Wigner functions and quantum kinetic theory of polarized photons

We derive the Wigner functions of polarized photons in the Coulomb gauge with the $\hbar$ expansion applied to quantum field theory, and identify side-jump effects for massless photons. We also discuss the photonic chiral vortical effect for the Chern-Simons current and zilch vortical effect for the zilch current in local thermal equilibrium. Moreover, using the real-time formalism, we construct the quantum kinetic theory (QKT) for polarized photons. By further adopting a specific power counting scheme for the distribution functions, we provide a more succinct form of an effective QKT. This photonic QKT involves quantum corrections associated with self-energy gradients in the collision term, which are analogous to the side-jump corrections pertinent to spin-orbit interactions in the chiral kinetic theory for massless fermions. The same theoretical framework can also be directly applied to weakly coupled gluons in the absence of background color fields.


I. INTRODUCTION
Quantum transport of circularly polarized photons is a fundamental issue in various areas of physics from optics, photonics, condensed matter physics, and nuclear physics to astrophysics. One well-known example is the spin-dependent deflection of photons due to the spin-orbit interaction, called the photonic spin Hall effect [1,2]. Another recently found example is the photonic helicity current induced by vorticity, called the photonic chiral vortical effect (CVE) [3][4][5][6][7]. However, the photonic CVE defined through the Chern-Simons (CS) current is not locally gauge-invariant. One can instead consider its gauge-invariant version called the zilch vortical effect (ZVE) [8,9] by making use of the so-called zilch [10][11][12] as an infinite set of conserved quantities in non-interacting Maxwell's theory. 1 The conventional approach to describe the quantum transport of photons out of equilibrium is based on the semi-classical equation of motion including the effects of the Berry phase [1,2,4,6]. In equilibrium, one may alternatively compute, e.g., the photonic helicity current from quantum field theory [3,[7][8][9]. To the best of our knowledge, however, the generic quantum kinetic theory (QKT) for non-equilibrium many-body photons with collisional effects has not been well established based on the underlying quantum field theory-quantum electrodynamics (QED).
It is thus tempting to explore a similar scenario for polarized photons. In Ref. [55], the Wigner functions of polarized photons and corresponding CVE and ZVE have been recently investigated. However, the Wigner functions were derived from the mixture of right and left-handed polarized photons via Maxwell's equations and gauge constraints therein. It is still desirable to obtain the Wigner functions constructed individually from right/left-handed polarized photons through a first-principle derivation from QED to make a direct comparison with fermionic Wigner functions in Weyl bases studied in Ref. [21]. Moreover, a corresponding QKT for tracking spin transport of polarized photons with collisions is also needed.
In this paper, by exploiting the spinor-helicity formalism to write the covariant form of polarization vectors for right/left-handed photons [56][57][58][59], we explicitly derive the Wigner functions up to O( ) in the Coulomb gauge, which manifest how the Berry connections are encoded in distribution functions as the case for Weyl fermions. We also compute the photonic CVE and ZVE in local thermal equilibrium and confirm that our results are consistent with the previous results in Refs. [8,9,55]. By using the real-time formalism and adopting a specific power counting scheme, we further construct the general form of the effective QKT for polarized photons with the collision term characterized by self-energies, similar to the fermionic case in Ref. [60].
This formalism can also be directly applied to weakly coupled gluons in the absence of background color fields. This would pave the way to future study of entangled spin transport of quarks and gluons in QGP. Motivated by recent experimental observations of global polarization of Λ hyperons in heavy ion collisions [61][62][63], this direction should be important to understand how the dynamical evolution of the quark spin will be converted to the local spin polarization of hadrons [60,[64][65][66][67][68][69][70] along the direction of the strong vorticity generated in peripheral collisions; see other theoretical works on developments of hydrodynamics with spin [71][72][73][74][75] and statistical quantum field theory [76][77][78], which also aim at exploring underlying mechanisms and reconciling the existing tension between theoretical predictions and experimental observations for local spin polarization in heavy ion collisions (see Ref. [79] for a recent review and more references therein).
The paper is organized as follows. In Sec. II, we briefly review the spinor-helicity formalism and introduce the polarization vectors in the Coulomb gauge. In Sec. III, we accordingly derive the Wigner functions for photons up to O( ). In Sec. IV, we analyze the Chern-Simons/zilch currents and CVE/ZVE for photons in local thermal equilibrium. In Sec. V, we derive the QKT for photons and its effective version with specific power counting. Finally, we make short summary and outlook in Sec. VI.

