From Chiral Kinetic Theory To Relativistic Viscous Spin Hydrodynamics
aa r X i v : . [ nu c l - t h ] A ug Relativistic Viscous Spin Hydrodynamics from Chiral Kinetic Theory
Shuzhe Shi, Charles Gale, and Sangyong Jeon
Department of Physics, McGill University, 3600 University Street, Montreal, Quebec H3A 2T8, Canada.
In this work, we start with chiral kinetic theory and construct the spin hydrodynamic frameworkfor a chiral spinor system. Using the 14-moment expansion formalism, we obtain the equationsof motion of second-order dissipative relativistic fluid dynamics with non-trivial spin polarizationdensity. In a chiral spinor system, the spin alignment effect could be treated in the same frameworkas the Chiral Vortical Effect (CVE). However, the quantum corrections due to fluid vorticity inducenot only CVE terms in the vector/axial charge currents but also corrections to the stress tensor. Inthis framework, viscous corrections to the hadron spin polarization are self-consistently obtained,which will be important for precise prediction of the polarization rate for the observed hadrons, e.g.Λ-hyperon.
I. INTRODUCTION
Relativistic heavy-ion collisions provide a special en-vironment to study the strong interaction. In such ex-periments, a new phase of matter — the Quark-GluonPlasma (QGP) — is created [1, 2]. Recently the STARCollaboration at the Relativistic Heavy Ion Collider re-ported measurement of a non-vanishing polarization ofΛ-hyperons [3, 4]. This result could imply an extremelyvortical fluid flow structure in the QGP produced in semi-central nucleus-nucleus collisions, and has attracted sig-nificant interest and generated wide enthusiasm. In ad-dition, detailed measurement of the spin polarization, inparticular the longitudinal polarization at different az-imuthal angles [5], disagrees with current theoretical ex-pectation [6–8].In theoretical attempts (e.g. [9–15]) to compute thehadron polarization rate, one typically assumes thathadrons are created according to the thermal equilib-rium distribution for particles in a locally rotating fluid,while the viscous corrections induced by off-equilibriumeffects are neglected. Also, studies generally assume thatthe spin degrees of freedom of either hadrons or partonshave negligible influences on the dynamical motion of themedium. A more sophisticated and self-consistent frame-work is required to understand the discrepancy alludedto above, and to describe the vortical structure of QGP.Consequently, we propose to develop a relativistic dis-sipative hydrodynamic theory with spin degrees of free-dom [16].While hydrodynamics is a macroscopic theory basedon conservation laws and the second law of thermody-namics, the evolution of dissipative quantities dependson the details of how the system approaches the thermaldistribution and needs the guidelines of kinetic theoryto correctly reflect microscopic processes. In a masslessfermion system, the microscopic transport processes aredescribed by the Chiral Kinetic Theory (CKT) [17–19].A convenient way to derive the CKT is the Wigner func-tion formalism [20–26]. For the pedagogical reason, wereview recent developments in the Wigner function for-malism of Chiral Kinetic Theory in Sec. II. With sucha tool, we derive ideal spin hydrodynamic equations for thermal equilibrium systems in Sec. III, and obtain vis-cous spin hydrodynamics in Sec. IV. In addition, we an-alyze the causality and stability of spin hydrodynamicequations against linear perturbations in App. A, andexplore the pseudo-gauge transformation to symmetrizethe stress-tensor in App. B. In the rest of the appendices,we include calculation details as supplemental material.In this paper, we take the mostly-negative conventionof metric g µν = diag(+ , − , − , − ), and adopt the followingnotation:∆ µν ≡ g µν − u µ u ν , (1)∆ µναβ ≡
12 ∆ µα ∆ νβ + 12 ∆ µβ ∆ να −
13 ∆ µν ∆ αβ , (2) ̟ µν ≡ (cid:16) ∂ ν u µ T − ∂ µ u ν T (cid:17) , (3) ω µ ≡ − T ǫ µνρσ u ν ̟ ρσ = 12 ǫ µνρσ u ν ∂ ρ u σ , (4)ˆd X ≡ u µ ∂ µ X , (5) θ ≡ ∂ µ u µ , (6) σ µν ≡ ∆ µναβ ∂ α u β . (7)In addition, we define the projected vector/tensor as: V h α i ≡ ∆ αµ V µ , (8) V h αβ i ≡ ∆ αβµν V µν , (9) V h α U β i ≡ ∆ αβµν V µ U ν . (10) II. CHIRAL KINETIC THEORY FROMWIGNER FUNCTION FORMALISM
Spin is an intrinsic quantum degree of freedom ofelementary particles. To describe the non-equilibriumcollective behavior of Dirac spinors taking into accountthe spin degrees of freedom, a natural framework is theWigner formalism: W ab ( x, p ) ≡ (cid:28)Z d y e i ¯ h p · y b ¯ ψ b ( x + y b ψ a ( x − y (cid:29) . (11)As a 4 × x and momen-tum p , it describes the phase space distribution for dif-ferent spin states, and can be decomposed in the Cliffordbasis { I, γ µ , γ ≡ iγ γ γ γ , γ γ µ , Σ µν ≡ i [ γ µ , γ ν ] } , W ≡ (cid:16) F + i P γ + V µ γ µ + A µ γ γ µ + 12 L µν Σ µν (cid:17) , (12)where the scalar F , pseudo-scalar P , vector V µ , axial-vector A µ , and tensor L µν are known as the Cliffordcomponents. With these components, one can expressthe thermodynamic quantities — the current, axial cur-rent, energy-momentum tensor, and the spin tensor cur-rent — respectively as: J µ ≡ h ¯ ψγ µ ψ i = Z d p (2 π ) V µ , (13) J µA ≡ h ¯ ψγ µ γ ψ i = Z d p (2 π ) A µ , (14) T µν ≡ h ¯ ψ ( iγ µ D ν ) ψ i = Z d p (2 π ) p ν V µ , (15) S λµν ≡ h ¯ ψ { γ λ , Σ µν } ψ i = 12 ǫ σλµν Z d p (2 π ) A σ . (16)In the absence of an external field, the equation ofmotion for the Wigner function can be obtained fromthe Dirac equation: γ µ ( p µ + 12 i ¯ h∂ µ ) W ( x, p ) = m W ( x, p ) , (17)which contains a set of coupled equations for the Cliffordcomponents. In the massless limit ( m = 0), the equationsare partially decoupled and the vector and axial-vectorcomponents, V µ and A µ , couple only with each other, butnot the scalar, pseudo-scalar, and tensor components: p µ V µ = 0 , p µ A µ = 0 , (18) ∂ µ V µ = 0 , ∂ µ A µ = 0 , (19) ¯ h ǫ µνρσ ∂ ρ V σ = p ν A µ − p µ A ν , (20) ¯ h ǫ µνρσ ∂ ρ A σ = p ν V µ − p µ V ν . (21)These equations can be further simplified by re-combining vector and axial-vector into left-handed (LH)and right-handed (RH) components, J µ ± ≡ ( V µ ± A µ ).They evolve independently: p µ J ± ,µ = 0 , (22) ∂ µ J ± ,µ = 0 , (23)¯ h ǫ µνρσ ∂ ρ J σ ± = ± ( p ν J ± ,µ − p µ J ± ,ν ) . (24)In Refs. [25, 26], the authors employ a semi-classical ex-pansion (i.e. ¯ h expansion) in the massless limit, and de-rive the CKT up to the leading order in ¯ h . In first orderCKT, the RH/LH components can be expressed as: J µ ± = (cid:16) p µ ± ¯ h ǫ µνρσ p ρ n σ p · n ∂ ν (cid:17) f ± , (25) where f ± are the RH/LH particle distribution functions,defined as the p µ -proportional section of correspondingchirality current J µ ± . Their equations of motion aredriven by the Chiral Kinetic Equation (CKE): h p µ ∂ µ ± ¯ h (cid:16) ∂ µ ǫ µνρσ p ρ n σ p · n (cid:17) ∂ ν i f ± = 0 . (26)In particular, n µ is a time-like arbitrary auxiliary vec-tor field, and could depend on space-time x µ in a non-trivial way. It is introduced to separate the p µ -paralleland p µ -perpendicular components. Noting that the mo-mentum p µ is a null vector hence self-perpendicular, theseparation