# Schwinger pair production correction in thermal system

Y. L. Wang  Key Laboratory of Beam Technology of the Ministry of Education,
and College of Nuclear Science and Technology, Beijing Normal University,
Beijing 100875, China  H. B. Sang  Key Laboratory of Beam Technology of the
Ministry of Education, and College of Nuclear Science and Technology, Beijing
Normal University, Beijing 100875, China  B. S. Xie<EMAIL_ADDRESS>Key
Laboratory of Beam Technology of the Ministry of Education, and College of
Nuclear Science and Technology, Beijing Normal University, Beijing 100875,
China  Beijing Radiation Center, Beijing 100875, China

###### Abstract

In this paper, we give formal results of Schwinger pair production correction
in thermal systems with external background field by using the evolution
operator method of thermo field dynamics, where especially tree level
correction of thermal photons is considered with linear response approaches by
an effective mass shift. We consider initial systems in two types of vacuums
as zero temperature and thermal vacuum, respectively, with correction of
thermal photons is or not included. As an example we give results of these
corrections to pair production for a constant external background electric
field.

###### pacs:

12.20.Ds, 12.20.-m, 11.10.Wx

##  I Introduction

In an early work of Schwinger, it shows that a very strong constant electric
field decays to real charged pairs Schwinger1951 . The produced charged pairs
are observable but not observed directly yet for the absence of strong
electric field near the critical field strength
$E_{c}=m^{2}_{e}c^{3}/e\hbar=1.3\times 10^{16}\mathrm{V/cm}$ in laboratories.
Recently, a series of papers show that direct tests of vacuum decay are
planing in the near future Gavrilov2019 ; Dunne2014 ; Piazza2012 . Besides,
some calculations indicate thermal enhancement of pair production Gould2017 ;
Monin2010 ; Brown2018 ; Draper2018 ; Medina2017 ; Korwar2018  and it is
similar to dynamically assisted Schwinger pair production Torgrimsson2019 ,
consequently experiment on thermal assisted pair production is proposed
Gould2019 .

Various methods obatin identical results at zero temperature Gelis2016  but
have distinct results at finite temperature for thermal pair production
studied in different mechanisms. Charged particles or external electromagnetic
fields are assumed to be thermalized initially, and represented in real- or
imaginary-time form. Early discussion Dittrich1979 ; Gies1999oneloop ;
Elmfors1995  combined with Schwinger’s approach proved that there is no
thermal contribution to imaginary part of an effective one-loop action. But
recent papers using worldline instants technique, where charged particles
absorb thermal photons to reach maximum decay at a specific temperature
Gould2017 ; Monin2010 ; Brown2018 ; Draper2018 ; Medina2017 ; Korwar2018 ,
prove that thermal effect enhances pair production. These also indicate that
there is a temperature threshold $T_{c}=qE/2\pi m$ Brown2018  and discrete
resonant peaks $T_{\ast}=nqE/2\pi m$ Medina2017  in a constant electric field
$E$, where $q$ and $m$ are charge and mass. Moreover, Kim’s works Kim2008
using operator method and Gavrilov’works Gavrilov2008 ; Gavrilov2008matrix
show that there are thermal corrections in thermal quantum electrodynamics
(QED) pair production. Here we should like to emphasize the following three
points. First, the Schwinger pair production is a nonequilibrium process
particularly when the field strength is greater than temperature. Second, in
imaginary-time form, the asymptotic structure of charged particles are changed
by inserting the thermal conditions in the lagrangian, the redefinition of
asymptotic states may cause counting problem Krekora2006 . Last, the relation
of decay rate, effective Lagrangian and pair production number
$\Gamma(A)=2\mathrm{Im}\mathcal{L}_{\mathrm{eff}}(A)=\pm\sum_{\mathbf{k}}\mathrm{ln}(1\pm\mathcal{N}_{\mathbf{k}})$
Schwinger1951  changes at finite temperature in real-time form Rose2013 .
Therefore, at finite temperature, the imaginary part of effective Lagrangian
may not equal to measured pair production rate. Furthermore, in general
thermal QED systems, the thermal photons are present, and usually are
considered in loop corrections but ignored in tree level calculations. We
notice that they also influence the structure of vacuum, consequently affect
the result of pair production.

Therefore to obtain a measurable pair production rate at finite temperature,
and to make a distinction among thermal charged particles, thermal photons and
external electromagnetic fields, we use the operator method of thermo field
dynamics (TFD) adding an effective mass shift approach. The core of our paper
is to combine Bogoliubov transformation which depends on the solution of
Klein-Gordon equation or Dirac equation under external fields Kim2008 ,
thermal transformation which depends on the choice of inequivalent thermal
vacuums, and effective mass corrections which concern the effect of thermal
photons. Our approaches adopt almost-independent approximation for particles,
consequently all interactions among charged particles are ignored, and the
single particle equations have same accuracy with one-loop effective
Lagrangian Kim2008  and the high order corrections are negligible Gies2017 .
In Dirac vacuum, negative energy(frequency) electron absorbs photons to become
electrons, and its in- and out- states depend on background fields, which are
assumed to consists of external space-time dependent field, $A^{ex}_{\mu}(x)$,
and temperature dependent field, $A^{\gamma}_{\mu}(T_{r})$ so that
$A_{\mu}=A^{ex}_{\mu}(x)+A^{\gamma}_{\mu}(T_{r})$. Assuming that thermal
fields are turned off when $t\rightarrow\infty$ (before measurement), and
employing semiclassical mass shift $m^{*2}=m^{2}-q^{2}\langle
A^{\gamma,\mu}A^{\gamma}_{\mu}\rangle$ to reflect the corrections of thermal
photon fields, we find that $T_{r}$ appears in the exponent which decreases
the threshold of tunneling so that it enhances the rate of pair production.

The paper is organized as follows. In Sec. II, we revisit the pair production
in vacuum then consider it in thermal charged particles, thermal photons and
their mixture. We will show their similar results only with change in thermal
photons and the mixture when temperature is small $T_{r}\ll m$. In Sec. III,
we apply the method to a constant electric field and give the integral form
and low-temperature polynomial form. In Sec. IV, we summarize our results, and
point out future directions. We use the natural units where
$c=\hbar=k_{B}=\varepsilon_{0}=1$. Inverse of temperature is denoted as
$\beta=1/T$. The charge and the mass of boson and fermion have unified marks
as $q$ and $m$.

