Two-point functions from chiral kinetic theory in magnetized plasma

It has been revealed that axial anomaly signifies a type of gapless collective excitation arising from the coupling of the the electric and chiral charge density waves, which is known as the chiral magnetic wave (CMW) Kharzeev:2010gd . These density waves are closely related to the anomalous transports of chiral magnetic effect (CME) Vilenkin:1980fu ; Kharzeev:2004ey ; Kharzeev:2007tn ; Fukushima:2008xe ; Son:2009tf ; Neiman:2010zi and chiral separation effect (CSE) Metlitski:2005pr ; Son:2004tq . For the quark-gluon plasma with a strong magnetic field, it has been demonstrated in a dimensionally reduced (1+1)D theory Kharzeev:2010gd that the CMW indeed stems from the connections $J_{0}=J_{1}^{5}$, $J_{0}^{5}=J_{1}$ between the vector $J$ and axial $J^{5}$ currents of CME and CSE. Besides the transports along magnetic field, axial anomaly also implies anomalous currents from chiral vortical effects (CVE) Vilenkin:1980zv ; Erdmenger:2008rm ; Banerjee:2008th ; Son:2009tf ; Neiman:2010zi ; Landsteiner:2011cp as responses to vorticity.

The chiral anomalous effects have been investigated in the framework of chiral kinetic theory (CKT) Son:2012wh ; Son:2012zy ; Stephanov:2012ki ; Gao:2012ix ; Pu:2010as ; Chen:2012ca ; Hidaka:2016yjf ; Manuel:2013zaa ; Manuel:2014dza ; Wu:2016dam ; Mueller:2017arw ; Mueller:2017lzw ; Huang:2018wdl ; Gao:2018wmr ; Carignano:2018gqt ; Lin:2019ytz ; Carignano:2019zsh ; Liu:2018xip ; Weickgenannt:2019dks ; Gao:2019znl ; Hattori:2019ahi ; Wang:2019moi ; Yang:2020hri ; Liu:2020flb ; Hayata:2020sqz ; chen2021equaltime , which can be viewed as an expansion in weak electromagnetic field and spacetime gradient. Furthermore, CKT with Landau level basis Lin:2019fqo ; Gao:2020ksg ; Hattori:2016lqx ; Sheng:2017lfu ; Fukushima:2019ugr has been derived in a different expansion scheme to study plasma magnetized by non-perturbative magnetic field. Particularly, in a strong magnetic field, the CKT in lowest Landau level approximation (LLLA) and collisionless limit has been applied to study photon self-energy Gao:2020ksg and magneto-vortical effect Lin:2021sjw . Recently, CKT in weak field expansion has been generalized to incorporate axial gauge field to derive correlation functions of axial currents and stress tensor Chen:2021azy . It would be interesting to study the transport phenomena in magnetized plasma as responses to perturbative vector and axial gauge fields with CKT.

In linear regime, the transport coefficients are related to the retarded two-point correlation functions via the Kubo formulas. The two-point function is important in its own right, e.g. the photon self-energy as a fundamental quantity characterizing the vacuum polarization by electromagnetic field has been intensively studied for magnetized plasma in field theory Danielsson:1995rh ; PhysRevD.83.111501 ; Chao:2014wla ; Fukushima:2015wck ; Chao:2016ysx ; Hattori:2017xoo ; Singh:2020fsj ; Wang:2021eud . On the other hand, as a contrast to the full non-perturbative effective Lagrangian of vacuum for constant electromagnetic field established by Heisenberg and Euler Heisenberg:1936nmg in the study of vacuum polarization, an effective action for the perturbative gauge fields in magnetized medium can also be constructed as the generating functional of the correlators.

