††institutetext: Physics Department, Boğaziçi University
34342 Bebek / Istanbul, Turkey

Pair Production in Real Proper Time and Unitarity without Borel Ambiguity

Cihan Pazarbaşı

Okinawa Institute of Science and Technology Graduate University,
1919-1 Tancha, Onna-son, Kunigami-gun, Okinawa, Japan 904-0495

<EMAIL_ADDRESS>

Abstract

The pair production of scalar particles in electromagnetic background fields
is analyzed using real proper time formulation of 1-loop effective action.
After explaining how real proper time formulation keeps unitarity of the
particle creation process in an unambiguous way and discussing the (lack of)
pair production in uniform (magnetic) electric backgrounds, we apply these
ideas to general electric field backgrounds. Our approach is based on a
recursive perturbative expansion of the proper time propagator $U(t)$ and we
show how the pair production probabilities can be obtained from its
singularities which arises upon a Pade summation of the expansion. We computed
the pair production probabilities for general space or time dependent electric
fields in locally constant approximation and showed they match with the well-
known worldline instanton calculations of pulse and periodic backgrounds.
Later, for one dimensional periodic electric and magnetic backgrounds, we
showed how the WKB integrals appear in the classical limit. We linked both
cases to specific WKB cycles in different spectral regions of the WKB problem
and computed the WKB actions by evaluating the WKB integrals as well as by
taking phase space integrals directly along Lefschetz thimbles. Finally, we
explain the equivalence of our construction with exact WKB method in the
classical limit by showing how the WKB cycles precisely contribute to the pair
production process which is in complete agreement with our time dependent
setting.

###### Contents

  1. 1 Introduction
  2. 2 Pair Production Problem and Resolution of Ambiguity
    1. 2.1 Ambiguity and its Resolution in Uniform Electromagnetic Background
  3. 3 Unambiguous Pair Production from Perturbative Expansions
    1. 3.1 Time Dependent Electric Fields:
    2. 3.2 Space Dependent Electric Fields:
    3. 3.3 Comments and Comparisons:
  4. 4 Connection to WKB Formalism
    1. 4.1 The WKB Integrals from the Propagator
    2. 4.2 Direct Computations via Lefschetz Thimbles
    3. 4.3 Connection to Exact WKB
  5. 5 Summary

##  1 Introduction

Particle pair production is one of the fundamental predictions of relativistic
quantum theories. Early on, it was noticed that in the presence of constant
electromagnetic fields, this can be explained by tunneling through a barrier
1931ZPhy…69..742S  as well as by quantum effects added to the classical
Lagrangian 1936ZPhy…98..714H ; 2006physics…5038H ; Weisskopf:1936hya ;
Miller:1994bs . Later, Schwinger Schwinger:1951nm  systematically formalized
the instability of vacuum and the particle pair production in connection with
the imaginary parts of the effective Lagrangian in the basis of gauge
theories, i.e. QED in his particular example. Since then, the particle
production in background electromagnetic fields has been investigated
thoroughly for different types of background potentials using different
methods. (See dunne2005heisenberg ; Gelis:2015kya ; Fedotov:2022ely  for
review on the subject.)

From a mathematical perspective, a consequence of the vacuum instability, so
that the particle creation, presents itself in the perturbation theory
PhysRev.85.631 : In QED, an expansion in the fine structure constant $\alpha$
should be divergent, as the theory is ill-defined for negative values of
$\alpha$. Although at first sight, a divergent series seems pathological, now
it is understood in the context of resurgence theory that such series encodes
all the physical information in a consistent way. (See Aniceto:2018bis  for a
comprehensive introductory discussion on the subject.)

A common way to decode the information beyond the perturbative expansion is
the Borel summation procedure. For exactly solvable QED background fields,
this approach was used at 1 loop Dunne:1999uy ; Dunne:2022esi  and 2 loops
Dunne:2002qg ; Dunne:2021acr  orders to obtain the imaginary contribution to
the effective action, so that the pair production probability. Standard Borel
summation method, however, possesses another pathology: The imaginary
contribution is multi-valued. This general feature of the Borel procedure
stems from the freedom of choice in the analytical continuation directions in
the presence of singularities. While these singularities are sources of non-
trivial information, which can not be probed by perturbative expansions, the
freedom of choice leads to multi-valued physical quantities.

This pathology has a resolution for problems with a stable vacuum where the
imaginary part should not exist in the full solution. Based on the works of
Bogomolny and Zinn-Justin in Bogomolny:1980ur ; Zinn-Justin:1981qzi , it has
been precisely shown in various settings Aniceto:2013fka ; Dunne_2014 ;
Cherman:2014ofa ; Kozcaz:2016wvy ; Dunne:2016jsr  that the ambiguous imaginary
contribution is canceled by inserting additional non-perturbative information.
In this way, the resulting quantity becomes real and, maybe more importantly,
unique.

In this paper, we will concentrate on pair production problems where vacua are
not always stable. Therefore, one can not rely on the cancellation mechanism
of Bogomolny and Zinn-Justin since the imaginary contributions should survive
whenever there is vacuum instability. In addition to that, multi-valued pair
production probability is not acceptable as one of these results violate the
unitarity of the theory. To see how this violation presents itself in the
effective action, let us look at in-out amplitude, i.e. $\mathcal{A}=\langle
0_{+}|0_{-}\rangle$, where $|0_{\pm}\rangle$ are the vacua at infinite past
and future. In general, the amplitude is a pure phase, i.e.,

$\mathcal{A}=e^{i\Gamma},$ (1)

where $\Gamma$ is the effective action of the theory. As long as
$\Gamma\in\mathbb{R}$, the transition probability $|\mathcal{A}|^{2}=1$ and
the vacuum is a stable one, i.e. there is no creation of particles. However,
when $\Gamma\in\mathbb{C}$, the transition probability becomes

$|\mathcal{A}|^{2}=e^{-2\mathrm{Im}\Gamma}.$ (2)

Obviously, whenever $\mathrm{Im}\Gamma\neq 0$, the vacuum is not stable and
there is a probability of particle creation which is defined as

$\mathcal{P}\sim 1-2\mathrm{Im}\Gamma\leq 1.$ (3)

Note that since the theory is unitary, $\mathrm{Im}\Gamma$ should not be
negative. At first, this seems to be a trivial statement. However, for
consistency and completeness, the $\mathrm{Im}\Gamma>0$ condition should be
implicit in methods we use, which should also cover the stable cases on an
equal footing.

To our knowledge, in the context of the pair production problem, the unitarity
problem was first noticed in Chadha:1977my  and it was shown using the exact
uniform backgroud field solution that the consistent treatment can be achieved
with the Schwinger proper time integral but only when the proper time is
chosen to be real. This approach was motivated from two observations:

  * •

Schwinger’s proper time integral at 1 loop order111 This fact is not limited
to the 1-loop order. See Dunne:2021acr  for an example at 2 loops. and the
Borel integral have the same form.

  * •

When the proper time is chosen real, possible analytical continuation
directions of the integral are defined by construction.

Let us elaborate on these observations: In general, the divergent series for
the perturbative effective action is found first, then the Borel procedure is
applied to tame the divergence and probe the singularity which leads to the
non-perturbative information, i.e the pair production probability in our case.
Instead of following this standard path, the first point suggests that it is
possible to probe these singularities before taking the proper time integral.
While the two approaches are equivalent, in this way, the crucial analytical
continuation information would be kept.

The predefined analytical continuation, on the other hand, stems from
utilization of the $i\varepsilon$ prescription. In addition to Chadha:1977my ,
its relation with the resolution of the Borel ambiguity was also discussed in
Olesen:1977ih ; Pazarbasi:2019web . In order to summarize the main idea
briefly, let us consider the Green’s operator $G(m^{2})=(m^{2}+H)^{-1}$ of a
relativistic quantum theory which is the derivative of the effective action,
i.e. $G(m^{2})=\frac{{\mathrm{d}}\Gamma(m^{2})}{{\mathrm{d}}m^{2}}$. It is not
well defined on the real $m^{2}$ line Taylor:72 . Instead, it is defined by an
analytical continuation on the upper or lower half of the complex plane by
redefining $G^{\pm}(m^{2})=\lim_{\varepsilon\rightarrow 0}(m^{2}\pm
i\varepsilon+H)^{-1}$. Then, any observable computed using these redefined
propagators have a certain predefined analytical continuation which prevents
any ambiguity in the limit $\varepsilon\rightarrow 0$. As we will discuss, in
proper time formalism, the analytical continuation information can also be
also transferred the proper-time integral contour of the 1 loop effective
action when the proper time is taken real instead of imaginary.

One of the main motivation of this paper is applying these ideas to general
electric backgrounds with arbitrary dimensional inhomogineities. For this
purpose, we investigated the pair-production of scalar particles in general
electromagnetic field backgrounds and compute the leading order corrections to
the uniform case in locally constant field approximation (LCFA).

For this purpose, we investigated the pair-production of scalar particles in
general electromagnetic field backgrounds and compute the leading order
corrections to the uniform case in locally constant field approximation
(LCFA). To a great extent, our results matched with Gusynin:1995bc ;
Gusynin:1998bt  where the authors generalized Schwinger’s approach to non-
uniform electromagnetic fields. However, in the case of spatially
inhomogeneous electric fields, we found a sign difference. This is a crucial
aspect of the space dependent electric fields since it is related to a
critical field strength which states a lower bound for the creation of real
particles. In addition to that, our approach is conceptually different as we
only relied on the perturbative expansion of the proper time propagator
$U^{\pm}(t)=e^{\mp iHt}$, i.e. Fourier transformation of $G^{\pm}(m^{2})$ and
the Pade summation of the resulting series. In this way, we obtained the non-
perturbative pair production rates of general background fields which arises
from a singularity of $U^{\pm}(t)$ located at complex $t\neq 0$ point without
any ambiguity.

Similar approaches were used in Dunne:1999uy ; Dunne:2022esi  where the
authors examines the single-pulse electric fields of form
$E(x\omega)=\mathcal{E}\sinh(\omega x)$ and relied on the existence of an
exact solution in this case. Moreover, since their focus was on the divergent
expansion of the effective action, the imaginary contribution to the effective
action upon a summation process suffers from the Borel ambiguity. Our approach
offers a way to generalize their procedures to more general electric
backgrounds while keeping the unitarity of the process without any ambiguity.
In order to get the perturbative expansions, we used a perturbative scheme
developed in Pazarbasi:2020god  for non-relativistic quantum mechanics and
adapted it to arbitrary background gauge potentials. Note that this method is
analogous to the small time expansion of heat kernels DeWitt:1975ys ;
Vassilevich:2003xt ; avramidi2019heat  and string inspired methods
Fliegner:1993wh ; Fliegner:1997rk ; Schubert:2001he ; Edwards:2019eby , but it
organizes the expansion via a recursion relation which enables us to compute
high enough orders to extract non-perturbative information out of it.

In addition to obtaining unitary results, another objective of the paper is
investigating how this perturbative approach is connected to two widely used
semi-classical approaches, i.e. Euclidean worldline instanton (WLI) and the
WKB method. A WLI appears as a periodic solution to the classical equation
motion in path integral formalism. In the presence of electric field
backgrounds, they are widely used in computations of the pair production
probabilities, which presents itself as the imaginary part of the effective
action Affleck:1981bma ; Dunne:2005sx ; Dunne:2006st ; Dunne:2006ur ;
Dumlu:2011cc ; Schneider:2014mla ; Dumlu:2015paa ; Ilderton:2015qda ;
Akal:2017sbs . It was shown Dietrich:2007vw ; Dunne:2008zza  that the WLI
method is equivalent to the semi-classical construction of Gutzwiller
Gutzwiller:1971fy ; Muratore-Ginanneschi:2002sjs , which is also linked to
Lefschetz thimbles Sueishi:2020rug . The thimble perspective was shown to be
an consistent way to define and compute real time path integrals in various
contexts Tanizaki:2014xba ; Feldbrugge:2017kzv ; Feldbrugge:2017fcc  including
the pair production problems Rajeev:2021zae .

The WKB method, on the other hand, is a connection problem based on the
Schrödinger type equations and non-trivial information is encoded in integrals
over closed paths which we call WKB cycles (or cycles in short). It was shown
in various settings Basar_2015 ; Codesido:2017dns ; Codesido:2017jwp ;
Basar_2017 ; Fischbach:2018yiu ; Raman:2020sgw  that different WKB cycles of a
quantum system are not independent of each other which is a sign of underlying
resurgent structure of the corresponding theory and closely related to the
Bogomolny Zinn-justin mechanism.

The application of WKB formalism to pair production problems Keldysh:1965ojf ;
Brezin:1970xf ; Nikishov:1970br ; 1972JETP…34..709P ; Kim:2000un ; Kim:2003qp
; Kim:2007pm ; Dumlu:2010ua ; Taya:2020dco , on the other hand, is based on
the Klein-Gordon equation for the mode functions. In its essence, non-trivial
information arises from the Stokes phenomenon and the exponential parts of the
pair production probabilities are linked to the integrals over WKB cycles.
This is often mentioned as tunneling description of the pair production
problems. An equivalence between the actions of the worldline instantons and
the WKB integrals for any background field with one dimensional
inhomogeneities was argued in Kim:2019yts . (See also Basar_2015  for an
example in the presence of periodic electric field backgrounds.) This
equivalence was also more precisely discussed in relation with the exact WKB
(EWKB) method in Taya:2020dco  where the authors made a connection between
specific WKB cycles in scattering region and WLI.