##  II QED pair production in thermal systems

This method is based on motion equation, and it is convenient to take Dirac
equation and Klein-Gordon equation into the form

$\begin{split}\left\\{(\partial_{\mu}-iqA_{\mu})^{2}\\!+\\!m^{2}\\!+\\!2q\left(\begin{matrix}(\mathbf{B}\\!+\\!i\mathbf{E})\cdot\mathbf{\mathbf{\sigma}_{s}}&\\\
&(\mathbf{B}\\!-\\!i\mathbf{E})\cdot\mathbf{\mathbf{\sigma}_{s}}\end{matrix}\right)\right\\}&\\\
\times\phi_{s}=0&,\end{split}$ (1)

where $\mathbf{B}$ and $\mathbf{E}$ are magnetic and electric fields, and
$\mathbf{\sigma_{s}}$ is half of pauli matrix for fermion and zero matrix for
boson. In this equation, the difference depends on their spin where spin
$s=\pm 1/2$ for spinor QED and spin $s=0$ for scalar QED .

###  II.1 QED pair production at zero temperature

The annihilation operator of incoming state
$\phi^{\small{(+)}}_{\mathbf{k},\mathrm{in}}$ is
$a_{\mathbf{k}}(t_{\mathrm{in}}=-\infty)$ for particle and
$b_{\mathbf{k}}(t_{\mathrm{in}}=-\infty)$ for antiparticle state
$\phi^{(-)}_{\mathbf{k},\mathrm{in}}$, and similarly the annihilation operator
of outgoing states $\phi^{(+)}_{\mathbf{k},\mathrm{out}}$ and
$\phi^{(-)}_{\mathbf{k},\mathrm{\mathrm{out}}}$ are defined by
$a_{\mathbf{k}}(t_{\mathrm{out}}=+\infty)$ and
$b_{\mathbf{k}}(t_{\mathrm{out}}=+\infty)$. The subscript $\mathbf{k}$ denotes
$(\mathbf{k},s)$ including momentum and spin, and the superscript $(+)$ and
$(-)$ distinguish the positive energy and negative energy states. The incoming
states and outgoing states depend on the gauge potential $A(x)$ that
determines the asymptotic solutions of Eq. (1).

The in-vacuum and out-vacuum $|0,\mathrm{in}\rangle$ and
$|0,\mathrm{out}\rangle$ are defined in the particle number representation as:

$|0,\mathrm{in}\rangle=\prod_{\mathbf{k}}\otimes|0_{\mathbf{k}},\mathrm{in}\rangle\,$
(2)

and

$\displaystyle|0,\mathrm{out}\rangle=\prod_{\mathbf{k}}U_{\mathbf{k}}(A^{ex})|0,\mathrm{in}\rangle=U(A^{ex})|0,\mathrm{in}\rangle.$
(3)

The $U(A^{ex})$ is an evolution operator and satisfies

$\displaystyle
a_{\mathbf{k},\mathrm{out}}=U_{\mathbf{k}}(A^{ex})a_{\mathbf{k},\mathrm{in}}U^{+}_{\mathbf{k}}(A^{ex}).$
(4)

The outgoing operators and ingoing operators are related through the
Bogoliubov transformation

$\displaystyle
a_{\mathbf{\mathbf{k}},\mathrm{out}}=\int\frac{d^{3}\mathbf{k^{{}^{\prime}}}}{2E_{\mathbf{k}}(2\pi)^{3}}\left[\langle\phi^{(+)}_{\mathbf{k},\mathrm{out}}|\phi^{(+)}_{\mathbf{k}^{{}^{\prime}},\mathrm{in}}\rangle
a_{\mathbf{k}^{{}^{\prime}}}+\langle\phi^{(+)}_{\mathbf{k},\mathrm{out}}|\phi^{(-)}_{\mathbf{k}^{{}^{\prime}},\mathrm{in}}\rangle
b^{+}_{\mathbf{k}^{{}^{\prime}}}\right],$ (5) $\displaystyle
b^{+}_{\mathbf{\mathbf{k}},\mathrm{out}}=\int\frac{d^{3}\mathbf{k^{{}^{\prime}}}}{2E_{\mathbf{k}}(2\pi)^{3}}\left[\langle\phi^{(-)}_{\mathbf{k},\mathrm{out}}|\phi^{(+)}_{\mathbf{k}^{{}^{\prime}},\mathrm{in}}\rangle
a_{\mathbf{k}^{{}^{\prime}}}+\langle\phi^{(-)}_{\mathbf{k},\mathrm{out}}|\phi^{(-)}_{\mathbf{k}^{{}^{\prime}},\mathrm{in}}\rangle
b^{+}_{\mathbf{k}^{{}^{\prime}}}\right].$

The four projections of in-basis and out-basis are Bogoliubov coefficients
that can be solved numerically by sloving Eq. (1) in iterative algorithm with
fast Fourier transform (FFT) Braun1999 ; Blinne2019  for arbitrary fields.
When Eq. (1) has an analytical solution in a certain field, Eq. (5) has a
compact form

$\left(\begin{aligned} &a_{\mathbf{\mathbf{k}},\mathrm{out}}\\\
&b^{+}_{\mathbf{\mathbf{k}},\mathrm{out}}\end{aligned}\right)=\left(\begin{matrix}\mu_{\mathbf{k}}&\nu^{*}_{\mathbf{k}}\\\
\eta\nu_{\mathbf{k}}&\mu^{*}_{\mathbf{k}}\end{matrix}\right)\left(\begin{aligned}
&a_{\mathbf{\mathbf{k}},\mathrm{in}}\\\
&b^{+}_{\mathbf{\mathbf{k}},\mathrm{in}}\end{aligned}\right),$ (6)

where either $\eta=-1$ for boson or $\eta=1$ for fermion, and the coefficients
have general form

$\mu_{\mathbf{k}}=\langle\phi^{(+)}_{\mathbf{k},\mathrm{out}}|\phi^{(+)}_{\mathbf{k},\mathrm{in}}\rangle,~{}\\\
\nu^{*}_{\mathbf{k}}=\langle\phi^{(+)}_{\mathbf{k},\mathrm{out}}|\phi^{(-)}_{\mathbf{k},\mathrm{in}}\rangle.$
(7)

These Bogoliubov coefficients satisfy the relation

$|\mu_{\mathbf{k}}|^{2}+\eta|\nu_{\mathbf{k}}|^{2}=1.$ (8)

Combining transformation relation Eq. (6) and number operator
$a^{+}_{\mathrm{in}}a_{\mathrm{in}}$ leads to the number of produced pairs
with $\mathbf{k}$ momentum111It is $\langle
0,\mathrm{in}|a^{+}_{\mathbf{k},\mathrm{out}}a_{\mathbf{k},\mathrm{out}}|0,\mathrm{in}\rangle$
in some literature but has no difference in result, since the transformation
is unitary.

