CERN-TH-2022-069

UMN-TH-4120/22

FTPI-MINN-22-11

KEK-TH-2415

Transient phenomena
in the axion assisted Schwinger effect

Valerie Domckea,b, Yohei Emac, Kyohei Mukaidad,e

a | _Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland_
---|---
b | _Laboratory for Particle Physics and Cosmology, Institute of Physics,_
| _School of Basic Sciences, EPFL, 1015 Lausanne, Switzerland_
c | _William I. Fine Theoretical Physics Institute, School of Physics and Astronomy,_
| _University of Minesota, Minneapolis, M 55455, USA_
d | _Theory Center, IPNS, KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan_
e | _Graduate University for Advanced Studies (Sokendai),_
| _1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan_

Particle production induced by a time-dependent background is well understood
as the projection of the time-evolved initial state onto a set of final
states. While the asymptotic initial and final states are well defined in the
usual way, the definition of particles and antiparticles at intermediate times
in the presence of external fields is ambiguous. These external fields
moreover induce divergences which require regularization. In this paper we
clarify some subtleties in the computation of transient effects in physical
quantities for fermions in a homogeneous axion background, including Schwinger
production in background electromagnetic fields. The presence of the axion
requires particular care as well as knowledge of the UV theory when regulating
the theory and computing the vacuum contribution to the fermion energy.

###### Contents

  1. 1 Introduction
  2. 2 Fermion production in a gauge field and axion background
    1. 2.1 Dirac equation with electric field and axion
    2. 2.2 Axion assisted Schwinger effect
  3. 3 Basis choice and Regularization
    1. 3.1 Basis choice and Hamiltonian
    2. 3.2 Regularization and anomaly equation
  4. 4 Transient phenomena and UV sensitivity
    1. 4.1 Transient effects and basis choice independence
    2. 4.2 Higher dimensional operator and UV sensitivity
  5. 5 Conclusions
  6. A Notation and conventions
  7. B Technical details on solutions of Dirac equation
    1. B.1 Dirac equation with electric field
    2. B.2 Dirac equation with electric and magnetic field
    3. B.3 Physical quantities

##  1 Introduction

The axion was first introduced as the angular degree of freedom of an
approximate global $U(1)$ symmetry to address the strong CP problem in QCD [1,
2, 3, 4]. In modern cosmology, this concept has been generalized to axion-like
particles which have been postulated to play a key role in string theory [5,
6, 7], inflation [8, 9], baryogenesis [10, 11], dark matter [12, 13, 14] and
in cosmological approaches to the hierarchy problem [15]. At the level of
quantum field theory, these axion-like particles (axions for short in
following) are pseudoscalars which couple to fermions and gauge fields only
through derivative couplings. Of particular phenomenological interest are
couplings to the fermions and gauge groups of the Standard Model (SM), in
particular the electromagnetic sector.\natural1\natural1$\natural$1 At
energies above the electroweak phase transition, the relevant Abelian gauge
group is hypercharge. For simplicity, we will use the notation familiar from
electromagnetism throughout this paper, though of course the discussion
applies to any Abelian gauge group with charged fermions. For non-Abelian
gauge groups, see _e.g._ [16, 17].  In this case, the gauge fields, charged
fermions and the axion form a coupled system linked by non-linear
interactions. For example, a non-vanishing axion velocity can trigger a
tachyonic instability in one of the gauge field helicity modes [18, 19],
Schwinger pair production of charged fermions in background gauge fields [20,
21] leads to an induced current which dampens this tachyonic instability [22,
23, 24], and the gauge fields in turn back-react as an effective friction to
the axion motion [25]. Moreover, it was recently pointed out that a non-
vanishing axion velocity has the intriguing property of boosting the Schwinger
pair production of charged fermions in an electric field [26]. In the axion-
assisted Schwinger effect [26] a sufficiently large constant axion velocity
can exponentially enhance the Schwinger production rate for finite fermion
momentum in the final state whereas the axion-driven pair production [27]
relies a non-vanishing acceleration of the axion field.

In the literature, particle production is usually discussed in terms of the
asymptotic initial and final states, where the particles and antiparticles are
well defined. Starting in a ground state in the asymptotic past, the
electromagnetic and/or axion background fields are switched on for some finite
time, and the final state in the asymptotic future, after switching off the
background fields, is determined by solving the equations of motion for the
Bogoliubov coefficients. The time-dependence of the background fields, _i.e._
the mismatch of the ground state in the asymptotic past and future, is the
origin of the fermion production. On the other hand, transient effects
occurring when the external fields are active are more subtle. At this
intermediate stage, the definition of particles and antiparticles is ambiguous
and the divergences associated with the external fields call for an
appropriate regularization scheme. In a practical computation, one may select
a particular basis of creation/annihilation operators, which however results
in seemingly very different transient effects for different choices of basis,
as is well known in the case of Schwinger production in time-dependent
electric fields [28, 29]. However, physical quantities such as the fermion
current and energy density should not depend on such unphysical
reparametrizations of the theory.

In this paper, we extend the existing results in a three-fold way. Firstly, we
point out that in order to compute backreaction effects in the coupled axion,
gauge field and fermion system, it is insufficient to focus on results for the
asymptotic future, it becomes instead crucial to correctly capture transient
phenomena. We demonstrate explicitly that including vacuum contributions and
an appropriate regularization scheme, the apparent differences caused by the
choice of basis are merely differences in interpretation, _i.e._ labeling
terms as vacuum contribution, time-dependent Bogoliubov coefficients or
higher-dimensional operators, while the physical observables are unaffected by
these different basis choices – as they should. We secondly use this
opportunity to compare different approaches in the existing literature, in
particular [26],[27] and [30]. In this context, we highlight the importance of
choosing a regularization scheme which respects the symmetries of the theory
and the benefits of making use of the Adler-Bell-Jackiw anomaly equation [31,
32] as a necessary verification of this. Thirdly, we numerically and
analytically compute the transient contributions to fermion production in the
presence of a background axion field. We show that _e.g._ the fermion energy
in the limit of large axion velocities can be exponentially enhanced, even in
the absence of electromagnetic background fields. However, we note that the
dominant contribution can be interpreted as dimension-eight operator in the
axion effective theory. This proves that the transient effects in the axion-
fermion system cannot be reliably computed without knowledge of the UV
properties of the theory, i.e. a particular realization of the Peccei Quinn
theory.

The remainder of this paper is organized as follows. After reviewing the key
ingredients of the axion assisted Schwinger effect in Section 2, we discuss in
some detail different basis choices and regularization schemes in Section 3.
The computation of transient phenomena in observable quantities is presented
in Section 4, including a discussion on the UV sensitivity. We conclude in
Sec. 5. Various technical details are relegated to the appendices. App. A
specifies our notation and conventions, while App. B gives the necessary
details on the solutions of the Dirac equation.

##  2 Fermion production in a gauge field and axion background

In this section we review the axion assisted Schwinger effect [26]. We
consider the following action:

$\displaystyle
S=\int\mathrm{d}^{4}x\left[\frac{1}{2}\left(\partial\phi\right)^{2}-V(\phi)-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}\left(i\not{D}-me^{2ic_{m}\phi/f_{a}\gamma_{5}}\right)\psi+c_{A}\frac{\alpha}{4\pi
f_{a}}\phi
F_{\mu\nu}\tilde{F}^{\mu\nu}+c_{5}\frac{\partial_{\mu}\phi}{f_{a}}\bar{\psi}\gamma^{\mu}\gamma_{5}\psi\right],$
(2.1)

where $\phi$ is the axion field with its potential $V(\phi)$ and its decay
constant $f_{a}$, $A_{\mu}$ is the U(1) gauge field with its field strength
$F_{\mu\nu}$, and $\psi$ is a fermion with the mass $m$ whose production we
will discuss in detail. The covariant derivative is defined by
$D_{\mu}=\partial_{\mu}+igQA_{\mu}$, where $g$ is the gauge coupling with
$\alpha=g^{2}/4\pi^{2}$ and $Q$ is the charge of the fermion. The dual field
strength tensor is defined as
$\tilde{F}_{\mu\nu}=\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}/2$ with
$\epsilon^{0123}=+1$. Finally, $c_{m}$, $c_{A}$ and $c_{5}$ are the coupling
constants between the axion and the gauge fields/fermions. The Dirac equation
reads

$\displaystyle\left[i\not{D}-me^{2i\theta_{m}\gamma_{5}}+\partial_{\mu}\theta_{5}\gamma^{\mu}\gamma_{5}\right]\psi=0,$
(2.2)

where we denote $\theta_{i}=c_{i}\phi/f_{a}$. We solve this equation in the
presence of background electromagnetic fields and a homogeneous axion field
with $\dot{\phi}\neq 0$.

It is well known that a strong electric field creates pairs of fermions and
anti-fermions through the Schwinger effect [20, 21]. The production rate
$\Gamma$ is proportional to

$\displaystyle\Gamma\propto\exp\left[-\frac{\pi
m_{T}^{2}}{g\left|{Q}\right|E}\right],\quad m_{T}^{2}=m^{2}+p_{T}^{2},$ (2.3)

where $p_{T}$ is the transverse momentum, _e.g._
$p_{T}^{2}=p_{x}^{2}+p_{y}^{2}$ if the electric field is in
$\hat{z}$-direction. Note that beyond the fermion mass suppression, the
production of the modes with large transverse momentum is exponentially
further suppressed. As however recently demonstrated in [26] this suppression
from the transverse momentum disappears once the axion velocity is large
enough, and the rate eventually becomes

$\displaystyle\Gamma\propto\exp\left[-\frac{\pi
m^{2}}{g\left|{Q}\right|E}\right]~{}~{}~{}~{}~{}\mathrm{for}~{}~{}~{}~{}~{}\dot{\theta}_{5+m}^{2}\gtrsim\frac{\pi
m^{2}p_{T}^{2}}{g\left|{Q}\right|E},$ (2.4)

where $\theta_{5+m}=(c_{5}+c_{m})\phi/f_{a}$. This enhances the Schwinger
effect for non-zero transverse momentum and was hence dubbed the axion
assisted Schwinger effect. The purpose of this section is to summarize the key
ingredients of this effect (see [26] and App. B for more details).

In anticipation of our later discussion, we note that Eq. (2.2) has a
redundancy. Indeed, the couplings shift under the chiral rotation $\psi\to
e^{ic\phi/f_{a}\gamma_{5}}\psi$ as

$\displaystyle c_{5}\to c_{5}-c,\quad c_{m}\to c_{m}+c.$ (2.5)

This indicates that only the combination $c_{5}+c_{m}$ is physical, and one
can eliminate either $c_{5}$ or $c_{m}$ by a chiral rotation. In this section
we focus on the $c_{5}=0$ basis following [26], and comment on the $c_{m}=0$
basis at the end of Sec. 2.2. A particular basis choice implies a natural
definition of particles and antiparticles by the corresponding positive and
negative frequency modes. Hence, throughout this paper, we identify the basis
choice with respect to the redundancy under a chiral rotation with a
particular choice of creation/annihilation operators, although these two
concepts are not equal, strictly speaking. This basis transformation is a key
topic of Sec. 3.

###  2.1 Dirac equation with electric field and axion

In this section, we focus on the case with a background electric field and
axion velocity, without any magnetic field. We consider background gauge and
axion fields parameterized by

$\displaystyle A^{\mu}=\left(0,0,0,A_{z}(t)\right),\qquad\phi=\phi(t),$ (2.6)

where the electric field is given by $E=-\dot{A}_{z}$. Without loss of
generality we take $E\geq 0$. We will take this background to be constant at
some initial and final time, $\dot{\phi}=0$, $\dot{A}=0$, with a time-
dependence at intermediate times. The Dirac equation is then given by

$\displaystyle 0$
$\displaystyle=\begin{pmatrix}i\partial_{0}+\Pi_{z}-\dot{\theta}_{5}&p_{x}-ip_{y}&-me^{2i\theta_{m}}&0\\\
p_{x}+ip_{y}&i\partial_{0}-\Pi_{z}-\dot{\theta}_{5}&0&-me^{2i\theta_{m}}\\\
-me^{-2i\theta_{m}}&0&i\partial_{0}-\Pi_{z}+\dot{\theta}_{5}&-(p_{x}-ip_{y})\\\
0&-me^{-2i\theta_{m}}&-(p_{x}+ip_{y})&i\partial_{0}+\Pi_{z}+\dot{\theta}_{5}\end{pmatrix}\psi.$
(2.7)

We solve this equation with the Bogoliubov coefficient method, which describes
the mixing of the positive and negative frequency modes. The Bogoliubov
coefficients $\alpha_{\lambda}$ and $\beta_{\lambda}$ evolve as (see App. B
for derivation)

$\displaystyle\begin{pmatrix}\dot{\alpha}_{1}\\\ \dot{\alpha}_{2}\\\
\dot{\beta}_{1}\\\ \dot{\beta}_{2}\end{pmatrix}=$
$\displaystyle\left[i\dot{\theta}_{5+m}\begin{pmatrix}-\frac{m}{m_{T}}\frac{\Pi_{z}}{\Omega}&\frac{p_{T}}{m_{T}}&\frac{m}{\Omega}e^{2i\Theta}&0\\\
\frac{p_{T}}{m_{T}}&\frac{m}{m_{T}}\frac{\Pi_{z}}{\Omega}&0&-\frac{m}{\Omega}e^{2i\Theta}\\\
\frac{m}{\Omega}e^{-2i\Theta}&0&\frac{m}{m_{T}}\frac{\Pi_{z}}{\Omega}&\frac{p_{T}}{m_{T}}\\\
0&-\frac{m}{\Omega}e^{-2i\Theta}&\frac{p_{T}}{m_{T}}&-\frac{m}{m_{T}}\frac{\Pi_{z}}{\Omega}\end{pmatrix}+\frac{m_{T}\dot{\Pi}_{z}}{2\Omega^{2}}\begin{pmatrix}0&0&-e^{2i\Theta}&0\\\
0&0&0&-e^{2i\Theta}\\\ e^{-2i\Theta}&0&0&0\\\
0&e^{-2i\Theta}&0&0\end{pmatrix}\right]\begin{pmatrix}{\alpha}_{1}\\\
{\alpha}_{2}\\\ {\beta}_{1}\\\ {\beta}_{2}\end{pmatrix},$ (2.8)

where $\Pi_{z}=p_{z}-gQA_{z}$, $p_{T}^{2}=p_{x}^{2}+p_{y}^{2}$,
$m_{T}^{2}=p_{T}^{2}+m^{2}$, $\Omega=\sqrt{\Pi_{z}^{2}+m_{T}^{2}}$, and
$\Theta=\int^{t}\mathrm{d}t\,\Omega$. We solve these equations with two
different initial conditions,

$\displaystyle\alpha_{1}^{(1)}$
$\displaystyle=1,\quad\alpha_{2}^{(1)}=\beta_{1}^{(1)}=\beta_{2}^{(1)}=0,\qquad\mathrm{and}\qquad\alpha_{2}^{(2)}=1,\quad\alpha_{1}^{(2)}=\beta_{1}^{(2)}=\beta_{2}^{(2)}=0.$
(2.9)

Using the Bogoliubov coefficients, we quantize the fermion as

$\displaystyle\psi$
$\displaystyle=\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}e^{i\vec{p}\cdot\vec{x}}e^{i\theta_{5}\gamma_{5}}\sum_{\lambda=1,2}\left[B_{\lambda}u_{\lambda}e^{-i\Theta}+D_{\lambda}^{\dagger}v_{\lambda}e^{+i\Theta}\right],$
(2.10)

where $u_{\lambda}$ and $v_{\lambda}$ are the solutions of the Dirac equation
for the constant background fields. Note that we have extracted the factor
$e^{i\theta_{5}\gamma_{5}}$ to eliminate the derivative coupling between the
axion and the fermion current, _i.e._ $c_{5}=0$ in this basis. The creation
and annihilation operators at the intermediate times, $B_{\lambda}$ and
$D_{\lambda}$, are expressed as

$\displaystyle B_{\lambda}$
$\displaystyle=\sum_{\lambda^{\prime}=1,2}\left[\alpha_{\lambda}^{(\lambda^{\prime})}b_{\lambda^{\prime}}-(-)^{\lambda+\lambda^{\prime}}{\beta_{\lambda}^{(\lambda^{\prime})}}^{*}d_{\lambda^{\prime}}^{\dagger}\right],\quad
D_{\lambda}^{\dagger}=\sum_{\lambda^{\prime}=1,2}\left[\beta_{\lambda}^{(\lambda^{\prime})}b_{\lambda^{\prime}}+(-)^{\lambda+\lambda^{\prime}}{\alpha_{\lambda}^{(\lambda^{\prime})}}^{*}d_{\lambda^{\prime}}^{\dagger}\right],$
(2.11)

where $b_{\lambda}$ and $d_{\lambda}$ are the creation and annihilation
operators in the infinite past. We impose the anti-commutation relation as

$\displaystyle\left\\{b_{\lambda_{1}}(\vec{p}),b_{\lambda_{2}}^{\dagger}(\vec{p}^{\prime})\right\\}$
$\displaystyle=\left\\{d_{\lambda_{1}}(\vec{p}),d_{\lambda_{2}}^{\dagger}(\vec{p}^{\prime})\right\\}=(2\pi)^{3}\delta_{\lambda_{1}\lambda_{2}}\delta^{(3)}(\vec{p}-\vec{p}^{\prime}),$
(2.12)
$\displaystyle\left\\{b_{\lambda_{1}}(\vec{p}),d_{\lambda_{2}}^{\dagger}(\vec{p}^{\prime})\right\\}$
$\displaystyle=\left\\{b_{\lambda_{1}}(\vec{p}),d_{\lambda_{2}}(\vec{p}^{\prime})\right\\}=0.$
(2.13)

It then follows from Eq. (2.8) that $B_{\lambda}$ and $D_{\lambda}$ satisfy
the same equal time anti-commutation relations, _i.e._ ,

$\displaystyle\left\\{B_{\lambda_{1}}(t,\vec{p}),B_{\lambda_{2}}^{\dagger}(t,\vec{p}^{\prime})\right\\}$
$\displaystyle=\left\\{D_{\lambda_{1}}(t,\vec{p}),D_{\lambda_{2}}^{\dagger}(t,\vec{p}^{\prime})\right\\}=(2\pi)^{3}\delta_{\lambda_{1}\lambda_{2}}\delta^{(3)}(\vec{p}-\vec{p}^{\prime}),$
(2.14)
$\displaystyle\left\\{B_{\lambda_{1}}(t,\vec{p}),D_{\lambda_{2}}^{\dagger}(t,\vec{p}^{\prime})\right\\}$
$\displaystyle=\left\\{B_{\lambda_{1}}(t,\vec{p}),D_{\lambda_{2}}(t,\vec{p}^{\prime})\right\\}=0.$
(2.15)

###  2.2 Axion assisted Schwinger effect

Figure 1: Numerical solutions for the total number of produced particles after
turning off the background fields. _Left:_ spectrum of produced particles for
different values of the background axion velocity $\dot{\theta}_{0}$ as a
function of the fermion momentum $p_{z}$. The gray dashed line corresponds to
the standard Schwinger effect formula, $n_{\psi}=2\exp(-\pi
m_{T}^{2}/g\left|{Q}\right|E_{0})$, where the factor 2 accounts for the
chirality. _Right:_ the height of the spectrum (at
$p_{z}=-50\sqrt{g\left|{Q}\right|E_{0}}$) as a function of
$\dot{\theta}_{0}^{2}$. The blue line corresponds to the numerical solution of
the time evolution equation of the Bogoliubov coefficients, while the gray
dashed line shows the analytical expression (2.17). In both figures, we take
$m=1.5$ and $p_{T}=2$ in units of $g\left|{Q}\right|E_{0}=1$.