In this line of the paper, we first observed that the pair production
probability arising from the analytical structure of $U^{\pm}(t)$ matches with
the WLI calculations in Dunne:2005sx ; Dunne:2006st . This shows that the
location of the singularity associated to the pair production process equal to
the values of the Euclidean WLI action and verifies an old conjecture by André
Voros voros94

Later, focusing on time dependent periodic background fields for both electric
and magnetic backgrounds, we showed how the leading order WKB integrals arises
from our real time perturbative decomposition in $\hbar\rightarrow 0$ limit.
In this way, we identified magnetic and electric backgrounds with the specific
WKB cycles of below and above barrier top regions of the periodic potential,
respectively. We showed that the WKB action corresponding to the electric
background can be linked to both of the WKB cycles above the barrier top while
the magnetic case is linked to the classically allowed region under the
barrier top. Then, we computed the WKB actions by two approaches: $i)$ Taking
the WKB integrals along corresponding cycles. $ii)$ Taking phase-space
integrals appear in $\hbar\rightarrow 0$ limit of $U^{\pm}$ along Lefschetz
thimbles, which presents a new way to obtain WKB actions.

The WKB actions we computed for both background fields yields compatible
results with their physical properties. In the magnetic case, we obtained a
real contribution indicating no real particle creation and it is related to
the analogous of Landau levels of the uniform magnetic fields. For the
electric case, on the other hand, we obtained both real and imaginary
contributions which arise from a linear combination of two WKB cycles above
the barrier top. The imaginary part matches with WLI action for the time
dependent periodic electric field; therefore, it is associated to the particle
creation222 Note that a similar observation was made in the direct quantum
mechanical setting in Basar_2015  where authors associated the WLI action with
the tunneling action around the barrier top while in our construction, the
electric case is precisely linked to the linear combination of WKB cycles
above the barrier top and the WLI action is given by the imaginary part of
this combination.. In order to have a complete picture, we also discussed the
periodic electric background potential starting from the corresponding Klein-
Gordon equation and show how exactly the linear combination of the WKB cycles
above the barrier top arises in EWKB approach. In this way, we verified the
computations in time dependent formulation based on the analytical
continuation and also provided a precise example on the connection between WLI
and EWKB method.

Outline of the paper: In Section 2, we first reformulated the real time
approach by providing an unambiguous definition of the pair production
probability which is compatible with the time dependent scattering amplitude
description we discussed above. This reformulation is based on the well-known
fact that $i\varepsilon$ prescription which is associated to the
forward/backward time flow directions in the scattering process and puts the
ideas in Chadha:1977my , where only one time direction for the electric case
was discussed, on a more rigorous basis. After discussing this reformulation,
we present its application to both uniform electric and magnetic backgrounds
in Section 2.1 and show that how the contributions emerging from the proper
time integrals are consistent with the unitarity condition and properties of
their associated vacua.

In Section 3, we extend our discussion to arbitrary background gauge
potentials in arbitrary dimension. We analyze general space dependent or time
dependent electric fields under very general assumptions and compute the first
two orders for the non-perturbative pair production probabilities in locally
constant field approximation (LCFA) limit.

In the second part of the paper, we focus on the connection to the semi-
classical methods. At the end of Section 3, we verify our results obtained by
the perturbative approach to WLI instanton actions of the periodic and single-
pulse electric field configurations Dunne:2005sx ; Dunne:2006st . Then, in
Section 4, considering both electric and magnetic periodic backgrounds, we
focus on the connection to the WKB approach. In Section 4.1, we first show how
the WKB loop integrals emerge in the real time formulation via the
decomposition of the effective action which we explain in Section 3 and
compute the corresponding actions via WKB integrals. Then, we provide an
alternative computations by direct integrations along Lefschetz thimbles in
Section 4.2. Finally, we finish with a discussion on the connection to EWKB
method in Section 4.3, where we obtain precise WKB cycles associated to the
real time electric background computations.

##  2 Pair Production Problem and Resolution of Ambiguity

In this paper, we only consider the pair production of bosonic matter as the
fermionic matter production can be handled equivalently. For bosonic matter
fields in the presence of an electromagnetic background, the effective action
at 1 loop order is expressed as

$\Gamma(m^{2})=i\ln\det(m^{2}+\mathcal{H}),$ (4)

where $m^{2}$ is the mass of bosonic matter particles and

$\mathcal{H}=\frac{1}{2}(p_{\mu}-eA_{\mu})^{2}$ (5)

is the Hamiltonian operator governing the motion of the bosonic particles.
Note that in this paper, we use $g_{\mu\nu}=\mathrm{diag}(-+++)$ metric
convention.

The effective action $\Gamma(m^{2})$ is related to the Green’s operator
$G^{\pm}(z)$ as

${\mathrm{Tr}}G^{\pm}(m^{2})=\frac{{\mathrm{d}}\Gamma^{\pm}(m^{2})}{{\mathrm{d}}m^{2}},$
(6)

which can also be written by an inverse derivative as

$\Gamma^{\pm}(m^{2})=\int^{m^{2}}{\mathrm{d}}z\,{\mathrm{Tr}}G^{\pm}(z),$ (7)

Integrating over singularities of $G^{\pm}(z)$ induces a discontinuity for the
effective action Eden:1966dnq  and we define a function in
$\varepsilon\rightarrow 0$ limit to represent this discontinuity

$\Delta\Gamma=\Gamma^{+}(m^{2})-\Gamma^{-}(m^{2})$ (8)

whose imaginary part, i.e. $\mathrm{Im}\Delta\Gamma(m^{2})$, is the total pair
production probability.

In the proper time dependent setting, the actions $\Gamma^{\pm}$ are written
by the Schwinger’s integrals as

$\displaystyle\Gamma^{\pm}(m^{2})$
$\displaystyle=\mp\int^{m^{2}}{\mathrm{d}}z\,\int_{0}^{\infty}{\mathrm{d}}t\,e^{\mp
im^{2}t}\,{\mathrm{Tr}}e^{\mp iHt},$ (9)
$\displaystyle=-i\int_{J^{\pm}_{0}}\frac{{\mathrm{d}}t}{t}\,e^{\mp
im^{2}t}\,{\mathrm{Tr}}e^{\mp iHt},$ (10)

where we analytically continued the integration contour rather than using
$i\varepsilon$ prescription and defined $J_{0}^{\pm}=\left[0,\infty e^{\mp
i\theta}\right]$. This is equivalent to $i\varepsilon$ prescription and we use
both analytical continuations in this paper when they are needed. Note that we
omitted $\theta\rightarrow 0^{+}$ limit in (10) for notational simplicity and
we will continue using this notation throughout the paper.

In (10), $t$ integral can be considered as a Borel-like integral. The
integrand has possible singularities at $t\neq 0$ containing non-perturbative
information which are the pair production probabilities in our context.
Contrary to the conventional Borel integral, on the other hand, the analytical
continuation directions are pre-determined. Thus, there is no Borel-like
ambiguity takes place for (10).

Note that there is also another pole at $t=0$, which corresponds to the UV
divergence in quantum field theories. In 4 dimensional QED,
${\mathrm{Tr}}e^{\mp iHt}$ has another singularity at $t=0$ which is handled
by the renormalization procedure but we don’t discuss them in this paper.

####  Remark:

${\mathrm{Tr}}e^{\mp iHt}$ is the proper time evolution operator for an
observer in particle’s rest frame as $H$ generates translations in the proper
time $t$. Then, $\Gamma^{+}$ and $\Gamma^{-}$ correspond to effective actions
for forward and backward proper-time evolutions respectively. Let us make some
formal manipulations and rewrite the latter one by setting $t\rightarrow-t$:

$\Gamma^{-}(m^{2})=i\int_{-\infty}^{0}\frac{{\mathrm{d}}t}{t}\,e^{-im^{2}t}\,{\mathrm{Tr}}e^{-iHt}.$
(11)

where $\theta\rightarrow 0^{+}$ limit is taken. Now, the action is described
by forward time evolution between $t=-\infty$ and $t=0$. Moreover, in this
form the gap equation in (8) becomes

$\Delta\Gamma(m^{2})=-i\int_{-\infty}^{\infty}\frac{{\mathrm{d}}t}{t}\,e^{-im^{2}t}\,{\mathrm{Tr}}e^{-iHt}$
(12)

which defines the effective action of a propagation between infinite past and
future of an observer in particles’ rest frame. Then, from (12) we observe
that the unambiguous definition in (8) assumes the proper time flow direction
from past infinity to future infinity. In this sense,
$\mathrm{Im}\Delta\Gamma$ is the pair production probability with correct sign
and the unitarity condition demands $\mathrm{Im}\Delta\Gamma>0$.

###  2.1 Ambiguity and its Resolution in Uniform Electromagnetic Background

Before considering general electromagnetic backgrounds, we are going to
discuss how the unambigous pair production rate emerges from the exact
effective action in both uniform electric and magnetic backgrounds and
precisely show that the real proper time approach leads to unambiguous results
for both cases, which are consistent with the unitarity condition and very
well known properties of the vacua of these backgrounds.

Figure 1: Complex $t$ planes. (Left) Real time cases: Regions allowed for
analytical continuations are represented by blue and disallowed regions are
represented by red. (Right) Imaginary time case: This is obtained by rotating
real time ones by $\frac{\pi}{2}$ in the respective allowed directions. Note
that blue and red regions on the right correspond to the same regions on the
right.

For completeness, let us first look at the imaginary time case and review how
the Borel ambiguity arises. The 1 loop effective action in Euclidean proper
time for the bosonic fields in a uniform electromagnetic background is written
as

$\Gamma_{\mathrm{Eucl}}(m)=\frac{{\mathcal{V}}_{4}}{16\pi^{2}}\int_{0}^{\infty}\frac{{\mathrm{d}}s}{s^{3}}\,\frac{e^{-sm^{2}}(se\mathcal{E})(se\mathcal{B})}{\sin\left(\frac{se\mathcal{E}}{2}\right)\sinh\left(\frac{se\mathcal{B}}{2}\right)}.$
(13)

where ${\mathcal{V}}_{4}=L^{3}T$ is the space-time volume factor.

The main problem here is the loss of the information provided by
$i\varepsilon$ prescription. In fact, when we start with the Euclidean time,
the exponential term makes the integral converging as the contour approaches
to infinity. However, when there is a singularity on the contour, there is no
guidance stating how to handle it. This is illustrated in Fig. 1 where the
allowed/forbidden regions for forward and backward time evolutions are merged
after proper Wick rotations. Therefore, the distinction between forward and
backward time evolutions vanishes and the expression (13) represents the Wick
rotated effective action for both cases. This also prevents from defining the
gap $\Delta\Gamma$ unambiguously as in (8). This also shows the origin of the
ambiguity in the Euclidean case and why its resolution needs a real time
approach.

We first review the standard picture for the pair production to emphasize its
problem. In (13), the poles related to the background electric field lie on
the real axis at $t=\pm\frac{n\pi}{e\mathcal{E}}$ while the poles
corresponding to magnetic background are on the imaginary axis $t=\pm
i\frac{n\pi}{e\mathcal{B}}$, where $n\in\mathbb{N}^{+}$ for both cases. It is
possible to compute the imaginary contributions using standard contour
integration. Common lore is that the electric poles lead to emergence of
$\mathrm{Im}\Gamma$ while the magnetic ones are not relevant since they do not
lie on the integration contour in (13).

Consider the contribution of the first electric pole. There are two possible
ways to analytically continue which leads to two distinct integration paths,
i.e. $J_{0}^{+}$ and $J_{0}^{-}$. (See Fig 2). There is no guideline for
choosing any of these paths; therefore we have two complex conjugate results:

$\mathrm{Im}\Gamma\sim\mp e^{-\frac{m^{2}\pi}{e\mathcal{E}}}.$ (14)

We know that the pair production probability is defined by $\mathcal{P}\sim
2\mathrm{Im}\Gamma$ and can only be positive. Therefore, we can just choose
the result with $+$ sign in (14). However, this is an ad-hoc approach and does
not hide the fact that the result is ambiguous. This is equivalent to the
Borel ambiguity and as we mentioned it can be overcome by real proper time
approach which provides us an ambiguous definition of the pair production
probability as we described above.

Figure 2:  Possible Borel integral contours on the real axis for uniform
electric field background in the Euclidean proper time and uniform magnetic
background in the real time case.

Let us now return to the real proper time case. The effective action in real
time is

$\Gamma^{\pm}(m)=-\frac{{\mathcal{V}}_{4}}{16\pi^{2}}\int_{J^{\pm}_{0}}\frac{{\mathrm{d}}t}{t^{3}}\,\frac{e^{\mp
itm^{2}}(te\mathcal{E})(te\mathcal{B})}{\sinh\left(\frac{te\mathcal{E}}{2}\right)\sin\left(\frac{te\mathcal{B}}{2}\right)}.$
(15)

Note that the location of the poles related to electric and magnetic fields
exchanged and for both $\Gamma^{\pm}$ the possible analytical continuation
directions are determined by definition. In the following, for simplicity of
discussion, we continue with pure electric field and pure magnetic field
backgrounds.