$\mathcal{N}_{\mathbf{k}}=\langle
0,\mathrm{out}|a^{+}_{\mathbf{k},\mathrm{in}}a_{\mathbf{k},\mathrm{in}}|0,\mathrm{out}\rangle=|\nu_{\mathbf{k}}|^{2}.$
(9)

Finally, the total number of produced pairs

$\mathcal{N}=\int\frac{d\mathbf{k}}{(2\pi)^{3}}|\nu_{\mathbf{k}}|^{2}.$ (10)

###  II.2 QED pair production in thermal charged particles

In thermal QED systems, real charged particles condense when spontaneous
vacuum breaking occurs even under external current. Here virtual particles are
ignored for we only consider one loop. Systems are assumed to be ideal
canonical ensemble, where chemical potential is ignored. The thermal vacuum is
defined as Das1997 ; Ojima1981

$|0,\beta_{q},\mathrm{in}\rangle=\sum_{\mathbf{k}}\sum_{n_{\mathbf{k}}}Z^{-1/2}_{\mathrm{in}}e^{-\beta_{q}n_{\mathbf{k}}\varepsilon_{\mathbf{k}}/2}|n_{\mathbf{k}},\tilde{n}_{\mathbf{k}},\mathrm{in}\rangle.$
(11)

where $n_{\mathbf{k}}$ and $\tilde{n}_{\mathbf{k}}$ are the number of particle
corresponding to $a_{\mathbf{k},\mathrm{in}}$ and
$\tilde{a}_{\mathbf{k},\mathrm{in}}$ respectively, and satisfy
$n_{\mathbf{k}}=\tilde{n}_{\mathbf{k}}$. The thermal vacuum can be treated as
thermal rotation of incoming vacuum that
$U(\theta)|0,\mathrm{in}\rangle=|0,\beta_{q},\mathrm{in}\rangle$ where

$U(\theta)=e^{-\sum_{\mathbf{k}}\theta_{\mathbf{k}}(\beta_{q})(\tilde{c}_{\mathbf{k},\mathrm{in}}c_{\mathbf{k},\mathrm{in}}-c^{+}_{\mathbf{k},\mathrm{in}}\tilde{c}^{+}_{\mathbf{k},\mathrm{in}})}.$
(12)

The relation between thermalized incoming particles and original incoming
particles is

$\left(\begin{aligned} &c_{\mathbf{k},\mathrm{in}}(\theta)\\\
&\tilde{c}^{+}_{\mathbf{k},\mathrm{in}}(\theta)\end{aligned}\right)=\left(\begin{matrix}\mathrm{cos(h)}\theta_{\mathbf{k}}&-\mathrm{sin(h)}\theta_{\mathbf{k}}\\\
\eta\mathrm{sin(h)}\theta_{\mathbf{k}}&\mathrm{cos(h)}\theta_{\mathbf{k}}\end{matrix}\right)\left(\begin{aligned}
&c_{\mathbf{k},\mathrm{in}}\\\
&\tilde{c}^{+}_{\mathbf{k},\mathrm{in}}\end{aligned}\right),$ (13)

where fermion picks trigonometric function
$\mathrm{sin}\theta/\mathrm{cos}\theta$ and boson picks hyperbolic function
$\mathrm{sinh}\theta/\mathrm{cosh}\theta$. These are represented as

$\mathrm{sin(h)}\theta_{\mathbf{k}}=\frac{e^{-\beta_{q}\varepsilon_{\mathbf{k}}/2}}{\sqrt{1+\eta
e^{-\beta_{q}\varepsilon_{\mathbf{k}}}}},~{}\mathrm{cos(h)}\theta_{\mathbf{k}}=\frac{1}{\sqrt{1+\eta
e^{-\beta_{q}\varepsilon_{\mathbf{k}}}}}.$ (14)

When thermal annihilation operators act on the thermal vacuum, it satisfies

$\begin{split}&c_{\mathbf{k},\mathrm{in}}(\theta)|0,\beta_{q},in\rangle=\tilde{c}_{\mathbf{k},\mathrm{in}}(\theta)|0,\beta_{q},\mathrm{in}\rangle=0,\\\
&c_{\mathbf{k}}|0,\mathrm{in}\rangle=\tilde{c}_{\mathbf{k}}|0,\mathrm{in}\rangle=0.\end{split}$
(15)

For all the dynamical observables are only made of non-tilde $c$ operators in
TFD, the number operator is just $c^{+}c$. Now we construct a doubling thermal
vacuum

$\begin{split}|0,0,\beta_{q},\mathrm{in}\rangle=\sum_{\mathbf{k}}\sum_{n_{\mathbf{k}}}\sum_{m_{\mathbf{k}}}\delta_{n_{\mathbf{k}},m_{\mathbf{k}}}Z^{-1/2}_{\mathrm{in}}e^{-\beta_{q}(n_{\mathbf{k}}+m_{\mathbf{k}})\varepsilon_{\mathbf{k}}/2}&\\\
\times|n_{\mathbf{k}},\tilde{n}_{\mathbf{k}},\mathrm{in}\rangle\oplus|m_{\mathbf{k}},\tilde{m}_{\mathbf{k}},\mathrm{in}\rangle&,\end{split}$
(16)

where $m_{\mathbf{k}}$ is the number of antiparticles corresponding to
$b_{\mathbf{k},\mathrm{in}}$ and $\delta_{n_{\mathbf{k}},m_{\mathbf{k}}}$
presents charge conservation in every pure states. We denote particles with
$\mathbf{a}$ and antiparticles with $\mathbf{b}$ where $\mathbf{\tilde{a}}$
and $\mathbf{\tilde{b}}$ are corresponding tilde particles. Because Eq. (15)
shows that a tilde particle is produced when a particle annihilates in thermal
vacuum, the relation between tilde annihilation operator $\tilde{a}$ and
$\tilde{b}$ has inverse results compared with Eq. (6):