In this paper, we introduce perturbative vector and axial gauge potentials to CKT in a strong background magnetic field to study the correlators among currents $J^{\mu},\,J_{5}^{\mu}$ and energy-momentum tensor $T^{{\mu}{\nu}}$. The CKT results, which express $J^{\mu},\,J_{5}^{\mu}$ and $T^{{\mu}{\nu}}$ in terms of the thermodynamic variables, give the constitutive relations that involve CMW mode as a consequence of axial anomaly and the presence of magnetic field. It is known that there is distinction between consistent and covariant anomalies Bardeen:1984pm , see also 2016AcPPB..47.2617L for a review, which is more explicit in the presence of axial gauge field. We find that CKT gives covariant currents, while the consistent currents are obtained by adding the Chern-Simons currents. The correlators of the consistent currents and stress tensor satisfy the derivative symmetry upon the interchange of the functional derivatives. It turns out that the derivative symmetry of correlators also fit the Onsager relations Hernandez:2017mch .

The paper is organized as follows: In Section (II), we discuss the property of current-current correlators in consistent and covariant anomalies with perturbative gauge potentials and strong background magnetic field. The density waves of vector and axial charges including CMW is obtained in this condition. In Section (III), we give a brief introduction of the covariant CKT with the same external field configuration. We then solve the chiral kinetic equations (CKE) to derive the constitutive relations of currents and stress tensor in Section (IV), where we reproduce the density waves in the currents and verify the property of the resulting correlators, whose generating functional is derived as the effective action for gauge fields.

Throughout this paper, we set ${\hbar}=1$ and $c=1$. We take positive charge $q_{f}=e$ for chiral fermions and absorb electric charge $e$ into the gauge potentials. We use the notations $(x^{\mu})=(x_{0},{\bf x}),\,(p^{\mu})=(p_{0},{\bf p})$ for four-vectors and adopt mostly minus signature. We also define $X^{[{\mu}}Y^{{\nu}]}\equiv X^{\mu}Y^{\nu}-X^{\nu}Y^{\mu}$ and $X^{\{{\mu}}Y^{{\nu}\}}\equiv X^{\mu}Y^{\nu}+X^{\nu}Y^{\mu}$ to be concise.

In the presence of axial gauge field, the difference between consistent current ${\cal J}^{\mu}$ and covariant current $J^{\mu}$ is nontrivial 2016AcPPB..47.2617L . With the explicitly conserved vector current, the consistent anomaly writes

	$\displaystyle\partial_{\mu}{\cal J}^{\mu}=$	$\displaystyle 0\,,$
	$\displaystyle\partial_{\mu}{\cal J}_{5}^{\mu}=$	$\displaystyle-\frac{1}{48{\pi}^{2}}{\epsilon}^{{\mu}{\nu}{\rho}{\lambda}}\left(3F_{{\mu}{\nu}}F_{{\rho}{\lambda}}+F_{{\mu}{\nu}}^{5}F_{{\rho}{\lambda}}^{5}\right)\,.$		(1)

For the gauge invariant currents, one has covariant anomaly,

	$\displaystyle\partial_{\mu}J^{\mu}=$	$\displaystyle-\frac{1}{8{\pi}^{2}}{\epsilon}^{{\mu}{\nu}{\rho}{\lambda}}F_{{\mu}{\nu}}F_{{\rho}{\lambda}}^{5}\,,$
	$\displaystyle\partial_{\mu}J_{5}^{\mu}=$	$\displaystyle-\frac{1}{16{\pi}^{2}}{\epsilon}^{{\mu}{\nu}{\rho}{\lambda}}\left(F_{{\mu}{\nu}}F_{{\rho}{\lambda}}+F_{{\mu}{\nu}}^{5}F_{{\rho}{\lambda}}^{5}\right)\,.$		(2)

The currents are related by the Chern-Simons currents as follows,

		$\displaystyle{\cal J}^{\mu}-\frac{1}{4{\pi}^{2}}{\epsilon}^{{\mu}{\nu}{\rho}{\lambda}}A_{\nu}^{5}F_{{\rho}{\lambda}}=J^{\mu}\,,$
		$\displaystyle{\cal J}_{5}^{\mu}-\frac{1}{12{\pi}^{2}}{\epsilon}^{{\mu}{\nu}{\rho}{\lambda}}A_{\nu}^{5}F_{{\rho}{\lambda}}^{5}=J_{5}^{\mu}\,.$		(3)

