Fermion propagator in a rotating environment
aa r X i v : . [ h e p - ph ] F e b Fermion propagator in a rotating environment
Alejandro Ayala,
1, 2
L. A. Hern´andez,
2, 3, 4
K. Raya,
1, 5 and R. Zamora
6, 7 Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, Apartado Postal 70-543, CdMx 04510, Mexico. Centre for Theoretical and Mathematical Physics, and Department of Physics,University of Cape Town, Rondebosch 7700, South Africa. Departamento de F´ısica, Universidad Aut´onoma Metropolitana-Iztapalapa,Av. San Rafael Atlixco 186, C.P, CdMx 09340, Mexico. Facultad de Ciencias de la Educaci´on, Universidad Aut´onoma de Tlaxcala, Tlaxcala, 90000, Mexico. School of Physics, Nankai University, Tianjin 300071, China. Instituto de Ciencias B´asicas, Universidad Diego Portales, Casilla 298-V, Santiago, Chile. Centro de Investigaci´on y Desarrollo en Ciencias Aeroespaciales (CIDCA),Fuerza A´erea de Chile, Casilla 8020744, Santiago, Chile.
We apply the exponential operator method to derive the propagator for a fermion immersed withina rigidly rotating environment with cylindrical geometry. Given that the rotation axis provides apreferred direction, Lorentz symmetry is lost and the general solution is not translationally invariantin the radial coordinate. However, under the approximation that the fermion is completely draggedby the vortical motion, valid for large angular velocities, translation invariance is recovered. Thepropagator can then be written in momentum space. The result is suited to be used applyingordinary Feynman rules for perturbative calculations in momentum space.
Keywords: Fermion propagator, rotation, polarization
I. INTRODUCTION
Collisions of heavy nuclei at high energies producedeconfined strongly interacting matter, dubbed as thequark-gluon plasma (QGP). When these collisions areoff-center, the inhomogeneity of the matter distributionin the transverse plane causes that the colliding regiondevelops an orbital angular velocity Ω directed along thenormal to the reaction plane [1, 2]. Estimates of thisangular velocity provide a value Ω ∼ s − [3].When the vortical motion is transferred to the particlesspin within the QGP, its effect can show, upon hadroniza-tion, as a global hadron polarization, namely, a preferreddirection of the spin of hadrons along the normal to thereaction plane. Recent measurements of the global Λ andΛ polarizations as functions of collision energy [3–5] showthat the Λ polarization rises more steeply than the Λ po-larization when the collision energy decreases. A suitableexplanation of this intriguing result motivates the searchfor the conditions to align the particle’s spin to the globalvortical motion. In particular, it is important to estab-lish how these conditions depend on parameters such asthe collision energy √ s NN , the impact parameter b , thetemperature T , the baryon chemical potential µ B andthe global angular velocity Ω.The problem has attracted a great deal of attentionover the last years [6–23]. In a recent work [24], we haveexplored the Λ and Λ polarization within a model wherethe overlap region in a peripheral heavy-ion collision con-sists of a dense core and a less dense corona, from wheredifferent Λ and Λ production mechanisms are at play [25].The calculation relies on the computation of the relax-ation time that a strange quark or antiquark takes toalign its spin to the global vorticity at finite temperatureand baryon