Chiral Kinetic Theory from Effective Field Theory Revisited

We revisit the chiral kinetic equation from high density effective theory approach, finding a chiral kinetic equation differs from counterpart derived from field theory in high order terms in the $O(1/\mu)$ expansion, but in agreement with the equation derived in on-shell effective field theory upon identification of cutoff. By using reparametrization transformation properties of the effective theory, we show that the difference in kinetic equations from two approaches are in fact expected. It is simply due to different choices of degree of freedom by effective theory and field theory. We also show that they give equivalent description of the dynamics of chiral fermions.

In fact the above mentioned effects are the most common anomalous transports discussed in the literature. More general anomalous transports have been discussed in two complimentary frameworks. One is anomalous hydrodynamics [8,17], whose basic degrees of freedom are fluid velocity, local energy density and charge density. The chiral effects appears as the anomalous transports to leading order in gradient. The anomalous hydrodynamics is ideal for strongly coupled system. The opposite limit is weakly coupled system. This is the regime where the other framework chiral kinetic theory (CKT) is most suitable [18][19][20][21][22][23][24][25][26][27]. Here the basic degree of freedom is distribution function of quasi-particles of fermions. It allows us to study system far from equilibrium. In CKT, chiral effects can in fact be obtained as linear response of the system to external magnetic field and vorticity field.
The chiral kinetic equation (CKE) has been derived in different ways. They can be categorized into two approaches. One approach is field theory or equivalently Dirac equation [19,[21][22][23][24][25][26][27]. The other approach is effective field theory (EFT), which includes high density EFT by Son and Yamamoto (SY) [20] and on-shell EFT by Carignano, Manuel and Torres-Rincon (CMTR) [31]. The EFT Lagrangian is defined with a cutoff. It is found by CMTR that the resulting chiral kinetic equation (CKE) differs from the counterpart obtained from field theory in higher order terms in expansion in the cutoff. We revisit the approach by SY and find agreement with CMTR. The key to understand the difference lie in the reparametrization properties of EFT. As we shall show, while the action and equation of motion is invariant under reparametrization. The Wigner function and operators acting on it are NOT. This leads to an ambiguity in formulating CKE. We fix the ambiguity with a simple scheme. More importantly, the reparamatrization properties dictate that a difference in CKE is actually expected: the difference can be attributed to the difference in the degree of freedom chosen by the two approaches. Nevertheless, both can give correct description of dynamics of chiral fermions.
The paper is organized as follows: In Section 2, we start with high density EFT and use it to derive Wigner function and its equation of motion. Section 3 is devoted to a discussion of reparametrization invariance of the action as well as the equation of motion in EFT approach. In Section 4, we elaborate on the ambiguity in formulating CKE and present a CKE with a simple scheme. In Section 5, we show that CKE from EFT approach is equivalent to CKE from field theory despite their apparent difference. We summarize and discuss possible extension to this framework in Section 6.

High Density Effective Theory and Chiral Kinetic Theory
High density effective theory (HDET) [32][33][34] is very useful in describing low energy dynamics. It is constructed in a simple manner by identifying the heavy degrees of freedom and integrating them out from the theory as irrelevant modes. This process generates a non-local effective Lagrangian, which can be expanded in terms of large momentum. By construction, HDET is valid for excitations near Fermi surface. The total momentum of a particle can be decomposed as: where v is a unit velocity vector (v 2 = 1) denoting a point on Fermi surface. In this division, particle energy and momentum can be simply given by p 0 = µ + l 0 and p = µv + l, with large Fermi momentum µv and small residual momentum l. We mention here that the choice of momentum decomposition is not unique and there is an ambiguity present in the parameter v, which is connected to the reparametrization transformation in the theory. The reparametrization transformation and its implications will be discussed in a great detail in the next sections.
The kinetic theory can be obtained from the high density effective Lagrangian by using the equations of motion for the gauge invariant Wigner function.
In the first subsection, we derive HDET Lagrangian for massless fermions valid in the vicinity of Fermi surface. This is not new but is included for completeness. Our second subsection is devoted to the derivation of Wigner function and its equation of motion, which eventually leads to dispersion relation and transport equation.