We study the magnetized chiral plasma with a constant fluid velocity $u^{\mu}$ in the strong magnetic field $B^{\mu}$. To this end, we take vanishing axial gauge field $A_{5}^{\mu}=0$ and $O(1)$ magnetic field $F_{{\mu}{\nu}}=\partial_{[{\mu}}A_{{\nu}]}={\epsilon}_{{\mu}{\nu}{\rho}{\sigma}}u^{\rho}B^{\sigma}$ as the constant backgrounds. In the perturbations, we turn on vector gauge field $a^{\lambda}$ and axial gauge field $a_{5}^{\lambda}$ symmetrically at $O(a)$ which give $f_{{\mu}{\nu}}=\partial_{[{\mu}}a_{{\nu}]}=E_{[{\mu}}u_{{\nu}]}+{\epsilon}_{{\mu}{\nu}{\rho}{\sigma}}u^{\rho}{\cal B}^{\sigma}$ and $f_{{\mu}{\nu}}^{5}=\partial_{[{\mu}}a_{{\nu}]}^{5}=E_{[{\mu}}^{5}u_{{\nu}]}+{\epsilon}_{{\mu}{\nu}{\rho}{\sigma}}u^{\rho}{\cal B}_{5}^{\sigma}$ at $O(\partial a)$. This is also the perturbation expansion scheme of the CKT in the next section.

To $O(\partial a)$, the Ward identities in (II) and (II) are reduced to

	$\displaystyle\partial_{\mu}{\cal J}^{\mu}=$	$\displaystyle\,\partial_{\mu}J^{\mu}+\frac{1}{8{\pi}^{2}}{\epsilon}^{{\mu}{\nu}{\rho}{\lambda}}F_{{\mu}{\nu}}f_{{\rho}{\lambda}}^{5}=0\,,$
	$\displaystyle\partial_{\mu}{\cal J}_{5}^{\mu}=$	$\displaystyle\,\partial_{\mu}J_{5}^{\mu}=-\frac{1}{8{\pi}^{2}}{\epsilon}^{{\mu}{\nu}{\rho}{\lambda}}F_{{\mu}{\nu}}f_{{\rho}{\lambda}}\,,$		(4)

and the currents are

	$\displaystyle{\cal J}^{\mu}=$	$\displaystyle J^{\mu}+\frac{1}{4{\pi}^{2}}{\epsilon}^{{\mu}{\nu}{\rho}{\lambda}}a_{\nu}^{5}F_{{\rho}{\lambda}}\,,$
	$\displaystyle{\cal J}_{5}^{\mu}=$	$\displaystyle J_{5}^{\mu}\,.$		(5)

In the regime of strong magnetic field, one can effectively perform a dimensional reduction from (3+1)D to (1+1)D, where the orthonormal vectors $u^{\mu}$ and $b^{\mu}=B^{\mu}/B$ give the directions of temporal and spatial axes severally. In this effective (1+1)D theory, the metric writes ${\hat{g}}^{{\mu}{\nu}}=u^{\mu}u^{\nu}-b^{\mu}b^{\nu}$, while the anti-symmetric tensor is ${\epsilon}^{{\mu}{\nu}}=u^{[{\mu}}b^{{\nu}]}$. Then, the Ward identities (II) to $O(\partial a)$ can be further written as