is not unique and depends on the choice of n µ . As explained in Ref. [25], such non-uniqueness leadsto the frame-dependence of the distribution function —also known the side-jump effect [27]. When choosing dif-ferent auxiliary field, e.g. u µ and v µ , the correspondingdistribution functions, f [ u ] , ± and f [ v ] , ± , differ at ¯ h -order: f [ u ] , ± − f [ v ] , ± = ∓ ¯ h ǫ µνρσ p µ u ν v ρ ∂ σ f (0) ± u · p )( v · p ) , (27)and consequently p µ f [ u ] , ± − p µ f [ v ] , ± = ∓ ¯ h (cid:16) ǫ µνρσ p ρ u σ p · u − ǫ µνρσ p ρ v σ p · v (cid:17) ∂ ν f (0) , ± , (28)so that the definition of J µ ± remains invariant. We re-fer the readers to [25] for detailed derivations. In theabove equations, f (0) ± is the classical ¯ h -order of chiral-ity density function and is frame-independent. As willbe discussed in Sec. III B, it shall be more natural tochoose n µ to be the local fluid velocity. For the sake ofgenerality, we keep n µ to be arbitrary at this point.Last but not the least, the conservation equation oftotal angular momentum0 = ∂ µ M µνλ ≡ ∂ µ ( L µνλ + ¯ hS µνλ ) ≡ ∂ µ ( T µλ x ν − T µν x λ ) + ¯ h∂ µ S µνλ = ( T νλ − T λν ) + ¯ h∂ µ S µνλ , (29)is satisfied automatically, which can be shown by tak-ing the momentum integral of Eq. (21), one of the equa-tions of motion for Wigner components. In a systemwith Dirac spinors, the conservation of total angular mo-mentum is not an extra constraint on the system evo-lution. The spin density current follows once the ax-ial charge density, accounting for the imbalance betweenright-handed (RH) and left-handed (LH) particles, is de-fined. III. SPIN HYDRODYNAMICS INEQUILIBRIUMA. Equilibrium Distribution
To connect kinetic theory with hydrodynamic theory,a natural starting point is the equilibrium limit of thedistribution function. This is non-trivial when rotationeffects are included: quantum corrections appear in thekinetic equation Eq.(26), therefore the equilibrium dis-tribution will also be modified. Here we derive the equi-librium distribution with vorticity corrections, f ± , eq , ina similar way as in Ref. [24]. We start from the principlethat equilibrium distribution f ± ( x, p ) ≡ f ± ( g ± ) shouldbe a function of the linear combination of the quantitiesconserved in collisions — namely, the particle number,the momentum, and the angular momentum, g ± = α ± + β λ p λ + ¯ hγ ± ,µν ǫ µναβ p α n β p · n , (30)where the coefficient α, β, γ are not arbitrary. They areconstrained by the CKE:0 = δ ( p ) h p µ ∂ µ ± ¯ h (cid:16) ∂ µ ǫ µνρσ p ρ n σ p · n (cid:17) ∂ ν i f ± ( g ± )= δ ( p ) d f ± d g ± h p µ ∂ µ ± ¯ h (cid:16) ∂ µ ǫ µνρσ p ρ n σ p · n (cid:17) ∂ ν i g ± . (31)To solve the coefficients, we take the semi-classical ex-pansion g ± = g (0) , ± + ¯ hg (1) , ± + O (¯ h )= (cid:16) α (0) , ± + p µ β µ (0) (cid:17) + ¯ h (cid:16) α (1) , ± + p µ β µ (1) + γ ± ,µν ǫ µναβ p α n β p · n (cid:17) + O (¯ h ) , (32)as well as f ± ( g ± ) = f (0) , ± ( g (0) , ± ) + ¯ hf ′ (0) , ± ( g (0) , ± ) (cid:16) α (1) , ± + p µ β µ (1) + γ ± ,µν ǫ µναβ p α n β p · n (cid:17) + O (¯ h ) . (33)From zeroth order CKE, one finds that ∂ µ α (0) , ± = 0 , ∂ µ β (0) ,ν + ∂ ν β (0) ,µ = ∂ · β (0) g µν . (34)On the other hand, physical quantities like J µ ± are inde-pendent of n µ , hence f [ u ] , ± − f [ v ] , ± = ∓ ¯ h ǫ µνρσ p µ u ν v ρ ∂ σ f (0) ± u · p )( v · p )= ∓ ¯ h ǫ µνρσ p µ u ν v ρ ∂ σ g (0) ± u · p )( v · p ) f ′ (0) , ± ( g (0) , ± ) + O (¯ h ) . (35)Comparing the above two equalities, one obtains that ∓ p λ ǫ µνρσ p µ u ν v ρ u · p )( v · p ) ∂ σ β (0) ,λ = γ ± ,µν (cid:16) ǫ µναβ p α u β p · u − ǫ µναβ p α v β p · v (cid:17) = 2 γ ± ,λσ p λ ǫ µνρσ p µ u ν v ρ u · p )( v · p ) . (36) Further noting the arbitrariness of u , v , and p , one gets γ ± ,µν = ± (cid:16) ∂ µ β (0) ,ν − ∂ ν β (0) ,µ (cid:17) . (37)Then we consider the first order CKE and find ∂ µ α (1) , ± = 0 , ∂ µ β (1) ,ν + ∂ ν β (1) ,µ = ∂ · β (1) g µν . (38)Consequently, one can respectively absorb α (1) , ± and β (1) ,µ into α (0) , ± and β (0) ,µ , and conclude that ∂ µ α ± = 0 , ∂ µ β ν + ∂ ν β µ = ∂ · β g µν ,γ µν ± = ±
14 ( ∂ µ β ν − ∂ ν β µ ) , (39)and f ± ( g ± ) = f ± ( α ± + p µ β µ ) ± ¯ h (cid:16) ∂ µ β ν − ∂ ν β µ × ǫ µναβ p α n β p · n (cid:17) f ′± ( α ± + p µ β µ ) + O (¯ h ) . (40)It is worth noting that compared to the derivation inRef. [24], we take into account the requirement that phys-ical quantities are independent of n µ , i.e. Eq. (35). Bydoing this, one would be able to rule out the ambiguousextra mode of γ ± ,µν pointed out in [24]. Additionally, theconditions (34) and (38) apply only for a system in globalequilibrium. They are not required in the derivation ofthe hydrodynamic equations.Comparing the general form with momentum-integrated thermodynamics quantities, one can find that α ± = µ ± /T corresponds to the RH/LH chemical po-tential, while β µ ≡ u µ /T corresponds to the flow veloc-ity and temperature. Particularly, the latter is indepen-dent of flavor or helicity. Combined with the Fermi-Diracdistribution, we can express the equilibrium distributionfunctions in a compact form: f eq , ± ( p ) = 1exp[ p · u − µ ± T ∓ ¯ h ǫ µνρσ ̟ µν p ρ n σ n · p ] + 1 , (41)where ̟ µν ≡ (cid:16) ∂ ν u µ T − ∂ µ u ν T (cid:17) is the thermal vorticity. B. Ideal Spin Hydrodynamics
With the thermal distribution obtained, now we moveon to construct the hydrodynamic quantities by tak-ing the equilibrium limit. For later convenience, wedefine the vorticity vector ω µ ≡ − T ǫ µνρσ u ν ̟ ρσ = ǫ µνρσ u ν ∂ ρ u σ , the vector/axial chemical potential µ V ≡ ( µ + + µ − ) / µ A ≡ ( µ + − µ − ) /
2, and denote the inte-gral R p ≡ R δ ( p )d p (2 π ) . By substituting equilibrium distri-bution in the definition, the equilibrium hydrodynamicquantities are: J µ eq , ± ≡ Z p p µ f eq , ± ± ¯ h ǫ µλσρ Z p p λ n σ n · p ∂ ρ f eq , ± = n ± u µ ± ¯ h (cid:18) ∂n ± ∂µ ± (cid:19) T,µ ∓ ω µ , (42) J µ eq ,V ≡ J µ eq , + + J µ eq , − = n V u µ + ¯ h (cid:18) ∂n A ∂µ V (cid:19) T,µ A ω µ , (43) J µ eq ,A ≡ J µ eq , + − J µ eq , − = n A u µ + ¯ h (cid:18) ∂n A ∂µ A (cid:19) T,µ V ω µ , (44) T µν eq ≡ Z p p µ p ν ( f eq , + + f eq , − )+¯ hǫ µλσρ Z p p ν p λ n σ n · p ∂ ρ ( f eq , + − f eq , − )= ε u µ u ν − P ∆ µν + ¯ h n A ω µ u ν + T ǫ µνσλ ̟ σλ ) , (45) S λµν eq ≡ ǫ λµνσ J eq ,A,σ (46)where n ± ≡ Z p ( u · p ) f eq , ± , ε = 3 P ≡ Z p ( u · p ) ( f eq , + + f eq , − ) . (47)We note that these are equivalent to the result inRef. [28], if implementing the equilibrium distributionfor both particle and anti-particle n ± = µ ± (cid:16) T + µ ± π (cid:17) ,ε = 7 π T
60 + T ( µ V + µ A )2 + µ V + 6 µ V µ A + µ A π . (48)Some comments are in order:(a) In the above equations, the quantum corrections tothe vector and axial currents are collectively known asthe Chiral Vortical Effect, (see e.g. [29]). In particu-lar, even in the purely neutral case µ V = µ A = 0, thequantum correction to the axial current, ¯ h ( T ω µ / u λ S λµν eq = ¯ hT ̟ µν /