density, which was computed introducing a phenomenological coupling between the quark spin andthe thermal vorticity [26, 27].However, in order to obtain a better estimate, it isimportant to set up the problem in terms of a first prin-ciples calculation to see whether the intuitive use of theabove mentioned phenomenological coupling is correct.In this work we take the first step toward achieving thisgoal and compute the propagator of fermions within arotating environment.Fermion and scalar propagators within a rotating sys-tem and subject to a thermal bath have been first com-puted in Ref. [28]. Given that rotation provides a pre-ferred direction, Lorentz symmetry is lost and the propa-gators are given by cumbersome expressions in the space-time representation. When these are used to compute therotation effects on meson masses, as is done in Ref. [29],including finite temperature as well as chemical potentialeffects, analytical results are not possible and the calcu-lation requires a numerical estimate. It is thus desirableto find an expression that, under suitable conditions, canbe approximated by a translationally invariant result. Aswe show in this work, this can be done provided we keepthe first non-trivial contribution in the angular velocityΩ, which is taken as a large quantity, effectively makingfermions partake of the rigid rotational motion. In thisapproximation the focus is on how the rotation influencesthe spin states rather than on the detailed dynamics ofthe fermion motion.To compute the fermion propagator subject to rota-tion, we follow the method introduced in Refs. [30, 31]that requires knowledge of the explicit set of solutions ofthe Dirac equation. These solutions have been studied byseveral authors imposing different boundary conditions.Working in a cylindrical geometry, in the pioneering workthat introduced the MIT bag model [32], these boundaryconditions are chosen such that the fermion current nor-mal to the cylinder surface vanishes. These conditionsare nowadays known as the MIT boundary conditions. Aslight modification of these conditions, known as the chi-ral MIT conditions, can also be imposed on the fermionmodes [33]. Bound and unbound solutions have also beenstudied in Ref. [34]. The solutions can also be found inthe presence of a magnetic field pointing in the same di-rection as the angular velocity, given that the geometryof the problem is not modified by the presence of thefield [33, 35, 36]. Thermal and rotating states were stud-ied in Ref. [37]. Lattice QCD has also been formulatedin rotating frames to study the angular momenta of glu-ons and quarks in a rotating QCD vacuum [38]. In allthese calculations, the causality condition, whereby theangular velocity and the cylinder radius R must satisfy R Ω <
1, is imposed.In this work we take a pragmatic approach. We findthe solutions to the Dirac equation for fermions rigidlyrotating inside a cylinder. In order to satisfy the causalitycondition for a given Ω, the solutions are taken as notexistent for r > R , but otherwise do not need to satisfy agiven boundary condition. The problem thus formulated,lends itself to attempt finding the fermion propagator inmomentum space, which is a useful quantity to employ inperturbative calculations using ordinary Feynman rules.The work is organized as follows: In Sec. II we formulatethe Dirac equation and find the solutions for a rigidlyrotating cylinder in unbound space. In Sec. III we findthe fermion propagator. We apply the approximationwhereby fermions are totally dragged by the rigid motionto find the expression of this propagator in momentumspace. We finally summarize and provide an outlook forthe use of these results in Sec. IV.