II. SPINOR-HELICITY FORMALISM FOR PHOTONS
Let us first briefly recapitulate essential parts of the so-called spinor-helicity formalism [56][57][58] (see, e.g., Ref. [59] for a review) that will be used in our following computations. The basic idea of this formalism is to express spin-one vector fields as bispinors since they transform in the (1/2, 1/2) representation of the Lorentz group. As an advantage of this formalism, we can avoid the redundancy to embed a massless photon with two physical degrees of freedom into a four-component vector field A µ (x).
In this formalism, the polarization vectors of photons are written with fermion spinors as [56][57][58] for the right-handed and left-handed helicity, respectively. Here, p is the momentum of a photon, while k is an auxiliary light-like vector such that p · k = 0 and p 2 = k 2 = 0. 2 The helicity eigenstates of massless fermions satisfy / p(1+γ 5 )u R (p)/2 = (p 0 −|p|)u R (p) and / p(1−γ 5 )u L (p)/2 = (p 0 − |p|)u L (p) in the Weyl representation. In general, one may replace p 0 by n · p and p by the component transverse to n µ which is a timelike vector specifying a Lorentz frame for the spin basis. Here we simply choose n µ = (1, 0). By using for on-shell photons, one can show We may now assign a proper k µ that meets the gauge choice. For example, we can take the Coulomb gauge ∂ ⊥α A α = 0, where a transverse projection is defined as v µ ⊥ ≡ (η µν − n µ n ν )v ν for an arbitrary vector v µ . 3 We also use shorthand notations |v| ≡ |v 2 ⊥ | andv ⊥µ ≡ v ⊥µ /|v|. Then, the polarization sum is supposed to have the form Comparing this expression with Eq.
(3), one should thus take which implies In terms of the two-component spinors c R,L (p), defined respectively as the lower and upper two-component fields of u R,L (p)/ 2|p|, we have Let us focus on ǫ R µ . In the frame n µ = (1, 0), the explicit forms of c R (p) and c R (k) are (see, e.g., Ref. [21]) Then, one can show that where I is a unit matrix. It turns out that c R (k) corresponds to the eigenvector of right-handed fermions with negative energy. Accordingly, we may denote c R (p) = c for convenience, where we use the indices "(±)" to represent the eigenvectors of right-handed fermions with positive and negative energies, respectively. We thus arrive at the expression

III. WIGNER FUNCTIONS FOR POLARIZED PHOTONS
Given the polarization vectors of right/left-handed photons, we are able to quantize the polarized gauge fields and compute the corresponding Wigner functions. Similar to the case for massless fermions, we can separate the right and left-handed sectors in the free theory. We start from the mode decomposition of a U(1) gauge field with the right-or left-handed helicity, where h = R, L represents the index of helicity, and a h p and a h † p are annihilation and creation operators, respectively, that satisfy the commutation relation We are interested in the lesser and greater propagators for right/left-handed photons, which are defined as [80] respectively, where Y = x − y and X = (x + y)/2. We may focus on the lesser propagator for right-handed photons, where p + = (p+p ′ )/2 and p − = p−p ′ . One can easily show that G h< µν (q, X) is a Hermitian matrix according to its definition. Carrying out the Wigner transformation with the p ± momenta, one finds where For brevity, we take = 1 above. In order to perform the p − integral analytically, we expand the integrand with respect to p − and retain the terms up to O(p − ) such as where We also have the expansions p 0 In the end, this expansion provides us with the Wigner functions up to O( ). Plugging those expressions into Eq. (15), we then find where we dropped the O(|p − | 2 ) terms in the integrand except for those contributing to the distribution functions. In the first term, we introduced the distribution function for right-handed photonsf In the second term, the commutation relation (12) leads to Combining those two cases, the distribution functionf R (q, X) is defined for the four-momentum q µ asf Note that the θ(−q 0 ) part in Eq. (19) characterizes the out-going photons.
To derive explicit forms of Π Here and below, we define A(M) = −i(M − M † )/2 for a matrix-valued quantity M and omit the arguments as c µν and Π (1) µνα are straightforward, and the details are given in Appendix A. The results are found to be where we have the Berry connections Note that a µ µνα (q), the Hermitian property of Wigner functions is maintained properly. According to these results, Eq. (19) reads up to O( ). Here, the standard polarization tensor in the Coulomb gauge and the spin tensor of photons are, respectively, given as Absorbing the Berry connections, we have defined a frame-dependent distribution function In fermionic systems, it has been shown that such a quantum correction is responsible for the so-called side jump effect [20,21]. Based on Eq. (28), we may also write down the lesser propagator for left-handed photons Combining G R< µν and G L< µν , we obtain the full lesser propagator for photons Note that the latter definition of f V is determined by the standard lesser propagator of photons at O( 0 ) so that f V reduces to the distribution function of unpolarized photons. Although G < µν itself is a gauge-dependent quantity, it can be utilized to calculate gauge-invariant quantities such as the energy-momentum tensor shown in Appendix. B. For the greater propagator, we simply have to replace f V by (1 + f V ) with keeping f A unchanged. More precisely, we have Since we will always work in the frame n µ = (1, 0), we hereafter omit the superscript "(n)" for the polarization tensor P µν in the following. We will also attach an index "γ" to the distribution function like f γ and similarly S γ µν for the spin tensor to stress that these are of photons.

