +z),
+then f = −2(x + z)3.
+(b) Concerning the condition f ̸= 0, it means (since char k = 0) that the quadrics Q1, Q2, Q3 are
+algebraically independent, hence linearly independent. Clearly, this may not occur; for example, for
+L (y, x, z) we have f = 0 because Q3 = −Q2.
+(c) We remark that any f as in Theorem 5.1 is necessarily reducible at least if k = C. This follows
+by the fact that a complex irreducible free divisor in 3 variables must have degree at least 5 (see [20,
+Theorem 2.8]).
+(d) A linear free divisor f in our fourth family can have an irreducible quadratic factor, at least over
+k = R (or eventually a suitable finite field extension of Q). Indeed, starting for example with the
+matrix L (0, x + z, y + z), we obtain
+f = −2xq := −2x(x2 + xy − y2 + 3xz − yz + z2).
+Forcing q to be the product of two linear forms with real coefficients yields a contradiction, hence q
+is irreducible over R. However, it should be pointed out that q is reducible over C, as the rank of
+its associated matrix is non-maximal. In fact we believe (but have no proof) that if k = C then a
+free cubic f as in Theorem 5.1 must necessarily be a product of linear forms. This would imply that
+the complex linear free divisor z(xz + y2) does not belong to our fourth family, which we have been
+unable to prove.
+(e) Our method does not work for higher degrees in general. Taking for example any of the matrices
+�
+x2
+y
+z
+y2
+z
+x
+�
+,
+�
+x
+y
+z
+z2
+x2
+y2
+�
+,
+�
+x2
+y2
+z2
+z2
+x2
+y2
+�
+,
+we are led (following the same recipe) to polynomials that are not free as their Jacobian ideals fail to
+be perfect. However, an interesting problem remains as to the possibility of producing free divisors
+by means of a similar technique, but with carefully chosen entries of higher degrees.
+
+FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD
+21
+(f) Our method does not work for higher dimensions in general.
+For example, over the ring
+k[x, y, z, w], consider the matrix
+
+
+x
+y
+z
+w
+x − y
+x + w
+y − z
+x + 3y
+2y − z
+3w
+x − w
+y + 2w
+
+ .
+The maximal minors are 4 cubics whose Jacobian matrix has a reduced determinant g ̸= 0.
+A
+computation shows that Jg is not perfect, so that g is not free.
+We also derive some additional features.
+Proposition 5.3. Let f be as in Theorem 5.1. The following assertions hold:
+(i) Jf is of linear type. In particular, r(Jf) = 0;
+(ii) R(Jf) (∼= SymRJf) is a complete intersection;
+(iii) reg Derk(R/(f)) = 1.
+Proof. (i) From the proof of the theorem, the 3 × 2 matrix ϕ is a minimal presentation matrix of Jf.
+It follows easily by the structure of ϕ – which in particular has only linear forms as entries – that
+the so-called G3 condition is satisfied (see the definition in the next section, right before Example
+6.5). Moreover, it is clear that Jf has projective dimension 1 (see also Lemma 1.3). Now, applying
+[42, Proposition 4.11] we obtain that Jf is of linear type.
+(ii) By the previous item we have R(Jf) ∼= SymRJf, and the latter is the quotient of R[s, t, u]
+(where s, t, u are variables over R) by the ideal generated by 2 linear forms ξ1, ξ2 in s, t, u, which are
+the entries of the matrix product [s t u] · ϕ. Saying that ht (ξ1, ξ2) = 1 means precisely
+ξ2 = λξ1,
+for some non-zero
+λ ∈ k,
+which is equivalent to the first column of ϕ being λ times the second column. Following the proof of
+the theorem, this would yield det Φ = 0, a contradiction. Therefore, ht (ξ1, ξ2) = 2.
+(iii) Since f is a free cubic, this follows from Lemma 1.7.
+We close the section with a working example which is, on the other hand, somewhat degenerated
+in the sense that two of the Li’s are equal.
+Example 5.4. Taking L (y, x, x) yields the line arrangement
+1
+2f = −x2y + y3 + x2z − y2z = (x + y)(x − y)(z − y),
+which is then free by Theorem 5.1. In this case, writing down the syzygy matrix ϕ of Jf as in the
+proof of the theorem and multiplying their columns by suitable non-zero scalars, we get the following
+simpler presentation matrix for Jf:
+
+
+y
+x
+x
+y
+x
+3y − 2z
+
+ .