We now turn to the fermion pair production in the presence of a background
electric field and axion velocity. We focus on the axion assisted Schwinger
effect, and thus take the axion velocity to be
constant.\natural2\natural2$\natural$2 If the axion velocity is exactly
constant, the axion field by itself cannot produce fermions, it can only
enhance the production driven by the electric field. In other words, the
production rate vanishes for $E=0$ for constant $\dot{\phi}$. If
$\ddot{\phi}\neq 0$, the axion can produce fermions independently of the
background gauge field, which includes the standard particle production by an
oscillating axion background field decaying into fermions as well as axion-
driven pair production.  It is well known that the notion of particles is ill-
defined in the time-dependent background, and thus we turn on and off the
external electric and axion fields adiabatically at $t_{\text{min}}$ and
$t_{\text{max}}$, respectively. To be specific, we take the background fields
as

$\displaystyle\dot{\theta}_{5+m}$
$\displaystyle=\frac{\dot{\theta}_{0}}{4}\left[1+\mathrm{tanh}\left(\frac{t-t_{\mathrm{min}}}{T}\right)\right]\left[1-\mathrm{tanh}\left(\frac{t-t_{\mathrm{max}}}{T}\right)\right],~{}~{}\dot{A}_{z}=-\frac{E_{0}}{4}\left[1+\mathrm{tanh}\left(\frac{t-t_{\mathrm{min}}}{T}\right)\right]\left[1-\mathrm{tanh}\left(\frac{t-t_{\mathrm{max}}}{T}\right)\right],$
(2.16)

and start to solve the system well before $t_{\mathrm{min}}$. Here we restrict
ourselves to evaluate the particle number
$n_{\psi}=\sum_{\lambda,\lambda^{\prime}}|\beta_{\lambda}^{(\lambda^{\prime})}|^{2}$
only well after $t_{\mathrm{max}}$. We will study physical observables at the
intermediate times in the subsequent sections.

In Fig. 1 we show a numerical solution of the equation of motion. We take
$t_{\mathrm{min}}=0$, $t_{\mathrm{max}}=100\sqrt{g\left|{Q}\right|E_{0}}$,
$T=\sqrt{50/g\left|{Q}\right|E_{0}}$, and evaluated the spectrum at
$t=1.5t_{\mathrm{max}}$. In the left panel, we show the spectrum for $Q=+1$,
$m=1.5$ and $p_{T}=2$ in the unit $g\left|{Q}\right|E_{0}=1$ for different
values of $\dot{\theta}_{0}$ (see [26] for several other values of $p_{T}$ and
$m$). The enhanced part of the spectrum corresponds to the modes that cross
$\Pi_{z}=0$. Since the gap size between the positive and negative frequency
modes is smallest at $\Pi_{z}=0$ (for $\dot{\theta}_{0}<p_{T}$), the modes are
most easily excited at this point. The height of the plateau is enhanced for
non-zero $\dot{\theta}_{0}$ compared to the standard Schwinger effect formula,
$n_{\psi}=2\exp(-\pi m_{T}^{2}/g\left|{Q}\right|E_{0})$ (which we show as the
dashed line), corresponding to the axion assisted Schwinger effect. In order
to take a closer look at this enhancement, in the right panel, we plot the
height of the plateau versus $\dot{\theta}_{0}^{2}$. As one can see, the
envelope eventually approaches to $\exp(-\pi m^{2}/g\left|{Q}\right|E_{0})$
for large enough $\dot{\theta}_{0}$. In other words, the suppression from the
transverse momentum is absent for a sufficiently large axion velocity. On top
of this enhancement, the height of the plateau strongly oscillates as a
function of $\dot{\theta}_{0}$. As demonstrated in [26] this result is well
reproduced by the analytical expression\natural3\natural3$\natural$3 Here we
assume that $\dot{\theta}_{0}$ is positive. If $\dot{\theta}_{0}$ is negative,
we should replace $\Omega_{-}$ by $\Omega_{+}$.

$\displaystyle
n_{\psi}=\left|{\exp\left[2i\int_{0}^{\Pi_{++}}\frac{\mathrm{d}\Pi_{z}}{g\left|{Q}\right|E_{0}}\Omega_{-}\right]}\right|^{2}+\left|{\exp\left[2i\int_{0}^{\Pi_{--}}\frac{\mathrm{d}\Pi_{z}}{g\left|{Q}\right|E_{0}}\Omega_{-}\right]}\right|^{2}-2\mathrm{Re}\left[\exp\left[2i\int_{\Pi_{-+}}^{\Pi_{++}}\frac{\mathrm{d}\Pi_{z}}{g\left|{Q}\right|E_{0}}\Omega_{-}\right]\right],$
(2.17)

which we show as the gray line in the right panel. Here we have defined

$\displaystyle\Omega_{\pm}$
$\displaystyle=\sqrt{\left(\sqrt{\Pi_{z}^{2}+p_{T}^{2}}\pm\dot{\theta}_{0}\right)^{2}+m^{2}},\quad\Pi_{\lambda\lambda^{\prime}}=\lambda\sqrt{(\dot{\theta}_{0}+i\lambda^{\prime}m)^{2}-p_{T}^{2}}.$
(2.18)

The latter corresponds to the solution of $\Omega_{-}(\Pi_{z})=0$ in the
complex $\Pi_{z}$-plane. This suggests that the dispersion relation
$\Omega_{-}$ is crucial for the axion assisted Schwinger effect. Indeed, as
shown in [26], this dispersion relation arises due to spin-momentum
interaction induced by the axion in the non-relativistic limit of the fermion.

Ref. [27] confirms this result in the $c_{m}=0$ basis. Indeed, one can
eliminate $c_{m}$ by the chiral rotation $\psi\to
e^{-i\theta_{m}\gamma_{5}}\psi$. The Bogoliubov coefficients in this basis
satisfy

$\displaystyle\begin{pmatrix}\dot{\tilde{\alpha}}_{+}\\\
\dot{\tilde{\alpha}}_{-}\\\ \dot{\tilde{\beta}}_{+}\\\
\dot{\tilde{\beta}}_{-}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}0&Ae^{i(\Theta_{+}-\Theta_{-})}&-B_{+}e^{2i\Theta_{+}}&-Ce^{i(\Theta_{+}+\Theta_{-})}\\\
-Ae^{-i(\Theta_{+}-\Theta_{-})}&0&-Ce^{i(\Theta_{+}+\Theta_{-})}&-B_{-}e^{2i\Theta_{-}}\\\
B_{+}e^{-2i\Theta_{+}}&Ce^{-i(\Theta_{+}+\Theta_{-})}&0&Ae^{-i(\Theta_{+}-\Theta_{-})}\\\
Ce^{-i(\Theta_{+}+\Theta_{-})}&B_{-}e^{-2i\Theta_{-}}&-Ae^{i(\Theta_{+}-\Theta_{-})}&0\end{pmatrix}\begin{pmatrix}\tilde{\alpha}_{+}\\\
\tilde{\alpha}_{-}\\\ \tilde{\beta}_{+}\\\ \tilde{\beta}_{-}\end{pmatrix},$
(2.19)

where

$\displaystyle\Theta_{\pm}=\int^{t}\Omega_{\pm}\mathrm{d}t.$ (2.20)

See App. B for the explicit forms of $A,B$ and $C$, which are irrelevant for
our purpose here. The key point is that the above equation explicitly involves
$\Omega_{\pm}$, and thus it is more straightforward to see the importance of
$\Omega_{\pm}$ on particle production with, _e.g._ the WKB method [27].
However, this basis has its own subtlety, related to the regularization and
renormalization, and demonstrating this point is our main goal in Sec. 3.
There we will see that each basis has its own advantages and pitfalls; the WKB
analysis of particle production is more straightforward in the $c_{m}=0$
basis, while the $c_{5}=0$ basis is more suitable for regularizing and
renormalizing the theory.

##  3 Basis choice and Regularization

As shown above, the axion assisted Schwinger effect may be more transparent in
the basis $c_{m}=0$ (which we will refer to as _the fermion current basis_)
than in the basis $c_{5}=0$ (which we will call _the Hamiltonian basis_).
However, there is a subtlety in the fermion current basis related to
regularization. Physical observables are independent of the choice of basis
and in principle one can of course work in any basis. In reality, however,
physical observables are often divergent and require regularization and
renormalization. A regularization scheme should preserve the symmetries of the
theory as much as possible, which is straightforward in the Hamiltonian basis
while can be more subtle in the fermion current basis. The key point is that
the Hamiltonian basis diagonalizes the Hamiltonian (as the name suggests),
while the fermion current basis does not. The purpose of this section is to
demonstrate this point in detail, by taking the anomaly equation as an
example.

We emphasize that the regularization and renormalization are not only of
academic interest. If we focus only on physical quantities in the asymptotic
future, after turning off the external fields (as we did in Sec. 2), there are
no complications. However, if we follow the time evolution of the physical
quantities at intermediate times, we necessarily have to deal with
divergences, regularization and renormalization. Since physical quantities
such as the induced current and the axial charge at intermediate times are
crucial to study, _e.g._ the backreaction of the fermion production to the
axion and gauge field dynamics, a proper treatment of the renormalization and
regularization is phenomenologically relevant.

###  3.1 Basis choice and Hamiltonian

A fermion chiral rotation $\psi\to e^{ic\gamma_{5}}\psi$ shifts the couplings
as $c_{5}\to c_{5}-c$ and $c_{m}\to c_{m}+c$. This redundancy corresponds to
the basis choice of the fermion that one works with. In Ref. [26] we take the
Hamiltonian basis ($c_{5}=0$), while Ref. [27] takes the fermion current basis
($c_{m}=0$).\natural4\natural4$\natural$4 More generally, we can take a basis
with both non-zero $c_{5}$ and $c_{m}$. This choice simply makes the equations
more involved and we do not see any benefit, and thus we focus only on the
Hamiltonian and fermion current basis in this paper.  Our goal here is to see
that the Hamiltonian is diagonalized only in the Hamiltonian basis, and not in
the fermion current basis, which is crucial for our study of the anomaly
equation in Sec. 3.2. Since we will discuss the anomaly equation, we now
include the magnetic field (anti-)parallel to the electric
field.\natural5\natural5$\natural$5 Gauge field production towards the end of
axion inflation provides this configuration [9], so the inclusion of the
magnetic field also has a phenomenological motivation.  We take the background
fields as

$\displaystyle A^{\mu}=\left(0,0,Bx,A_{z}\right),\qquad\phi=\phi(t),$ (3.1)

which corresponds to $\vec{E}=E\hat{z}=-\dot{A}_{z}\hat{z}$ and
$\vec{B}=B\hat{z}$, where the magnetic field $B$ is taken constant. This
configuration has

$\displaystyle F_{\mu\nu}\tilde{F}^{\mu\nu}=-4\vec{E}\cdot\vec{B}\neq 0,$
(3.2)

which makes the anomaly equation non-trivial at the intermediate times.

In the presence of the magnetic field, a charged particle follows a circular
trajectory in the plane orthogonal to the magnetic field. This transverse
motion is quantized in quantum mechanics, resulting in discrete Landau levels.
The mode without transverse motion is called the lowest Landau level, while
those with transverse motion are called higher Landau levels. Therefore, in
the presence of a magnetic field, the fermion is quantized as

$\displaystyle\psi=$
$\displaystyle\int\frac{\mathrm{d}p_{y}\mathrm{d}p_{z}}{(2\pi)^{2}}e^{i(p_{y}y+p_{z}z)}e^{i\gamma_{5}\theta_{5}}\left[B_{0}u_{0}e^{-i\Theta_{0}}+D_{0}^{\dagger}v_{0}e^{i\Theta_{0}}+\sum_{n,\lambda=1,2}\left[B_{n,\lambda}u_{n,\lambda}e^{-i\Theta_{n}}+D_{n,\lambda}^{\dagger}v_{n,\lambda}e^{i\Theta_{n}}\right]\right],$
(3.3)

where

$\displaystyle B_{0}$
$\displaystyle=\alpha_{0}b_{0}-\beta_{0}^{*}d_{0}^{\dagger},\quad
B_{n,\lambda}=\sum_{\lambda^{\prime}=1,2}\left[\alpha_{n,\lambda}^{(\lambda^{\prime})}b_{n,\lambda^{\prime}}-\left(-1\right)^{\lambda+\lambda^{\prime}}{\beta_{n,\lambda}^{(\lambda^{\prime})}}^{*}d_{n,\lambda^{\prime}}^{\dagger}\right],$
(3.4) $\displaystyle D_{0}^{\dagger}$
$\displaystyle=\beta_{0}b_{0}+\alpha_{0}^{*}d_{0}^{\dagger},\quad
D_{n,\lambda}^{\dagger}=\sum_{\lambda^{\prime}=1,2}\left[\beta_{n,\lambda}^{(\lambda^{\prime})}b_{n,\lambda^{\prime}}+\left(-1\right)^{\lambda+\lambda^{\prime}}{\alpha_{n,\lambda}^{(\lambda^{\prime})}}^{*}d_{n,\lambda^{\prime}}^{\dagger}\right],$
(3.5)

in the Hamiltonian basis, and

$\displaystyle\psi=$
$\displaystyle\int\frac{\mathrm{d}p_{y}\mathrm{d}p_{z}}{(2\pi)^{2}}e^{i(p_{y}y+p_{z}z)}e^{-i\gamma_{5}\theta_{m}}\left[\tilde{B}_{0}\tilde{u}_{0}e^{-i\Theta_{0,s}}+\tilde{D}_{0}^{\dagger}\tilde{v}_{0}e^{i\Theta_{0,s}}+\sum_{n,\lambda=\pm}\left[\tilde{B}_{n,\lambda}\tilde{u}_{n,\lambda}e^{-i\Theta_{n,\lambda}}+\tilde{D}_{n,\lambda}^{\dagger}\tilde{v}_{n,\lambda}e^{i\Theta_{n,\lambda}}\right]\right],$
(3.6)

where

$\displaystyle\tilde{B}_{0}$
$\displaystyle=\tilde{\alpha}_{0}{b}_{0}-\tilde{\beta}_{0}^{*}{d}_{0}^{\dagger},\quad\tilde{B}_{n,\lambda}=\sum_{\lambda^{\prime}=\pm}\left[\tilde{\alpha}_{n,\lambda}^{(\lambda^{\prime})}{b}_{n,\lambda^{\prime}}-\tilde{\beta}_{n,\lambda}^{(\lambda^{\prime})*}{d}_{n,\lambda^{\prime}}^{\,\dagger}\right],$
(3.7) $\displaystyle\tilde{D}_{0}^{\dagger}$
$\displaystyle=\tilde{\beta}_{0}{b}_{0}+\tilde{\alpha}_{0}^{*}{d}_{0}^{\dagger},\quad\tilde{D}_{n,\lambda}^{\dagger}=\sum_{\lambda^{\prime}=\pm}\left[\tilde{\beta}_{n,\lambda}^{(\lambda^{\prime})}{b}_{n,\lambda^{\prime}}+\tilde{\alpha}_{n,\lambda}^{(\lambda^{\prime})*}{d}_{n,\lambda^{\prime}}^{\,\dagger}\right],$
(3.8)

in the fermion current basis, respectively (see App. B for definitions and the
derivation). The integer $n$ labels the discrete Landau levels, $n=0$
corresponding to the lowest Landau level while $n=1,2,\cdots$ refers to the
higher Landau levels. Notice that the higher Landau levels are analogous to
the situation without a magnetic field; the only difference is whether the
transverse momentum is quantized or not. The frequencies in the decomposition
are different in the two bases. They are obtained as

$\displaystyle\Omega_{0}$
$\displaystyle=\sqrt{\Pi_{z}^{2}+m^{2}},\quad\Omega_{n}=\sqrt{\Pi_{z}^{2}+m_{T}^{2}},$
(3.9)

with $m_{T}=\sqrt{m^{2}+m_{B}^{2}}$, $m_{B}=\sqrt{2ng|QB|}$,
$\Theta_{0}=\int^{t}\mathrm{d}t\,\Omega_{0}$ and
$\Theta_{n}=\int^{t}\mathrm{d}t\,\Omega_{n}$ in the Hamiltonian basis, and

$\displaystyle\Omega_{0,s}=\sqrt{(\Pi_{z}-s\dot{\theta}_{5+m})^{2}+m^{2}},\quad\Omega_{n,\lambda}=\sqrt{(\Pi+\lambda\dot{\theta}_{5+m})^{2}+m^{2}},$
(3.10)

with $\Pi=\sqrt{\Pi_{z}^{2}+m_{B}^{2}}$, $s=\text{sign}(QB)$,
$\Theta_{0,s}=\int^{t}\mathrm{d}t\,\Omega_{0,s}$ and
${\Theta}_{n,\lambda}=\int^{t}\mathrm{d}t\,\Omega_{n,\lambda}$ in the fermion
current basis, respectively. Finally the creation and annihilation operators
satisfy

$\displaystyle\left\\{b_{n_{1},\lambda_{1}}(\vec{p}),b_{n_{2},\lambda_{2}}^{\dagger}(\vec{p}^{\prime})\right\\}$
$\displaystyle=\left\\{d_{n_{1},\lambda_{1}}(\vec{p}),d_{n_{2},\lambda_{2}}^{\dagger}(\vec{p}^{\prime})\right\\}=(2\pi)^{2}\delta_{\lambda_{1}\lambda_{2}}\delta_{n_{1}n_{2}}\delta^{(2)}(\vec{p}-\vec{p}^{\prime}),$
(3.11)
$\displaystyle\left\\{b_{n_{1},\lambda_{1}}(\vec{p}),d_{n_{2},\lambda_{2}}^{\dagger}(\vec{p}^{\prime})\right\\}$
$\displaystyle=\left\\{b_{n_{1},\lambda_{1}}(\vec{p}),d_{n_{2},\lambda_{2}}(\vec{p}^{\prime})\right\\}=0,$
(3.12)

and $B$, $D$, $\tilde{B}$ and $\tilde{D}$ satisfy the same equal-time anti-
commutators. As usual, we assume that there is no external field in the
infinite past. This allows us to uniquely define the vacuum as
$b_{0}|0\rangle=b_{n,\lambda}|0\rangle=d_{0}|0\rangle=d_{n,\lambda}|0\rangle=0$.
The physical quantities are then evaluated as the expectation value of the
corresponding operators with respect to this vacuum.

