Consistent Relativistic Chiral Kinetic Theory: a derivation from OSEFT
aa r X i v : . [ h e p - ph ] O c t Consistent relativistic chiral kinetic theory: A derivation fromon-shell effective theory
Stefano Carignano ∗ and Cristina Manuel † Instituto de Ciencias del Espacio (ICE, CSIC)C. Can Magrans s.n., 08193 Cerdanyola del Vall`es, Catalonia, Spain andInstitut d’Estudis Espacials de Catalunya (IEEC)C. Gran Capit`a 2-4, Ed. Nexus, 08034 Barcelona, Spain
Juan M. Torres-Rincon ‡ Department of Physics and Astronomy, Stony Brook University,Stony Brook, New York 11794-3800, USA. (Dated: October 16, 2018)We formulate the on-shell effective field theory (OSEFT) in an arbitrary frameand study its reparametrization invariance, which ensures that it respects Lorentzsymmetry. In this formulation the OSEFT Lagrangian looks formally equivalent tothe sum over lightlike velocities of soft collinear effective field theory in the Abelianlimit, but differences remain in the scale of the gauge fields involved in the two effec-tive theories. We then use the OSEFT Lagrangian expanded in inverse powers of theon-shell energy to derive how the classical transport equations for charged masslessfermions are corrected by quantum effects, as derived from quantum field theory. Weprovide a formulation in a full covariant way and explain how the consistent form ofthe chiral anomaly equation can be recovered from our results. We also show howthe side-jump transformation of the distribution function associated with masslesscharged fermions can be derived from the reparametrization invariance transforma-tion rules of the OSEFT quantum fields. Finally, we discuss the differences in ourresults with respect to others found in the literature. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] . INTRODUCTION
In this paper we use the so-called on-shell effective field theory (OSEFT) [1–3] to provide aderivation of the transport equations obeyed by charged chiral fermions beyond the classicallimit approximation.A formulation of transport theory for chiral fermions has been developed in Refs. [4–7],starting with the action of a point particle modified by the Berry curvature, together with amodified Poisson bracket structure. Other alternative approaches to derive the same transportequation can be found in the literature [1, 8–17].The first derivation of chiral kinetic theory (CKT) from quantum field theory was madein Ref. [6] for systems at finite density and zero temperature, using the so-called high densityeffective field theory (HDET) [18]. OSEFT was actually proposed to provide a similar deriva-tion that could be valid also in a thermal background, where antifermions are also relevantdegrees of freedom. Regardless of the background, transport equations describe the propa-gation of an on-shell quasiparticles, and therefore it seems natural to use for their derivationan effective field theory approach that describes only the propagation of on-shell degrees offreedom, as OSEFT, while off-shell modes are integrated out. Let us stress that the notionof on-shell quasiparticle depends on the energy scales one is looking at in the system underconsideration. It is well known that for plasmas at finite temperature T only the high energymodes of order T can be considered as quasiparticles and their evolution studied with classicaltransport equations [19–21], while the same picture does not apply to lower energy modes.To get corrections to the classical point-particle picture described above from quantum fieldtheory, one simply has to study how the off-shell modes modify the evolution of the highlyenergetic modes. These corrections are taken into account in the OSEFT Lagrangian, andexpressed as operators of increasing dimension over powers of the on-shell energy scale, so thatthese modifications can be described with the accuracy one desires. The OSEFT Lagrangiancan then be used to derive how the classical transport picture is modified, by using for theirderivation 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 equationsin full covariant form, and derive their properties under Lorentz transformations. While theinitial proposals of CKT were not given in a covariant form, it was soon realized that itwould have peculiar properties under Lorentz transformations [22, 23], especially seen whenformulating two-body collisions, but also expressed in the so-called side-jump behavior of the2istribution function of CKT, that expresses that it is frame dependent.We present in this paper 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 thateffective field theory. While previous formulations of OSEFT were given in the preferred frameof the thermal bath, we generalize it to an arbitrary frame, introducing the frame vector u µ . The resulting OSEFT Lagrangian then looks formally equivalent to that correspondingto a sum over velocities of the so-called soft collinear effective field theory (SCET) [24–27],although there are some differences, as will be discussed in the following. We further studythe reparametrization invariance of OSEFT, that ensures that our formalism is respectful ofLorentz invariance.We compute both the vector current and axial current in the OSEFT, by taking functionalderivatives to the action, and take these expressions to deduce the corresponding values in thetransport framework, which requires a Wigner transformation of a two-point function, togetherwith a gradient expansion. As very clearly explained in the review [28], such a definition canonly lead to the consistent form of the chiral anomaly, rather than the covariant form. Wecheck from our expressions that this is indeed the case.Our final form of the relativistic chiral transport equation mainly differs from that intro-duced in Refs. [9, 10, 13], in pieces that may be subleading when considering effects close tothermal equilibrium, but that might be relevant for studies off equilibrium, and also in the gra-dient terms of the gauge fields. It also differs, when fixing the frame, with the chiral transportequation obtained from the modified form of the one-point particle action.Our paper is organized 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 fieldtheory. In Sec. III we study the reparametrization invariance of this effective field