Consistent Relativistic Chiral Kinetic Theory: a derivation from OSEFT

We formulate the on-shell effective field theory (OSEFT) in an arbitrary frame and study its reparametrization invariance (RI), which ensures that it is respectful of Lorentz symmetry. In this formulation the OSEFT Lagrangian looks formally equivalent to the sum over light-like velocities of soft collinear effective field theory in the Abelian limit, differences remain in the scale of the gauge fields involved in the two effective theories. We then use the OSEFT Lagrangian expanded in powers of the on-shell energy to derive how the classical transport equations for charged massless fermions are corrected by quantum effects, as derived from quantum field theory. We provide a formulation in a full covariant way, and explain how the consistent form of the chiral anomaly equation can be recovered from our results. We also show how the side jump transformation of the distribution function associated with massless charged fermions can be derived from the RI transformation rules of the OSEFT quantum fields. Finally, we discuss differences in our results with others found in the literature.


I. INTRODUCTION
In this manuscript we use the so called on-shell effective field theory (OSEFT) [1][2][3] to provide a derivation of the transport equations obeyed by charged chiral fermions beyond the classical limit approximation.
The first derivation of chiral kinetic theory (CKT) from quantum field theory was made in Ref. [6] for systems at finite density and zero temperature, using the so called high density effective field theory (HDET) [18]. OSEFT was actually proposed to provide a similar derivation that could be valid also in a thermal background, where antifermions are also relevant degrees of freedom. Regardless of the background, transport equations describe the propagation of on-shell quasiparticles, and therefore it seems natural to use for their derivation an effective field theory approach that describes only the propagation of on-shell degrees of freedom, as OSEFT, while off-shell modes are integrated out. Let us stress that the notion of on-shell quasiparticle depends on the energy scales one is looking at in the system under consideration.
It is well-known, for example, that for plasmas at finite temperature T only the high energy modes of order T can be considered as quasiparticles and their evolution studied with classical transport equations [19][20][21], while the same picture does not apply to lower energy modes.
To get corrections to the classical point-particle picture described above from quantum field theory, one simply has to study how the off-shell modes modify the evolution of the highly energetic modes. These corrections are taken into account in the OSEFT Lagrangian, and expressed as operators of increasing dimension over powers of the on-shell energy scale, so that these modifications can be described with the accuracy one desires. The OSEFT Lagrangian can then be used to derive how the classical transport picture is modified, by using for their derivation an increasing number of terms in the high energy expansion.
One of the advantages of our formulation is that it may allow us to derive transport equations in full covariant form, and derive their properties under Lorentz transformations. While the initial proposals of CKT were not given in a covariant form, it was soon realised that it would have peculiar properties under Lorentz transformations [22,23], very specially seen when formulating two-body collisions, but also expressed in the so called side-jump behaviour of the distribution function of CKT, that expresses that it is frame dependent.
We present in this manuscript a derivation of CKT in a covariant way, as derived from OSEFT, and explain how the side jump effects can be deduced from the same symmetries of that effective field theory. While previous formulations of OSEFT were given in the preferred frame of the thermal bath, we generalise it to an arbitrary frame, introducing the frame vector u µ . The resulting OSEFT Lagrangian then looks formally equivalent to that corresponding to a sum over velocities of the so called soft collinear effective field theory (SCET) [24][25][26][27], although there are some differences, as will be discussed in the following. We further study the reparametrization invariance of OSEFT, that ensures that our formalism is respectful of Lorentz invariance.
We compute both the vector current and axial current in the OSEFT, by taking functional derivatives to the action, and take these expressions to deduce the corresponding values in the transport framework, which requires a Wigner transformation of a two-point function, together with a gradient expansion. As very clearly explained in the review [28], such a definition can only lead to the consistent form of the chiral anomaly, rather than the covariant form. We check from our expressions that this is indeed the case.
Our final form of the relativistic chiral transport equation mainly differs from that introduced in Refs. [9,10,13], in pieces that may be subleading when considering effects close to thermal equilibrium, but that might be relevant for studies off-equilibrium, and also in the gradient terms of the gauge fields. It also differs, when fixing the frame, with the chiral transport equation obtained from the modified form of the one-point particle action.