$\left(\begin{aligned} &\tilde{b}_{\mathbf{\mathbf{k}},\mathrm{out}}\\\
&\tilde{a}^{+}_{\mathbf{\mathbf{k}},\mathrm{out}}\end{aligned}\right)=\left(\begin{matrix}\mu_{\mathbf{k}}&\nu^{*}_{\mathbf{k}}\\\
\eta\nu_{\mathbf{k}}&\mu^{*}_{\mathbf{k}}\end{matrix}\right)\left(\begin{aligned}
&\tilde{b}_{\mathbf{\mathbf{k}},\mathrm{in}}\\\
&\tilde{a}^{+}_{\mathbf{\mathbf{k}},\mathrm{in}}\end{aligned}\right).$ (17)

Then combining Eqs. (6), (13) and (17), the transformation of incoming
particles under thermal rotation and external electromagnetic fields is

$\left(\begin{aligned} &a_{\mathbf{k},\mathrm{out}}(\theta)\\\
&\tilde{a}^{+}_{\mathbf{k},\mathrm{out}}(\theta)\\\
&b^{+}_{\mathbf{k},\mathrm{out}}(\theta)\\\
&\tilde{b}_{\mathbf{k},\mathrm{out}}(\theta)\end{aligned}\right)=\left(\begin{matrix}\mu_{\mathbf{k}}\mathrm{cos(h)}\theta_{\mathbf{k}}&-\mu_{\mathbf{k}}\mathrm{sin(h)}\theta_{\mathbf{k}}&\nu^{*}_{\mathbf{k}}\mathrm{cos(h)}\theta_{\mathbf{k}}&-\nu^{*}_{\mathbf{k}}\mathrm{sin(h)}\theta_{\mathbf{k}}\\\
\eta\mu^{*}_{\mathbf{k}}\mathrm{sin(h)}\theta_{\mathbf{k}}&\mu^{*}_{\mathbf{k}}\mathrm{cos(h)}\theta_{\mathbf{k}}&\nu_{\mathbf{k}}\mathrm{sin(h)}\theta_{\mathbf{k}}&\eta\nu_{\mathbf{k}}\mathrm{cos(h)}\theta_{\mathbf{k}}\\\
\eta\nu_{\mathbf{k}}\mathrm{cos(h)}\theta_{\mathbf{k}}&-\eta\nu_{\mathbf{k}}\mathrm{sin(h)}\theta_{\mathbf{k}}&\mu^{*}_{\mathbf{k}}\mathrm{cos(h)}\theta_{\mathbf{k}}&-\mu^{*}_{\mathbf{k}}\mathrm{sin(h)}\theta_{\mathbf{k}}\\\
\eta\nu^{*}_{\mathbf{k}}\mathrm{sin(h)}\theta_{\mathbf{k}}&\nu^{*}_{\mathbf{k}}\mathrm{cos(h)}\theta_{\mathbf{k}}&\eta\mu_{\mathbf{k}}\mathrm{sin(h)}\theta_{\mathbf{k}}&\mu_{\mathbf{k}}\mathrm{cos(h)}\theta_{\mathbf{k}}\end{matrix}\right)\left(\begin{aligned}
&a_{\mathbf{k},\mathrm{in}}\\\ &\tilde{a}^{+}_{\mathbf{k},\mathrm{in}}\\\
&b^{+}_{\mathbf{k},\mathrm{in}}\\\
&\tilde{b}_{\mathbf{k},\mathrm{in}}\end{aligned}\right).$ (18)

This transformation can be expressed as
$T_{AB,\mathrm{out}}(\theta)=U(\theta)U(A^{ex})T_{AB,\mathrm{in}}U^{+}(A^{ex})U^{+}(\theta)$.
Such that we can calculate pair production in thermal vacuum, when
$t\rightarrow\infty$, the number is

$\begin{split}\mathcal{N}_{\mathbf{k},\mathrm{out}}(\beta_{q})=&\langle
0,\beta_{q},\mathrm{out}|a^{+}_{\mathbf{k},\mathrm{in}}a_{\mathbf{k},\mathrm{in}}|0,\beta_{q},\mathrm{out}\rangle\\\
=&\langle
0,\mathrm{in}|U^{+}(\theta)U^{+}(A^{ex})a^{+}_{\mathbf{k},\mathrm{in}}a_{\mathbf{k},\mathrm{in}}U(A^{ex})U(\theta)|0,\mathrm{in}\rangle\\\
=&|\mu_{\mathbf{k}}|^{2}\mathrm{sin(h)}^{2}\theta_{\mathbf{k}}+|\nu_{\mathbf{k}}|^{2}\mathrm{cos(h)}^{2}\theta_{\mathbf{k}}.\end{split}$
(19)

And there are particles condensing in initial thermal vacuum when
$t\rightarrow-\infty$, where

$\mathcal{N}_{\mathbf{k},\mathrm{in}}(\beta_{q})=\langle
0,\beta_{q},\mathrm{in}|a^{+}_{\mathbf{k},\mathrm{in}}a_{\mathbf{k},\mathrm{in}}|0,\beta_{q},\mathrm{in}\rangle=\mathrm{sin(h)}^{2}\theta_{\mathbf{k}}.$
(20)

So the number of created pairs is

$\Delta\mathcal{N}_{\mathbf{k}}(T_{q})=\mathcal{N}_{\mathbf{k},\mathrm{out}}(\beta_{q})-\mathcal{N}_{\mathbf{k},\mathrm{in}}(\beta_{q})=\left\\{\begin{aligned}
&|\nu_{\mathbf{k}}|^{2}(1+2n^{B}_{\mathbf{k}})\\\
&|\nu_{\mathbf{k}}|^{2}(1-2n^{F}_{\mathbf{k}})\end{aligned}\right.$ (21)

Where $n^{B/F}_{\mathbf{k}}$ denotes either the Bose-Einstein distribution
$n^{B}_{\mathbf{k}}=1/(e^{\beta_{q}\varepsilon_{\mathbf{k}}}-1)$ for scalar
QED or $n^{F}_{\mathbf{k}}=1/(e^{\beta_{q}\varepsilon_{\mathbf{k}}}+1)$ for
spinor QED. This is consistent with Kim2009 . In Eq. (21), factor
$|\nu_{\mathbf{k}}|^{2}$ response to the transformation under external fields
and $2|\nu_{\mathbf{k}}|^{2}\overline{n}^{B/F}_{\mathbf{k}}$ arises due to the
present thermal charged particles. It conforms to the Bose-Einstein
condensation for boson and the Pauli blocking for fermion.