Let us now study the Hamiltonian. The conjugate momentum follows from Eq.
(2.1) as

$\displaystyle\pi_{\phi}$
$\displaystyle=\dot{\phi}+\frac{c_{5}}{f_{a}}\bar{\psi}\gamma^{0}\gamma_{5}\psi,\quad\pi_{\psi}=i\bar{\psi}\gamma^{0},\quad\pi_{A_{i}}=-F^{0i}+c_{A}\frac{\alpha}{\pi
f_{a}}\phi\,\epsilon^{0ijk}F_{jk},$ (3.13)

and thus the Hamiltonian density operator is given by

$\displaystyle\mathcal{H}=\frac{1}{2}\dot{\phi}^{2}+V(\phi)+\psi^{\dagger}\left(i\partial_{0}+\dot{\theta}_{5}\gamma_{5}\right)\psi+\frac{1}{2}\left(\vec{E}^{2}+\vec{B}^{2}\right).$
(3.14)

where we have assumed that the fermion satisfies the Dirac equation, and
substituted our configuration of the axion and gauge field. Thus we define the
fermionic part of the Hamiltonian as

$\displaystyle
H_{\psi}\equiv\frac{1}{2}\int\mathrm{d}^{3}x\left[\psi^{\dagger},\left(i\partial_{0}+\dot{\theta}_{5}\gamma_{5}\right)\psi\right],$
(3.15)

where we have anti-symmetrized the operator. Note that ${H}_{\psi}$ contains
not only the time derivative of the fermion but also $\dot{\theta}_{5}$
explicitly. In the Hamiltonian basis, by substituting Eq. (3.3), we obtain

$\displaystyle H_{\psi}$
$\displaystyle=\frac{1}{2}\int\frac{\mathrm{d}p_{y}\mathrm{d}p_{z}}{(2\pi)^{2}}\left[\Omega_{0}\left(\left[B_{0}^{\dagger},B_{0}\right]+\left[D_{0}^{\dagger},D_{0}\right]\right)+\sum_{n,\lambda}\Omega_{n}\left(\left[B_{n,\lambda}^{\dagger},B_{n,\lambda}\right]+\left[D_{n,\lambda}^{\dagger},D_{n,\lambda}\right]\right)\right].$
(3.16)

This shows that the Hamiltonian basis diagonalizes the Hamiltonian even at the
intermediate times for a time-dependent background, as the name already
indicated. In particular, $\Omega_{0}$ and $\Omega_{n}$ are the energy
eigenvalues, which we can then use for regularization without spoiling any
symmetry. On the other hand, in the fermion current basis , the Hamiltonian is
expressed by substituting Eq. (3.6) as

$\displaystyle H_{\psi}=$
$\displaystyle\frac{1}{2}\int\frac{\mathrm{d}p_{y}\mathrm{d}p_{z}}{(2\pi)^{2}}\left\\{\frac{\Pi_{z}^{2}+m^{2}-s\dot{\theta}_{5+m}\Pi_{z}}{\Omega_{0,s}}\left(\left[\tilde{B}_{0}^{\dagger},\tilde{B}_{0}\right]+\left[\tilde{D}_{0}^{\dagger},\tilde{D}_{0}\right]\right)-\frac{m\dot{\theta}_{5+m}}{\Omega_{0,s}}\left(e^{2i\Theta_{0,s}}\left[\tilde{B}_{0}^{\dagger},\tilde{D}_{0}^{\dagger}\right]+(\mathrm{h.c.})\right)\right.$
$\displaystyle\left.+\sum_{n,\lambda}\left[\frac{\Pi_{B}^{2}+m^{2}+\lambda\dot{\theta}_{5+m}\Pi_{B}}{\Omega_{n,\lambda}}\left(\left[\tilde{B}_{n,\lambda}^{\dagger},\tilde{B}_{n,\lambda}\right]+\left[\tilde{D}_{n,\lambda}^{\dagger},\tilde{D}_{n,\lambda}\right]\right)+\frac{m\dot{\theta}_{5+m}}{\Omega_{n,\lambda}}\left(e^{2i\Theta_{n,\lambda}}\left[\tilde{B}_{n,\lambda}^{\dagger},\tilde{D}_{n,\lambda}^{\dagger}\right]+(\mathrm{h.c.})\right)\right]\right\\}.$
(3.17)

This shows that the fermion current basis _does not_ diagonalize the
Hamiltonian. The off-diagonal parts (the second and fourth terms) originate
from the second term in Eq. (3.15). This in particular means that
$\Omega_{0,s}$ and $\Omega_{n,\pm}$ are _not_ energy eigenvalues. We may then
anticipate that the use of regulator functions based on these frequencies will
fail to produce the correct regularized theory. We will investigate this point
in Sec. 3.2, using the anomaly equation as an explicit example.

###  3.2 Regularization and anomaly equation

It is well-known that the chiral rotation is anomalous in the presence of an
gauge field. As a result, the divergence of the axial current $J^{\mu}_{5}$
satisfies the anomaly equation:

$\displaystyle\partial_{\mu}J^{\mu}_{5}=-\frac{g^{2}Q^{2}}{8\pi^{2}}F_{\mu\nu}\tilde{F}^{\mu\nu}+2m\,\bar{\psi}e^{2i\theta_{m}\gamma_{5}}i\gamma_{5}\psi,$
(3.18)

where $J^{\mu}_{5}=\bar{\psi}\gamma^{\mu}\gamma_{5}\psi$. The anomaly equation
is an operator relation and thus must hold in any background field
configuration. This relation provides a non-trivial check of the computation,
in particular of the regularization procedure, since an improper
regularization easily destroys the anomaly equation.

As demonstrated in Ref. [26] the regularization based on the Hamiltonian basis
correctly reproduces the anomaly equation, as we briefly review here. We
consider the spatially-averaged version of the anomaly equation. With our
gauge field configuration (3.1) this is given by

$\displaystyle\dot{q}_{5}=\frac{g^{2}Q^{2}EB}{2\pi^{2}}+2m\left\langle\bar{\psi}e^{2i\theta_{m}\gamma_{5}}i\gamma_{5}\psi\right\rangle,$
(3.19)

where we have replaced $F_{\mu\nu}$ by the electric and magnetic fields, and

$\displaystyle q_{5}$
$\displaystyle\equiv\frac{1}{2\mathrm{Vol}\left(\mathbb{R}_{3}\right)}\int\mathrm{d}^{3}x\left\langle\left[{\psi}^{\dagger},\gamma_{5}\psi\right]\right\rangle,\quad\left\langle\bar{\psi}e^{2i\theta_{m}\gamma_{5}}i\gamma_{5}\psi\right\rangle\equiv\frac{1}{2\mathrm{Vol}\left(\mathbb{R}_{3}\right)}\int\mathrm{d}^{3}x\left\langle\left[\bar{\psi},i\gamma_{5}e^{2i\theta_{m}\gamma_{5}}\psi\right]\right\rangle.$
(3.20)

The lowest Landau level contribution to the chiral charge is given by

$\displaystyle\left.q_{5}\right|_{\mathrm{LLL}}$
$\displaystyle=\frac{g\left|{QB}\right|}{4\pi^{2}}\int\mathrm{d}p_{z}\left[\frac{s{\Pi}_{z}}{\Omega_{0}}\left(2\left\lvert\beta_{0}\right\rvert^{2}-R\right)+\frac{m}{\Omega_{0}}\left(\alpha_{0}\beta_{0}^{*}e^{-2i\Theta_{0}}+\alpha^{*}_{0}\beta_{0}e^{2i\Theta_{0}}\right)\right],$
(3.21)

where we have introduced the regulator function $R=R(\Omega_{0})$. In
particular, normal ordering corresponds to $R=0$. The time derivative is given
by

$\displaystyle\left.\dot{q}_{5}\right|_{\mathrm{LLL}}=\frac{g\left|{QB}\right|}{4\pi^{2}}\int\mathrm{d}p_{z}\left[s\frac{m^{2}\dot{{\Pi}}_{z}}{\Omega_{0}^{3}}\left(1-R\right)-s\frac{{\Pi}_{z}}{\Omega_{0}}\dot{R}-2im\left(\alpha_{0}\beta_{0}^{*}e^{-2i\Theta_{0}}-\alpha_{0}^{*}\beta_{0}e^{2i\Theta_{0}}\right)\right],$
(3.22)

where we have used the time evolution equation of the Bogoliubov coefficients.
We thus obtain

$\displaystyle\left.\dot{q}_{5}\right|_{\mathrm{LLL}}=\frac{gQB}{4\pi^{2}}\int\mathrm{d}p_{z}\left[\frac{m^{2}\dot{{\Pi}}_{z}}{\Omega_{0}^{3}}\left(1-R\right)-\frac{{\Pi}_{z}}{\Omega_{0}}\dot{R}\right]+2m\left.\left\langle\bar{\psi}e^{2i\theta_{m}\gamma_{5}}i\gamma_{5}\psi\right\rangle\right|_{\mathrm{LLL}}.$
(3.23)

The integral does not depend on the explicit form of $R=R(\Omega_{0})$, and we
obtain

$\displaystyle\left.\dot{q}_{5}\right|_{\mathrm{LLL}}=\frac{g^{2}Q^{2}EB}{2\pi^{2}}+2m\left.\left\langle\bar{\psi}e^{2i\theta_{m}\gamma_{5}}i\gamma_{5}\psi\right\rangle\right|_{\mathrm{LLL}}.$
(3.24)

This shows that the anomalous term is supplied solely by the lowest Landau
level. Indeed one can see that the higher Landau level satisfies

$\displaystyle\left.\dot{q}_{5}\right|_{\mathrm{HLL}}$
$\displaystyle=2m\left.\left\langle\bar{\psi}_{n}e^{2i\gamma_{5}\theta_{m}}i\gamma_{5}\psi_{n}\right\rangle\right|_{\mathrm{HLL}}.$
(3.25)

We have thus proven that the anomaly equation (3.19) indeed holds.

We now study the anomaly equation in the fermion current basis. We saw that
the fermion current basis does not diagonalize the Hamiltonian. It is then
naturally expected that the frequencies in this basis, $\Omega_{0,s}$ and
$\Omega_{n,\lambda}$, are not suitable for regularization as they are not
related to the energy eigenvalues. Indeed, in the following, we will see that
the anomaly equation _does not_ hold if one uses $\Omega_{0,s}$ and
$\Omega_{n,\lambda}$ as the argument of the regulator function. This
highlights the main point of this section; the regularization is
straightforward in the Hamiltonian basis, but is more tricky in the fermion
current basis .

If we use $\Omega_{0,s}$ and $\Omega_{n,\lambda}$ for regularization, the
lowest Landau level contribution may be written as

$\displaystyle\left.q_{5}\right|_{\mathrm{LLL}}$
$\displaystyle=\frac{g\left|{QB}\right|}{4\pi^{2}}\int\mathrm{d}p_{z}\left[\frac{s\Pi_{z}-\dot{\theta}_{5+m}}{\Omega_{0,s}}\left(2\left|{\tilde{\beta}_{0}}\right|^{2}-\tilde{R}(\Omega_{0,s})\right)+\frac{2m}{\Omega_{0,s}}\mathrm{Re}\left[\tilde{\alpha}_{0}^{*}\tilde{\beta}_{0}e^{2i\Theta_{0,s}}\right]\right],$
(3.26)

with a regulator function $\tilde{R}$. Notice that this includes the normal
ordering in the fermion current basis as a special case for $\tilde{R}=0$. By
taking the time derivative and using the time evolution equation of the
Bogoliubov coefficients, we obtain

$\displaystyle\left.\dot{q}_{5}\right|_{\mathrm{LLL}}$
$\displaystyle=\frac{gQB}{4\pi^{2}}\int\mathrm{d}p_{z}\left[\frac{m^{2}(\dot{\Pi}_{z}-s\ddot{\theta}_{5+m})}{\Omega_{0,s}}\left(1-\tilde{R}\right)-\frac{{\Pi}_{z}-s\dot{\theta}_{5+m}}{\Omega_{0,s}}\dot{\tilde{R}}\right]+2m\left.\left\langle\bar{\psi}e^{2i\theta_{m}\gamma_{5}}i\gamma_{5}\psi\right\rangle\right|_{\mathrm{LLL}}.$
(3.27)

The integral again does not depend on the explicit form of $\tilde{R}$ as long
as $\tilde{R}=\tilde{R}(\Omega_{0,s})$, and we obtain

$\displaystyle\left.\dot{q}_{5}\right|_{\mathrm{LLL}}$
$\displaystyle=\frac{gQB}{2\pi^{2}}\left(gQE-s\ddot{\theta}_{5+m}\right)+2m\left.\left\langle\bar{\psi}e^{2i\theta_{m}\gamma_{5}}i\gamma_{5}\psi\right\rangle\right|_{\mathrm{LLL}}.$
(3.28)

Note the additional term proportional to $\ddot{\theta}_{5+m}$. One can check
(after a somewhat tedious computation) that the contributions from the higher
Landau levels do not cancel this additional term. Therefore, the
regularization based on the frequencies in the fermion current basis,
$\Omega_{0,s}$ and $\Omega_{n,\lambda}$, _does not_ reproduce the anomaly
equation. This is understandable because the fermion current basis does not
diagonalize the Hamiltonian, and thus the regularization with $\Omega_{0,s}$
and $\Omega_{n,\lambda}$ does not necessarily preserve all the symmetries that
one wants to preserve. This is in contrast to the Hamiltonian basis; the
Hamiltonian is invariant under all the symmetries by definition, and thus the
regularization with the energy eigenvalues, _i.e._ , the frequencies in the
Hamiltonian basis, automatically preserves all the symmetries. Needless to
say, if we use the same regularization as the Hamiltonian basis, we correctly
reproduce the anomaly equation in the fermion current basis as well, as the
physics is independent of the basis choice. The point here is that the
regularization is straightforward in the Hamiltonian basis, but requires
special care in the fermion current basis. One may prefer to work in the
fermion current basis to study particle production (see the end of Sec. 2.2),
but may prefer to work in the Hamiltonian basis to regularize and renormalize
the theory.

Before closing this section, we would like to comment on Ref. [30], which
computed the fermion contribution to the divergence of the axial current in
the fermion current basis with dimensional regularization. The authors
obtained a term with derivatives acting on the axion that does not vanish even
in the massless fermion limit, on top of the standard Chern-Simons
contribution. From the point of view presented here, this is simply because
the regularization scheme employed in [30] does not respect all the
symmetries. Indeed, as the authors noted in the paper, the terms that they got
can be canceled by adding local counter terms to the action, called an
irrelevant anomaly, indicating that there is a better regularization scheme.
This cancellation is not a coincidence or fine tuning, but a reflection of the
fact that the contribution from $c_{5}+c_{m}$ vanishes in the massless limit
due to the chiral rotation invariance of the theory. Therefore, there is a
special reason to choose, among others, the regularization that preserves this
property. Nevertheless, we agree that there exists a subtlety on higher
dimensional operators related to the UV completion, and we discuss this point
in the next section.

##  4 Transient phenomena and UV sensitivity

In Sec. 3 we demonstrated that the regularization is straightforward in the
Hamiltonian basis but requires a special care in the fermion current basis,
giving the anomaly equation as an example. Armed with the proper understanding
of the regularization, we now study another interesting property of the
physical quantities at the intermediate times: exponentially enhanced
transient fermion energy density production for finite fermion masses.

As Eq. (2.4) suggests, although the axion assisted Schwinger effect lifts the
suppression from the transverse momentum, the production rate is still
suppressed exponentially by the fermion mass $m$ for $m^{2}\gg
g\left|{Q}\right|E$. However, Eq. (2.4) describes the spectrum only at the
infinite future, after turning off the external fields, and the physical
quantities at the intermediate times do not necessarily feature this
suppression factor. Indeed, as we will see, if the axion velocity is large
enough, the fermion energy density develops a component that is not
exponentially suppressed even for $m^{2}\gg g\left|{Q}\right|E$. With the
proper regularization, we will check that the same result is obtained in both
the Hamiltonian and the fermion current bases (as it should), although their
appearance is very different; the transient feature originates from the
Bogoliubov coefficients in the Hamiltonian basis while it comes from the
vacuum contribution in the fermion current basis.

This phenomena is related to higher dimensional operators in the axion
effective theory after integrating out fermions. At the intermediate times,
the physical quantities often contain components that are suppressed only by a
power, not an exponential, of the fermion mass in the heavy fermion limit.
These components are in general identified with higher dimensional operators
that arise after integrating out the fermion, such as the Euler-Heisenberg
Lagrangian, as described in detail in Refs. [33, 23]. We will see that the
transient effect that we study here also corresponds to higher dimensional
operators in the axion effective theory.

Identified with higher dimensional operators, this transient effect is
actually degenerate with a choice of the UV theory. The effective theory with
the axion is non-renormalizable, and thus there is in general no reason to
prohibit adding higher dimensional operators by hand. These higher dimensional
operators contribute in the same way as the transient effect that we discuss
here, and thus giving any physical meaning to the transient effect requires
specifying the UV theory. We believe that it is worth clarifying these points,
also to avoid any possible future confusion in literature (see, _e.g._ Refs.
[34, 35, 36, 37, 38, 39, 40] for a similar confusion on the induced current
from the Schwinger effect, which turns out to originate merely from the Euler-
Heisenberg Lagrangian [33, 23]). Therefore, we explain all the above points in
detail in this section.

###  4.1 Transient effects and basis choice independence

We now study the fermion production for a large axion velocity. For our
purpose in this section, we do not actually need any external electric nor
magnetic fields, and thus we set $E=B=0$ for simplicity. In this case there is
no particle production at the infinite future, nevertheless the physical
quantities show non-trivial transient behaviors at the intermediate times as
we will see. One can verify that these transient effects also occur if the
electromagnetic fields are present. Without external electromagnetic fields,
the equations of the Bogoliubov coefficients are simplified as

$\displaystyle\begin{pmatrix}\dot{\alpha}_{1}\\\ \dot{\alpha}_{2}\\\
\dot{\beta}_{1}\\\ \dot{\beta}_{2}\end{pmatrix}=$ $\displaystyle
i\dot{\theta}_{5+m}\begin{pmatrix}-\frac{m}{m_{T}}\frac{p_{z}}{\Omega}&\frac{p_{T}}{m_{T}}&\frac{m}{\Omega}e^{2i\Theta}&0\\\
\frac{p_{T}}{m_{T}}&\frac{m}{m_{T}}\frac{p_{z}}{\Omega}&0&-\frac{m}{\Omega}e^{2i\Theta}\\\
\frac{m}{\Omega}e^{-2i\Theta}&0&\frac{m}{m_{T}}\frac{p_{z}}{\Omega}&\frac{p_{T}}{m_{T}}\\\
0&-\frac{m}{\Omega}e^{-2i\Theta}&\frac{p_{T}}{m_{T}}&-\frac{m}{m_{T}}\frac{p_{z}}{\Omega}\end{pmatrix}\begin{pmatrix}{\alpha}_{1}\\\
{\alpha}_{2}\\\ {\beta}_{1}\\\ {\beta}_{2}\end{pmatrix},$ (4.1)

in the Hamiltonian basis, and

$\displaystyle\dot{\tilde{\alpha}}_{\lambda}$
$\displaystyle=-\frac{m\ddot{\theta}_{5+m}}{2\Omega_{\lambda}^{2}}\exp\left[2i\Theta_{\lambda}\right]\tilde{\beta}_{\lambda},\quad\dot{\tilde{\beta}}_{\lambda}=\frac{m\ddot{\theta}_{5+m}}{2\Omega_{\lambda}^{2}}\exp\left[-2i\Theta_{\lambda}\right]\tilde{\alpha}_{\lambda},$
(4.2)

in the fermion current basis, respectively, where we replace $\Pi_{z}\to
p_{z}$ and $\Pi\to p$. We first study the system in the Hamiltonian basis, and
then in the fermion current basis.