High Density Effective Theory
We start with the Lagrangian for right-handed chiral fermions with finite density µ and zero temperature with D µ = ∂ µ + iA µ as a covariant derivative. Here we consider the Weyl representation for massless fermions in which, ψ(x) is a two component spinor with γ µ = (1, σ). The energy spectrum for the above Lagrangian is given by the corresponding Hamiltonian as with E ± representing the energy of particles and anti-particles: E ± = −µ ± |p|. At low energy, particles near the Fermi surface with E + ∼ 0 are the relevant degrees of freedom while anti-particles with E − ∼ −2µ are identified as heavy modes and integrated out from the theory. In integrating out the heavy mode, we decompose the energy and momentum of fermions as p 0 = µ + l 0 and p = µv + l with l 0 , l ≪ µ. The decomposition of large and small momenta is done by taking the Fourier transform as The fermion field is represented as a sum over different patches of Fermi surface, with large Fermi momentum factored out in the transformation, leaving ψ ±v (x) as the velocity dependent fields carrying the residual momentum l. Further, we define the projection operators for massless fermions: P ± = (1 ± σ · v)/2, with the properties P ± ψ ±v = ψ ±v and P ± ψ ∓v = 0. They will be used to project the positive and negative energy states ψ ±v (x) from state ψ(x), respectively.
For the particle near Fermi surface, Lagrangian (1), in terms of the new variables, can be expressed as ). In the limit l/µ → 0, irrelevant degrees of freedom or heavy mode (ψ −v ) can be integrated out by using the classical equation of Putting the expression of ψ −v in (4) and collecting all the terms up to O(1/µ 2 ), we get the effective Lagrangian, which depends only on ψ +v field, as:

Wigner Function and Equation of Motion
We are interested in deriving the chiral kinetic theory and higher order corrections to the dispersion relation for the Weyl fermion. Thus, we construct the two-point function corresponding to the effective field. For homogeneous system in thermal equilibrium, two-point function G v (x, y) depends only on the relative coordinates.
For inhomogeneous system, it is convenient to work with relative and central coordinates s µ = x µ − y µ and X µ = (x µ + y µ )/2, respectively. For the derivation of dispersion relation and transport equation, we use the Wigner function formalism [35][36][37]. We define the Fourier transform of the two-point function with respect to relative variable s µ : where l µ denotes the residual momentum. It is to be mentioned here that the Wigner function has similar hermiticity property as the two-point function but it is not gauge invariant.
Thus, in the presence of gauge field, it is difficult to make the physical interpretation of Wigner function as a quantum analogue of distribution function. To maintain gauge invariance, Wigner transform is multiplied by the linking operator as follows In the above U (y, x) is the Wilson line given by with path ordering P from y to x. EOM emerging from the effective Lagrangian is satisfied by the bare two-point function as here operator D is given by D = D (0) + D (1) + D (2) . We also note that function G v (x, y) satisfies the properties: P − G v (x, y) = 0, G v (x, y) P − = 0, which are known as the projection conditions. Considering the above, we can construct the following expressions by summing and subtracting the two terms in eq. (10) We will express (11) by gauge invariant Wigner function in the following. To proceed, we consider system with small inhomogeneity, the above equation can then be simplified by using the gradient expansion and rewriting the derivatives as ∂ . We perform the gradient expansion by neglecting the terms which involves higher order derivatives ∂ X to obtain We assume the following hierarchy of scales: ∂ X ≪ l, spacetime disturbance is much slower than momentum so that we can ignore higher order terms in ∂ X ; l ≪ µ, this is needed to justify HDET, which describes low energy dynamics. Finally, making (11) gauge invariant and collecting the contributions from Wilson line, we obtain the following results: where we have defined ∆ µ = ∂ µ − F µν ∂ ∂lν .l µ = l µ − A µ is the kinetic momentum of particle. In the following, we will suppress the bar for notational simplicity. Details of the calculation is collected in appendix. From the I (n) + terms, together with the projection conditions, we deduce the following form ofG ṽ where P + and n v (X, l) are the projection operators and distribution function, respectively.
We point out that (13) agrees with SY [20] up to all order of O(1/µ). At order 1/µ 2 , we get different coefficients in last two terms of both I − , which is consistent with CMTR [31] upon identifying the cutoffs in the two effective theories. As we shall show below, the difference is crucial for understanding the kinetic equation.
The delta function in (14) is usually interpreted as dispersion relation, which naively should be invariant under reparametrization. In other words, the dispersion should not depend on v when converting to original momentum p 0 and p. A quick exercise shows that this is not the case, which hints an ambiguity in the resulting CKE! We postpone writing down CKE, but investigate the reparametrization transformation more closely in the next section. The results will guide us to write down unambiguously CKE in EFT approach. In addition, it provides a resolution to the discrepancy between CKE from EFT approach and field theory approach.