IV. HELICITY CURRENTS AND CHIRAL/ZILCH VORTICAL EFFECT
We may now utilize G h< µν to evaluate the CS currents for polarized photons, is convenient for computations in terms of Wigner functions as in the case for energymomentum tensors of fermions discussed in, e.g., Ref. [41]. In light of Eq. (34), the CS currents can be derived from the Wigner functions as Apparently, only the anti-symmetric and imaginary component of Wigner functions contributes to the CS currents. Substituting Eqs. (28) and (31) into Eq. (35), the CS currents read Accordingly, the full CS current becomes where the phase-space CS current density is given aŝ Despite the gauge dependence, the CS current could be regarded as the spin component of the angular momentum for photons [81,82]. Spin-one nature of photons manifests itself in the sidejump term associated with S µν γ in Eq. (38) that is twice larger than the corresponding term in the axial charge current characterizing the spin polarization of massless fermions.
Nevertheless, due to the gauge dependence of the CS current, it is tempting to also introduce the zilch current as a gauge-invariant quantity delineating the helicity currents of photons [10][11][12][13]. We may make comparisons with the related studies of the zilch vortical effect in Refs. [8,9,55]. Let us now consider the spin-3 zilch To construct the zilch in terms of the Wigner function, we rewrite the zilch current with the gauge field which yields where Although the general form of the zilch is involved, it may be simplified with the aid of a specific power counting. Analogous to the power-counting scheme proposed in Ref. [60] for fermions, we assume that f γ V = O( 0 ) and f γ A = O( 1 ). We confirm, a posteriori, that these assumptions are satisfied when a nonzero f γ A is induced by a vorticity in local thermal equilibrium (see below). When this counting is applied, the leading-order contribution to the zilch current reads where q 0 = q · n and we use subindices such as F q to represent an arbitrary function F (q, X) in the phase space. The zilch current can be defined as Z α (X) ≡ d 4 qẐ α (q, X)/(2π) 4 witĥ Z α (q, X) ≡ Θ αµ (n) n ν n ρ Z µνρ (q, X) = 2π where Θ αµ (n) ≡ η αµ − n α n µ . We may further investigate the CS and zilch currents in a local thermal equilibrium with nonzero fluid vorticity ω µ = ǫ µναβ u ν ∂ α u β /2, where u µ is the fluid four velocity. Although it is in principle not possible to determine the equilibrium distribution functions without solving the kinetic theory with collisional effects, we may physically expect that the equilibrium distribution functions take the Bose-Einstein form with the correction due to the spin-vorticity coupling where U µ = βu µ with β = 1/T and T being a temperature. The spin-vorticity coupling term is given by the helicity of photons λ γ and the thermal vorticity Ω µν ≡ ∂ [µ U ν] /2. This yields the leading-order terms in : where N(q · u) ≡ e βq·u − 1 −1 andΩ ρσ ≡ ǫ ρσαβ Ω αβ /2. Indeed, these leading-order expressions with the spin-vorticity coupling meet our assumptions that f γ V = O( 0 ) and f γ A = O( 1 ), unless the magnitude of the thermal vorticity compensates the smallness of . In more general cases, a stronger thermal vorticity could induce a larger f γ A beyond our counting scheme. By analogy to the case of fermions where the helicity is λ f = 1/2 [20,21], we may anticipate λ γ = 1 for photons. As shown around Eq. (52), we derive λ γ = 1 by demanding a frame independence of the zilch. Using Eq. (45), the zilch current in the equilibrium state is given by Here, we assume a small flow velocity and thus u µ ≈ n µ . Then, we usedΩ ασ n σ ≈ βω α which follows fromΩ When taking λ γ = 1 as determined just below, one finds This equilibrium zilch current agrees with the results in Refs. [8,9,55]. On the other hand, for the equilibrium CS current, one finds When taking λ γ = 1, the CS current is also consistent with the one in Ref. [55].
For the frame-dependent term in Eq. (50) to be absent, the helicity parameter is required to be Then, the equilibrium zilch current should read In fact, the frame and gauge invariances of the zilch current Z µνρ eq (q, X) can be succeeded tô Z α eq (q, X) ≡ Θ αµ (u) u ν u ρ Z µνρ eq (q, X) if one defines the projection with the flow vector u µ instead of that in Eq. (43) with n µ . Nevertheless, the final form of the equilibrium zilch current remains the same within the current working regime where u µ ≈ n µ .