II. FERMIONS IN A RIGIDLY ROTATINGCYLINDER
The physics within a relativistic rotating frame is mosteasily described in terms of a metric tensor resemblingthat of a curved space-time. For our purposes, we con-sider that the interaction region after a relativistic heavy-ion collision can be thought of as a rigid cylinder rotat-ing around the ˆ z -axis with constant angular velocity Ω.Therefore, the metric tensor is given by g µν = − ( x + y )Ω y Ω − x Ω 0 y Ω − − x Ω 0 − − . (1)A fermion with mass m within the cylinder is describedby the Dirac equation [33, 35][ i ( γ µ ∂ µ + Γ µ ) − m ] ψ = 0 , (2) where Γ µ corresponds to the affine connection, deter-mined from the equationsΓ µ = − i ω µij σ ij ,ω µij = g αβ e αi ( ∂ µ e βj + Γ βµν e νj ) , (3)where the commutator σ ij = i γ i , γ j ] (4)corresponds to the fermion spin and, the Christoffel sym-bols are given in terms of the metric tensor byΓ λµν = 12 g λσ ( g σν,µ + g µσ,ν − g µν,σ ) . (5)Greek indices ( µ, ν, . . . = t, x, y, z ) refer to the generalcoordinates in the moving frame, while Latin indices( i, j, . . . = 0 , , ,
3) refer to the Cartesian coordinates inthe local rest frame. Notice that γ µ = e µi γ i correspondto the Dirac matrices in curved space-time, which satisfythe usual anti-commutation relations { γ µ , γ ν } = 0 . (6)The tetrad e µi is written in the Cartesian gauge [37], suchthat it connects the general coordinates with the Carte-sian coordinates in the local rest frame as x µ = e µi x i .Explicitly, e t = e x = e y = e z = 1 ,e t = y Ω ,e t = − x Ω , (7)with the rest of the components being equal to zero. Thenon-zero components of the Christoffel symbols, Eq. (5),are Γ ytx = Γ yxt = Ω , Γ xty = Γ xyt = − Ω , Γ xtt = − x Ω , Γ ytt = − y Ω . (8)Thus, given the above results and Eq. (3), it is straight-forward to see that Γ µ merely reduces toΓ µ → Γ t = − i σ . (9)Subsequently, the gamma matrices in the rotating frameare expressed, in terms of the usual gamma matrices, as γ t = γ , γ x = γ + y Ω γ ,γ z = γ , γ y = γ − x Ω γ . (10)Therefore, Eq. (2) becomes h iγ (cid:18) ∂ t − x Ω ∂ y + y Ω ∂ x − i σ (cid:19) + iγ ∂ x + iγ ∂ y + iγ ∂ z − m i ψ = 0 . (11)In the Dirac representation, σ = (cid:18) σ σ (cid:19) , (12)where σ = diag(1 , −
1) is the Pauli matrix associatedwith the third component of the spin. In consequence,Eq. (11) can be conveniently rewritten as h γ ( i∂ t + Ω ˆ J z ) + i~γ · ~ ∇ − m i ψ = 0 , (13)where ˆ J z ≡ ˆ L z + ˆ S z = − i ( x∂ y − y∂ x ) + 12 σ . (14)ˆ J z defines ˆ z -component of the total angular momentumoperator, such that the first term is associated with theorbital angular momentum ( ˆ L z ), while the second one isrelated with the spin ( ˆ S z ). As usal, − i~ ∇ is the momen-tum operator. The solution of Eq. (13) has been studiedin many works, e.g. [33–35, 37, 39]. For instance, Ref. [34]thoroughly discusses the bound and unbound solutions.At this stage, one could in principle be temptedto write Eq. (13) already in cylindrical coordinates.Nevertheless, in such case, the relevant Dirac matri-ces become coordinate dependent and use of some not-uniquely determined unitary transformations would