12, such a quantum correc-tion term induces the spin-vorticity alignment.(b) On top of existing chiral-hydro that includes anoma-lous transport terms in current and axial current, ourderivation also indicates new terms in the stress tensoraccounting for the feedback to energy and momentumflow. Quantum correction introduces an anti-symmetricterm ∝ ¯ h (4 ω µ u ν − ω ν u µ + T ǫ µνσλ ̟ σλ ), together with asymmetric correction ∝ h ( ω µ u ν + ω ν u µ ). These termsare proportional to chirality imbalance, and vanish if µ A = 0, i.e. equal amount of RH and LH particles atany spatial and temporal points.(c) As a first-order derivative term ω µ appears in thehydrodynamic equations, it is non-trivial to show their causality and stability. With details in App. A, theseequations are shown to be causal and stable against linearperturbations, which follow from the fact that ∂ µ ω µ =(1 / ǫ µνρσ ( ∂ µ u ν )( ∂ ρ u σ ) does not contain second-orderderivatives of the velocity, such as ∂ α ∂ β u µ .(d) It might be worth noting that we take the canoni-cal definition of energy-momentum tensor T µν and spindensity S λµν . There have been discussions on the equiv-alence of evolution equations when taking other defini-tions, differing by a pseudo-gauge transformation [30–32].In App. B, we derive the explicit form of the pseudo-gauge transformation to symmetrize T µν . We emphasizethat such a pseudo-gauge transformation does not causean ambiguity, as the microscopic distribution, f ± ( p ), isinvariant under such a transformation. Physical observ-ables, including the spin polarization vector, are con-structed based on the distribution functions, hence theyare not influenced by the pseudo-gauge transformation.(e) Last but not the least, one can find that all these hy-drodynamic quantities are independent of the choice ofauxiliary field n , but the distribution functions, f ± , de-pends on the explicit form of n µ . We obtain the physicalchoice of such an auxiliary field as follows. We denotethe spin correction term in distribution function (41) asΣ µν [ n ] ≡ ǫ µναβ p α n β n · p . (49)Noting that n µ Σ µν [ n ] = 0 transforms as a vector underLorentz transformation, Σ ′ µν [ n ] = ǫ µνρ p ′ ρ E ′ p only containsspatial part in the frame satisfying n ′ µ = { , , , } atspace-time point ( t ′ , x ′ , y ′ , z ′ ). It represents the polariza-tion tensor ǫ ijk ˆ p k / − ǫ ijk ˆ p k /
2, which isaccounted by the sign difference in the current term andequilibrium distribution function. Consequently, Σ µν [ n ] serves as the spin tensor in the frame co-moving with n µ . To correctly reflect the spin polarization in the dis-tribution function, it is more natural to take n µ = u µ tobe the flow velocity. We adopt this choice for the rest ofthis paper. IV. HYDRODYNAMICS NEAR EQUILIBRIUM
In this section we extend the discussion to non-equilibrium systems, and derive second order spin hy-drodynamics from CKT. To describe non-equilibrium hy-drodynamics evolution, we start with the chiral kineticequations with collision terms. The quantum correctionterm in CKE could be further simplified, see Eq. (G3) inApp. G. Taking n µ = u µ , the equations become p µ ∂ µ f ± ± ¯ h (cid:16) ǫ µνρσ p ν ( ∂ ρ u σ )4 u · p (cid:17) ∂ µ f ± = C ± [ f + , f − ] , (50)where C + ( p ) = Z k , p ′ , k ′ h W (cid:16) ˜ f + ( p ′ ) ˜ f + ( k ′ ) f + ( p ) f + ( k ) − ˜ f + ( p ) ˜ f + ( k ) f + ( p ′ ) f + ( k ′ ) (cid:17) + W (cid:16) ˜ f + ( p ′ ) ˜ f − ( k ′ ) f + ( p ) f − ( k ) − ˜ f + ( p ) ˜ f − ( k ) f + ( p ′ ) f − ( k ′ ) (cid:17)i , (51) C − ( p ) = Z k , p ′ , k ′ h W (cid:16) ˜ f − ( p ′ ) ˜ f − ( k ′ ) f − ( p ) f − ( k ) − ˜ f − ( p ) ˜ f − ( k ) f − ( p ′ ) f − ( k ′ ) (cid:17) + W (cid:16) ˜ f − ( p ′ ) ˜ f + ( k ′ ) f − ( p ) f + ( k ) − ˜ f − ( p ) ˜ f + ( k ) f − ( p ′ ) f + ( k ′ ) (cid:17)i , (52)are the collision kernels. For later convenience, we recastthe CKE to be: h ( u · p ) ∓ ¯ h ω · p u · p i ˆd f ± − C ± [ f + , f − ]= − p µ ∇ µ f ± ∓ ¯ h (cid:16) ǫ µνρσ p ν ( ∂ ρ u σ )4 u · p (cid:17) ∇ µ f ± , (53)where ˆd X ≡ u µ ∂ µ X , ∇ µ ≡ ∆ µν ∂ ν . In the 14-momentexpansion formalism, we expand the non-equilibrium cor-rection to be moments of p h α · · · p β i , and truncate termsup to p order: f ± ≡ f ± eq + f ± eq (1 − f ± eq ) (cid:20) + λ ± Π Π + λ ± ν ν µ ± p µ + λ ± π π µν p µ p ν (cid:21) = f ± + f ± (1 − f ± ) (cid:20) ∓ ¯ h T ω · pu · p + λ ± Π Π + λ ± ν ν µ ± p µ + λ ± π π µν p µ p ν (cid:21) , (54)where f , ± ( p ) = 1exp[ u · p − µ ± T ] + 1 , (55) f eq , ± ( p ) = 1exp[ u · p − µ ± T ± ¯ h T ω · pu · p ] + 1= f , ± ∓ f , ± (1 − f , ± ) ¯ h T ω · pu · p + O (¯ h ) . (56)Noting that the equilibrium form of polarization vector ω µ is a first-order derivative term, we keep up to firstorder in viscous expansion. This is consistent with theorder of quantum corrections.It is worth noting that in above expressions, T and µ ± are the effective temperature and chemical potentials, re-spectively. In principle, these quantities are well defined only in thermal systems; while in practice, one can de-fine them for non-equilibrated systems by matching theenergy and particle densities ǫ ≡ Z p ( u · p ) [ f + ( p ) + f − ( p )] , (57) n ± ≡ Z p ( u · p ) f ± ( p ) , (58)with their corresponding equilibrium expectations: ǫ = ǫ eq ≡ Z p ( u · p ) [ f eq , + ( p ) + f eq , − ( p )] , (59) n ± = n eq , ± ≡ Z p ( u · p ) f eq , ± ( p ) . (60)With these, one can separate the pressure into twoparts — the thermal pressure P , and the bulk pressureΠ being the non-equilibrium correction: P ≡ − Z p ∆ µν p µ p ν [ f eq , + ( p ) + f eq , − ( p )] , (61)Π ≡ − Z p ∆ µν p µ p ν [ δf + ( p ) + δf − ( p )] , (62)where δf ± ≡ f ± − f ± eq denotes the non-equilibrium sectorof the distribution functions. Implementing the energymatching relation (59), one can re-express Eq.(62) asΠ = − Z p ( p µ p µ )[ δf + ( p ) + δf − ( p )]= − m Z p [ δf + ( p ) + δf − ( p )] . (63)In the massless limit m = 0, the bulk viscous pressurevanishes, hence the scalar corrections λ ± Π Π disappear.Besides, one can further define the non-equilibriumcorrections to hydrodynamics — the dissipative quan-tities: π µν ≡ Z p ∆ µναβ p α p β [ f + ( p ) + f − ( p )] , (64) ν µ ± ≡ Z p ∆ µα p α δf ± ( p ) . (65)From the relations in Eq.(57 - 65), one can fix the coef-ficients in non-equilibrium distribution function: λ ± π = 14 J ± , , λ ± ν = J ± , ( u · p ) − J ± , D ± , . (66)Detailed derivations can be found in Appendix D.Substituting the distribution function in the defini-tion (13-15), we find the RH and LH particle currentsand energy-momentum stress tensor: J µ ± = n ± u µ + ν µ ± ± ¯ h ∂n ± ∂µ ± ω µ ± ¯ h ǫ µρσλ u ρ ∂ σ (cid:16) G (1) , ± , D ± , ν ± ,λ (cid:17) ± ¯ hJ ± , J ± , (cid:16) ǫ µρσλ u ρ σ σξ π λξ − π µλ ω λ (cid:17) ≡ n ± u µ + ν µ ± + ¯ h J µ quantum , ± , (67) T µν = ε u µ u ν − P ∆ µν + π µν + 4¯ h ω µ ( ν ν + − ν ν − )+ ¯ h n A ω µ u ν + T ǫ µνσλ ̟ σλ )+ ¯ h ǫ µρσλ u ρ ∆ νξ ∂ σ (cid:20)(cid:16) J +3 , J +4 , − J − , J − , (cid:17) π λξ (cid:21) + ¯ h ǫ µρσλ u ρ u ν ∂ σ ( ν + λ − ν − λ ) − ¯ h ǫ µνρσ u ρ ( ∂ σ u λ )( ν + λ − ν − λ )+ 2¯ h ǫ µλρσ u ρ ( ∂ σ u ν )( ν + λ − ν − λ ) ≡ ε u µ u ν − P ∆ µν + π µν + ¯ h T µν quantum . (68)Together with classical dissipation terms π µν and ν µ ± , vis-cous corrections also modifies the quantum T µν quantum and J µ quantum , ± , from their equilibrium form. In this work,we take the Landau frame and define flow velocity u µ asthe time-like left-eigenvector of stress tensor, with energydensity ǫ being the eigenvalue: u µ T µν classical = ǫ u ν . (69)Finally, we derive the equations of motion for dissipa-tive terms, ruled by:∆ µνρσ ˆd π ρσ ≡ Z p ∆ µναβ p α p β (cid:16) ˆd δf + + ˆd δf − (cid:17) , (70)∆ µν ˆd ν ± ,ν ≡ Z p ∆ µα p α ˆd δf ± , (71)while the equation of motion for δf ± is derived fromEq. (53):ˆd δf ± − (cid:16) u · p ± ¯ h ω · p u · p ) (cid:17) C ± [ f + , f − ]= − ˆd f eq , ± − p µ ∇ µ f ± u · p ∓ ¯ hǫ µνλσ p ν p ρ u λ ( ∂ ρ u σ − ∂ σ u ρ )4( u · p ) ∇ µ f ± . (72)Putting the lengthy calculations in App. E and keepingup to second-order terms, the relaxation equations for all the dissipative terms are∆ αβρσ ˆd π ρσ − ( A (2)+ , + A (2) − , ) π αβ − ¯ h X + , +2 , − − X − , +2 , − )∆ αβρσ ω ρ ν σ + + ¯ h X − , − , − − X + , − , − )∆ αβρσ ω ρ ν σ − = 85 P σ αβ − θ π αβ + 87 ∆ αβ σ µν π µν − σ αµ π βµ − σ βµ π αµ − π αµ ǫ βµνρ u ν ω ρ − π βµ ǫ αµνρ u ν ω ρ + 2¯ h