Reparametrization Invariance and Frame Dependence
Reparametrization is a redundancy in a theory which manifests that the physical implications do not change upon choosing a slightly different parameter. Reparametrization invariance (RI) have been discussed extensively in the heavy quark effective theory (HQET) [38][39][40][41][42] as well as soft collinear effective theory [43][44][45][46]. The symmetry greatly constrains the form of the Lagrangian for the effective theories, which is particularly useful for higher order terms. Here we closely follow the discussion of HQET, in which the field describing the particles are velocity dependent. HQET is constructed by dividing the particle momentum into small and large momentum part p µ = mv µ + l µ , where m and l µ are the heavy quark mass and small residual momentum with v 2 = 1. However, it is to be noted that this decomposition is not unique. We can have a different momentum decomposition by making an infinitesimal change in parameter v µ as with a constraint v · δv = 0 emerging from the condition v 2 = 1. The HQET is invariant under the above reparametrization. HDET is also reparametrization invariant. In HDET, we have a large chemical potential µ in place of mass parameter m and the decomposition of total momentum in large and small parts has condition v 2 = 0 with v µ = (1, v).

Reparametrization of Classical Action
In this subsection, we show the reparametrization invariance of the Lagrangian for massless fermions with finite density. For this purpose, let us start with effective Lagrangian, which is non-local due to the presence of operators in the denominator This Lagrangian is essentially (6), but we keep terms to all order in O(1/µ) expansion and replace ψ +v with ψ v . It is important to mention here that the introduction of variable v µ breaks the Lorentz invariance of the Lagrangian. Under the reparametrization v → v ′ = v + δv, the original spinor field ψ(x) does not change, but the field ψ v (x) does. The transformation of ψ v can be worked out using the following representation Noting that ψ(x) can be expressed in terms of ψ v (x) by using classical EOM of ψ −v (x) (5), with P ′ + defined in terms of v ′ and plugging (18) into (17), we obtain Note the sign flip from δv ·x = −δv·x with δv = (0, δv). The last equality holds in the sense that integration by part is used. Each term in the transformation of ψ v can be understood as follows: the second term in (19) arises due to a change in the Fermi momentum appearing in the Fourier decomposition. The term 1 in the bracket follows from the change of Dirac structure in the projection operator. The last term in the bracket is reminiscent of the anti-particle contribution, because the way of integrating out anti-particle field depends on choice of v. We will loosely refer to this as anti-particle contribution. We also point out that the reparametrization transformation connects terms at different orders in the O(1/µ) expansion of the Lagrangian (16). We will see in showing the RI, we get some mixed terms at different orders which ultimately cancel each other.

Now let us focus on RI of Lagrangian and denote
variation of Lagrangian under reparametrization is given as it is to be noted that the operator A has a velocity dependence thus its transformation is the following: We mention here that due to the Dirac structure and property of projection operators P + σ ⊥ P + = 0, variation δA gives On the other hand, from the transformation of the other part of Lagrangian we get in the above, we have used the constraint v · δv = 0. Using properties of Dirac structure and eqns. (22) and (23) we finally get Thus, the classical action remains invariant under the reparametrization to all order in 1/µ.

Reparametrization of EOM
It follows that the two-point function We are interested in the reparametrization transformation of gauge invariant Wigner function which is defined as given below It is to be noted that the gauge link U (y, x) is invariant whereas, the residual momentum l µ changes as l µ′ = l µ − µ δv µ . Thus, together with (26) we have Now, representing the variation δG v (x, y) by the central and relative coordinates, we use the gradient expansion to obtain the following with definition ∆ ij = δ ij −v i v j . According to (14), the distribution function can be obtained by taking the trace ofG v (X, l). Note that the first two terms in (29) simply vanishes, giving rise to the following Note that from (30), RT of gauge invariant Wigner function comes entirely from the antiparticle contribution.
Let us focus on RI of summed and subtracted parts of equations of motion, from which dispersion relation and transport equation emerge. These terms are the following where n = 0, 1, 2, ... For the notational simplicity, let us denote D ± = D x ± D † y . The transformation of EOM yields the following it can be easily seen that the terms in the above come from variation of l µ , D ± and G v (x, y) it is very easy to observe that the above commutator vanishes due to the constraint v·δv = 0.
We have the following terms coming from O(1) which cancel with each other s e il·s [D moreover, there is one more term at this order as: It vanishes upon taking the trace. Now, at O(1/µ), all the terms in (32) contribute. The first three terms of (32) are given as s e il·s D In the above, we have already taken the trace by substituting σ m → v m and dropped the vanishing terms from the equation. The fourth and fifth terms of (32) arise due to change in Dirac structure, with their contribution given as: The last two terms of (32) at O(1/µ) from the anti-particle contribution are given as follows if we naively substitute ∂ sµ → −il µ , we would conclude that this contribution is of higher order, since v µ l µ = l 0 − l = O(1/µ). However, we note that ∆ i = ∂ i + is ν F iν , thus the term from ∂ s acting on ∆ j should be kept: in the above, upon taking the trace, some of the terms vanish and we do not consider those terms. Finally, combining all the contributions from (37), (38) and (39), we get where in the above equation, ν = 0 is allowed.
which cancel each other. Further, there is one more term at O(1) which is: D in the above, trace over the sigma matrices has been taken and we have dropped vanishing terms. Moreover, we should note that the index ν = 0 is also allowed. The fourth and fifth terms of (32) at this order are given as The remaining last two terms at O(1/µ) is from the anti-particle contribution, which are given as the first term is ignored because ∆ 2 is of higher order. Now, sum of terms (43), (44) and Before closing this section, we point out one difference with CMTR: In CMTR, the reparametrization transformation of the Wigner function (distribution function) is interpreted as side jump effect [31], which has been discussed extensively as the transformation of distribution function under Lorentz boost [24,[47][48][49]. However, in our case reparametrization corresponds to an infinitesimal rotation in the parameter v. The RT of distribution function arises due to contribution of anti-particle in the EFT, which does not have a direct connection with side jump.