berequired [40, 41]. To avoid this issue, we follow a differ-ent strategy. Consider a solution of the form ψ ( x ) = h γ ( i∂ t + Ω ˆ J z ) + i~γ · ~ ∇ + m i φ ( x ) . (15)Then, Eq. (13) implies that φ ( x ) obeys the second orderdifferential equation h ( i∂ t + Ω ˆ J z ) + ∂ x + ∂ y + ∂ z − m i φ ( x ) = 0 . (16)Since the above equation does not contain gamma ma-trices, to find solutions consistent with the backgroundgeometry it now becomes convenient to work in cylindri-cal coordinates, ( t, x, t, z ) → ( t, ρ sin ϕ, ρ cos ϕ, z ). Thus,Eq. (16) becomes (cid:20) ( i∂ t + Ω ˆ J z ) + (cid:18) ∂ ρ + 1 ρ ∂ ρ + 1 ρ ∂ ϕ (cid:19) + ∂ z − m (cid:21) φ ( x ) = 0 . (17)In these coordinatesˆ L z = − i∂ ϕ ⇒ ˆ J z = − i∂ ϕ + ˆ S z . (18)Assuming that the solution of Eq. (17) admits a separa-tion of variables, we can write φ ( x ) = e − iEt + ik z z u ( ρ, ϕ ) . (19)Due to the form of ˆ S z , Eqs. (12) and (14), the spin op-erator will produce eigenvalues s = ± /
2. Consequently, total angular momentum conservation ( j = ℓ + s ) de-mands solutions with ℓ (for s = +1 /
2) and ℓ + 1 (for s = − / φ = φ φ φ φ , (20)Eq. (17) becomes h (cid:0) E + ( l + )Ω (cid:1) + (cid:16) ∂ ρ + ρ ∂ ρ − ℓ ρ (cid:17) − k z − m i φ , ( x ) = 0 , (21) h (cid:0) E + ( l + 1 − )Ω (cid:1) + (cid:16) ∂ ρ + ρ ∂ ρ − ( ℓ +1) ρ (cid:17) − k z − m i φ , ( x ) = 0 . (22)The above correspond to Bessel equations h ρ ∂ ρ + ρ∂ ρ + ( ρ k ⊥ − ℓ ) i φ , = 0 , (23) h ρ ∂ ρ + ρ∂ ρ + ( ρ k ⊥ − ( ℓ + 1) ) i φ , = 0 , (24)where k ⊥ = ˜ E − k z − m , (25)is the transverse momentum squared and we have defined˜ E ≡ E + j Ω, which represents the fermion energy asseen from the inertial frame. The solutions of Eqs. (23)and (24) that are finite for ρ → u ( ρ, ϕ ) = e iϕℓ J ℓ ( k ⊥ ρ ) , for φ , , (26) u ( ρ, ϕ ) = e iϕ ( ℓ +1) J ℓ +1 ( k ⊥ ρ ) , for φ , . (27)Therefore, the solution of Eq. (15) can be explicitly writ-ten as φ ( x ) = J ℓ ( k ⊥ ρ ) J ℓ +1 ( k ⊥ ρ ) e iϕ J ℓ ( k ⊥ ρ ) J ℓ +1 ( k ⊥ ρ ) e iϕ e − iEt + ik z z + iℓϕ . (28)Having determined the solutions φ , Eq. (15) can beused to find the spinor wave functions which become ψ ( x ) = ˜ E + m − k z −P − E + m −P + k z k z P − − ˜ E + m P + − k z − ˜ E + m × J ℓ ( k ⊥ ρ ) J ℓ +1 ( k ⊥ ρ ) e iϕ J ℓ ( k ⊥ ρ ) J ℓ +1 ( k ⊥ ρ ) e iϕ e − iEt + ik z z + iℓϕ , (29)where P ± = k x ± ik y . In cylindrical coordinates, P ± = − ie ± iϕ ( ∂ ρ ± iρ − ∂ ϕ ) . (30) P ± act on the wave functions as ladder operators [34],namely P ± e iℓϕ J ℓ ( k ⊥ ρ ) = ± ik ⊥ e i ( ℓ ± ϕ J ℓ ± ( k ⊥ ρ ) . (31)Thus, combining Eqs. (29)-(31), the explicit result for ψ ( x ) reads as ψ ( x ) = [ ˜ E + m − k z + ik ⊥ ] J ℓ ( k ⊥ ρ )[ ˜ E + m + k z − ik ⊥ ] J ℓ +1 ( k ⊥ ρ ) e iϕ [ − ˜ E + m + k z − ik ⊥ ] J ℓ ( k ⊥ ρ )[ − ˜ E + m − k z + ik ⊥ ] J ℓ +1 ( k ⊥ ρ ) e iϕ × e − i ˜ Et + ik z z + iℓϕ . (32)Armed with the explicit expressions, we now follow theapproach discussed in Refs. [30, 31] to find the fermionpropagator. III. FERMION PROPAGATOR IN A RIGIDLYROTATING CYLINDER