Our manuscript is organised as follows. In Sec. II we formulate OSEFT in an arbitrary frame, introducing a frame vector, and showing its formal equivalence with soft collinear effective field theory. In Sec. III we study the reparametrization invariance of this effective field theory, a basic ingredient to show that it is respectful of Lorentz symmetry. In Sec. IV we introduce the basic two-point function in the OSEFT that will be used to derive the basic set of transport equations. The main content of the paper is in Sec. V, with the derivation of the collisionless transport equation, first using the OSEFT variables in Sec. V A, and then expressed in terms of the QED original variables in Sec. V B. In Sec. VI we derive both the vector and axial current obtained in the OSEFT approach, and check that they obey the consistent form of the quantum anomalies. In Sec. VII we derived the side jump transformation of the distribution function from the reparametrization invariance transformations of the OS-EFT quantum fields. We conclude in Sec. VIII, where we summarise our main findings, and give a possible interpretation of the origin of the discrepancy of our results with alternative approaches. In Appendix A we give some details of our computations.

II. OSEFT IN AN ARBITRARY FRAME AND SCET
Let us review the OSEFT as originally formulated [1,2] introducing the basic fields and notation. Let us recall that the propagation of an on-shell massless fermion is described by its energy p, with p > 0, and the four light-like velocity v µ = (1, v), where v is three-dimensional unit vector, and thus its four momentum is p µ = pv µ . However, for a fermion close to be on-shell, its four momentum can be expressed as where k µ is the residual momentum (k µ ≪ p), i.e. the part of the momentum which makes q µ off-shell. A similar decomposition of the momentum for almost on-shell antifermions can be done as follows The Dirac field can be written as where the basic OSEFT quantum fields obey and the particle/antiparticle projectors are expressed as It is possible to integrate out the H (1,2) fields of the QED Lagrangian [1], to have an effective theory for the fields χ v and ξṽ only.
If we assume that the physical phenomena we aim to describe is dominated by the contribution of on-shell particles, then the corresponding OSEFT Lagrangian can be written as a sum over the different values of the on-shell momenta as where the precise meaning of the sum displayed in Eq. (7) is not needed at this stage (we will come back to this point later on, see also Ref. [2]), and where is minus the transverse projector to v, written in covariant form. Note that with our conven- From now on, and as done in Ref. [2], whenever we write a tensor with the symbol ⊥, it means that a transverse projector applies to all its Lorentz indices. If only the transverse projector is applied to one of the indices, we will write ⊥ only affecting that index. Thus, σ µν ⊥ = P µα ⊥ P νβ ⊥ σ αβ , while σ µ ⊥ ν = P µα ⊥ g νβ σ αβ . In the original formulation of the OSEFT a choice of frame was made [1,2]. The energies of the on-shell particles in Eq. (1) are measured in the same frame where, for example, the thermal bath is defined. If we want to express the same OSEFT Lagrangian in an arbitrary frame, we will then have to introduce a time-like vector u µ which defines that frame. Then one could write all the above different equations simply by replacing With our specific choice of variables v µ andṽ µ , then it is not difficult to see that Note that in OSEFT u µ is not an independent vector, once v µ andṽ µ have been defined. While in the static frame we chose a particular definition of the vectors v µ andṽ µ , which implicitly assumed that u µ = (1, 0, 0, 0), in an arbitrary frame we will only ask that these light-like vectors obey Thus u · v = 1 and u 2 = 1 are automatically fulfilled.
In our formulation of the OSEFT in an arbitrary frame we will sometimes useṽ µ , and sometimes we will use u µ . The last option is convenient, as in kinetic theory it may appear also in the thermal equilibrium distribution associated with the massless particles.
As for the the particle/antiparticle projectors in an arbitrary frame we will write them as where we used that The OSEFT Lagrangian in a general frame is then written down as where where we have used that It is noteworthy that Eq.(16) formally looks similar to the Lagrangian of soft-collinear effective field theory [24][25][26][27]. The corresponding projectors Eqs. (13,14) are also those used in SCET. We note that the explicit form of the OSEFT and SCET Lagrangians differ because of our different convention in defining the quantum fluctuating fields: in SCET the exponential terms of Eq. (3) have been included in the quantum fields of the effective theory. We also explicitly separate the contribution of particles and antiparticles. Further, we recall that we are considering an effective field theory for QED, while SCET is an effective field theory for QCD.