#### Hamiltonian basis.

Figure 2: _Left:_ Time evolution of the occupation number $n_{\psi}$ in the
Hamiltonian basis for a large axion velocity $\dot{\theta}_{0}$ and vanishing
electromagnetic fields. Although approaching to zero at the asymptotic future,
$n_{\psi}$ is $\mathcal{O}(1)$ at the intermediate time for large
$\dot{\theta}_{0}$. We fix $p_{z}=0$, and show the results for several
different values of $m$ and $\dot{\theta}_{0}$ in the unit $p_{T}=1$. _Right:_
Time evolution of the regularized (differential) energy density
$\mathrm{d}^{2}\rho_{\psi}^{(\mathrm{reg})}/\mathrm{d}p_{x}\mathrm{d}p_{y}$ in
the Hamiltonian basis with a large axion velocity, where we integrate over
$p_{z}$ in the range of $-150$ to $150$. This clearly shows that the transient
effect in $n_{\psi}$ is reflected in the physical energy density.

We show our numerical results for
$n_{\psi}=\sum_{\lambda,\lambda^{\prime}}|\beta_{\lambda}^{(\lambda^{\prime})}|^{2}$
as a function of time in the left panel of Fig. 2. We fix $p_{z}=0$,
$t_{\mathrm{min}}=0$, $t_{\mathrm{max}}=100$ and $T=\sqrt{50}$, and take
several different values of $m$ and $\dot{\theta}_{0}$ in the unit $p_{T}=1$.
As one can see, even though the fermion mass is large and there is no external
electric field, $n_{\psi}$ becomes of $\mathcal{O}(1)$ in the intermediate
times. This is the transient effect that we study in this section. Although
$n_{\psi}$ itself is not a physical quantity, the same feature appears in the
physical quantities such as the energy density. The fermion energy density in
the Hamiltonian basis is expressed as

$\displaystyle\rho_{\psi}$
$\displaystyle\equiv\frac{1}{\mathrm{Vol}(\mathbb{R}_{3})}\langle
H_{\psi}\rangle=\rho_{\psi}^{(\mathrm{reg})}+\rho_{\psi}^{\mathrm{(vac)}},$
(4.3)

where the regularized part $\rho_{\psi}^{\mathrm{(reg)}}$ and the vacuum part
$\rho_{\psi}^{\mathrm{(vac)}}$ are respectively given by

$\displaystyle\rho_{\psi}^{\mathrm{(reg)}}$
$\displaystyle=\int\frac{\mathrm{d}^{3}p}{\left(2\pi\right)^{3}}\,2\Omega\sum_{\lambda,\lambda^{\prime}}\left|{\beta_{\lambda}^{(\lambda^{\prime})}}\right|^{2},\quad\rho^{\mathrm{(vac)}}=-2\int\frac{\mathrm{d}^{3}p}{\left(2\pi\right)^{3}}\,\Omega.$
(4.4)

We may employ normal ordering to renormalize the energy density, corresponding
to dropping $\rho_{\psi}^{\mathrm{(vac)}}$. Recall that the normal ordering in
the Hamiltonian basis indeed correctly reproduces the anomaly equation (see
Sec. 3.2). In the right panel of Fig. 2, we plot the time evolution of the
differential energy density
$\mathrm{d}^{2}\rho_{\psi}^{(\mathrm{reg})}/\mathrm{d}p_{x}\mathrm{d}p_{y}$
(integrated from $p_{z}=-150$ to $150$). We see that the energy density shows
the same transient effect as $n_{\psi}$.

#### Fermion current basis.

Figure 3: _Left:_ Time evolution of the occupation number $\tilde{n}_{\psi}$
in the fermion current basis with a large axion velocity. Even for a large
axion velocity, $\tilde{n}_{\psi}$ is well suppressed and does not show the
transient effect that shows up in $n_{\psi}$ in the Hamiltonian basis.
_Right:_ Time evolution of the vacuum contribution to the energy density
$\mathrm{d}^{2}\rho_{\psi}^{(\mathrm{diff})}/\mathrm{d}p_{x}\mathrm{d}p_{y}$,
again integrated over $p_{z}=-150$ to $150$ in the unit $p_{T}=1$, in the
fermion current basis with a large axion velocity. This component features the
transient effect, making the total energy density basis-independent. The
parameters are the same as Fig. 2.

We now study the same phenomena in the fermion current basis. In the left
panel of Fig. 3, we plot the time evolution of
$\tilde{n}_{\psi}=\sum_{\lambda,\lambda^{\prime}}|\tilde{\beta}_{\lambda}^{(\lambda^{\prime})}|^{2}$,
with the parameters as in Fig. 2. It is clear that $\tilde{n}_{\psi}$ is well
suppressed and does not show this transient
effect.\natural6\natural6$\natural$6 The peaks at $t\sim t_{\mathrm{min}}$ and
$t_{\mathrm{max}}$ are induced by non-zero $\ddot{\theta}_{5+m}$, which
depends on the details of turning on and off the axion velocity and is not in
the focus of our interest here. Note that the spectrum is well below
$\mathcal{O}(1)$ even at the peaks.  Since $\tilde{n}_{\psi}$ is not a
physical quantity, this by itself does not imply any contradiction in our
computation. In order to investigate the origin of the transient effect in the
fermion current basis, we take a closer look at the renormalization of the
energy density. In the fermion current basis, we may decompose the fermion
energy density (before the renormalization) as

$\displaystyle\rho_{\psi}$
$\displaystyle=\tilde{\rho}_{\psi}^{\mathrm{(reg)}}+\rho_{\psi}^{(\mathrm{diff})}+\rho_{\psi}^{(\mathrm{vac})},$
(4.5)

where

$\displaystyle\tilde{\rho}_{\psi}^{\mathrm{(reg)}}$
$\displaystyle=\sum_{\lambda,\lambda^{\prime}}\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\left[\frac{p^{2}+m^{2}+\lambda
p\dot{\theta}_{5+m}}{\Omega_{\lambda}}\left(\left|{\tilde{\beta}_{\lambda}^{(\lambda^{\prime})}}\right|^{2}-\left|{\tilde{\alpha}_{\lambda}^{(\lambda^{\prime})}}\right|^{2}+\delta_{\lambda\lambda^{\prime}}\right)-2\dot{\theta}_{5+m}\frac{m}{\Omega_{\lambda}}\mathrm{Re}\left[e^{2i\Theta_{\lambda}}\tilde{\alpha}_{\lambda}^{(\lambda^{\prime})*}\tilde{\beta}_{\lambda}^{(\lambda^{\prime})}\right]\right],$
(4.6) $\displaystyle\rho_{\psi}^{\mathrm{(diff)}}$
$\displaystyle=-\sum_{\lambda}\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\left(\frac{p^{2}+m^{2}+\lambda
p\dot{\theta}_{5+m}}{\Omega_{\lambda}}-\Omega\right).$ (4.7)

and $\rho_{\psi}^{\text{(vac)}}$ is as in Eq. (4.4). Here
$\tilde{\rho}_{\psi}^{\mathrm{(reg)}}$ corresponds to the normal ordering
contribution in the fermion current basis, while
$\rho_{\psi}^{\mathrm{(diff)}}$ corresponds to the difference of the normal
orderings in the Hamiltonian and the fermion current basis. After
renormalizing the energy density by dropping $\rho^{(\mathrm{vac})}_{\psi}$,
this difference $\rho_{\psi}^{\mathrm{(diff)}}$ still contributes to the
energy density. In fact it is this term that gives the transient effect in the
fermion current basis, as one can see in the right panel of Fig. 3. Note that,
as we saw in Sec. 3.2, the normal ordering in the fermion current basis does
not correctly reproduce the anomaly equation, and thus one needs to take
$\rho_{\psi}^{\mathrm{(diff)}}$ into account. Therefore, the transient effect
arises independent of the basis choice, providing further indication that our
regularization and renormalization procedure is consistent.

The momentum integral of $\rho_{\psi}^{\mathrm{(diff)}}$ can be done exactly,
and we obtain

$\displaystyle\rho_{\psi}^{\mathrm{(diff)}}$
$\displaystyle=\frac{m^{2}\dot{\theta}_{5+m}^{2}}{4\pi^{2}}\log\left(\frac{\Lambda^{2}}{m^{2}}\right)+\frac{\dot{\theta}_{5+m}^{4}}{4\pi^{2}},$
(4.8)

where we have used a hard UV cut-off $\Lambda$ for the momentum integral. The
first term corresponds to the fermion contribution of the axion wavefunction
renormalization: Including the axion contribution to the energy density, we
have

$\displaystyle\rho_{\phi}+\rho_{\psi}^{\mathrm{(diff)}}$
$\displaystyle=\frac{1}{2}Z_{\phi}\dot{\phi}^{2}+V(\phi)+\frac{m^{2}\dot{\theta}_{5+m}^{2}}{4\pi^{2}}\log\left(\frac{\Lambda^{2}}{m^{2}}\right)+\frac{\dot{\theta}_{5+m}^{4}}{4\pi^{2}},$
(4.9)

where we include the axion wavefunction renormalization $Z_{\phi}$. By
expanding $Z_{\phi}=1+\delta_{\phi}$, we obtain the divergent part of the
counter term as

$\displaystyle\left.\delta_{\phi}\right|_{\mathrm{div.}}=-\frac{(c_{5}+c_{m})^{2}}{2\pi^{2}}\frac{m^{2}}{f_{a}^{2}}\log\left(\frac{\Lambda^{2}}{m^{2}}\right).$
(4.10)

This agrees with the standard Feynman diagrammatic computation (see, _e.g._
Ref. [30]). The second term in (4.8) is an axion dimension-eight effective
operator. It is trivial to check that the second term reproduces the transient
effects that we observed numerically.\natural7\natural7$\natural$7 There are
other higher dimensional operators that arise from
$\tilde{\rho}_{\psi}^{(\mathrm{reg})}$, but they are subdominant in the
present case.  Thus, we conclude that the transient effect can be interpreted
as an axion higher dimensional operator. Armed with this understanding, in the
next subsection, we discuss the resulting UV sensitivity of this transient
effect.

#### Massless limit.

As an aside, it is probably appropriate here to comment on the massless limit
$m\to 0$, before discussing the UV sensitivity. In this limit, all the effects
of $\dot{\theta}_{5+m}$ should disappear as the axion-fermion coupling is
unphysical in this limit.

In the Hamiltonian basis, this is trivial to check. The Bogoliubov coefficient
$\beta$ simply vanishes in the limit $m\to 0$ which we have checked
numerically. The situation is more non-trivial in the fermion current basis,
as Eq. (4.8) in the massless limit reads

$\displaystyle\lim_{m\to 0}\left[\rho_{\psi}^{\mathrm{(diff)}}\right]$
$\displaystyle=\frac{\dot{\theta}_{5+m}^{4}}{4\pi^{2}},$ (4.11)

which does not vanish. In the massless limit, there is however another
contribution from $\tilde{\rho}_{\psi}^{(\mathrm{reg})}$, and we will see in
the following that this contribution cancels the contribution of Eq. (4.11).
By noting that

$\displaystyle\lim_{m\to
0}\frac{m}{\Omega_{\lambda}^{2}}=\pi\,\delta(p+\lambda\dot{\theta}_{5+m}),$
(4.12)

we see that Eq. (4.2) is non-trivial only for
$|{p+\lambda\dot{\theta}_{5+m}}|\lesssim m$ as $m\rightarrow 0$. In this
region, the phase factors in Eq. (4.2) are approximately constant and hence
irrelevant for\natural8\natural8$\natural$8 To see this, let us take
$\dot{\theta}_{5+m}$ as a time variable, _i.e._ ,
$\int\mathrm{d}t\,\Omega_{\lambda}=\int\mathrm{d}\dot{\theta}_{5+m}\,\Omega_{\lambda}/\ddot{\theta}_{5+m}$.
For $|{p+\lambda\dot{\theta}_{5+m}}|\lesssim m$, we have $\Omega_{\lambda}\sim
m$ and $\dot{\theta}_{5+m}$ varies by $\Delta\dot{\theta}_{5+m}\sim m$. Thus
the change in the phase factor during $|{p+\lambda\dot{\theta}_{5+m}}|\lesssim
m$ is roughly
$\Delta\dot{\theta}_{5+m}\times\Omega_{\lambda}/\ddot{\theta}_{5+m}\sim
m^{2}/\ddot{\theta}_{5+m}$.

$\displaystyle m^{2}\lesssim|{\ddot{\theta}_{5+m}}|.$ (4.13)

Therefore in the limit $m^{2}\ll|\ddot{\theta}_{5+m}|$, we can separate the
time domain into two parts, $|{p+\lambda\dot{\theta}_{5+m}}|\gtrsim m$ and
$|{p+\lambda\dot{\theta}_{5+m}}|\lesssim m$. In the former case
$\tilde{\alpha}_{\lambda}$ and $\tilde{\beta}_{\lambda}$ do not evolve in
time. A nontrivial evolution happens only for the latter case, where the
equations are given by

$\displaystyle\dot{\tilde{\alpha}}_{\lambda}$
$\displaystyle=-\frac{\ddot{\theta}_{5+m}}{2}\frac{m}{(p+\lambda\dot{\theta}_{5+m})^{2}+m^{2}}\tilde{\beta}_{\lambda},\quad\dot{\tilde{\beta}}_{\lambda}=\frac{\ddot{\theta}_{5+m}}{2}\frac{m}{(p+\lambda\dot{\theta}_{5+m})^{2}+m^{2}}\tilde{\alpha}_{\lambda},$
(4.14)

where we have dropped the phase factors. It is convenient to first assume that
$\dot{\theta}_{5+m}$ is a monotonic function. Later we will see that we can
drop this assumption. If $\dot{\theta}_{5+m}$ is monotonic, we can change the
time variable from $t$ to $\dot{\theta}_{5+m}$. The equations can be then
solved analytically. With the initial condition that
$\tilde{\alpha}_{\lambda}=1$, $\tilde{\beta}_{\lambda}=0$, and
$\dot{\theta}_{5+m}=0$, the solutions in the massless limit are

$\displaystyle\tilde{\alpha}_{\lambda}=\Theta(p+\lambda\dot{\theta}_{5+m}),\quad\tilde{\beta}_{\lambda}=\Theta(-\lambda\dot{\theta}_{5+m}-p).$
(4.15)

We now come back to the assumption that $\dot{\theta}_{5+m}$ is monotonic.
Even if $\dot{\theta}_{5+m}$ is not monotonic, we can consider the time
domains between $\ddot{\theta}_{5+m}=0$ separately. In each domain
$\dot{\theta}_{5+m}$ is a good time variable and thus we can repeat the same
computation as above. It is then clear that, for a given mode with fixed $p$,
the interchange of $\tilde{\alpha}_{\lambda}=1,\tilde{\beta}_{\lambda}=0$ and
$\tilde{\alpha}_{\lambda}=0,\tilde{\beta}_{\lambda}=1$ happens every time
$p+\lambda\dot{\theta}_{5+m}$ crosses zero. If $p+\lambda\dot{\theta}_{5+m}$
crosses zero an even number of times, then $\tilde{\alpha}_{\lambda}=1$ and
$\tilde{\beta}_{\lambda}=0$ in the end, while for an odd number of times we
have $\tilde{\alpha}_{\lambda}=0$ and $\tilde{\beta}_{\lambda}=1$. We assume
that $\dot{\theta}_{5+m}=0$ at the beginning, then the mode with
$p+\lambda\dot{\theta}_{5+m}>0$ at a given time should cross the point
$p+\lambda\dot{\theta}_{5+m}=0$ an even number of times before the given time,
while the one with $p+\lambda\dot{\theta}_{5+m}<0$ should cross the point odd
times, independently of the detailed time evolution of $\dot{\theta}_{5+m}$.
Thus we conclude that Eq. (4.15) is correct independent of the time dependence
of $\dot{\theta}_{5+m}$ in the massless limit. We have also checked that the
numerical solution agrees well with Eq. (4.15) in the massless limit. Assuming
$\dot{\theta}_{5+m}>0$ for definiteness, we obtain

$\displaystyle\lim_{m\to 0}\left[\tilde{\rho}_{\psi}^{(\mathrm{reg})}\right]$
$\displaystyle=2\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\frac{p-\dot{\theta}_{5+m}}{\lvert
p-\dot{\theta}_{5+m}\rvert}p\,\Theta(\dot{\theta}_{5+m}-p)=-\frac{\dot{\theta}_{5+m}^{4}}{4\pi^{2}}.$
(4.16)

We thus see that

$\displaystyle\lim_{m\to
0}\left[\tilde{\rho}_{\psi}^{(\mathrm{reg})}+{\rho}_{\psi}^{(\mathrm{diff})}\right]=0,$
(4.17)

so that the total energy density vanishes in the massless limit in the fermion
current basis as well, providing further evidence for the consistency of our
regularization and renormalization. As soon as the fermion mass becomes
finite, the contribution from $\tilde{\rho}_{\psi}^{(\mathrm{reg})}$ is well
suppressed and ${\rho}_{\psi}^{(\mathrm{diff})}$ dominates, as we saw in Fig.
3. Thus, contrary to Ref. [30], we find the expected decoupling of the
fermions in the massless limit without the need to introduce any counter terms
(see also discussion below).

###  4.2 Higher dimensional operator and UV sensitivity

In the previous subsection, we have seen that the transient effect in the
fermion energy density can be identified with a higher dimensional operator.
In the fermion current basis, this arises due to the difference of the vacuum
contribution with respect to the normal ordering prescription in the
Hamiltonian basis and takes the form

$\displaystyle\rho_{\psi}^{\mathrm{(diff)}}$
$\displaystyle=\frac{m^{2}\dot{\theta}_{5+m}^{2}}{4\pi^{2}}\log\left(\frac{\Lambda^{2}}{m^{2}}\right)+\frac{\dot{\theta}_{5+m}^{4}}{4\pi^{2}}.$
(4.18)

Identifying the second term with a higher dimensional operator, this is
actually degenerate with the choice of the UV theory. The low energy effective
field theory describing the axion is non-renormalizable, and thus there is a
priori no constraint to adding higher dimensional operators by hand, on top of
the transient effect discussed here. Note that the situation is different for
the Euler-Heisenberg Lagrangian. In the case of the Euler-Heisenberg
Lagrangian, the theory is renormalizable before integrating out a heavy
particle,\natural9\natural9$\natural$9 Strictly speaking there exists the
Landau pole which suggests a new scale, but we ignore this subtlety here.  and
it makes sense to set all the higher dimensional operators to vanish before
integrating out the heavy particle. One can then uniquely determine the
coefficients of the higher dimensional operators after integrating out the
heavy particle.