+It follows by Proposition 5.3(ii) that the Rees algebra is the complete intersection ring
+R(Jf) ∼= R[s, t, u]/(ys + x(t + u), xs + yt + (3y − 2z)u).
+6. Maximal analytic spread and an application to homaloidness
+Consider the standard graded polynomial ring R = k[x1, . . . , xn] = k ⊕ R+, n ≥ 3, and let f ∈ R
+be a non-zero reduced homogeneous polynomial of degree d ≥ 3. Recall that the Jacobian ideal Jf
+can be minimally generated by the derivatives fx1, . . . , fxn since by convention f is not allowed to
+be a cone. Moreover, ht Jf ≥ 2 as f is reduced.
+
+22
+R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS
+6.1. When does the Jacobian ideal have maximal analytic spread? Our goal in this part is
+to answer this question by means of various characterizations. Recall that
+ht Jf ≤ ℓ(Jf) ≤ n,
+so here we are specifically interested in the property ℓ(Jf) = n, which holds if for example Jf is of
+linear type; indeed, in this case we can write F(Jf) = SymRJf/R+SymRJf ∼= Symk(Jf/R+Jf) ∼= R
+as k-algebras.
+Example 6.1. The Jacobian ideal of the (non-free) cubic
+f = xyz + w3 ∈ k[x, y, z, w]
+can be shown to be of linear type, and so ℓ(Jf) = 4.
+On the other hand, as we have seen in Proposition 2.3(2), even for linear free divisors in at least
+5 variables the analytic spread of Jf can be arbitrarily smaller than the number n of variables.
+Some of the characterizations to be given here are of cohomological nature, and some rely on the
+asymptotic behavior of depth. One of the ingredients is a suitable auxiliary module, which we now
+introduce. As usual, we denote the gradient vector of a polynomial g ∈ R by ∇g = (gx1, . . . , gxn) ∈
+Rn. Given f as above, we set
+Cf := Rn/
+� n
+�
+i=1
+R ∇fxi
+�
+.
+A few preparatory concepts are in order before stating our result.
+Let E be a finitely generated module over a Noetherian ring A and let G Φ→ F → E → 0 be an
+A-free presentation of E. Consider the dual map HomA(Φ, A): F → G. The Auslander transpose
+(or Auslander dual) of E is the A-module
+Tr E = coker HomA(Φ, A),
+which is unique up to projective summands. We refer to [3].
+Now, suppose A = �
+i≥0 Ai is standard graded over a field A0 and let A+ = �
+i≥1 Ai be the
+homogeneous maximal ideal of A. Assume that the A-module E is graded as well. Then, given an
+integer j ≥ 0, the j-th local cohomology module of E is the limit
+Hj
+A+(E) = lim
+−→ Extj
+A(A/As
++, E).
+Saying that E ∼= E′ as graded A-modules means, as usual, that there is a degree zero isomorphism
+between E and E′.
+Finally, given r ≥ 1, recall that the Noetherian ring A is said to satisfy (Serre’s) condition Sr if
+depth Ap ≥ min {r, ht p}
+for all
+p ∈ Spec A.
+This clearly holds (for all r) if A is Cohen-Macaulay.
+Back to the polynomial setup, our result here is as follows.
+Theorem 6.2. Given f ∈ R as before, the following assertions are equivalent:
+(i) ℓ(Jf) = n;
+(ii) dim Cf = n − 1;
+(iii) ∇fx1, . . . , ∇fxn are R-linearly independent;
+(iv) Ext1
+R(Cf, R) ∼= Cf(d − 2) as graded R-modules;
+(v) Hn−1
+R+ (Cf) ∼= HomR(Cf, k)(n − d + 2) as graded R-modules;
+(vi) depth R/Jm
+f = 0 for some m ≥ 1, where the bar denotes integral closure;
+(vii) depth R/Jm
+f = 0 for all m ≫ 0.
+Moreover, if R(Jf) satisfies the S2 condition, then these assertions are also equivalent to the following
+ones:
+
+FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD
+23
+(viii) depth R/Jm
+f = 0 for all m ≫ 0;
+(ix) depth R/Jm
+f = 0 for some m ≥ 1.