After noticing the above formal similarities of SCET and OSEFT when the latter is formulated in an arbitrary frame, it has to be stressed that they are still different effective field theories. SCET was originally formulated to describe the physics associated with highly energetic jets in vacuum, and there are only two light-like vector in the theory, v µ andṽ µ , fixed by the direction of the jet. In SCET the covariant derivatives are associated with collinear and ultrasoft gauge fields. OSEFT was in principle developed to describe many body particle systems, close to thermal equilibrium, where one can consider to have many on-shell particles and their propagation in the background of soft gauge fields. Thus for a fixed value of the energy there might be particles moving in all arbitrary (light-like) directions, and a sum over v µ is displayed in the final Lagrangian, which is absent in SCET. In OSEFT the covariant derivatives we use mainly contain soft gauge fields.
OSEFT also uses a different notation, which makes clear that its main goal is to make an analytical expansion in powers of the inverse of the on-shell energy 1/E. At finite temperature and/or density we will obtain different expressions multiplied by a particle distribution function. After integration over momenta, this expansion on the inverse of the on-shell energy will turn out to give an expansion in powers of the inverse of the temperature and/or chemical potential [2,3].
After mentioning the explicit similarities and differences of these two effective field theories, it is possible to use some of the results obtained in SCET to learn about some properties of OSEFT, such as that of reparametrization invariance, which will be discussed in the following section.

III. REPARAMETRIZATION INVARIANCE OF OSEFT
Reparametrization invariance (RI) is the symmetry associated with the ambiguity of the decomposition of the momentum q µ performed in Eq. (1). If M µν defines the 6 Lorentz generators of SO(3, 1), the decomposition of Eq. (1) suggests an apparent breaking of five is possible to show that the OSEFT Lagrangian is RI invariant, which is equivalent to say that is Lorentz invariant. Let us stress that this reduces to study RI of SCET for every sector of the theory defined by the vectors v µ andṽ µ , something which has been extensively investigated [29]. The fact that the covariant derivatives displayed in SCET and OSEFT contain gauge fields of different scales does not however affect the proof of RI, which turns out to be formally equivalent in the two effective field theories.
Let us review how this effectively works. The Dirac field defined in Eq. (3) should be same independent of the choice of the parameters used to define the effective field theory, thus As in SCET, we will see that the effective field theory action remains invariant under infinitesimal changes of the vectors v µ andṽ µ that preserve their basic properties expressed in Eq. (12). It is possible to show that the OSEFT Lagrangian is invariant under the following Please note that the transformation rule of the vector u µ can be deduced from Eq. (11).
Just to have a flavour of the meaning of the above symmetries, let us imagine one fixes the values of the two light-like vectors as v µ = (1, 0, 0, 1) andṽ µ = (1, 0, 0, −1). Then apparently there are five broken generators in the OSEFT, which are Q ± 1 = J 1 ± K 2 , Q ± 2 = J 2 ± K 1 , and K 3 , where J i and K i are the generators of rotations and boosts, respectively. Then type I refer to the combined action of an infinitesimal boost in the x(y) direction and a rotation around the y(x) axis, such thatṽ µ is left invariant, with generators (Q − 1 , Q + 2 ). Type II transformations are similar but (Q + 1 , Q − 2 ) leave v µ invariant, while type III is a boost along the direction 3, K 3 . It is also worth to note that the the generators (Q − 1 , Q + 2 , J 3 ) obey the SE(2) Lie algebra, that is the symmetry group of the two-dimensional Euclidean plane. They correspond to what is known as the Wigner little group associated with the vector p µ = pv µ [30], see also [31][32][33].
Similarly, the generators (Q + 1 , Q − 2 , J 3 ) correspond to the Wigner's little group associated with p µ = −pṽ µ (antiparticles). As discussed in Ref. [30] these Wigner translations are associated with shifts of the trajectory of finite wave-packets of massless particles proportional to the particle's helicity.
