Causality and stability in relativistic viscous non-resistive magneto-fluid dynamics
Rajesh Biswas, Ashutosh Dash, Najmul Haque, Shi Pu, Victor Roy
PPrepared for submission to JHEP
Causality and stability in relativistic viscousmagneto-fluid dynamics
Rajesh Biswas a , Ashutosh Dash a , Najmul Haque a , Shi Pu b , Victor Roy a, a School of Physical Sciences, National Institute of Science Education and Research, HBNI, 752050,Jatni , India. b Department of Modern Physics, University of Science and Technology of China, Hefei 230026,China.
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
We investigate the causality and the stability of the relativistic viscousmagneto-hydrodynamics in the framework of the Israel-Stewart (IS) second-order theory,and also within a modified IS theory which incorporates the effect of magnetic fields in therelaxation equations of the viscous stress. We compute the dispersion relation by perturb-ing the fluid variables around their equilibrium values. In the ideal magnetohydrodynamicslimit, the linear dispersion relation yields the well-known propagating modes: the Alfvénand the magneto-sonic modes. In the presence of bulk viscous pressure, the causality boundis found to be independent of the magnitude of the magnetic field. The same bound alsoremains true, when we take the full non-linear form of the equation using the method ofcharacteristics. In the presence of shear viscous pressure, the causality bound is indepen-dent of the magnitude of the magnetic field for the two magneto-sonic modes. The causalitybound for the shear-Alfvén modes, however, depends both on the magnitude and the di-rection of the propagation. For modified IS theory in the presence of shear viscosity, newnon-hydrodynamic modes emerge but the asymptotic causality condition is the same as thatof IS. In summary, although the magnetic field does influence the wave propagation in thefluid, the study of the stability and asymptotic causality conditions in the fluid rest frameshows that the fluid remains stable and causal given that they obey certain asymptoticcausality condition. Corresponding author. a r X i v : . [ nu c l - t h ] J u l ontents A defined in section 4.3 and the characteristic velocities 26 In relativistic heavy-ion collisions experiment, two fast moving charged nuclei collide witheach other and generate a deconfined state of matter known as Quark-Gluon-Plasma (QGP).In non-central collisions an extremely strong magnetic field ( ∼ - Gauss) is alsoproduced in the initial stages refs. [1–5] mostly due to the spectator protons.The huge magnetic fields induce many novel quantum transport phenomena. One of themost interesting and important phenomena is the Chiral Magnetic Effect (CME) refs. [6–8],which means a charge current will be induced and be parallel to the magnetic fields in achiraly imbalanced system. Along with the CME, it was also theoretically predicted thatmassless fermions with the same charge but different chirality will be separated, knownas chiral separation effect (CSE). The electric fields may also cause the chiral separationeffects or chiral Hall effects refs. [9–12]. There are many discussions on other high ordernon-linear chiral transport phenomena refs. [13–15]. One theoretical framework for studyingthese quantum transport phenomena is the chiral kinetic theory refs. [16–31] and numerical– 1 –imualations based on this framework can be found in refs. [32–39]. Recently, the chiralparticle production is found to be connected to the famous Schwinger mechanism ref. [40],and is proved through the world-line formalism ref. [41] and Wigner functions ref. [42].There are also many theoretical studies of CME from the quantum field theory refs. [43–47]and the chiral charge fluctuation refs. [48, 49]. The strong magnetic field might also inducesanisotropic transport of momentum which results in the anisotropic transport coefficientsrefs. [50, 51]. In refs. [52, 53] relativistic Boltzmann equation was used to study the effectof electromagnetic fields in heavy-ion collisions. For the recent developments, one can seethe reviews refs. [54–62] and references therein.The charge separation in Au+Au collisions are claimed to be observed by the STARcollaboration refs. [63–65]. However, it is still a challenge to extract the CME signals fromthe huge backgrounds caused by the collective flows refs. [66–68]. Therefore, it requires thesystematic and quantitative studies of the evolution of the QGP coupled with the electro-magnetic fields for the discovery of CME. It is widely accepted that the QGP produced inhigh energy heavy-ion collisions behaves as almost ideal fluid (i.e., possess very small shearand bulk viscosity). This conclusion was made primarily based on the success of relativisticviscous hydrodynamics simulations in explaining a multitude of experimental data with avery small specific shear viscosity ( η /s) as an input refs. [69–76]. Most of these theoreticalstudies use IS second-order causal viscous hydrodynamics formalism or some variant of it.The fact that the QGP is composed of electrically charged quarks indicates that it shouldhave finite electrical conductivity which is corroborated by the lattice-QCD calculationsrefs. [77–79] and perturbative QCD calculations refs. [80, 81]. The electrical conductivityof the QGP and the hadronic phase was also calculated by various other groups (mostly us-ing the Boltzmann transport equation) see refs. [82–96]. It is then natural to expect that theappropriate equation of motion of the high temperature QGP and low temperature hadronicphase under large magnetic fields is given by the relativistic viscous magneto-hydrodynamicframework. As mentioned earlier the IS second-order theory of causal dissipative fluid dy-namics, although successful, known to allow superluminal signal propagation (and henceacausal) under certain circumstances refs. [97–100]. It is then important to know underwhat physical conditions the theory remains causal and stable in presence of a magneticfield which is also important for the numerical MHD studies of heavy-ion collisions.Relativistic magnetohydrodynamics (MHD) is a self-consistent macroscopic frameworkwhich describe the evolution of any charged fluid in the presence of electromagnetic fieldsrefs. [101–107]. In ref. [4], we have computed the ratio of the magnetic field energy to thefluid energy density in the transverse plane of Au-Au collisions at √ s NN = 200 GeV inthe event-by-event simulations. Our results imply that the magnetic field energy is notnegligible. In ref. [101], we have derived the analytic solutions of a longitudinal Bjorkenboost invariant MHD with transverse electromagnetic fields in the ideal limit. We havefound that the transverse magnetic fields will decay as ∼ /τ with τ being the propertime. Later, in ref. [102], we have studied the corrections from the magnetization effectsand extended the discussion to ( )-dimensional ideal MHD refs. [108, 109]. We havealso investigated the effects of large magnetic fields on (2 + 1) -dimensional reduced MHDat √ s NN = 200 GeV ref. [110]. Very recently, we have derived the analytic solutions of– 2 –HD in the presence of finite electric conductivity, CME and chiral anomaly ref. [106] andextended the results to cases with the transverse and longitudinal electric conductivitiesref. [107]. For numerical simulations of ideal MHD, one can see refs. [104, 105].As mentioned earlier in the ordinary relativistic hydrodynamics, the widely used frame-work is the second order IS theory [111]. The pioneering studies on the instabilities of firstorder hydrodynamics are shown in refs. [97, 112]. Later, the systematic studies for the dis-sipative fluid dynamics have been done earlier with bulk viscous pressure [99], shear viscousstress [98] and heat currents [113], also see refs. [100, 114]. There have been several recentstudies on casualty and stability of ideal MHD in refs. [115–117] and reference therein.We aim to study the stability and causality of the IS theory for MHD, whose form isderived by the complete moment expansion as done in refs. [118, 119]. First, we analyze thepropagating modes in ideal non-resistive MHD. Next, we discuss the causality and stabilityof the relativistic MHD with dissipative effects. To analyse the causality and stability ofthe relativistic viscous fluid, we linearise the relevant equations by using a small sinusoidalperturbation around the local equilibrium and study the corresponding dispersion relationsin line with the studies in refs. [97–99, 112] .The manuscript is organized as follows: in section 2 we briefly discuss the energy-momentum tensor of fluid for ideal MHD case and the modified IS theory. In section 3we revisit the standard analysis of causality and stability of a system without magneticfields. Then, in section 4 we show the stability and causality of an ideal MHD and carryout the analysis of characteristic velocities in section 5. In section 6 we consider the newlydeveloped IS theory for non-resistive MHD. Finally, we conclude our work in section 7.Throughout the paper, we use the natural unit and the flat space-time metric g µν = diag (+1 , − , − , − . The fluid velocity satisfies u µ u µ = 1 and the projection operatorperpendicular to u µ is ∆ µν = g µν − u µ u ν . In this work we consider the causal relativistic second order theory for relativistic fluidsby Israel-Stewart (IS) and also a modified form of the IS theory in presence of a magneticfield given in ref. [118], for later use we define it as NRMHD-IS theory (here NRMHDcorresponds to non resistive magneto-hydrodynamics). The total energy momentum tensorof the fluid can be written as, T µν = (cid:0) ε + P + Π + B (cid:1) u µ u ν − (cid:18) P + Π + B (cid:19) g µν − B µ B ν + π µν , (2.1)where ε , P are fluid energy density, pressure , u µ is the fluid four velocity and Π , π µν arebulk viscous pressure and shear viscous tensor, respectively. The magnetic and electric fourvectors are defined as B µ = 12 (cid:15) µναβ u ν F αβ , E µ = F µν u ν , (2.2)where F µν = ( ∂ µ A ν − ∂ ν A µ ) is the field strength tensor. The space-time evolution of thefluid and magnetic fields are described by the energy-momentum conservation ∂ µ T µν = 0 , (2.3)– 3 –oupled with Maxwell’s equations ∂ µ F µν = j ν ,(cid:15) µναβ ∂ β F να = 0 . (2.4)The non-resistance limit means the electric conductivity σ e is infinite. In this limit, in orderto keep the charge current j µ = σ e E µ be finite, the E µ → . Then, the relevant Maxwell’sequations which govern the evolution of magnetic fields in the fluid is, ∂ ν ( B µ u ν − B ν u µ ) = 0 . (2.5)For simplicity, we will also neglect the magnetisation of the QGP, which implies an isotropicpressure and no change in the Equation of Sate (EoS) of the fluid due to magnetic field (e.g.see ref. [102]).In the original IS theory the viscous stresses Π , π µν are considered as an independentdynamical variables given by the following equations (e.g. see refs. [120–122]), Π = Π NS − τ Π ˙Π+ τ Π q q · ˙ u − (cid:96) Π q ∂ · q − ζ ˆ δ Π θ + λ Π q q · ∇ α + λ Π π π µν σ µν (2.6) π µν = π µν NS − τ π ˙ π <µν> +2 τ πq q <µ ˙ u ν> + 2 (cid:96) πq ∇ <µ q ν> + 2 τ π π <µλ ω ν>λ − η ˆ δ π µν θ − τ π π <µλ σ ν>λ − λ πq q <µ ∇ ν> α + 2 λ π Π Π σ µν , (2.7)where ζ and η are bulk and shear viscosity, respectively. The coefficients τ Π and τ π are therelaxation times for the bulk and shear viscosity, respectively and ω µν ≡ ∆ µα ∆ νβ ( ∂ α u β − ∂ β u α ) is the vorticity tensor. The subscript NS means the Navier-Stokes values, Π NS = − ζθ = − ζ∂ µ u µ ,π µνNS = 2 ησ µν , (2.8)where σ µν = ∇ <µ u ν> = 12 ( ∇ µ u ν + ∇ ν u µ ) −
13 ∆ µν ∂ α u α . (2.9)Note that all of these coefficients are functions of baryon chemical potential ( µ ) and tem-perature ( T ). Equation (2.7) can be derived from the kinetic theory via complete momentexpansion, one can see refs. [123–125] for more details.For further simplification, we also ignore the coupling of viscosity with other dissipativeforces and concentrate on the following terms, Π = Π NS − τ Π ˙Π , (2.10) π µν = π µν NS − τ π ˙ π <µν> . (2.11)We note that in principle the magnetic field may cause viscous tensor to be anisotropicas shown in ref. [126] but in this work we consider zero magnetisation and hence useeqs. (2.10), (2.11) for simplicity. – 4 – Dispersion relation in the absence of magnetic field
As is known, IS theory is a consistent fluid dynamical prescription which preserves causalityprovides that the relaxation time associated with the dissipative quantities (such as shearand bulk viscous stresses) are not too small refs. [97–100, 112–114]. Here we aim to studythe stability and causality of a relativistic viscous fluid (governed by the IS equations) in anexternal magnetic field by linearising the governing equations under a small perturbation.Before discussing the causality and stability of a relativistic viscous fluid in a magneticfield, for the sake of completeness, let us summaries here the findings without the magneticfield. We note that the following results are not new and most of them can be found inrefs. [98, 99, 114].