15 ∆ αβµν ω µ ∇ ν n A + ¯ h n A ∆ αβµν ∇ µ ω ν − h n A ε + P ∆ αβµν ω µ ∇ ν P + ¯ h n A ε + P (cid:16) σ βµ ǫ µαλσ u λ ∇ σ P + σ αµ ǫ µβλσ u λ ∇ σ P (cid:17) , (73)and ∆ αβ ˆd ν ± β − A (1) ± , ν α ± − B (1) ± , ν α ∓ ± ¯ h T W (1) ± , ω α + ¯ h (cid:16) A (2)+ , − − A (2) − , − (cid:17) π αβ ω α = D ± , J ± , ∇ α µ ± T + D ± , J ± , J ± , ∆ αρ ∇ µ π µρ − π αµ ∇ µ J ± , J ± , − θν α ± − σ αµ ν ± µ − ǫ αµνγ u µ ν ν ± ω γ ∓ ¯ h ω α ˆd I ± , ∓ ¯ h T D ± , J ± , ∆ αβ ˆd ω β ± h n ± ε ± + P ± (cid:16) θω α + σ αµ ω µ (cid:17) ∓ ¯ h I ± , (cid:16) θω α + 45 σ αµ ω µ (cid:17) ± ¯ h ǫ µαλσ u λ ˆd u σ ( ∇ µ I ± , ) , (74)where A , B , W , X are integrals of collision kernel definedin App. F. They are functions of temperature T andchemical potentials µ ± . We note that there have beensimilar attempts to derive the dissipative spin hydrody-namics from relaxation time approximation [28, 33], i.e.the collision kernel is approximated by ( f − f eq ) /τ eq . Weemphasize that by taking 14-moment formalism with con-crete collision kernel, we are able to obtain the exact formof transport coefficients and relaxation times. In this pa-per, we aim to construct a theoretical framework basedon general form of collision terms. More realistic studies— focusing on relativistic heavy-ion collisions — of therelaxation time can be found in [34–36].We end by discussing the viscous correction to the spindegrees of freedom. At the macroscopic level, the spindensity at the fluid co-moving frame is S µν ≡ u λ S λµν = 12 ǫ σλµν Z p u λ A σ = 12 ǫ σλµν u λ ( J + ,σ − J − ,σ )= ¯ h T (cid:16) ∂n + ∂µ + + ∂n − ∂µ − (cid:17) ∆ µα ∆ νβ ̟ αβ + 12 ǫ µνσλ ν A,σ u λ + ¯ h ǫ µνσλ ǫ σαβγ u λ u α ∂ β (cid:18) G (1) , +4 , D +3 , ν γ + + G (1) , − , D − , ν γ − (cid:19) + ¯ h (cid:16) J +2 , J +4 , + J − , J − , (cid:17) ( π µξ σ ν ξ − π νξ σ µξ ) − ¯ h (cid:16) J +2 , J +4 , + J − , J − , (cid:17) ǫ µνσλ u λ π σα ω α . (75)Especially, in the equilibrium limit that all viscous cor- rections are turned-off, i.e. ν µ → π µν →
0, the spindensity S µν ∝ ∆ µα ∆ νβ ̟ αβ is proportional to the spatialcomponents of thermal vorticity tensor.At the microscopic level, one would be interested inthe polarization rate for individual particles, especiallyfor final hadrons. The momentum-dependent mean spinvector for each hadron can be obtained as follows (seee.g. [37]), S µ ( p ) = − ǫ µνρσ p ν R dΣ fo ,λ tr[ { γ λ , Σ ρσ } W ( x, p )] R dΣ fo ,λ p λ tr[ W ( x, p )]= 14 m H ǫ µνρσ p ν R dΣ λ fo ǫ λρσδ A δ ( x, p ) R dΣ fo ,λ V λ ( x, p )= 12 m H R dΣ λ fo p λ A µ ( x, p ) R dΣ λ fo V λ ( x, p ) , (76)where Σ fo ,λ represents the freeze-out hyper-surface. As-suming that hadrons take the same distribution as the14-moment formalism (54), we find S µ ( p ) = 12 m H (cid:26)h Z Σ f V, i + Z Σ f V, (1 − f V, )( λ ν ν α p α + λ π π αβ p α p β ) (cid:27) − × (cid:26)h − ¯ h ǫ µνρσ Z Σ p ν ̟ ρσ f V, (1 − f V, ) i + Z Σ p µ f V, (1 − f V, ) µ A T + Z Σ p µ f V, (1 − f V, ) (cid:16) λ ν ν αA p α + λ + ν − λ − ν ν α p α + λ + π − λ − π π αβ p α p β (cid:17)(cid:27) + O (¯ h ) , (77)where f V, ≡ [ e ( u · p − µ ) /T + 1] − is the Fermi-Dirac dis-tribution, R Σ ( · · · ) ≡ R Σ dΣ λ fo p λ ( · · · ) is the integral overfreeze-out hyper-surface, and µ A ≡ ( µ + − µ − ) / , µ ≡ ( µ + + µ − ) / ,ν µA ≡ ν µ + − ν µ − , ν µ ≡ ν µ + + ν µ − . (78)In the expression of the mean spin vector per particle(77),if keeping terms in [ · · · ] only, one can repeat the equilib-rium result in Ref. [37], while the other terms are cor-rections. Among them, there is a term proportional to µ A /T , which is a leading order contribution, in both gra-dient expansion and semi-classical expansion. It acts op-positely for Λ and ¯Λ hyperons, and might help to explainthe measured difference in their polarization rate [3]. Therest of the terms are viscous corrections: the ones in thedenominator, {· · · } − , are corrections to spin-averagedparticle distribution; while the ones in the numerator arecorrections directly to the spin distribution. The lattermight be related to the sign difference between theoryand experiment results on azimuthal angle distribution oflongitudinal polarization. Last but not the least, notingthat for systems starting with zero chirality imbalance,all quantities proportional to the difference between rightand left, i.e. µ A and ν µA , appear because of chiral trans- port, hence are proportional to ¯ h . In other words, thecorrections in the numerator are not “infinitely large”compared to the equilibrium expression. V. SUMMARY AND OUTLOOK
In this work, we start from a 14-moment expansionformalism and obtain the second-order viscous spin hy-drodynamics from a system of massless Dirac spinors. Insuch a system, the spin alignment effect could be treatedin the same framework as for chiral hydrodynamics, butwith non-trivial quantum corrections to the stress ten-sor. We further obtain the non-equilibrium correctionto the spin polarization vector, and find a potential newsource of the difference in the polarization rate of Λ and¯Λ hyperons.We construct a hydrodynamic theory that self-consistently solves the evolution of systems contain-ing spin degrees of freedom and includes the viscous-corrections in the hadron spin polarization rate, and theexplicit form of the hydrodynamics quantities and equa-tions are shown in Eqs. (67, 68, 73, 74). This frameworkwill be implemented in future numerical hydrodynamicsimulations to precisely quantify both global and localpolarization rates of final-state hadrons created in heavy-ion collisions.We end by noting that hydrodynamic theory is amacroscopic theory that can be derived from conserva-tion laws and the second law of thermodynamics. A hy-drodynamic theory containing the spin degrees of free-dom has been constructed based on such macroscopicprinciples in Ref. [38]. It is particularly interesting tocompare the results of this paper, derived from a micro-scopic approach, to those of [38]. By such comparison, we find extra terms could be added to the results of [38]without violating conservation laws and entropy produc-tion law. Those results will be reported in a separatepublication.