	$\displaystyle\partial_{\mu}{\cal J}^{\mu}=$	$\displaystyle\,\partial_{\mu}J^{\mu}-\frac{B}{4{\pi}^{2}}{\epsilon}^{{\rho}{\lambda}}f_{{\rho}{\lambda}}^{5}=0\,,$
	$\displaystyle\partial_{\mu}{\cal J}_{5}^{\mu}=$	$\displaystyle\,\partial_{\mu}J_{5}^{\mu}=\frac{B}{4{\pi}^{2}}{\epsilon}^{{\rho}{\lambda}}f_{{\rho}{\lambda}}\,,$		(6)

which can be understood as the genuine (1+1)D anomalies $\frac{1}{2{\pi}}{\epsilon}^{{\rho}{\lambda}}f_{{\rho}{\lambda}}^{5}$ and $\frac{1}{2{\pi}}{\epsilon}^{{\rho}{\lambda}}f_{{\rho}{\lambda}}$ multiplied by the Landau-level degeneracy $\frac{B}{2{\pi}}$. Likewise, the currents in (II) write

	$\displaystyle{\cal J}^{\mu}=$	$\displaystyle J^{\mu}-\frac{B}{2{\pi}^{2}}{\epsilon}^{{\mu}{\nu}}a_{\nu}^{5}\,,$
	$\displaystyle{\cal J}_{5}^{\mu}=$	$\displaystyle J_{5}^{\mu}\,.$		(7)

Meanwhile, the dimensional reducd vector and axial currents are connected by the leading order constitutive relation,

$\displaystyle J_{5}^{\mu}=-{\epsilon}^{{\mu}{\nu}}J_{\nu}\,.$

(8)

Note we have used the covariant currents in the above gauge invariant identity which will be confirmed by the CKT solution. While the consistent currents are not gauge invariant in general due to the additional gauge dependent Chern-Simons currents, a nice characteristic thereof is that they can be interpreted as the functional derivatives of effective action. The consequential current-current correlators are constrained by the derivative symmetry since the order of variations does not matter. Concretely, we have

	$\displaystyle{\cal G}_{VV}^{{\mu}{\nu}}\equiv\frac{{\delta}{\cal J}^{\mu}}{{\delta}a_{\nu}}=\frac{{\delta}^{2}{\cal L}}{{\delta}a_{\nu}\,{\delta}a_{\mu}}\,,\qquad{\cal G}_{AA}^{{\mu}{\nu}}\equiv\frac{{\delta}{\cal J}_{5}^{\mu}}{{\delta}a_{\nu}^{5}}=\frac{{\delta}^{2}{\cal L}}{{\delta}a_{\nu}^{5}\,{\delta}a_{\mu}^{5}}\,,$
	$\displaystyle{\cal G}_{VA}^{{\mu}{\nu}}\equiv\frac{{\delta}{\cal J}^{\mu}}{{\delta}a_{\nu}^{5}}=\frac{{\delta}^{2}{\cal L}}{{\delta}a_{\nu}^{5}\,{\delta}a_{\mu}}\,,\qquad{\cal G}_{AV}^{{\mu}{\nu}}\equiv\frac{{\delta}{\cal J}_{5}^{\mu}}{{\delta}a_{\nu}}=\frac{{\delta}^{2}{\cal L}}{{\delta}a_{\nu}\,{\delta}a_{\mu}^{5}}\,,$		(9)

where we have defined the correlators with effective Lagrangian density. In momentum space, the derivative symmetry relations write

$\displaystyle{\cal G}_{VV}^{{\mu}{\nu}}(q)={\cal G}_{VV}^{{\nu}{\mu}}(-q)\,,\qquad{\cal G}_{AA}^{{\mu}{\nu}}(q)={\cal G}_{AA}^{{\nu}{\mu}}(-q)\,,\qquad{\cal G}_{VA}^{{\mu}{\nu}}(q)={\cal G}_{AV}^{{\nu}{\mu}}(-q)\,,$

(10)

which hold beyond (1+1)D. Though the covariant currents can not be thought of as functional derivatives of effective action, we can still define the correlators in form as,

	$\displaystyle G_{VV}^{{\mu}{\nu}}\equiv\frac{{\delta}J^{\mu}}{{\delta}a_{\nu}}\,,\qquad G_{AA}^{{\mu}{\nu}}\equiv\frac{{\delta}J_{5}^{\mu}}{{\delta}a_{\nu}^{5}}\,,$
	$\displaystyle G_{VA}^{{\mu}{\nu}}\equiv\frac{{\delta}J^{\mu}}{{\delta}a_{\nu}^{5}}\,,\qquad G_{AV}^{{\mu}{\nu}}\equiv\frac{{\delta}J_{5}^{\mu}}{{\delta}a_{\nu}}\,.$		(11)