Thus, our conclusion is that a computation of transient effects in the axion
effective theory requires the specification of the UV theory. In the limit
$\dot{\theta}_{5+m}\ll f_{a}$, our computation of the transient effects within
the axion effective field theory is valid. In the opposite case, the axion
effective field theory expansion is invalid. One then must go back to the
original UV theory and study the dynamics within the UV theory.

Before concluding, let us briefly comment on other regularization schemes. In
particular, in the fermion current basis, we can evaluate the axion effective
action

$\displaystyle\Gamma[\phi]=-i\mathrm{Tr}\log\left[i\not{\partial}-m+\partial_{\mu}\theta_{5+m}\gamma^{\mu}\gamma_{5}\right],$
(4.19)

by expanding in powers of $\dot{\theta}_{5+m}/m$ and evaluating the resulting
terms using dimensional regularization. Treating the $\gamma_{5}$ matrices
using the t’Hooft-Veltman prescription [41], we reproduce the first term in
Eq. (4.8) but obtain a vanishing coefficient for the term proportional to
$\dot{\theta}_{5+m}^{4}$, in contrast to the second term in Eq. (4.8). This
may also be related to somewhat surprising results in the literature, such as
the non-restoration of the axial symmetry in the massless fermion limit found
in [30] using dimensional regularization. In fact, the latter result
demonstrates that this regularization scheme does not respect the chiral
symmetry of the theory in the massless limit (adding to the list of
difficulties of dimensional regularization for chiral theories [41]), thus
violating a key requirement of a ‘good’ regularization scheme.

##  5 Conclusions

In this paper, we have studied fermion production in the presence of a
background axion field with non-vanishing velocity and/or electromagnetic
fields. We have included general dimension-five axion-fermion couplings and
thus considered the Dirac equation of the form

$\displaystyle\left[i\not{D}-me^{2i\theta_{m}\gamma_{5}}+\partial_{\mu}\theta_{5}\gamma^{\mu}\gamma_{5}\right]\psi=0,$
(5.1)

where $\theta_{i}=c_{i}\phi/f_{a}$ with $\phi$ being the axion. While the
spectrum of the produced fermion at the infinite future, after turning off the
external fields, is studied in detail in [26], here we have put more focus on
physical quantities at intermediate times, obtained before turning off the
external fields. These physical quantities at the intermediate times are of
great phenomenological interest. For instance, in the context of axion
inflation, the fermion production backreacts to the axion-gauge field system
in the form of physical quantities such as the induced current, which in turn
determines the production of gauge fields, gravitational waves and primordial
density fluctuations.

In general, physical quantities are divergent in the presence of the external
fields, and thus a proper understanding of regularization and renormalization
is necessary to evaluate the physical quantities at the intermediate times. In
particular, a proper regularization scheme must preserve the symmetries of the
theory as much as possible, such as the chiral rotation invariance as
reflected in the Adler-Bell-Jackiw anomaly equation, and it can be highly non-
trivial to find such a proper regularization scheme (see _e.g._ [30] for a
related discussion). The main observation of this paper is that the theory is
most straightforwardly regularized and renormalized in the $c_{5}=0$ basis (or
the Hamiltonian basis). A chiral rotation $\psi\to
e^{ic\phi/f_{a}\gamma_{5}}\psi$ shifts the couplings as $c_{5}\to c_{5}-c$ and
$c_{m}\to c_{m}+c$, and a different choice of $c$ corresponds to a different
choice of the fermion basis that one works with. This rotation is merely a
redundancy of the theory and the final result should not be affected by this
choice. Nevertheless the Hamiltonian basis is most preferable for regularizing
and renormalizing the theory, as the fermionic part of the Hamiltonian is
diagonalized only in the Hamiltonian basis and not in other bases including
the $c_{m}=0$ basis (or the fermion current basis) employed in [27]. We have
supported our observation by showing that the regularization using a regulator
function depending on the frequencies in the Hamiltonian basis (corresponding
to the energy eigenvalues) correctly reproduces the anomaly equation, while
the regularization using the frequencies appearing in the fermion current
basis does not.

With the proper understanding of the regularization and the renormalization,
we have studied transient effects in the fermion energy density with a large
axion velocity. We have seen that, in the presence of a large axion velocity,
the fermion energy density $\rho_{\psi}$ at intermediate times is no longer
exponentially suppressed by the fermion mass, but is significantly enhanced
even in the absence of external electromagnetic fields and even for constant
axion velocity. This should be compared with the usual formula for the
Schwinger effect, $\rho_{\psi}\propto\exp(-\pi m^{2}/g\left|{Q}\right|E)$,
which indicates an exponential suppression of the energy density at the
infinite future. We have confirmed that the same result follows in both the
Hamiltonian and fermion current bases although its appearance is very
different; the transient effect originates from the Bogoliubov coefficients in
the Hamiltonian basis, while it stems from a vacuum contribution in the
fermion current basis. The latter expression allows us to identify the
transient effect with an axion higher dimensional operator. We stress that
although this operator is degenerate with the choice of UV completion, one can
study the transient effect using our prescription for regularization and
renormalization once one specifies the UV completion of the theory. We did not
discuss in any detail phenomenological applications of these transient
effects. One obvious obstacle is that a significant enhancement of the fermion
energy density, as observed in Fig. 2, requires a very large axion velocity.
This is not only difficult to achieve in realistic models, but also per se
indicates a breakdown of the axion effective theory. A phenomenological
analysis of this transient effect should thus be conducted in the framework of
the original Peccei-Quinn theory, whenever it is relevant. On the other hand,
our results also show that in the limit of small axion velocities,
$\dot{\theta}_{5+m}\ll f_{a}$, transient effects can be reliably computed in
the axion effective field theory.

#### Acknowledgements

It is a pleasure to thank Peter Adshead, Hiroyuki Kitamoto and Masaki Yamada
for helpful discussions related to this project. Y.E. is supported in part by
U.S. Department of Energy Grant No. de-sc0011842. K.M. is supported by MEXT
Leading Initiative for Excellent Young Researchers Grant No. JPMXS0320200430.

##  Appendix A Notation and conventions

Here we summarize our notation and conventions in this paper. We take the
mostly minus convention and define the anti-commutator of the gamma matrices
as

$\displaystyle\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}$
$\displaystyle=\eta^{\mu\nu},\quad\eta_{\mu\nu}=\mathrm{diag}(1,-1,-1,-1).$
(A.1)

It then follows that $\gamma^{0}$ is hermitian while $\gamma^{i}$ is anti-
hermitian. Unless specified otherwise, we work with the Weyl representation,
given by

$\displaystyle\gamma^{0}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\gamma^{i}=\begin{pmatrix}0&\sigma_{i}\\\
-\sigma_{i}&0\end{pmatrix},$ (A.2)

where $\sigma_{i}$ denotes the Pauli matrices, given by

$\displaystyle\sigma_{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\sigma_{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\quad\sigma_{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$
(A.3)

We define $\gamma_{5}$ as

$\displaystyle\gamma_{5}=-\frac{i}{4!}\epsilon^{\mu\nu\rho\sigma}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3},$
(A.4)

where $\epsilon^{\mu\nu\rho\sigma}$ is the totally anti-symmetric tensor with
$\epsilon^{0123}=-\epsilon_{0123}=1$. In the Weyl representation $\gamma_{5}$
is given by

$\displaystyle\gamma_{5}$ $\displaystyle=\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}.$ (A.5)

The covariant derivative is given by

$\displaystyle iD_{\mu}=i\partial_{\mu}-gQA_{\mu}.$ (A.6)

The field strength is defined as
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, and its dual as

$\displaystyle\tilde{F}_{\mu\nu}=\frac{\epsilon_{\mu\nu\rho\sigma}}{2}F^{\rho\sigma}.$
(A.7)

##  Appendix B Technical details on solutions of Dirac equation

In this appendix we summarize the technical details omitted in the main text.
In particular, we explain in detail the solutions of the Dirac equation in the
Hamiltonian and the fermion current bases.

###  B.1 Dirac equation with electric field

We first consider the case with only the electric field and the axion. The
Dirac equation is given by

$\displaystyle\left[i\not{D}-me^{2i\theta_{m}\gamma_{5}}+\partial_{\mu}\theta_{5}\gamma^{\mu}\gamma_{5}\right]\psi=0,$
(B.1)

and we take the background fields as

$\displaystyle A^{\mu}=\left(0,0,0,A_{z}(t)\right),\qquad\phi=\phi(t).$ (B.2)

Since the background fields do not depend on the space coordinate, we can move
to Fourier space as

$\displaystyle\psi(x,t)$
$\displaystyle=\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}e^{i\vec{p}\cdot\vec{x}}\psi(p,t),$
(B.3)

where $\vec{p}=(p_{x},p_{y},p_{z})$. The Dirac equation then reads

$\displaystyle 0$
$\displaystyle=\begin{pmatrix}i\partial_{0}+\Pi_{z}-\dot{\theta}_{5}&p_{x}-ip_{y}&-me^{2i\theta_{m}}&0\\\
p_{x}+ip_{y}&i\partial_{0}-\Pi_{z}-\dot{\theta}_{5}&0&-me^{2i\theta_{m}}\\\
-me^{-2i\theta_{m}}&0&i\partial_{0}-\Pi_{z}+\dot{\theta}_{5}&-(p_{x}-ip_{y})\\\
0&-me^{-2i\theta_{m}}&-(p_{x}+ip_{y})&i\partial_{0}+\Pi_{z}+\dot{\theta}_{5}\end{pmatrix}\psi.$
(B.4)

We solve this equation in the Hamiltonian basis $c_{5}=0$ and the fermion
current basis $c_{m}=0$.

In the Hamiltonian basis, we eliminate $c_{5}$ by $\psi\to
e^{i\theta_{5}\gamma_{5}}\psi$. The solutions of the Dirac equation for
constant $\phi$ and $A_{z}$ are then given by

$\displaystyle u_{1}$
$\displaystyle=\frac{e^{-i\gamma_{5}\theta_{5+m}}}{N}\begin{pmatrix}-(p_{x}-ip_{y})m_{T}\\\
\left(\Omega+\Pi_{z}\right)\left(m+m_{T}\right)\\\
(p_{x}-ip_{y})\left(\Omega+\Pi_{z}\right)\\\
m_{T}\left(m+m_{T}\right)\end{pmatrix},\quad
u_{2}=\frac{e^{-i\gamma_{5}\theta_{5+m}}e^{-i\varphi_{p}}}{N}\begin{pmatrix}m_{T}\left(m+m_{T}\right)\\\
-(p_{x}+ip_{y})\left(\Omega+\Pi_{z}\right)\\\
\left(\Omega+\Pi_{z}\right)\left(m+m_{T}\right)\\\
(p_{x}+ip_{y})m_{T}\end{pmatrix},$ (B.5) $\displaystyle v_{1}$
$\displaystyle=\frac{e^{-i\gamma_{5}\theta_{5+m}}}{N}\begin{pmatrix}-(p_{x}-ip_{y})\left(\Omega+\Pi_{z}\right)\\\
-m_{T}\left(m+m_{T}\right)\\\ -(p_{x}-ip_{y})m_{T}\\\
\left(\Omega+\Pi_{z}\right)\left(m+m_{T}\right)\end{pmatrix},\quad
v_{2}=\frac{e^{-i\gamma_{5}\theta_{5+m}}e^{-i\varphi_{p}}}{N}\begin{pmatrix}\left(\Omega+\Pi_{z}\right)\left(m+m_{T}\right)\\\
(p_{x}+ip_{y})m_{T}\\\ -m_{T}\left(m+m_{T}\right)\\\
(p_{x}+ip_{y})\left(\Omega+\Pi_{z}\right)\end{pmatrix},$ (B.6)

with

$\displaystyle
N=2\sqrt{\Omega\left(\Omega+\Pi_{z}\right)\left(m+m_{T}\right)m_{T}},\quad
p_{x}-ip_{y}=p_{T}e^{-i\varphi_{p}},\quad m_{T}=\sqrt{p_{T}^{2}+m^{2}}.$ (B.7)

Once we turn on the time dependence of $A_{z}$ and $\phi$, the Bogoliubov
coefficients evolve as

$\displaystyle\begin{pmatrix}\dot{\alpha}_{1}\\\ \dot{\alpha}_{2}\\\
\dot{\beta}_{1}\\\ \dot{\beta}_{2}\end{pmatrix}=$
$\displaystyle\left[i\dot{\theta}_{5+m}\begin{pmatrix}-\frac{m}{m_{T}}\frac{\Pi_{z}}{\Omega}&\frac{p_{T}}{m_{T}}&\frac{m}{\Omega}e^{2i\Theta}&0\\\
\frac{p_{T}}{m_{T}}&\frac{m}{m_{T}}\frac{\Pi_{z}}{\Omega}&0&-\frac{m}{\Omega}e^{2i\Theta}\\\
\frac{m}{\Omega}e^{-2i\Theta}&0&\frac{m}{m_{T}}\frac{\Pi_{z}}{\Omega}&\frac{p_{T}}{m_{T}}\\\
0&-\frac{m}{\Omega}e^{-2i\Theta}&\frac{p_{T}}{m_{T}}&-\frac{m}{m_{T}}\frac{\Pi_{z}}{\Omega}\end{pmatrix}+\frac{m_{T}\dot{\Pi}_{z}}{2\Omega^{2}}\begin{pmatrix}0&0&-e^{2i\Theta}&0\\\
0&0&0&-e^{2i\Theta}\\\ e^{-2i\Theta}&0&0&0\\\
0&e^{-2i\Theta}&0&0\end{pmatrix}\right]\begin{pmatrix}{\alpha}_{1}\\\
{\alpha}_{2}\\\ {\beta}_{1}\\\ {\beta}_{2}\end{pmatrix},$ (B.8)

where $\Theta=\int^{t}\mathrm{d}t\,\Omega$. We solve these equations with two
different initial conditions,

$\displaystyle\alpha_{1}^{(1)}$
$\displaystyle=1,\quad\alpha_{2}^{(1)}=\beta_{1}^{(1)}=\beta_{2}^{(1)}=0,\qquad\mathrm{and}\qquad\alpha_{2}^{(2)}=1,\quad\alpha_{1}^{(2)}=\beta_{1}^{(2)}=\beta_{2}^{(2)}=0.$
(B.9)

The former corresponds to the mode which is initially $u_{1}$, while the
latter is the one which is initially $u_{2}$. The Bogoliubov coefficients then
describe the mixing of these two positive frequency modes with the others at a
later time. In terms of the original field in the coordinate space, the
fermion is thus quantized as

$\displaystyle\psi$
$\displaystyle=\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}e^{i\vec{p}\cdot\vec{x}}e^{i\theta_{5}\gamma_{5}}\sum_{\lambda,\lambda^{\prime}=1,2}\left[u_{\lambda}e^{-i\Theta}\left(\alpha_{\lambda}^{(\lambda^{\prime})}b_{\lambda^{\prime}}-(-)^{\lambda+\lambda^{\prime}}{\beta_{\lambda}^{(\lambda^{\prime})}}^{*}d_{\lambda^{\prime}}^{\dagger}\right)+v_{\lambda}e^{+i\Theta}\left(\beta_{\lambda}^{(\lambda^{\prime})}b_{\lambda^{\prime}}+(-)^{\lambda+\lambda^{\prime}}{\alpha_{\lambda}^{(\lambda^{\prime})}}^{*}d_{\lambda^{\prime}}^{\dagger}\right)\right].$
(B.10)

In the fermion current basis, we instead eliminate $c_{m}$ by $\psi\to
e^{-i\theta_{m}\gamma_{5}}\psi$. In this case we may define the solutions of
the Dirac equation for constant $A_{z}$ and $\dot{\phi}$ as

$\displaystyle\tilde{u}_{+}$
$\displaystyle=\frac{1}{N_{+}}\begin{pmatrix}-(p_{x}-ip_{y})(\Omega_{+}+\Pi+\dot{\theta}_{5+m})\\\
(\Pi+\Pi_{z})(\Omega_{+}+\Pi+\dot{\theta}_{5+m})\\\ -(p_{x}-ip_{y})m\\\
(\Pi+\Pi_{z})m\end{pmatrix},\quad\tilde{u}_{-}=\frac{1}{N_{-}}\begin{pmatrix}(p_{x}-ip_{y})m\\\
(\Pi-\Pi_{z})m\\\ (p_{x}-ip_{y})(\Omega_{-}+\Pi-\dot{\theta}_{5+m})\\\
(\Pi-\Pi_{z})(\Omega_{-}+\Pi-\dot{\theta}_{5+m})\end{pmatrix},$ (B.11)
$\displaystyle\tilde{v}_{+}$
$\displaystyle=\frac{1}{N_{+}}\begin{pmatrix}(p_{x}-ip_{y})m\\\
-(\Pi+\Pi_{z})m\\\ -(p_{x}-ip_{y})(\Omega_{+}+\Pi+\dot{\theta}_{5+m})\\\
(\Pi+\Pi_{z})(\Omega_{+}+\Pi+\dot{\theta}_{5+m})\end{pmatrix},\quad\tilde{v}_{-}=\frac{1}{N_{-}}\begin{pmatrix}-(p_{x}-ip_{y})(\Omega_{-}+\Pi-\dot{\theta}_{5+m})\\\
-(\Pi-\Pi_{z})(\Omega_{-}+\Pi-\dot{\theta}_{5+m})\\\ (p_{x}-ip_{y})m\\\
(\Pi-\Pi_{z})m\end{pmatrix},$ (B.12)

with

$\displaystyle\Pi$
$\displaystyle=\sqrt{\Pi_{z}^{2}+p_{x}^{2}+p_{y}^{2}},\quad\Omega_{\pm}=\sqrt{(\Pi\pm\dot{\theta}_{5+m})^{2}+m^{2}},\quad
N_{\pm}=2\sqrt{\Pi(\Pi\pm\Pi_{z})\Omega_{\pm}(\Omega_{\pm}+\Pi\pm\dot{\theta}_{5+m})},$
(B.13)

and $\Theta_{\pm}=\int^{t}\Omega_{\pm}\mathrm{d}t$. Once we turn on the time
dependence of the background fields, the Bogoliubov coefficients satisfy