#### Uniform Electric Background:

In $\mathcal{B}\rightarrow 0$ limit, the effective action (15) becomes

$\Gamma_{\mathrm{E}}^{\pm}(m)=-\frac{{\mathcal{V}}_{4}}{4\pi^{3}}\int_{J^{\pm}_{0}}\frac{{\mathrm{d}}t}{t^{3}}\,\frac{e^{\mp
itm^{2}}(te\mathcal{E})}{2\sinh(\frac{te\mathcal{E}}{2})}$ (16)

and the integrand has poles at

$t_{\mathrm{E}}=\pm\frac{2\pi in_{\mathrm{E}}}{e\mathcal{E}}\quad,\quad
n_{E}\in\mathbb{N}^{+}.$

While all the poles can be treated collectively, the pair production rate of
first particle and anti-particle pair is only linked to the first order non-
perturbative term Cohen:2008wz . Therefore, only this part needs to satisfy
the unitarity condition without any ambiguity and we will consider the first
poles at $t_{\mathrm{E}}=\pm\frac{2\pi i}{e\mathcal{E}}$ in our analysis.

Both $\Gamma^{+}_{\mathrm{E}}$ and $\Gamma^{-}_{\mathrm{E}}$ contain poles at
$\pm\frac{2\pi i}{e\mathcal{E}}$ in the complex $t$ plane. However, since the
analytical continuation of $t$ is allowed only in one direction by
construction, $\Gamma^{+}_{\mathrm{E}}$ and $\Gamma^{-}_{\mathrm{E}}$ can only
get contributions from one of the poles, i.e., $-\frac{2\pi i}{e\mathcal{E}}$
or $+\frac{2\pi i}{e\mathcal{E}}$ respectively.

(a)

(b)

Figure 3: Contours for the real time integrals in Electric case. 3(a):
Contours for $\Gamma^{+}_{\mathrm{E}}$. 3(b): Contours for
$\Gamma^{-}_{\mathrm{E}}$.

First, we consider $\Gamma^{+}_{\mathrm{E}}(m)$. To get the contribution from
the pole at $-\frac{2\pi i}{e\mathcal{E}}$, we simply rotate in clockwise
direction by $\frac{\pi}{2}$ degree as in Fig. 3(b) and using
$\int_{J_{0}}+\int_{-J_{-\pi/2}^{+}}=0$, we re-write the effective action as

$\displaystyle\Gamma^{+}_{\mathrm{E}}(m^{2})$
$\displaystyle=-{\mathcal{V}}_{4}\frac{e\mathcal{E}}{8\pi^{2}}\int_{J^{+}_{-\frac{\pi}{2}}}\frac{{\mathrm{d}}t}{t^{2}}\frac{e^{-im^{2}t}}{\sinh\left(\frac{te\mathcal{E}}{2}\right)}$
(17)

Note that the analytical continuation of the contour $J_{\pi/2}^{-}$ around
the singularity at $t=-\frac{2\pi i}{e\mathcal{E}}$ is fixed by the rotation
direction of the initial contour on the real axis. Then, we obtain the
imaginary part of $\Gamma^{+}_{\mathrm{E}}$ with a fixed sign as

$\mathrm{Im}\Gamma^{+}_{\mathrm{E}}(m^{2})=+{\mathcal{V}}_{4}\frac{(e\mathcal{E})^{2}}{16\pi^{2}}e^{-\frac{2\pi
m^{2}}{e\mathcal{E}}}.$ (18)

In the same way, we can extract the imaginary part of
$\Gamma^{-}_{\mathrm{E}}(m^{2})$ from the pole at $t=\frac{2\pi
i}{e\mathcal{E}}$. This time, we rotate the contour in counter-clockwise
direction as in Fig. 3(a) and re-write the effective action as

$\displaystyle\Gamma^{-}_{\mathrm{E}}(m^{2})$
$\displaystyle=-{\mathcal{V}}_{4}\frac{e\mathcal{E}}{8\pi^{2}}\int_{J^{-}_{\frac{\pi}{2}}}\frac{{\mathrm{d}}t}{t^{2}}\frac{e^{itm^{2}}}{\sinh(\frac{te\mathcal{E}}{2})}$
(19)

and get the imaginary part of $\Gamma^{-}_{\mathrm{E}}$ as

$\mathrm{Im}\Gamma^{-}_{\mathrm{E}}(m^{2})=-{\mathcal{V}}_{4}\frac{(e\mathcal{E})^{2}}{16\pi^{2}}e^{-\frac{2\pi
m^{2}}{e\mathcal{E}}}.$ (20)

Combining these two results, we get the imaginary part of the gap
$\Delta\Gamma=\Gamma^{+}-\Gamma^{-}$, so that the pair production rate as

$\mathcal{P}=\mathrm{Im}\Delta\Gamma_{\mathrm{E}}(m^{2})={\mathcal{V}}_{4}\frac{(e\mathcal{E})^{2}}{8\pi^{2}}e^{-\frac{2\pi
m^{2}}{eE}},$ (21)

which recovers the standard pair production probability with an unambigous
sign. Note that along with the individual results for
$\mathrm{Im}\Gamma^{\pm}$, the unambigous definition of $\Delta\Gamma$ is also
a primary factor for the unambiguous end result. Without this definition, the
ambiguity would be persistent in the real time case in the same way with the
Euclidean case.

#### Uniform Magnetic Background:

Although the uniform magnetic case does not yield pair production, it is
important to see how the unambiguous definition of $\Delta\Gamma$ fits in this
case. In $\mathcal{E}\rightarrow 0$ limit, (15) reduces to

$\Gamma^{\pm}_{\mathrm{B}}(m^{2})=-\frac{{\mathcal{V}}_{4}}{4\pi^{2}}\int_{J^{\pm}_{0}}\frac{{\mathrm{d}}t}{t^{3}}\frac{e^{\mp
itm^{2}}teB}{2\sin\left(\frac{teB}{2}\right)}.$ (22)

Now the poles are on the real axis and the treatment is similar to the
standard Borel integral. However, again as in the electric background case,
the restriction on the analytical continuation directions for both
$\Gamma^{+}_{\mathrm{B}}$ and $\Gamma^{-}_{\mathrm{B}}$ prevents ambiguous
results. Taking the integrals using the appropriate contours as pictured in
Fig. 2, we find the gap
$\Delta\Gamma_{\mathrm{B}}=\Gamma^{+}_{\mathrm{B}}-\Gamma^{-}_{\mathrm{B}}$
due to the pole at $t_{\mathrm{B}}=\frac{2\pi}{e\mathcal{B}}$ as

$\Delta\Gamma_{\mathrm{B}}=-\frac{eB}{8\pi^{2}}\left[\left(-i\pi\frac{eB}{2\pi}\right)e^{-\frac{2\pi
im^{2}}{eB}}-\left(i\pi\frac{eB}{2\pi}\right)e^{\frac{2\pi
im^{2}}{eB}}\right]=i\frac{(eB)^{2}}{8\pi^{3}}\cos\left(\frac{2\pi
m^{2}}{eB}\right)$ (23)

This is again imaginary contribution with a sign that keeps the theory unitary
but now it has an oscillatory behaviour. This is fine as the it indicates that
the stable quantum vacuum is not completely empty and there are successive
particle creations and annihilations as expected from any relativistic quantum
theory but effectively this process leads to no effective pair creation.
Therefore, there is no need for any cancellation of this contribution by
Bogomolny Zinn-Justin mechanism.

##  3 Unambiguous Pair Production from Perturbative Expansions

In physics, exact solutions are very rare. Therefore, while the discussion in
Section 2.1 explicitly show how the pair production rate in the presence of a
uniform electromagnetic field emerges in a way that the unitarity is
preserved, it is also important to show how the same information can be
achieved when an exact solution is not accessible.

In absence of an exact solution, perturbative techniques allow us to compute
the coefficients in a perturbation series, which form generically divergent
series. The non-perturbative information is encoded in this divergent series
and can be extracted using Borel-Pade techniques Costin:2019xql ;
Costin:2020hwg ; Costin:2020pcj ; Florio:2019hzn . However, the Borel-Pade
summation leads to the same ambiguity that we discussed in the Euclidean case
in the previous section and whenever there is a persistent instability of the
vacuum. As in the uniform case, to get an unambiguous result, we use the real
time approach and probe the poles of the propagator directly, i.e. we need to
sum the perturbative expansion of the propagator
${\mathrm{Tr}}U^{\pm}(t)={\mathrm{Tr}}e^{\mp itH}$ in (10) before taking the
proper time integral.

For this reason, we adapt the recursive perturbative scheme introduced in
Pazarbasi:2020god  which is based on the construction in Moss:1993rc ;
Moss_1999  to problems with background gauge potentials. In Pazarbasi:2020god
, it was shown that the non-commutativity of phase-space variables acts as a
source for a derivative expansion which is identified as the semi-classical
expansion. In the present context, the non-commutativity again produces a
derivative expansion which corresponds to the deviation from the uniform
field. At each order of the derivative expansion, there is also another
expansion which depends on the electromagnetic field strength and using this
one the non-perturbative pair production probability at each order in the
derivative expansion can be obtained.

With the perturbative scheme that we will explain shortly, we compute the pair
production probability in arbitrary temporal or spatial electric field
backgrounds up to the $2^{\mathrm{nd}}$ order in derivative expansion.
Throughout this section we rely on the perturbative expansions and compute the
non-perturbative results via the summation of the series expansions. Our
series expansions are compatible with the exact results found in
Gusynin:1995bc ; Gusynin:1998bt  but since our approach relies on purely
perturbative calculations and free from the complications of the exact
computations. In addition to that the pair production probability we compute
at the $2^{\mathrm{nd}}$ order slightly differs from Gusynin:1998bt  in
presence of spatial inhomogeneities.

Note that the non-perturbative pair production problem based on the summation
of the perturbative expansions has been discussed for an exactly solvable one
dimensional electric background in Dunne:1999uy ; Dunne:2022esi . The content
of this section can be seen as a first step of the generalization of these
results to general background potentials in arbitrary dimensions while keeping
the unitarity of the physical quantities without any ad-hoc choices.

Recursive Expansion: In this section, we will investigate pure electric
fields. The Hamiltonian is written in general as

$H=-\frac{1}{2}(p_{0}-eA_{0}({\mathbf{x}}))^{2}+\frac{1}{2}({\mathbf{p}}-e{\mathbf{A}}(x_{0}))^{2}\equiv-\frac{1}{2}\pi^{2}_{0}(p_{0},{\mathbf{x}})+\frac{1}{2}\bm{\pi}^{2}({\mathbf{p}},x_{0}).$
(24)

We assume that the time like and space like parts of the gauge potential are
in the following forms respectively:

$A_{0}(x_{0},{\mathbf{x}})=-\frac{\mathcal{E}}{\omega}H_{0}(\omega\,{\mathbf{x}})\quad,\quad{\mathbf{A}}=-\frac{\mathcal{E}}{\omega}\mathbf{H}(\omega
x_{0}).$ (25)

Then, the electric field is

${\mathbf{E}}=\frac{\mathcal{E}}{\omega}\left(\frac{{\mathrm{d}}\mathbf{H}(\omega
x_{0})}{{\mathrm{d}}x_{0}}+\nabla H_{0}(\omega\,{\mathbf{x}})\right).$

Since we are interested in the locally uniform fields, we assume that both
$H_{0}(\omega{\mathbf{x}})$ and $\mathbf{H}(\omega x_{0})$ are slowly varying
functions of ${\mathbf{x}}$ and $x_{0}$ respectively. For the same reason, we
also assume that $m\gg e\mathcal{E}\gg\omega$ throughout this paper. This will
guide us in our calculations.

The effective action for the general Hamiltonian in (24) is written as

$\Gamma^{\pm}(m^{2})=-i\int_{J^{\pm}_{0}}\frac{{\mathrm{d}}t}{t}e^{\mp
im^{2}t}\int\frac{{\mathrm{d}}^{4}p}{(2\pi)^{4}}\,\langle
p_{0},{\mathbf{p}}|e^{\mp
it\left(-\frac{1}{2}\pi_{0}^{2}(p_{0},{\mathbf{x}})+\frac{1}{2}\bm{\pi}^{2}({\mathbf{p}},x_{0})\right)}|{\mathbf{p}},p_{0}\rangle.$
(26)

Note that since $\pi_{0}$ and $\bm{\pi}$ do not commute with each other, the
propagator does not factorize trivially. To get the perturbative expansion, we
can either redefine the propagator $U(t)={\mathrm{Tr}}e^{\mp itH}$ as

$U(t)=e^{\pm\frac{it\pi^{2}_{0}}{2}}\tilde{U}(t)$ (27)

or

$U(t)=e^{\mp\frac{it\bm{\pi}^{2}}{2}}\tilde{U}(t).$ (28)

As shown in Pazarbasi:2020god , both choices are equivalent. Solving the time
dependent Schrodinger equation for redefined propagators and following the
steps in Pazarbasi:2020god , we reach a recursive expansion for $\tilde{U}(t)$
for each case and express the effective action in terms of this recursive
expansion.