When the density of particles and antiparticles is unequal or does not reach
equilibrium density, the system can be in thermal but not in chemical
equilibrium. We can redefine the distribution functions by a replacement
$f^{\pm}=\lambda^{\pm}n^{\pm}$ as Ref. Biro1993 , where superscript $\pm$
represent particles/antiparticles and $\lambda^{\pm}$ are fugacity factors for
describing the deviation from chemical equilibrium, and add the chemical
potential in density functions
$n^{\pm}=1/(e^{\beta_{q}(\varepsilon\mp\mu)}+\eta)$. Taking these replacements
into Eq. (14) and Eq. (18), we get

$\Delta\mathcal{N}_{\mathbf{k}}(T_{q},\kappa)=|\nu_{\mathbf{k}}|^{2}(1-\eta(\lambda^{-}n^{-}_{\mathbf{k}}+\lambda^{+}n^{+}_{\mathbf{k}}))=\left\\{\begin{aligned}
&|\nu_{\mathbf{k}}|^{2}(1+(\lambda^{-}n^{B,-}_{\mathbf{k}}+\lambda^{+}n^{B,+}_{\mathbf{k}}))\\\
&|\nu_{\mathbf{k}}|^{2}(1-(\lambda^{-}n^{F,-}_{\mathbf{k}}+\lambda^{+}n^{F,+}_{\mathbf{k}}))\end{aligned}\right.$
(22)

When $\lambda^{+}=\lambda^{-}=1$, total charges are conservative, and the
presence of the chemical potential enhances the thermal correction for
$n^{-}_{\mathbf{k}}+n^{+}_{\mathbf{k}}>2n_{\mathbf{k}}$. The fugacity factors
$\lambda^{\pm}$ depend on the ratio of the experimental to equilibrium density
Thoma2009  and the non-ideal density $n^{\pm}$ also can be established as
Gavrilov2008 ; Umezawa1995 .

Note that what we calculate is the number of incoming particles
$a_{\mathrm{in}}$ which is not thermalized particles
$a_{\mathrm{in}}(\theta)$. Because the thermal transformation is equivalent to
Kubo-Martin-Schwinger condition, the thermal particles $a(\theta)$ satisfy
corresponding condition, i.e. $\phi(t)=\pm\phi(t-i\beta)$. However, it is a
basic condition in early imaginary-time effective lagrangian approaches at
finite temperature, where the asymptotic particles are also assumed to meet
this condition. So the number of thermalized particles is

$\langle
0,\beta_{q},\mathrm{out}|a^{+}_{\mathbf{k},\mathrm{in}}(\theta)a_{\mathbf{k},\mathrm{in}}(\theta)|0,\beta_{q},\mathrm{out}\rangle=|\nu_{\mathbf{k}}|^{2}.$
(23)

This is just the result at zero temperature, so it may indirectly explain why
some early discussions infer the absence of thermal corrections in pair
production in one loop at finite temperature.

###  II.3 Tree level correction of thermal photons

Once charged pairs appear, thermal photons emerge as heat mediums in
equilibrium QED systems. Inspired by paper Kohlfurst2014  which discusses mass
shift when pairs produce in asymptotic periodic oscillating electromagnetic
fields, we find these thermal Abelian gauge fields also change the asymptotic
structure of charged particles, which cause inequivalent vacuum differed from
original one. In this consideration, we assume that the total background
fields consist of external fields and thermal fields, or more generally, any
electromagnetic fields can be divided into:

$\displaystyle A_{\mu}(x,T)=A_{\mu}^{ex}(x)+A^{r}_{\mu}(T_{r}),$ (24)

with an assumption

$[A_{\mu}^{ex}(x),A^{r}_{\mu}(T)]=0.$ (25)

Which means that the two fields are both classical fields and only tree level
Feynman diagrams are included. We assume that the external field is strong and
thermal field is weak, so the two kinds of fields are absolutely separated and
nonlinear terms are ignored.

Substituting Eq. (24) to Eq. (1) and averaging it, all linear terms will be
offset for thermal photons are isotropic. Only $-q^{2}\langle
A^{\gamma,\mu}A^{\gamma}_{\mu}\rangle$ is left and is equivalent to a mass
shift in Eq. (1)

$m^{*2}=m^{2}-q^{2}\left<A^{\gamma,\mu}A^{\gamma}_{\mu}\right>.$ (26)

This $A^{r}(T)$ is assumed to be the sum of linear free photon fields, so the
square term is proportional to invariant density $n_{\omega}/2\omega$. After
integrating, see Appendix A in detail, we obtain

$m^{*2}=m^{2}+\frac{q^{2}T^{2}}{12}.$ (27)

The square root of this shift part was also obtained in Loewe1992  and called
effective mass of thermal photons Kibble1966 . Before the further discussions,
we denote that $m^{*}_{+}=m^{*}$ and $m^{*2}_{-}=m^{2}-q^{2}T^{2}/12$. Next we
will indicate that different asymptotic time conditions of thermal fields
would lead to opposite thermal corrections.

Now we constrain a condition $A^{r}(t=\pm\infty)\neq 0$ meaning thermal fields
are always existing and can be treated as a static heat bath. Hence
$|0,\mathrm{in};m^{*}_{+}\rangle$ and $|0,\mathrm{out};m^{*}_{+}\rangle$ are
the dressed in-vacuum and out-vacuum. The $m^{*}_{+}$ in the kets denotes that
naked mass $m$ of charged particles is directly substituted by dressed mass
$m^{*}_{+}$. The corresponding number operator is
$\mathcal{N}_{\mathbf{k}}(T_{r})=a^{+}_{\mathbf{k},\mathrm{in};m^{*}_{+}}a_{\mathbf{k},\mathrm{in};m^{*}_{+}}$,
so what we calculate is the number of dressed particles

$N_{\mathbf{k}}(T_{r})=\langle
0,\mathrm{out};m^{*}_{+}|\mathcal{N}_{\mathbf{k}}(T_{r})|0,\mathrm{out};m^{*}_{+}\rangle=|\nu_{\mathbf{k}}|^{2}_{m\rightarrow
m^{*}_{+}}.$ (28)

It means the threshold has increased($E^{*}_{+}=2m^{*}_{+}c^{2}>2mc^{2}$) and
more energy is demanded to counteract oscillating force turning into potential
energy of dressed particles.