We consider a perturbation around the static quantities X , X = X + δ ˜ X, δ ˜ X = δXe i ( ωt − k · r ) , (3.1)where we choose five independent variables X = ( ε, u x , u y , u z , Π) . Here, we only considerthe system in the local rest frame, i.e. u µ = (1 , ) . Then, we linearise eq. (2.3), (2.10)in vanishing magnetic fields and shear viscous tensor limit and obtain a cubic polynomialequation of the form given in eq. (A.2) with X i ’s are X = iτ Π αk , X = − (cid:18) α + 1 b (cid:19) k , X = − iτ Π , (3.2)and the other two roots being zero. The solutions of this cubic polynomial are obtainedfrom eq. (A.3). Here, we introduce a constant α = c s , where c s is speed of sound.We adopt the following parametrisation of the bulk viscosity coefficient and the relax-ation time refs. [99, 114]: ζ = a s, (3.3) τ Π = ζε + P b = a b T , (3.4)where s and T are the entropy density and the temperature, respectively. The parameters a and b characterize the magnitudes of the viscosity and the relaxation time, respectively.In the small wave-number limit, the dispersion relation is ω = (cid:40) iτ Π , ± k √ α. (3.5)Whereas the asymptotic forms of the dispersion relation in this case for large k are ω = i αb τ Π (1+ αb ) , ± k (cid:113) α + b + i τ Π (1+ αb ) . (3.6)Note that one of the roots is a pure imaginary which is also known as the non-hydrodynamicmode because it is independent of k in the k → limit.– 5 – .2 Dispersion relation for shear viscosity We use the following parametrization taken from ref. [98] for the shear viscous coefficientand the corresponding relaxation time: η = as, (3.7) τ π = ηε + P b = abT . (3.8)Again we linearise eqs. (2.3), (2.11) (the magnetic field and the bulk viscous pressure aretaken to be zero) and obtain a set of equations with nine independent variables. Two of theroots are non-hydrodynamic with corresponding dispersion relation is ω = i/τ π . Anotherfour roots are ω = 12 τ π (cid:18) i ± (cid:114) ητ π ε + P k − (cid:19) , (3.9)where each roots are double degenerate, they are known as the shear modes. The remainingthree modes are obtained from a cubic polynomial of the form given in eq. (A.2) with X i ’sare X = iτ π αk , X = − (cid:18) α + 43 b (cid:19) k , X = − iτ π . (3.10)These modes called sound modes as given in ref. [98]. In the small k limit, the dispersionrelation for the sound modes are ω = (cid:40) iτ π , ± k √ α. (3.11)And in the large k limit, the dispersion relations are ω = i αbτ π (4+3 αb ) , ± k (cid:113) α + b + i τ π (4+3 αb ) . (3.12)For details see ref. [98]. We extend our studies to explore the cases in a non-vanishing magnetic field. In thissection, we will investigate the dispersion relation and the speed of sound in a viscous fluidin the presence of a homogeneous magnetic field. We will derive the physical conditionsof causality and stability. To achieve this goal, we carry out a systematic study for thefollowing cases, (i) non-resistive ideal MHD, (ii) viscous MHD with bulk viscosity only, (iii)with shear viscosity only, (iv) with both bulk and shear viscosity.
For an ideal non-resistive fluid in magnetic field the energy-momentum tensor eq. (2.1)takes the following form T µν = (cid:0) ε + P + B (cid:1) u µ u ν − (cid:18) P + B (cid:19) g µν − B b µ b ν . (4.1)– 6 –ere, we define b µ ≡ B µ B , (4.2)which is normalized to b µ b µ = − and orthogonal to u µ i.e, b µ u µ = 0 .Again we consider the similar perturbation as eq. (3.1) around the equilibrium config-uration in the local rest frame ( u µ = (1 , ) ). Ignoring the second and higher-order termsfor the perturbations in ε, P, u µ and B µ , the perturbed energy-momentum tensor can beexpressed as, δ ˜ T µν = (cid:0) ε + P + B (cid:1) ( u µ δ ˜ u ν + δ ˜ u µ u ν ) + (cid:16) δ ˜ ε + δ ˜ P + 2 B δ ˜ B (cid:17) u µ u ν − (cid:16) δ ˜ P + B δ ˜ B (cid:17) g µν − B (cid:16) b µ δ ˜ b ν + δ ˜ b µ b ν (cid:17) − B δ ˜ Bb µ b ν . (4.3)Next, using the above δ ˜ T µν in the energy-momentum conservation equation and notingthat ∂ µ δ ˜ T µν = 0 we get the following four equations, iωδ ˜ ε − ik x hδ ˜ u x − ik y hδ ˜ u y − ik z hδ ˜ u z + ik z B δ ˜ b t + iωB δ ˜ B = 0 , (4.4) − ik x αδ ˜ ε + iωhδ ˜ u x + ik z B δ ˜ b x − ik x B δ ˜ B = 0 , (4.5) − ik y αδ ˜ ε + iωhδ ˜ u y + ik z B δ ˜ b y − ik y B δ ˜ B = 0 , (4.6) − ik z αδ ˜ ε + iωhδ ˜ u z − iωB δ ˜ b t + ik x B δ ˜ b x + ik y B δ ˜ b y + ik z B δ ˜ B = 0 . (4.7)Here, we define h = ε + P + B , and use δ ˜ P = αδ ˜ ε . The relevant Maxwell’s equationswhich govern the evolution of magnetic fields in the fluid is (cid:15) µναβ ∂ β F να = 0 , which can alsobe written in the following form ∂ ν ( B µ u ν − B ν u µ ) = 0 . (4.8)Linearizing the above Maxwell’s equations lead to the following set of equations, ik x B δ ˜ b x + ik y B δ ˜ b y + ik z δ ˜ B = 0 , (4.9) ik z B δ ˜ u x + iωB δ ˜ b x = 0 , (4.10) ik z B δ ˜ u y + iωB δ ˜ b y = 0 , (4.11) − ik x B δ ˜ u x − ik y B δ ˜ u y + iωδ ˜ B = 0 . (4.12)The equations of motion are the energy-momentum conservation equations [Eqs. (4.4)-(4.7)]and the Maxwell’s equations [eq. (4.9)-(4.12)]. However, we notice that eq. (4.9) does notinclude a time-derivative and it is a constraint equation for δ ˜ B , δ ˜ b x and δ ˜ b y . This constraintis consistently propagated to the remaining system of equations of motion. After replacing δ ˜ B by δ ˜ b x and δ ˜ b y , these equations become, Aδ ˜ X T = 0 , (4.13)where, δ ˜ X = (cid:16) δ ˜ ε, δ ˜ u x , δ ˜ u y , δ ˜ u z , δ ˜ b x , δ ˜ b y (cid:17) , (4.14)– 7 –nd A is a × matrix of the following form, A = iω − ik x h − ik y h − ik z ( ε + P ) − i k x k z ωB − i k y k z ωB − iαk x iωh ik z B (cid:16) k x + k z k z (cid:17) i k x k y k z B − iαk y iωh i k x k y k z B ik z B (cid:16) k y + k z k z (cid:17) − iαk z iω ( ε + P ) 0 00 ik z B iωB
00 0 ik z B iωB . (4.15)In deriving the above equations, we have also used the following condition δ ˜ u µ b µ + u µ δ ˜ b µ = 0 ,for changing the variable from δ ˜ b t to δ ˜ u z .Without loss of generality, we consider the magnetic magnetic field b µ along the z -axisand k µ lies in the x - z plane and making an angle θ with the magnetic field, i.e., b µ = (0 , , , ,k µ = ( ω, k sin θ, , k cos θ ) . (4.16)The dispersion relations are obtained by solving det( A ) = 0 . (4.17)which gives us six hydrodynamic modes. Two of these modes are the called Alfvén modeswhose dispersion relation are given as, ω = ± kv A cos θ, v A = B h , (4.18)where, v A is the speed of Alfvén wave. The fluid displacement is perpendicular to thebackground magnetic field in this case and the Alfvén modes can be thought of as the usualvibrational modes that travel down a stretched string.The rest four modes correspond to the magneto-sonic modes with the following disper-sion relations ω = ± v M k, (4.19)where v M is the speed of the magneto-sonic waves, v M = 12 (cid:20) v A + α (cid:0) − v A sin θ (cid:1) ± (cid:113)(cid:8) v A + α (cid:0) − v A sin θ (cid:1)(cid:9) − αv A cos θ (cid:21) . (4.20)The ± sign before the square-root term is for the “fast” and the “slow” magneto-sonic waves,respectively. From eq. (4.20), it is clear that when the propagation of the perturbationis parallel to the background magnetic fields ( θ = 0 ), then the fast magneto-sonic modespropagate with the speed of sound ( c s = √ α ) and the slow magneto-sonic modes propagateswith the speed of the Alfvén waves ( v A ). Whereas for θ = π , the velocity of the slowmagneto-sonic mode becomes zero and the velocity of the fast magneto-sonic wave is v f = v A + α (cid:0) − v A (cid:1) . (4.21)More discussions can be found in refs. [116, 117].– 8 – .2 MHD with bulk viscosity Next, we consider QGP with finite bulk viscosity and a non-zero magnetic field. Usually,the bulk viscosity is proportional to the interaction measure ( ε − P ) /T of the system andhence supposed to be zero for a conformal fluid. Lattice calculation as in refs. [127, 128]shows that the interaction measure has a peak around the QGP to hadronic phase cross-overtemperature T co . For the sake of simplicity, here we take ζ/s = constant in the followingcalculation. The energy-momentum tensor in this case takes the following form, T µν = (cid:0) ε + P + Π + B (cid:1) u µ u ν − (cid:18) P + Π + B (cid:19) g µν − B b µ b ν . (4.22)As before, we can decompose the energy-momentum tensor into two parts: an equilibriumand a perturbation around the equilibrium i.e, T µν = T µν + δ ˜ T µν . (4.23)Here, the perturbed energy-momentum tensor takes the following form, δ ˜ T µν = (cid:0) ε + P + B (cid:1) ( u µ δ ˜ u ν + δ ˜ u µ u ν ) + (cid:16) δ ˜ ε + δ ˜ P + δ ˜Π + 2 B δ ˜ B (cid:17) u µ u ν − (cid:16) δ ˜ P + δ ˜Π + B δ ˜ B (cid:17) g µν − B (cid:16) b µ δ ˜ b ν + δ ˜ b µ b ν (cid:17) − B δ ˜ Bb µ b ν . (4.24)We choose the independent variables as, δ ˜ X = (cid:16) δ ˜ ε, δ ˜ u x , δ ˜ u y , δ ˜ u z , δ ˜ b x , δ ˜ b y , δ ˜Π (cid:17) . (4.25)These conservation equations can be cast into the form Aδ ˜ X T = 0 and setting det A = 0 ,we get, ω − v A k cos θ = 0 , (4.26) ω + X ω + X ω + X ω + X ω + X = 0 , (4.27)where X = − iτ Π αv A k cos θ, X = (cid:18) α + 1 b (cid:19) v A k cos θ, X = iτ Π Y k , X = − (cid:18) Y + 1 b (cid:0) − v A sin θ (cid:1)(cid:19) k , X = − iτ Π , Y = v A + α (cid:0) − v A sin θ (cid:1) . (4.28)The solution of eq. (4.26) gives the following dispersion relation, ω = ± v A k cos θ. (4.29)These two solutions of eq. (4.29) correspond to the Alfvén modes where v A is the Alfvénvelocity. The rest five modes obtained from eq. (4.27) correspond to the magneto-sonic– 9 –odes. Generally, quintic equations cannot be solved algebraically. Fortunately, we findsolutions for some special cases discussed below.For θ = 0 , we find that two modes coincides with the Alfvén modes in eq. (4.29) andthe remaining three modes are obtained from a third-order polynomial of the form given ineq. (A.2), with the coefficients X , X , X given as X = iτ Π αk , X = − (cid:18) α + 1 b (cid:19) k , X = − iτ Π . (4.30)The solutions of this cubic polynomial can be written as ω l = 13 (cid:32) − ξ − ( l − ∆ C − ξ ( l − C − X (cid:33) (4.31)where l = 1 , , , ξ is the primitive cubic root of unity, i.e., ξ = − √− and the othervariables C, ∆ etc. are given in eq. (A.4).For θ = π/ , the eq. (4.27) reduces to a third-order polynomial of the form eq. (A.2),where X i ’s are given as, X = iτ Π v f k , X = − (cid:18) v f + ζhτ Π (cid:19) k , X = − iτ Π , (4.32)where v f is the group velocity for the fast magneto-sonic waves defined in eq.(4.21) and theother two roots are zero.Note that all three roots in eq. (4.31) are complex because the coefficients of eq. (4.30)are complex and hence the phase velocity of any perturbations may contains a damping orgrowing and an oscillatory component. The left panel of Fig. 1 shows the imaginary part ofthe normalised ω as a function of the k/T and the right panel shows the group velocity as afunction of k/T for different values of magnetic fields. Note that the imaginary part of thenon-propagating mode increases and imaginary part of the propagating modes decreaseswhen the magnetic field increases. But it is clear that (cid:61) ( ω ) always lies in the upper half ofthe complex plane for the parameters considered here. This implies that any perturbationwill always decay and the fluid is always stable. Also, for this parameter set-up the groupvelocity v g ≤ , so the wave propagation is causal.If we take the small k limit, Eqs. (4.26) and (4.27) yield the following modes: ω = iτ Π , ± kv A cos θ, ± kv M . (4.33)For this case the group velocity is observed to be same as the velocity for the ideal MHD.We analyse the causality of the system by following ref. [98] where it was shown that toguarantee the causality requires that the asymptotic value of the group velocity should beless than the speed of light. Alfvén mode in eq. (4.26) remains unaffected due to the bulkviscosity and hence always remain causal. For the magneto-sonic waves in the large k limit,– 10 – k / T () / T (a) qB = 0 m qB = 5 m qB = 20 m k / T v g (b) qB = 0 m qB = 5 m qB = 20 m Figure 1 . (Color online) The imaginary parts of the dispersion relations obtain from eq. (4.27)for θ = π with different magnetic fields denoted by different colors. The blue, green and redcolors correspond to B = m π and m π , respectively. In left panel the solid lines are for thepropagating modes ( ω , ) and the dashed lines are for the non-propagating mode ( ω ) . The otherparameters used are a = 0 . , α = 1 / , T = 200 MeV, τ Π =0.985 fm − and kept fixed for all thecurves. θ b θ b Figure 2 . (Color online) Contour plot showing various causal regions, obtained from eq. (4.36),for fast (left panel) and slow (right panel) branches. The red contour is the critical line of causality,denoting v L = 1 . The region above the red line is causal for the fast magneto-sonic waves andacausal below. The slow branch is causal throughout the parameter space. The magnitude ofthe magnetic field has been fixed to qB = 10 m π and the other parameters used are α = 1 / , T = 200 MeV. – 11 –e take the following ansatz ω = v L k in eq. (4.27) and collect terms in the leading-order of k , this yields v L − xv L + y = 0 , (4.34)where, x = v A + (cid:18) α + 1 b (cid:19) (cid:0) − v A sin θ (cid:1) ,y = (cid:18) α + 1 b (cid:19) v A cos θ. (4.35)The velocities v L are v L = 12 (cid:16) x ± (cid:112) x − y (cid:17) . (4.36)Here, we see that unlike the small k limit, at large k the group velocity is affected by thetransport coefficients. In order to have causal propagation, one demands v L ≤ , whichyields a causal parameter-set for the two branches, which correspond to the fast or slowmagneto-sonic modes,fast: (0 < y < ∧ (2 √ y ≤ x < y + 1) , slow: [(0 < y < ∧ ( x ≥ √ y )] ∨ [( y ≥ ∧ ( x > y + 1)] (4.37)Contour plot of the various causal regions is shown in Fig. 2, where b is defined in eq. (3.4).For the fast branch, we find that, although the asymptotic velocities depend on the mag-nitude of the magnetic field and the direction θ , the critical value, i.e., b = 1 . (red solidline), is independent of them. The slow branch is similarly B and θ dependent but moreoveris causal throughout the parameter space. Many theoretical studies indicate that shear viscosity over entropy η/s has a minimum nearthe crossover temperature T co and rises as a function of temperature on both sides of T co ref. [129]. In this section, we consider a fluid with a non-zero shear viscosity but vanishingbulk viscosity. We may expect that the present scenario is applicable for the initial stateof the QGP phase where T ∼ (4 - T co (for top RHIC and LHC energy in Au+Au andPb+Pb collisions) and the bulk viscosity is vanishingly small in that temperature range.The energy-momentum tensor for a fluid with zero bulk and non-zero shear viscosityin a magnetic field takes the following form, T µν = (cid:0) ε + P + B (cid:1) u µ u ν − (cid:18) P + B (cid:19) g µν − B b µ b ν + π µν . (4.38)According to the IS second-order theories of relativistic dissipative fluid dynamics, the space-time evolutions of the shear stress tensor are given by eq. (2.11). For a given perturbationin the fluid, the energy-momentum tensor and the shear stress tensor can be decomposedas, T µν = T µν + δ ˜ T µν , (4.39) π µν = π µν + δ ˜ π µν . (4.40)– 12 –here the perturbed energy-momentum tensor is, δ ˜ T µν = (cid:0) ε + P + B (cid:1) ( u µ δ ˜ u ν + δ ˜ u µ u ν ) + (cid:16) δ ˜ ε + δ ˜ P + 2 B δ ˜ B (cid:17) u µ u ν − (cid:16) δ ˜ P + B δ ˜ B (cid:17) g µν − B (cid:16) b µ δ ˜ b ν + δ ˜ b µ b ν (cid:17) − B δ ˜ Bb µ b ν + δ ˜ π µν . (4.41)As usual, to solve the set of equations eq. (2.11), the conservation of the perturbed energy-momentum tensor (eq. (4.41)), and eqs. (4.9)-(4.12) for obtaining the dispersion relationwe write them in a matrix form Aδ ˜ X T = 0 , (4.42)where, δ ˜ X = ( δ ˜ ε, δ ˜ u x , δ ˜ u y , δ ˜ u z , δ ˜ b x , δ ˜ b y , δ ˜ π xx , δ ˜ π xy , δ ˜ π xz , δ ˜ π yy , δ ˜ π yz ) and the matrix A givenin eq. (B.1). The det( A ) = 0 gives, (1 + iωτ π ) = 0 , (4.43) ω − iτ π ω − (cid:18) v A cos θ + ηhτ π (cid:19) k ω + iτ π k v A cos θ = 0 , (4.44) ω + X ω + X ω + X ω + X ω + X ω + X = 0 , (4.45)where, X = − iτ π , X = − τ π − (cid:20) Y + 13 b (cid:8) − v A (cid:0) θ (cid:1)(cid:9)(cid:21) k , X = iτ π (cid:20) Y + 13 b (cid:8) − v A (cid:0) θ (cid:1)(cid:9)(cid:21) k , X = Y τ π k + (cid:20) α (cid:18) v A cos θ + ηhτ π (cid:19) + 13 b (cid:26) ηhτ π + v A (cid:0) θ (cid:1)(cid:27)(cid:21) k , X = − iτ π (cid:20) α (cid:18) v A cos θ + ηhτ π (cid:19) + v A b (cid:0) θ (cid:1)(cid:21) k , X = − ατ π v A k cos θ, Y = v A + α (cid:0) − v A sin θ (cid:1) . (4.46)From eq. (4.43) we get two non-propagating and stable modes, ω = iτ π . (4.47)Eq. (4.44) is a third-order polynomial equation and the analytic solution for this type ofequations are discussed in appendix A. Eq. (4.45) is a sixth-order polynomial equationwhich is impossible to solve analytically. We can still gain some insight for a few specialcases which are discussed below.For θ = 0 , eq. (4.44) still remains a third-order polynomial equation and the coefficientsof that polynomial can easily be obtained from eq. (4.44) as X = iτ π v A k , X = − (cid:18) v A + ηhτ π (cid:19) k , X = − iτ π (4.48)– 13 –n the other hand, eq. (4.45) can be factorized into two third-order polynomial equations.The coefficients of one of such the third-order polynomial equation are X = iατ π k , X = − (cid:18) α + 43 b (cid:19) k , X = − iτ π , (4.49)whereas the coefficients of the remaining other third-order polynomial equation from eq. (4.45)are same as eq. (4.48)The roots of these third order polynomial equations are discussed in appendix A withthe given X i s. We checked that the dispersion relations obtained from these equations withthe coefficients given in eq. (4.49) are same as the sound mode in ref. [98].For θ = π/ , one root of eq. (4.44) vanish and other two roots are of the form, ω = 12 τ π (cid:32) i ± (cid:114) ητ π h k − (cid:33) . (4.50)From eq. (4.45), one of the root vanish and other two roots are of the form, ω = 12 τ π (cid:18) i ± (cid:114) ητ π ε + P k − (cid:19) . (4.51)The remaining three modes from eq. (4.45), are obtained from a cubic polynomial with X i ’sgiven as: X = iτ π v f k , X = − (cid:18) v f + 4 η hτ π (cid:19) k , X = − iτ π . (4.52)The corresponding roots can be calculated using the formula given in appendix A.The left panel of Fig. 3 shows the dependence of the imaginary parts of ω as a functionof k/T and the right panel shows the group velocity as a function of k/T for differentvalues of magnetic field for θ = 0 . Various lines corresponds to different magnetic fields: qB = 0 (blue lines), qB = 5 m π (green lines), qB = 20 m π (red lines). Fig. 4 shows the samething but for θ = π (eq. (4.52)).In the small k limit the dispersion relations that we get from eqs. (4.43)-(4.45) are ω = iτ π , ± kv A cos θ, ± kv M . (4.53)Note that, the first root have a degeneracy five.In the large k limit we use the ansatz ω = v L k and keep only the leading-order termsin k , then the velocities v L are v L = (cid:40) v A cos θ + ηhτ π , (cid:104) x ± (cid:112) x − y (cid:105) , (4.54)where, x = v A + α (cid:0) − v A sin θ (cid:1) + 13 b (cid:8) − v A (cid:0) θ (cid:1)(cid:9) ,y = α (cid:18) v A cos θ + ηhτ π (cid:19) + 13 b (cid:26) v A (cid:0) θ (cid:1) + 4 ηhτ π (cid:27) . (4.55)– 14 – k / T () / T ( a ) qB = 0 m qB = 5 m qB = 20 m k / T v g ( b ) qB = 0 m qB = 5 m qB = 20 m Figure 3 . (Color online) The left panel shows the imaginary parts of the dispersion relations and theright panel shows the group velocities obtained from a cubic polynomial with the coefficients given ineq. (4.48) for θ = 0 . The other parameters used are a = 0 . , α = 1 / , T = 200 MeV , τ π = 0 . fm − and their values are kept fixed for all the curves. In the left panel, the solid lines are for (cid:61) ( ω , ) which are degenerate. The dash-dotted lines correspond to (cid:61) ( ω ) . k / T () / T (a) qB = 0 m qB = 5 m qB = 20 m k / T v g ( b ) qB = 0 m qB = 5 m qB = 20 m Figure 4 . (Color online) The left panel shows the imaginary parts of ω and the right panel showsthe group velocities obtained from the cubic polynomial with the coefficient given in eq. (4.52) for θ = π . The other parameters used are a = 0 . , α = 1 / , T = 200 MeV , τ π = 0 . fm − and arekept fixed for all the curves.In the left panel, the dash-dotted lines represent (cid:61) ( ω ) and the solidlines are for (cid:61) ( ω , ) which are also degenerate. The asymptotic causality condition for the shear-Alfvén mode can readily be obtained as,shear-Alfvén: v A cos θ + ( ε + P ) hb ≤ , (4.56)where b is defined in eq. (3.8). We observe that in this case the wave velocity and thecausality conditions depend on both the magnitude of the magnetic field and direction of– 15 – .160.32 0.480.640.8 0.96 1.121.28 1.44 1.6 θ b θ b θ b Figure 5 . (Color online) Contour plot showing various causal regions, obtained from eq. (4.54),for Alfvén mode (top left) and the set of fast (top right) and slow modes (bottom) from eq. (4.54).The red contour is the critical line of causality, denoting v L = 1 . The region above the red line iscausal and below it corresponds to the acausal zone. The magnitude of the magnetic field has beenfixed to qB = 10 m π and the other parameters used are α = 1 / , T = 200 MeV. propagation of the perturbation. To explore the inter-dependency we show various causalregions as a function of b and θ as a contour plot in Fig. 5 (top left). We notice thatthe critical value of b at θ = 0 is b c = 1 and this value is independent of magnitude ofthe magnetic field. In the other extreme, i.e. for θ = π/ , the critical value is b c =[1+ B / ( ε + P )] − , i.e., b c decreases with increasing magnetic field. In the limit of vanishingmagnetic field it has been found in ref. [98] that for causal propagation of the shear modes– 16 – ≥ should be satisfied. In the presence of magnetic field we found that, this constraintcan be relaxed to even smaller values of b , given that the waves