It is possible to check easily that our Lagrangian is invariant under the above three RI transformations [29], which formally is equivalent to say that it is Lorentz invariant. Let us discuss these briefly, as they are the same RI symmetries of SCET. We will mainly focus now on what our different notation implies. We will concentrate in the following on the χ v fields and the Lagrangian L E,v , as for the antiparticles ξṽ things works analogously, after trivial changes (namely, u · p → −u · p and v µ ↔ṽ µ ). We will also see that the type II symmetry will allow us to generate the side jumps that were discussed in the framework of chiral kinetic theory in Ref. [22]. This point will be discussed in Sec. VII.
Let us first start with type I symmetry. The change in the vector v µ implies a relabelling of what is called on-shell and residual parts of the momentum defined in Eq. (1). After a type I symmetry the on-shell part and residual momenta change as respectively. This implies that under a type I transformation the covariant derivatives acting on the fluctuating fields also transform.
Type II symmetry implies that the new on-shell and residual momenta changes as while the type III transformation lead to the changes in the on-shell and residual momenta, respectively.
In Table I we summarise the transformation rules under all three types of transformations. The OSEFT Lagrangian is invariant under these three RI transformations [29] In explicit computations of Feynman diagrams, or derivations of transport equations, we will expand the Lagrangian in power series of 1/E. While Eq. (25) is exact to all orders in a 1/E expansion, in a perturbative analysis in 1/E it is important to note that RI invariance implies that different terms in the expansion are connected by symmetry. This comes from the fact that the covariant derivatives, or the fields, transform with terms proportional to E.
For completeness, we will also mention other discrete symmetries of the OSEFT. Under parity, charge conjugation and time reversal the basic OSEFT field transform as . There is also a spin symmetry, which is not a SU(2) symmetry, but a U(1) symmetry, which corresponds to helicity [33].

IV. WIGNER FUNCTION IN THE OSEFT
We focus here our attention on the basic Wigner function used in the following part of the manuscript for the derivation of the transport equations from OSEFT. We will use the Keldysh-Schwinger (KS) formulation, allowing the time variables to take complex values, and define the two-point Green's functions of the OSEFT on the closed time-path contour. These are represented by a 2 × 2 matrix where T denotes time ordering, andT anti-time ordering.
We will focus on one of the entries only, namely S < E,v , as this two-point function depends only on medium effects, while the diagonal entries of Eq. (29) do also contain vacuum contributions.
We will drop the superindex < in what follows to make the notation lighter.
A similar two-point function can be introduced for the antiparticle quantum fluctuations.
From now on we will focus on the particle's sector, as the antiparticle's transport equations may be derived similarly, and only involve some few changes to the particle's derivation (E → −E, and v µ ↔ṽ µ ). However, we will have to take into account both degrees of freedom when computing physical observables.
In order to make contact with transport theory, one defines the (gauge-covariantly modified) Wigner transform of the the above two-point functions. If X = 1 2 (x + y) and s = x − y define the center of mass and relative coordinates, respectively, then where U is the Wilson line and P denotes path-ordering along the path γ from x to y. Using that then one can show that the introduction of the Wilson line allows us to define the Wigner function in terms of the kinetic momentumk µ = k µ − eA µ (X). From now on, we will denote the kinetic momentum without the bar to keep the notation light.
We will focus on the construction of the transport equation associated with the vector and axial vector components of the the above two-point function, and define where χ is an index that indicates the helicity/chirality of the particle, and is a chirality projector. Now, simply by using that one can decompose Further, for the constraint / vχ v = 0 for particles, one can deduce that H χ E,v = 0. One can also show that χ v (x)γ ⊥ µ χ v (x) = 0, and thus, J µ,χ (E,v),⊥ (X, k) = 0.
We will thus write our transport equations in terms of the two-point function and its (gauge-covariantly modified) Wigner transform.
A basic ingredient to derive classical or semiclassical transport equations is to perform the gradient expansion, which assumes By doing this, we will neglect consistently gradients of the gauge fields. This does not mean that we are considering only situations of constant background fields, but rather that their variation is consistently neglected, as we won't take into account second order derivatives on X of the two-point Green function.