Acknowledgments —
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B , 100-106 (2019)[arXiv:1901.06615 [hep-th]]. Appendix A: Stability and Causality of Spin FluidDynamics
A unique feature of spin hydrodynamics is the emer-gence of vorticity vector ω µ terms at ideal order, whichis a first-order derivative of velocity u µ . Given this, onemay be concerned by the numerical stability and rela-tivistic causality of the theory. Generally speaking this isnot an issue, as the definition of vorticity vector containsanti-symmetric Levi-Civita tensor, hence neither ∂ µ ω µ nor ω µ ∂ µ X contain second-order derivative terms, noteven the product of first order terms with respect to thesame variable. To see this, we examine the linear per-turbation on top of a homogenous-constant background.Without loss of generality, we take the co-moving frameof the background fluid velocity, hence the full velocityis u µ = (1 , , ,
0) + (0 , δu x , δu y , δu z ). Similarly, the fullenergy density becomes ε + δε , while number density is n V + δn V , axial number density is n A + δn A . Then theevolution of perturbative quantities is governed by: ∂ µ δJ µV = 0 , ∂ µ δJ µA = 0 , ∂ µ δT µν = 0 . (A1)Expanding the hydrodynamic equations for linear per-turbations, one finds:0 = ∂ t δn V + n V ∂ i δu i , (A2)0 = ∂ t δn A + n A ∂ i δu i , (A3)0 = ∂ t δε + H∂ i δu i , (A4)0 = H∂ t δu x + ∂ x δP + ¯ h n A ∂ t ( ∂ y δu z − ∂ z δu y ) , (A5)0 = H∂ t δu y + ∂ y δP + ¯ h n A ∂ t ( ∂ z δu x − ∂ x δu z ) , (A6)0 = H∂ t δu z + ∂ z δP + ¯ h n A ∂ t ( ∂ x δu y − ∂ y δu z ) . (A7)where H ≡ ε + P is the enthalpy. Compared to the “spin-less” hydro, the dispersion relations contains second- order derivative terms (¯ h n A / ∂ t ∂ i δu i . However, thisdoes not mean instability or acausality, as one of themmust be time derivative. To see it explicitly, we solve theplane-wave eigenmodes δεδn V δn A δu x δu y δu z = exp[ i ( ωt + k x x + k y y + k z z )] δε δn V δn A δu x δu y δu z , (A8)and the equations becomes ω a k x a k y a k z ω b k x b k y b k z ω c k x c k y c k z d k x e k x f k x ω g ∗ ω k z g ω k y d k y e k y f k y g ω k z ω g ∗ ω k x d k z e k z f k z g ∗ ω k y g ω k x ω · δεδn V δn A δu x δu y δu z = 0 , (A9)where a ≡ ε + P, b ≡ n V , c ≡ n A ,d ≡ ε + P ∂P∂ε , e ≡ ε + P ∂P∂n V ,f ≡ ε + P ∂P∂n A , g ≡ i · ¯ h n A ε + P ) . (A10)Particularly, g is purely imaginary, and g ∗ = − g . Thesolution of perturbation field would be trivial unless thedeterminant of the coefficient matrix vanishes:0 = ω (1 − g k )[( ad + be + cf ) k − ω ] , (A11)which yields ω = 0 or ω = ± p ad + be + cf k . (A12)Particularly, the speed of sound c s ≡ ∂ω∂k = (cid:16) ∂P∂ε + n V ε + P ∂P∂n V + n A ε + P ∂P∂n A (cid:17) / (A13)is determined by the equation of state and takes the sameformula as the “spin-less” hydro. From the above analy-sis one can see that spin hydrodynamics equations remaincausal and is stable for linear perturbations, even thoughthey contain the derivative term ω µ . Appendix B: Pseudo-Gauge Transformation toSymmetrize The Energy-Momentum Tensor
It is worth noting that in this work we take the canon-ical definition of the energy-momentum tensor: T µν = Z d p (2 π ) p ν V µ = Z p p µ p ν f V + ¯ h ǫ µλσρ Z p p ν p λ n σ n · p ∂ ρ f A , (B1)0which contains quantum correction which is not neces-sary symmetric. However, in this appendix we show howto symmetrize the stress tensor without changing anyphysical observables. In principle, without changing anyphysical observables or the evolution of thermodynamicquantities. In one can alter the form of the stress tensorby adding divergenceless term T µν Φ ≡ T µν + 12 ∂ λ (Φ λµν + Φ µνλ + Φ νλµ ) , (B2)while the spin density becomes S λµν Φ ≡ S λµν − Φ λµν , inorder to maintain angular momentum conservation. Suchtransformation is referred to as a pseudo-gauge trans-formation in Refs. [30–32], In practice, we employ theSchouten identity (G1) and separate the quantum correc-tion of the stress tensor into symmetric and divergence-less anti-symmetric components:¯ hǫ µλσρ Z p p ν p λ n σ n · p ∂ ρ f A = ¯ h Z p (cid:16) ǫ µλσρ p ν + ǫ νλσρ p µ (cid:17) p λ n σ n · p ∂ ρ f A + ¯ h Z p (cid:16) ǫ µλσρ p ν − ǫ νλσρ p µ (cid:17) p λ n σ n · p ∂ ρ f A = ¯ h Z p (cid:16) ǫ µλσρ p ν + ǫ νλσρ p µ (cid:17) p λ n σ n · p ∂ ρ f A + ¯ h Z p (cid:16) ǫ σρµν p λ + ǫ ρµνλ p σ + ǫ µνλσ p ρ (cid:17) p λ n σ n · p ∂ ρ f A = ¯ h Z p (cid:16) ǫ µλσρ p ν + ǫ νλσρ p µ (cid:17) p λ n σ n · p ∂ ρ f A + ¯ h ǫ µνλρ ∂ λ Z p p ρ f A + O (¯ h ) . (B3)Especially, the anti-symmetric term vanishes after takingthe divergence, ¯ h ǫ µνλρ ∂ µ ∂ λ R p p ρ f A = 0, and does notcontribute to the conservation equation. This identityalso yields the explicit form of the pseudo-gauge trans-formation: Φ λµν ≡ − ¯ h ǫ λµνρ Z p p ρ f A (B4)so that T µν sym ≡ T µν can + 12 ∂ λ (Φ λµν + Φ µνλ + Φ νλµ )= Z p p µ p ν f V + ¯ h Z p (cid:16) ǫ µλσρ p ν + ǫ νλσρ p µ (cid:17) p λ n σ n · p ∂ ρ f A (B5)is symmetric. Using such a definition, the equilibriumform of stress tensor becomes T µν sym , eq = ε u µ u ν − P ∆ µν + ¯ h n A ( ω µ u ν + ω ν u µ ) . (B6)It is worth mentioning that the pseudo-gauge transforma-tion does not bring any ambiguity in our framework, be-cause of the following two reasons. First, the additional term is divergenceless by definition, hence it does not al-ter the evolution of the system. Second, although thepseudo-gauge transformation modifies the definition of“spin density” S λµν , the spin/chirality dependent distri-bution function remain the same. In other words, physi-cal observables, such as spin polarization vector as shownin Eq. (77), are independent of the choice of pseudo-gauge. Appendix C: Thermodynamic Integrals andOrthogonal Polynomials