move obliquely.The causality constraint of the fast and slow waves in eq. (4.54) can be written in theform of (4.37). The simplified expression for the magneto-sonic modes can be written asfast: (0 < y < ∧ (2 √ y ≤ x < y + 1) , slow: [(0 < y < ∧ ( x ≥ √ y )] ∨ [( y ≥ ∧ ( x > y + 1)] . (4.57)We show various causal regions as a function of b and θ as a contour plot in Fig. 5 (topright and bottom). The critical value of b , i.e., b c = 2 (obtained from (4.54)) is independentof the angle θ and the magnitude of magnetic field for the fast magneto-sonic mode. Inthe absence of a magnetic field this value coincides with that obtained for the sound modein ref. [98]. Similarly, the slow magneto-sonic mode yields the critical value of b c = 1 ,independent of the angle θ and the magnitude of magnetic field B . It is still interesting tosee that although the critical values of the fast and slow modes are B and θ independent, theasymptotic velocities are nevertheless dependent. Increasing the magnetic field, increasesthe asymptotic group velocities but the causal region always remains causal no matter howlarge the magnetic field becomes. In this subsection, we investigate the stability and causality of a viscous fluid with finiteshear and the bulk viscosity in a magnetic field.In heavy-ion collisions the initial magnetic field is very large and both shear and bulkviscosities are non-zero for the temperature range achieved in these collisions, hence thepresent case is most relevant to the actual heavy-ion collisions at top RHIC and LHCenergies. The energy-momentum tensor is, T µν = (cid:0) ε + P + Π + B (cid:1) u µ u ν − (cid:18) P + Π + B (cid:19) g µν − B b µ b ν + π µν . (4.58)The small variation of the energy-momentum tensor due to the perturbed fields is, δ ˜ T µν = (cid:0) ε + P + B (cid:1) ( u µ δ ˜ u ν + δ ˜ u µ u ν ) + (cid:16) δ ˜ ε + δ ˜ P + δ ˜Π + 2 B δ ˜ B (cid:17) u µ u ν − (cid:16) δ ˜ P + δ ˜Π + B δ ˜ B (cid:17) g µν − B (cid:16) b µ δ ˜ b ν + δ ˜ b µ b ν (cid:17) − B δ ˜ Bb µ b ν + δ ˜ π µν . (4.59)Following the same procedure, as discussed in the previous two sections, we obtain thedispersion relations for the following independent variables, δ ˜ X = ( δ ˜ ε, δ ˜ u x , δ ˜ u y , δ ˜ u z , δ ˜ b x , δ ˜ b y , δ ˜ π xx , δ ˜ π xy , δ ˜ π xz , δ ˜ π yy , δ ˜ π yz , δ ˜Π) T . (4.60)Following the usual procedure of linearisation we get a × dimensional square matrix A . By setting det A = 0 we have the following equations which subsequently give thedispersion relations, (1 + iωτ π ) = 0 , (4.61) ω − iτ π ω − (cid:18) v A cos θ + ηhτ π (cid:19) k ω + iτ π k v A cos θ = 0 , (4.62) ω + X ω + X ω + X ω + X ω + X ω + X ω + X = 0 , (4.63)– 17 –ere, X = − i (cid:18) τ Π + 2 τ π (cid:19) , X = − τ π (cid:18) τ Π + 1 τ π (cid:19) − (cid:20) v A + (cid:18) α + 1 b (cid:19) (cid:0) − v A sin θ (cid:1) + 13 b (cid:8) − v A (cid:0) θ (cid:1)(cid:9)(cid:21) k , X = iτ π τ Π + i (cid:20)(cid:18) τ Π + 2 τ π (cid:19) Y + 2 b τ π (cid:0) − v A sin θ (cid:1) + 13 b (cid:18) τ π + 1 τ Π (cid:19) (cid:110) − v A (cid:0) θ (cid:1) (cid:111)(cid:21) k , X = (cid:20) b τ π (cid:0) − v A sin θ (cid:1) + 13 bτ π τ Π (cid:8) − v A (cid:0) θ (cid:1)(cid:9) + (cid:18) τ Π τ π + 1 τ π (cid:19) Y (cid:21) k + (cid:20)(cid:18) α + 1 b (cid:19) (cid:18) v A cos θ + ηhτ π (cid:19) + 13 b (cid:26) ( v A (cid:0) θ (cid:1) + η hτ π (cid:27)(cid:21) k , X = − iτ π τ Π Y k − i (cid:34) b τ π (cid:18) v A cos θ + ηhτ π (cid:19) + 13 bτ Π (cid:26) v A (cid:18) τ Π + 1 τ π (cid:19) (cid:0) θ (cid:1) + 4 η hτ π (cid:27) + α (cid:40) (cid:0) − v A (cid:1) b (cid:18) τ Π + 1 τ π (cid:19) + v A (cid:18) τ Π + 2 τ π (cid:19)(cid:41)(cid:35) k , X = − (cid:20) v A b τ π cos θ + v A bτ π τ Π (cid:0) θ (cid:1) + αηhτ π τ Π + αv A (cid:18) τ π τ Π + 1 τ π (cid:19) cos θ (cid:21) k , X = αv A τ π τ Π k cos θ, Y = v A + α (cid:0) − v A sin θ (cid:1) . (4.64)First, we find that eq. (4.61) gives two non-propagating modes of frequency ω = iτ π . Now,the eq. (4.62) is a third-order polynomial and can be solved analytically as discussed previ-ously whereas the eq. (4.63) is a seventh-order polynomial equation and can not be solvedanalytically, therefore we lookout for the solution of these equations for some special casesdiscussed below.For θ = 0 , we obtain two cubic and a single quartic equations. The X i ’s of the twocubic polynomials are same as in eq. (4.48). The dispersion relations for these cases arealready discussed in the previous section, hence we will not repeat them here. The X i ’s forthe fourth-order polynomial equation are X = − i (cid:18) τ π + 1 τ Π (cid:19) , X = − τ π τ Π − (cid:18) α + 1 b + 43 b (cid:19) k ,X = i (cid:20) α (cid:18) τ π + 1 τ Π (cid:19) + 1 b τ π + 43 bτ Π (cid:21) k , X = ατ π τ Π k , (4.65)and the corresponding roots can be calculated using the formula given in Appendix A.For another case, we choose θ = π , this time two of the roots turned out to be zero,and another two roots are the same as eq. (4.51). As before, we call these four modes as– 18 –hear mode. The X i ’s for the fourth-order polynomial equation are X = 1 τ π τ Π v f k ,X = i (cid:20)(cid:18) τ π + 1 τ Π (cid:19) v f + 1 hτ π τ Π (cid:18) ζ + 43 η (cid:19)(cid:21) k ,X = − τ π τ Π − (cid:20) v f + 1 h (cid:18) ζτ Π + 4 η τ π (cid:19)(cid:21) k ,X = − i (cid:18) τ π + 1 τ Π (cid:19) , (4.66)and the corresponding roots can be calculated using the formula given in Appendix A. k / T () / T (a) qB = 0 m qB = 5 m qB = 20 m k / T v g (b) qB = 0 m qB = 5 m qB = 20 m Figure 6 . (Color online) In the left panel (cid:61) ( ω ) /T versus k/T and in the right panel group velocityas a function of k/T are plotted for different magnetic fields for θ = π . v g is obtained from aquartic equation with the coefficients eq. (4.66). The solid lines in the left panel corresponds tothe propagating modes, the dashed lines and the dash-dotted lines correspond the non-propagatingmodes. The other parameters used here are a = a = 0 . , T = 200 MeV , τ Π = 0 . fm − and τ π =0 . fm − and kept constants for all the curves. Note that the imaginary part of the propagating modes (obtained from eq. (4.66)) aredegenerate and hence not shown separately in Fig. 6. The dash-dotted lines in the left panelof Fig. 6 correspond to the non-propagating modes generated due to the bulk viscosity, thisis because in the small k limit they reduce to iτ Π , and in the same logic the dotted linecorresponds to the non-propagating mode due to the shear viscosity. In general, we findthat the (cid:61) ( ω ) is always positive for our set-up. So, for this parameter set, the fluid isalways stable under small perturbation for non-zero bulk and shear viscosity. Also, we noteanother interesting point, when the magnetic field is increased the imaginary part of thepropagating mode tends to zero i.e, the damping of the perturbation diminishes.– 19 –n the small k limit the dispersion relations from eqs. (4.61)-(4.63) become ω = iτ π , iτ Π , ± kv A cos θ, ± kv M , (4.67)here also the first root have degeneracy of five. Similarly, in the large k limit using theansatz ω = v L k , we obtain the asymptotic group velocities v L as: v L = (cid:40) v A cos θ + ηhτ π , (cid:104) x ± (cid:112) x − y (cid:105) , (4.68)where, x = (cid:20) v A + (cid:18) α + 1 b (cid:19) (cid:0) − v A sin θ (cid:1) + 13 b (cid:8) − v A (cid:0) θ (cid:1)(cid:9) (cid:21) ,y = (cid:20)(cid:18) α + 1 b (cid:19) v A cos θ + (cid:18) α + 1 b + 43 b (cid:19) ηhτ π + v A b (cid:0) θ (cid:1)(cid:21) . (4.69)Now we are ready to explore the causality of a fluid in magnetic field. For this, we againcheck whether the asymptotic group velocity has super or sub luminal speed. We foundthat the theory as a whole is causal if the fluid satisfy the following asymptotic causalityconditions for magneto-sonic waves:fast: (0 < y < ∧ (2 √ y ≤ x < y + 1) , slow: [(0 < y < ∧ ( x ≥ √ y )] ∨ [( y ≥ ∧ ( x > y + 1)] . (4.70)From eq. (4.68) we find that a larger magnetic field gives a larger v L , but always remainsub-luminal given b and b are larger than their corresponding critical values (discussedearlier). It is also clear from eq. (4.68) the asymptotic group velocity for non-zero bulk andshear viscosity is larger than the individual shear and bulk viscous cases.In Fig. 7 we show the contour plot of various causal regions as a function of b and θ .The critical line (red line) of the fast magneto-sonic mode (left panel) show that b and b areinversely proportional. On the other hand, the causality condition for the slow magneto-sonic waves is independent of b . The critical value of b for the slow magneto-sonic modeis b c = 1 . The characteristic curves can be seen as the lines along which any information is transportedin the fluid, for example small perturbations, discontinuities, defects or shocks etc travelalong one of these characteristic curves refs. [130–132]. Here we take the effect of non-linearity in the propagation speed which is ignored in the linearisation procedure discussedearlier. Without the loss of generality we consider the ( )-dimensional case with only– 20 – .96 1.081.2 1.321.441.56 1.681.8 1.922.04 b b b b Figure 7 . (Color online) Contour plot showing various causal regions, obtained from eq. (4.70),for fast (left panel) and slow (right panel) branches. The red contour is the critical line of causality,denoting v L = 1 . The region above the red line is causal for the slow magneto-sonic waves andacausal below similarly for the fast magneto-sonic wave right side of red line is causal region andleft side is acausal region. The magnitude of the magnetic field has been fixed to qB = 10 m π andthe other parameters used are α = 1 / , T = 200 MeV. bulk viscosity (shear viscosity can be added in the similar way) and write the energy-momentum conservation equation, Maxwell’s Equation and the IS equation in the standardform for studying the characteristic velocities as, P βmn ∂ β Q n + R m = 0 , (5.1)here, Q n = ( ε, u x , u y , b x , B, Π) and R m = (0 , , , , , Π) . We parametrize the fluid velocityas u µ = (cosh θ, sinh θ cos φ, sinh θ sin φ, and the b µ = (sinh θ, cosh θ cos φ, cosh θ sin φ, .The matrix elements of P tmn , P xmn , P ymn are given in Appendix B.We find the characteristic velocities ( v chx , v chy ) by solving the following equations, det (cid:16) v chx P t − P x (cid:17) = 0 , (5.2) det (cid:16) v chy P t − P y (cid:17) = 0 . (5.3)For simplicity, here we take fluid in the LRF i.e, u µ = (1 , , , and the magnetic filedalong the y -axis b µ = (0 , , , . Then the characteristic velocities are, v chx = ± (cid:115) B + α ( ε + P + Π)( h + Π) + ζτ Π ( h + Π) , (5.4)– 21 – chy = ± B √ ( h +Π) , ± (cid:113) α + ζτ Π ( ε + P +Π) , (5.5)here h = ε + P + B and the other roots are zero. The characteristic velocities obtain ineqs. (5.4), (5.5) are same with the eq. (4.36) for θ = π and θ = 0 , respectively provide Π = 0 . So we conclude that, the asymptotic group velocity obtained by linearizing theMHD-IS equations is same as the characteristic velocities.