Transport Equation
As we show in the previous section, the Wigner function and differential operator acting on the choice of parameter. This is expected as we explained in the previous section that the degree of freedom corresponding to the Wigner function is effective particle plus a suppressed anti-particle, whose contribution is dependent on choice of v. This is analogous to renormalization scheme dependence in field theory. There is natural choice of scheme: l v, or equivalently l = l, l ⊥ = 0. This scheme has been used in [50].
Within this scheme, the plus and minus equations (13) simplify significantly as: We can combine the plus equations as This gives the dispersion relation l 0 = l − B·v 2µ + B·vl 2µ 2 . It is simpler in terms of original momentum p 0 = p − B·p 2p . Noting that the Wigner function satisfies P +Gv =G v P + =G v , we can parametrizeG v as Here n v is the distribution function, which depends on coordinate X and spatial momentum l. The dependence on l 0 is entirely in the delta function. The transport equation follows from the minus equations. With the parametrization, it is easy to see the differential operators pass through the delta function. Thus we obtain The structure of transport equation is simpler if we write in terms of full momentum p = µv + l: This is in agreement with CMTR. However, field theory approach gives a slightly different form of transport equation [24]: The difference in the transport equations is in fact expected. In (51), the distribution function n corresponds to particle with positive energy, while in (50), the distribution function n v is somewhat unconventional. It is clear from our derivation that it corresponds to an effective degree of freedom: particle plus a suppressed contribution of anti-particle.
Since the difference comes from suppressed anti-particle contribution, it is not surprising that the difference only shows up in high order terms in 1/µ expansion. We will show in the next section that they indeed give equivalent description of the same dynamics as expected.

Constitutive Equation
Let us now express physical quantities in terms of the effective distribution function n v . We restrict ourselves to vector current only: j µ = ψ † σ µ ψ. We wish to express it in terms of ψ v .
Using (18), we obtain where the first three orders are given by We proceed order by order in the evaluation of the current. At zeroth order, we simply where we have made the substitution σ µ → v µ becauseG v ∝ P + . At first order, time component of the current vanishes by the trace property trσ i P + = 0. For spatial components, we first substitute σ by σ ⊥ by trace the property P + σ i ⊥ σ j P + = P + σ i ⊥ σ j ⊥ P + : The limit needs to be taken carefully. We use the following identities which holds in taking trace withG v , we obtain where we have used the scheme condition l ⊥ = 0 to simplify the expression. The second order is more complicated. For time component, only one term contributes We use the trick in (56) to evaluate / D ⊥x G v (x, y)U (x, y) / D † ⊥y . Dropping O(∂ 2 X ) terms, we obtain Spatial components of second order current contain contributions from all three terms in (53). The evaluation of the first term can be simplified by the identity This amounts to the replacement σ k → −v k , making the evaluation parallels the case of n (2) . It follows that The other two terms can be written as After lengthy algebra, we end up with The total current is the sum of (61) and (63):