theory, abasic ingredient to show that it is respectful of Lorentz symmetry. In Sec. IV, we introduce thebasic two-point function in the OSEFT that will be used to derive the basic set of transportequations. The main content of the paper is in Sec. V, with the derivation of the collisionlesstransport equation, first using the OSEFT variables in Sec. V A, and then expressed in terms ofthe QED original variables in Sec. V B. In Sec. VI, we derive both the vector and axial currentobtained in the OSEFT approach, and check that they obey the consistent form of the quantumanomalies. In Sec. VII we derived the side-jump transformation of the distribution functionfrom the reparametrization invariance transformations of the OSEFT quantum fields. We3onclude in Sec. VIII, where we summarize our main findings, and give a possible interpretationof the origin of the discrepancy of our results with alternative approaches. In Appendix A wegive some details of our computations, while in Appendix B we show how to obtain the chiralmagnetic effect from our formulation.We use natural units ~ = c = k B = 1 and metric conventions g µν = (1 , − , − , − II. OSEFT IN AN ARBITRARY FRAME AND SCET
Let us review the OSEFT as originally formulated [1, 2], introducing the basic fields andnotation. Let us recall that the propagation of an on-shell massless fermion is described by itsenergy p , with p >
0, and the lightlike 4-velocity v µ = (1 , v ), where v is three-dimensional unitvector, and thus its 4-momentum is p µ = pv µ . However, for a fermion close to being on-shell,its 4-momentum can be expressed as q µ = pv µ + k µ , (1)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 bedone as follows, q µ = − p ˜ v µ + k µ , (2)where ˜ v µ = (1 , − v ) .The Dirac field can be written as ψ v, ˜ v = e − ipv · x (cid:16) P v χ v ( x ) + P ˜ v H (1)˜ v ( x ) (cid:17) + e ip ˜ v · x (cid:0) P ˜ v ξ ˜ v ( x ) + P v H (2) v ( x ) (cid:1) , (3)where the basic OSEFT quantum fields obey P v χ v = χ v , P ˜ v χ v = 0 , (4) P ˜ v ξ ˜ v = ξ ˜ v , P v ξ ˜ v = 0 , (5)and the particle/antiparticle projectors are expressed as P v = 12 γ · v γ , P ˜ v = 12 γ · ˜ v γ . (6)It is possible to integrate out the H (1 , fields of the QED Lagrangian [1], to have an effectivetheory for the fields χ v and ξ ˜ v only. 4f we assume that the physical phenomena we aim to describe are dominated by the con-tribution of on-shell particles, then the corresponding OSEFT Lagrangian can be written as asum over the different values of the on-shell momenta as L = X p, v L p, v , (7)where the precise meaning of the sum displayed in Eq. (7) is not needed at this stage (we willcome back to this point later on; see also Ref. [2]), and L p, v = L p,v + e L p, ˜ v = ¯ χ v ( x ) (cid:18) i v · D + i /D ⊥ p + i ˜ v · D i /D ⊥ (cid:19) γ χ v ( x )+ ¯ ξ ˜ v ( x ) (cid:18) i ˜ v · D + i /D ⊥ − p + iv · D i /D ⊥ (cid:19) γ ξ ˜ v ( x ) , (8)where D µ = ∂ µ + ieA µ is the covariant derivative, /D ⊥ = P µν ⊥ γ µ D ν and P µν ⊥ = g µν −
12 ( v µ ˜ v ν + v ν ˜ v µ ) , (9)is minus the transverse projector to v , written in covariant form. Note that with our conven-tions k ⊥ = P µν ⊥ k µ k ν = − k ⊥ . From now on, and as done in Ref. [2], whenever we write a tensorwith the symbol ⊥ , it means that a transverse projector applies to all its Lorentz indices. Ifonly the transverse projector is applied to one of the indices, we will write ⊥ only affectingthat 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 energiesof the on-shell particles in Eq. (1) are measured in the same frame where, for example, thethermal bath is defined. If we want to express the same OSEFT Lagrangian in an arbitraryframe, we will then have to introduce a timelike vector u µ which defines that frame. Then onecould write all the above different equations simply by replacing p → u µ p µ ≡ E , γ → γ µ u µ . (10)With our specific choice of variables v µ and ˜ v µ , then it is not difficult to see that u µ = v µ + ˜ v µ . (11)Note that in OSEFT u µ is not an independent vector, once v µ and ˜ v µ have been defined. Whilein the static frame we chose a particular definition of the vectors v µ and ˜ v µ , which implicitly5ssumed that u µ = (1 , , , v = ˜ v = 0 , v · ˜ v = 2 . (12)Thus, u · v = 1 and u = 1 are automatically fulfilled.In our formulation of the OSEFT in an arbitrary frame, we will sometimes use ˜ v µ , andsometimes we will use u µ . The last option is convenient, as in kinetic theory it may appearalso 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 P v = 12 /v /u = 14 /v / ˜ v (13) P ˜ v = 12 / ˜ v /u = 14 / ˜ v /v , (14)where we used that /v /v = / ˜ v / ˜ v = 0.The OSEFT Lagrangian in a general frame is then written down as L = X E,v ( L E,v + L − E, ˜ v ) , (15)where L E,v + L − E, ˜ v = ¯ χ v ( x ) (cid:18) i v · D + i /D ⊥ E + i ˜ v · D i /D ⊥ (cid:19) / ˜ v χ v ( x )+ ¯ ξ ˜ v ( x ) (cid:18) i ˜ v · D + i /D ⊥ − E + iv · D i /D ⊥ (cid:19) /v ξ ˜ v ( x ) . (16)where we have used that /vχ v = 0 , / ˜ vξ ˜ v = 0 . It is noteworthy that Eq.(16) formally looks similar to the Lagrangian of soft-collineareffective field theory [24–27]. The corresponding projectors Eqs. (13, 14) are also those used inSCET. We note that the explicit forms of the OSEFT and SCET Lagrangians differ because ofour different convention in defining the quantum fluctuating fields: in SCET, the exponentialterms of Eq. (3) have been included in the quantum fields of the effective theory. We alsoexplicitly separate the contribution of particles and antiparticles. Further, we recall that weare considering an effective field theory for QED, while SCET is an effective field theory forQCD.After noticing the above formal similarities of SCET and OSEFT when the latter is for-mulated in an arbitrary frame, it has to be stressed that they are still different effective field6heories. SCET was originally formulated to describe the physics associated with highly ener-getic jets in vacuum, and there are only two lightlike vectors in the theory, v µ and ˜ v µ , fixedby the direction of the jet. In SCET, the covariant derivatives are associated with collinearand ultrasoft gauge fields. OSEFT was in principle developed to describe many body particlesystems, close to thermal equilibrium, where one can consider having many