So far all the results we discussed were obtained for viscous fluid in a magnetic field withinthe frame-work of the IS theory. However, in a recent work ref. [118] a modified form ofthe IS theory due to the magnetic field was derived. The modified theory which we callas NRMHD-IS from now on shows that the relaxation equation for the shear-stress tensorcontains additional terms, here we neglected most of the terms and only keep the termwhich couples magnetic field and the shear viscosity. The simplified NRMHD-IS equationtakes the following form τ π ddτ π <µν> + π µν = 2 ησ µν − δ πB Bb αβ ∆ µνακ g λβ π κλ . (6.1)Where δ πB is a new coefficient appearing only due to the magnetic field and b αβ = − (cid:15) αβγδ u γ b δ is an anti-symmetric tensor which satisfy b µν u ν = b µν b ν = 0 . The rank-fourtraceless and symmetric projection operator is defined as ∆ µνακ = (∆ µα ∆ νκ + ∆ νκ ∆ µα ) − ∆ µν ∆ ακ .Before proceeding further, a few comments on the NRMHD-IS equations are in order.It is well known that in the presence of a magnetic field, the transport coefficients split intoseveral components, namely three bulk components and five shear components refs. [50, 116–118]. The information of these anisotropic transport coefficients are hidden inside thenew coupling terms of the modified IS theory eq. (6.1). Note that the first-order termson the right-hand sides are proportional to the usual shear-viscosity. These terms canbe combined with the first-order terms on the left-hand side and, after inversion of therespective coefficient matrices, will lead to the various anisotropic transport coefficients. Onthe other hand, when solving the full second-order equations of the modified IS theory, onedoes not need to replace the standard viscosity with the anisotropic transport coefficients,since the effect of the magnetic field, is already taken into account by the new terms in theseequations. Regarding modified second-order theory with finite bulk viscosity, we would liketo mention that, there is still no existing theory that yields three distinct bulk componentsin Navier-Stokes limit (for details see ref. [118]) and it is still an open issue.The last term of eq. (6.1) is the only non-trivial term added to the conventional IStheory for which we already discussed the results in previous sections. So, here we onlyconsider the last term of eq. (6.1) and calculate the corresponding correction to the oldresults. – 22 –irst, we add a perturbation to the new term which contributes to δ ˜ π µν , δ ˜ I µν = δ πB B b αβ ∆ µνακ g λβ δ ˜ π κλ . (6.2)While calculating eq. (6.2) we use the fact that in the local rest frame the unperturbedshear stress tensor vanishes i.e., π µν = 0 , and as a consequence δ ˜ B, δ ˜ b µν terms are absentin eq. (6.2). For later use we define the projection of a four-vector A µ as A <µ> = ∆ µν A ν ,which is orthogonal to u µ .Using these new definitions we write the eq. (6.2) in a more simplified form as, δ ˜ I µν = δ πB B (cid:16) b <µ>λ δ ˜ π <ν>λ + b <ν>λ δ ˜ π <µ>λ (cid:17) − δ πB B µν b <κ>λ δ ˜ π <κ>λ . (6.3)In the LRF, the following components of the b µν are found to be non-zero b xy = 1 , b yx = − , b xy = − , b yx = 1 , b xy = b xy = 1 and b yx = b yx = − , where b µ taken as (0 , , , .For the (3 + 1) dimensional case there are five independent equations for the shear stressaccording to the IS theory. For each five equations there are corresponding componentsof the δ ˜ I µν which for our case are δ ˜ I xx = − δ πB B δ ˜ π xy , δ ˜ I xy = δ πB B ( δ ˜ π xx − δ ˜ π yy ) , δ ˜ I xz = − δ πB B δ ˜ π yz , δ ˜ I yy = δ πB B δ ˜ π yx and δ ˜ I yz = δ πB B δ ˜ π xz . We include these newterms to the corresponding IS equations that we previously derived in section (4.3). Herealso we get a × matrix. As usual, we derive the dispersion relations from det( A ) = 0 which is a eleventh-order polynomial equation. Since finding the analytic solution of thispolynomial equation is not possible, here we investigate some special cases.In the hydrodynamical-limit i.e, in the small k limit we get the following modes ω = iτ π , iτ π ± B δ πB τ π , iτ π ± B δ πB τ π , ± v A k cos θ, ± v M k. (6.4)Note that the frequency of a few non-hydrodynamic modes are changed due to the newcoupling terms appearing in the NRMHD-IS theory.For the large k limit we use the ansatz ω = v L k and take only the leading order termsin k which yields the following velocities v L = (cid:40) v A cos θ + ηhτ π , (cid:104) x ± (cid:112) x − y (cid:105) , (6.5)here, x = v A + α (cid:0) − v A sin θ (cid:1) + 13 b (cid:8) − v A (cid:0) θ + 4 sin θ (cid:1)(cid:9) ,y = α (cid:18) v A cos θ + ηhτ π (cid:19) + 13 b (cid:26) v A (cid:0) θ + 3 sin θ (cid:1) + 4 ηhτ π (cid:27) , (6.6)and the remaining roots are zero. Since the causality of the fluid depends on the asymptoticcausality condition which here is given in eq. (6.5) and turned out to be the same aseq. (4.54). So it is clear that the causality condition remains same as eq. (4.56) whereasthe dispersion relations gets modified. – 23 – Conclusions
The current work goes beyond the previous results of refs. [116, 117] which used first orderviscous MHD. As is well known the first order gradient terms in the energy-momentumtensor breaks causality, which is reflected from the existence of the superluminal mode.This prohibits the application of viscous MHD in relativistic systems and it is necessaryto have rigorous treatment which the present work aims. The remedy is to go beyondthe first viscous corrections in hydrodynamics, and to include second order terms as well.We studied here the stability and causality of the relativistic dissipative fluid dynamicswithin the framework of the standard and modified IS theories in the magnetic field. Bylinearising the energy-momentum conservation equation, relaxation equations for viscousstresses, and the Maxwell’s equations we obtain the dispersion relations for various cases.In the absence of viscous stresses, the dispersion relation yields the well-known collectivemodes namely the Alfvén, slow and fast magnetosonic modes. For the bulk viscous casethe Alfvén mode turned out to be independent of the bulk viscosity. The asymptoticcausality constraint for the magneto-sonic modes is independent of the magnetic field andthe angle of propagation. For the fast mode, the causality condition is the same as thatpreviously derived in ref. [99] in the absence of magnetic field. The slow mode, on the otherhand, remains causal throughout the parameter space. We also derived the causality boundwith finite bulk viscosity using the full non-linear set of the equation using the method ofcharacteristics and found that it agrees with the result obtained using small perturbations.In the presence of shear viscosity, the causality constraint for the two magnetosonic modesis found to be independent of the magnetic field and the angle of propagation. Shear-Alfvénmodes, on the other hand, do depend on them. We found that the causality constraint ischanged in presence of a magnetic field. For the modified IS theory in the presence ofshear viscosity, new non-hydrodynamic modes emerge but the causality constraint remainsunaltered. Finally, in the presence of both shear and bulk viscosity, we have deduced thecausal region of parameter space.There are many possible directions for future work, namely, the study of causalitybounds:(i) in resistive, second-order dissipative MHD where the electric field is non-zeroand contributes in the equations of motion. ref. [119] (ii) theories which have spin degreesof freedom allows to include effects of polarization and magnetization ref. [133]. These andother interesting questions will be addressed in the future.