Let us consider pure time dependent and pure space dependent electric fields.
For the time dependent case, where $A_{0}({\mathbf{x}})=0$ and
$\pi_{0}=p_{0}$, the perturbative expansion is written as

$\displaystyle\Gamma^{\pm}_{\mathrm{time}}$
$\displaystyle=-i\sum_{m=0}\int_{J^{\pm}_{0}}\frac{{\mathrm{d}}t}{t}e^{\mp
im^{2}t}\int\frac{{\mathrm{d}}^{4}p\,{\mathrm{d}}^{4}x}{(2\pi)^{4}}\,e^{\mp\frac{i\bm{\pi}^{2}t}{2}}\,\sum_{k=1}^{m}\tilde{U}^{\pm}_{m,k}(t)\,e^{\pm\frac{ip_{0}^{2}t}{2}},$
(29)

where $\tilde{U}^{\pm}$ is given by the recursion relation

$\displaystyle\tilde{U}_{m,k}^{\pm}(t)$
$\displaystyle=\frac{1}{2}\sum_{l=1}^{m-k+1}\frac{1}{l!}\frac{\partial^{l}\bm{\pi}^{2}}{\partial
x_{0}^{l}}\int_{0}^{t}{\mathrm{d}}t_{1}\,\,\mathsf{b}_{\mp}^{l}(p_{0},\partial_{p_{0}},t_{1})\tilde{U}_{m-l,k-1}^{\pm}(t_{1}),$
(30)

with the initial value $\tilde{U}^{\pm}_{0,0}=e^{\pm\frac{ip_{0}^{2}t}{2}}$
and the operator valued function

$\mathsf{b}_{\mp}(p_{0},\partial_{p_{0}},t_{i})=i\partial_{p_{0}}\mp
p_{0}\,t_{i}\pm p_{0}\,t.$ (31)

Note that equations (29) - (31) indicate that the problem is effectively one
dimensional regardless the details of the background gauge field.

If the background field, on the other hand, is space dependent, then the
problem might be a multi-dimensional one. In this case,
${\mathbf{A}}(x_{0})=0$ , $\bm{\pi}={\mathbf{p}}$ and the perturbative
expansion becomes

$\Gamma^{\pm}_{\mathrm{space}}=-i\sum_{m=0}\int_{J^{\pm}_{0}}\frac{{\mathrm{d}}t}{t}e^{\mp
im^{2}t}\int\frac{{\mathrm{d}}^{4}p\,{\mathrm{d}}^{4}x}{(2\pi)^{4}}\,e^{\pm\frac{i\pi_{0}t}{2}}\,\sum_{k=1}^{m}\tilde{U}^{\pm}_{m,k}(t)\,e^{\mp\frac{i{\mathbf{p}}^{2}t}{2}},$
(32)

where $\tilde{U}^{\pm}$ is

$\displaystyle\tilde{U}_{m,k}^{\pm}(t)$
$\displaystyle=\frac{1}{2}\sum_{l=1}^{m-k+1}\frac{1}{l!}\nabla^{l}\pi_{0}^{2}\int_{0}^{t}{\mathrm{d}}t_{1}\,\,\mathsf{b}_{\pm}^{l}({\mathbf{p}},\nabla_{{\mathbf{p}}},t_{1})\,\tilde{U}_{m-l,k-1}^{\pm}(t_{1}),$
(33)

with $\tilde{U}_{0,0}^{\pm}=e^{\mp\frac{i{\mathbf{p}}^{2}t}{2}}$ and

$\mathsf{b}_{\pm}({\mathbf{p}},\nabla_{{\mathbf{p}}},t_{i})=i\nabla_{{\mathbf{p}}}\pm{\mathbf{p}}\,t_{i}\mp{\mathbf{p}}\,t.$
(34)

Structure of the Perturbative Expansions: The expressions (30) and (33) shows
that each expansion is organized as a derivative expansion on $\bm{\pi}^{2}$
and $\pi^{2}_{0}$. This suggests that the perturbative expansions of the
proper-time propagator in a double expansion in $\omega$ and $e\mathcal{E}$ as

$\frac{{\mathrm{Tr}}e^{\mp
itH}}{t}=\sum_{m,n}\alpha_{m,n}(t)\,\omega^{2m}(e\mathcal{E})^{2n}.$ (35)

Note that in (35), the order of $\omega$ expansion reflects the number of
derivatives applied to the gauge potentials $A_{0}({\mathbf{x}})$ or
${\mathbf{A}}(x_{0})$ at a given order in $e\mathcal{E}$ expansion, while the
order of $e\mathcal{E}$ expansion is equal to the number of gauge potentials
at that order. Then, for both time dependent and space dependent backgrounds,
at a fixed order in $\omega$, the expansion in $e\mathcal{E}$ can be organized
such that the series consists of
$\left(e\mathcal{E}\partial_{0}{\mathbf{A}}(x_{0})\right)^{2}$ and
$\left(e\mathcal{E}\nabla A({\mathbf{x}})\right)^{2}$ terms respectively,
while higher order derivatives on the background field appear as pre-factors
to these series.

We determine the per-factors for the first two order in $\omega$ expansion as
in the following. First observe that since the coefficients $\alpha_{m,n}$ are
computed upon integrating the series, which obtained recursively via (30) or
(33), over 4-momenta $(p_{0},{\mathbf{p}})$, all the terms with odd powers
$p_{0}$ or ${\mathbf{p}}$ will vanish. For example, $\tilde{U}_{1,1}^{\pm}$
term for the time dependent case reads from (30) as

$\displaystyle\tilde{U}_{1,1}^{\pm}(t)$
$\displaystyle=\frac{\partial_{0}\bm{\pi}^{2}}{2}\int_{0}^{t}{\mathrm{d}}t_{1}\,\mathsf{b}_{\mp}(p_{0},\partial_{p_{0}},t_{1})\,e^{\pm\frac{ip_{0}^{2}t}{2}}$
(36)
$\displaystyle=\mp\frac{p_{0}t^{2}}{2}\left({\mathbf{p}}-{\mathbf{A}}(x_{0})\right)\partial_{0}{\mathbf{A}}(x_{0})\,e^{\pm\frac{ip_{0}^{2}t}{2}}$
(37)

which is linear in both $p_{0}$ and ${\mathbf{p}}$ and through the 4-momentum
integration its contribution to (29) would vanish. Similarly for space
dependent case, we get

$\displaystyle\tilde{U}_{1,1}^{\pm}(t)$
$\displaystyle=\frac{\nabla\pi_{0}^{2}}{2}\int_{0}^{t}{\mathrm{d}}t_{1}\,\mathsf{b}_{\pm}({\mathbf{p}},\nabla_{\mathbf{p}},t_{1})\,e^{\mp\frac{i{\mathbf{p}}^{2}t}{2}}$
(38)
$\displaystyle=\pm\frac{{\mathbf{p}}\,t^{2}}{2}\left(p_{0}-A_{0}({\mathbf{x}})\right)\nabla
A_{0}({\mathbf{x}})\,e^{\mp\frac{i{\mathbf{p}}^{2}t}{2}}$ (39)

The first non-trivial contribution arises from $\tilde{U}_{2,1}^{\pm}(t)$
term. It is written as

$\displaystyle\tilde{U}_{2,1}^{\pm}(t)$
$\displaystyle=\frac{\partial_{0}^{2}\bm{\pi}^{2}}{2}\int_{0}^{t}{\mathrm{d}}t_{1}\,\mathsf{b}_{\mp}^{2}(p_{0},\partial_{p_{0}},t_{1})e^{\pm\frac{ip_{0}^{2}t}{2}}$
(40)
$\displaystyle=\left(\partial_{0}{\mathbf{A}}\cdot\partial_{0}{\mathbf{A}}-({\mathbf{p}}-{\mathbf{A}})\partial^{2}{\mathbf{A}}\right)Q^{\pm}_{\mathrm{time}}(p_{0},t)$
(41)

for the time dependent case and

$\displaystyle\tilde{U}_{2,1}^{\pm}(t)$
$\displaystyle=\frac{\nabla^{2}\pi_{0}^{2}}{2}\int_{0}^{t}{\mathrm{d}}t_{1}\,\mathsf{b}_{\pm}^{2}({\mathbf{p}},\nabla_{\mathbf{p}},t_{1})e^{\mp\frac{i{\mathbf{p}}^{2}t}{2}}$
(42) $\displaystyle=\left(\nabla A_{0}\cdot\nabla
A_{0}-(p_{0}-A_{0})\nabla^{2}A_{0}\right)Q^{\pm}_{\mathrm{space}}({\mathbf{p}},t)$
(43)

for the space dependent case. For both cases, only the first terms contribute
to the effective action (29) and (32), while the second terms vanish upon
momentum integrals and $\tilde{U}_{2,1}^{\pm}$ contributes only to the leading
order in LCFA, i.e. $\omega^{0}$. Higher order terms in $e\mathcal{E}$
expansion can be obtained in a similar manner and throughout the series all
the space dependence manifests itself as $|\partial_{0}{\mathbf{A}}|^{2n}$ and
$|\nabla A_{0}|^{2n}$ while odd terms vanishes through momentum integrals as
in (37) and (39).

First non-vanishing corrections to the leading order expansion arises from
$\tilde{U}_{4,2}^{\pm}$ term as

$\displaystyle\tilde{U}_{4,2}^{\pm}(t)$
$\displaystyle=\sum_{l=1}^{3}\partial^{l}_{0}\bm{\pi}^{2}\int_{0}^{t}{\mathrm{d}}t_{1}\,\mathsf{b}^{l}_{\mp}(p_{0},\partial_{p_{0}},t_{1})\,U_{3,l}$
(44)
$\displaystyle=\sum_{l=0}^{2}\left(\partial_{0}^{l+1}\bm{\pi}^{2}\cdot\partial_{0}^{3-l}\bm{\pi}^{2}\right)Q^{\pm,(l)}_{\mathrm{time}}(p_{0},t)$
(45)

for time dependent fields and

$\displaystyle\tilde{U}_{4,2}^{\pm}(t)$
$\displaystyle=\sum_{l=1}^{3}\nabla^{l}\pi_{0}^{2}\int_{0}^{t}{\mathrm{d}}t_{1}\,\mathsf{b}^{l}_{\pm}({\mathbf{p}},\nabla_{{\mathbf{p}}},t_{1})\,U_{3,l}$
(46)
$\displaystyle=\sum_{l=0}^{2}\left(\nabla^{l+1}\pi_{0}^{2}\cdot\nabla^{3-l}\pi_{0}^{2}\right)Q^{\pm,(l)}_{\mathrm{space}}({\mathbf{p}},t)$
(47)

for space dependent fields.

The spatial dependence in (45) and (47) appears as
$(\partial_{0}{\mathbf{A}}\cdot\partial_{0}^{3}{\mathbf{A}})$,
$(\partial_{0}^{2}{\mathbf{A}}\cdot\partial_{0}^{2}{\mathbf{A}})$ and $(\nabla
A_{0}\cdot\nabla^{3}A_{0})$ and $(\nabla^{2}A_{0}\cdot\nabla^{2}A_{0})$
respectively. At order $\omega^{2}$, all the higher order terms in
$e\mathcal{E}$ expansion contain these terms along with the terms appear at
$\omega^{0}$ order, i.e. $|\partial_{0}{\mathbf{A}}|^{2n}$ and $|\nabla
A_{0}|^{2n}$. This will help us to organized the series expansions before
performing the Pade summation.

Phase Space Integrals: Before summing the series at each $\omega$ order, we
first take the momentum integrals. At this point, it is appropriate to shift
the relevant parts of the canonical momentum as
${\mathbf{p}}\rightarrow{\mathbf{p}}+{\mathbf{A}}(x_{0})$ in the temporal case
and $p_{0}\rightarrow p_{0}+A_{0}({\mathbf{x}})$ in the spatial case. In this
way, we eliminate all the terms with momentum and spatial variables which are
coupled to each other after the initial expansion. Then, all momentum
integrals become Gaussian and it is straightforward to compute them order by
order in (29) and (32). Note that each Gaussian integral will have the same
pre-factor:

$\frac{1}{\left(\sqrt{\pm 2\pi it}\right)^{3}\sqrt{\mp 2\pi
it}}=\pm\frac{1}{4\pi^{2}it^{2}},$ (48)

where $(+)$ sign is for the forward proper-time evolution, while $(-)$ sign is
for the backward evolution. This is different than the standard prefactor of
$4$ dimensional Gaussian integrals, i.e. $\left(\frac{1}{\sqrt{\pm 2\pi
it^{2}}}\right)^{4}=-\frac{1}{4\pi^{2}t^{2}}$. The difference is due to the
sign change in the Lorentzian metric and overall imaginary factor is important
as it contributes to the effective action directly.

Summation: The poles at $t\neq 0$, which are the sources of the pair
production probability as we discussed in the constant field case, are related
to the summation of (35). In the limit $e\mathcal{E}\gg\omega$, which we are
interested in this paper, it is convenient to sum $e\mathcal{E}$ series at
each order of the $\omega$ expansion. Therefore, after the momentum integrals,
we will sum $e\mathcal{E}$ expansion via Pade method for orders
$O(\omega^{0})$ and $O(\omega^{2})$ separately to compute the imaginary
contributions to the effective action which are the non-perturbative pair
production probabilities at the corresponding orders up to the spatial
integral which we will deal last.