+Proof. Let H be the graded Hessian map of f, i.e. the degree zero homomorphism Rn(−(d − 2)) →
+Rn whose matrix in the canonical bases is the Hessian matrix of f. The image of H is the submodule
+of Rn generated by the homogeneous vectors ∇fx1, . . . , ∇fxn. Thus, Cf is the cokernel of H, i.e. it
+has a graded R-free presentation
+(22)
+Rn(−(d − 2))
+H
+−→ Rn −→ Cf −→ 0.
+Dualizing this sequence, and denoting by E∗ (resp. ϕ∗) the R-dual of an R-module E (resp. an
+R-module homomorphism ϕ), we get an exact sequence
+(23)
+0 −→ C ∗
+f −→ Rn
+H∗
+−→ Rn(d − 2) −→ Cf(d − 2) −→ 0,
+where we observe that, since the Hessian matrix is symmetric, H∗ = H ⊗ 1R(d−2) so that, indeed,
+coker H∗ = (coker H)(d − 2) = Cf(d − 2).
+Now, ℓ(Jf) is the dimension of the special fiber ring F(Jf) = k[fx1, . . . , fxn], which by Lemma
+1.9 can be computed as the rank of the Hessian matrix of f. Thus,
+ℓ(Jf) = rank H = rank H∗,
+and hence ℓ(Jf) = n if and only if H is injective (this is of course equivalent to Rn(−(d − 2)) ∼=
+�n
+i=1 R ∇fxi via H, which thus proves (i)⇔ (iii)), if and only if H∗ is injective. The latter property
+means C ∗
+f = 0.
+Therefore, in order to prove (i)⇔ (iv), it suffices to verify that C ∗
+f = 0 if and only if (iv) holds.
+Suppose C ∗
+f = 0. Then, as we have seen, H is injective. Dualizing (22) (which is now a short exact
+sequence) and comparing with (23), we obtain (iv). Conversely, assume that (iv) takes place. Thus
+Cf ∼= Ext1
+R(Cf, R)(2 − d) ∼= Ext1
+R(Cf(d − 2), R).
+Now recall that the R-torsion τR(Cf) of Cf coincides with the kernel of the canonical biduality map
+Cf → C ∗∗
+f , and so, by [3, Proposition 2.6(a)], we have
+τR(Cf) ∼= Ext1
+R(Tr Cf, R).
+But (23) gives Tr Cf = Cf(d − 2). Putting these facts together, we obtain
+C ∗
+f ∼= Ext1
+R(Cf(d − 2), R)∗ ∼= Ext1
+R(Tr Cf, R)∗ ∼= τR(Cf)∗ = 0.
+Next, let us prove that (iv)⇔ (v). The graded canonical module of the standard graded polynomial
+ring R is ωR = R(−n), so
+Ext1
+R(Cf, ωR) ∼= Ext1
+R(Cf, R)(−n).
+Also recall that, in the present setting, the Matlis duality functor is given by HomR(−, k). Thus, by
+graded local duality (see [7, Example 13.4.6]), we can write
+(24)
+Hn−1
+R+ (Cf) ∼= HomR(Ext1
+R(Cf, R)(−n), k) ∼= HomR(Ext1
+R(Cf, R), k)(n).
+If (iv) holds, then Hn−1
+R+ (Cf) ∼= HomR(Cf(d − 2), k)(n) ∼= HomR(Cf, k)(n − d + 2).
+Conversely,
+suppose (v). Using (24), we get
+HomR(Cf, k)(n − d + 2) ∼= HomR(Ext1
+R(Cf, R), k)(n),
+which is the same as an isomorphism HomR(Cf(−n + d − 2), k) ∼= HomR(Ext1
+R(Cf, R)(−n), k).
+Taking Matlis duals and tensoring with R(n), we obtain (iv).
+We proceed to show that (i)⇔(ii). First note that, by (22), the 0-th Fitting ideal of Cf is the
+principal ideal generated by the determinant h of the Hessian matrix of f, so we have
+�
+0 :R Cf =
+�
+(h)
+
+24
+R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS
+and hence dim Cf = dim R/(h). It follows that dim Cf = n−1 if and only if h ̸= 0, i.e. H is injective,
+which as seen above is equivalent to (i).
+We clearly have ℓ(Jf) = ℓ((Jf)R+) and ht R+ = n. Thus, by the general characterization given in
+[30, Proposition 4.1], we have that ℓ(Jf) = n if and only if R+ ∈ AssRR/Jm
+f for all m ≫ 0, which is
+tantamount to saying that (vii) holds. This proves the equivalence (i)⇔(vii).