Recall that in order to find the solution for an equa-tion describing the Green’s function G ( x, x ′ ) of a givendifferential operator H ( ∂ x , x ), namely H ( ∂ x , x ) G ( x, x ′ ) = δ ( x − x ′ ) , (33)the Green’s function can be represented as G ( x, x ′ ) = ( − i ) Z −∞ dτ U ( x, x ′ ; τ ) , (34)where τ is known as a proper-time parameter and U ( x, x ′ ; τ ) is an evolution operator in this proper-time.This operator satisfies i∂ τ U ( x, x ′ ; τ ) = H ( ∂ x , x ) U ( x, x ′ ; τ ) , (35)together with the boundary conditions U ( x, x ′ ; −∞ ) = 0 ,U ( x, x ′ ; 0) = δ ( x − x ′ ) , (36)from where the solution is readily found as U ( x, x ′ ; τ ) = exp[ − iτ H ( ∂ x , x )] δ ( x − x ′ ) . (37)In order to find the precise form of the proper-timeevolution operator, we can use that, when the eigenfunc-tions φ λ ( x ) of the operator H ( ∂ x , x ) are known, the Diracdelta-function can be expressed in terms of the closurerelation obeyed by the eigenfunctions φ λ ( x ), namely X λ φ λ ( x ) φ † λ ( x ′ ) = δ ( x − x ′ ) . (38)Therefore, an exact expression for the proper-time evo-lution operator can be written as U ( x, x ′ ; τ ) = X λ exp[ − iτ λ ] φ λ ( x ) φ † λ ( x ′ ) , (39) where we have used the eigenvalue equation H ( ∂ x , x ) φ λ ( x ) = λφ λ ( x ) . (40)Using Eqs. (34) and (39), the propagator G ( x, x ′ ) can bewritten as G ( x, x ′ ) = ( − i ) Z −∞ dτ X λ exp[ − iτ λ ] φ λ ( x ) φ † λ ( x ′ ) . (41)It is easy to show that the solutions in Eq. (28) satisfythe closure relation ∞ X l = −∞ Z dEdk z dk ⊥ k ⊥ (2 π ) φ ( x ) φ † ( x ′ ) = δ ( x − x ′ ) , (42)where we have taken the quantum numbers E, k ⊥ , k z , ℓ asindependent, namely, the on-shell restriction of Eq. (25)is not imposed, as corresponds for a procedure to find thepropagator. Furthermore, notice that k ⊥ is taken in thecontinuous domain 0 ≤ k ⊥ ≤ ∞ and thus, no boundaryrestriction is required on the space variable ρ .To obtain the fermion propagator, we notice that, inthe same manner that the solutions of the Dirac equa-tion are obtained from the solutions to the second orderdifferential equation, Eq. (15), the fermion propagator S ( x, x ′ ) can be derived [30] from S ( x, x ′ ) = h γ ( i∂ t + Ω ˆ J z ) + i~γ · ~ ∇ + m i G ( x, x ′ ) , (43)where G ( x, x ′ ) = ( − i ) Z −∞ dτ e − iτ ( ˜ E − k ⊥ − k z − m + iǫ ) × ∞ X ℓ = −∞ Z dEdk z dk ⊥ k ⊥ (2 π ) φ ( x ) φ † ( x ′ ) . (44)Therefore, substituting Eq. (28) into Eq. (44) and per-forming the integral over τ , the expression for the fermionpropagator can be written as S ( x, x ′ ) = ∞ X ℓ = −∞ Z dEdk z k ⊥ dk ⊥ (2 π ) Φ( ρ, ρ ′ ) × e − i ( E − ( ℓ +1 / t − t ′ ) e ik z ( z − z ′ ) e iℓ ( ϕ − ϕ ′ ) E − k z − m − k ⊥ + iǫ , (45)whereΦ( ρ, ρ ′ ) ≡ diag[( E − k z + m + ik ⊥ ) J ℓ ( k ⊥ ρ ) J ℓ ( k ⊥ ρ ′ ) , ( E + k z + m − ik ⊥ ) J ℓ +1 ( k ⊥ ρ ) J ℓ +1 ( k ⊥ ρ ′ ) e i ( ϕ − ϕ ′ ) , ( − E + k z + m − ik ⊥ ) J ℓ ( k ⊥ ρ ) J ℓ ( k ⊥ ρ ′ ) , ( − E − k z + m + ik ⊥ ) J ℓ +1 ( k ⊥ ρ ) J ℓ +1 ( k ⊥ ρ ′ ) e i ( ϕ − ϕ ′ ) ] , (46)and we have implemented the change of variable E → E +Ω( ℓ +1 / t − t ′ ),( z − z ′ ) and ( ϕ − ϕ ′ ).The expression for the propagator can be further re-duced. Let us focus on one of the elements, the compo-nent S ( x, x ′ ). We use the partial translation invarianceto write S ( ρ, ρ ′ , ϕ, z, t ) = ∞ X ℓ = −∞ Z dEdk z k ⊥ dk ⊥ (2 π ) × ( E − k z + m + ik ⊥ ) J ℓ ( k ⊥ ρ ) J ℓ ( k ⊥ ρ ′ ) × e − i ( E − ( ℓ +1 / t e ik z z e iℓϕ E − k z − m − k ⊥ + iǫ . (47)In order to calculate the sum over ℓ , we use the integralrepresentation of the Bessel functions J ℓ ( x ) = 12 π Z π − π e i ( x sin( τ ) − ℓτ ) dτ, (48)thus arriving at S ( ρ, ρ ′ , ϕ, z, t ) = Z dEdk z k ⊥ dk ⊥ (2 π ) e − i ( E − Ω / t e ik z z × ( E − k z + m + ik ⊥ ) E − k z − m − k ⊥ + iǫ × Z π − π dτ e ik ⊥ ρ ′ sin( τ ) × ∞ X ℓ = −∞ J ℓ ( k ⊥ ρ ) e iℓ ( ϕ +Ω t − τ ) . (49)We now use the Jacobi-Anger expansion ∞ X ℓ = −∞ J ℓ ( x ) e iℓy = e ix sin( y ) , (50)supplemented by the change of variable ρ, ρ ′ → R, r givenby ρ ′ = R − r/ ,ρ = R + r/ , (51) we get S ( R, r, ϕ, z, t ) = Z dEdk z k ⊥ dk ⊥ (2 π ) e − i ( E − Ω / t e ik z z × ( E − k z + m + ik ⊥ ) E − k z − m − k ⊥ + iǫ × Z π − π dτ e − ik ⊥ r (sin τ − sin θ ) / × e ik ⊥ R (sin τ +sin θ ) , (52)where we have defined θ ≡ ϕ + Ω t − τ .We now make the approximation whereby the fermionis totally dragged by the vortical motion such that theangular position is determined by the product of the an-gular velocity and the time, namely ϕ + Ω t = 0. This isa very good approximation for large Ω, as in a relativis-tic heavy-ion collision. In this way, sin θ → − sin τ andthus the last factor in Eq. (52) becomes 1. Notice that,under this approximation, the function depends only onrelative coordinates making it translationally invariant.Using the identity Z π − π dτ e ik ⊥ r sin τ = (2 π ) J ( k ⊥ r ) , (53)we obtain S ( r, ϕ, z, t ) = Z dEdk z k ⊥ dk ⊥ (2 π ) e − i ( E − Ω / t e ik z z × ( E − k z + m + ik ⊥ ) E − k z − m − k ⊥ + iǫ J ( k ⊥ r ) . (54)We now use that the function depends only on relativecoordinates to introduce the Fourier transform S ( p ) = Z d xe ip · x S ( x ) , (55)and obtain S ( p ) = p + Ω / − p z + m + ip ⊥ ( p − Ω / − ~p − m + iǫ . (56)The rest of the terms in the propagator can be workedin a similar fashion and thus we get S ( p ) = p +Ω / − p z + m + ip ⊥ ( p − Ω / − ~p − m + iǫ p − Ω / p z + m − ip ⊥ ( p +Ω / − ~p − m + iǫ − ( p +Ω / p z + m − ip ⊥ ( p − Ω / − ~p − m + iǫ
00 0 0 − ( p − Ω / − p z + m + ip ⊥ ( p +Ω / − ~p − m + iǫ . (57)The result can be further simplified by introducing theoperators O ± ≡ (cid:2) ± iγ γ (cid:3) , (58) such that the propagator looks like S ( p ) = p + Ω / − p z + m + ip ⊥ ( p + Ω / − ~p − m + iǫ γ O + + p − Ω / p z + m − ip ⊥ ( p − Ω / − ~p − m + iǫ γ O − . (59)Equation (59) is our main result. We emphasize that thispropagator is obtained under the approximation wherebythe fermion is dragged by the vortical motion. In thismanner we have traded the detailed description of thefermion motion in favor of accounting for the way the an-gular velocity translates into an influence on the fermionspin degrees of freedom. IV. SUMMARY AND OUTLOOK
In this work we have derived the propagator for afermion immersed within a rigidly rotating environment.The motivation stems from the search of the descriptionfrom first principles to study how the fermion spin is af-fected by the overall rotational motion. The method weused has been recently put forward in Refs. [30, 31] andit has been applied to re-deriving the propagator for elec-trically charged bosons, fermions and even gauge bosonsin the presence of a magnetic field. To our knowledge,this is the first time the method is used in the context of fermions immersed in a rotating environment.We found that the propagator is diagonal in Lorentzspace and the general expression is not translationallyinvariant in the transverse radial coordinate. However,under the approximation that the fermion is completelydragged by the overall vortical motion, translation invari-ance is recovered, which allows us to find the expressionfor the propagator in momentum space.The propagator thus found is now suited to be used inperturbative calculations using ordinary Feynman rulesin momentum space. Work in this direction is currentlybeing pursued and will be soon reported elsewhere.