Space-time Integrals: After the summation, the spatial integrals can be also
computed perturbatively or, in some special cases, exactly when the gauge
potentials $A_{0}({\mathbf{x}})$ or ${\mathbf{A}}(x_{0})$ are known. In the
remaining of this section, we will consider general gauge potentials for both
temporal and spatial cases under some general assumptions which we will state
in each case separately. Then, we will handle the spatial integrals via the
saddle point approximation which will be possible due to the relation $m\gg
e\mathcal{E}\gg\omega$. Note that in the temporal case, the spatial integral
is one dimensional as ${\mathbf{A}}(x_{0})$ has one dimensional space-time
dependence. On the other hand, the dimension $D$ of the space integral depends
on the dimensionality of $A_{0}({\mathbf{x}})$. Even though it does not change
the exponential part of the pair production probability, it plays a role in
prefactors. We will elaborate this, when we discuss the space dependent
fields.

###  3.1 Time Dependent Electric Fields:

We first consider time dependent electric fields for

${\mathbf{A}}(x_{0})=\frac{\mathcal{E}}{\omega}\mathbf{H}(\omega x_{0}).$ (49)

As we stated above, this is effectively a one-dimensional problem and in
principle it can be handled by exact WKB methods as in Taya:2020dco . Here
using the recursion relation and the Pade summation of perturbative expansion
of the propagator, we will obtain unambiguous version of the pair production
probability. Our formulation also indicates a possible relationship between
Schwinger’s formalism and the WKB formulation. We will return this relation
shortly at the end of this paper.

Notation: As we mentioned before and it is apparent from (49), the order in
$\omega$ expansion is equal to the number of derivative acting on
${\mathbf{A}}(x_{0})$. To follow the order of $\omega$ properly, we introduce
the following notation for the derivative of the gauge potential in (25):

$\frac{\partial^{n}{\mathbf{A}}(x_{0})}{\partial
x_{0}^{n}}=\mathcal{E}\omega^{n-1}\mathbf{H}^{(n)}(\omega x_{0})$ (50)

where with $\mathbf{H}^{(n)}(\omega x_{0})$ we mean the vector corresponding
to the $n^{\mathrm{th}}$ derivative evaluated at $x_{0}$. Along with the
assumption $m\gg e\mathcal{E}\gg\omega$ , throughout this subsection, we will
assume that $\mathbf{H}^{(1)}(\omega x_{0})$ is slowly varying and it takes
the following values at any of its saddle point which we denote as
$x_{0}^{\\{i\\}}$:

$i)\,\mathbf{H}^{(1)}(\omega
x_{0}^{\\{i\\}})=\mathbf{h}^{\\{i\\}}_{1}\quad,\quad
ii)\,\mathbf{H}^{(2)}(\omega x_{0}^{\\{i\\}})=0\quad,\quad
iii)\,\mathbf{H}^{(3)}(\omega
x_{0}^{\\{i\\}})=-\omega^{2}\mathbf{h}_{3}^{\\{i\\}},$ (51)

where both $\mathbf{h}_{1}^{\\{i\\}}$ and $\mathbf{h}_{3}^{\\{i\\}}$ are
vectors evaluated at $x_{0}^{\\{i\\}}$. Note that first two assumptions
indicates the locally constant behaviour of $\mathbf{H}(\omega x_{0})$ and
states that the electric field is ${\mathbf{E}}=\mathcal{E}=\mathrm{constant}$
at the saddle points $x_{0}^{\\{i\\}}$, while third assumption implies the
saddle points $x_{0}^{\\{i\\}}$ are non-degenerate.

#### Leading Order:

Using the recursion formula (30), we obtained high order expansion at order
$O(\omega^{0})$ as

$\sum_{n=0}^{\infty}\alpha^{\pm}_{0,n}(te\mathcal{E}|\mathbf{H}^{(1)}|)^{2n}$
(52)

where $|\mathbf{H}^{(1)}|=\sqrt{\frac{\partial\mathbf{H}}{\partial
x_{0}}\cdot\frac{\partial\mathbf{H}}{\partial x_{0}}}$. Before summing the
series, it is convenient to re-scale
$t\rightarrow\frac{t}{e\mathcal{E}|\mathbf{H}^{(1)}|}$ and express the
effective action at order $O(\omega^{0})$, as

$\Gamma^{\pm}_{\mathrm{E},\omega^{0}}=-{\mathcal{V}}_{3}\frac{(e\mathcal{E}|\mathbf{H}^{(1)}|)^{2}}{4\pi^{2}}\int_{\mathbb{R}}{\mathrm{d}}x_{0}\,\int_{J^{\pm}_{0}}\frac{{\mathrm{d}}t}{t^{3}}\,e^{\mp
i\frac{m^{2}t}{e\mathcal{E}|\mathbf{H}^{(1)}|}}\,\mathcal{S}_{\omega^{0}}^{\pm}(t)$
(53)

where ${\mathcal{V}}_{3}$ is the 3-dimensional volume associated to the
integration over ${\mathbf{x}}$ and

$\mathcal{S}_{\omega^{0}}^{\pm}=\sum_{n=0}^{\infty}\alpha^{\pm}_{0,n}t^{2n}$
(54)

is the series with the coefficients $\alpha_{0,n}^{\pm}$ being rational
numbers. First 7 terms of the series are

$\left\\{1,-\frac{1}{24},\frac{7}{5760},-\frac{31}{967680},\frac{127}{154828800},-\frac{73}{3503554560},\frac{1414477}{2678117105664000}\right\\}$
(55)

This expansion is equal to the expansion of the denominator of the integrand
in (16) as expected. However, since the gauge potential is a space dependent
function, the final expression for the pair production probability will have a
small but significant difference with the uniform case probability in (21).

First two terms in the expansion in (53) are singular at $t=0$. They are the
UV divergent terms and can be handled by renormalization methods. On the other
hand, the series also contains non-singular term which are associated to
singularities at $t\neq 0$ upon their summation. Note that this series is a
convergent one. It becomes divergent if we take the proper time integral
directly. This is in fact the divergence predicted by Dyson in PhysRev.85.631
and it would need to be treated by the Borel procedure. Instead, similar to
the uniform case in Section 2.1, it is possible to probe the singularity
structure represented by this series directly by using the Pade summation
before taking the proper time integral.

In computations of the pair-production probability, it is possible to ignore
the UV divergent terms and perform Pade approximation when probing the poles
at finite $t$. However, we observe that they affect the pre-factor as the
$\frac{1}{t^{3}}$ term effects the residue around the singularities at $t\neq
0$. For this we keep them in the following calculations. As in the constant
field case, a proper way to consider this residue problem would be
regularizing the proper-time integral in (53) by a cut-off in its lower limit.
Then, residues so that the pair-production probabilities can be computed
without dealing with the UV divergence.

Computing the series $S_{\omega^{0}}^{\pm}$ up to $O\left(t^{50}\right)$ and
summing it via Pade method with precision $O(10^{-8})$ leads to 11
singularities for both cases at

$t^{\pm}_{\star}=\pm 2\pi in\quad,\quad n={1,\dots,11}.$ (56)

which matches with the exact result. Note that for $n>11$ the locations of the
singularities start to deviate from the exact result and needs high order term
for better approximations.

As we indicated in the previous section, only the first pole is linked to the
pair production probability. Therefore, we are only interested in the residue
of the leading poles at $t^{\pm}_{\star}=\pm 2\pi i$. In addition to this, as
in the uniform case, the sign of the imaginary parts are fixed by the
analytical continuation directions of the proper-time contour and
$\mathrm{Im}\Gamma^{+}$ gets contribution only from $t_{\star}^{-}$ while
$\mathrm{Im}\Gamma^{-}$ arises from $t_{\star}^{+}$. The residues for both
cases are the same:

$R^{\pm}_{\omega^{0}}=0.02533029\simeq\frac{1}{4\pi^{2}}.$ (57)

and taking the analytical continuation directions into account, we find the
imaginary part as

$\mathrm{Im}\Gamma^{\pm}_{\mathrm{E},\omega^{0}}=\pm{\mathcal{V}}_{3}\int_{\mathbb{R}}{\mathrm{d}}x_{0}\,\frac{\left(e\mathcal{E}|\mathbf{H}^{(1)}|\right)^{2}}{16\pi^{3}}\,e^{-\frac{2\pi
m^{2}}{e\mathcal{E}|\mathbf{H}^{(1)}|}}.$ (58)

This is the general expression of the leading order derivative expansion for a
time dependent electric field and compatible with the general result in
Gusynin:1995bc ; Gusynin:1998bt . The uniform case in Section 2.1 is just the
special case where $|\mathbf{H}^{(1)}|=1$.

The remaining integral in (58) can be evaluated exactly for specific cases.
(See e.g. Dunne:1999uy .) For general background fields, on the other hand, it
is possible to handle the integral by saddle point approximation, which is
appropriate since $m\gg e\mathcal{E}$ and $\mathbf{H}(\omega x_{0})$ is chosen
to be slowly varying. Then, using the assumptions given in (51), the leading
order saddle point approximation leads to the leading order pair production
probability as

$\mathrm{Im}\Delta\Gamma_{\mathrm{E},\omega^{0}}(m^{2})\simeq{\mathcal{V}}_{3}\sum_{i}\frac{(e\mathcal{E})^{5/2}\,|\mathbf{h}^{i}_{1}|^{3}}{8\pi^{3}m\,\omega}\sqrt{\frac{|\mathbf{h}_{1}^{\\{i\\}}|}{\mathbf{h}_{1}^{\\{i\\}}\cdot\mathbf{h}_{3}^{\\{i\\}}}}\,e^{-\frac{2\pi
m^{2}}{e\mathcal{E}|\mathbf{h}^{(i)}_{1}|}},$ (59)

where $\mathbf{h}^{i}_{1}$ and $\mathbf{h}^{i}_{3}$ can be determined for
specific background fields. Note that as we stated before, the prefactor in
(59) differs from the expression in uniform electric field case in (21) and
these two results are not equal to each other in $\omega\rightarrow 0$,
$\mathbf{h}_{1}^{\\{i\\}}\rightarrow 0$ and
$\mathbf{h}_{3}^{\\{i\\}}\rightarrow 0$ limits as (59) diverges in these
limits. Instead, one should take the limit before handling the space integral
in (58).

#### Next to Leading Order:

At the order $O(\omega^{2})$, the expansion (35) can be organized in two
parts:

$(e\mathcal{E}\omega)^{2}\,\mathbf{H}^{(3)}\cdot\mathbf{H}^{(1)}\sum_{n=0}^{\infty}\alpha^{\pm,{(1)}}(te\mathcal{E}\mathbf{H}^{(1)}\cdot
te\mathcal{E}\mathbf{H}^{(1)})^{n}$ (60)

and

$(e\mathcal{E}\omega)^{2}\,\mathbf{H}^{(2)}\cdot\mathbf{H}^{(2)}\sum_{n=0}^{\infty}\alpha^{\pm,{(2)}}\left(te\mathcal{E}\mathbf{H}^{(1)}\cdot
te\mathcal{E}\mathbf{H}^{(1)}\right)^{n}$ (61)

Then, re-scaling $t\rightarrow\frac{t}{e\mathcal{E}|\mathbf{H}^{(1)}|}$ as
before, we express $O(\omega^{2})$ correction to the effective actions as

$\displaystyle\Gamma^{\pm}_{\mathrm{E},\omega^{2}}=-i\,\omega^{2}{\mathcal{V}}_{3}\int{\mathrm{d}}x_{0}\,\int_{J^{\pm}_{0}}\frac{{\mathrm{d}}t}{4\pi^{2}}\,e^{\mp\frac{im^{2}t}{e\mathcal{E}|\mathbf{H}^{(1)}|}}\left[\frac{e\mathcal{E}\mathbf{H}^{(3)}\cdot\mathbf{H}^{(1)}}{|\mathbf{H}^{(1)}|}\,\mathcal{S}^{\pm}_{\omega^{2},1}(t)+\frac{e\mathcal{E}|\mathbf{H}^{(2)}|^{2}}{|\mathbf{H}^{(1)}|}\,\mathcal{S}_{\omega^{2},2}^{\pm}(t)\right]$
(62)

where

$\mathcal{S}_{\omega^{2},1}^{\pm}(t)=\sum_{n=0}^{\infty}\alpha_{2,n}^{\pm,(1)}t^{2n}\quad,\quad\mathcal{S}_{\omega^{2},2}^{\pm}(t)=\sum_{n=0}^{\infty}\alpha^{\pm,(2)}_{2,n}t^{2n}$
(63)

with $\alpha_{2,n}^{\pm,(1)}$ and $\alpha_{2,n}^{\pm,(2)}$ being rational
numbers. Again we provide first few terms for $\alpha_{2,n}^{\pm,(1)}$:

$\pm\left\\{\frac{1}{120},-\frac{13}{20160},\frac{157}{4838400},\frac{577}{425779200},-\frac{2844701}{55794106368000},\frac{2401579}{1339058552832000}\right\\}$
(64)

and for $\alpha_{2,n}^{\pm,(2)}$:

$\pm\left\\{\frac{1}{160},-\frac{103}{80640},\frac{2039}{19353600},-\frac{1163}{189235200},\frac{66417431}{223176425472000},-\frac{6241699}{486930382848000}\right\\}.$
(65)

Using the terms up to $O(t^{48})$, we approximated the singularity structure
associated to both series and found the leading poles again at
$t^{\pm}_{\star}=\pm 2\pi i$ with $O(10^{-8})$ precision. Note that contrary
to the leading order poles, these are not simple poles. Instead, the
singularity for associated to $\mathcal{S}^{\pm}_{\omega^{2},1}(t)$ is a
$3^{\mathrm{rd}}$ order pole while the one for
$\mathcal{S}^{\pm}_{\omega^{2},2}(t)$ is a $4^{\mathrm{th}}$ order one. Then,
together with the prefactors, we get the leading order behaviour of the
residues as