V. QUANTUM KINETIC THEORY FOR PHOTONS
In this section, we would like to construct the QKT for on-shell photons. We may first start with the free-streaming case for simplicity as also shown in Ref. [55] and then move to a more sophisticated construction incorporating collisions. The primary goal is to derive the generic form of collision term with quantum correction at O( 1 ) in terms of lesser/greater self-energies. In such a formalism as recently constructed for fermions [21,60], given the information of classical collision term at O( 0 ) from, e.g., diagrammatic calculations, one can systematically obtain the corresponding quantum correction in collisions. 4

A. Free-streaming case
Considering the equation of motion for free photons, one finds which results in after the Wigner transformation. We may decompose the lesser propagator into the symmetric and anti-symmetric parts, According to the Hermitian property of photon Wigner functions, G h<(ρν) and G h<[ρν] are purely real and imaginary, respectively. In the Coulomb gauge such that ∂ µ x⊥ A ν (y)A µ (x) = 0 and ∂ ν y⊥ A ν (y)A µ (x) = 0, we should have the following constraints for Wigner functions in phase space, and hence, which are indeed satisfied by Eqs. (28) and (31). The real and imaginary parts of Eq. (55) read 4 A concrete example for the application of QKT has been shown in Ref. [48] on the neutrino transport in core-collapse supernovae. Given the neutrino self-energies at O( 0 ) obtained through the weak interaction with thermal nucleons, the O( ) correction in the collision term of neutrino QKT is derived, which explicitly reveals the influence from vorticity and magnetic fields.
which further reduce to by using the constraints from the Coulomb gauge in Eq. (58). Due to the symmetric and anti-symmetric properties of G h<µν S and G h<µν A , we thus find the master equations dictating the dynamics of G h<µν S/A read where Eq. (61) In such a case, the free-streaming part is unmodified by the corrections for polarized photons analogous to the case for fermions in the absence of background fields, whereas the collision term is more involved.

B. Collisions
We may follow the standard approach based on the Dyson-Schwinger equation and the realtime formalism to systematically incorporate the collision term for the kinetic theory of photons in light of a similar derivation for fermions [21,83]. However, such a derivation is technically more involved due to the tensor structure of photon Wigner functions as shown in Appendix C. Because of the axial part of Wigner functions and the involvement of corrections, the derivation of QKT for photons is nontrivial as opposed to the derivation for the classical kinetic theory in Ref. [84]. Eventually, for the full photon propagator, we obtain a master equation up to O( 3 ) This master equation then gives rise to the kinetic equations and on-shell constraint equations up to O( ). Here Σ ≶ µν denote the lesser/greater self-energies of photons. We also introduced the retarded and advanced propagators in Eq. (C8) with a similar definition applied to the retarded and advanced self-energies for photons. Moreover, we define Σ + σρ ≡ (Σ ret σρ + Σ adv σρ )/2 and G + σρ ≡ (G ret σρ + G adv σρ )/2. In light of the derivation for CKT of fermions, we will take Σ + σρ = G + σρ = 0, which corresponds to the vanishing real part of the retarded self-energy and of the retarded propagator for the fermionic case. The terms in Eq. (63) are essential to satisfy the gauge constraint (57).
We may decompose the self-energies into real and imaginary parts, and the constraint equations dictating the on-shell conditions up to O( ).