+Evidently, (vii)⇒(vi), and the converse follows once we recall the chain (see [30, Proposition 3.4])
+AssRR/Jf ⊂ AssRR/J2
+f ⊂ AssRR/J3
+f ⊂ . . .
+Thus, we have proved that the statements (i), . . . ,(vii) are equivalent.
+Now we point out that the implication (vii)⇒(viii) holds regardless of R(Jf) satisfying S2. Indeed,
+condition (vii) means that the irrelevant ideal R+ belongs to the limit value A
+∗(Jf) of the function
+m �→ AssRR/Jm
+f ,
+which is known to eventually stabilize (see, e.g., [30, Proposition 3.4]). There is also the set A ∗(Jf)
+defined analogously as the stable set of asymptotic prime divisors with respect to the usual filtration
+given by the powers of Jf. By [30, Proposition 3.17], we have
+A
+∗(Jf) ⊂ A ∗(Jf)
+and hence R+ ∈ A ∗(Jf), which gives (viii). Notice that (viii)⇒(ix) trivially.
+It remains to show (ix)⇒(i), under the hypothesis that R(Jf) satisfies S2. In this case, by [14,
+Remark 2.16], the extended Rees algebra
+R[Jft, t−1] =
+�
+i∈Z
+Iiti ⊂ R[t, t−1]
+(where, by convention, Ii = R whenever i ≤ 0) must satisfy S2 as well.
+Note that (ix) means
+R+ ∈ AssRR/Jm
+f
+for some m ≥ 1. Now we are in a position to apply [14, Proposition 4.1] in order
+to conclude that ℓ(Jf) = n, as needed.
+Remark 6.3. With the aid of [30, Proposition 3.26 and Proposition 3.20], the assertions (i), . . . ,(vii)
+of Theorem 6.2 are also seen to be equivalent to each of the following ones:
+(a) R+ ∈ A
+∗(IJf) for any non-zero R-ideal I, i.e.,
+depth R/ImJm
+f
+= 0
+for all
+m ≫ 0;
+(b) For some j, the integral closure of R[fx1/fxj, . . . , fxn/fxj] ⊂ k(x1, . . . , xn) contains a prime
+Q of height 1 such that Q ∩ R = R+.
+While, in Theorem 6.2, the implication (ix)⇒(i) (and consequently the implication (viii)⇒(i))
+holds if the Rees ring R(Jf) satisfies S2, we do not know whether this hypothesis can be dropped.
+Thus the following question becomes natural (see also Question 6.17 in the next subsection).
+Question 6.4. Suppose depth R/Jm
+f = 0 for all m ≫ 0. Is it true that ℓ(Jf) = n ? Does this hold
+if we only assume that depth R/Jm
+f = 0 for some m ≥ 1 ?
+Now let f ∈ R be a linear free divisor. As we will see later, if n ≤ 4 then ℓ(Jf) = n. The converse
+is known to be false, and in Example 6.5 below we show in addition that there is a linear free divisor
+f such that ℓ(Jf) = n for any prescribed n.
+The following well-known notion will be useful (we state it over R). A non-zero homogeneous ideal
+I of R, minimally generated by ν elements, is said to satisfy the Gs condition for a given s ≥ 0 if
+ht Iν−j(ϕ) ≥ j + 1
+for
+j = 1, . . . , s − 1.
+Here, ϕ denotes a minimal presentation matrix (or syzygy matrix) of I, and note that there is no
+dependence on the choice of ϕ because each Iν−j(ϕ) is just a Fitting ideal of I.
+
+FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD
+25
+Example 6.5. Given an arbitrary n, consider the normal crossing divisor
+f = x1 · · · xn ∈ R = k[x1, . . . , xn],
+which is a well-known linear free divisor.
+The ideal Jf is simply the ideal generated by all the
+products of distinct n − 1 indeterminates, and satisfies
+depth R/Jm
+f
+= max {0, n − m − 1}
+for all
+m ≥ 1.
+In particular, depth R/Jm
+f
+= 0 for all m ≥ n − 1. We now claim that R(Jf) is Cohen-Macaulay
+(hence it has the S2 property). Notice first that a syzygy matrix of Jf is given by
+ϕ =
+
+
+x1
+0
+. . .