A. Computation using the OSEFT variables
For our derivation, we substantially follow the approach of Ref. [6], where a chiral transport equation valid for Fermi systems at T = 0 was derived from HDET [18]. Actually, one of the motivations to develop OSEFT in Ref. [1] was to extend the validity of the same derivation at finite temperature, where also antiparticles have to be taken into account. While in a system at finite density and vanishing temperature the Fermi sea provides a natural privileged frame, our derivation will be valid for an arbitrary frame. With some minor technical differences (use of Dirac rather than Weyl fermions, use of local field redefinitions, consideration of non-homogeneous distribution functions), we will find the final form of the chiral transport equation in an arbitrary frame, respectful of reparametrization invariance, and therefore, Lorentz invariance. We will point out an important difference with Ref. [6] in our final results.
We start by considering the equations obeyed by the two-point Green's functions, as follows from the OSEFT Lagrangian. To derive the collisionless transport equation it is enough to consider the tree level equations. These can be expressed as where from the OSEFT Lagrangian we can extract [38] O (0) and we limit our study to operators up to 1/E 2 in the energy expansion.
It is convenient to introduce local field redefinitions, as in Ref. [2], as these simplify quite a lot the computations at higher orders. Local field redefinitions might not be respectful of RI if one considers off-shell quantities, but they will not affect the result of on-shell quantities.
Thus, after the field redefinition the second order differential operator becomes We have checked that these two forms of the second-order Lagrangian lead to an equivalent form of the (on-shell) transport equation.
We now combine the sum and difference of Eqs. (39) and (40), and compute their Wigner transform. For every order in the energy expansion we define however, note that these are matrix equations in the Dirac subspace of the particles. In order to recover the transport equation we trace the above equations We can also derive separate equations for each helicity by mutliplying by the appropriate chiral projector.
Furthermore, from Eqs. (33) and (36) one can write We leave for the Appendix A some details of the computations, and present here our final results. For n = 0 for n = 1 while for n = 2 one gets and We can check that, when computed in the static frame defined by fixing the frame vector as u µ = (1, 0, 0, 0), and using Eq. (11), our results agree with those computed from HDET in Ref. [6] if we replace the chemical potential µ by the energy E, except in what follows.
With the local field redefinition, the factor multiplying the time derivative in the transport equation is 1, while without it one gets a non-trivial factor. We have checked that the same equation is obtained if we normalise the transport equation of Ref. [6] so as to obtain the same normalisation of the time derivative term. We however disagree in the numerical factor of the piece proportional to F νρṽ ρ in Eqs. (53) and (54), in what it is apparently an algebraic mistake.
The numerical factors found above turn out to be essential to derive both the proper form of the dispersion relation, and the consistent form of the anomaly equation.

B. Going backwards to the original variables
Having derived the relevant equations in terms of the OSEFT variables, let us now go back and express them in terms of the original momenta of the full theory.

Dispersion relation
The dispersion relation is fixed after imposing which suggests that the Wigner function can be written as where f χ E,v (X, k) is the particle distribution function, and we have introduced a (2π) factor in order to reproduce, to leading order, the expected density in a QED plasma. We keep the labels E and v in the distribution function, as this function will depend on the on-shell variables, see for example Ref. [2], where it was explicitly seen that close to equilibrium the on-shell energy acts as a sort of chemical potential for the residual momentum. The function K χ fixes then the dispersion relation, to the order considered, and can be read from the I χ,+ functions. In particular, up to order n = 2, Note that we could replace ǫ αβµνṽ β v α = 2ǫ αβµν u β v α in the above expression. The on-shell constraint can be solved to different orders in the energy expansion. To leading order it is while at the following order showing that (v · k) turns out to be subleading in the 1/E expansion when taken on-shell.