$\displaystyle\begin{pmatrix}\dot{\tilde{\alpha}}_{+}\\\
\dot{\tilde{\alpha}}_{-}\\\ \dot{\tilde{\beta}}_{+}\\\
\dot{\tilde{\beta}}_{-}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}0&Ae^{i(\Theta_{+}-\Theta_{-})}&-B_{+}e^{2i\Theta_{+}}&-Ce^{i(\Theta_{+}+\Theta_{-})}\\\
-Ae^{-i(\Theta_{+}-\Theta_{-})}&0&-Ce^{i(\Theta_{+}+\Theta_{-})}&-B_{-}e^{2i\Theta_{-}}\\\
B_{+}e^{-2i\Theta_{+}}&Ce^{-i(\Theta_{+}+\Theta_{-})}&0&Ae^{-i(\Theta_{+}-\Theta_{-})}\\\
Ce^{-i(\Theta_{+}+\Theta_{-})}&B_{-}e^{-2i\Theta_{-}}&-Ae^{i(\Theta_{+}-\Theta_{-})}&0\end{pmatrix}\begin{pmatrix}\tilde{\alpha}_{+}\\\
\tilde{\alpha}_{-}\\\ \tilde{\beta}_{+}\\\ \tilde{\beta}_{-}\end{pmatrix},$
(B.14)

where

$\displaystyle A$
$\displaystyle=\frac{gQE\,p_{T}m}{4\Pi^{2}\sqrt{\Omega_{+}\Omega_{-}}}\left[\frac{\Omega_{+}+\Omega_{-}+2\Pi}{\sqrt{\left(\Omega_{+}+\Pi+\dot{\theta}_{5+m}\right)\left(\Omega_{-}+\Pi-\dot{\theta}_{5+m}\right)}}\right],$
(B.15) $\displaystyle B_{\pm}$
$\displaystyle=\pm\frac{m}{2\Omega_{\pm}^{2}}\left(\frac{\Pi_{z}}{\Pi}gQE\pm\ddot{\theta}_{5+m}\right),\quad
C=\frac{gQE\,p_{T}\left[\left(\Omega_{+}+\Pi+\dot{\theta}_{5+m}\right)\left(\Omega_{-}+\Pi-\dot{\theta}_{5+m}\right)-m^{2}\right]}{4\Pi^{2}\sqrt{\Omega_{+}\Omega_{-}\left(\Omega_{+}+\Pi+\dot{\theta}_{5+m}\right)\left(\Omega_{-}+\Pi-\dot{\theta}_{5+m}\right)}}.$
(B.16)

We solve these equations with two different initial conditions,

$\displaystyle\tilde{\alpha}_{+}^{(+)}$
$\displaystyle=1,\quad\tilde{\alpha}_{-}^{(+)}=\tilde{\beta}_{+}^{(+)}=\tilde{\beta}_{-}^{(+)}=0,\qquad\mathrm{and}\qquad\tilde{\alpha}_{-}^{(-)}=1,\quad\tilde{\alpha}_{+}^{(-)}=\tilde{\beta}_{+}^{(-)}=\tilde{\beta}_{-}^{(-)}=0.$
(B.17)

In this basis the Dirac field is quantized as

$\displaystyle\psi=\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}e^{i\vec{p}\cdot\vec{x}}e^{-i\theta_{m}\gamma_{5}}\sum_{\lambda\lambda^{\prime}=\pm}\left[\tilde{u}_{\lambda}e^{-i\Theta_{\lambda}}\left(\tilde{\alpha}_{\lambda}^{(\lambda^{\prime})}{b}_{\lambda^{\prime}}-\tilde{\beta}_{\lambda}^{(\lambda^{\prime})*}{d}_{\lambda^{\prime}}^{\,\dagger}\right)+\tilde{v}_{\lambda}e^{i\Theta_{\lambda}}\left(\tilde{\beta}_{\lambda}^{(\lambda^{\prime})}{b}_{\lambda^{\prime}}+\tilde{\alpha}_{\lambda}^{(\lambda^{\prime})*}{d}_{\lambda^{\prime}}^{\,\dagger}\right)\right],$
(B.18)

where we use that
$(\tilde{\alpha}_{+},\tilde{\alpha}_{-},\tilde{\beta}_{+},\tilde{\beta}_{-})^{T}$
and
$(-\tilde{\beta}_{+},-\tilde{\beta}_{-},\tilde{\alpha}_{+},\tilde{\alpha}_{-})^{\dagger}$
satisfy the same equation. We do not put tildes on the creation and
annihilation operators here. Since these are the operators at the infinite
past where we assume there is no external field, these are equivalent to those
in the Hamiltonian basis (up to linear transformation in the case of
degenerate eigenvalues).

The creation and annihilation operators at the infinite past are quantized as

$\displaystyle\left\\{b_{\lambda_{1}}(\vec{p}),b_{\lambda_{2}}^{\dagger}(\vec{p}^{\prime})\right\\}$
$\displaystyle=\left\\{d_{\lambda_{1}}(\vec{p}),d_{\lambda_{2}}^{\dagger}(\vec{p}^{\prime})\right\\}=(2\pi)^{3}\delta_{\lambda_{1}\lambda_{2}}\delta^{(3)}(\vec{p}-\vec{p}^{\prime}),$
(B.19)
$\displaystyle\left\\{b_{\lambda_{1}}(\vec{p}),d_{\lambda_{2}}^{\dagger}(\vec{p}^{\prime})\right\\}$
$\displaystyle=\left\\{b_{\lambda_{1}}(\vec{p}),d_{\lambda_{2}}(\vec{p}^{\prime})\right\\}=0,$
(B.20)

It then follows from the time evolution of the Bogoliubov coefficients that
$B_{\lambda}$, $D_{\lambda}$, $\tilde{B}_{\lambda}$ and $\tilde{D}_{\lambda}$
satisfy the same equal time anti-commutation relations. The proof is rather
lengthy, and we refer interested readers to App. D of [26], where it is shown
explicitly that the creation and annihillation operators in the Hamiltonian
basis, $B_{\lambda}$ and $D_{\lambda}$, satisfy the same equal-time anti-
commutators as $b_{\lambda}$ and $d_{\lambda}$. One can then show that
$\tilde{B}_{\lambda}$ and $\tilde{D}_{\lambda}$ satisfy the same equal-time
anti-commutators by expressing them in terms of $B_{\lambda}$ and
$D_{\lambda}$, with the help of the completeness condition

$\displaystyle\sum_{\lambda}\left[u_{\lambda}u^{\dagger}_{\lambda}+v_{\lambda}v^{\dagger}_{\lambda}\right]=\mathbbm{1}_{4}.$
(B.21)

###  B.2 Dirac equation with electric and magnetic field

We now include the magnetic field. We take the background fields as

$\displaystyle A^{\mu}=\left(0,0,Bx,A_{z}\right),\qquad\phi=\phi(t).$ (B.22)

The background fields do not explicitly depend on $y$ and $z$, and thus we
perform the Fourier transformation as

$\displaystyle\psi=\int\frac{\mathrm{d}p_{y}\mathrm{d}p_{z}}{(2\pi)^{2}}e^{i(p_{y}y+p_{z}z)}\psi(x,p_{y},p_{z},t).$
(B.23)

By using the explicit forms of the gamma matrices, we obtain

$\displaystyle 0=$
$\displaystyle\left[i\mathbbm{1}_{4}\partial_{0}+i\sqrt{2g\left|{QB}\right|}\begin{pmatrix}-S_{-s}\hat{a}+S_{s}\hat{a}^{\dagger}&0\\\
0&S_{-s}\hat{a}-S_{s}\hat{a}^{\dagger}\end{pmatrix}+\Pi_{z}\begin{pmatrix}\sigma_{3}&0\\\
0&-\sigma_{3}\end{pmatrix}-m\begin{pmatrix}0&e^{2i\theta_{m}}\\\
e^{-2i\theta_{m}}&0\end{pmatrix}+\dot{\theta}_{5}\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}\right]\psi,$ (B.24)

where

$\displaystyle\bar{x}_{s}=\sqrt{g\left|{QB}\right|}\left(x-s\frac{p_{y}}{g\left|{QB}\right|}\right),\quad
S_{\pm}=\frac{1}{2}\left(\sigma_{1}\pm
i\sigma_{2}\right),\quad\hat{a}=\frac{1}{\sqrt{2}}\left(\partial_{\bar{x}_{s}}+\bar{x}_{s}\right),\quad\hat{a}^{\dagger}=\frac{1}{\sqrt{2}}\left(-\partial_{\bar{x}_{s}}+\bar{x}_{s}\right),$
(B.25)

and $s=\mathrm{sgn}(QB)$. We expand the modes as

$\displaystyle\psi=\sum_{n,s^{\prime},\lambda}\psi_{n,s^{\prime}}^{\left(\lambda\right)}h_{n}(\bar{x}_{s})\chi_{s^{\prime}}^{\left(\lambda\right)},$
(B.26)

where $h_{n}$ is related to the Hermite polynomial $H_{n}$ as

$\displaystyle h_{n}(\bar{x}_{s})$
$\displaystyle=\left(\frac{g\left|{QB}\right|}{\pi}\right)^{1/4}\left(\frac{1}{2^{n}n!}\right)^{1/2}e^{-\bar{x}_{s}^{2}/2}H_{n}(\bar{x}_{s}),$
(B.27)

and satisfies

$\displaystyle\hat{a}h_{n}=\sqrt{n}\,h_{n-1},\quad\hat{a}^{\dagger}h_{n}=\sqrt{n+1}\,h_{n+1},\quad\int\mathrm{d}xh_{n}(\bar{x}_{s})h_{n^{\prime}}(\bar{x}_{s})=\delta_{nn^{\prime}}.$
(B.28)

The spinor $\chi$ is given by

$\displaystyle\chi_{+}^{\left({L}\right)}=(1~{}~{}0~{}~{}0~{}~{}0)^{T},\quad\chi_{-}^{\left({L}\right)}=(0~{}~{}1~{}~{}0~{}~{}0)^{T},\quad\chi_{+}^{\left({R}\right)}=(0~{}~{}0~{}~{}1~{}~{}0)^{T},\quad\chi_{-}^{\left({R}\right)}=(0~{}~{}0~{}~{}0~{}~{}1)^{T}.$
(B.29)

The operator $S_{s}\hat{a}^{\dagger}$ changes $s^{\prime}=-s\to s$ and $n\to
n+1$, while $S_{-s}\hat{a}$ changes $s^{\prime}=s\to-s$ and $n\to n-1$, and
the combination $2n+1-ss^{\prime}$ is invariant under both operations. It then
follows that only the modes with the same value of $2n+1-ss^{\prime}$ mix with
each other. The mode with $2n+1-ss^{\prime}=0$, i.e., $n=0$ and $s^{\prime}=s$
corresponds to the lowest Landau level, while the others are the higher Landau
levels.

#### Lowest Landau level.

The equation of motion of the lowest Landau level is given by

$\displaystyle 0$
$\displaystyle=\begin{pmatrix}i\partial_{0}+s\Pi_{z}-\dot{\theta}_{5}&-me^{2i\theta_{m}}\\\
-me^{-2i\theta_{m}}&i\partial_{0}-s\Pi_{z}+\dot{\theta}_{5}\end{pmatrix}\begin{pmatrix}\psi^{(L)}_{0,s}\\\
\psi^{(R)}_{0,s}\end{pmatrix}.$ (B.30)

In the Hamiltonian basis, we eliminate $c_{5}$ by a chiral rotation. The
solutions of the Dirac equation for constant $\phi$ and $A_{z}$ are then given
by

$\displaystyle
u_{0}=\frac{e^{-i\gamma_{5}\theta_{5+m}}}{\sqrt{2\Omega_{0}\left(\Omega_{0}+s{\Pi}_{z}\right)}}\left[m\chi_{s}^{(L)}+\left(\Omega_{0}+s\Pi_{z}\right)\chi_{s}^{(R)}\right]h_{0},\quad
v_{0}=\frac{e^{-i\gamma_{5}\theta_{5+m}}}{\sqrt{2\Omega_{0}\left(\Omega_{0}+s{\Pi}_{z}\right)}}\left[\left(\Omega_{0}+s\Pi_{z}\right)\chi_{s}^{(L)}-m\chi_{s}^{(R)}\right]h_{0},$
(B.31)

where $\Omega_{0}=\sqrt{\Pi_{z}^{2}+m^{2}}$. After turning on the background
fields, the Bogoliubov coefficients evolve as

$\displaystyle\dot{\alpha}_{0}$
$\displaystyle=i\dot{\theta}_{5+m}\frac{s{\Pi}_{z}}{\Omega_{0}}\alpha_{0}-\left(s\frac{m\dot{{\Pi}}_{z}}{2\Omega_{0}^{2}}+i\dot{\theta}_{5+m}\frac{m}{\Omega_{0}}\right)e^{2i\Theta_{0}}\beta_{0},$
(B.32) $\displaystyle\dot{\beta}_{0}$
$\displaystyle=-i\dot{\theta}_{5+m}\frac{s{\Pi}_{z}}{\Omega_{0}}\beta_{0}+\left(s\frac{m\dot{{\Pi}}_{z}}{2\Omega_{0}^{2}}-i\dot{\theta}_{5+m}\frac{m}{\Omega_{0}}\right)e^{-2i\Theta_{0}}\alpha_{0},$
(B.33)

where $\Theta_{0}=\int^{t}\mathrm{d}t\,\Omega_{0}$. We solve this equation
with $\alpha_{0}=1$ and $\beta_{0}=0$ as the initial condition. The lowest
Landau level is then quantized in the Hamiltonian basis as

$\displaystyle\psi_{0}$
$\displaystyle=\sum_{\lambda}\psi_{0,s}^{\left(\lambda\right)}h_{0}\chi_{s}^{\left(\lambda\right)}=e^{i\gamma_{5}\theta_{5}}\left[\left(\alpha_{0}b_{0}-\beta_{0}^{*}d_{0}^{\dagger}\right)u_{0}e^{-i\Theta_{0}}+\left(\beta_{0}b_{0}+\alpha_{0}^{*}d_{0}^{\dagger}\right)v_{0}e^{i\Theta_{0}}\right].$
(B.34)

In the fermion current basis, the solution of the Dirac equation for constant
$\dot{\phi}$ and $A_{z}$ is given by

$\displaystyle\tilde{u}_{0}$
$\displaystyle=\frac{m\chi_{s}^{(L)}+\left(\Omega_{0,s}+s\Pi_{z}-\dot{\theta}_{5+m}\right)\chi_{s}^{(R)}}{\sqrt{2\Omega_{0,s}\left(\Omega_{0,s}+s\Pi_{z}-\dot{\theta}_{5+m}\right)}}h_{0},\quad\tilde{v}_{0}=\frac{\left(\Omega_{0}+s{\Pi}_{z}-\dot{\theta}_{5+m}\right)\chi_{s}^{(L)}-m\chi_{s}^{(R)}}{\sqrt{2\Omega_{0,s}\left(\Omega_{0,s}+s\Pi_{z}-\dot{\theta}_{5+m}\right)}}h_{0},$
(B.35)

where $\Omega_{0,s}=\sqrt{(\Pi_{z}-s\dot{\theta}_{5+m})^{2}+m^{2}}$. The
Bogoliubov coefficients evolve as

$\displaystyle\dot{\tilde{\alpha}}_{0}$
$\displaystyle=-\frac{m\left(s\dot{{\Pi}}_{z}-\ddot{\theta}_{5+m}\right)}{2\Omega_{0,s}^{2}}e^{2i\Theta_{0,s}}\tilde{\beta}_{0},\quad\dot{\tilde{\beta}}_{0}=+\frac{m\left(s\dot{{\Pi}}_{z}-\ddot{\theta}_{5+m}\right)}{2\Omega_{0,s}^{2}}e^{-2i\Theta_{0,s}}\tilde{\alpha}_{0},$
(B.36)

where $\Theta_{0,s}=\int^{t}\mathrm{d}t\,\Omega_{0,s}$. The fermion is then
quantized as

$\displaystyle\psi_{0}$
$\displaystyle=e^{-i\gamma_{5}\theta_{m}}\left[\left(\tilde{\alpha}_{0}{b}_{0}-\tilde{\beta}_{0}^{*}d_{0}^{\dagger}\right)\tilde{u}_{0}e^{-i\Theta_{0,s}}+\left(\tilde{\beta}_{0}{b}_{0}+\tilde{\alpha}_{0}^{*}{d}_{0}^{\dagger}\right)\tilde{v}_{0}e^{i\Theta_{0,s}}\right].$
(B.37)

The creation and annihilation operators satisfy

$\displaystyle\left\\{b_{0}(\vec{p}),b_{0}^{\dagger}(\vec{p}^{\prime})\right\\}$
$\displaystyle=\left\\{d_{0}(\vec{p}),d_{0}^{\dagger}(\vec{p}^{\prime})\right\\}=(2\pi)^{2}\delta^{(2)}(\vec{p}-\vec{p}^{\prime}),$
(B.38)
$\displaystyle\left\\{b_{0}(\vec{p}),d_{0}^{\dagger}(\vec{p}^{\prime})\right\\}$
$\displaystyle=\left\\{b_{0}(\vec{p}),d_{0}(\vec{p}^{\prime})\right\\}=0,$
(B.39)

and $B_{0}$, $D_{0}$, $\tilde{B}_{0}$ and $\tilde{D}_{0}$ satisfy the same
equal-time anti-commutators, which one can show by noting that

$\displaystyle\left|{\alpha_{0}}\right|^{2}+\left|{\beta_{0}}\right|^{2}=\left|{\tilde{\alpha}_{0}}\right|^{2}+\left|{\tilde{\beta}_{0}}\right|^{2}=1.$
(B.40)

#### Higher Landau levels.

The equation of motion of the higher Landau levels is given by

$\displaystyle
0=\begin{pmatrix}i\partial_{0}+s\Pi_{z}-\dot{\theta}_{5}&im_{B}&-me^{2i\theta_{m}}&0\\\
-im_{B}&i\partial_{0}-s\Pi_{z}-\dot{\theta}_{5}&0&-me^{2i\theta_{m}}\\\
-me^{-2i\theta_{m}}&0&i\partial_{0}-s\Pi_{z}+\dot{\theta}_{5}&-im_{B}\\\
0&-me^{-2i\theta_{m}}&im_{B}&i\partial_{0}+s\Pi_{z}+\dot{\theta}_{5}\end{pmatrix}\begin{pmatrix}\psi_{n,s}^{(L)}\\\
\psi_{n-1,-s}^{(L)}\\\ \psi_{n,s}^{(R)}\\\ \psi_{n-1,-s}^{(R)}\end{pmatrix},$
(B.41)

for $n=1,2,...$ and $m_{B}=\sqrt{2ng\left|{QB}\right|}$. In the Hamiltonian
basis, the solutions of the Dirac equation for constant $\phi$ and $A_{z}$ are
given by