$R^{\pm}_{\omega^{2},1}\simeq-i\,0.11936621\,\frac{m^{4}}{(e\mathcal{E}|\mathbf{H}^{(1)}|)^{2}}e^{-\frac{2\pi
m^{2}}{e\mathcal{E}|\mathbf{H}^{(1)}|}}\simeq-\frac{3im^{4}}{8\pi\,(e\mathcal{E}|\mathbf{H}^{(1)}|)^{2}}\,e^{-\frac{2\pi
im^{2}}{e\mathcal{E}|\mathbf{H}^{(1)}|}}$ (66)

for the series $\mathcal{S}_{\omega^{2},1}^{\pm}(t)$ and

$R^{\pm}_{\omega^{2},2}\simeq\mp
0.50000000\,\frac{m^{6}}{(e\mathcal{E}|\mathbf{H}^{(1)}|)^{3}}\,e^{-\frac{2\pi
m^{2}}{e\mathcal{E}|\mathbf{H}^{(1)}|}}\simeq\mp\frac{m^{6}}{2(e\mathcal{E}|\mathbf{H}^{(1)}|)^{3}}\,e^{-\frac{2\pi
m^{2}}{e\mathcal{E}|\mathbf{H}^{(1)}|}}$ (67)

for the series $\mathcal{S}_{\omega^{2},2}^{\pm}$. Finally, considering the
proper analytical continuation directions, we obtained the discontinuity for
the effective action at order $O(\omega^{2})$ up $x_{0}$ integral as

$\mathrm{Im}\Delta\Gamma_{\mathrm{E},\omega^{2}}\simeq{\mathcal{V}}_{3}\frac{m^{6}\omega^{2}}{4\pi(e\mathcal{E})^{2}}\int{\mathrm{d}}x_{0}\,\frac{\mathbf{H}^{(2)}\cdot\mathbf{H}^{(2)}}{|\mathbf{H}^{(1)}|^{4}}\,e^{-\frac{2\pi
m^{2}}{e\mathcal{E}|\mathbf{H}^{(1)}|}}$ (68)

which only arises from the pole associated to the series
$S^{\pm}_{\omega^{2},2}$.

As in the computation of the leading order term, we are going to handle the
$x_{0}$ integral via the saddle point approximation based on the assumptions
$m\gg e\mathcal{E}\mathbf{H}^{(1)}(\omega x_{0})$ and the behaviour of
$\mathbf{H}^{(1)}(\omega x_{0})$ around its saddle points given in (51). The
first order saddle point approximation to (68) vanishes since
$\mathbf{H}^{(2)}(\omega x_{0}^{i})=0$ for any saddle point $x_{0}^{i}$ and
first non-zero term comes from the second order in the saddle point
approximation. Although the second order approximation is more complicated
than the first order (see e.g. bender1999advanced ), due to our assumption in
(51), only one term contributes to
$\mathrm{Im}\Delta\Gamma^{(2)}_{\mathrm{E},\omega^{2}}$ and we found

$\mathrm{Im}\Delta\Gamma^{(2)}_{\mathrm{E},\omega^{2}}\simeq{\mathcal{V}}_{3}\sum_{i}\frac{\omega
m^{3}}{8\pi^{2}\sqrt{e\mathcal{E}}}\frac{|\mathbf{h}_{3}^{\\{i\\}}|^{2}\,|\mathbf{h}_{1}^{\\{i\\}}|^{1/2}}{\left(\mathbf{h}_{3}^{\\{i\\}}\cdot\mathbf{h}_{1}^{\\{i\\}}\right)^{3/2}}\,e^{-\frac{2\pi
m^{2}}{e\mathcal{E}|\mathbf{h}_{0}^{i}|}}.$ (69)

Note that contrary to singular behaviour of the leading order imaginary
contribution in (59), the expression in (69) vanishes in the uniform case.

###  3.2 Space Dependent Electric Fields:

Now, we will consider space-dependent pure electric backgrounds for

$A_{0}({\mathbf{x}})=\frac{\mathcal{E}}{\omega}H_{0}(\omega{\mathbf{x}}).$
(70)

Since the effective dimensionality of this case depends on the details of the
function $H_{0}$, its treatment with WKB methods is not always tractable.
Therefore, the (proper) time dependent method we present here is much more
convenient for applications.

Notation: As in the time dependent case, the number of derivatives acting on
$A_{0}({\mathbf{x}})$ counts the order in $\omega$ expansion. However, now,
due to the multi-dimensional nature of $A_{0}({\mathbf{x}})$, a tensorial
notation will be convenient when we tackle the saddle point approximation. For
this reason we introduce the following notation:

$\frac{\partial^{n}\nabla A_{0}({\mathbf{x}})}{\partial x_{k_{1}}\partial
x_{k_{2}}\dots\partial
x_{k_{n}}}=\mathcal{E}\omega^{n}\partial_{k_{1}}\partial_{k_{2}}\dots\partial_{k_{n}}\nabla
H_{0}(\omega{\mathbf{x}})$ (71)

where $\partial_{k_{1}}\partial_{k_{2}}\dots\partial_{k_{n}}\nabla
H_{0}(\omega{\mathbf{x}})$ is $n^{\mathrm{th}}$ order directional derivatives
of $\nabla H_{0}$ evaluated at ${\mathbf{x}}$. Note that we keep the first
order derivative in vector notation rather than tensor notation since only
$|\nabla H_{0}|$ appear in the exponent of the spatial integrals. Similar to
the time-dependent case, we assume the following properties for the
directional derivatives of $\nabla H_{0}$

$\displaystyle i)\,\nabla H_{0}(\omega{\mathbf{x}}^{i})$
$\displaystyle={\mathbf{f}}^{\\{i\\}}\quad,\quad ii)\,\partial_{k_{1}}\nabla
H_{0}(\omega{\mathbf{x}}^{i})=0$ (72) $\displaystyle iii)$
$\displaystyle\,\partial_{k_{1}}\partial_{k_{2}}\nabla
H_{0}(\omega{\mathbf{x}}^{i})={\mathbf{f}}_{k_{1},k_{2}}^{\\{i\\}}.$ (73)

where ${\mathbf{f}}^{\\{i\\}}$ is a scalar corresponding to the value of
$\nabla H_{0}$ at a saddle point ${\mathbf{x}}^{i}$ and
${\mathbf{f}}_{k_{1},k_{2}}$ is a vector corresponding to an element of the
matrix333To be clear, the $2^{{\mathrm{nd}}}$ derivatives of $\nabla
H_{0}(\omega{\mathbf{x}})$ actually form a third rank tensor but since we will
only need the vector valued function $f^{\\{i\\}}$ in future calculations, we
kept the matrix form. of $\nabla H_{0}$ evaluated at the same point.

#### Leading Order:

At the leading order, using the recursive relations (30), we get the coupling
expansion at order $O(\omega^{0})$ as

$\sum_{n=0}^{\infty}\alpha_{0,n}^{\pm}(te\mathcal{E}\nabla H_{0}\cdot
te\mathcal{E}\nabla H_{0})^{n}$ (74)

where the coefficients are the same with the ones in (55).

Then, as in the time dependent case, rescaling
$t\rightarrow\frac{t}{e\mathcal{E}|\nabla H_{0}|}$, we express the effective
actions $\Gamma^{\pm}_{\mathrm{E},\omega^{0}}$ as

$\Gamma^{\pm}_{\mathrm{E},\omega^{0}}=-{\mathcal{V}}_{\tilde{D}}\frac{(e\mathcal{E}|\nabla
H_{0}|)^{2}}{4\pi^{2}}\int_{\mathbb{R}^{D}}{\mathrm{d}}^{D}x\,\int_{J^{\pm}_{0}}\frac{{\mathrm{d}}t}{t^{3}}\,e^{\mp\frac{im^{2}t}{e\mathcal{E}|\nabla
H_{0}|}}\,\mathcal{S}_{\omega^{0}}^{\pm}(t).$ (75)

where $D=1,2,3$ depending to the potential $H_{0}(\omega{\mathbf{x}})$ and
$\tilde{D}=4-D$ is the dimension of transverse directions including the time
direction. Since the expansion is the same with the time dependent case,
approximating it via Pade method leads to the leading pole at
$t_{\star}^{\pm}=\pm 2\pi in$ with residue

$R^{\pm}_{\omega^{0}}\simeq\frac{1}{4\pi^{2}}.$

Then, taking $t$ integral with proper analytical continuations leads to the
same expression with the time dependent case up to the spatial dependence

$\mathrm{Im}\Gamma^{\pm}_{\mathrm{E},\omega^{0}}=\pm{\mathcal{V}}_{\tilde{D}}\int_{\mathbb{R}^{D}}{\mathrm{d}}^{D}x\,\frac{\left(e\mathcal{E}|\nabla
H_{0}|\right)^{2}}{16\pi^{3}}\,e^{-\frac{2m^{2}\pi}{e\mathcal{E}|\nabla
H_{0}|}}.$ (76)

Finally, we handle the $D$-dimensional space integral via the saddle point
approximation which leads to

$\mathrm{Im}\Delta\Gamma_{\mathrm{E},\omega^{0}}\simeq{\mathcal{V}}_{\tilde{D}}\sum_{i}\frac{(e\mathcal{E}|{\mathbf{f}}^{\\{i\\}}|)^{2}}{8\pi^{3}}\left(\frac{e\mathcal{E}|{\mathbf{f}}^{\\{i\\}}|^{3}}{m^{2}\omega^{2}}\right)^{D/2}\,e^{-\frac{2m^{2}\pi}{e\mathcal{E}|{\mathbf{f}}^{\\{i\\}}|}}\left(\det\bm{\eta}^{\\{i\\}}\right)^{-1/2}$
(77)

where

$\bm{\eta}^{\\{i\\}}_{k_{1},k_{2}}={\mathbf{f}}_{k_{1},k_{2}}^{\\{i\\}}\cdot{\mathbf{f}}^{\\{i\\}}$
(78)

is the matrix elements of $|\nabla H_{0}|=\sqrt{\nabla H_{0}\cdot\nabla
H_{0}}$ which are evaluated at a saddle point $x=x^{\\{i\\}}$.

#### Next to Leading Order:

As in the time dependent case, we organized the expansions at order
$O(\omega^{2})$ in two parts:

$(e\mathcal{E}\omega)^{2}\,\nabla^{3}H_{0}\cdot\nabla
H_{0}\sum_{n=0}^{\infty}i\alpha^{\mp,{(1)}}(te\mathcal{E}\nabla H_{0}\cdot
te\mathcal{E}\nabla H_{0})^{n}$ (79)

and

$(e\mathcal{E}\omega)^{2}\,\left(\nabla^{2}\,H_{0}\right)^{2}\sum_{n=0}^{\infty}i\alpha^{\mp,{(2)}}\left(te\mathcal{E}\nabla
H_{0}\cdot te\mathcal{E}\nabla H_{0}\right)^{n}$ (80)

and again re-scaling $t\rightarrow\frac{t}{e\mathcal{E}|\nabla H_{0}|}$, we
get

$\displaystyle\Gamma^{\pm}_{\mathrm{E},\omega^{2}}=-i\omega^{2}{\mathcal{V}}_{\tilde{D}}\int{\mathrm{d}}^{D}x\,\int_{0}^{\infty}\frac{{\mathrm{d}}t}{4\pi^{2}}\,e^{\mp\frac{im^{2}t}{e\mathcal{E}|\nabla
H_{0}|}}\left[\frac{e\mathcal{E}\nabla(\nabla^{2}H_{0})\cdot\nabla
H_{0}}{|\nabla
H_{0}|}\,\mathcal{S}^{\mp}_{\omega^{2},1}(t)+\frac{e\mathcal{E}\nabla^{2}H_{0}}{|\nabla
H_{0}|}\,\mathcal{S}_{\omega^{2},2}^{\mp}(t)\right]$ (81)

where $\mathcal{S}_{\omega^{2},1}$ and $\mathcal{S}_{\omega^{2},2}$ are
written in (63) with the same coefficients in (64) and (65) respectively but
opposite signs. The latter leads to the sign difference in the correction to
the leading order contribution with respect to the time dependent case.