on-shell particlesand their propagation in the background of soft gauge fields. Thus, for a fixed value of theenergy there might be particles moving in all arbitrary (lightlike) directions, and a sum over v µ is displayed in the final Lagrangian, which is absent in SCET. In OSEFT, the covariantderivatives we use mainly contain soft gauge fields.OSEFT also uses a different notation, which makes clear that its main goal is to make ananalytical expansion in powers of the inverse of the on-shell energy 1 /E . At finite tempera-ture and/or density we will obtain different expressions multiplied by a particle distributionfunction. After integration over momenta, this expansion on the inverse of the on-shell energywill turn out to give an expansion in powers of the inverse of the temperature and/or chemicalpotential [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 ofOSEFT, such as that of reparametrization invariance, which will be discussed in the followingsection. III. REPARAMETRIZATION INVARIANCE OF OSEFT
Reparametrization invariance (RI) is the symmetry associated with the ambiguity of thedecomposition of the momentum q µ performed in Eq. (1). If M µν defines the six Lorentzgenerators of SO (3 , { v µ M µν , u µ M µν } , or, equivalently, { v µ M µν , ˜ v µ M µν } . However, itis possible to show that the OSEFT Lagrangian is RI invariant, which is equivalent to sayingthat is Lorentz invariant. Let us stress that this reduces to the study of the RI of SCETfor every sector of the theory defined by the vectors v µ and ˜ v µ , something which has beenextensively investigated [29]. The fact that the covariant derivatives displayed in SCET andOSEFT contain gauge fields of different scales does not, however, affect the proof of RI, whichturns 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 the7ame independent of the choice of the parameters used to define the effective field theory; thus, ψ v, ˜ v ( x ) = ψ ′ v ′ , ˜ v ′ ( x ) . (17)As in SCET, we will see that the effective field theory action remains invariant underinfinitesimal changes of the vectors v µ and ˜ v µ that preserve their basic properties expressed inEq. (12). It is possible to show that the OSEFT Lagrangian is invariant under the followingsymmetries(I) ( v µ → v µ + λ µ ⊥ ˜ v µ → ˜ v µ (II) ( v µ → v µ ˜ v µ → ˜ v µ + ǫ µ ⊥ (III) ( v µ → (1 + α ) v µ ˜ v µ → (1 − α )˜ v µ (18)where { λ µ ⊥ , ǫ µ ⊥ , α } are five infinitesimal parameters, and v · λ ⊥ = v · ǫ ⊥ = ˜ v · λ ⊥ = ˜ v · ǫ ⊥ = 0.Please note that the transformation rule of the vector u µ can be deduced from Eq. (11).Just to have a flavor of the meaning of the above symmetries, let us imagine one fixes thevalues of the two lightlike vectors as v µ = (1 , , ,
1) and ˜ v µ = (1 , , , − Q ± = J ± K , Q ± = J ± K , and K ,where J i and K i are the generators of rotations and boosts, respectively. Then, type I refers tothe combined action of an infinitesimal boost in the x ( y ) direction and a rotation around the y ( x ) axis, such that ˜ v µ is left invariant, with generators ( Q − , Q +2 ). Type II transformationsare similar but ( Q +1 , Q − ) leave v µ invariant, while type III is a boost along the direction 3, K .It is also worth it to note that the the generators ( Q +1 , Q − , J ) obey the SE (2) Lie algebra,that is the symmetry group of the two-dimensional Euclidean plane. They correspond to whatis known as the Wigner little group associated with the vector p µ = pv µ [30], see also Refs. [31–33]. Similarly, the generators ( Q − , Q +2 , J ) correspond to Wigner’s little group associated with p µ = − p ˜ v µ (antiparticles). As discussed in Ref. [30] these Wigner translations are associatedwith shifts of the trajectory of finite wave packets of massless particles proportional to theparticle’s helicity.It is possible to check easily that our Lagrangian is invariant under the above three RItransformations [29], which formally is equivalent to say that it is Lorentz invariant. Let usdiscuss these briefly, as they are the same RI symmetries of SCET. We will mainly focus now onwhat our different notation implies. We will concentrate in the following in the particle sector,as for antiparticles things works analogously after trivial changes (namely, u · p → − u · p and v µ ↔ ˜ v µ ). We will also see that the type II symmetry will allow us to generate the side-jumpsthat were discussed in the framework of chiral kinetic theory in Ref. [22]. This point will bediscussed in Sec. VII. 8et us first start with type I symmetry. The change in the vector v µ implies a relabeling ofwhat is called on-shell and residual parts of the momentum defined in Eq. (1). After a type Isymmetry the on-shell part and residual momenta change as( u · p ) v µ → ( u · p ) v µ + 12 ( λ ⊥ · p ) v µ + ( u · p ) λ µ ⊥ , (19) k µ → k µ −
12 ( λ ⊥ · p ) v µ − ( u · p ) λ µ ⊥ , (20)respectively. This implies that under a type I transformation the covariant derivatives actingon the fluctuating fields also transform.Type II symmetry implies that the new on-shell and residual momenta change as( u · p ) v µ → ( u · p ) v µ + 12 ( p · ǫ ⊥ ) v µ , (21) k µ → k µ −
12 ( p · ǫ ⊥ ) v µ , (22)while the type III transformation leads to the changes( u · p ) v µ → ( u · p ) v µ (1 + 2 α ) − α (˜ v · p ) v µ , (23) k µ → k µ − αEv µ + α (˜ v · p ) v µ , (24)in the on-shell and residual momenta, respectively.In Table I, we summarize the transformation rules under all three types of transformations. Type I Type II Type III v µ v µ + λ µ ⊥ v µ v µ (1 + α )˜ v µ ˜ v µ ˜ v µ + ǫ µ ⊥ ˜ v µ (1 − α ) u µ u µ + λ µ ⊥ u µ + ǫ µ ⊥ u µ (1 − α ) + αv µ E E + λ ⊥ · p E + ( ǫ ⊥ · p ) E (1 + α ) − α (˜ v · p ) D µ D µ + iEλ ⊥ µ + i ( λ ⊥ · p ) v µ D µ + i ( ǫ ⊥ · p ) v µ D µ + 2 iαE v µ − iα (˜ v · p ) v µ ( v · D ) ( v · D ) + λ ⊥ · D ⊥ ( v · D ) ( v · D )(1 + α )(˜ v · D ) (˜ v · D ) + iλ ⊥ · p (˜ v · D ) + i ǫ ⊥ · p + ǫ ⊥ · D ⊥ (˜ v · D )(1 − α ) + 4 iEα − iα (˜ v · p ) D ⊥ µ D ⊥ µ − λ ⊥ µ (˜ v · D ) − ˜ v µ λ ⊥ · D ⊥ + iEλ ⊥ µ D ⊥ µ − ǫ ⊥ µ ( v · D ) − v µ ǫ ⊥ · D ⊥ D ⊥ µ P v P v + /λ ⊥ / ˜ v P v − /ǫ ⊥ /v P v χ v ( x ) (cid:0) /λ ⊥ / ˜ v (cid:1) χ v ( x ) (cid:16) /ǫ ⊥ E + i ˜ v · D i /D ⊥ (cid:17) χ v ( x ) χ v ( x ) TABLE I: Transformation rules in OSEFT under RI transformations of types I, II and III .