ACKNOWLEDGMENTS
This work was supported in part by UNAM-DGAPA-PAPIIT grant number IG100219 and by Consejo Na-cional de Ciencia y Tecnolog´ıa grant numbers A1-S-7655and A1-S-16215. R. Zamora acknowledges support fromFONDECYT (Chile) under grant No. 1200483. [1] F. Becattini, F. Piccinini, andJ. Rizzo, Phys. Rev. C , 024906 (2008),arXiv:0711.1253 [nucl-th].[2] F. Becattini, G. Inghirami, V. Rolando, A. Beraudo,L. Del Zanna, A. De Pace, M. Nardi, G. Pagliara, andV. Chandra, Eur. Phys. J. C , 406 (2015), [Erratum:Eur.Phys.J.C 78, 354 (2018)], arXiv:1501.04468 [nucl-th].[3] L. Adamczyk et al. (STAR), Nature , 62 (2017),arXiv:1701.06657 [nucl-ex].[4] B. I. Abelev et al. (STAR),Phys. Rev. C , 024915 (2007), [Erratum: Phys.Rev.C95, 039906 (2017)], arXiv:0705.1691 [nucl-ex].[5] J. Adam et al. (STAR), Phys. Rev. C , 014910 (2018),arXiv:1805.04400 [nucl-ex].[6] F. Becattini, L. Csernai, and D. J. Wang,Phys. Rev. C , 034905 (2013), [Erratum: Phys.Rev.C93, 069901 (2016)], arXiv:1304.4427 [nucl-th].[7] Y. Xie, R. C. Glastad, and L. P.Csernai, Phys. Rev. C , 064901 (2015),arXiv:1505.07221 [nucl-th].[8] A. Sorin and O. Teryaev,Phys. Rev. C , 011902 (2017),arXiv:1606.08398 [nucl-th].[9] Y. Sun and C. M. Ko, Phys. Rev. C , 024906 (2017),arXiv:1706.09467 [nucl-th].[10] D. Suvarieva, K. Gudima, and A. Zinchenko,EPJ Web Conf. , 08003 (2017).[11] Y. L. Xie, M. Bleicher, H. St¨ocker, D. J. Wang,and L. P. Csernai, Phys. Rev. C , 054907 (2016),arXiv:1610.08678 [nucl-th].[12] H. Li, L.-G. Pang, Q. Wang, and X.-L. Xia, Phys. Rev. C , 054908 (2017),arXiv:1704.01507 [nucl-th].[13] X.-L. Xia, H. Li, and Q. Wang,PoS CPOD2017 , 023 (2018),arXiv:1712.02677 [nucl-th]. [14] I. Karpenko and F. Becat-tini, Nucl. Phys. A , 519 (2019),arXiv:1811.00322 [nucl-th].[15] D. Suvarieva, K. Gudima, and A. Zinchenko,Phys. Part. Nucl. Lett. , 182 (2018).[16] Z.-Z. Han and J. Xu, Phys. Lett. B , 255 (2018),arXiv:1707.07262 [nucl-th].[17] X.-L. Xia, H. Li, Z.-B. Tang, andQ. Wang, Phys. Rev. C , 024905 (2018),arXiv:1803.00867 [nucl-th].[18] M. Baznat, K. Gudima, A. Sorin, andO. Teryaev, Phys. Rev. C , 041902 (2018),arXiv:1701.00923 [nucl-th].