In this appendix, we discuss some mathematical rela-tions related to the thermodynamics integrals R p ( · · · ) f and R p ( · · · ) f (1 − f ), and construct the orthogonal poly-nomials used in the main text. • Integration by Part: In the main text, integration bypart is frequently employed to derive/simplify the ther-mal integrals. Noting thatdd p f = − pE p T f (1 − f ) , dd p f (1 − f ) = − pE p T f (1 − f )(1 − f ) , (C1)and applying integration by part, one can find Z d p (2 π ) E p f (1 − f ) F [ E p , p ]= T Z d p (2 π ) E p f E p p dd p ( pF [ E p , p ]) , (C2) Z d p (2 π ) E p f (1 − f )(1 − f ) F [ E p , p ]= T Z d p (2 π ) E p f (1 − f ) E p p dd p ( pF [ E p , p ]) . (C3) • Orthogonality in Thermodynamic Integrals: For anarbitrary function of co-moving energy F = F ( u · p ),angular dependence yields the orthogonal property: Z d p F (2 π ) E p p h µ · · · p µ m i p h ν · · · p ν n i = m ! δ mn (2 m + 1)!! ∆ µ ··· µ m ν ··· ν m Z d p F (2 π ) E p (∆ αβ p α p β ) m . (C4) • Orthogonal Polynomials: we start by defining somethermodynamic integrals as I n,q ≡ Z d p ( − ∆ µν p µ p ν ) q ( u · p ) n − q (2 π ) E p (2 q + 1)!! f , (C5) J n,q ≡ Z d p ( − ∆ µν p µ p ν ) q ( u · p ) n − q (2 π ) E p (2 q + 1)!! f (1 − f ) , (C6) G ( q ) n,m ≡ J n,q J m,q − J n − ,q J m +1 ,q , (C7) G n,m ≡ G (0) n,m = J n, J m, − J n − , J m +1 , , (C8) D n,q ≡ J n +1 ,q J n − ,q − J n,q . (C9)1Then we construct the polynomials P ( ℓ ) m as functions ofthe co-moving energy E p ≡ ( u · p ). They are defined tosatisfy the orthonormal relation: δ mn = Z d p (2 π ) E p ω ( ℓ ) P ( ℓ ) m P ( ℓ ) n , (C10)where the weight function ω ( ℓ ) = ( − ( ℓ ) (∆ µν p µ p ν ) ℓ (2 ℓ + 1)!! J ℓ,ℓ f ( p )(1 − f ( p )) , (C11)satisfies the normalization relation1 = Z d p (2 π ) E p ω ( ℓ ) . (C12)For each ℓ , we explicitly write down the 0 th -, 1 st -, and2 nd -order polynomials as P ( ℓ )0 = 1 , (C13) P ( ℓ )1 = J ℓ +1 ,ℓ p D ℓ +1 ,ℓ − J ℓ,ℓ p D ℓ +1 ,ℓ ( u · p ) , (C14) P ( ℓ )2 = D ℓ +2 ,ℓ − G ( ℓ )2 ℓ +3 , ℓ ( u · p ) + D ℓ +1 ,ℓ ( u · p ) √ N ℓ . (C15)where the normalization factor is N ℓ ≡ D ℓ +1 ,ℓ J ℓ,ℓ (cid:16) J ℓ +2 ,ℓ D ℓ +2 ,ℓ − J ℓ +3 ,ℓ G ( ℓ )2 ℓ +3 , ℓ + J ℓ +4 ,ℓ D ℓ +1 ,ℓ (cid:17) . (C16)We further define F [ X ] , ± r,q ≡ ( − q q !(2 q + 1)!! Z p f , ± (1 − f , ± ) ( − ∆ αβ p α p β ) q ( u · p ) r λ ± X , (C17)with X being Π, ν , π , or Ω. In particular, matchingrelations ensures that F [Π] , ± , = − m , F [Π] , ±− , = 0 , F [Π] , ±− , = 0 , F [ π ] , ± , = 1 / , F [ ν ] , ± , = 1 , F [ ν ] , ±− , = 0 , F [Ω] , ± , = 0 , F [Ω] , ± , = − . (C18)Similarly, we have I ± , = J ± , /T = n ± , I ± , = ǫ ± ,J ± , = T ( ǫ ± + P ± ) , J ± , = ∂n ± ∂α ± . (C19)From the definition and after integration by parts, onecan find J n,q = ∂I n,q ∂α (cid:12)(cid:12)(cid:12) β , (C20) J n,q = − ∂I n − ,q ∂β (cid:12)(cid:12)(cid:12) α , (C21) J n,q = ( n + 1) T I n − ,q . (C22) • Simplification of Thermodynamic Integrals: Employ-ing the on-shell condition ( − ∆ µν p µ p ν ) = ( u · p ) − m ,one can find I n,q = q !(2 q + 1)!! q X k =0 ( − k m k k !( q − k )! I n − k, , (C23) J n,q = q !(2 q + 1)!! q X k =0 ( − k m k k !( q − k )! J n − k, , (C24) F [ X ] , ± r,q = ( − q ( q !) (2 q + 1)!! q X k =0 ( − k m k k !( q − k )! F [ X ] , ± r +2 k − q, . (C25)These expressions can be further simplified when takingthe massless limit m = 0, I n,q = 1(2 q + 1)!! I n, , (C26) J n,q = 1(2 q + 1)!! J n, , (C27) D n,q = h q + 1)!! i D n, , (C28) G ( q ) n,m = h q + 1)!! i G n,m , (C29) F [ X ] , ± r,q = ( − q q !(2 q + 1)!! F [ X ] , ± r − q, . (C30) Appendix D: Coefficients in Dissipative Quantities
In this appendix, we show the full details of computingthe coefficients λ X obtained from matching dissipativequantities with non-equilibrium distribution functions.In the moment expansion formalism, we expand the dis-tribution functions near their equilibrium forms: f ± ≡ f ± eq + f ± eq (1 − f ± eq ) (cid:20) + λ ± Π Π + λ ± ν ν µ ± p µ + λ ± π π µν p µ p ν (cid:21) , (D1)where the non-equilibrium corrections can be expressedas λ ± Π Π = c ± , P (0)0 + c ± , P (0)1 + c ± , P (0)2 , (D2) λ ± ν ν α ± = c α ± , P (1)0 + c α ± , P (1)1 , (D3) λ ± π π αβ = c αβ ± , P (2)0 , (D4)In above equations, P ( ℓ ) n are orthogonal polynomi-als of co-moving energy ( u · p ), and their ex-plicit form can be found in Sec. C. Additionally,( c ± , , c ± , , c ± , , c α ± , , c α ± , , c αβ ± , ) are coefficients that de-pend on temperature T , chemical potential µ ± , fluid ve-locity u µ , but not on momentum p . In addition, thecoefficients are orthogonal to velocity: c µ ± ≡ ∆ µα c α ± , c µν ± ≡ ∆ µναβ c αβ ± . (D5)2It might be worth mentioning that although it has beenshown in the main text that Π as well as the scalar cor-rection λ Π Π vanish for massless system, we formally keepthese terms in this appendix, for the convenience of fu-ture extensions.To determine the coefficients, we first denote δf ± ≡ f ± − f ± eq , and compute the integrals: Z p δf ± = J ± , c ± , , (D6) Z p ( u · p ) δf ± = J ± , c ± , − q D ± , c ± , , (D7) Z p ( u · p ) δf ± = J ± , c ± , − G ± , q D ± , c ± , + q J ± , D ± , − J ± , G ± , + J ± , D ± , q D ± , /J ± , c ± , , (D8) Z p ∆ µα p α δf ± = − J ± , c µ ± , , (D9) Z p ( u · p )∆ µα p α δf ± = − J ± , c µ ± , + q D ± , c µ ± , , (D10) Z p ∆ µναβ p α p β δf ± = 2 J ± , c µν ± , . (D11)Keeping up to ¯ h -order, we find Z p p µ u · p f ± ( p ) = − (cid:16) J ± , c µ ± , + D ± , q D ± , c µ ± , (cid:17) + (cid:16) I ± , + J ± , c ± , (cid:17) u µ , (D12) Z p p µ p ν ( u · p ) f ± ( p ) = I ± , u µ u ν − I ± , ∆ µν − J ± , ( u µ c ν ± , + u ν c µ ± , ) + 2 J ± , c µν ± , , (D13) Z p p µ p ν u · p f ± ( p ) = I ± , u µ u ν − I ± , ∆ µν − J ± , ( u µ c ν ± , + u ν c µ ± , ) + 2 J ± , c µν ± , , (D14) Z p p h µ i p h ν i p λ ( u · p ) f ± ( p ) = − I ± , ∆ µν u λ + 2 J ± , c µν ± , u λ + (cid:18) ∆ µν ∆ λα + ∆ µλ ∆ να + ∆ λν ∆ µα (cid:19) × (cid:18) J ± , c α ± , + J ± , J ± , − J ± , J ± , q D ± , c α ± , (cid:19) . (D15) Then, the matching relations of Eqs. (57 - 65) require c ± , = − m J ± , , c ± , = J ± , q D ± , c ± , ,c ± , = D ± , q J ± , /D ± , q J ± , D ± , − J ± , G ± , + J ± , D ± , c ± , ,c µ ± , = − ν µ ± J ± , , c µ ± , = J ± , q D ± , c µ ± , ,c µν ± , = π µν J ± , . (D16)Finally, substituting the coefficients in Eqs.