+0
+0
+x2
+. . .
+0
+...
+...
+...
+...
+0
+0
+. . .
+xn−1
+−xn
+−xn
+. . .
+−xn
+
+
+.
+Here we have
+ht In−j(ϕ) = j + 1
+for
+j = 1, . . . , n − 1,
+so that Jf satisfies the Gn property. Moreover, because f is free, Jf has projective dimension 1 over
+R (see Lemma 1.3). It follows by [42, Proposition 4.11] that R(Jf) is Cohen-Macaulay, as claimed.
+Now we are in a position to apply Theorem 6.2 to conclude that ℓ(Jf) = n.
+In addition, the theorem gives us that Cf has projective dimension 1 over R (because the gradient
+vectors of the natural generators of Jf generate a free module) and dimension n − 1, hence Cf is a
+Cohen-Macaulay module, which yields
+Hi
+R+(Cf) ∼=
+�
+HomR(Cf, k)(2), i = n − 1
+0
+, i ̸= n − 1
+In Example 6.5, another way to confirm that ℓ(Jf) = n is by showing that the (monomial) ideal Jf
+is of linear type. This fact and lots of other experiments suggest a more restrictive question as well
+as a conjecture about the interplay between maximal analytic spread and the linear type property;
+as already seen, the latter implies the former.
+Question 6.6. For arbitrary n ≥ 3 (the number of variables), does there exist a free divisor f, with
+Jf not of linear type, such that ℓ(Jf) = n ?
+The case of interest is n ≥ 4. Indeed, if n = 3 then any member of the family of (non-linear) free
+divisors given in Section 4 yields an affirmative answer to this question.
+Conjecture 6.7. If f is a linear free divisor such that ℓ(Jf) = n, then Jf is of linear type.
+We have not been able to solve this conjecture for n ≥ 5. It is true in n ≤ 4 variables, as we can
+verify using the classification of linear free divisors given in [27, 6.4, p. 837].
+Next we furnish more examples.
+Example 6.8. Consider the so-called Gordan-Noether cubic
+f = xw2 + ytw + zt2 ∈ R = k[x, y, z, w, t].
+In this case, while the symmetric algebra SymRJf = B/S = R[t1, t2, t3, t4, t5]/S has dimension
+6 and depth 5, a calculation shows that R(Jf) is Cohen-Macaulay (in particular, it has the S2
+condition). Indeed, if ϕ is a minimal presentation matrix of Jf, then the saturation
+S :B I4(ϕ)∞,
+
+26
+R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS
+which by [31, Lemma 2.11] defines R(Jf) in the ring B (note that I4(ϕ) defines the non-principal
+locus of Jf), is perfect of codimension 4. Now, further computations show that depth R/Jf = 2 and
+depth R/Jm
+f
+= 1
+for all
+m ≥ 2.
+Therefore, using Theorem 6.2, we conclude that ℓ(Jf) < 5. More precisely, the special fiber ring can
+be expressed as F(Jf) = k[t1, t2, t3, t4, t5]/(t2
+2 − t1t3), which yields ℓ(Jf) = 4.
+Example 6.9. Consider the quintic
+f = 2w4u + xu4 + ywu3 + zw2u2 ∈ R = k[x, y, z, w, u].
+This is the case n = 5 of Theorem 2.2, hence f is a linear free divisor (here it should be mentioned,
+for completeness, that f/u is not free and its Jacobian ideal is not even linearly presented). By
+Proposition 2.3(ii), we have ℓ(Jf) = 4 < 5, and Theorem 2.5 ensures that R(Jf) is Cohen-Macaulay.
+Therefore, by Theorem 6.2, we conclude that
+depth R/Jm
+f
+> 0
+for all
+m ≥ 1.
+In fact, for such f it can be verified that depth R/Jf = 3, depth R/J2
+f = 2, and depth R/Jm
+f = 1 for
+all m ≥ 3. An interesting consequence of the non-vanishing of the asymptotic depth of Jf concerns
+the higher conormal modules Jm
+f /Jm+1
+f
+. Indeed, in this situation the (also well-defined) conormal
+asymptotic depth of Jf must be positive as well, since by [6] we can write
+lim
+m→∞ depth Jm
+f /Jm+1
+f
+≥
+lim
+m→∞ depth R/Jm
+f
+> 0.
+It follows that R+ /∈ AssRJm
+f /Jm+1
+f
+for all m ≫ 0.