[19] E. E. Kolomeitsev, V. D. Toneev, andV. Voronyuk, Phys. Rev. C , 064902 (2018),arXiv:1801.07610 [nucl-th].[20] Y. Xie, D. Wang, and L. P.Csernai, Eur. Phys. J. C , 39 (2020),arXiv:1907.00773 [hep-ph].[21] Y. Guo, S. Shi, S. Feng, andJ. Liao, Phys. Lett. B , 134929 (2019),arXiv:1905.12613 [nucl-th].[22] H.-B. Li and X.-X. Ma,Phys. Rev. D , 076007 (2019),arXiv:1907.01151 [hep-ph].[23] J. I. Kapusta, E. Rrapaj, andS. Rudaz, Phys. Rev. C , 064911 (2020),arXiv:2004.14807 [hep-th].[24] A. Ayala et al. , Phys. Lett. B , 135818 (2020),arXiv:2003.13757 [hep-ph].[25] A. Ayala, E. Cuautle, G. Herrera, and L. M. Montano,Phys. Rev. C , 024902 (2002), arXiv:nucl-th/0110027.[26] A. Ayala, D. De La Cruz, S. Hern´andez-Ort´ız, L. A. Hern´andez, and J. Sali-nas, Phys. Lett. B , 135169 (2020),arXiv:1909.00274 [hep-ph]. [27] A. Ayala, D. de la Cruz, L. A. Hern´andez,and J. Salinas, Phys. Rev. D , 056019 (2020),arXiv:2003.06545 [hep-ph].[28] A. Vilenkin, Phys. Rev. D , 2260 (1980).[29] M. Wei, Y. Jiang, and M. Huang, (2020),arXiv:2011.10987 [hep-ph].[30] S. N. Iablokov and A. V. Kuznetsov,J. Phys. Conf. Ser. , 012078 (2019).[31] S. N. Iablokov and A. V. Kuznetsov,Phys. Rev. D , 096015 (2020),arXiv:2008.07890 [hep-th].[32] A. Chodos, R. L. Jaffe, K. Johnson, C. B. Thorn, andV. F. Weisskopf, Phys. Rev. D , 3471 (1974).[33] M. N. Chernodub and S. Gongyo, JHEP , 136 (2017),arXiv:1611.02598 [hep-th].[34] V. E. Ambrus and E. Winstan-ley, Phys. Rev. D , 104014 (2016),arXiv:1512.05239 [hep-th].[35] H.-L. Chen, K. Fukushima, X.-G. Huang, andK. Mameda, Phys. Rev. D , 104052 (2016), arXiv:1512.08974 [hep-ph].[36] S. Ebihara, K. Fukushima, andK. Mameda, Phys. Lett. B , 94 (2017),arXiv:1608.00336 [hep-ph].[37] V. E. Ambru¸s and E. Winstan-ley, Phys. Lett. B , 296 (2014),arXiv:1401.6388 [hep-th].[38] A. Yamamoto and Y. Hirono,Phys. Rev. Lett. , 081601 (2013),arXiv:1303.6292 [hep-lat].[39] Y. Jiang and J. Liao,Phys. Rev. Lett. , 192302 (2016),arXiv:1606.03808 [hep-ph].[40] M. Loewe, F. Marquez, and R. Zamora,J. Phys. A , 465303 (2012), arXiv:1112.6402 [hep-ph].[41] E. Ley-Koo and R. Wang, Rev. Mex. Phys.34