If we adopt a weaker condition $A^{r}(t=+\infty)=0$, the number operator
should be a unaltered quantity which corresponds to final measurement.
Similarly, the out-vacuum is supposed to in an equivalent measured Fock space
Das1997 , or has the same mass term in Lagrangian compared with original one,
and is denoted by $|0,\mathrm{out}\rangle$, corresponding the vacuum after
evolution under $A^{ex}$ and $A^{r}$. In this scenario, as long as $A^{r}$ is
turned off, the dressed particles roll into original particles and the vacuum
restore into original vacuum, so the dressing thermal photons turn into energy
of produced particles that enhance pair production. Next we verify it in an
evolutive approach instead of standard background method. The out-vacuum is

$\begin{split}|0,\mathrm{out}\rangle&\approx e^{i\int dt\int
d^{3}x\overline{\mathcal{L}}(A^{ex},A^{r})-\mathcal{L}(A^{ex})}e^{i\int
d^{4}x\mathcal{L}(A^{ex})}|0,\mathrm{in};m^{*}_{-}\rangle\\\
&=U(A^{r}|A^{ex})U(A^{ex})|0,\mathrm{in};m^{*}_{-}\rangle.\end{split}$ (29)

In the first line, we use Eq. (27) and an assumption that $A^{r}(T_{r})$ is
small. The $\overline{\mathcal{L}}$ is Lagrangian density under external
fields $A^{ex}$ and thermal fields $A^{r}$, where the overline means that the
$A^{r}$ has been averaged and will vanish when $t\rightarrow\infty$. The
$\mathcal{L}(A^{ex})$ is just a normal Lagrangian density. Their detailed
expressions depend on what vacuum they act on, and we do not need to give a
concrete expression here. But we divide the evolution operator into two parts:
the $U(A^{ex})$ which undertakes main evolution of vacuum and
$U(A^{r}|A^{ex})$ which changes the asymptotic structure of vacuum. It is
obvious that

$\begin{split}&U(A^{ex})|0,\mathrm{in};m^{*}_{-}\rangle=|0,\mathrm{out};m^{*}_{-}\rangle,\\\
&U(A^{r}|A^{ex})|0,\mathrm{out};m^{*}_{-}\rangle=|0,\mathrm{out}\rangle.\end{split}$
(30)

And the commutative relation $[U(A^{r}|A^{ex}),U(A^{ex})]=0$ holds due to Eq.
(25), therefore the evolutionary order does not affect the final result.
Furthermore, because the result of
$\overline{\mathcal{L}}(A^{ex},A^{r})-\mathcal{L}(A^{ex})$ is a mass term,
$U(A^{r}|A^{ex})$ is an unitary operator. Applying above analysis, we get that

$U^{+}(A^{r}|A^{ex})a_{\mathbf{k},\mathrm{in}}U(A^{r}|A^{ex})~{}U^{+}(A^{r}|A^{ex})|0,\mathrm{in}\rangle=a_{\mathbf{k},\mathrm{in};m^{*}_{-}}|0,\mathrm{in};m^{*}_{-}\rangle=0.$
(31)

Now with above relations, we can obtain the number of produced pairs in this
scenario as

$\begin{split}N_{\mathbf{k}}(T_{r})=&\langle
0,\mathrm{out}|a^{+}_{\mathbf{k},\mathrm{in}}a_{\mathbf{k},\mathrm{in}}|0,\mathrm{out}\rangle\\\
=&\langle
0,\mathrm{in};m^{*}_{-}|U^{+}(A^{ex})U^{+}(A^{r}|A^{ex})a^{+}_{\mathbf{k},\mathrm{in}}a_{\mathbf{k},\mathrm{in}}U(A^{r}|A^{ex})U(A^{ex})|0,\mathrm{in};m^{*}_{-}\rangle\\\
=&\langle
0,\mathrm{out};m^{*}_{-}|a^{+}_{\mathbf{k},\mathrm{in};m^{*}_{-}}a_{\mathbf{k},\mathrm{in};m^{*}_{-}}|0,\mathrm{out};m^{*}_{-}\rangle\\\
=&|\nu_{\mathbf{k}}|^{2}_{m\rightarrow m^{*}_{-}}.\end{split}$ (32)

In this result, Eq. (29) and Eq. (31) are used. Note that when $U(A^{ex})$
acts on vacuum, the result depends on the asymptotic structure of the vacuum,
so does $U(\theta)$. This result means in order to take tree level correction
of thermal photons into pair production, we only need to do a mass shift in
Bogoliubov coefficient, so the threshold has gone
down($E^{*}_{-}=2m^{*}_{-}c^{2}<2mc^{2}$), which verifies our guess that
dressing thermal photons enhance pair production. In subsequent discussions,
we will adopt this approach.

More generally, stable equilibrium QED systems should contain both thermal
charged particles and thermal photons. Following above approaches, the thermal
out-vacuum also is an equivalent measured thermal vacuum

$|0,0,\beta_{q},\mathrm{out}\rangle=U(A^{r}|A^{ex})U(A^{ex})U(\theta)|0,0,\mathrm{in};m^{*}_{-}\rangle.$
(33)

Then we do similar calculations as Eq. (32), and pair production in this
situation is obtained as

$N_{\mathbf{k}}(T_{q},T_{r})=[|\nu_{\mathbf{k}}|^{2}(1\pm
n^{B/F}_{\mathbf{k}})]_{m\rightarrow m^{*}_{-}}.$ (34)

This result is just Eq. (21) with a negative mass shift, which means that the
tunnelling particles and thermal particles all have absorbed thermal photons.
A remark is that the $\overline{n}^{F/B}_{\mathbf{k}}$ in Eq. (34) depends on
ont only charged particles temperature $T_{q}$ in Boltzmann factor
$e^{\varepsilon_{\mathbf{k}}/T_{q}}$, but also thermal photons temperature
$T_{r}$ in
$\varepsilon_{\mathbf{k}}=\sqrt{m^{2}-q^{2}T^{2}_{r}/12+\mathbf{k}^{2}}.$ Here
we ignored any nonlinear and high terms due to the assumation that $T_{r}$ is
small $q^{2}T^{2}/12\ll m^{2}$ that $m^{*2}_{-}$ is always positive.