Pade approximation to (81) leads to a similar expression with (68)

$\mathrm{Im}\Delta\Gamma_{\mathrm{E},\omega^{2}}\simeq-{\mathcal{V}}_{\tilde{D}}\frac{m^{6}\omega^{2}}{4\pi(e\mathcal{E})^{2}}\int{\mathrm{d}}^{D}x\,\frac{|\nabla^{2}H_{0}|}{|\nabla
H_{0}|^{4}}\,e^{-\frac{2\pi m^{2}}{e\mathcal{E}|\nabla H_{0}|}}.$ (82)

Finally, in view of the assumptions in (73), we get the first non-vanishing
term as

$\mathrm{Im}\Delta\Gamma_{\mathrm{E},\omega^{2}}^{(2)}\simeq{\mathcal{V}}_{\tilde{D}}\frac{m^{4}\omega^{2}\,}{8\pi^{2}e\mathcal{E}|{\mathbf{f}}^{\\{i\\}}|}\left(\frac{e\mathcal{E}|{\mathbf{f}}^{\\{i\\}}|^{3}}{m^{2}\omega^{2}}\right)^{D/2}\frac{\bm{\Phi}^{\\{i\\}}\cdot(\bm{\eta}^{\\{i\\}})^{-1}}{\left(\det\bm{\eta}^{\\{i\\}}\right)^{1/2}}\,e^{\frac{-2m^{2}\pi}{e\mathcal{E}|{\mathbf{f}}^{\\{i\\}}|}},$
(83)

where $\eta$ is the matrix defined in (78) and $\tilde{\bm{\Phi}}^{\\{i\\}}$
is a matrix whose elements are given as

$\bm{\Phi}^{\\{i\\}}=\sum_{l}{\mathbf{f}}^{\\{i\\}}_{lk_{1}}\cdot{\mathbf{f}}^{\\{i\\}}_{lk_{2}}.$

###  3.3 Comments and Comparisons:

Now combining our results in (59) and (69), the non-perturbative pair-
production probability, i.e.
$\mathcal{P}_{\mathrm{E}}=\frac{\Delta\Gamma_{\mathrm{E}}}{{\mathcal{V}}_{\tilde{D}}}$,
for a general time dependent electric field is written as

$\displaystyle\mathcal{P}_{\mathrm{time}}$
$\displaystyle\simeq\sum_{i}\frac{(e\mathcal{E})^{5/2}}{8\pi^{3}m\omega}\sqrt{\frac{|\mathbf{h}_{1}^{\\{i\\}}|}{\mathbf{h}_{1}^{\\{i\\}}\cdot\mathbf{h}_{3}^{\\{i\\}}}}\left[|\mathbf{h}^{i}_{1}|^{3}+\frac{m^{4}\omega^{2}\pi}{(e\mathcal{E})^{3}}\frac{|\mathbf{h}_{3}^{\\{i\\}}|^{2}}{\mathbf{h}_{3}^{\\{i\\}}\cdot\mathbf{h}_{1}^{\\{i\\}}}\right]e^{-\frac{2m\pi}{e\mathcal{E}|\mathbf{h}_{1}^{i}|}}.$
(84)

Similarly for the space dependent case, from (77) and (83) we get,

$\displaystyle\mathcal{P}_{\mathrm{space}}$
$\displaystyle\simeq\sum_{i}\frac{(e\mathcal{E})^{2}}{8\pi^{3}|{\mathbf{f}}^{\\{i\\}}|}\left(\frac{e\mathcal{E}|{\mathbf{f}}^{\\{i\\}}|}{m^{2}\omega^{2}}\right)^{D/2}\,\frac{|{\mathbf{f}}^{\\{i\\}}|^{3}-\frac{m^{4}\omega^{2}\pi}{(e\mathcal{E})^{3}}\bm{\Phi}^{\\{i\\}}\cdot(\bm{\eta}^{\\{i\\}})^{-1}}{\left(\det\bm{\eta}^{\\{i\\}}\right)^{-1/2}}\,e^{-\frac{2\pi
m^{2}}{e\mathcal{E}|{\mathbf{f}}^{\\{i\\}}|}}$ (85)

As a consistency check, we consider the one dimensional limits of both cases.
Let the background electric field to be one dimensional and is directed along
the $x_{1}$-axis. Then, the gauge fields are simplified to

${\mathbf{A}}(x_{0})=\frac{\mathcal{E}}{\omega}\left(H_{1}(\omega
x_{0}),0,0\right)$

and

$A_{0}({\mathbf{x}})=\frac{\mathcal{E}}{\omega}H_{0}(\omega x_{1}).$

Then, setting
$\mathbf{h}_{1}^{\\{i\\}}={\mathbf{f}}^{\\{i\\}}=h_{1}^{\\{i\\}}$ and
$\mathbf{h}_{3}^{\\{i\\}}={\mathbf{f}}_{k_{1}k_{2}}^{\\{i\\}}=h_{1}^{\\{i\\}}$
in the one dimensional limit (84) and (85) simplifies to

$\displaystyle\mathcal{P}^{\pm}_{1{\mathrm{D}}}\simeq\sum_{i}\frac{(e\mathcal{E})^{5/2}}{8\pi^{3}m\omega\sqrt{h_{3}^{\\{i\\}}}}\left[(h_{1}^{\\{i\\}})^{3}\pm\frac{m^{4}\omega^{2}\pi}{2(e\mathcal{E})^{3}}\frac{h_{3}^{\\{i\\}}}{h_{1}^{\\{i\\}}}\right]\,e^{-\frac{2\pi
m^{2}}{e\mathcal{E}h_{1}^{\\{i\\}}}},$ (86)

where $\mathcal{P}^{+}_{1{\mathrm{D}}}$ and $\mathcal{P}_{1{\mathcal{D}}}^{-}$
refers to time and space dependent cases respectively.

#### Comparison to Worldline Instantons:

We can compare (86) with the well-known worldline instanton calculations
Dunne:2005sx ; Dunne:2006st  for the periodic and single-pulse background
potentials: For the periodic case, i.e.
$A(x)=\frac{\mathcal{E}}{\omega}\sin(\omega x)$, the saddle point relevant to
the pair production rate is at $x=0$ where $h_{1}=1$ and $h_{3}=1$. Then, we
get

$\mathcal{P}_{\mathrm{periodic}}\simeq\frac{(e\mathcal{E})^{5/2}}{8\pi^{3}m\omega}\left(1\pm\frac{m^{4}\omega^{2}\pi}{2(eE)^{3}}\right)e^{-\frac{2\pi
m^{2}}{e\mathcal{E}}}.$ (87)

For the single-pulse case, i.e. $A(x)=\frac{\mathcal{E}}{\omega}\tanh(\omega
x)$, again the saddle point approximation is dominated by $x=0$ where
$h_{1}=1$ and $h_{3}=2$ and we get

$\mathcal{P}_{\mathrm{pulse}}\simeq\frac{(e\mathcal{E})^{5/2}}{8\pi^{3}m\omega\sqrt{2}}\left(1\pm\frac{m^{4}\omega^{2}\pi}{(eE)^{3}}\right)e^{-\frac{2\pi
m^{2}}{e\mathcal{E}}}.$ (88)

Let us compare these results from the WL instantons for both periodic and
single-pulse cases which were first computed in Dunne:2005sx ; Dunne:2006st
for Euclidean proper-time case. In our convention of the Hamiltonian (5), we
get the imaginary part of the effective action for the periodic background as

$\mathrm{Im}\Gamma^{\mathrm{WL}}_{\mathrm{periodic}}\simeq{\mathcal{V}}_{3}\frac{\sqrt{2\pi}(e\mathcal{E})^{3/2}}{32\pi^{2}}\frac{\left(1\pm
2\gamma_{\mathrm{E}}^{2}\right)^{3/4}}{\mathbb{K}\left(\frac{\pm
2\gamma_{\mathrm{E}}^{2}}{1\pm
2\gamma_{\mathrm{E}}^{2}}\right)}\,\frac{\mathrm{exp}\left\\{-\frac{e\mathcal{E}}{\sqrt{2}\omega^{2}}S_{\mathrm{I}}\right\\}}{\mathbb{K}\left(\frac{\pm
2\gamma_{\mathrm{E}}^{2}}{1\pm
2\gamma_{\mathrm{E}}^{2}}\right)-\mathbb{E}\left(\frac{\pm
2\gamma_{\mathrm{E}}^{2}}{1\pm 2\gamma_{\mathrm{E}}^{2}}\right)}$ (89)

where

$\displaystyle S_{\mathrm{I}}$ $\displaystyle=4\sqrt{2}\sqrt{1\pm
2\gamma_{\mathrm{E}}^{2}}\left[\mathbb{K}\left(\frac{\pm
2\gamma_{\mathrm{E}}^{2}}{1\pm
2\gamma_{\mathrm{E}}^{2}}\right)-\mathbb{E}\left(\frac{\pm
2\gamma_{\mathrm{E}}^{2}}{1\pm 2\gamma_{\mathrm{E}}^{2}}\right)\right]$ (90)
$\displaystyle=-4\sqrt{2}\bigg{[}\mathbb{E}(\mp
2\gamma_{\mathrm{E}}^{2})-(1\pm 2\gamma_{\mathrm{E}}^{2})\mathbb{K}(\mp
2\gamma_{\mathrm{E}}^{2})\bigg{]}$ (91)

and single-pulse background as

$\mathrm{Im}\Gamma_{\mathrm{pulse}}^{\mathrm{WL}}\simeq{\mathcal{V}}_{3}\frac{(e\mathcal{E})^{5/2}}{8\pi^{3}m\omega}\left(1\pm
2\gamma_{\mathrm{E}}^{2}\right)^{5/4}\mathrm{exp}\left\\{-\frac{m^{2}\pi}{e\mathcal{E}}\left(\frac{2}{1+\sqrt{1\pm
2\gamma_{\mathrm{E}}^{2}}}\right)\right\\}.$ (92)

In both cases, $\gamma_{\mathrm{E}}=\frac{m\omega}{e\mathcal{E}}$ and it is
called the Keldysh parameter. The expansions in (87) and (88) match exactly
with the first two order expansions of (89) and (92) respectively.

Note that the importance of the sign difference between the spatial and
temporal cases are apparent in (89) and (92): Both equations vanishes in
$2\gamma_{\mathrm{E}}\rightarrow 1$ limit for the spatial case while for the
temporal case, they are always positive for any values of
$\gamma_{\mathrm{E}}^{2}>0$. This features are related to (lack of) a critical
field strength which sets a lower bound for the creation of real particles for
(temporal) spatial inhomogeneities.

In this sense, $O(\omega^{2})$ terms in (87) and (88) are the first
corrections in the derivative expansion of the leading order fluctuations of
the worldline instantons. While on the one side, there are corrections to the
derivative expansion, on the other side, there are also higher order
corrections to the fluctuations which forms a series in
$\frac{e\mathcal{E}}{2\pi m^{2}}$. Then, as in the recursive perturbative
expansion that we discussed in the beginning of this chapter, the perturbative
expansion around the worldline instantons is also a double expansion and the
expansions in (87) and (88) corresponds to the first two terms in $o\ll
e\mathcal{E}$ limit. It seems it is possible to obtain more information about
other limits of these double expansions. This requires handling the recursive
expansion in different way and utilizing the Pade method with a conformal
transformation. (See e.g Dunne:2022esi .) We postpone this problem for future
work.

##  4 Connection to WKB Formalism

At the end of the previous section, we discussed that the non-perturbative
pair production probabilities we obtained from the leading singularity of the
Schrodinger kernel ${\mathrm{Tr}}e^{\mp iHt}$ matches with the WLI
computations in Dunne:2005sx ; Dunne:2006st . From this equivalence we also
observe, as it was conjectured by André Voros voros94 , a one to one
correspondence between the location of the singularities and the values of WLI
actions. Another semi-classical method frequently used in the pair production
problems is the WKB formalism where the information about the particle
creation is obtained from the WKB loop integrals which are shown to be
equivalent to WLI action Kim:2019yts ; Taya:2020dco . In the following, we
discuss the connection between the time dependent setting we have been using
in this paper and WKB formalism. This also helps us to elaborate on the
connection of our discussion to WLI.

####  Setup:

In this section, for concreteness, we focus on a specific background potential
and choose the periodic magnetic and (temporal) electric background fields in
$x_{3}$ direction, i.e.

$A^{\mathrm{B}}_{\mu}=\left(0,0,-\frac{\mathcal{E}}{\omega}\sin(\omega
x_{1}),0\right)$ (93)

and

$A^{\mathrm{E}}_{\mu}=\left(0,0,0,-\frac{\mathcal{E}}{\omega}\sin(\omega
x_{0})\right)$ (94)

In addition to this, we only keep $p_{0}$ and $p_{1}$ for the magnetic case
and $p_{0}$ for the electric case non-zero and set the other components of
momenta to zero as they only play a role in the pre-factor. Then, the
corresponding Hamiltonians are

$\displaystyle H_{\mathrm{B}}$
$\displaystyle=-\frac{p_{0}^{2}}{2}+\frac{p_{1}^{2}}{2}+\frac{g_{\mathrm{B}}^{2}}{2\omega^{2}}\sin^{2}(\omega
x_{1})\quad,\quad g_{\mathrm{B}}=e\mathcal{B}$ (95) $\displaystyle
H_{\mathrm{E}}$
$\displaystyle=-\frac{p_{0}^{2}}{2}+\frac{g_{\mathrm{E}}^{2}}{2\omega^{2}}\sin^{2}(\omega
x_{0})\qquad\;\quad,\quad g_{\mathrm{E}}=e\mathcal{E}.$ (96)

In the magnetic case, we also defined a new variable as
$-k^{2}=m^{2}-\frac{p_{0}^{2}}{2}<0$. Then, at the leading order in the
recursive expansions (29) and (32), the action for the effectively one
dimensional systems are written as

$\displaystyle\Gamma^{\pm}_{\mathrm{B},0}(k^{2})$
$\displaystyle=\mp\int^{-k^{2}}{\mathrm{d}}z\,\int_{J^{\pm}_{0}}\frac{{\mathrm{d}}t\,e^{\mp
izt}}{2\pi}\,\int_{C^{\pm}_{\mathrm{B}}}{\mathrm{d}}x\,e^{\mp
it\frac{e^{2}{\mathcal{B}}^{2}}{2\omega^{2}}\sin^{2}(\omega
x)}\int_{C_{p}^{\pm}}{\mathrm{d}}p\,e^{\mp\frac{itp^{2}}{2}},$ (97)
$\displaystyle\Gamma^{\pm}_{\mathrm{E},0}(m^{2})$
$\displaystyle=\mp\int^{m^{2}}{\mathrm{d}}z\,\int_{J^{\pm}_{0}}\frac{{\mathrm{d}}t\,e^{\mp
izt}}{2\pi}\int_{C^{\pm}_{\mathrm{E}}}{\mathrm{d}}x\,e^{\mp
it\frac{e^{2}\mathcal{E}^{2}}{2\omega^{2}}\sin^{2}(\omega
x)}\int_{C_{p}^{\mp}}{\mathrm{d}}p\,e^{\pm\frac{itp^{2}}{2}},$ (98)

where we dropped the space-time indices for simplicity.