C. Effective QKT for photons
As mentioned in Sec. IV, we may further apply the power-counting scheme such that The effective kinetic theory then becomes with the constraint equations Note that contracting the first equation in Eq. (67) with −η µν /2 leads to the ordinary Boltzmann equation for photons, where we introduced the shorthand notationŝ From Eq. (69), one can read out the self-energies by comparing the collision term above with scattering cross section. Since by using Eq. (69), we may rewrite the kinetic equation for G <µν One may wonder whether the possible corrections to the tensor structure ofĜ <µν A from collisions should be included in Eq. (71). Nonetheless, such corrections will be at higher orders in our power counting. Due to the gauge constraint (57), the right-hand side of Eq. (72) should be transverse to both n µ and q µ ⊥ . Except for ǫ µναβ q α n β , the only possible tensor structure is given by A [µ B ν] , where A µ and B µ are two new timelike vectors characterizing anisotropy of systems such that A · n = A · q ⊥ = 0 and so does B µ . In addition, A µ and B µ should accordingly serve as the source for non-vanishing f γ A . However, according to our power counting, A µ and B µ have to be led by gradient terms and at O( ). Consequently, we conclude that A [µ B ν] = O( 2 ) and the right-hand side of Eq. (72) is proportional to S µν γ for systems we considered. In practice, it is sometime more convenient to rewrite Also, we may analyze the last term involving δ ′ (q 2 ) in Eq. (72). This term can be more explicitly written as up to O( ). Although (q ·∂Σ ≷ Reρσ ) above could be off-shell, both P µρ and P σν therein are on-shell. Provided (q·∂Σ ≷ Reρσ ) are symmetric as their generic property, we find P [µρ (q · ∂Σ Re )Ĝ S ν] ρ δ ′ (q 2 ) = 0. Then the effective QKT for on-shell photons reduces to Finally, we may rewrite Eq. (75) as a scalar kinetic equation, where we used S γ µν S µν γ = 2 for on-shell photons. Eventually, Eqs. (69) and (76) jointly delineate the evolution of f γ V/A in phase space. As mentioned in the beginning of this section, in practice, one could read out the structure of self-energies through the collision term in Eq. (69) as the standard Boltzmann equation. Inputting the Wigner functions with corrections, one is able to evaluate the collision term in Eq. (76), which systematically captures the quantum corrections.

VI. SUMMARY AND OUTLOOK
In this paper, we have derived the Wigner functions and QKT of polarized photons in the Coulomb gauge up to O( ) based on QED. We found that the Wigner functions incorporate anti-symmetric and imaginary components characterizing the helicity distribution in phase space, which are also responsible for the CS and zilch currents. In particular, the derivation analogous to the fermionic case reveals the absorption of Berry connections into distribution functions of polarized photons, which hence manifests itself in the frame-dependence of the distribution functions. We also discussed the photonic CVE and ZVE triggered by fluid vorticity and our findings are in agreements with some of previous studies. This QKT enables us to track both the number (or energy) and spin densities of photons (dictated by f γ V and f γ A more precisely) with collisions in terms of the self-energies. Adopting suitable power counting, the QKT boils down to simplified scalar kinetic equations for practical applications.
There are several future directions. First, combined with the effective QKT for fermions recently developed in Ref. [60], we can further investigate the intertwined spin transport between electrons and photons in QED. 5 In particular, it is curious whether the ZVE derived by de-manding the explicit frame independence in thermal equilibrium in the present study remains unchanged when considering the helicity transfer from electrons through collisions. Second, similarly to the case of massless Dirac fermions [28], it would be interesting to extend our formalism in curved spacetime, where an effective refractive index should lead to the photonic spin Hall effect. Third, since the present formalism of photons can be directly applied to weakly coupled gluons in the absence of background color fields, we may study a similar scenario for entangled spin transport between quarks and gluons in QCD. Although more complicated scattering processes are involved, this direction would be crucial to understand the dynamical spin polarization pertinent to heavy-ion experiments. We may also generalize the present formalism to the gluonic case with the inclusion of background color fields, which could have potential applications to the chirality transfer in QCD at the early stage of the relativistic heavy-ion collisions. Finally, our formalism may also be applied to other non-equilibrium systems involving polarized photons in condensed matter physics and astrophysics. have Then, we use and where a µ ± (q) are the Berry connections given in Eq. (27). Plugging those expressions, we obtain Inserting κ αµβν given above, this can be further written as Eq. (26).