Equivalence of Chiral Kinetic Theories
To show the equivalence of (50) and (51), we try to express n in terms of n v . Note that n and n v are nothing but the coefficient of delta functions inG andG v , which are defined bỹ Using the representation of ψ in terms of ψ v in (18), we obtain Plugging (67) into (65) and taking the trace for extracting distribution function, we obtain The rest of the terms vanish by the identity trσ i ⊥ P + = 0. The first term in the bracket is simply trG v (X, l). The second term is the higher order correction, which precisely compensate the difference in differential operator in transport equations as we shall see.
Let us evaluate the second term using the following trick The trace can be evaluated as Plugging this into (69), we find the symmetric terms are either O(∂ 2 X ) or O(l 2 ) thus are neglected. Keeping the anti-symmetric term, we end up with Plugging (71) into (51), we find the correction give rise to two additional terms upto O( 1 µ 2 ), which are from ∆ 0 + v i ∆ i acting on the correction. To see them more explicitly, we expand The l-derivative terms give rise to The generated terms precisely match the difference between (50) and (51). Therefore we have shown the equivalence of the two transport equations.

Summary
We revisit the high density effective theory approach to CKT. We find the resulting CKE differs from the counterpart from field theory approach in high order terms in the 1/µ expansion. Our CKE from high density effective theory is formally the same as the counterpart obtained from on-shell effective field theory upon identifying the expansion parameters in the two theories. We further show that despite different forms of kinetic equations obtained from field theory approach and effective theory approach, the two equations are equivalent, with the difference being in the choice of degree of freedoms. CKE from field theory uses particle as degree of freedom, while CKE from effective field theory uses effective particle as degree of freedom, which follows from integrating out anti-particle contribution.
The way of integrating out the anti-particle contribution depends on the choice of a parameter in the EFT. In high density effective theory, this parameter is the Fermi velocity v. Both distribution function and CKE transform under reparametrization of v. Making a specific choice of v l leads to our CKE. Similar reparametrization transformation also exists on-shell effective theory. In the latter case, reparametrization corresponds to Lorentz boost, and transformation of distribution function is identified as side-jump effect. In our case reparametrization corresponds to rotational transformation instead. It would be interesting to explore possible relation with side jump. We leave it for future work.
It is worth noting that our current study does not include a collision term for fermions, therefore we do not have a mechanism for relaxation. The collision term for fermions can be included by making external gauge field dynamical. This would also introduce gauge field degree of freedom into the CKE.
Finally it is interesting to speculate possible extensions with effective theory approach to CKT. Field theory approach essentially assumes an expansion in deriving CKE. It is complicated to go to higher order term in expansion. Effective field theory assumes a different expansion in the cutoff of the EFT. In the absence of collision term, the CKE we obtain also stops at order O( ). In principle it is possible to go to higher order term in with effective field theory. It would be interesting to further investigate this point.

Acknowledgments
We are grateful to Y. Hidaka, J-.H. Gao A Evaluation of I (n) ± (with n = 0, 1, 2) using the Gradient Expansion In the appendix, we explicitly show the derivation of I (n) ± terms at different orders of 1/µ. As we mentioned before, form of the terms I (n) ± is following with D (n) = D (0) + D (1) + D (2) . Considering the central and relative coordinates (X µ , s µ ), with definition of derivative and gauge field: ∂ x = 1 2 ∂ X +∂ s , ∂ y = 1 2 ∂ X −∂ s and A µ (X ± s 2 ) = A µ (X) ± 1 2 s · ∂ X A µ (X), we write covariant derivatives as D µ (x) = 1 2 ∂ Xµ + ∂ sµ + iA µ (X) + i 2 s · ∂ X A µ (X), Using the above form of covariant derivatives and expressing G v (x, y) = U (x, y)G v (x, y) = e −is·AG v (x, y), we can write terms in I ± as iv µ D xµ G e −is·A = iv µ 1 2 it is to be mentioned that the product σ ⊥i σ ⊥j = δ ij −v i v j −i ε jkm v i v k σ m −iε ikm v j v k σ m + i ε ijk σ k . We note thatG v ∝ P + for right handed fermions and to obtain the physical interpretations, we have to take the trace by replacing σ k → v k . Thus, the product yields the following: σ ⊥i σ ⊥j = ∆ ij +i ε ijk v k with ∆ ij = δ ij −v i v j . Finally, the sum and difference of EOM at order O(1/µ) are I (1) + = s e i(l−A)·s 1 2µ in the above higher order terms ∂ Xi ∂ Xj and F mi ∂ Xj have been ignored. Now, the I ± terms at order O(1/µ 2 ) can be simplified in the similar manner to result into the following: where we have kept terms to the lowest order in O(∂ X ) and O(l). It should be noted that in the above equation index µ = 0, 1, 2, 3 withv µ = (1, −v). We take the trace over the sigma matrices and simplify the above equation which yields From the above equation, at order O(1/µ 2 ), we can clearly see the difference by a numerical factor in I ± with SY.