By noting the connection (II) to $O(\partial a)$, ${\cal G}$ and $G$ are related as

	$\displaystyle{\cal G}_{VV}^{{\mu}{\nu}}=$	$\displaystyle G_{VV}^{{\mu}{\nu}}\,,\qquad{\cal G}_{VA}^{{\mu}{\nu}}=G_{VA}^{{\mu}{\nu}}-\frac{B}{2{\pi}^{2}}u^{[{\mu}}b^{{\nu}]}\,,$
	$\displaystyle{\cal G}_{AA}^{{\mu}{\nu}}=$	$\displaystyle G_{AA}^{{\mu}{\nu}}\,,\qquad{\cal G}_{AV}^{{\mu}{\nu}}=G_{AV}^{{\mu}{\nu}}\,.$		(12)

Before closing this section, we compute the response to the perturbative $a^{\mu}$ and $a_{5}^{\mu}$ in the effective (1+1)D theory. In such condition, there are only temporal and longitudinal components in the currents,

$\displaystyle J^{\mu}=n\,u^{\mu}+n_{5}\,b^{\mu}\,,\qquad J_{5}^{\mu}=n_{5}\,u^{\mu}+n\,b^{\mu}\,.$

(13)

Using constitutive relation (8) together with the Fourier transformation of the anomaly equations (II), one easily finds

$\displaystyle n=\frac{{\Delta}{\mu}}{2{\pi}^{2}}B\,,\qquad n_{5}=\frac{{\Delta}{\mu}_{5}}{2{\pi}^{2}}B\,,$

(14)

where we have defined

	$\displaystyle{\Delta}{\mu}\equiv$	$\displaystyle\frac{{\bar{q}}_{3}\left({\bar{q}}_{3}{\bar{a}}_{0}-{\bar{q}}_{0}{\bar{a}}_{3}\right)+{\bar{q}}_{0}\left({\bar{q}}_{3}{\bar{a}}_{0}^{5}-{\bar{q}}_{0}{\bar{a}}_{3}^{5}\right)}{{\bar{q}}_{0}^{2}-{\bar{q}}_{3}^{2}}\,,$
	$\displaystyle{\Delta}{\mu}_{5}\equiv$	$\displaystyle\frac{{\bar{q}}_{0}\left({\bar{q}}_{3}{\bar{a}}_{0}-{\bar{q}}_{0}{\bar{a}}_{3}\right)+{\bar{q}}_{3}\left({\bar{q}}_{3}{\bar{a}}_{0}^{5}-{\bar{q}}_{0}{\bar{a}}_{3}^{5}\right)}{{\bar{q}}_{0}^{2}-{\bar{q}}_{3}^{2}}\,.$		(15)

Here ${\bar{a}}$ and ${\bar{q}}$ are the gauge potentials and momentum in the local rest frame of the fluid, which can be obtained by the Lorentz tansformation defined in Appendix (A) . We interpret ${\Delta}{\mu}$ and ${\Delta}{\mu}_{5}$ as the change of chemical potential ${\mu}$ and axial chemical potential ${\mu}_{5}$ characterizing the redistribution of the chiral plasma induced by the gauge fields $a^{\mu}$ and $a_{5}^{\mu}$. Correspondingly, $n_{5}$ and $n$ give the Fourier transformed densities of electric and axial charges, which are related to the anomalous transports of CME and CSE. In coordinate space, ${\Delta}{\mu}$ and ${\Delta}{\mu}_{5}$ write