+Example 6.10. Consider the plane sextic
+f = x6 − 2x3y2z + y4z2 + y6 ∈ R = k[x, y, z].
+This is the case α = 3 and β = 2 of Theorem 4.1(i), hence f is a free divisor (which is no longer
+linear). By Theorem 4.1(ii), the ring R(Jf) is Cohen-Macaulay. It can be verified that
+depth R/Jm
+f
+= 0
+for all
+m ≥ 2.
+Applying Theorem 6.2 we obtain that ℓ(Jf) = 3. Now from Theorem 4.1(iii) we know that Jf cannot
+be of linear type. To see this explicitly, one of the minimal generators of the defining ideal of the
+Rees algebra in the ring R[t1, t2, t3] is the following polynomial which is not linear in the ti’s:
+3xyt1t2 + 2xzt1t3 − 3yzt2t3 − 3y2t2
+3 − 2z2t2
+3.
+Furthermore, the theorem yields Ext1
+R(Cf, R) ∼= Cf(4) and
+Hj
+R+(Cf) ∼=
+�
+HomR(Cf, k)(−1), j = 2
+0
+, j ̸= 2
+Example 6.11. Consider the quartic
+f = x4 − xyz2 + z3w ∈ R = k[x, y, z, w],
+which is not free as Jf is not perfect. Let us also mention that the ideal Jf is not of linear type,
+since the polynomial
+4xt2
+2 − 4zt1t4 − yt2
+4 ∈ R[t1, t2, t3, t4]
+is one of the minimal generators of the defining ideal of R(Jf). On the other hand, it is not hard to
+verify that the associated graded ring of Jf – i.e. the graded algebra �
+s≥0 Js
+f/Js+1
+f
+– satisfies the
+S1 property; since in addition ht Jf ≥ 2, we get by [14, Remark 2.16] that R(Jf) satisfies S2 (it can
+be shown that in fact R(Jf) is Cohen-Macaulay). Furthermore, depth R/Ji
+f = 1 for i = 1, 2, 3, while
+depth R/Jm
+f
+= 0
+for all
+m ≥ 4.
+
+FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD
+27
+Applying Theorem 6.2, we conclude that ℓ(Jf) = 4. We also get Ext1
+R(Cf, R) ∼= Cf(2), and
+Hj
+R+(Cf) ∼=
+�
+HomR(Cf, k)(2), j = 3
+0
+, j ̸= 3
+6.2. Application: Criterion for homaloidness. As above let f ∈ R = k[x1, . . . , xn], n ≥ 3, be a
+non-zero reduced homogeneous polynomial. In this subsection, we assume additionally that the field
+k is algebraically closed. To the form f we can associate the rational map
+Pf = (fx1 : · · · : fxn) : Pn−1 ��� Pn−1,
+the so-called polar map defined by f. Thus the base locus of Pf is the singular locus of the projective
+hypersurface V (f) ⊂ Pn−1.
+Definition 6.12. ([21]) The polynomial f is homaloidal if Pf is birational (hence a Cremona
+transformation).
+Over k = C, this definition can be translated by saying that Pf has degree 1 (taking into account
+an appropriate notion of degree in this context), and according to [19, Corollary 2] the property of
+being homaloidal depends only on fred.
+The following is a preliminary fact connecting this class of polynomials to the class of free divisors.
+It can be also seen as a first source of examples of homaloidal divisors (examples in higher dimensions
+can be found, e.g., in [13] and [33]). Recall that in general the dimension of the image of the polar
+map Pf is given by ℓ(Jf) − 1 (see the proof of Proposition 6.14).
+Proposition 6.13. (k = C) If n ≤ 4 then every linear free divisor is homaloidal.
+Proof.
+Let f ∈ R be a linear free divisor.
+Recall we are supposing that f is not a cone (see
+Subsection 1.1). Thus, by [25, Proposition 2.4 and Proposition 2.5], f has a non-zero Hessian, so
+that ℓ(Jf) = n. Hence, the dimension of the image of Pf is n − 1. As the linear rank of the gradient
+ideal Jf is maximal, it follows by [22, Theorem 3.2] that Pf is birational.
+Notice that this proposition fails if n ≥ 5. Indeed, if in this case we take f as being a linear free
+divisor as described in Theorem 2.2, then by Proposition 2.3(ii) the analytic spread of Jf is 4, hence
+the image of Pf has dimension at most n − 2 and so this map cannot be birational.