Appendix B: Energy-momentum tensor of photons
We consider the Belinfante energy-momentum (EM) tensor of photons, which is a traceless and symmetric tensor. The first term can be written as which yields the following form after the Wigner transformation: Here, we defined a derivative operator By further implementing the Coulomb-gauge constraint (57), we find Inserting the lesser propagator (32), we find where q 0 = q · n. Recall that we took n µ = (1, 0). Accordingly, one finds 2 F αβ F αβ (q, X) = 0, and thus the phase-space electromagnetic energy density is given by In the case of our counting scheme in which f γ A = O( ), the quantum correction will be at O( 2 ) and hence can be neglected. and considering the SK contour, we obtain which can be further written as where t 0 and t 0 − iβ correspond to the initial time and final time in the SK contour. G <µν (x, y) should be independent of t 0 at t 0 → −∞, so that we drop t 0 dependence by taking t 0 → −∞. The last integrals along the imaginary-time axis in (C7) thus vanish. For the remaining integrals with respect to real time, we may rewrite the upper bounds of the integral as ∞ since the integrations from x 0 to ∞ (or from y 0 to ∞) do not contribute. Now, by introducing the retarded and advanced propagators and similar definitions for the self-energy, Eq. (C7) becomes Introducing G +µν (x, y) = [G ret (x, y) + G adv (x, y)]/2, and Σ +µν (x, y) = [Σ ret (x, y) + Σ adv (x, y)]/2, we can express G µν ret/adv and Σ µν ret/adv as Using these expressions, we find Eq. (C9) become We may subsequently implement the Wigner transformation and work in the Coulomb gauge. Note that following the Wigner transformation in our setup, where δP µν (q) = − ∂ qα P µν (q) ∂ α is determined by the structure of P µν (q). One may check the term in P µν (q) is essential to satisfy the gauge constraint.
To be more precise, by taking ∂ x ⊥µ of Eq. (C4), the gauge constraint yields ∂ x ⊥µ ǫ µρ (x)Σ ν ρ (x, y) = 0 withΣ ν ρ (x, y) = d 4 zΣ ρσ (x, z)G σν (z, y). Given ǫ µρ (x) is an operator, we find q ⊥µ + i ∂ ⊥µ /2 P µρ (q)Σ ν ρ (q, X) = 0 withΣ ν ρ (q, X) being the dual function ofΣ ν ρ (x, y) after the Wigner transformation. One can directly show q ⊥µ + i ∂ ⊥µ /2 P µρ (q) = 0 and thus the gauge constraint is always satisfied. We assume a weak-coupling theory, so that we, hereafter, assign to the self-energy. The Wigner transformation of (C12) takes the form where G ≶,+ µν and Σ ≶,+ µν are now functions of X and q, and We also introduced the Moyal product such that A(q, X) ⋆ B(q, X) = d 4 Y e iq·Y / d 4 zA(X + Y /2, z)B(z, X − Y /2) , which can be expanded in terms of as A(q, X) ⋆ B(q, X) = A(q, X)B(q, X) + i 2 A(q, X) * B(q, X) where A * B ≡ (∂ qα A)(∂ X α B) − (∂ X α A)(∂ qα B) is a shorthand notation of the Poisson bracket. The term Σ (δ) σρ + Σ + ρσ in Eq. (C14) gives the self-energy correction, which we drop since we are interested in the collisional effects. We also drop the term proportional to P µρ Σ < ρσ ⋆ G +σν because its contribution is negligible compared with G < for on-shell photons. Eventually, Eq. (C14) reduces to We may now decompose Eq. (C18) into the real and imaginary parts. By decomposing the greater/lesser self-energies into the real and imaginary parts as Σ ρσ = (Σ Re ) ρσ + i(Σ Im ) ρσ , we find where we introduced shorthand notations, (AB) ν ρ ≡ A ρσ B σν and ( AB) ν ρ ≡ A > ρσ B <σν −A < ρσ B >σν . By further separating the symmetric and anti-symmetric parts, we derive the constraint and kinetic equations up to O( ), The terms in the kinetic theory are essential to satisfy the gauge constraint. For example, we may show that Eq. (C18) as a master equation of the kinetic theory satisfies the gauge constraint. Contracting the left/right-hand sides of the kinetic equation (C18) with q ⊥µ , we have and RHS = − 2 q ⊥µ P µρ i( ΣG) ν ρ + respectively. It turns out that the LHS is equal to the RHS up to O( 2 ). The proof above also justifies the inclusion of necessary correction in Eq. (C13).