Acknowledgments
RB and VR acknowledge financial support from the DST Inspire faculty research grant(IFA-16-PH-167), India. AD, NH, and VR are supported by the DAE, Govt. of India.NH is also supported in part by the SERB-SRG under Grant No. SRG/2019/001680. Wewould also like to thank Ze-yu Zhai for pointing out some typographical errors.– 24 –
Solutions of dispersion relations
In general, the hydrodynamic dispersion relations arise as solutions to P n ( X , X , ..., X n − ) = 0 , (A.1)where P = det A , is a n th order polynomial obtained from the determinant of matrix A afterlinearising the MHD equations. In this appendix, we enlist the roots of certain polynomials P n that we will encounter throughout this work. For n = 3 , the polynomial P is of theform ω + X ω + X ω + X = 0 , (A.2)and the corresponding roots are given as ω k ( X , X , X ) = 13 (cid:32) − ξ − ( k − ∆ C − ξ ( k − C − X (cid:33) . (A.3)Here k = 1 , , , ξ is the primitive cubic root of unity, i.e., ξ = − √− and the othervariables are defined C = (cid:118)(cid:117)(cid:117)(cid:116) ∆ + (cid:113)(cid:0) ∆ − (cid:1) , ∆ = X − X , ∆ = 2 X − X X + 27 X . (A.4)Similarly, for n = 4 , the polynomial P is of the form ω + X ω + X ω + X ω + X = 0 , (A.5)and the corresponding roots are given as ω , ( X , X , X , X ) = ± (cid:114)(cid:16) − p + qS − S (cid:17) − S − X ω , ( X , X , X , X ) = ± (cid:114)(cid:16) − p − qS − S (cid:17) + S − X , (A.6)where p = 18 (cid:0) X − X (cid:1) ,q = 18 (cid:0) X − X X + 8 X (cid:1) ,S = 12 (cid:115)(cid:18) (cid:18) ∆ Q + Q (cid:19) + 112 (cid:0) X − X (cid:1)(cid:19) ,Q = (cid:118)(cid:117)(cid:117)(cid:116) ∆ + (cid:113)(cid:0) ∆ − (cid:1) , ∆ = X + 12 X − X X , ∆ = 2 X − X X − X X X + 27 (cid:0) X + X X (cid:1) . (A.7)– 25 – Details of matrix A defined in section 4.3 and the characteristic ve-locities By linearising the energy-momentum conservation equations, Maxwell’s equations and ISequation for shear viscosity, we write these in the matrix form as eq. (4.42). Here the formof matrix A is iω − ik x h − ik y h − ik z ( ε + P ) − i k x k z ωB − i k y k z ωB − iαk x iωh ik z B (cid:16) k x + k z k z (cid:17) i k x k y k z B − ik x − ik y − ik z − iαk y iωh i k x k y k z B ik z B (cid:16) k y + k z k z (cid:17) − ik x − ik y − ik z − iαk z iω ( ε + P ) 0 0 ik z − ik x ik z − ik y iB k z iωB iB k z iωB − iηk x iηk y iηk z f − iηk y − iηk x f − iηk z − iηk x f iηk x − iηk y iηk z f
00 0 − iηk z − iηk y f , (B.1)where f = 1 + iωτ π . Similarly we can write the matrix A for the modified IS theory, alsofor both the bulk and shear viscosity case.In section 5 we derive the characteristic velocities for the MHD with the bulk viscosityonly. For simplicity we consider ( )-dimensional case and write the energy-momentumconservation equation, Maxwell’s equation and the IS equation for bulk in the form ofeq. (5.1). The matrix elements of P tmn are P t = (1 + α ) cosh θ − α, P t = 2 ( ε + P + Π) sinh θ cos φ,P t = 2 ( ε + P + Π) sinh θ sin φ P t = B,P t = sinh θ, P t = (1 + α ) sinh θ cosh θ cos φ,P t = sin(2 φ )2 cosh θ (cid:2) ( ε + P + Π) sinh θ − B (cid:3) , P t = − B sinh θ,P t = sinh θ cosh θ cos φ, P t = (1 + α ) sinh θ cosh θ sin φ,P t = B sinh θ cot φ, P t = sinh θ cosh θ sin φ,P t = B sinh θ, P t = − B cosh θ,P t = − cos φ, P t = − B sinh θ cot φ,P t = B cosh θ cot φ, P t = − sin φ,P t = ζ tanh θ cos φ, P t = ζ tanh θ sin φ,P t = τ Π cosh θ, – 26 – t = 12 cosh θ (cid:2) ε + P + Π) (cid:0) cosh θ + sinh θ cos φ (cid:1) + B (cid:8) cosh(2 θ ) − cos(2 φ ) (cid:9)(cid:3) ,P t = cot φ θ (cid:2) ε + P + Π) sinh θ sin φ − B (cid:8) (cosh(2 θ ) − cos(2 φ ) (cid:9)(cid:3) ,P t = cos φ cosh θ (cid:2) ( ε + P + Π) (cid:0) cosh θ + sinh θ sin φ (cid:1) + B (cid:3) , The matrix elements of P xmn are P x = (1 + α ) sinh θ cosh θ cos φ, P x = sin(2 φ )2 cosh θ (cid:2) ( ε + P + Π) sinh θ − B (cid:3) ,P x = − B sinh θ, P x = sinh θ cosh θ cos φ,P x = (1 + α ) sinh θ cos φ + α, P x = 2 ( h + Π) sinh θ cos φ,P x = − B cosh θ cos φ, P x = − B cos(2 φ ) ,P x = 1 + sinh θ cos φ, P x = (1 + α ) sinh θ sin φ cos φ,P x = ( ε + P + Π) sinh θ cos φ, P x = B cosh θ cos(2 φ ) csc φ,P x = − B sin(2 φ ) , P x = sinh θ sin φ cos φ,P x = − B sin φ cosh θ (cid:2) θ csc φ (cid:3) , P x = B cosh θ cos φ,P x = B sinh θ csc φ, P x = ζ,P x = τ Π sinh θ cos φ,P x = 12 cosh θ (cid:2) ε + P + Π) (cid:0) cosh θ + sinh θ cos φ (cid:1) + B (cid:8) cosh(2 θ ) − cos(2 φ ) (cid:9)(cid:3) ,P x = ( ε + P + Π) sinh θ sin φ − B sinh θ cos(2 φ ) csc φ, The matrix elements of P ymn are P y = (1 + α ) sinh θ cosh θ sin φ, P y = B sinh θ cot φ,P y = sinh θ cosh θ sin φ, P y = (1 + α ) sinh θ sin φ cos φ,P y = ( ε + P + Π) sinh θ cos φ P y = B cosh θ cos(2 φ ) csc φ,P y = − B sin(2 φ ) , P y = sinh θ sin φ cos φ,P y = (1 + α ) sinh θ sin φ + α, P y = − B sinh θ cos φ,P y = 2 ( ε + P + Π) sinh θ sin φ, P y = 2 B cosh θ cos φ,P y = B cos(2 φ ) , P y = 1 + sinh θ sin φ,P y = B sin φ cosh θ (cid:2) θ csc φ (cid:3) , P y = − B cosh θ cos φ,P y = − B sinh θ csc φ, P y = ζ,P y = τ Π sinh θ sin φ. – 27 – y = cot φ θ (cid:2) ε + P + Π) sinh θ sin φ − B (cid:8) (cosh(2 θ ) − cos(2 φ ) (cid:9)(cid:3) ,P y = cos φ cosh θ (cid:2) ( ε + P + Π) (cid:0) cosh θ + sinh θ sin φ (cid:1) + B (cid:3) ,P y = ( ε + P + Π) sinh θ sin φ − B sinh θ cos(2 φ ) csc φ, and all the other coefficients are zero. References [1] A. Bzdak and V. Skokov,
Event-by-event fluctuations of magnetic and electric fields inheavy ion collisions , Phys. Lett.
B710 (2012) 171–174, [ arXiv:1111.1949 ].[2] W.-T. Deng and X.-G. Huang,
Event-by-event generation of electromagnetic fields inheavy-ion collisions , Phys. Rev.
C85 (2012) 044907, [ arXiv:1201.5108 ].[3] K. Tuchin,
Particle production in strong electromagnetic fields in relativistic heavy-ioncollisions , Adv. High Energy Phys. (2013) 490495, [ arXiv:1301.0099 ].[4] V. Roy and S. Pu,
Event-by-event distribution of magnetic field energy over initial fluidenergy density in √ s NN = 200 GeV Au-Au collisions , Phys. Rev. C92 (2015) 064902,[ arXiv:1508.03761 ].[5] H. Li, X.-l. Sheng, and Q. Wang,
Electromagnetic fields with electric and chiral magneticconductivities in heavy ion collisions , Phys. Rev.
C94 (2016), no. 4 044903,[ arXiv:1602.02223 ].[6] D. Kharzeev,
Parity violation in hot QCD: Why it can happen, and how to look for it , Phys.Lett.
B633 (2006) 260–264, [ hep-ph/0406125 ].[7] D. E. Kharzeev, L. D. McLerran, and H. J. Warringa,
The Effects of topological chargechange in heavy ion collisions: ’Event by event P and CP violation’ , Nucl. Phys.
A803 (2008) 227–253, [ arXiv:0711.0950 ].[8] K. Fukushima, D. E. Kharzeev, and H. J. Warringa,
The Chiral Magnetic Effect , Phys. Rev.
D78 (2008) 074033, [ arXiv:0808.3382 ].[9] X.-G. Huang and J. Liao,
Axial Current Generation from Electric Field: Chiral ElectricSeparation Effect , Phys. Rev. Lett. (2013), no. 23 232302, [ arXiv:1303.7192 ].[10] S. Pu, S.-Y. Wu, and D.-L. Yang,
Holographic Chiral Electric Separation Effect , Phys. Rev.
D89 (2014), no. 8 085024, [ arXiv:1401.6972 ].[11] S. Pu, S.-Y. Wu, and D.-L. Yang,
Chiral Hall Effect and Chiral Electric Waves , Phys. Rev.
D91 (2015), no. 2 025011, [ arXiv:1407.3168 ].[12] Y. Jiang, X.-G. Huang, and J. Liao,
Chiral electric separation effect in the quark-gluonplasma , Phys. Rev. D (2015), no. 4 045001, [ arXiv:1409.6395 ].[13] D. Satow, Nonlinear electromagnetic response in quark-gluon plasma , Phys. Rev. D (2014), no. 3 034018, [ arXiv:1406.7032 ].[14] J.-W. Chen, T. Ishii, S. Pu, and N. Yamamoto, Nonlinear Chiral Transport Phenomena ,Phys. Rev.
D93 (2016), no. 12 125023, [ arXiv:1603.03620 ].[15] S. Ebihara, K. Fukushima, and S. Pu,
Boost invariant formulation of the chiral kinetictheory , Phys. Rev.
D96 (2017), no. 1 016016, [ arXiv:1705.08611 ]. – 28 –
16] M. A. Stephanov and Y. Yin,
Chiral Kinetic Theory , Phys. Rev. Lett. (2012) 162001,[ arXiv:1207.0747 ].[17] D. T. Son and N. Yamamoto,
Kinetic theory with Berry curvature from quantum fieldtheories , Phys. Rev.
D87 (2013), no. 8 085016, [ arXiv:1210.8158 ].[18] J.-W. Chen, S. Pu, Q. Wang, and X.-N. Wang,
Berry Curvature and Four-DimensionalMonopoles in the Relativistic Chiral Kinetic Equation , Phys. Rev. Lett. (2013), no. 26262301, [ arXiv:1210.8312 ].[19] C. Manuel and J. M. Torres-Rincon,
Kinetic theory of chiral relativistic plasmas and energydensity of their gauge collective excitations , Phys. Rev.
D89 (2014), no. 9 096002,[ arXiv:1312.1158 ].[20] C. Manuel and J. M. Torres-Rincon,
Chiral transport equation from the quantum DiracHamiltonian and the on-shell effective field theory , Phys. Rev.
D90 (2014), no. 7 076007,[ arXiv:1404.6409 ].[21] J.-Y. Chen, D. T. Son, M. A. Stephanov, H.-U. Yee, and Y. Yin,
Lorentz Invariance inChiral Kinetic Theory , Phys. Rev. Lett. (2014), no. 18 182302, [ arXiv:1404.5963 ].[22] J.-Y. Chen, D. T. Son, and M. A. Stephanov,
Collisions in Chiral Kinetic Theory , Phys.Rev. Lett. (2015), no. 2 021601, [ arXiv:1502.06966 ].[23] Y. Hidaka, S. Pu, and D.-L. Yang,
Relativistic Chiral Kinetic Theory from Quantum FieldTheories , Phys. Rev.
D95 (2017), no. 9 091901, [ arXiv:1612.04630 ].[24] N. Mueller and R. Venugopalan,
The chiral anomaly, Berry’s phase and chiral kinetictheory, from world-lines in quantum field theory , Phys. Rev.
D97 (2018), no. 5 051901,[ arXiv:1701.03331 ].[25] Y. Hidaka, S. Pu, and D.-L. Yang,
Nonlinear Responses of Chiral Fluids from KineticTheory , Phys. Rev.
D97 (2018), no. 1 016004, [ arXiv:1710.00278 ].[26] Y. Hidaka, S. Pu, and D.-L. Yang,
Non-Equilibrium Quantum Transport of Chiral Fluidsfrom Kinetic Theory , Nucl. Phys.
A982 (2019) 547–550, [ arXiv:1807.05018 ].[27] A. Huang, S. Shi, Y. Jiang, J. Liao, and P. Zhuang,
Complete and Consistent ChiralTransport from Wigner Function Formalism , Phys. Rev.
D98 (2018), no. 3 036010,[ arXiv:1801.03640 ].[28] J.-H. Gao, Z.-T. Liang, Q. Wang, and X.-N. Wang,
Disentangling covariant Wignerfunctions for chiral fermions , Phys. Rev.