Up to now, we have not used any particular external fields, and it is for the
first time to obtain formal results of Schwinger pair production at finite
temperature with counting tree level correction of real thermal photons,
compared with previous resultsGould2017 ; Monin2010 ; Brown2018 ; Draper2018 ;
Medina2017 ; Gould2019 ; Dittrich1979 ; Gies1999oneloop ; Elmfors1995 .

##  III Constant electric field

In this section, we apply the formal results above to an external constant
electric field. Before considering finite temperature, we review the solutions
of Eq. (1) in a constant electric field Kim2008 . In the time-dependent gauge
$A_{z}=-Et$, Eq. (1) takes the form

$\left[\partial^{2}_{t}+m^{2}+\mathbf{k}^{2}_{\bot}+(k_{z}-qEt)^{2}+2isqE\right]\phi_{\omega,\mathbf{k}}(t)=0.$
(35)

When $t=-\infty$, the solution of the parabolic cylinder function is expressed
as

$\phi_{\omega,\mathbf{k}}(t)=D_{p}(z).$ (36)

And when $t=\infty$, it turns to

$D_{p}(z)=e^{-ip\pi}D_{p}(-z)+\frac{\sqrt{2\pi}}{\Gamma(-P)}e^{-i(p+1)\pi/2}D_{-p-1}(iz),$
(37)

where

$\begin{split}z&=\sqrt{\frac{2}{qE}}e^{i\pi/4}(k_{z}-qEt),\\\
p&=-\frac{1}{2}-i\frac{m^{2}+\mathbf{k}^{2}_{\bot}+2isqE}{2qE}.\end{split}$
(38)

The Bogoliubov coefficients are

$\mu_{\mathbf{k}}=\frac{\sqrt{2\pi}}{\Gamma(-p)}e^{-i(p+1)\pi/2},\quad\nu_{\mathbf{k}}=e^{ip\pi}.$
(39)

So the total number of produced pair is

$\begin{split}N(E,T_{q}=T_{r}=0)&=\sum_{\mathbf{k}}|\nu_{\mathbf{k}}|^{2}\\\
&=\frac{qE}{4\pi(1-|s|)}\int\frac{d^{2}k_{\perp}}{(2\pi)^{2}}e^{-\frac{\pi(m^{2}+k_{\perp}^{2})}{qE}}\\\
&=\frac{q^{2}E^{2}}{16(1-|s|)\pi^{3}}e^{-\frac{\pi m^{2}}{qE}},\end{split}$
(40)

where, $\frac{qE}{4\pi(1-|s|))}$ is the number of states along the
$z$-direction and Eq. (40) is consistent with Schwinger’s result Schwinger1951
when $|s|=1/2$ for spinor.

In the following, thermal gauge fields are assumed to be turned off when
$t\rightarrow\infty$ as long as they are considered. Meanwhile, we assume that
all systems are in equilibrium initially although an external constant field
is not good to get asymptotically free states. The time scale of pair
production is also assumed to be much shorter than the time scale of thermal
dissipation of produced pairs, so the back-reactions are neglected, where low
temperature and low density limit also been assumed to accord with it. In a
word, systems are thermalized initially but process of producing pairs are
dynamic.

###  III.1 $T_{q}=0,T_{r}\neq 0$

Now, we consider a pure thermal photon system with a constant electric field,
where the low temperature limit $q^{2}T^{2}_{r}/12\ll m^{2}$ is reasonable
since no real particles exist initially due to low quantum fluctuation. But
the tunneling particles are also dressed by thermal photons. In this way, the
number of produced pairs is obtained by substituting $m^{2}$ in Eq. (40) with
$m^{2}-q^{2}T^{2}_{r}/12$ according to Eq. (32)

$N(E,T_{r})=\frac{(qE)^{2}}{16(1-|s|)\pi^{3}}e^{\frac{-\pi
m^{2}}{qE}}e^{\frac{\pi qT^{2}_{r}}{12E}}.$ (41)

At small $T_{r}$, this result has a series expanded form

$\begin{split}N(E,T_{r})=&N(E,T_{q}=T_{r}=0)\\\ &\times\left(1+\frac{\pi
qT^{2}_{r}}{12E}+\frac{1}{2}(\frac{\pi
q}{12E})^{2}T^{4}_{r}+\cdots\right).\end{split}$ (42)

The exponent $T^{2}_{r}/E$ means that the more stronger electric field is, the
less effect fixed thermal photons cause. In Ref. Gies2000twoloop , thermal
photons are considered by two-loop calculation and lowest order thermal
correction term contains $T^{4}$, but we get $T^{2}$. This is because that
thermal fields are treated as external current in our discussion, while those
are internal in Gies2000twoloop  where more double vertexes in diagram
expansion exist.

###  III.2 $T_{q}\neq 0,T_{r}=0$

In this physical situation, the temperature $T_{q}$ of thermal charged
particles is not supposed to be high enough to produce real photons. Then,
with a constant electric field, the result is obtained directly by combining
Eq. (21) and Eq. (40), that is

$\begin{split}N(E,T_{q})=&\frac{qE}{4\pi(1-|s|)}\int\frac{d^{2}k_{\perp}}{(2\pi)^{2}}e^{-\frac{\pi(m^{2}+k_{\perp}^{2})}{qE}}\\\
&\times\left(1-\frac{2\eta}{e^{\sqrt{m^{2}+k_{\perp}^{2}}/T_{q}}+\eta}\right).\end{split}$
(43)

Here, the longitudinal momentum of initial particles is assumed to be zero,
where an extra force is constructed to neutralize the electric field force. It
returns to (40) when $T_{q}\rightarrow 0$, but tends to either infinity for
bosons or zero for fermions when $T_{q}\rightarrow\infty$. When return to
$T_{q}<<m$, the number of produced pairs (43) has an approximate form

$\begin{split}N(E,T_{q})=&\frac{(qE)^{2}}{16(1-|s|)\pi^{3}}e^{-\pi
m^{2}/(qE)}\\\ &\times\left(1-\eta\frac{qE}{4\pi^{2}}e^{-\pi
m^{2}/(qE)-m/T_{q}}\right).\end{split}$ (44)