+Our application regarding homaloidness is the following ideal-theoretic, also homological, version
+of the criterion given in [22, Theorem 3.2]. It is not as practical or effective as the original one, but
+in our view it adds some flavor to the classical – typically geometric – theory and, moreover, helps
+linking to different algebraic tools and invariants.
+Proposition 6.14. Given f ∈ R as before, let ϕ1 be the submatrix of a minimal syzygy matrix of
+the ideal Jf consisting of its linear syzygies, and suppose In−1(ϕ1) ̸= (0). Assume any one of the
+following situations:
+(i) projdim Jm
+f = n − 1 for some m ≥ 1;
+(ii) R(Jf) satisfies S2, and projdim Jm
+f = n − 1 for some m ≥ 1.
+Then f is homaloidal.
+Proof. First, in either case, our Theorem 6.2 (together with the Auslander-Buchsbaum formula)
+ensures that ℓ(Jf) = n. On the other hand,
+dim(image Pf) = dim Proj k[fx1, . . . , fxn] = dim Proj F(Jf) = ℓ(Jf) − 1 = n − 1.
+Now [22, Theorem 3.2] ensures that Pf is birational, as needed.
+Example 6.15. Let us first point out that not all homaloidal polynomials satisfy the condition
+In−1(ϕ1) ̸= (0). Indeed, consider the cubic
+f = xw2 + yzw + z3 ∈ k[x, y, z, w].
+Then it can be checked that:
+
+28
+R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS
+(a) f is an irreducible homaloidal polynomial. This is indeed the first member of the family of
+irreducible homaloidal hypersurfaces described in [28, p. 1264];
+(b) The Jacobian ideal Jf is not linearly presented, and moreover has not enough linear syzygies.
+More precisely, only 2 columns of a minimal presentation matrix ϕ are linear syzygies, and
+hence obviously I3(ϕ1) = (0);
+(c) Jf is not of linear type;
+(d) Quite interestingly, Jf satisfies the conditions present in part (ii) of our Proposition 6.14. In
+particular, it can be even shown that the Rees algebra of Jf is Cohen-Macaulay.
+We now remark that if f is a linear free divisor then the condition In−1(ϕ1) ̸= (0) is automatically
+satisfied as in this case ϕ1 = ϕ and In−1(ϕ) = Jf by the Hilbert-Burch theorem. We thus record the
+following corollary.
+Corollary 6.16. If f is a linear free divisor satisfying either condition (i) or (ii) of Proposition
+6.14, then f is homaloidal.
+Before giving the first illustration, we raise the following question. We remark that the answer
+is yes if the second part of Question 6.4 has an affirmative answer as well.
+Also note that, by
+Proposition 6.13, the case of interest is n ≥ 5.
+Question 6.17. (n ≥ 5) Let f be a linear free divisor satisfying projdim Jm
+f = n−1 for some m ≥ 1.
+Must f be homaloidal?
+Example 6.18. The simplest example in arbitrary dimension is the normal crossing divisor f =
+x1 · · · xn ∈ R studied in Example 6.5. Then f is a linear free divisor and we have seen in particular
+that depth R/Jn−1
+f
+= 0, i.e., projdim Jn−1
+f
+= n − 1. Since Jf is the ideal generated by all squarefree
+monomials of degree n − 1, we get by [50, Proposition 7.4.5] that all powers of Jf are integrally
+closed; in particular,
+Jn−1
+f
+= Jn−1
+f
+.
+It follows by Corollary 6.16 (or Proposition 6.14(i)) that f is homaloidal, thus retrieving the well-
+known fact that the rational map Pn−1 ��� Pn−1 given by
+(x1 : . . . : xn) �→ (x2x3 · · · xn : x1x3 · · · xn : . . . : x1x2 · · · xn−1)
+is birational – the so-called Cremona involution on Pn−1.
+Below we illustrate Proposition 6.14 in the situation where f is not free, and in both reducible
+and irreducible cases.
+Example 6.19. (n = 6) Consider the hyperplane-quadric arrangement
+f = xw(yz + zt + tu) ∈ R = k[x, y, z, w, t, u].
+In this case, f is non-free because Jf is not perfect, while on the other hand this ideal (which is
+linearly presented, so that ϕ1 = ϕ) satisfies I5(ϕ1) ̸= (0) and
+projdim J3
+f = 5.