$\displaystyle u_{n,1}$
$\displaystyle=\frac{e^{-i\gamma_{5}\theta_{5+m}}}{N}\left[im_{B}\left(-m_{T}\chi_{s}^{(L)}+\left(\Omega_{n}+s\Pi_{z}\right)\chi_{s}^{(R)}\right)h_{n}+\left(m+m_{T}\right)\left(\left(\Omega_{n}+s\Pi_{z}\right)\chi_{-s}^{(L)}+m_{T}\chi_{-s}^{(R)}\right)h_{n-1}\right],$
(B.42) $\displaystyle u_{n,2}$
$\displaystyle=\frac{ie^{-i\gamma_{5}\theta_{5+m}}}{N}\left[\left(m+m_{T}\right)\left(m_{T}\chi_{s}^{(L)}+\left(\Omega_{n}+s\Pi_{z}\right)\chi_{s}^{(R)}\right)h_{n}+im_{B}\left(\left(\Omega_{n}+s\Pi_{z}\right)\chi_{-s}^{(L)}-m_{T}\chi_{-s}^{(R)}\right)h_{n-1}\right],$
(B.43) $\displaystyle v_{n,1}$
$\displaystyle=\frac{ie^{-i\gamma_{5}\theta_{5+m}}}{N}\left[-im_{B}\left(\left(\Omega_{n}+s\Pi_{z}\right)\chi_{s}^{(L)}+m_{T}\chi_{s}^{(R)}\right)h_{n}+\left(m+m_{T}\right)\left(-m_{T}\chi_{-s}^{(L)}+\left(\Omega_{n}+s\Pi_{z}\right)\chi_{-s}^{(R)}\right)h_{n-1}\right],$
(B.44) $\displaystyle v_{n,2}$
$\displaystyle=\frac{ie^{-i\gamma_{5}\theta_{5+m}}}{N}\left[\left(m+m_{T}\right)\left(\left(\Omega_{n}+s\Pi_{z}\right)\chi_{s}^{(L)}-m_{T}\chi_{s}^{(R)}\right)h_{n}-im_{B}\left(m_{T}\chi_{-s}^{(L)}+\left(\Omega_{n}+s\Pi_{z}\right)\chi_{-s}^{(R)}\right)h_{n-1}\right],$
(B.45)

where $m_{T}^{2}=m^{2}+m_{B}^{2}$ and
$\Omega_{n}=\sqrt{\Pi_{z}^{2}+m_{T}^{2}}$. The normalization factor is given
by

$\displaystyle
N=2\sqrt{\Omega_{n}\left(\Omega_{n}+s\Pi_{z}\right)m_{T}\left(m+m_{T}\right)}.$
(B.46)

After turning on the time dependence of the background fields, the time
evolution of the Bogoliubov coefficients is given by

$\displaystyle\begin{pmatrix}\dot{\alpha}_{n,1}\\\ \dot{\alpha}_{n,2}\\\
\dot{\beta}_{n,1}\\\ \dot{\beta}_{n,2}\end{pmatrix}=$
$\displaystyle\left[i\dot{\theta}_{5+m}\begin{pmatrix}-\frac{m}{m_{T}}\frac{s\Pi_{z}}{\Omega_{n}}&\frac{m_{B}}{m_{T}}&\frac{m}{\Omega_{n}}e^{2i\Theta_{n}}&0\\\
\frac{m_{B}}{m_{T}}&\frac{m}{m_{T}}\frac{s\Pi_{z}}{\Omega_{n}}&0&-\frac{m}{\Omega_{n}}e^{2i\Theta_{n}}\\\
\frac{m}{\Omega_{n}}e^{-2i\Theta_{n}}&0&\frac{m}{m_{T}}\frac{s\Pi_{z}}{\Omega_{n}}&\frac{m_{B}}{m_{T}}\\\
0&-\frac{m}{\Omega_{n}}e^{-2i\Theta_{n}}&\frac{m_{B}}{m_{T}}&-\frac{m}{m_{T}}\frac{s\Pi_{z}}{\Omega_{n}}\end{pmatrix}+\frac{sm_{T}\dot{\Pi}_{z}}{2\Omega_{n}^{2}}\begin{pmatrix}0&0&-e^{2i\Theta_{n}}&0\\\
0&0&0&-e^{2i\Theta_{n}}\\\ e^{-2i\Theta_{n}}&0&0&0\\\
0&e^{-2i\Theta_{n}}&0&0\end{pmatrix}\right]\begin{pmatrix}{\alpha}_{n,1}\\\
{\alpha}_{n,2}\\\ {\beta}_{n,1}\\\ {\beta}_{n,2}\end{pmatrix},$ (B.47)

where $\Theta_{n}=\int^{t}dt\,\Omega_{n}$. Notice that this is equivalent to
the case without the magnetic field after replacing $m_{B}\to p_{T}$. Finally
the fermion is quantized as

$\displaystyle\psi_{n}$
$\displaystyle=\sum_{\lambda}\left[\psi_{n,s}^{\left(\lambda\right)}h_{n}\chi_{s}^{\left(\lambda\right)}+\psi_{n-1,-s}^{\left(\lambda\right)}h_{n-1}\chi_{-s}^{\left(\lambda\right)}\right]$
$\displaystyle=\sum_{\lambda,\lambda^{\prime}=1,2}\left[u_{n,\lambda}e^{-i\Theta_{n}}\left(\alpha_{n,\lambda}^{(\lambda^{\prime})}b_{n,\lambda^{\prime}}-\left(-1\right)^{\lambda+\lambda^{\prime}}{\beta_{n,\lambda}^{(\lambda^{\prime})}}^{*}d_{n,\lambda^{\prime}}^{\dagger}\right)+v_{n,\lambda}e^{i\Theta_{n}}\left(\beta_{n,\lambda}^{(\lambda^{\prime})}b_{n,\lambda^{\prime}}+\left(-1\right)^{\lambda+\lambda^{\prime}}{\alpha_{n,\lambda}^{(\lambda^{\prime})}}^{*}d_{n,\lambda^{\prime}}^{\dagger}\right)\right].$
(B.48)

Here the superscript indicates the initial conditions for $\alpha$ and
$\beta$, that is,

$\displaystyle\alpha_{n,1}^{(1)}$
$\displaystyle=1,\quad\alpha_{n,2}^{(1)}=\beta_{n,1}^{(1)}=\beta_{n,2}^{(1)}=0,$
(B.49) $\displaystyle\alpha_{n,2}^{(2)}$
$\displaystyle=1,\quad\alpha_{n,1}^{(2)}=\beta_{n,1}^{(2)}=\beta_{n,2}^{(2)}=0,$
(B.50)

at the initial time.

In the fermion current basis, the solutions of the Dirac equation for constant
$\dot{\phi}$ and $A_{z}$ are given by

$\displaystyle\tilde{u}_{n,+}$
$\displaystyle=\frac{1}{N_{+}}\left[-im_{B}\left(\left(\Omega_{n,+}+\Pi_{B}+\dot{\theta}_{5+m}\right)\chi_{s}^{(L)}+m\chi_{s}^{(R)}\right)h_{n}+\left(\Pi_{B}+s\Pi_{z}\right)\left(\left(\Omega_{n,+}+\Pi_{B}+\dot{\theta}_{5+m}\right)\chi_{-s}^{(L)}+m\chi_{-s}^{(R)}\right)h_{n-1}\right],$
(B.51) $\displaystyle\tilde{u}_{n,-}$
$\displaystyle=\frac{1}{N_{-}}\left[im_{B}\left(m\chi_{s}^{(L)}+\left(\Omega_{n,-}+\Pi_{B}-\dot{\theta}_{5+m}\right)\chi_{s}^{(R)}\right)h_{n}+\left(\Pi_{B}-s\Pi_{z}\right)\left(m\chi_{-s}^{(L)}+\left(\Omega_{n,-}+\Pi_{B}-\dot{\theta}_{5+m}\right)\chi_{-s}^{(R)}\right)h_{n-1}\right]$
(B.52) $\displaystyle\tilde{v}_{n,+}$
$\displaystyle=\frac{1}{N_{+}}\left[im_{B}\left(m\chi_{s}^{(L)}-\left(\Omega_{n,+}+\Pi_{B}+\dot{\theta}_{5+m}\right)\chi_{s}^{(R)}\right)h_{n}+\left(\Pi_{B}+s\Pi_{z}\right)\left(-m\chi_{-s}^{(L)}+\left(\Omega_{n,+}+\Pi_{B}+\dot{\theta}_{5+m}\right)\chi_{-s}^{(R)}\right)h_{n-1}\right],$
(B.53) $\displaystyle\tilde{v}_{n,-}$
$\displaystyle=\frac{1}{N_{-}}\left[im_{B}\left(-\left(\Omega_{n,-}+\Pi_{B}-\dot{\theta}_{5+m}\right)\chi_{s}^{(L)}+m\chi_{s}^{(R)}\right)h_{n}+\left(\Pi_{B}-s\Pi_{z}\right)\left(-\left(\Omega_{n,-}+\Pi_{B}-\dot{\theta}_{5+m}\right)\chi_{-s}^{(L)}+m\chi_{-s}^{(R)}\right)\right],$
(B.54)

where $\Pi_{B}=\sqrt{\Pi_{z}^{2}+m_{B}^{2}}$,
$\Omega_{n,\pm}=\sqrt{(\Pi_{B}\pm\dot{\theta}_{5+m})^{2}+m^{2}}$, and
$N_{\pm}=2\sqrt{\Pi_{B}\left(\Pi_{B}\pm
s\Pi_{z}\right)\Omega_{n,\pm}\left(\Omega_{n,\pm}+\Pi_{B}\pm\dot{\theta}_{5+m}\right)}$.
The time evolution equations for the Bogoliubov coefficients are given by

$\displaystyle\begin{pmatrix}\dot{\tilde{\alpha}}_{n,+}\\\
\dot{\tilde{\alpha}}_{n,-}\\\ \dot{\tilde{\beta}}_{n,+}\\\
\dot{\tilde{\beta}}_{n,-}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}0&A_{n}e^{i(\Theta_{n,+}-\Theta_{n,-})}&-B_{n,+}e^{2i\Theta_{n,+}}&-C_{n}e^{i(\Theta_{n,+}+\Theta_{n,-})}\\\
-A_{n}e^{-i(\Theta_{n,+}-\Theta_{n,-})}&0&-C_{n}e^{i(\Theta_{n,+}+\Theta_{n,-})}&-B_{n,-}e^{2i\Theta_{n,-}}\\\
B_{n,+}e^{-2i\Theta_{n,+}}&C_{n}e^{-i(\Theta_{n,+}+\Theta_{n,-})}&0&A_{n}e^{-i(\Theta_{n,+}-\Theta_{n,-})}\\\
C_{n}e^{-i(\Theta_{n,+}+\Theta_{n,-})}&B_{n,-}e^{-2i\Theta_{n,-}}&-A_{n}e^{i(\Theta_{n,+}-\Theta_{n,-})}&0\end{pmatrix}\begin{pmatrix}\tilde{\alpha}_{n,+}\\\
\tilde{\alpha}_{n,-}\\\ \tilde{\beta}_{n,+}\\\
\tilde{\beta}_{n,-}\end{pmatrix},$ (B.55)

where

$\displaystyle A_{n}$
$\displaystyle=s\frac{gQEm_{B}m}{4\Pi_{B}^{2}\sqrt{\Omega_{n,+}\Omega_{n,-}}}\left[\frac{\Omega_{n,+}+\Omega_{n,-}+2\Pi_{B}}{\sqrt{\left(\Omega_{n,+}+\Pi_{B}+\dot{\theta}_{5+m}\right)\left(\Omega_{n,-}+\Pi_{B}-\dot{\theta}_{5+m}\right)}}\right],$
(B.56) $\displaystyle B_{n,\pm}$
$\displaystyle=\pm\frac{m}{2\Omega_{n,\pm}^{2}}\left(\frac{\Pi_{z}}{\Pi_{B}}gQE\pm\ddot{\theta}_{5+m}\right),\quad
C_{n}=s\frac{gQEm_{B}\left[\left(\Omega_{n,+}+\Pi_{B}+\dot{\theta}_{5+m}\right)\left(\Omega_{n,-}+\Pi_{B}-\dot{\theta}_{5+m}\right)-m^{2}\right]}{4\Pi_{B}^{2}\sqrt{\Omega_{n,+}\Omega_{n,-}\left(\Omega_{n,+}+\Pi_{B}+\dot{\theta}_{5+m}\right)\left(\Omega_{n,-}+\Pi_{B}-\dot{\theta}_{5+m}\right)}}.$
(B.57)

We solve these equations with two different initial conditions,

$\displaystyle\tilde{\alpha}_{n,+}^{(+)}$
$\displaystyle=1,\quad\tilde{\alpha}_{n,-}^{(+)}=\tilde{\beta}_{n,+}^{(+)}=\tilde{\beta}_{n,-}^{(+)}=0,\qquad\mathrm{and}\qquad\tilde{\alpha}_{n,-}^{(-)}=1,\quad\tilde{\alpha}_{n,+}^{(-)}=\tilde{\beta}_{n,+}^{(-)}=\tilde{\beta}_{n,-}^{(-)}=0.$
(B.58)

The Dirac field is quantized as

$\displaystyle\psi_{n}=e^{-i\theta_{m}\gamma_{5}}\sum_{\lambda\lambda^{\prime}=\pm}\left[\tilde{u}_{n,\lambda}e^{-i\Theta_{n,\lambda}}\left(\tilde{\alpha}_{n,\lambda}^{(\lambda^{\prime})}{b}_{n,\lambda^{\prime}}-\tilde{\beta}_{n,\lambda}^{(\lambda^{\prime})*}{d}_{n,\lambda^{\prime}}^{\,\dagger}\right)+\tilde{v}_{n,\lambda}e^{i\Theta_{n,\lambda}}\left(\tilde{\beta}_{n,\lambda}^{(\lambda^{\prime})}{b}_{n,\lambda^{\prime}}+\tilde{\alpha}_{n,\lambda}^{(\lambda^{\prime})*}{d}_{n,\lambda^{\prime}}^{\,\dagger}\right)\right].$
(B.59)

Finally, the creation and annihilation operators satisfy

$\displaystyle\left\\{b_{n_{1},\lambda_{1}}(\vec{p}),b_{n_{2},\lambda_{2}}^{\dagger}(\vec{p}^{\prime})\right\\}$
$\displaystyle=\left\\{d_{n_{1},\lambda_{1}}(\vec{p}),d_{n_{2},\lambda_{2}}^{\dagger}(\vec{p}^{\prime})\right\\}=(2\pi)^{2}\delta_{\lambda_{1}\lambda_{2}}\delta_{n_{1}n_{2}}\delta^{(2)}(\vec{p}-\vec{p}^{\prime}),$
(B.60)
$\displaystyle\left\\{b_{n_{1},\lambda_{1}}(\vec{p}),d_{n_{2},\lambda_{2}}^{\dagger}(\vec{p}^{\prime})\right\\}$
$\displaystyle=\left\\{b_{n_{1},\lambda_{1}}(\vec{p}),d_{n_{2},\lambda_{2}}(\vec{p}^{\prime})\right\\}=0,$
(B.61)

and $B$, $D$, $\tilde{B}$ and $\tilde{D}$ satisfy the same equal-time anti-
commutators, which one can show in the same as the case without the magnetic
field.

#### Summary.

In summary, in the presence of the magnetic field, the fermion is quantized as

$\displaystyle\psi=$
$\displaystyle\int\frac{\mathrm{d}p_{y}\mathrm{d}p_{z}}{(2\pi)^{2}}e^{i(p_{y}y+p_{z}z)}e^{i\gamma_{5}\theta_{5}}\left[\left(\alpha_{0}b_{0}-\beta_{0}^{*}d_{0}^{\dagger}\right)u_{0}e^{-i\Theta_{0}}+\left(\beta_{0}b_{0}+\alpha_{0}^{*}d_{0}^{\dagger}\right)v_{0}e^{i\Theta_{0}}\right.$
$\displaystyle\left.+\sum_{n,\lambda,\lambda^{\prime}}\left[u_{n,\lambda}e^{-i\Theta_{n}}\left(\alpha_{n,\lambda}^{(\lambda^{\prime})}b_{n,\lambda^{\prime}}-\left(-1\right)^{\lambda+\lambda^{\prime}}{\beta_{n,\lambda}^{(\lambda^{\prime})}}^{*}d_{n,\lambda^{\prime}}^{\dagger}\right)+v_{n,\lambda}e^{i\Theta_{n}}\left(\beta_{n,\lambda}^{(\lambda^{\prime})}b_{n,\lambda^{\prime}}+\left(-1\right)^{\lambda+\lambda^{\prime}}{\alpha_{n,\lambda}^{(\lambda^{\prime})}}^{*}d_{n,\lambda^{\prime}}^{\dagger}\right)\right]\right],$
(B.62)

in the Hamiltonian basis, and

$\displaystyle\psi=$
$\displaystyle\int\frac{\mathrm{d}p_{y}\mathrm{d}p_{z}}{(2\pi)^{2}}e^{i(p_{y}y+p_{z}z)}e^{-i\gamma_{5}\theta_{m}}\left[\left(\tilde{\alpha}_{0}{b}_{0}-\tilde{\beta}_{0}^{*}{d}_{0}^{\dagger}\right)\tilde{u}_{0}e^{-i\Theta_{0,s}}+\left(\tilde{\beta}_{0}{b}_{0}+\tilde{\alpha}_{0}^{*}{d}_{0}^{\dagger}\right)\tilde{v}_{0}e^{i\Theta_{0,s}}\right.$
$\displaystyle\left.+\sum_{n,\lambda,\lambda^{\prime}}\left[\tilde{u}_{n,\lambda}e^{-i\Theta_{n,\lambda}}\left(\tilde{\alpha}_{n,\lambda}^{(\lambda^{\prime})}{b}_{n,\lambda^{\prime}}-\tilde{\beta}_{n,\lambda}^{(\lambda^{\prime})*}{d}_{n,\lambda^{\prime}}^{\,\dagger}\right)+\tilde{v}_{n,\lambda}e^{i\Theta_{n,\lambda}}\left(\tilde{\beta}_{n,\lambda}^{(\lambda^{\prime})}{b}_{n,\lambda^{\prime}}+\tilde{\alpha}_{n,\lambda}^{(\lambda^{\prime})*}{d}_{n,\lambda^{\prime}}^{\,\dagger}\right)\right]\right],$
(B.63)

in the fermion current basis, respectively.

###  B.3 Physical quantities

For convenience, we summarize the physical quantities expressed in terms of
the Bogoliubov coefficients in both the Hamiltonian and the fermion current
bases in this appendix. We focus on the energy density, the induced current,
the axial charge and the chiral mass operator, which are defined as

$\displaystyle\rho_{\psi}$
$\displaystyle\equiv\frac{1}{2\mathrm{Vol}(\mathbb{R}_{3})}\int\mathrm{d}^{3}x\left\langle\left[\psi^{\dagger},\left(i\partial_{0}+\dot{\theta}_{5}\gamma_{5}\right)\psi\right]\right\rangle,\quad\left\langle
J_{z}\right\rangle\equiv\frac{1}{2\mathrm{Vol}(\mathbb{R}_{3})}\int\mathrm{d}^{3}x\left\langle\left[\bar{\psi},\gamma^{3}\psi\right]\right\rangle,$
(B.64) $\displaystyle q_{5}$
$\displaystyle\equiv\frac{1}{2\mathrm{Vol}\left(\mathbb{R}_{3}\right)}\int\mathrm{d}^{3}x\left\langle\left[{\psi}^{\dagger},\gamma_{5}\psi\right]\right\rangle,\quad\left\langle\bar{\psi}e^{2i\theta_{m}\gamma_{5}}i\gamma_{5}\psi\right\rangle\equiv\frac{1}{2\mathrm{Vol}\left(\mathbb{R}_{3}\right)}\int\mathrm{d}^{3}x\left\langle\left[\bar{\psi},i\gamma_{5}e^{2i\theta_{m}\gamma_{5}}\psi\right]\right\rangle.$
(B.65)

Below give the expressions both cases, without and with the magnetic field,
separately.