(D2 - D4),one eventually obtain: λ ± Π ≡ − m J ± , P (0) , ± + J ± , q D ± , P (0) , ± + D ± , q J ± , /D ± , P (0) , ± q J ± , D ± , − J ± , G ± , + J ± , D ± , ! , (D17) λ ± ν ≡ − J ± , P (1) , ± + J ± , P (1) , ± q D ± , ! = J ± , ( u · p ) − J ± , D ± , , (D18) λ ± π ≡ P (2) , ± J ± , = 14 J ± , . (D19)With these, we have F [ π ] , ± r,q = ( − q q ! J ± q − r,q J ± , , (D20) F [ ν ] , ± r,q = ( − q q ! J ± , J ± q − r +1 ,q − J ± , J ± q − r,q D ± , . (D21)3 Appendix E: Equation of motion for Dissipative Quantities
In this appendix, we derive the equations of motion for dissipative terms, ruled by:∆ µνρσ ˆd π ρσ ≡ Z p ∆ µναβ p α p β (cid:16) ˆd δf + + ˆd δf − (cid:17) , (E1)∆ µν ˆd ν ± ,ν ≡ Z p ∆ µα p α ˆd δf ± . (E2)where δf ± ≡ f ± − f ± eq , andˆd δf ± − (cid:16) u · p ± ¯ h ω · p u · p ) (cid:17) C ± [ f + , f − ] = − ˆd f eq , ± − p µ ∇ µ f ± u · p ∓ ¯ hǫ µνλσ p ν p ρ u λ ( ∂ ρ u σ − ∂ σ u ρ )4( u · p ) ∇ µ f ± . (E3)Although it has been proven that the bulk viscous pressure Π vanishes for massless system, we keep it for laterconvenience f ± = f ± + f ± (1 − f ± ) (cid:20) ∓ ¯ h T ω · pu · p + λ ± Π Π + λ ± ν ν µ ± p µ + λ ± π π µν p µ p ν (cid:21) , (E4) λ ± π = 14 J ± , , λ ± ν = J ± , ( u · p ) − J ± , D ± , . (E5)From conservation equations one can find: − ˆd n ± = n ± θ + ∂ µ ν µ ± ± ¯ h ∂ µ ( I ± , ω µ ) , (E6) − ˆd ǫ = ( ǫ + P ) θ − π αβ σ αβ + ¯ h n A u ν ˆd ω ν + 3¯ h ∂ µ ( n A ω µ ) , (E7) − ˆd u ν = 1 ǫ + P (cid:16) − ∇ ν P + ∆ να ∂ β π αβ + ¯ h n A ∆ να ˆd ω α + 3¯ h n A ω µ ∇ µ u ν (cid:17) . (E8)Then, we obtain the equation of motion for the shear viscous tensor:∆ αβρσ ˆd π ρσ − ( A (2)+ , + A (2) − , ) π αβ − ¯ h X + , +2 , − − X − , +2 , − )∆ αβρσ ω ρ ν σ + + ¯ h X − , − , − − X + , − , − )∆ αβρσ ω ρ ν σ − = − Z p p h α p β i ˆd f eq , + − Z p p h α p β i p µ u · p ∇ µ f + − ¯ h ǫ µνλσ u λ ( ∂ ρ u σ − ∂ σ u ρ ) Z p p h α p β i p ν p ρ ( u · p ) ∇ µ f + − Z p p h α p β i ˆd f eq , − − Z p p h α p β i p µ u · p ∇ µ f − + ¯ h ǫ µνλσ u λ ( ∂ ρ u σ − ∂ σ u ρ ) Z p p h α p β i p ν p ρ ( u · p ) ∇ µ f − = 85 P σ αβ − θ π αβ + 87 ∆ αβ σ µν π µν − σ αµ π βµ − σ βµ π αµ − π αµ ǫ βµνρ u ν ω ρ − π βµ ǫ αµνρ u ν ω ρ + 2¯ h
15 ∆ αβµν ω µ ∇ ν n A + ¯ h n A ∆ αβµν ∇ µ ω ν − h n A ∆ αβµν ω µ ˆd u ν + ¯ h n A h σ βµ ǫ µαλσ u λ (ˆd u σ ) + σ αµ ǫ µβλσ u λ (ˆd u σ ) i = 85 P σ αβ − θ π αβ + 87 ∆ αβ σ µν π µν − σ αµ π βµ − σ βµ π αµ − π αµ ǫ βµνρ u ν ω ρ − π βµ ǫ αµνρ u ν ω ρ + 2¯ h
15 ∆ αβµν ω µ ∇ ν n A + ¯ h n A ∆ αβµν ∇ µ ω ν − h n A ε + P ∆ αβµν ω µ ∇ ν P + ¯ h n A ε + P (cid:16) σ βµ ǫ µαλσ u λ ∇ σ P + σ αµ ǫ µβλσ u λ ∇ σ P (cid:17) , (E9)4for dissipative currents:∆ αβ ˆd ν ± β − A (1) ± , ν α ± − B (1) ± , ν α ∓ ± ¯ h T W (1) ± , ω α ± ¯ h U (1) ± , Ω α + ± ¯ h V (1) ± , Ω α − + ¯ h (cid:16) A (2)+ , − − A (2) − , − (cid:17) π αβ ω α = − Z p p h α i ˆd f eq , ± − Z p p h α i p µ u · p ∇ µ f ± ∓ ¯ h ǫ µνλσ u λ ( ∂ ρ u σ − ∂ σ u ρ ) Z p p h α i p ν p ρ ( u · p ) ∇ µ f ± = h − n ± ˆd u α ∓ ¯ h αβ ˆd( I ± , ω β ) i + h ∇ α n ± − J ± , J ± , ∆ αρ ∇ µ π µρ − π αµ ∇ µ J ± , J ± , − θν α ± − σ αµ ν ± µ − ǫ αµνγ u µ ν ν ± ω γ ∓ ¯ h I ± , (cid:16) θω α + 35 σ αµ ω µ (cid:17)i ± h ¯ h ǫ µαλσ u λ ˆd u σ ∇ µ I ± , − ¯ h I ± , (cid:16) σ αµ ω µ − θ ω α (cid:17)i = D ± , J ± , ∇ α µ ± T + D ± , J ± , J ± , ∆ αρ ∇ µ π µρ − π αµ ∇ µ J ± , J ± , − θν α ± − σ αµ ν ± µ − ǫ αµνγ u µ ν ν ± ω γ ∓ ¯ h ω α ˆd I ± , ∓ ¯ h T D ± , J ± , ∆ αβ ˆd ω β ± h n ± ε ± + P ± (cid:16) θω α + σ αµ ω µ (cid:17) ∓ ¯ h I ± , (cid:16) θω α + 45 σ αµ ω µ (cid:17) ± ¯ h ǫ µαλσ u λ ˆd u σ ( ∇ µ I ± , ) , (E10)The following relations are useful in the calculations above Z p p α f ± = n ± u α + ν α ± ± ¯ h J ± , T ω α , (E11) Z p p α p β f ± = ǫ ± u α u β − P ± ∆ αβ + π αβ ± ± ¯ h n ± ( u α ω β + u β ω α ) , (E12) Z p p α p β ( u · p ) f ± = n ± u α u β − I ± , ∆ αβ + F [ π ] , ± , π αβ + u α ν β ± + u β ν α ± ± ¯ h J ± , T ( u α ω β + u β ω α ) , (E13) Z p p α p β p γ ( u · p ) f ± = ǫ ± u α u β u γ − P ± (cid:16) u α ∆ βγ + u β ∆ αγ + u γ ∆ αβ (cid:17) + (cid:16) u α π βγ + u β π αγ + u γ π αβ (cid:17) + F [ ν ] , ± , (cid:16) ∆ βγ ν α ± + ∆ αγ ν β ± + ∆ αβ ν γ ± (cid:17) ± ¯ h n ± (cid:16) u α u β ω γ + u α u γ ω β + u β u γ ω α (cid:17) ∓ ¯ h J ± , T (cid:16) ∆ βγ ω α + ∆ αγ ω β + ∆ αβ ω γ (cid:17) , (E14) Z p p α p β p γ ( u · p ) f ± = n ± u α u β u γ − I ± , (cid:16) u α ∆ βγ + u β ∆ αγ + u γ ∆ αβ (cid:17) + F [ π ] , ± , (cid:16) u α π βγ + u β π αγ + u γ π αβ (cid:17) + (cid:16) u α u β ν γ ± + u α u γ ν β ± + u β u γ ν α ± (cid:17) + F [ ν ] , ± , (cid:16) ∆ βγ ν α ± + ∆ αγ ν β ± + ∆ αβ ν γ ± (cid:17) ± ¯ h J ± , T (cid:16) u α u β ω γ + u α u γ ω β + u β u γ ω α (cid:17) ∓ ¯ h J ± , T (cid:16) ∆ βγ ω α + ∆ αγ ω β + ∆ αβ ω γ (cid:17) , (E15) Z p p h α p β i p h γ i p h δ i ( u · p ) f ± = 2 I ± , ∆ αβµν g µγ g νδ + F [ π ] , ± , (cid:16) g ρσ ∆ αβµρ ∆ γδσν π µν + 79 ∆ γδ π αβ (cid:17) . (E16)5The following integrals of equilibrium distributions are also used: Z p p α p β p γ ( u · p ) f , ± = I ± , h u α u β u γ − (cid:16) u α g βγ + u β g αγ + u γ g αβ (cid:17)i , (E17) Z p p α p β p γ p δ ( u · p ) f , ± = n ± h u α u β u γ u δ + 115 (cid:16) g αβ g γδ + g αγ g βδ + g αδ g βγ (cid:17) − (cid:16) u α u β g γδ + u α u γ g βδ + u α u δ g βγ + u β u γ g αδ + u β u δ g αγ + u γ u δ g αβ (cid:17)i , (E18) Z p p α p β p γ p δ ( u · p ) f , ± = I ± , h u α u β u γ u δ + 115 (cid:16) g αβ g γδ + g αγ g βδ + g αδ g βγ (cid:17) − (cid:16) u α u β g γδ + u α u γ g βδ + u α u δ g βγ + u β u γ g αδ + u β u δ g αγ + u γ u δ g αβ (cid:17)i , (E19) Z p p α p β p γ p δ p ρ ( u · p ) f , ± = n ± h u α u β u γ u δ u ρ + 115 (cid:16) u α g βγ g δρ + [14 other rotation terms] (cid:17) − (cid:16) u α u β u γ g δρ + [9 other rotation terms] (cid:17)i . (E20) Appendix F: Collision Kernels