D98 (2018), no. 3 036019, [ arXiv:1802.06216 ].[29] Y.-C. Liu, L.-L. Gao, K. Mameda, and X.-G. Huang,
Chiral kinetic theory in curvedspacetime , Phys. Rev.
D99 (2019), no. 8 085014, [ arXiv:1812.10127 ].[30] S. Lin and A. Shukla,
Chiral Kinetic Theory from Effective Field Theory Revisited , JHEP (2019) 060, [ arXiv:1901.01528 ].[31] S. Lin and L. Yang, Chiral kinetic theory from Landau level basis , Phys. Rev. D (2020),no. 3 034006, [ arXiv:1909.11514 ].[32] Y. Sun, C. M. Ko, and F. Li,
Anomalous transport model study of chiral magnetic effects inheavy ion collisions , Phys. Rev.
C94 (2016), no. 4 045204, [ arXiv:1606.05627 ].[33] Y. Sun and C. M. Ko,
Chiral vortical and magnetic effects in the anomalous transportmodel , Phys. Rev.
C95 (2017), no. 3 034909, [ arXiv:1612.02408 ]. – 29 –
34] Y. Sun and C. M. Ko, Λ hyperon polarization in relativistic heavy ion collisions from achiral kinetic approach , Phys. Rev. C96 (Aug, 2017) 024906, [ arXiv:1706.09467 ].[35] Y. Sun and C. M. Ko,
Chiral kinetic approach to the chiral magnetic effect in isobariccollisions , Phys. Rev.
C98 (2018), no. 1 014911, [ arXiv:1803.06043 ].[36] Y. Sun and C. M. Ko,
Azimuthal angle dependence of the longitudinal spin polarization inrelativistic heavy ion collisions , Phys. Rev.
C99 (2019), no. 1 011903, [ arXiv:1810.10359 ].[37] W.-H. Zhou and J. Xu,
Simulating the Chiral Magnetic Wave in a Box System , Phys. Rev.
C98 (2018), no. 4 044904, [ arXiv:1810.01030 ].[38] W.-H. Zhou and J. Xu,
Simulating chiral anomalies with spin dynamics , Phys. Lett.
B798 (2019) 134932, [ arXiv:1904.01834 ].[39] S. Y. F. Liu, Y. Sun, and C. M. Ko,
Spin polarizations in a covariant angular momentumconserved chiral transport model , arXiv:1910.06774 .[40] K. Fukushima, D. E. Kharzeev, and H. J. Warringa, Real-time dynamics of the ChiralMagnetic Effect , Phys. Rev. Lett. (2010) 212001, [ arXiv:1002.2495 ].[41] P. Copinger, K. Fukushima, and S. Pu,
Axial Ward identity and the Schwinger mechanism– Applications to the real-time chiral magnetic effect and condensates , Phys. Rev. Lett. (2018), no. 26 261602, [ arXiv:1807.04416 ].[42] X.-L. Sheng, R.-H. Fang, Q. Wang, and D. H. Rischke,
Wigner function and pair productionin parallel electric and magnetic fields , Phys. Rev.
D99 (2019), no. 5 056004,[ arXiv:1812.01146 ].[43] B. Feng, D.-f. Hou, H. Liu, H.-c. Ren, P.-p. Wu, and Y. Wu,
Chiral Magnetic Effect in aLattice Model , Phys. Rev. D (2017), no. 11 114023, [ arXiv:1702.07980 ].[44] Y. Wu, D. Hou, and H.-c. Ren, Field theoretic perspectives of the Wigner functionformulation of the chiral magnetic effect , Phys. Rev. D (2017), no. 9 096015,[ arXiv:1601.06520 ].[45] S. Lin and L. Yang, Mass correction to chiral vortical effect and chiral separation effect ,Phys. Rev. D (2018), no. 11 114022, [ arXiv:1810.02979 ].[46] M. Horvath, D. Hou, J. Liao, and H.-c. Ren, Chiral magnetic response to arbitrary axialimbalance , Phys. Rev. D (2020), no. 7 076026, [ arXiv:1911.00933 ].[47] B. Feng, D.-F. Hou, and H.-C. Ren,
QED radiative corrections to chiral magnetic effect ,Phys. Rev. D (2019), no. 3 036010, [ arXiv:1810.05954 ].[48] D.-f. Hou and S. Lin, Fluctuation and Dissipation of Axial Charge from Massive Quarks ,Phys. Rev. D (2018), no. 5 054014, [ arXiv:1712.08429 ].[49] S. Lin, L. Yan, and G.-R. Liang, Axial Charge Fluctuation and Chiral Magnetic Effect fromStochastic Hydrodynamics , Phys. Rev. C (2018), no. 1 014903, [ arXiv:1802.04941 ].[50] A. Dash, S. Samanta, J. Dey, U. Gangopadhyaya, S. Ghosh, and V. Roy, Anisotropictransport properties of Hadron Resonance Gas in magnetic field , arXiv:2002.08781 .[51] M. Kurian, V. Chandra, and S. K. Das, Impact of longitudinal bulk viscous effects to heavyquark transport in a strongly magnetized hot QCD medium , Phys. Rev. D (2020), no. 9094024, [ arXiv:2002.03325 ]. – 30 –
52] V. Voronyuk, V. Toneev, W. Cassing, E. Bratkovskaya, V. Konchakovski, and S. Voloshin, (Electro-)Magnetic field evolution in relativistic heavy-ion collisions , Phys. Rev. C (2011) 054911, [ arXiv:1103.4239 ].[53] M. Greif, I. Bouras, C. Greiner, and Z. Xu, Electric conductivity of the quark-gluon plasmainvestigated using a perturbative QCD based parton cascade , Phys. Rev. D (2014), no. 9094014, [ arXiv:1408.7049 ].[54] D. E. Kharzeev, J. Liao, S. A. Voloshin, and G. Wang, Chiral magnetic and vortical effectsin high energy nuclear collisions: A status report , Prog. Part. Nucl. Phys. (2016) 1–28,[ arXiv:1511.04050 ].[55] J. Liao, Anomalous transport effects and possible environmental symmetry âĂŸviolationâĂŹin heavy-ion collisions , Pramana (2015), no. 5 901–926, [ arXiv:1401.2500 ].[56] V. A. Miransky and I. A. Shovkovy, Quantum field theory in a magnetic field: Fromquantum chromodynamics to graphene and Dirac semimetals , Phys. Rept. (2015)1–209, [ arXiv:1503.00732 ].[57] X.-G. Huang,
Electromagnetic fields and anomalous transports in heavy-ion collisions — Apedagogical review , Rept. Prog. Phys. (2016), no. 7 076302, [ arXiv:1509.04073 ].[58] K. Fukushima, Extreme matter in electromagnetic fields and rotation , Prog. Part. Nucl.Phys. (2019) 167–199, [ arXiv:1812.08886 ].[59] A. Bzdak, S. Esumi, V. Koch, J. Liao, M. Stephanov, and N. Xu,
Mapping the Phases ofQuantum Chromodynamics with Beam Energy Scan , arXiv:1906.00936 .[60] J. Zhao and F. Wang, Experimental searches for the chiral magnetic effect in heavy-ioncollisions , Prog. Part. Nucl. Phys. (2019) 200–236, [ arXiv:1906.11413 ].[61] Y.-C. Liu and X.-G. Huang,
Anomalous chiral transports and spin polarization in heavy-ioncollisions , Nucl. Sci. Tech. (2020), no. 6 56, [ arXiv:2003.12482 ].[62] J.-H. Gao, G.-L. Ma, S. Pu, and Q. Wang, Recent developments in chiral and spinpolarization effects in heavy ion collisions , arXiv:2005.10432 .[63] STAR
Collaboration, B. Abelev et al.,
Azimuthal Charged-Particle Correlations andPossible Local Strong Parity Violation , Phys. Rev. Lett. (2009) 251601,[ arXiv:0909.1739 ].[64]
STAR
Collaboration, B. Abelev et al.,
Observation of charge-dependent azimuthalcorrelations and possible local strong parity violation in heavy ion collisions , Phys. Rev. C (2010) 054908, [ arXiv:0909.1717 ].[65] ALICE
Collaboration, B. Abelev et al.,
Charge separation relative to the reaction plane inPb-Pb collisions at √ s NN = 2 . TeV , Phys. Rev. Lett. (2013), no. 1 012301,[ arXiv:1207.0900 ].[66]
CMS
Collaboration, V. Khachatryan et al.,
Observation of charge-dependent azimuthalcorrelations in p -Pb collisions and its implication for the search for the chiral magneticeffect , Phys. Rev. Lett. (2017), no. 12 122301, [ arXiv:1610.00263 ].[67] CMS
Collaboration, A. M. Sirunyan et al.,
Constraints on the chiral magnetic effect usingcharge-dependent azimuthal correlations in p Pb and PbPb collisions at the CERN LargeHadron Collider , Phys. Rev. C97 (2018), no. 4 044912, [ arXiv:1708.01602 ]. – 31 – CMS
Collaboration, A. M. Sirunyan et al.,
Probing the chiral magnetic wave in pP b andPbPb collisions at √ s NN =5.02TeV using charge-dependent azimuthal anisotropies , Phys.Rev. C100 (2019), no. 6 064908, [ arXiv:1708.08901 ].[69] C. Shen, S. A. Bass, T. Hirano, P. Huovinen, Z. Qiu, H. Song, and U. Heinz,
The QGPshear viscosity: Elusive goal or just around the corner? , J. Phys.
G38 (2011) 124045,[ arXiv:1106.6350 ].[70] M. Luzum and P. Romatschke,
Conformal Relativistic Viscous Hydrodynamics:Applications to RHIC results at s(NN)**(1/2) = 200-GeV , Phys. Rev.
C78 (2008) 034915,[ arXiv:0804.4015 ]. [Erratum: Phys. Rev.C79,039903(2009)].[71] U. Heinz and R. Snellings,
Collective flow and viscosity in relativistic heavy-ion collisions ,Ann. Rev. Nucl. Part. Sci. (2013) 123–151, [ arXiv:1301.2826 ].[72] P. Bozek and I. Wyskiel-Piekarska, Particle spectra in Pb-Pb collisions at sqrtS N N = 2 . TeV , Phys. Rev.
C85 (2012) 064915, [ arXiv:1203.6513 ].[73] V. Roy, A. K. Chaudhuri, and B. Mohanty,
Comparison of results from a 2+1D relativisticviscous hydrodynamic model to elliptic and hexadecapole flow of charged hadrons measuredin Au-Au collisions at √ s NN = 200 GeV , Phys. Rev. C86 (2012) 014902,[ arXiv:1204.2347 ].[74] U. Heinz, C. Shen, and H. Song,
The viscosity of quark-gluon plasma at RHIC and theLHC , AIP Conf. Proc. (2012), no. 1 766–770, [ arXiv:1108.5323 ].[75] H. Niemi, G. S. Denicol, P. Huovinen, E. Molnar, and D. H. Rischke,
Influence of atemperature-dependent shear viscosity on the azimuthal asymmetries of transversemomentum spectra in ultrarelativistic heavy-ion collisions , Phys. Rev.
C86 (2012) 014909,[ arXiv:1203.2452 ].[76] B. Schenke, S. Jeon, and C. Gale,
Higher flow harmonics from (3+1)D event-by-eventviscous hydrodynamics , Phys. Rev.
C85 (2012) 024901, [ arXiv:1109.6289 ].[77] S. Gupta,
The Electrical conductivity and soft photon emissivity of the QCD plasma , Phys.Lett. B (2004) 57–62, [ hep-lat/0301006 ].[78] G. Aarts, C. Allton, A. Amato, P. Giudice, S. Hands, and J.-I. Skullerud,
Electricalconductivity and charge diffusion in thermal QCD from the lattice , JHEP (2015) 186,[ arXiv:1412.6411 ].[79] A. Amato, G. Aarts, C. Allton, P. Giudice, S. Hands, and J.-I. Skullerud, Electricalconductivity of the quark-gluon plasma across the deconfinement transition , Phys. Rev. Lett. (2013), no. 17 172001, [ arXiv:1307.6763 ].[80] P. B. Arnold, G. D. Moore, and L. G. Yaffe,
Transport coefficients in high temperaturegauge theories. 2. Beyond leading log , JHEP (2003) 051, [ hep-ph/0302165 ].[81] J.-W. Chen, Y.-F. Liu, S. Pu, Y.-K. Song, and Q. Wang, Negative off-diagonalconductivities in a weakly coupled quark-gluon plasma at the leading-log order , Phys. Rev.