It turns out convenient to express the on-shell constraint in terms of the original momentum q µ . Then one can check that it leads to the constraint where S µν χ is the spin tensor defined as if solved up to order 1/E 2 in the OSEFT variables. To see this, we can express Eq. (60) in terms of on-shell and residual momenta. Using and also that we can write for the residual momentum then the spin tensor can be written as We can then easily obtain where in the last expression we used Eq. (59) and the fact that we are considering expansions in powers of 1/E. Furthermore, employing once again the decomposition in Eq. (35) both for k α and F µν , we can express S µν χ F µν in terms of the OSEFT variables Finally, we can replace above the vector u β byṽ β /2, the difference being a higher 1/E effect. Thus, in returning to the original variables we will identify, to order n = 2 accuracy in the where we have defined When the Wigner function is expressed in terms of the original variables, there is still an E dependence. In explicit computations of physical parameters, such as the vector current (see Sec. VI), this E dependence disappears when one finally expresses the whole current in terms of the original variables.

Transport equation
The transport equation is obtained from We will express the transport equation in terms of the original momentum q µ . Let us define which satisfies u · v q = 1. In the absence of gauge fields this vector can be written as If we further consider the on-shell condition at lowest order v · k = 0, then then it is not difficult to realise that v q µ o.s.
If we now we include the gauge fields, after using Eq. (59) we then get v q µ o.s.
which is the combination that appears in the I χ,− functions.
If we define one can write the transport equation in terms of the original variables as where we have used thatṽ ρ = 2u ρ − v ρ q in the last term only. In the absence of the 1/E q corrections, Eq. (76) corresponds to a classical transport equation of a charged fermion in the collisionless limit [39].
After taking into account the on-shell condition Eq. (76) is similar, but not identical, to the one proposed in Ref. [10], see also Refs. [9,13], if we identify their frame vector n µ with our u µ . For homogeneous backgrounds, Eq. (76) contains a term, the piece proportional to S µν χ F νρ v ρ q , which is absent in Eq. (11) of Ref. [10]. It could be eliminated by introducing a new term in the OSEFT Lagrangian at order 1/E 2 , namely, the same that appears in Eq. (43), but changing the (ṽ · D) by (v · D). However, this could only be done at the expense of breaking reparametrization invariance, and ultimately, Lorentz invariance.
For non-homogeneous backgrounds, Eq. (11) of Ref. [10] kept some gradient terms of the gauge fields, and frame vector. The gradient expansion used to reach to the above transport equation was made neglecting gradients of the electromagnetic fields (see Appendix A), which would otherwise naturally emerge in the computations of the functions I χ,− : thus not all the gradient terms were kept in Refs. [9,13], and in a close to thermal equilibrium situation it might be non-consistent to keep those gradient terms while neglecting ∂ 2 X G. Let us consider now our covariant relativistic equation, and write in the frame u µ = (1, 0, 0, 0). Using F i0 = E i , F ij = −ǫ ijk B k , and also that in this frame After considering the on-shell condition, it is not difficult to arrive to where we have defined B i ⊥,q ≡ B i −q i (B ·q). This equation differs from Eq. (13) of Ref. [9], which for homogeneous backgrounds reads Eq. (78) also differs from the transport equation described in Sec. IIB of Ref. [6], although that equation leads to the covariant chiral anomaly equation, while ours lead to the consistent form of the chiral anomaly equation, as we discuss in the following section.

VI. CONSISTENT CURRENT AND CHIRAL ANOMALY EQUATION
In this section we compute both the consistent electromagnetic and chiral currents. For the computation of the latter, the best option is to introduce an artificial chiral gauge field A 5 µ and an artificial gauge field tensor F 5 µν , which are finally sent to zero, as advocated in Ref. [28], and in Ref. [14], for example. Thus we assume that the original QED Lagrangian reads One can proceed with the same derivation of the OSEFT Lagrangian in the presence of the chiral field. After introducing the chiral projectors, it is not difficult to realise that all our equations remain valid if we replace in all our final formulas, in agreement with the prescription of Ref. [14].
The electromagnetic and chiral currents are obtained from the OSEFT action, simply by performing the functional derivatives respectively. Alternatively, one could start with the QED currents, and plug the explicit expression of the Dirac fields in Eq. (3) to finally write the current in terms of the OSEFT fields. For example, considering only the contribution of the particles Using the expression of the H (1) v of Ref. [1] generalised to an arbitrary frame, we find where we have to take into account the local field redefinition Eq. (44), so as to compute the current in the same way as the corrections to the transport equations. A completely analogous computation can be carried out for the chiral current.