	$\displaystyle{\Delta}{\mu}=$	$\displaystyle\frac{1}{2}{\cal D}(x-y)\left[b\cdot\partial\left({\epsilon}^{{\mu}{\nu}}f_{{\mu}{\nu}}\right)+u\cdot\partial\left({\epsilon}^{{\mu}{\nu}}f_{{\mu}{\nu}}^{5}\right)\right]\,,$
	$\displaystyle{\Delta}{\mu}_{5}=$	$\displaystyle\frac{1}{2}{\cal D}(x-y)\left[u\cdot\partial\left({\epsilon}^{{\mu}{\nu}}f_{{\mu}{\nu}}\right)+b\cdot\partial\left({\epsilon}^{{\mu}{\nu}}f_{{\mu}{\nu}}^{5}\right)\right]\,,$		(16)

with $\hat{\partial}^{\mu}\equiv{\hat{g}}^{{\mu}{\nu}}\partial_{\nu}$ and the Green’s function ${\cal D}(x-y)$ being the inverse of the operator $\hat{\partial}_{\mu}\hat{\partial}^{\mu}$,

$\displaystyle\hat{\partial}_{\mu}\hat{\partial}^{\mu}{\cal D}(x-y)=\int d^{4}y\,{\delta}^{(4)}(x-y)\,.$

(17)

The non-local factor $\frac{1}{{\bar{q}}_{0}^{2}-{\bar{q}}_{3}^{2}}$ in $n$ and $n_{5}$ indicates there are density waves of electric and axial charges propagate with frequency ${\bar{q}}_{0}$ and momentum ${\bar{q}}_{3}$ along the direction of the magnetic field. The poles ${\bar{q}}_{0}\pm{\bar{q}}_{3}=0$ give the dispersion relation of the gapless collective excitation in the chiral plasma, which describes the CMW propagating with the speed of light in the limit of strong magnetic field. The poles are independent of the limit $a^{\mu}\to 0$ and $a_{5}^{\mu}\to 0$, which means the existence of CMW does not rely on the perturbation of the gauge potentials.

The Lorentz covariant CKE with Landau level states have been given by Lin:2021sjw in the collisonless limit,

	$\displaystyle p_{{\mu}}j_{s}^{{\mu}}$	$\displaystyle=0\,,$		(18)
	$\displaystyle{\Delta}_{{\mu}}j_{s}^{{\mu}}$	$\displaystyle=0\,,$		(19)
	$\displaystyle p^{[{\mu}}j_{s}^{{\nu}]}$	$\displaystyle=-\frac{s}{2}{\epsilon}^{{\mu}{\nu}{\rho}{\sigma}}{\Delta}_{{\rho}}j_{{\sigma}}^{s}\,,$		(20)

where ${\Delta}_{\mu}=\partial_{\mu}-\frac{\partial}{\partial p_{\nu}}\left(F_{{\mu}{\nu}}+f_{{\mu}{\nu}}^{s}\right)$ with helicity $s=\pm 1$ for right/left-handed chiral fermions and $f_{{\mu}{\nu}}^{s}=f_{{\mu}{\nu}}+sf_{{\mu}{\nu}}^{5}$, which means the perturbative left-handed and right-handed gauge fields, or equivalently, vector and axial gauge fields are taken on the equal footing. We note that in Gao:2020ksg , it has been verified that the transport equation agrees with the anomalous Ward identity. Repeating the same procedure, one readily reproduces the covariant anomaly (II) from (19). This implies that CKT gives covariant currents, which will be checked in the final solutions.

The CKE (18) to (20) are valid to $O(\partial a)$ expansion. Therefore we seek solution of $j_{s}^{\mu}$ to the first order in perturbation $a$ and gradient $\partial a$,

$\displaystyle j_{s}^{\mu}=j_{(0)s}^{\mu}+j_{a\,s}^{\mu}+\sum_{{\cal A}s}j_{{\cal A}s}^{\mu}\,,$

(21)

where $j_{(0)s}^{\mu}$ is the $O(1)$ background solution while $j_{a\,s}^{\mu}$ is solution for the $O(a)$ perturbations $a_{s}^{\mu}$. The last term $\sum j_{{\cal A}s}^{\mu}$ is the solution for the $O(\partial a)$ perturbations, where ${\cal A}_{s}=E_{\parallel}^{s},E_{\perp}^{s},{\cal B}_{\parallel}^{s},{\cal B}_{\perp}^{s}$ denote parallel/perpendicular electric/magnetic field respectively as is detailed in Appendix (B).