D88 (2013), no. 8 085039, [ arXiv:1308.2945 ].[82] J. Dey, S. Satapathy, A. Mishra, S. Paul, and S. Ghosh,
From Non-interacting toInteracting Picture of Quark Gluon Plasma in presence of magnetic field and its fluidproperty , arXiv:1908.04335 .[83] J. Dey, S. Satapathy, P. Murmu, and S. Ghosh, Shear viscosity and electrical conductivity ofrelativistic fluid in presence of magnetic field: a massless case , arXiv:1907.11164 . – 32 –
84] A. Das, H. Mishra, and R. K. Mohapatra,
Transport coefficients of hot and dense hadrongas in a magnetic field: a relaxation time approach , Phys. Rev. D (2019), no. 11114004, [ arXiv:1909.06202 ].[85] A. Harutyunyan and A. Sedrakian,
Electrical conductivity of a warm neutron star crust inmagnetic fields , Phys. Rev. C (2016), no. 2 025805, [ arXiv:1605.07612 ].[86] B. Kerbikov and M. Andreichikov, Electrical Conductivity of Dense Quark Matter withFluctuations and Magnetic Field Included , Phys. Rev. D (2015), no. 7 074010,[ arXiv:1410.3413 ].[87] S.-i. Nam, Electrical conductivity of quark matter at finite T under external magnetic field ,Phys. Rev. D (2012) 033014, [ arXiv:1207.3172 ].[88] X.-G. Huang, A. Sedrakian, and D. H. Rischke, Kubo formulae for relativistic fluids instrong magnetic fields , Annals Phys. (2011) 3075–3094, [ arXiv:1108.0602 ].[89] K. Hattori, S. Li, D. Satow, and H.-U. Yee,
Longitudinal Conductivity in Strong MagneticField in Perturbative QCD: Complete Leading Order , Phys. Rev. D (2017), no. 7 076008,[ arXiv:1610.06839 ].[90] M. Kurian, S. Mitra, S. Ghosh, and V. Chandra, Transport coefficients of hot magnetizedQCD matter beyond the lowest Landau level approximation , Eur. Phys. J. C (2019),no. 2 134, [ arXiv:1805.07313 ].[91] M. Kurian and V. Chandra, Effective description of hot QCD medium in strong magneticfield and longitudinal conductivity , Phys. Rev. D (2017), no. 11 114026,[ arXiv:1709.08320 ].[92] B. Feng, Electric conductivity and Hall conductivity of the QGP in a magnetic field , Phys.Rev. D (2017), no. 3 036009.[93] K. Fukushima and Y. Hidaka, Electric conductivity of hot and dense quark matter in amagnetic field with Landau level resummation via kinetic equations , Phys. Rev. Lett. (2018), no. 16 162301, [ arXiv:1711.01472 ].[94] A. Das, H. Mishra, and R. K. Mohapatra,
Electrical conductivity and Hall conductivity of ahot and dense hadron gas in a magnetic field: A relaxation time approach , Phys. Rev. D (2019), no. 9 094031, [ arXiv:1903.03938 ].[95] A. Das, H. Mishra, and R. K. Mohapatra, Electrical conductivity and Hall conductivity of ahot and dense quark gluon plasma in a magnetic field: A quasiparticle approach , Phys. Rev.D (2020), no. 3 034027, [ arXiv:1907.05298 ].[96] S. Ghosh, A. Bandyopadhyay, R. L. Farias, J. Dey, and G. Krein,
Anisotropic electricalconductivity of magnetized hot quark matter , arXiv:1911.10005 .[97] W. A. Hiscock and L. Lindblom, Generic instabilities in first-order dissipative relativisticfluid theories , Phys. Rev. D (1985) 725–733.[98] S. Pu, T. Koide, and D. H. Rischke, Does stability of relativistic dissipative fluid dynamicsimply causality? , Phys. Rev. D (2010) 114039, [ arXiv:0907.3906 ].[99] G. Denicol, T. Kodama, T. Koide, and P. Mota, Stability and Causality in relativisticdissipative hydrodynamics , J. Phys. G (2008) 115102, [ arXiv:0807.3120 ].[100] S. Floerchinger and E. Grossi, Causality of fluid dynamics for high-energy nuclearcollisions , JHEP (2018) 186, [ arXiv:1711.06687 ]. – 33 – Analytic Bjorken flow in one-dimensionalrelativistic magnetohydrodynamics , Phys. Lett.
B750 (2015) 45–52, [ arXiv:1506.06620 ].[102] S. Pu, V. Roy, L. Rezzolla, and D. H. Rischke,
Bjorken flow in one-dimensional relativisticmagnetohydrodynamics with magnetization , Phys. Rev.
D93 (2016), no. 7 074022,[ arXiv:1602.04953 ].[103] M. Hongo, Y. Hirono, and T. Hirano,
Anomalous-hydrodynamic analysis ofcharge-dependent elliptic flow in heavy-ion collisions , Phys. Lett. B (2017) 266–270,[ arXiv:1309.2823 ].[104] G. Inghirami, L. Del Zanna, A. Beraudo, M. H. Moghaddam, F. Becattini, and M. Bleicher,
Numerical magneto-hydrodynamics for relativistic nuclear collisions , Eur. Phys. J.
C76 (2016), no. 12 659, [ arXiv:1609.03042 ].[105] G. Inghirami, M. Mace, Y. Hirono, L. Del Zanna, D. E. Kharzeev, and M. Bleicher,
Magnetic fields in heavy ion collisions: flow and charge transport , arXiv:1908.07605 .[106] I. Siddique, R.-j. Wang, S. Pu, and Q. Wang, Anomalous magnetohydrodynamics withlongitudinal boost invariance and chiral magnetic effect , Phys. Rev.
D99 (2019), no. 11114029, [ arXiv:1904.01807 ].[107] R.-j. Wang, P. Copinger, and S. Pu,
Anomalous magnetohydrodynamics with constantanisotropic electric conductivities , in28th International Conference on Ultrarelativistic Nucleus-Nucleus Collisions, 4, 2020. arXiv:2004.06408 .[108] S. Pu and D.-L. Yang,
Transverse flow induced by inhomogeneous magnetic fields in theBjorken expansion , Phys. Rev.
D93 (2016), no. 5 054042, [ arXiv:1602.04954 ].[109] S. Pu and D.-L. Yang,
Analytic Solutions of Transverse Magneto-hydrodynamics underBjorken Expansion , EPJ Web Conf. (2017) 13021, [ arXiv:1611.04840 ].[110] V. Roy, S. Pu, L. Rezzolla, and D. H. Rischke,
Effect of intense magnetic fields onreduced-MHD evolution in √ s NN = 200 GeV Au+Au collisions , Phys. Rev. C (2017),no. 5 054909, [ arXiv:1706.05326 ].[111] W. Israel and J. Stewart, Transient relativistic thermodynamics and kinetic theory , AnnalsPhys. (1979) 341–372.[112] W. Hiscock and L. Lindblom,
Stability and causality in dissipative relativistic fluids , AnnalsPhys. (1983) 466–496.[113] S. Pu, T. Koide, and Q. Wang,
Causality and stability of dissipative fluid dynamics withdiffusion currents , AIP Conf. Proc. (2010), no. 1 186–192.[114] G. Denicol, T. Kodama, T. Koide, and P. Mota,
Shock propagation and stability in causaldissipative hydrodynamics , Phys. Rev. C (2008) 034901, [ arXiv:0805.1719 ].[115] K. Dionysopoulou, D. Alic, C. Palenzuela, L. Rezzolla, and B. Giacomazzo, General-Relativistic Resistive Magnetohydrodynamics in three dimensions: formulation andtests , Phys. Rev. D (2013) 044020, [ arXiv:1208.3487 ].[116] S. s. Grozdanov, D. M. Hofman, and N. Iqbal, Generalized global symmetries and dissipativemagnetohydrodynamics , Phys. Rev. D (2017), no. 9 096003, [ arXiv:1610.07392 ].[117] J. Hernandez and P. Kovtun, Relativistic magnetohydrodynamics , JHEP (2017) 001,[ arXiv:1703.08757 ]. – 34 – Nonresistive dissipative magnetohydrodynamics from the Boltzmannequation in the 14-moment approximation , Phys. Rev.
D98 (2018), no. 7 076009,[ arXiv:1804.05210 ].[119] G. S. Denicol, E. MolnÃąr, H. Niemi, and D. H. Rischke,
Resistive dissipativemagnetohydrodynamics from the Boltzmann-Vlasov equation , Phys. Rev.
D99 (2019), no. 5056017, [ arXiv:1902.01699 ].[120] R. Baier, P. Romatschke, D. T. Son, A. O. Starinets, and M. A. Stephanov,
Relativisticviscous hydrodynamics, conformal invariance, and holography , JHEP (2008) 100,[ arXiv:0712.2451 ].[121] B. Betz, D. Henkel, and D. Rischke, From kinetic theory to dissipative fluid dynamics , Prog.Part. Nucl. Phys. (2009) 556–561, [ arXiv:0812.1440 ].[122] B. Betz, D. Henkel, and D. Rischke, Complete second-order dissipative fluid dynamics , J.Phys. G (2009) 064029.[123] G. Denicol, H. Niemi, E. Molnar, and D. Rischke, Derivation of transient relativistic fluiddynamics from the Boltzmann equation , Phys. Rev. D (2012) 114047,[ arXiv:1202.4551 ]. [Erratum: Phys.Rev.D 91, 039902 (2015)].[124] G. Denicol, E. MolnÃąr, H. Niemi, and D. Rischke, Derivation of fluid dynamics fromkinetic theory with the 14-moment approximation , Eur. Phys. J. A (2012) 170,[ arXiv:1206.1554 ].[125] E. MolnÃąr, H. Niemi, G. Denicol, and D. Rischke, Relative importance of second-orderterms in relativistic dissipative fluid dynamics , Phys. Rev. D (2014), no. 7 074010,[ arXiv:1308.0785 ].[126] X.-G. Huang, M. Huang, D. H. Rischke, and A. Sedrakian, Anisotropic Hydrodynamics,Bulk Viscosities and R-Modes of Strange Quark Stars with Strong Magnetic Fields , Phys.Rev. D (2010) 045015, [ arXiv:0910.3633 ].[127] S. Borsanyi, Z. Fodor, C. Hoelbling, S. D. Katz, S. Krieg, and K. K. Szabo, Full result forthe QCD equation of state with 2+1 flavors , Phys. Lett. B (2014) 99–104,[ arXiv:1309.5258 ].[128]
HotQCD
Collaboration, A. Bazavov et al.,
Equation of state in ( 2+1 )-flavor QCD , Phys.Rev. D (2014) 094503, [ arXiv:1407.6387 ].[129] P. Kovtun, D. T. Son, and A. O. Starinets, Viscosity in strongly interacting quantum fieldtheories from black hole physics , Phys. Rev. Lett. (2005) 111601, [ hep-th/0405231 ].[130] A. Sommerfeld, Partial differential equations in physics : lectures : on theoretical physics.Levent Books„ indian reprint ed., 2004.[131] R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. 1. Wiley, New York,1989.[132] L. Rezzolla and O. Zanotti, Relativistic Hydrodynamics. OUP Oxford, 2013.[133] W. Israel, The Dynamics of Polarization , Gen. Rel. Grav. (1978) 451–468.(1978) 451–468.