The OSEFT Lagrangian is invariant under these three RI transformations: [29] δ (I) L E,v = δ (II) L E,v = δ (III) L E,v = 0 . (25)9n explicit computations of Feynman diagrams, or derivations of transport equations, wewill expand the Lagrangian in power series of 1 /E . While Eq. (25) is exact to all orders in a1 /E expansion, in a perturbative analysis in 1 /E it is important to note that RI invarianceimplies that different terms in the expansion are connected by symmetry. This comes fromthe 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. Underparity, charge conjugation and time reversal, the basic OSEFT fields transform as χ v ( x ) → γ χ ˜ v (˜ x P ) , ξ ˜ v ( x ) → γ ξ v (˜ x P ) (26) χ v ( x ) → − iγ ξ ∗ v ( x ) , ξ ˜ v ( x ) → − iγ χ ∗ ˜ v ( x ) (27) χ v ( x ) → − γ γ χ ˜ v ( − ˜ x T ) , ξ ˜ v ( x ) → − γ γ ξ v ( − ˜ x T ) (28)respectively, where if x µ = ( x , x ), then ˜ x µP = ( x , − x ), and ˜ x µT = ( − x , x ).There is also a spin symmetry, which is not a SU (2) symmetry but a U (1) symmetry, whichcorresponds to helicity [33]. IV. WIGNER FUNCTION IN THE OSEFT
We focus our attention here on the basic Wigner function used in the following part of thepaper for the derivation of the transport equations from OSEFT. We will use the Keldysh-Schwinger 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 representedby a 2 × S E,v ( x, y ) = S cE,v ( x, y ) S For our derivation, we substantially follow the approach of Ref. [6], where a chiral transportequation valid for Fermi systems at T = 0 was derived from HDET [18]. Actually, one of themotivations to develop OSEFT in Ref.[1] was to extend the validity of the same derivation atfinite temperature, where also antiparticles have to be taken into account. While in a systemat 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(the use of Dirac rather than Weyl fermions, use of local field redefinitions, and considerationof nonhomogeneous distribution functions), we will find the final form of the chiral trans-port equation in an arbitrary frame, respectful of reparametrization invariance, and therefore,Lorentz invariance. We will point out an important difference from Ref.[6] in our final results.We start by considering the equations obeyed by the two-point Green’s functions, as followsfrom the OSEFT Lagrangian. To derive the collisionless transport equation it is enough toconsider the tree level equations. These can be expressed as X n =0 (cid:0) O ( n ) x (cid:1) S E,v ( x, y ) = 0 , (39)and X n =0 S E,v ( x, y )( O ( n ) y ) † = 0 , (40)12here from the OSEFT Lagrangian we can extract [42] O (0) x = i v · D / ˜ v , (41) O (1) x = − E (cid:16) D ⊥ + e σ µν ⊥ F µν (cid:17) / ˜ v , (42) O (2) x = − E i /D ⊥ ( i ˜ v · D ) i /D ⊥ / ˜ v E (cid:16) (cid:2) /D ⊥ , (cid:2) i ˜ v · D , /D ⊥ (cid:3)(cid:3) + (cid:8) ( /D ⊥ ) , i ˜ v · D (cid:9) (cid:17) / ˜ v , and we limit our study to operators up to 1 /E in the energy expansion.It is convenient to introduce local field redefinitions to eliminate the temporal derivative inEq. (43), as in Ref. [2], as these simplify quite a lot the computations at higher orders [43].Local field redefinitions might not be respectful of RI if one considers off-shell quantities, butthey will not affect the result of on-shell quantities. Thus, after the field redefinition χ v → χ ′ v = /D ⊥ E ! χ v , (44)the second order differential operator becomes O (2) x, LFR = 18 E (cid:16) (cid:2) /D ⊥ , (cid:2) i ˜ v · D , /D ⊥ (cid:3)(cid:3) − n D ⊥ + e σ µν ⊥ F µν , ( iv · D − i ˜ v · D ) o (cid:17) / ˜ v . (45)We have checked that these two forms of the second-order Lagrangian lead to an equivalentform of the (on-shell) transport equation.We now combine the sum and difference of Eqs. (39) and (40), and compute their Wignertransform. For every order in the energy expansion we define I ( n ) ± = Z d se ik · s (cid:0) O ( n ) x U ( x, y ) S E,v ( x, y ) ± S E,v ( x, y ) U ( x, y ) O ( n ) † y (cid:1) , (46)however, note that these are matrix equations in the Dirac subspace of the particles. In orderto recover the transport equation we trace the above equationsTr( I ( n ) ± ) = X χ = ± I ( n ) χ, ± . (47)We can also derive separate equations for each helicity by multiplying by the appropriatechiral projector.Furthermore, from Eqs. (33) and (36) one can write G χE,v ( X, k ) = 12 (˜ v · J χE,v )( X, k ) . (48)13e leave for the Appendix A some details of the computations, and present here our finalresults. For n = 0, I (0) χ, + = 4 k · v G χE,v ( X, k ) , (49) I (0) χ, − = 2 iv µ [ ∂ µX − eF µν ( X ) ∂ k,ν ] G χE,v ( X, k ) , (50)for n = 1, I (1) χ, + = 2 E (cid:16) k ⊥ − eχ ǫ αβµν ˜ v β v α F ⊥ µν (cid:17) G χE,v ( X, k ) , (51) I (1) χ, − = 2 iE k µ ⊥ [ ∂ X,µ − eF µν ∂ νk ] G χE,v ( X, k ) , (52)while for n = 2, one gets I (2) χ, + = − E (cid:18)h k ⊥ − eχ ǫ αβµ ⊥ ν ⊥ ˜ v β v α F µν i ˜ v · k − v · k eχ ǫ αβµ ⊥ ν ⊥ ˜ v β v α F νρ ˜ v ρ k µ (cid:19) G χE,v ( X, k ) , (53)and I (2) χ, − = 2 E (cid:18) − k µ ⊥ ˜ v · k − v · k h k ⊥ − eχ ǫ αβδγ ˜ v β v α F ⊥ δγ i ( v µ − ˜ v µ ) − eχ ǫ αβµ ⊥ ν ⊥ ˜ v β v α F νρ ˜ v ρ (cid:19) × i [ ∂ X,µ − eF σµ ( X ) ∂ k,σ ] G χE,v ( X, k ) . (54)We can check that, when computed in the static frame defined by fixing the frame vectoras u µ = (1 , , , µ by the energy E , except in what follows.With the local field redefinition, the factor multiplying the time derivative in the transportequation is 1, while without it one gets a nontrivial factor. We have checked that the sameequation is obtained if we normalize the transport equation of Ref. [6] so as to obtain the samenormalization of the time derivative term. We, however, disagree in the numerical factor of thepiece proportional to F νρ ˜ v 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 deriving both the proper formof the dispersion relation, and the consistent form of the anomaly equation. B. Going backward to the original variables Having derived the relevant equations in terms of the OSEFT variables, let us now go backand express them in terms of the original momenta of the full theory.14 . Dispersion relation The dispersion relation is fixed after imposing I (0) χ, + + I (1) χ, + + I (2) χ, + = 0 , (55)which suggests that the Wigner function can be written as G χE,v ( X, k ) = 2 πδ ( K χ ) f χE,v ( X, k ) , (56)where f χE,v ( X, k ) is the particle distribution function, and we have introduced a (2 π ) factorin order to reproduce, to leading order, the expected density in a QED plasma. We keepthe labels E and v in the distribution function, as this function will depend on the on-shellvariables; see for example Ref.