The $O(1)$ background solution for $j_{(0)s}^{\mu}$ in LLL has been given as

	$\displaystyle j_{(0)s}^{{\mu}}=$	$\displaystyle(u+sb)^{{\mu}}{\delta}({\hat{p}}\cdot(u+sb))f_{s}({\hat{p}}\cdot u)e^{\frac{p_{T}^{2}}{B}}$
	$\displaystyle=$	$\displaystyle(u+sb)^{{\mu}}{\delta}(p\cdot(u+sb))f_{s}(p\cdot u)e^{\frac{p_{T}^{2}}{B}}\equiv(u+sb)^{{\mu}}j_{s}\,,$		(22)

where ${\hat{p}}^{\mu}$ is the (1+1)D momentum and $p_{T}^{\mu}$ is the rest part transverse to both $u^{\mu}$ and $b^{\mu}$. They are defined by the (1+1)D metric ${\hat{g}}^{{\mu}{\nu}}=u^{{\mu}}u^{{\nu}}-b^{{\mu}}b^{{\nu}}$ and transverse projector $P^{{\mu}{\nu}}\equiv{\hat{g}}^{{\mu}{\nu}}-g^{{\mu}{\nu}}$ as ${\hat{p}}^{\mu}\equiv{\hat{g}}^{{\mu}{\nu}}p_{\nu}=(p\cdot u)u^{\mu}-(p\cdot b)b^{\mu}$ and $p_{T}^{\mu}\equiv-P^{{\mu}{\nu}}p_{\nu}=p^{\mu}-{\hat{p}}^{\mu}$. These projections confine the LLL background currents to be effective (1+1)D since the $p_{T}$ dependent factor $e^{\frac{p_{T}^{2}}{B}}$ only gives Landau-level degeneracy $\frac{B}{2{\pi}}$. The distribution function involving fermion energy is defined as

$\displaystyle f_{s}(p\cdot u)=\frac{2}{(2{\pi})^{3}}\sum_{r=\pm}\frac{r{\theta}(rp\cdot u)}{e^{r\left(p\cdot u-{\mu}_{s}\right)/T}+1}\,,$

(23)

where ${\mu}_{s}={\mu}+s{\mu}_{5}$ is the chemical potential for chiral fermions and $T$ is temperature, which are taken as constant for simplicity. Also, to be concise, when solving the CKE, we focus on the right-handed fermions and suppress helicity $s$ in the notation, which will be restored in the final solution.

To begin with, we examine the structure of the CKE by analyzing perturbative solution at $O(a)$ and $O(\partial a)$ separately. Substituting the perturbative expansion (21) into CKE (18) to (20) and noting the background solution (III) automatically satisfies the CKE, at $O(a)$ we obtain

	$\displaystyle p_{{\mu}}j_{a}^{{\mu}}$	$\displaystyle=0\,,$		(24)
	$\displaystyle F_{{\mu}{\nu}}\frac{\partial}{\partial p_{\nu}}j_{a}^{{\mu}}$	$\displaystyle=0\,,$		(25)
	$\displaystyle p^{[{\mu}}j_{a}^{{\nu}]}-\frac{1}{2}{\epsilon}^{{\mu}{\nu}{\rho}{\sigma}}F_{{\rho}{\lambda}}\frac{\partial}{\partial p_{\lambda}}j_{{\sigma}}^{a}$	$\displaystyle=0.$		(26)

Since (24) through (26) are homogeneous equations, which are generally satisfied by the solution with a structure similar to the LL states (III), we take the ansatz $j_{a}^{\mu}=(u+b)^{\mu}j_{a}$ with $j_{a}\propto{\delta}(p\cdot(u+b))e^{\frac{p_{T}^{2}}{B}}$ as $O(a)$ redistribution of the LLL state.