At leading order in the energy expansion, one can immediately express the current in terms of the two-point function. After a Wigner transform one finds We can use now the explicit form of the Wigner function at order n = 0, see Eq. (56). If we further make the identification [2,35] E,v then, at leading order the current is expressed as where we have approximated Ev µ ≈ q µ at leading order, and it is understood that the on-shell condition is taken to leading order, thus, without the gauge field contribution. Similarly, the axial current at leading order reads At the following orders in the energy expansion, and due to the presence of derivative terms in the explicit expression of the current, a point-splitting regularisation is needed. This means that we take the fieldχ v at the value y. We then perform the (gauge covariantly modified) Wigner transform, together with the derivative expansion, to finally take the limit y → x. Note that this point-splitting regularization is only needed to properly define the Wigner transform (see, for example the scalar QED example explained in Ref. [34] for the proper definition of the current), and not to regulate ultraviolet problems, which are absent in the two-point function we are studying.
If one considers corrections up to order n = 2 then the vector current reads which, if converted to the original momentum reads For the axial current we get the same expression but the whole integral is multiplied by χ.
In order to get the complete current, the antiparticle contribution has to be added. As mentioned in Sec. IV this can be recovered from the OSEFT particle contribution Eq. (89) by simply replacing v µ ↔ṽ µ and E → −E.
Let us consider the current associated with one single value of the chirality. Using the transport equation (76) and the antisymmetry of the spin tensor it is not difficult to deduce To deduce the form of the chiral anomaly, we will now consider the frame u µ = (1, 0, 0, 0), as then the analysis simplifies quite a lot. We will also consider the situation where, to leading order, the distribution function corresponds to a thermal distribution function, with a chemical potential that depends on the chirality: that is, there is a fermion chiral imbalance in the system. The proof, however, can also be extended to distribution functions which, when the on-shell condition to leading order is considered, are parity invariant. One can express the integral in the r.h.s. of Eq. (91), after taking into account the on-shell condition, as a surface integral. As the distribution function vanishes for |q| → ∞, the only non-vanishing contribution arises for low values of the momenta, where the quasiparticle picture breaks down.
We proceed as in Ref. [5], and Refs. [1,8], and define a sphere centered in |q| = 0 of radius R and then compute the only non-vanishing surface integral At this point, we should consider the contribution of all the chiralities, of both fermions and antifermions so as to obtain the full complete contribution to the axial and vector currents.
We thus assume the following fermion and antifermion distribution functions respectively, to obtain the non-conservation of the chiral current The vector current also has a quantum anomaly also in the presence of chiral gauge fields Eq. (94) gives account of the consistent form of the chiral anomaly equation, rather than its covariant form. We refer the reader to the excellent review [28] that gives very clear explanations about these two different forms of the quantum anomaly. After defining our currents as a functional derivatives of the action, one cannot get anything else than the consistent current.
Unfortunately, the vector current is also non-conserved. It is possible to add the so-called Bardeen counterterms [36] to the quantum action with the choice c 1 = 1 12π 2 and c 2 = 0, and then one can get a vector conserved current [28]. Previous approaches to CKT have shown to provide both the covariant currents and also the covariant form of the chiral anomaly [1][2][3], see also Ref. [14]. One can relate the consistent and covariant currents by adding Chern-Simons currents [28]. It is actually easy to show that under the type I and type III symmetries of RI the distribution function in the OSEFT remains invariant. For example, under type I symmetry the basic two-point function transforms as (see Table I) where we have used that / λ ⊥ / v = −/ v / λ ⊥ , and / v/ v = 0. It then follows that under a type I transformation. Similarly, it is possible to show that the distribution function does not change under a type III transformation.