###  III.3 $T_{q}=T_{r}\neq 0$

Finally, we assume that the initial systems are in absolute equilibrium where
thermal photons and charged particles have same temperature $T_{q}=T_{r}=T$
with an external constant electric field: the system is actually an ideal
particle-antiparticle-photon gas. Combining Eq. (43) and Eq. (34), and the
number of produced pairs is obtained as

$\begin{split}N(E,T_{q}=T_{r}=T)=&\frac{qE}{4\pi(1-|s|)}\int\frac{d^{2}k_{\perp}}{(2\pi)^{2}}e^{-\frac{\pi(m^{2}+k_{\perp}^{2})}{qE}+\frac{\pi
qT^{2}}{12E}}\\\
&\times\left(1-\frac{2\eta}{e^{\beta\sqrt{m^{2}-q^{2}T^{2}/12+k_{\perp}^{2}}}+\eta}\right),\end{split}$
(45)

which has an approximate form at small $T$

$\begin{split}N(E,T_{q}=T_{r}=T\ll m)=&\frac{qE}{4(1-|s|)\pi}e^{\frac{-\pi
m^{2}}{qE}}e^{\frac{\pi qT^{2}}{12E}}\\\
&\times\left(1-\eta\frac{qE}{4\pi^{2}}e^{-\sqrt{m^{2}/T^{2}-q^{2}/12}}\right).\end{split}$
(46)

So far, we have not use any limit on electric field $E$, but low temperature
limit for $T_{r}$ is necessary for our linear assumption in Sec. II. C. In
fact, we expect the external field is much stronger than temperature so that
the thermal dissipation and back-reactions can be neglected.

##  IV Discussion

In this paper, to calculate Schwinger pair production at finite temperature,
we generalize the evolution operator method of TFD to generic thermal QED
systems that contain not only thermal charged particles but also photons. Then
we get thermal transformation relation of outgoing operators and incoming
operators under external fields. What we have done on the basis of Eq. (1) is
similar to Sauter’s calculation on Dirac equation Sauter1931 , except for the
part where we add thermal distribution. Then we use an effective substitute
$m^{*2}_{+}=m^{2}+q^{2}T^{2}/12$ to represent the tree level correction of
thermal photons, but that has an inverse substitution when the turning off of
thermal fields is considered. Lastly, applying thermal average approach
$\langle\mathrm{out-vacuum}|a^{+}_{\mathrm{in}}a_{\mathrm{in}}|\mathrm{out-
vacuum}\rangle$, we get QED pair production in thermal systems. In our
results, $|\nu_{\mathbf{k}}|^{2}$ corresponds to decaying part of incoming
states under external field, and density $n^{B/F}_{\mathbf{k}}$ arises for
initial thermal charged particles and $m^{*2}_{+}$ represents effective mass
of both thermal particles and tunneling particles dressed by thermal photons.
In an external constant electric field, the precise integral results and the
approximate polynomial results are obtained and they both recover Schwinger’s
result at $T=0$.

Back to our results, we haven’t used any particular fields but only the
Bogoliubov coefficient $\nu_{\mathbf{k}}$ in our formal results, so it can be
applied to arbitrary fields as long as we obtain the coefficient. There are no
poles or infinity to be renormalized, since the number of produced pairs is an
observable quantity. If charged particles or photons are not in ideal
equilibrium initially, it can be completed by using a corresponding
distribution function to replace ideal one as we have down in Sec. II. B. It
also can be generalized to thermal Bethe-Heitler process $(\gamma Z\rightarrow
e^{+}e^{-}Z)$ and thermal trident mechanism $(e^{-}Z\rightarrow
e^{-}e^{+}e^{-}Z)$ where thermal massive charged particle $Z$ exists
Maltsev2019 , and classical Coulomb field can be taken into background field
as equivalent approach, but the regions need to be distinguished for the Debye
shielding. So it may be helpful in relativistic plasma and heavy ion
collision.

On the experimental side, it is proposed to observe thermal Schwinger pair
production by a thermal bath of photons in a constant electric field Rose2013
, and our results in a constant $E$ supplement neglected correction. What more
is that our formal results can be applied to general laser fields by combining
numerical simulation Krekora2005 . In addition, momentum spectrum distribution
of production particles is given in our results, and spatial spectral
distribution also can be obtained by Fourier transform. These results are
available by comparisons with that in possible future experiments.

Again, our results mainly work in low temperature limit $T_{r}\ll m$, where
back-reactions and dissipations have not be considered. With temperature
increasing, full equilibrium description is more appropriate, more detailed
summary and review one can refer to Gould2019 . By the way to generalize
temperature to arbitrary scale, the effective Lagrangian approaches are more
proper, which is beyond the present study and is still an open topic problem
for future research.

##  Appendix A effective mass of thermal photons

In Sec. II. B, we illustrate the use of mass shift with thermal photons, that
is

$m^{*2}=m^{2}+\frac{q^{2}T^{2}}{12},$ (47)

where $q^{2}T^{2}/12,$ is square of real thermal photon effective mass, which
is different from the effective photon mass, $m_{eff}=qT/3$ which is
calculated in one-loop polarization Weldon1982 . Now, we derive it with
semiclassical approaches, begin with

$m^{*2}=m^{2}-q^{2}\langle A^{r,\mu}A^{r}_{\mu}\rangle,$ (48)

where $A_{r}$ is the linear sum of all thermal photons and $\langle\rangle$
indicates averaging over time and space. We assume that all thermal photons
are free and satisfy $\langle A^{f,\mu}A^{f}_{\mu}\rangle=-1/2\omega$, and
integrate this relation by

$\langle
A^{r,\mu}A^{r}_{\mu}\rangle=-\int^{\infty}_{0}d\omega\frac{n(\omega)}{2\omega}.$
(49)

Where $n(\omega)/2w$ is the invariant density and $n(\omega)$ equal to

$n(\omega)=\frac{g(\omega)}{e^{\beta\omega}-1}=\frac{\omega^{2}}{\pi^{2}(e^{\beta\omega}-1)},$
(50)

where $g(\omega)$ is the energy density and it’s value is inserted directly
here. Combining Eqs. (48), (49) and (50), we obtain Eq. (47).

###### Acknowledgements.

We thank Dr. Z. L. Li for helpful discussions and M. Ababekri for his reading
of the paper and modification to language. This work was supported by the
National Natural Science Foundation of China (NSFC) under Grants No. 11875007
and No. 11935008.

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