At $O(\partial a)$, the CKE with constant fluid velocity can be grouped into four cases for ${\cal A}=E_{\parallel},E_{\perp},{\cal B}_{\parallel},{\cal B}_{\perp}$,

	$\displaystyle p_{\mu}j_{{\cal A}}^{\mu}=$	$\displaystyle 0\,,$		(27)
	$\displaystyle F_{{\mu}{\nu}}\frac{\partial}{\partial p_{\nu}}j_{{\cal A}}^{\mu}=$	$\displaystyle\partial_{\mu}^{\cal A}j_{a}^{{\mu}}-f_{{\mu}{\nu}}^{{\cal A}}\frac{\partial}{\partial p_{\nu}}j_{(0)}^{\mu}\,,$		(28)
	$\displaystyle p^{[{\mu}}j_{{\cal A}}^{{\nu}]}-\frac{1}{2}{\epsilon}^{{\mu}{\nu}{\rho}{\sigma}}F_{{\rho}{\lambda}}\frac{\partial}{\partial p_{\lambda}}j_{\sigma}^{{\cal A}}=$	$\displaystyle-\frac{1}{2}{\epsilon}^{{\mu}{\nu}{\rho}{\sigma}}\left(\partial_{\rho}^{\cal A}j_{{\sigma}}^{a}-f_{{\rho}{\lambda}}^{{\cal A}}\frac{\partial}{\partial p_{\lambda}}j_{\sigma}^{(0)}\right)\,,$		(29)

where $\partial_{\mu}^{\cal A}j_{a}$ are defined by splitting the gradient of $j_{a}$ into four terms in accordance to ${\cal A}$,

	$\displaystyle\partial_{\mu}j_{a}=$	$\displaystyle\left(u_{\mu}\bar{\partial}_{0}+w_{\mu}^{k}\bar{\partial}_{k}+b_{\mu}\bar{\partial}_{3}\right)\left({\bar{a}}_{0}\frac{\partial j_{a}}{\partial{\bar{a}}_{0}}+{\bar{a}}_{k}\frac{\partial j_{a}}{\partial{\bar{a}}_{k}}+{\bar{a}}_{3}\frac{\partial j_{a}}{\partial{\bar{a}}_{3}}\right)$
	$\displaystyle=$	$\displaystyle\left(u_{\mu}\bar{\partial}_{0}+b_{\mu}\bar{\partial}_{3}\right)\left({\bar{a}}_{0}\frac{\partial j_{a}}{\partial{\bar{a}}_{0}}+{\bar{a}}_{3}\frac{\partial j_{a}}{\partial{\bar{a}}_{3}}\right)+w_{\mu}^{k}\bar{\partial}_{k}{\bar{a}}_{i}\frac{\partial j_{a}}{\partial{\bar{a}}_{i}}$
		$\displaystyle\quad+\left(u_{\mu}\bar{\partial}_{0}{\bar{a}}_{k}\frac{\partial j_{a}}{\partial{\bar{a}}_{k}}+w_{\mu}^{k}\bar{\partial}_{k}{\bar{a}}_{0}\frac{\partial j_{a}}{\partial{\bar{a}}_{0}}\right)+\left(w_{\mu}^{k}\bar{\partial}_{k}{\bar{a}}_{3}\frac{\partial j_{a}}{\partial{\bar{a}}_{3}}+b_{\mu}\bar{\partial}_{3}{\bar{a}}_{k}\frac{\partial j_{a}}{\partial{\bar{a}}_{k}}\right)$
	$\displaystyle\equiv$	$\displaystyle\partial_{\mu}^{E\parallel}j_{a}+\partial_{\mu}^{{\cal B}\parallel}j_{a}+\partial_{\mu}^{E\perp}j_{a}+\partial_{\mu}^{{\cal B}\perp}j_{a}\,.$		(30)