#### Without magnetic field.

The averaged energy density of the fermion is given as the expectation value
of the Hamiltonian as $\rho_{\psi}\equiv{\left\langle
H_{\psi}\right\rangle}/{\mathrm{Vol}(\mathbb{R}_{3})}$. This is given by

$\displaystyle\rho_{\psi}$
$\displaystyle=\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\Omega\sum_{\lambda,\lambda^{\prime}}\left[\left|{\beta_{\lambda}^{(\lambda^{\prime})}}\right|^{2}-\left|{\alpha_{\lambda}^{(\lambda^{\prime})}}\right|^{2}\right]$
$\displaystyle=\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\sum_{\lambda,\lambda^{\prime}}\left[\frac{\Pi^{2}+m^{2}+\lambda\dot{\theta}_{5+m}\Pi}{\Omega_{\lambda}}\left(\left|{\tilde{\beta}_{\lambda}^{(\lambda^{\prime})}}\right|^{2}-\left|{\tilde{\alpha}_{\lambda}^{(\lambda^{\prime})}}\right|^{2}\right)+\frac{2m\dot{\theta}_{5+m}}{\Omega_{\lambda}}\mathrm{Re}\left[\tilde{\alpha_{\lambda}^{(\lambda^{\prime})}}^{*}\tilde{\beta}_{\lambda}^{(\lambda^{\prime})}e^{2i\Theta_{\lambda}}\right]\right],$
(B.66)

where here and in the following, the first line is computed in the Hamiltonian
basis and the second line is computed in the fermion current basis. The
induced current is given by

$\displaystyle\langle J_{z}\rangle=\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}$
$\displaystyle\sum_{\lambda,\lambda^{\prime}}\left[\frac{\Pi_{z}}{\Omega}\left(\left|{\beta_{\lambda}^{(\lambda^{\prime})}}\right|^{2}-\left|{\alpha_{\lambda}^{(\lambda^{\prime})}}\right|^{2}\right)+\frac{2m_{T}}{\Omega}\mathrm{Re}\left[\alpha_{\lambda}^{(\lambda^{\prime})*}\beta_{\lambda}^{(\lambda^{\prime})}e^{2i\Theta}\right]\right]$
$\displaystyle=\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}$
$\displaystyle\sum_{\lambda^{\prime}}\left\\{\sum_{\lambda}\left[\frac{\lambda\Pi_{z}}{\Pi}\frac{\Pi+\lambda\dot{\theta}_{5+m}}{\Omega_{\lambda}}\left(\left|{\tilde{\beta}_{\lambda}^{(\lambda^{\prime})}}\right|^{2}-\left|{\tilde{\alpha}_{\lambda}^{(\lambda^{\prime})}}\right|^{2}\right)+\frac{2\Pi_{z}}{\Pi}\frac{m}{\Omega_{\lambda}}\mathrm{Re}\left[\tilde{\alpha}_{\lambda}^{(\lambda^{\prime})*}\tilde{\beta}_{\lambda}^{(\lambda^{\prime})}\right]\right]\right.$
$\displaystyle~{}~{}~{}~{}~{}\left.+\frac{p_{T}}{\Pi}\frac{m(\Omega_{+}-\Omega_{-}+2\dot{\theta}_{5+m})}{\tilde{N}}\mathrm{Re}\left[e^{i(\Theta_{+}-\Theta_{-})}\left(\tilde{\beta}_{-}^{(\lambda^{\prime})*}\tilde{\beta}_{+}^{(\lambda^{\prime})}-\tilde{\alpha}_{+}^{(\lambda^{\prime})*}\tilde{\alpha}_{-}^{(\lambda^{\prime})}\right)\right]\right.$
$\displaystyle~{}~{}~{}~{}~{}\left.+\frac{2p_{T}}{\Pi}\frac{(\Omega_{+}+\Pi+\dot{\theta}_{5+m})(\Omega_{-}+\Pi-\dot{\theta}_{5+m})+m^{2}}{\tilde{N}}\mathrm{Re}\left[e^{i(\Theta_{+}+\Theta_{-})}\left(\tilde{\alpha}_{+}^{(\lambda^{\prime})*}\tilde{\beta}_{-}^{(\lambda^{\prime})}+\tilde{\alpha}_{-}^{(\lambda^{\prime})*}\tilde{\beta}_{+}^{(\lambda^{\prime})}\right)\right]\right\\},$
(B.67)

where

$\displaystyle\tilde{N}=\sqrt{\Omega_{+}\Omega_{-}(\Omega_{+}+\Pi+\dot{\theta}_{5+m})(\Omega_{-}+\Pi-\dot{\theta}_{5+m})}.$
(B.68)

The axial charge is given by

$\displaystyle q_{5}$
$\displaystyle=\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\sum_{\lambda}\left[\frac{m}{m_{T}}\frac{\Pi_{z}}{\Omega}\left(\left|{\alpha_{1}^{(\lambda^{\prime})}}\right|^{2}-\left|{\beta_{1}^{(\lambda^{\prime})}}\right|^{2}-\left|{\alpha_{2}^{(\lambda)}}\right|^{2}+\left|{\beta_{2}^{(\lambda)}}\right|^{2}\right)\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.-\frac{2p_{T}}{m_{T}}\mathrm{Re}\left[\alpha_{1}^{(\lambda)*}\alpha_{2}^{(\lambda)}+\beta_{1}^{(\lambda)*}\beta_{2}^{(\lambda)}\right]-\frac{2m}{\Omega}\mathrm{Re}\left[\left(\alpha_{1}^{(\lambda)*}\beta_{1}^{(\lambda)}-\alpha_{2}^{(\lambda)*}\beta_{2}^{(\lambda)}\right)e^{2i\Theta}\right]\right]$
$\displaystyle=\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\sum_{\lambda,\lambda^{\prime}}\left[\frac{\lambda\Pi+\dot{\theta}_{5+m}}{\Omega_{\lambda}}\left(\left|{\tilde{\alpha}_{\lambda}^{(\lambda^{\prime})}}\right|^{2}-\left|{\tilde{\beta}_{\lambda}^{(\lambda^{\prime})}}\right|^{2}\right)-\frac{2m}{\Omega_{\lambda}}\mathrm{Re}\left[\tilde{\alpha}_{\lambda}^{(\lambda^{\prime})}\tilde{\beta}_{\lambda}^{(\lambda^{\prime})}e^{2i\Theta_{\lambda}}\right]\right].$
(B.69)

Finally the chiral mass operator is given by

$\displaystyle\left\langle\bar{\psi}e^{2i\theta_{m}\gamma_{5}}i\gamma_{5}\psi\right\rangle$
$\displaystyle=2\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\sum_{\lambda}\mathrm{Im}\left[\left(\alpha_{1}^{(\lambda)*}\beta_{1}^{(\lambda)}-\alpha_{2}^{(\lambda)*}\beta_{2}^{(\lambda)}\right)e^{2i\Theta}\right]$
$\displaystyle=2\int\frac{\mathrm{d}^{3}p}{(2\pi)^{3}}\sum_{\lambda,\lambda^{\prime}}\mathrm{Im}\left[\tilde{\alpha}_{\lambda}^{(\lambda^{\prime})*}\tilde{\beta}_{\lambda}^{(\lambda^{\prime})}e^{2i\Theta_{\lambda}}\right].$
(B.70)

Notice that we do not perform the regularization and renormalization of these
quantities in this appendix.

#### With magnetic field.

The averaged energy density of the fermion is given
by\natural10\natural10$\natural$10 We note that
$\int_{-\infty}^{\infty}\mathrm{d}x\mathrm{d}p_{y}\left|{h_{n}(\bar{x}_{s})}\right|^{2}=\int_{-\infty}^{\infty}\mathrm{d}p_{y}=g\left|{QB}\right|\int_{-\infty}^{\infty}\mathrm{d}x$.

$\displaystyle\rho_{\psi}=\frac{g\left|{QB}\right|}{4\pi^{2}}\int\mathrm{d}p_{z}$
$\displaystyle\left[\Omega_{0}\left(\left|{\beta_{0}}\right|^{2}-\left|{\alpha_{0}}\right|^{2}\right)+\sum_{n,\lambda,\lambda^{\prime}}\Omega_{n}\left(\left|{\beta_{n,\lambda}^{(\lambda^{\prime})}}\right|^{2}-\left|{\alpha_{n,\lambda}^{(\lambda^{\prime})}}\right|^{2}\right)\right]$
$\displaystyle=\frac{g\left|{QB}\right|}{4\pi^{2}}\int\mathrm{d}p_{z}$
$\displaystyle\left\\{\frac{\Pi_{z}^{2}+m^{2}-s\dot{\theta}_{5+m}\Pi_{z}}{\Omega_{0,s}}\left(\left|{\tilde{\beta}_{0}}\right|^{2}-\left|{\tilde{\alpha}_{0}}\right|^{2}\right)+\frac{2m\dot{\theta}_{5+m}}{\Omega_{0,s}}\mathrm{Re}\left[\tilde{\alpha_{0}}^{*}\tilde{\beta}_{0}e^{2i\Theta_{0,s}}\right]\right.$
$\displaystyle\left.+\sum_{n,\lambda,\lambda^{\prime}}\left[\frac{\Pi_{B}^{2}+m^{2}+\lambda\dot{\theta}_{5+m}\Pi_{B}}{\Omega_{n,\lambda}}\left(\left|{\tilde{\beta}_{n,\lambda}^{(\lambda^{\prime})}}\right|^{2}-\left|{\tilde{\alpha}_{n,\lambda}^{(\lambda^{\prime})}}\right|^{2}\right)+\frac{2m\dot{\theta}_{5+m}}{\Omega_{n,\lambda}}\mathrm{Re}\left[\tilde{\alpha_{n,\lambda}^{(\lambda^{\prime})}}^{*}\tilde{\beta}_{n,\lambda}^{(\lambda^{\prime})}e^{2i\Theta_{n,\lambda}}\right]\right]\right\\}.$
(B.71)

The induced current is given by

$\displaystyle\langle J_{z}\rangle=\frac{gQB}{4\pi^{2}}\int\mathrm{d}p_{z}$
$\displaystyle\left\\{\frac{s\Pi_{z}}{\Omega_{0}}\left[\left|{\beta_{0}}\right|^{2}-\left|{\alpha_{0}}\right|^{2}\right]+\frac{2m}{\Omega_{0}}\mathrm{Re}\left[\alpha_{0}^{*}\beta_{0}e^{2i\Theta_{0}}\right]\right.$
$\displaystyle\left.+\sum_{n,\lambda,\lambda^{\prime}}\left[\frac{s\Pi_{z}}{\Omega_{n}}\left[\left|{\beta_{n,\lambda}^{(\lambda^{\prime})}}\right|^{2}-\left|{\alpha_{n,\lambda}^{(\lambda^{\prime})}}\right|^{2}\right]+\frac{2m_{T}}{\Omega_{n}}\mathrm{Re}\left[\alpha_{n,\lambda}^{(\lambda^{\prime})*}\beta_{n,\lambda}^{(\lambda^{\prime})}e^{2i\Theta_{0}}\right]\right]\right\\}$
$\displaystyle=\frac{gQB}{4\pi^{2}}\int\mathrm{d}p_{z}$
$\displaystyle\left\\{\frac{s\Pi_{z}-\dot{\theta}_{5+m}}{\Omega_{0,s}}\left[\left|{\tilde{\beta}_{0}}\right|^{2}-\left|{\tilde{\alpha}_{0}}\right|^{2}\right]+\frac{2m}{\Omega_{0,s}}\mathrm{Re}\left[\tilde{\alpha}_{0}^{*}\tilde{\beta}_{0}e^{2i\Theta_{0,s}}\right]\right.$
$\displaystyle\left.+\sum_{n,\lambda^{\prime}}\left[\sum_{\lambda}\left[\lambda\frac{s\Pi_{z}}{\Pi}\frac{\Pi+\lambda\dot{\theta}_{5+m}}{\Omega_{\lambda}}\left(\left|{\tilde{\beta}_{n,\lambda}^{(\lambda^{\prime})}}\right|^{2}-\left|{\tilde{\alpha}_{n,\lambda}^{(\lambda^{\prime})}}\right|^{2}\right)+\frac{2s\Pi_{z}}{\Pi}\frac{m}{\Omega_{n,\lambda}}\mathrm{Re}\left[\tilde{\alpha}_{n,\lambda}^{(\lambda^{\prime})*}\tilde{\beta}_{n,\lambda}^{(\lambda^{\prime})}\right]\right]\right.\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.\left.+\frac{m_{B}}{\Pi}\frac{m(\Omega_{n,+}-\Omega_{n,-}+2\dot{\theta}_{5+m})}{\tilde{N}_{n}}\mathrm{Re}\left[e^{i(\Theta_{n,+}-\Theta_{n,-})}\left(\tilde{\beta}_{n,-}^{(\lambda^{\prime})*}\tilde{\beta}_{n,+}^{(\lambda^{\prime})}-\tilde{\alpha}_{n,+}^{(\lambda^{\prime})*}\tilde{\alpha}_{n,-}^{(\lambda^{\prime})}\right)\right]\right.\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.\left.+\frac{2m_{B}}{\Pi}\frac{(\Omega_{n,+}+\Pi+\dot{\theta}_{5+m})(\Omega_{n,-}+\Pi-\dot{\theta}_{5+m})+m^{2}}{\tilde{N}_{n}}\mathrm{Re}\left[e^{i(\Theta_{n,+}+\Theta_{n,-})}\left(\tilde{\alpha}_{n,+}^{(\lambda^{\prime})*}\tilde{\beta}_{n,-}^{(\lambda^{\prime})}+\tilde{\alpha}_{n,-}^{(\lambda^{\prime})*}\tilde{\beta}_{n,+}^{(\lambda^{\prime})}\right)\right]\right]\right\\},$
(B.72)

where

$\displaystyle\tilde{N}_{n}=\sqrt{\Omega_{n,+}\Omega_{n,-}(\Omega_{n,+}+\Pi+\dot{\theta}_{5+m})(\Omega_{n,-}+\Pi-\dot{\theta}_{5+m})}.$
(B.73)

The axial charge is expressed as

$\displaystyle q_{5}$
$\displaystyle=\left.q_{5}\right|_{\mathrm{LLL}}+\left.q_{5}\right|_{\mathrm{HLL}},$
(B.74)

where

$\displaystyle\left.q_{5}\right|_{\mathrm{LLL}}$
$\displaystyle=\frac{g\left|{QB}\right|}{4\pi^{2}}\int\mathrm{d}p_{z}\left[\frac{s\Pi_{z}}{\Omega_{0}}\left(\left|{\beta_{0}}\right|^{2}-\left|{\alpha_{0}}\right|^{2}\right)+\frac{2m}{\Omega_{0}}\mathrm{Re}\left[\alpha_{0}^{*}\beta_{0}e^{2i\Theta_{0}}\right]\right]$
$\displaystyle=\frac{g\left|{QB}\right|}{4\pi^{2}}\int\mathrm{d}p_{z}\left[\frac{s\Pi_{z}-\dot{\theta}_{5+m}}{\Omega_{0,s}}\left(\left|{\tilde{\beta}_{0}}\right|^{2}-\left|{\tilde{\alpha}_{0}}\right|^{2}\right)+\frac{2m}{\Omega_{0,s}}\mathrm{Re}\left[\tilde{\alpha}_{0}^{*}\tilde{\beta}_{0}e^{2i\Theta_{0,s}}\right]\right],$
(B.75)

and

$\displaystyle\left.q_{5}\right|_{\mathrm{HLL}}$
$\displaystyle=\frac{g\left|{QB}\right|}{4\pi^{2}}\int\mathrm{d}p_{z}\sum_{n,\lambda}\left[\frac{m}{m_{T}}\frac{s\Pi_{z}}{\Omega_{n}}\left(\left|{\alpha_{n,1}^{(\lambda^{\prime})}}\right|^{2}-\left|{\beta_{n,1}^{(\lambda^{\prime})}}\right|^{2}-\left|{\alpha_{n,2}^{(\lambda)}}\right|^{2}+\left|{\beta_{n,2}^{(\lambda)}}\right|^{2}\right)\right.$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.-\frac{2m_{B}}{m_{T}}\mathrm{Re}\left[\alpha_{n,1}^{(\lambda)*}\alpha_{n,2}^{(\lambda)}+\beta_{n,1}^{(\lambda)*}\beta_{n,2}^{(\lambda)}\right]-\frac{2m}{\Omega_{n}}\mathrm{Re}\left[\left(\alpha_{n,1}^{(\lambda)*}\beta_{n,1}^{(\lambda)}-\alpha_{n,2}^{(\lambda)*}\beta_{n,2}^{(\lambda)}\right)e^{2i\Theta_{n}}\right]\right]$
$\displaystyle=\frac{g\left|{QB}\right|}{4\pi^{2}}\int\mathrm{d}p_{z}\sum_{n,\lambda,\lambda^{\prime}}\left[\frac{\lambda\Pi_{B}+\dot{\theta}_{5+m}}{\Omega_{n,\lambda}}\left(\left|{\tilde{\alpha}_{n,\lambda}^{(\lambda^{\prime})}}\right|^{2}-\left|{\tilde{\beta}_{n,\lambda}^{(\lambda^{\prime})}}\right|^{2}\right)-\frac{2m}{\Omega_{n,\lambda}}\mathrm{Re}\left[\tilde{\alpha}_{n,\lambda}^{(\lambda^{\prime})}\tilde{\beta}_{n,\lambda}^{(\lambda^{\prime})}e^{2i\Theta_{n,\lambda}}\right]\right].$
(B.76)

Finally the chiral mass operator is expressed as

$\displaystyle\left\langle\bar{\psi}e^{2i\theta_{m}\gamma_{5}}i\gamma_{5}\psi\right\rangle$
$\displaystyle=\frac{g\left|{QB}\right|}{2\pi^{2}}\int\mathrm{d}p_{z}\,\mathrm{Im}\left[-\alpha_{0}^{*}\beta_{0}e^{2i\Theta_{0}}+\sum_{n,\lambda}\left(\alpha_{n,1}^{(\lambda)*}\beta_{n,1}^{(\lambda)}-\alpha_{n,2}^{(\lambda)*}\beta_{n,2}^{(\lambda)}\right)e^{2i\Theta_{n}}\right]$
$\displaystyle=\frac{g\left|{QB}\right|}{2\pi^{2}}\int\mathrm{d}p_{z}\,\mathrm{Im}\left[-\tilde{\alpha}_{0}^{*}\tilde{\beta}_{0}e^{2i\Theta_{0,s}}+\sum_{n,\lambda,\lambda^{\prime}}\tilde{\alpha}_{n,\lambda}^{(\lambda^{\prime})*}\tilde{\beta}_{n,\lambda}^{(\lambda^{\prime})}e^{2i\Theta_{n,\lambda}}\right].$
(B.77)

Again we do not perform the regularization and renormalization of these
quantities here.

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