[2], where it was explicitly seen that close to equilibrium theon-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, K χ = 2 k · v + 1 E (cid:16) k ⊥ − eχ ǫ αβµν ˜ v β v α F ⊥ µν (cid:17) − E (cid:18)h k ⊥ − eχ ǫ αβµ ⊥ ν ⊥ ˜ v β v α F µν i ˜ v · k − v · k eχ ǫ αβµ ⊥ ν ⊥ ˜ v β v α F νρ ˜ v ρ k µ (cid:19) . (57)Note that we could replace ǫ αβµν ˜ v β v α = 2 ǫ αβµν u β v α in the above expression. The on-shellconstraint can be solved to different orders in the energy expansion. To leading order it issimply 2 k · v = 0 , (58)while at the following order,2 k · v + 1 E (cid:16) k ⊥ − eχ ǫ αβµν ˜ v β v α F ⊥ µν (cid:17) = 0 , (59)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 q − eS µνχ F µν = 0 , (60)where S µνχ is the spin tensor defined as S µνχ = χ ǫ αβµν u β q α q · u ) , (61)15f solved up to order 1 /E in the OSEFT variables. To see this, we can express Eq. (60) interms of on-shell and residual momenta. Using E q ≡ q · u = E + k · u , (62)and also that we can write for the residual momentum k µ = k µ ⊥ + 12 ( v · k )˜ v µ + 12 (˜ v · k ) v µ , k = k ⊥ + ( v · k )(˜ v · k ) , (63)then the spin tensor can be written as S µνχ = χ ǫ αβµν u β (cid:18) v α + k ⊥ α E (cid:19) + O (cid:18) E (cid:19) . (64)We can then easily obtain q − eS µνχ F µν = 2 E (cid:20) v · k + 12 E (cid:0) k ⊥ − eS µνχ F µν (cid:1) (cid:18) − (˜ v · k )2 E (cid:19)(cid:21) + O (cid:18) E (cid:19) , (65)where in the last expression we used Eq. (59) and the fact that we are considering expansionsin powers of 1 /E . Furthermore, employing once again the decomposition in Eq. (35) both for k α and F µν , we can express S µνχ F µν in terms of the OSEFT variables S µνχ F µν = χ ǫ αβµν u β (cid:18) v α + k ⊥ α E (cid:19) (cid:0) F ⊥ µν + F µ ⊥ ρ ˜ v ρ v ν + F µ ⊥ ρ v ρ ˜ v ν (cid:1) + O (cid:18) E (cid:19) (66)Finally, we can replace the above vector u β by ˜ v β / 2, the difference being a higher 1 /E effect.This can be checked by noticing that v µ A µ ≪ ˜ v µ A µ . Note that the condition Eq. (58) involvesthe kinetic, rather than canonical, momentum, which implies that not all the vector gaugefield components are equally relevant in the 1 /E expansion.Under these conditions one can then check that Eq. (65) becomes exactly EK χ . Eq. (55)thus enforces the on-shell condition Eq. (60), as anticipated.Thus, in returning to the original variables, we will identify, to order n = 2 accuracy in the1 /E expansion, G χE,v ( X, k ) = (2 π ) δ ( K χ ) f χE,v ( X, k ) = (2 π ) E δ ( EK χ ) f χE,v ( X, k ) = πE δ + ( Q χ ) f χ ( X, q ) , (67)where we have defined δ + ( Q χ ) = δ (cid:0) q − eS µνχ F µν (cid:1) θ ( E q ) . (68)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 (seeSec. VI), this E dependence disappears when one finally expresses the whole current in termsof the original variables. 16 . Transport equation The transport equation is obtained from I (0) χ, − + I (1) χ, − + I (2) χ, − = 0 . (69)We will express the transport equation in terms of the original momentum q µ . Let us definethe vector v qµ ≡ q µ E q = EE q v µ + k µ E q , (70)which satisfies u · v q = 1. In the absence of gauge fields this vector can be written as v qµ = v µ + k µ − v µ ( k · u ) E − ( k · u ) k µ − v µ ( k · u ) E + · · · (71)If we further consider the on-shell condition at lowest order v · k = 0, then k µ − v µ ( k · u ) (cid:12)(cid:12)(cid:12) o . s . = k µ ⊥ , (72)and it is not difficult to realize that v qµ (cid:12)(cid:12)(cid:12) o . s . = v µ + k µ ⊥ E − ( k · ˜ v ) k µ ⊥ E + v µ − ˜ v µ E k ⊥ + O ( 1 E ) . (73)If we now we include the gauge fields, after using Eq. (59) we then get v qµ (cid:12)(cid:12)(cid:12) o . s . = v µ + k µ ⊥ E − ( k · ˜ v ) k µ ⊥ E + v µ − ˜ v µ E (cid:16) k ⊥ − eχ ǫ αβµν ˜ v β v α F ⊥ µν (cid:17) + O ( 1 E ) (74)which is the combination that appears in the I χ, − functions.If we define ∆ µ ≡ ∂ µX − eF µν ( X ) ∂ q,ν , (75)one can write the transport equation in terms of the original variables as (cid:18) v qµ − e E q S µνχ F νρ (cid:0) u ρ − v ρq (cid:1)(cid:19) ∆ µ f ( X, q ) δ + ( Q ) = 0 , (76)where we have used that ˜ v ρ = 2 u ρ − v ρq 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 thecollisionless limit [44].After taking into account the on-shell condition, Eq. (76) is similar, but not identical, tothe one proposed in Ref. [10], see also Refs. [9, 13], if we identify their frame vector n µ withour u µ . For homogeneous backgrounds, Eq. (76) contains a term, the piece proportional to17 µνχ F νρ v ρq , which is absent in Eq. (11) of Ref. [10]. It could be eliminated by introducing a newterm in the OSEFT Lagrangian at order 1 /E , namely, the same that appears in Eq. (43), butchanging the (˜ v · D ) by ( v · D ). However, this could only be done at the expense of breakingreparametrization invariance and, ultimately, Lorentz invariance.For nonhomogeneous backgrounds, Eq. (11) of Ref. [10] kept some gradient terms of thegauge fields and frame vector. The gradient expansion used to