The Green function Eq. (37) used in our derivation of the transport equation has however a non-trivial transformation under type II symmetry. Using the transformation rules of Table I we obtain In OSEFT χ v (y)γ µ ⊥ χ v (x) = 0. After the Wigner transform, together with the gradient expansion, we end up with Taking into account the definition of the two-point function at order 1/E involves the current density that might be computed (see the integrand of Eq. (89) at order 1/E) this implies that the distribution function should change as under a type II transformation.
In terms of the original variables, one then gets Taking into account that ǫ µ ⊥ /2 = u ′µ − u µ , we see that Eq. (103) agrees with the infinitesimal form of the side jump transformation first discussed in Ref. [23] in the absence of gauge fields, later generalised in the presence of the gauge fields in Ref. [9].

VIII. DISCUSSION
We have derived from OSEFT the corrections to the classical transport equations associated with on-shell massless charged fermions and antifermions. We have seen how from the proposed equations one can derive the consistent form of the chiral anomaly equation when considering a chiral imbalance system in thermal equilibrium. Our formulation turns out to be the proper generalisation of the HDET approach to chiral transport theory of Ref. [6], but valid also for finite temperature systems and formulated in an arbitrary frame. The study of reparametrization invariance of the theory allows us to claim that the results are consistent with Lorentz symmetry, even if the kinetic equation depends on a frame vector. We have also deduced the side jumps of the distribution function of the theory from the transformation rule under RI of the OSEFT quantum fields.
Let us insist that when we consider the frame vector as u µ = (1, 0), our equations almost agree with those of Ref. [6], except in a couple of factors, in what apparently was an algebraic mistake. It is however important to stress that the results obtained either in Ref. [6] or in this manuscript do not match exactly with the transport equation in Sec. IIB of Ref. [6], which were obtained starting with a corrected form of the classical point-particle action, with modified Poisson brackets. This starting point can be justified by performing a Foldy-Wouthuysen diagonalization of the quantum Dirac Hamiltonian, as seen in Ref. [1]. However, the same exact form of the transport equation is not obtained if the starting point is a quantum field theory. Let us stress that in such a formulation one obtains the covariant form of the chiral anomaly, as the chiral current is not defined by performing a functional derivative of an action, but from the equation obeyed by the current in the transport approach.
The question remains whether there can be more than one possible transport equation describing equally well the same system. The Foldy-Wouthuysen diagonalization used in Ref. [1] suggests that the starting quantum fields used there or those used in our OSEFT approach are not the same beyond the classical limit approximation. Thus, probably it is not so surprising that one does not end up with the same exact form of the corresponding kinetic equations, while the two approaches give an equivalent description of the system. It is probably more surprising the discrepancies we obtained from the results of Refs. [9,10,13], obtained from massless QED, assuming homogenous gauge field backgrounds. OSEFT only helps in organising the quantum field theory computation at large energies, as it has already been checked in the computation of Feynman diagrams at high T [2,37]. We cannot comment on the possible origin of these discrepancies, although it seems that the approach should also lead to the consistent form of the chiral anomaly, rather than its covariant form, as claimed in Ref. [10].
While in this manuscript we have focused our attention to the collisionless form of the transport equation, a much more challenging task is to derive the collision terms from OSEFT, such that the Lorentz symmetry is respected, and the side jumps are properly described. This will be the subject of a different project.
Acknowledgments: We are indebted to J. Soto for many discussions during the evolution of this project. We are also specially thankful to K. Landsteiner, for different discussions on the difference between the consistent and covariant forms of the chiral anomaly. We ackkowledge interesting discussions with M. Beneke  We provide in this Appendix some details of the computation of the I χ,± functions. We take here e = 1 for simplicity.
We start from the equation of motion for quantum fields χ v and similarly its hermitian conjugate for y. By adding and subtracting them, we can build equations for the two-point function. For each piece we isolate the different possible Dirac structures, so we write then taking the trace of Eq. (46) one gets x,µν ± β (n) * y,µν Tr σ µν / v 2 S E,v (x, y) , For the α and β coefficients we find (after neglecting terms of higher order in the gradient expansion like ∂ X α F µν ) We now perform the change of variables to the center of mass and relative coordinates X, s.
As an example, we can work out the lowest order function. If here k µ denotes the canonical momentum then where nowk µ = k µ − A µ is the canonical momentum.