+Moreover, as in Example 6.11, the associated graded ring of Jf has the S1 property and hence R(Jf)
+satisfies S2. By Proposition 6.14(ii), f is homaloidal. i.e., the rational map P5 ��� P5 given by
+(x : y : z : w : t : u) �→ (w(yz + zt + tu) : xzw : xw(y + t) : x(yz + zt + tu) : xwu : xwt)
+is Cremona.
+
+FREE DIVISORS, BLOWUP ALGEBRAS, AND ANALYTIC SPREAD
+29
+Example 6.20. (n = 5) Consider the irreducible cubic
+f = xt2 + yzt + z3 + w2t ∈ R = k[x, y, z, w, t].
+The ideal Jf is perfect but f is non-free as ht Jf = 3. It also satisfies I4(ϕ1) ̸= (0) and
+projdim J3
+f = 4.
+Moreover, the associated graded ring of Jf is Cohen-Macaulay; in particular, R(Jf) satisfies S2. By
+Proposition 6.14(ii), f is homaloidal. Explicitly, the rational map P4 ��� P4 given by
+(x : y : z : w : t) �→ (t2 : zt : z2 + 1
+3yt : wt : yz + w2 + 2xt)
+is Cremona.
+Next, we provide a couple of additional observations and questions that, in our view, are interesting
+and potentially motivating for future research. First, note that if we write the homaloidal quartic f
+of Example 6.19 as
+f = xg,
+g = w(yz + zt + tu),
+then a further calculation shows (using again our proposition) that g is homaloidal as well. This
+fact, among other examples, led us to suggest the following “addition-deletion” problem inspired by
+well-known investigations in free divisor theory (see [43], also [1] and [39]).
+Question 6.21. (Addition-deletion for homaloidal divisors.) For polynomials f, g ∈ R, with f homa-
+loidal, when is the product fg homaloidal? If fg is homaloidal, when is f or g homaloidal?
+Now let f ∈ R = k[x, y, z, w, t, u, v] stand for the 2-catalecticant determinant
+f = det
+
+
+x
+y
+z
+z
+w
+t
+t
+u
+v
+
+ .
+According to [33, Proposition 3.25(b)]), this cubic is homaloidal. Then, for such f, we have detected
+an intriguing, curious fact: the determinant h(f) of the Hessian matrix of f is a linear free divisor
+– in particular, h(f) is already reduced. In the situation where h(f) is not reduced, we naturally
+consider h(f)red, which likewise can be a linear free divisor. For example, let
+g = det
+
+
+x
+w
+z
+y
+y
+x
+w
+z
+w
+z
+y
+x
+z
+y
+x
+w
+
+
+in the ring R = k[x, y, z, w]. Note g is in fact a linear free divisor, and using Corollary 6.16 it is not
+hard to see that g is also homaloidal. Here,
+h(g) = λg2
+for some non-zero
+λ ∈ k,
+and therefore h(g)red is free.
+As expected, this phenomenon does not take place in general. For instance, if once again we take
+f as the homaloidal quartic of Example 6.19, then a calculation shows h(f) = 3x2w2f 2, so that
+h(f)red = 3f is not free.
+The facts above led us to raise the following question, which reconnects us to the central topic of
+freeness and closes the paper.
+Question 6.22. Let f be a homaloidal polynomial. When is h(f)red a (linear) free divisor? If h(f)
+is reduced and not a cone, must it be a (linear) free divisor?
+
+30
+R. BURITY, C. B. MIRANDA-NETO, AND Z. RAMOS
+Acknowledgements. The second-named author was supported by the CNPq grants 301029/2019-9
+and 406377/2021-9. The third-named author was supported by the CNPq grants 305860/2019-4 and
+425752/2018-6.
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+Departamento de Matem´atica, Universidade Federal da Para´ıba, 58051-900 Jo˜ao Pessoa, Para´ıba,
+Brazil.
+Email address: [email protected]
+Departamento de Matem´atica, Universidade Federal da Para´ıba, 58051-900 Jo˜ao Pessoa, Para´ıba,
+Brazil.
+Email address: [email protected]
+Departamento de Matem´atica, CCET, Universidade Federal de Sergipe, 49100-000 S˜ao Cristov˜ao,
+SE, Brazil.
+Email address: [email protected]
+

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