reach to the above transportequation was made by neglecting gradients of the electromagnetic fields (see Appendix A),which would otherwise naturally emerge in the computations of the functions I χ, − ; thus, notall the gradient terms were kept in Refs. [9, 13], and in a close to thermal equilibrium situation,it might be nonconsistent to keep those gradient terms while neglecting ∂ X G .Let us consider now our covariant relativistic equation and write it in the frame u µ =(1 , , , F i = E i , F ij = − ǫ ijk B k , and also S µνχ → S ijχ = χ ǫ ijk q k q , S µνχ F µν = − χ B · q q . (77)After considering the on-shell condition, it is not difficult to arrive at ∆ + ˆq i (cid:18) eχ B · ˆq q (cid:19) ∆ i + eχ ǫ ijk E j ˆ q k − B i ⊥ , q q ∆ i ! f χ ( X, q ) = 0 , (78)where we have defined B i ⊥ , q ≡ B i − ˆq i ( B · ˆq ). This equation differs from Eq. (13) of Ref. [9],which for homogeneous backgrounds reads (cid:18) ∆ + ˆq i (cid:18) eχ B · ˆq q (cid:19) ∆ i + eχ ǫ ijk E j ˆ q k q ∆ i (cid:19) f χ ( X, q ) = 0 . (79)Eq. (78) also differs from the transport equation described in Sec. IIB of Ref. [6], althoughthat equation leads to the covariant chiral anomaly equation, while ours leads to the consistentform 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. Forthe computation of the latter, the best option is to introduce an artificial chiral gauge field A µ and an artificial gauge field tensor F 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 L = X E,v (cid:0) ¯ ψ v, ˜ v ( x ) iγ µ (cid:0) ∂ µ + ieA µ + ieγ A µ (cid:1) ψ v, ˜ v ( x ) (cid:1) . (80)18ne can proceed with the same derivation of the OSEFT Lagrangian in the presence of thechiral field. After introducing the chiral projectors, it is not difficult to realize that all ourequations remain valid if we replace A µ → A µ + χA µ , F µν → F µν + χF µν , (81)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 byperforming the functional derivatives j µ ( x ) = − δ S δA µ ( x ) , j µ ( x ) = − δ S δA µ ( x ) , (82)respectively. Alternatively, one could start with the QED currents, and plug the explicitexpression of the Dirac fields in Eq. (3) to finally write the current in terms of the OSEFTfields. For example, considering only the contribution of the particles¯ ψ v, ˜ v ( x ) γ µ ψ v, ˜ v ( x ) → (cid:16) ¯ χ v ( x ) + ¯ H (1)˜ v ( x ) (cid:17) γ µ (cid:16) χ v ( x ) + H (1)˜ v ( x ) (cid:17) ≡ j µ ( x ) (83)Using the expression of the H (1)˜ v of Ref. [1] generalized to an arbitrary frame, we find j µ ( x ) = v µ ¯ χ v / ˜ v χ v + 12 E (cid:18) ¯ χ v γ µ ⊥ i /D ⊥ / ˜ v χ v + ¯ χ v ( i ←− /D ) ⊥ γ µ ⊥ / ˜ v χ v (cid:19) − ˜ v µ E (cid:18) ¯ χ v ( i ←− /D ) ⊥ ( i /D ) ⊥ / ˜ v χ v (cid:19) + v µ E (cid:18) ¯ χ v ( ←− /D ) ⊥ / ˜ v χ v + ¯ χ v ( /D ) ⊥ / ˜ v χ v (cid:19) − E (cid:18) ¯ χ v ( i ˜ v · D ) γ µ ⊥ ( i /D ) ⊥ / ˜ v χ v + ¯ χ v ( i ←− /D ) ⊥ ( i ˜ v · ←− D ) γ µ ⊥ / ˜ v χ v (cid:19) + O ( 1 E ) , (84)where we have to take into account the local field redefinition, Eq. (44), so as to compute thecurrent in the same way as the corrections to the transport equations. A completely analogouscomputation can be carried out for the chiral current.At leading order in the energy expansion, one can immediately express the current in termsof the two-point function. After a Wigner transform, one finds j µ (0) ( X ) = e X E,v,χ Z d k (2 π ) v µ G χE,v ( X, k ) . (85)We can use now the explicit form of the Wigner function at order n = 0; see Eq. (56). If wefurther make the identification [2, 36] X E,v Z d k (2 π ) ≡ Z d q (2 π ) , (86)19hen, at leading order, the current is expressed as j µ (0) ( X ) = e X χ = ± Z d q (2 π ) θ ( E q ) δ ( q ) q µ f χ ( X, q ) , (87)where we have approximated Ev µ ≈ q µ at leading order, and it is understood that the on-shellcondition is taken to leading order, thus, without the gauge field contribution. Similarly, theaxial current at leading order reads j µ , (0) ( X ) = e X χ = ± χ Z d q (2 π ) θ ( E q ) δ ( q ) q µ f χ ( X, q ) . (88)At the following orders in the energy expansion, and due to the presence of derivative termsin the explicit expression of the current, a point-splitting regularization is needed. This meansthat we take the field ¯ χ 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 . Notethat this point-splitting regularization is only needed to properly define the Wigner transform(see, for example, the scalar QED example explained in Ref.[35] for the proper definition of thecurrent) and not to regulate ultraviolet problems, which are absent in the two-point functionwe are studying.If one considers corrections up to order n = 2, then the vector current reads j µ (2) ( X ) = e X E,v,χ Z d k (2 π ) ( (cid:18) v µ + k µ ⊥ E − ( k · ˜ v ) k µ ⊥ E + v µ − ˜ v µ E (cid:16) k ⊥ − eχ ǫ αβµν ˜ v β v α F ⊥ µν (cid:17)(cid:19) − χ E (cid:18) ǫ µναβ ˜ v α v β − ( k · ˜ v )2 E ǫ µναβ ˜ v α v β (cid:19) [ ∂ Xν − eF νσ ∂ σk ] + χ E ǫ µναβ ˜ v α v β k ν ˜ v ρ [ ∂ Xρ − eF ρσ ∂ σk ]+ eχ E ǫ µραβ ˜ v α v β F ρσ ˜ v σ ) G χE,v ( X, k ) , (89)which, if converted to the original momentum, reads j µ (2) ( X ) = e X χ = ± Z d q (2 π ) ( q µ + S µνχ ∆ ν − e E q S µνχ F νρ (2 u ρ − v ρq ) ) f χ ( X, q ) δ + ( Q χ ) . (90)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. Asmentioned in Sec. IV this can be recovered from the OSEFT particle contribution, Eq. (89),by simply replacing v µ ↔ ˜ v µ and E → − E . 