In this section, we compute the collision kernels for distribution: f ± = f ± + f ± (1 − f ± ) φ ± [ p ] , (F1) φ ± [ p ] ≡ (cid:20) ∓ ¯ h T ω · pu · p + λ ± Π Π + λ ± ν ν µ ± p µ + λ ± π π µν p µ p ν (cid:21) . (F2)One shall keep in mind that λ Π and λ ν are still functions of energy E p .Noticing that ˜ f , ± ( p ) = f , ± ( p ) · exp( E p /T − µ ± /T ) , (F3)one could find ˜ f , + ( p ′ ) ˜ f , + ( k ′ ) f , + ( p ) f , + ( k ) = ˜ f , + ( p ) ˜ f , + ( k ) f , + ( p ′ ) f , + ( k ′ ) , (F4)˜ f , − ( p ′ ) ˜ f , − ( k ′ ) f , − ( p ) f , − ( k ) = ˜ f , − ( p ) ˜ f , − ( k ) f , − ( p ′ ) f , − ( k ′ ) , (F5)˜ f , + ( p ′ ) ˜ f , − ( k ′ ) f , + ( p ) f , − ( k ) = ˜ f , + ( p ) ˜ f , − ( k ) f , + ( p ′ ) f , − ( k ′ ) . (F6)In general, we express the ℓ -indices kernel as: C h µ ··· µ ℓ i + ,r ≡ Z p p h µ · · · p µ ℓ i E rp C + [ f + , f − ]= Z p Z p ′ Z k Z k ′ p h µ · · · p µ ℓ i E rp h W (cid:0) ˜ f + ( p ′ ) ˜ f + ( k ′ ) f + ( p ) f + ( k ) − ˜ f + ( p ) ˜ f + ( k ) f + ( p ′ ) f + ( k ′ ) (cid:1) + W (cid:0) ˜ f + ( p ′ ) ˜ f − ( k ′ ) f + ( p ) f − ( k ) − ˜ f + ( p ) ˜ f − ( k ) f + ( p ′ ) f − ( k ′ ) (cid:1)i (F7)= Z p Z p ′ Z k Z k ′ p h µ · · · p µ ℓ i E rp h W ˜ f , + ( p ′ ) ˜ f , + ( k ′ ) f , + ( p ) f , + ( k ) (cid:16) φ + [ p ] + φ + [ k ] − φ + [ p ′ ] − φ + [ k ′ ] (cid:17) + W ˜ f , + ( p ′ ) ˜ f , − ( k ′ ) f , + ( p ) f , − ( k ) (cid:16) φ + [ p ] + φ − [ k ] − φ + [ p ′ ] − φ − [ k ′ ] (cid:17)i . (F8)6Then the relevant terms are C + ,r − = Π Z p Z p ′ Z k Z k ′ E r − p × h W ˜ f , + ( p ′ ) ˜ f , + ( k ′ ) f , + ( p ) f , + ( k ) (cid:16) λ +Π [ E p ] − λ +Π [ E ′ p ] + λ +Π [ E k ] − λ +Π [ E ′ k ] (cid:17) + W ˜ f , + ( p ′ ) ˜ f , − ( k ′ ) f , + ( p ) f , − ( k ) (cid:16) λ +Π [ E p ] − λ +Π [ E ′ p ] + λ − Π [ E k ] − λ − Π [ E ′ k ] (cid:17)i ≡ A (0)+ ,r Π , (F9) C h µ i + ,r − = ν µ + Z p Z p ′ Z k Z k ′ ∆ αβ p β E r − p × h W ˜ f , + ( p ′ ) ˜ f , + ( k ′ ) f , + ( p ) f , + ( k ) (cid:16) λ + ν [ E p ] p α − λ + ν [ E ′ p ] p ′ α + λ + ν [ E k ] k α − λ + ν [ E ′ k ] k ′ α (cid:17) + W ˜ f , + ( p ′ ) ˜ f , − ( k ′ ) f , + ( p ) f , − ( k ) (cid:16) λ + ν [ E p ] p α − λ + ν [ E ′ p ] p ′ α (cid:17)i + ν µ − Z p Z p ′ Z k Z k ′ ∆ αβ p β E r − p × h W ˜ f , + ( p ′ ) ˜ f , − ( k ′ ) f , + ( p ) f , − ( k ) (cid:16) λ − ν [ E k ] k α − λ − ν [ E ′ k ] k ′ α (cid:17)i + ¯ h ω µ T Z p Z p ′ Z k Z k ′ ∆ αβ p β E r − p × h W ˜ f , + ( p ′ ) ˜ f , + ( k ′ ) f , + ( p ) f , + ( k ) (cid:16) p α E p − p ′ α E ′ p + k α E k − k ′ α E ′ k (cid:17) + W ˜ f , + ( p ′ ) ˜ f , − ( k ′ ) f , + ( p ) f , − ( k ) (cid:16) p α E p − p ′ α E ′ p − k α E k + k ′ α E ′ k (cid:17)i ≡ A (1)+ ,r ν µ + + B (1)+ ,r ν µ − + ¯ h T W (1)+ ,r ω µ , (F10) C h µν i + ,r − = π µν Z p Z p ′ Z k Z k ′ ∆ αβα ′ β ′ p α ′ p β ′ E r − p × h W ˜ f , + ( p ′ ) ˜ f , + ( k ′ ) f , + ( p ) f , + ( k ) p α p β − p ′ α p ′ β + k α k β − k ′ α k ′ β J +4 , + W ˜ f , + ( p ′ ) ˜ f , − ( k ′ ) f , + ( p ) f , − ( k ) (cid:16) p α p β − p ′ α p ′ β J +4 , + k α k β − k ′ α k ′ β J − , (cid:17)i ≡ A (2)+ ,r π µν . (F11)The following term is also needed:¯ h ω γ ∆ µναβ Z p p α p β p γ E r − p C + = ¯ h µνρσ ω ρ ν σ + · Z p Z p ′ Z k Z k ′ (∆ α ′ β ′ p α ′ p β ′ )∆ αβ p β E r − p × h W ˜ f , + ( p ′ ) ˜ f , + ( k ′ ) f , + ( p ) f , + ( k ) (cid:16) λ + ν [ E p ] p α − λ + ν [ E ′ p ] p ′ α + λ + ν [ E k ] k α − λ + ν [ E ′ k ] k ′ α (cid:17) + W ˜ f , + ( p ′ ) ˜ f , − ( k ′ ) f , + ( p ) f , − ( k ) (cid:16) λ + ν [ E p ] p α − λ + ν [ E ′ p ] p ′ α (cid:17)i + ¯ h µνρσ ω ρ ν σ − · Z p Z p ′ Z k Z k ′ (∆ α ′ β ′ p α ′ p β ′ )∆ αβ p β E r − p × h W ˜ f , + ( p ′ ) ˜ f , − ( k ′ ) f , + ( p ) f , − ( k ) (cid:16) λ − ν [ E k ] k α − λ − ν [ E ′ k ] k ′ α (cid:17)i ≡ X + , +2 ,r ¯ h µνρσ ω ρ ν σ + + X + , − ,r ¯ h µνρσ ω ρ ν σ − . (F12)7 Appendix G: Other Mathematical Relations
In this appendix, we list some of the mathematicalrelations employed in the derivation. • Schouten identity — in this paper, we frequentlyemploy the following identity:0 = p µ ǫ νρσλ + p ν ǫ ρσλµ + p ρ ǫ σλµν + p σ ǫ λµνρ + p λ ǫ µνρσ . (G1) • Projector:∆ αβγµνλ = 16 (cid:16) ∆ αµ ∆ βν ∆ γλ + ∆ αν ∆ βλ ∆ γµ + ∆ αλ ∆ βµ ∆ γν +∆ αµ ∆ βλ ∆ γν + ∆ αν ∆ βµ ∆ γλ + ∆ αλ ∆ βν ∆ γµ (cid:17) − (cid:16) ∆ αβ ∆ µν ∆ γλ + ∆ αβ ∆ νλ ∆ γµ + ∆ αβ ∆ λµ ∆ γν +∆ βγ ∆ µν ∆ αλ + ∆ βγ ∆ νλ ∆ αµ + ∆ βγ ∆ λµ ∆ αν +∆ γα ∆ µν ∆ βλ + ∆ γα ∆ νλ ∆ βµ + ∆ γα ∆ λµ ∆ βν (cid:17) . (G2) • Simplifying quantum correction term in CKE:¯ hδ ( p ) (cid:16) ∂ µ ǫ µνρσ p ρ u σ p · u (cid:17) ∂ ν f = ¯ hδ ( p ) (cid:16) ǫ µνρσ p ρ ∂ µ u σ p · u − ǫ µνρσ p λ p ρ u σ ∂ µ u λ p · u ) (cid:17) ∂ ν f = ¯ hδ ( p ) (cid:16) ǫ µνρσ p ρ ∂ µ u σ p · u − ǫ µνρσ p λ p ρ u σ ( ∂ µ u λ + ∂ λ u µ )4( p · u ) − ǫ µνρσ p λ p ρ u σ ( ∂ µ u λ − ∂ λ u µ )4( p · u ) (cid:17) ∂ ν f = ¯ hδ ( p ) (cid:16) ǫ µνρσ p ρ ∂ [ µ u σ ] p · u − ǫ µνρσ p λ p ρ u σ ∂ [ µ u λ ] p · u ) (cid:17) ∂ ν f = ¯ hδ ( p ) (cid:16) ǫ µνρσ p ρ ∂ [ µ u σ ] p · u + ( − ǫ µνρσ p λ − ǫ λνρσ p µ + ǫ ρσλµ p ν + ǫ σλµν p ρ + ǫ λµνρ p σ ) p ρ u σ ∂ [ µ u λ ] p · u ) (cid:17) ∂ ν f = ¯ hδ ( p ) (cid:16) ǫ µνρσ p ν ( ∂ ρ u σ )4 p · u (cid:17) ∂ µ f + O (¯ h ) ..