20et us consider the current associated with one single value of the chirality. Using thetransport equation (76) and the antisymmetry of the spin tensor, it is not difficult to deduce ∂ µ j µχ ( X ) = e Z d q (2 π ) ( q µ + S µνχ ∆ ν − e E q S µνχ F νρ (2 u ρ − v ρq ) ) F µλ ∂∂q λ ( f χ δ + ( Q χ )) . (91)To deduce the form of the chiral anomaly, we will now consider the frame u µ = (1 , , , | q | → ∞ , the only nonvanishing contributionarises for low values of the momenta, where the quasiparticle picture breaks down. We proceedas in Ref. [5], and Refs. [1, 8], and define a sphere centered in | q | = 0 of radius R and thencompute the only nonvanishing surface integral ∂ µ j µχ ( X ) = − e χ lim R → (cid:18)Z d S R (2 π ) · E ˆq · B R f χ ( | q | = R ) − Z d S R (2 π ) · ˆq R E · B f χ ( | q | = R ) (cid:19) = e χ E · B π f χ ( | q | = 0) . (92)At this point, we should consider the contribution of all the chiralities, of both fermions andantifermions so as to obtain the full complete contribution to the axial and vector currents.We thus assume the following fermion and antifermion distribution functions, f χ ( | q | ) = 1 e ( | q |− µ χ ) /T + 1 , ˜ f χ ( | q | ) = 1 e ( | q | + µ χ ) /T + 1 , (93)respectively, to obtain the nonconservation of the chiral current ∂ µ J µ ( X ) = 13 e π ( E · B + E · B ) . (94)The vector current also has a quantum anomaly also in the presence of chiral gauge fields ∂ µ J µ ( X ) = 13 e π ( E · B + E · B ) . (95)Eq. (94) gives account of the consistent form of the chiral anomaly equation, rather thanits covariant form. We refer the reader to the excellent review [28] that gives very clearexplanations about these two different forms of the quantum anomaly. After defining our21urrents as functional derivatives of the action, one cannot get anything else than the consistentcurrents. Unfortunately, the vector current is also nonconserved. It is possible to add the so-called Bardeen counterterms [37] to the quantum action e Z d x ǫ µνρλ A µ A ν (cid:0) c F ρλ + c F ρλ (cid:1) , (96)with the choice c = π and c = 0, and then one can get a vector conserved current [28].Previous approaches to CKT have shown to provide both the covariant currents and alsothe covariant form of the chiral anomaly [1–3]; see also Ref. [14]. One can relate the consistentand covariant currents by adding Chern-Simons currents [28]. VII. SIDE JUMPS DERIVED FROM REPARAMETRIZATION INVARIANCE OFTHE OSEFT Once we know how the fields of the OSEFT behave under the three types of RI transfor-mations, we can deduce how the different two-point functions behave under the same trans-formations. Then, after performing the (gauge-covariantly modified) Wigner transform and agradient expansion, we can deduce how the distribution function behaves under the same sortof transformations.It is actually easy to show that under the type I and type III symmetries of RI the distri-bution function in the OSEFT remains invariant. For example, under type I symmetry thebasic two-point function transforms as (see Table I) h ¯ χ v ( y ) / ˜ v χ v ( x ) i ′ → h ¯ χ v ( y ) (cid:18) / ˜ v/λ ⊥ (cid:19) / ˜ v (cid:18) /λ ⊥ / ˜ v (cid:19) χ v ( x ) i = h ¯ χ v ( y ) / ˜ v χ v ( x ) i , (97)where we have used that /λ ⊥ / ˜ v = − / ˜ v /λ ⊥ , and / ˜ v/ ˜ v = 0. It then follows that( f χE,v ( X, k )) ′ = f χE,v ( X, k ) . (98)under a type I transformation. Similarly, it is possible to show that the distribution functiondoes not change under a type III transformation.The Green function (37) used in our derivation of the transport equation has, however, anontrivial transformation under type II symmetry. Using the transformation rules of Table I,we obtain 22 ¯ χ v ( y ) / ˜ v χ v ( x ) i ′ → h ¯ χ v ( y ) / ˜ v + /ǫ ⊥ χ v ( x ) i (99)+ h ¯ χ v ( y ) ( i ←− /D ⊥ ,y ) † /ǫ †⊥ E / ˜ v χ v ( x ) i + h ¯ χ v ( y ) / ˜ v (cid:18) /ǫ ⊥ i /D ⊥ ,x E (cid:19) χ v ( x ) i + O ( 1 E ) . In OSEFT h ¯ χ v ( y ) γ µ ⊥ χ v ( x ) i = 0. After the Wigner transform, together with the gradientexpansion, we end up with( G χE,v ( X, k )) ′ → G χE,v ( X, k ) − E k ⊥ · ǫ ⊥ G χE,v ( X, k ) − χE ǫ µ ⊥ ν ⊥ αβ v α ˜ v β ǫ ⊥ ν ( ∂ Xµ − eF µλ ∂ λk ) G χE,v ( X, k ) , (100)Taking into account the definition of the two-point function at order 1 /E involves thecurrent density that might be computed [see the integrand of Eq. (89) at order 1 /E ] as G χE,v ( X, k ) = 12 ˜ v µ · ( v µ + k µ ⊥ E + · · · )(2 π ) f χE,v ( X, k ) δ + ( K χ ) ; (101)this implies that the distribution function should change as( f χE,v ( X, k )) ′ → f χE,v ( X, k ) − χE ǫ µ ⊥ ν ⊥ αβ v α ˜ v β ǫ ⊥ ν ( ∂ Xµ − eF µλ ∂ λk ) f χE,v ( X, k ) , (102)under a type II transformation.In terms of the original variables, one then gets( f χ ( X, q )) ′ → f χ ( X, q ) − E q S µνχ ǫ ⊥ ν ∆ µ f χ ( X, q ) + O (cid:18) ǫ ⊥ , E q (cid:19) . (103)Taking into account that ǫ µ ⊥ / u ′ µ − u µ , we see that Eq. (103) agrees with the infinitesimalform of the side-jump transformation first discussed in Ref. [23] in the absence of gauge fields,later generalized in the presence of the gauge fields in Ref.[9]. VIII. DISCUSSION We have derived from OSEFT the corrections to the classical transport equations associ-ated with on-shell massless charged fermions and antifermions. We have seen how from theproposed equations one can derive the consistent form of the chiral anomaly equation whenconsidering a chiral imbalance system in thermal equilibrium. Our formulation turns out tobe the proper generalization of the HDET approach to chiral transport theory of Ref. [6], butvalid also for finite temperature systems and formulated in an arbitrary frame. The study of23eparametrization invariance of the theory allows us to claim that the results are consistentwith Lorentz symmetry, even if the kinetic equation depends on a frame vector. We have alsodeduced the side-jumps of the distribution function of the theory from the transformation ruleunder RI of the OSEFT quantum fields.Let us insist that when we consider the frame vector as u µ = (1 , ), our equations almostagree with those of Ref. [6], except in a couple of factors, in what apparently was an algebraicmistake. It is, however, important to stress that the transport equation obtained either inRef. [6] or in this paper do not match exactly with the transport equation in Sec. IIB ofRef. [6], which were obtained starting with a corrected form of the classical point-particleaction, with modified Poisson brackets. This starting point can be justified by performinga 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 isa quantum field theory. Let us stress that in such a formulation one obtains the covariant formof the chiral anomaly, as the chiral current is not defined by performing a functional derivativeof 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 de-scribing the same system equally well. The Foldy-Wouthuysen diagonalization used in Ref. [1]suggests that the starting quantum fields used there or those used in our OSEFT approach arenot the same beyond the classical limit approximation. Thus, probably it is not so surprisingthat 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.Probably more surprising are the discrepancies we obtained from the results of Refs. [9, 10,13], obtained from massless QED, assuming homogenous gauge field backgrounds. OSEFTonly helps in organizing the quantum field theory computation at large energies, as it hasalready been checked in the computation of Feynman diagrams at high T [2, 38]. We cannotcomment on the possible origin of these discrepancies, although it seems that the approachshould also lead to the consistent form of the chiral anomaly, rather than its covariant form,as claimed in Ref. [10].Let us, however, stress that discrepancies of our results with others published in the lit-erature only appear at order n = 2 in the energy expansion both in the transport equationand the current. Let us mention that since the chiral magnetic effect, as well as other chiraltransport effects, appear already at order n = 1 our formulation gives the same description as24hat of other formulations (see Appendix B for the computation of the chiral magnetic effect).While in this paper we have focused our attention to the collisionless form of the transportequation, a much more challenging task is to derive the collision terms from OSEFT, such thatthe Lorentz symmetry is respected, and the side-jumps are properly described. This will bethe subject of a different project. Acknowledgments: We are indebted to J. Soto for many discussions during the evolutionof this project. We are also especially thankful to K. Landsteiner, for different discussionson the difference between the consistent and covariant forms of the chiral anomaly. We ac-knowledge interesting discussions with M. Beneke and T. Schaefer. We have been supportedby the MINECO (Spain) under Project No. FPA2016-81114-P. This work was also supportedby the COST Action CA15213 THOR. J.M.T.-R. was supported by the U.S. Department ofEnergy under Contract No. DE-FG-88ER40388. S. C. acknowledges financial support by the“Fondazione Angelo Della Riccia.” Appendix A: Derivation of the I χ, ± functions We provide in this Appendix some details of the computation of the I χ, ± functions. Wetake here e = 1 for simplicity.We start from the equation of motion for quantum fields χ v , (cid:0) O (0) x + O (1) x + O (2) x (cid:1) χ v ( x ) = 0 , (A1)and similarly its Hermitian conjugate for y . By adding and subtracting them, we can buildequations for the two-point function. For each piece, we isolate the different possible Diracstructures, so we write O ( n ) x = (cid:0) α ( n ) x + β ( n ) x,µν σ µν ⊥ (cid:1) / ˜ v I ( n ) ± ) = Z d se ik · s (cid:26)(cid:0) α ( n ) x ± α ( n ) ∗ y (cid:1) Tr (cid:20) / ˜ v S E,v ( x, y ) (cid:21) + (cid:0) β ( n ) x,µν ± β ( n ) ∗ y,µν (cid:1) Tr (cid:20) σ µν / ˜ v S E,v ( x, y ) (cid:21)(cid:27) , (A3)For the α and β coefficients, we find (after neglecting terms of higher order in the gradient25xpansion like ∂ Xα F µν ) α (0) = iv · D , β (0) µν = 0 , (A4) α (1) = − E D ⊥ , β (1) µν = − E F µν , (A5) α (2) = 14 E ( v α − ˜ v α ) (cid:16) F µα D µ − iD α D ⊥ (cid:17) , β (2) µν = i E (cid:18) F µα ˜ v α D ν − F µν ( v · D − ˜ v · D ) (cid:19) . (A6)We now perform the change of variables to the center of mass and relative coordinates X, s .The recurring combinations will be D xα − ( D yα ) ∗ = 2 ( ∂ sα + iA α ( X )) , D xα + ( D yα ) ∗ = ∂ Xα + is β ∂ β A α ( X ) , (A7)together with( D x ⊥ ) + (( D y ⊥ ) ∗ ) = 2 (cid:16) ∂ X · ∂ s + i ( ∂ X · A ( X ) + A ( X ) · ∂ X ) + is β ∂ Xβ A α ( X ) ( ∂ sα + iA α ( X )) (cid:17) , (A8)( D x ⊥ ) + (( D y ⊥ ) ∗ ) = 2 (cid:16) ∂ s + 2 iA ( X ) · ∂ s − A ( X ) (cid:17) , (A9)We also use thatTr (cid:20) / ˜ v S E,v (cid:21) = 2 X χ = ± G χE,v , Tr (cid:20) σ µν / ˜ v S E,v (cid:21) = − X χ = ± χǫ µναρ ˜ v α J χ ( E,v ) ,ρ , (A10)where G and J are defined in Eq. (36) and Eq. (37), respectively.For an example, we can work out the lowest order function. If here k µ denotes the canonicalmomentum then I (0)+ = Z d se ik · s iv · ( D x − D ∗ y ) X χ = ± G χE,v ( X, s ) e − iAs = Z d se ik · s iv · − ik + iA ( X )) X χ = ± G χE,v ( X, s ) e − iAs = 4( v · ¯ k ) Z d se i ¯ k · s X χ = ± G χE,v ( X, s ) = 4( v · ¯ k ) X χ = ± G χE,v ( X, ¯ k ) (A11)where now ¯ k µ = k µ − A µ is the canonical momentum.26 ppendix B: Chiral magnetic effect In this Appendix we briefly show how from our formulation one can reproduce the chiralmagnetic effect. For this, we start from the current Eq. (90) and focus on its spatial componentsin the local rest frame u µ = (1 , , , q integration, we get j i ( X ) = e X χ = ± Z d q (2 π ) (cid:16) q i E q + S ijχ ∆ j E q − e E q S ijχ F jσ ˜ v σ (cid:17) f χ ( X, q ) (cid:12)(cid:12)(cid:12) q = E q , (B1)with the dispersion relation in this frame given by q = E q = | q | (cid:18) − eχ B · ˆ q | q | (cid:19) . 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