In both (97) and (98), the $t$ variable is analytically continued in
$\mathrm{Im}t<0$ and $\mathrm{Im}t>0$ for $\Gamma^{+}$ and $\Gamma^{-}$. This
affects the region where each expression is well-defined. For example the $t$
integral for $\Gamma_{\mathrm{B},0}^{\pm}$ is well-defined if

$\mathrm{Re}\left[-k^{2}+\frac{p^{2}}{2}+\frac{e^{2}\mathcal{B}^{2}}{2\omega^{2}}\sin^{2}(\omega
x)\right]\geq 0,$ (99)

while in the electric case, the condition becomes

$\mathrm{Re}\left[m^{2}-\frac{p^{2}}{2}+\frac{e^{2}\mathcal{E}^{2}}{2\omega^{2}}\sin^{2}(\omega
x)\right]\geq 0.$ (100)

Note that in the limits when the left hand side of these expressions are zero,
we encounter with a singularity of the classical limits of the Green’s
operators444Here, with the classical limit, we mean that in the effective
field theory perspective, there is no quantum correction to the classical
potential $V(x)=\sin^{2}x$ which changes the location of the singularities.
$G_{\mathrm{B},0}=(z+H_{\mathrm{B}})^{-1}$ and
$G_{\mathrm{E},0}=\left(z+H_{\mathrm{E}}\right)^{-1}$ respectively. From the
classical mechanics point of view, $z$ can be viewed as energy variable. Then,
in this sense, these singularities are associated to the (possibly complex)
turning points of the classical motion which is governed by the corresponding
Hamiltonian. As we will see these points are the sources of the non-trivial
information when we express $\Gamma^{\pm}_{\mathrm{B},0}$ and
$\Gamma^{\pm}_{\mathrm{E},0}$ in terms of the WKB integrals.

Note that these singular limits also helps us to understanding the classical
dynamics of each problem: Since both $k^{2}>0$ and $m^{2}>0$, for the magnetic
case, we have the equality
$k^{2}=\frac{p^{2}}{2}+\frac{(e\mathcal{B})^{2}}{2\omega^{2}}\sin^{2}x$ which
is associated the motion of a particle inside a well of the inverse potential
$V(x)=-\frac{(e\mathcal{B})^{2}}{2\omega^{2}}\sin^{2}x$. On the other hand,
for the electric case, we have
$m^{2}-\frac{p^{2}}{2}=\frac{(e\mathcal{E})^{2}}{2\omega^{2}}\sin^{2}x$ which
is linked to a scattering problem above the top of the potential barrier.
These are compatible with the known properties of the magnetic and electric
backgrounds. In the following, we will explain how these features are linked
to the WKB formulation.

In order to make the connection with the WKB formalism clearer, we make same
adjustments in our parameters. First, we introduce the Keldysh (adiabaticity)
parameters as $\gamma_{\mathrm{B}}=\frac{k\omega}{e\mathcal{B}}$ and
$\gamma_{\mathrm{E}}=\frac{m\omega}{e\mathcal{E}}$. Then, we re-scale

$t\rightarrow\frac{2\omega^{2}}{g^{2}}t,\quad
x\rightarrow\frac{x}{\omega},\quad p\rightarrow\frac{g}{\sqrt{2}\omega}p,\quad
z\rightarrow\pm\frac{g^{2}}{2\omega^{2}}z,$

where $g=e\mathcal{B}$ for the magnetic case and $g=e\mathcal{E}$ for the
electric case and $\pm$ signs in the last scaling are for electric and
magnetic cases respectively.

$\displaystyle\Gamma^{\pm}_{\mathrm{B},0}(2\gamma^{2}_{\mathrm{B}})$
$\displaystyle=\pm\frac{e{\mathcal{B}}}{\sqrt{2}\omega^{2}}\int^{2\gamma^{2}_{\mathrm{B}}}{\mathrm{d}}z\,\int_{J^{\mp}_{0}}\frac{{\mathrm{d}}t\,e^{\pm
izt}}{2\pi}\,\int_{C^{\pm}_{\mathrm{B}}}{\mathrm{d}}x\,e^{\mp
it\sin^{2}x}\int_{C_{p}^{\mp}}{\mathrm{d}}p\,e^{\mp\frac{itp^{2}}{2}},$ (101)
$\displaystyle\Gamma^{\pm}_{\mathrm{E},0}(2\gamma^{2}_{\mathrm{E}})$
$\displaystyle=\mp\frac{e\mathcal{E}}{\sqrt{2}\omega^{2}}\int^{2\gamma^{2}_{\mathrm{E}}}{\mathrm{d}}z\,\int_{J^{\pm}_{0}}\frac{{\mathrm{d}}t\,e^{\mp
izt}}{2\pi}\,\int_{C_{\mathrm{E}}^{\pm}}{\mathrm{d}}x\,e^{\mp
it\sin^{2}x}\int_{C_{p}^{\pm}}{\mathrm{d}}p\,e^{\pm\frac{itp^{2}}{2}}.$ (102)

In the rest of this section,

  * •

We will first reduce $\eqref{QuantumAction_magneticScaled}$ and (102) to WKB
integrals. Then, we are going to associate each case to suitable WKB cycles
and compute $\Delta\Gamma_{\mathrm{B},0}(2\gamma_{\mathrm{B}}^{2})$ and
$\Delta\Gamma_{\mathrm{E},0}(2\gamma_{\mathrm{E}}^{2})$.

  * •

We will verify our results via reduction to the WKB formalism, we will also
compute $\Delta\Gamma_{\mathrm{E},0}$ and $\Delta\Gamma_{\mathrm{B},0}$, by
directly taking the spatial integral along the corresponding Lefschetz
thimbles.

  * •

Finally, we will finish with making a precise connection between our
discussion on the electric case and EWKB approach.

###  4.1 The WKB Integrals from the Propagator

In this subsection, we will carefully map (101) and (102) to WKB integrals by
associating WKB cycles to electric and magnetic cases. For this reason, we
will change the order of $t$ and $x$ integrals. This exchange is perfectly
valid when the spatial contours $C_{\mathrm{B}}^{\pm}$ and
$C_{\mathrm{E}}^{\pm}$ are chosen such that the integrals converges.

Let us start with the electric case. In order to make the connection with WKB
formalism clearer, we make the following manipulation

$z+\sin^{2}x=\kappa-\cos^{2}x=\kappa-\sin^{2}x,$

where we defined $\kappa=z+1$ and shifted $x\rightarrow x+\frac{\pi}{2}$ to
obtain the last expression. Then, the condition in (100) becomes

$\displaystyle\mathrm{Re}\left[\kappa-\frac{p^{2}}{2}\right]\geq\mathrm{Re}\left[\sin^{2}x\right].$
(103)

In the case of $\kappa<1$, (103) indicates the classically allowed region of
below the barrier top of potential $V(x)=\sin^{2}x$. In this case, all the
turning points are real. In our original problem, on the other hand, we are
interested in the scattering region i.e. $\kappa>1$, where the turning points
become complex. We illustrated both cases in Fig. 4.

Figure 4: Possible WKB cycles for $V(x)=\sin^{2}x$. Left: Bounded region under
the barrier top. Right: Scattering region over the barrier top. In both cases,
conventional WKB cycles are shown. They are named as  $A$ and  $B$ cycles for
the bounded region and  $A^{\prime}$ and  $B^{\prime}$ cycles for the
scattering region. Wavy lines are branch cuts. In the main text, we observe
that a linear combination of  $A^{\prime}$ and  $B^{\prime}$ cycles contribute
to $\Delta\Gamma_{\mathrm{E},0}$, while $\Delta\Gamma_{\mathrm{B},0}$ is
encoded by  $A$-cycle.

In order to obtain the WKB integrals, we take the momentum555The proper way to
take the momentum integrals is complexifying $p$ variable and rotating the
initial contour to the directions where the integrand decays exponentially.
This is a very special version of the Lefschetz thimble approach to the saddle
point approximation. We will briefly discuss Lefschetz thimbles when we deal
with the spatial integral in Section 4.2. and time integrals in (102), we
write $\Gamma^{\pm}_{\mathrm{E},0}$ as

$\displaystyle\Gamma^{\pm}_{\mathrm{E},0}(2\gamma^{2}_{\mathrm{E}})$
$\displaystyle=\frac{e\mathcal{E}}{\sqrt{2}\omega^{2}}\int^{1+2\gamma^{2}_{\mathrm{E}}}{\mathrm{d}}\kappa\,G_{\mathrm{E},0}^{\pm}(\kappa),$
(104)

where

$\displaystyle G^{\pm}_{\mathrm{E},0}(\kappa)$
$\displaystyle=\mp\int_{\tilde{C}_{\mathrm{E}}^{\pm}}{\mathrm{d}}x\,\left[\lim_{\theta\rightarrow
0}\int_{0}^{\infty}\frac{{\mathrm{d}}t\,e^{\pm i\theta}\,}{\sqrt{\mp 2\pi
it\,e^{\pm i\theta}}}\,e^{-t\left[\pm ie^{\pm
i\theta}\left(\kappa-\sin^{2}x\right)\right]}\right]$ (105)
$\displaystyle=\mp\frac{1}{\sqrt{2}}\int_{\tilde{C}_{\mathrm{E}}^{\pm}}\frac{{\mathrm{d}}x}{\sqrt{\kappa-\sin^{2}x}}.$
(106)

Note that $\tilde{C}_{\mathrm{E}}$ is the contour for the shifted potential.

The integrand has a branch cut associated to the square root term. We choose
the cuts to be in the regions where $\kappa<\sin^{2}x$. Then, for $\kappa<1$,
the integration contour $\tilde{C}_{\mathrm{E}}$ is well defined between two
neighbouring cuts and we place it to the interval
$\tilde{C}_{\mathrm{E}}^{\pm}=\left[-x_{\mathrm{E}},x_{\mathrm{E}}\right]$,
where $x_{\mathrm{E}}=\sin^{-1}\sqrt{\kappa}$. (See Fig. 4 (Left) where $\pm
x_{\mathrm{E}}$ are illustrated by green points.) In this sense, the
discontinuity $\Delta
G_{\mathrm{E},0}=G^{\pm}_{\mathrm{E},0}-G^{-}_{\mathrm{E},0}$ can be expressed
by the integration along a WKB cycle in this region, which we call $A$-cycle.
To see this connection, let us express $\Delta G_{\mathrm{E},0}$ as

$\displaystyle\Delta G_{\mathrm{E},0}$
$\displaystyle=-\frac{2}{\sqrt{2}}\int_{-x_{\mathrm{E}}}^{x_{\mathrm{E}}}\frac{{\mathrm{d}}x}{\sqrt{\kappa-\sin^{2}x}}$
(107)
$\displaystyle=-\frac{1}{\sqrt{2}}\left[\int_{-x_{\mathrm{E}}}^{x_{\mathrm{E}}}\frac{{\mathrm{d}}x}{\sqrt{\kappa-\sin^{2}x}}-\int_{x_{\mathrm{E}}}^{-x_{\mathrm{E}}}\frac{{\mathrm{d}}x\;(-1)}{\sqrt{\kappa-\sin^{2}x}}\right]$
(108)

Then, re-writing the second part in the $2^{\mathrm{nd}}$ Riemann sheet, we
absorb $(-1)$ term into $\sqrt{\kappa-\sin^{2}x}$ and obtain

$\Delta
G_{\mathrm{E},0}(\kappa)=-\frac{1}{\sqrt{2}}\oint_{A}\frac{{\mathrm{d}}x}{\sqrt{\kappa-\sin^{2}x}},$
(109)

where $A$ is chosen in clockwise direction. In the WKB formalism, this
quantity is called the “period” associated to the underlying oscillatory
classical motion along the $A$-cycle.

Finally, integrating over $\kappa$, we get the discontinuity
$\Delta\Gamma_{\mathrm{E},0}=\Gamma_{\mathrm{E},0}^{+}-\Gamma_{\mathrm{E},0}^{-}$
as

$\Delta\Gamma_{\mathrm{E},0}=-\frac{e\mathcal{E}}{\sqrt{2}\omega^{2}}\,\sqrt{2}\oint_{A}{\mathrm{d}}x\,\sqrt{1+2\gamma^{2}_{\mathrm{E}}-\sin^{2}x}$
(110)

where the loop integral along with the pre-factor $\sqrt{2}$ is indeed the
leading order WKB integral for the semi-classical “action” of the quantum
mechanical system. (See e.g. Basar_2017  for the definitions that we use in
this section for the period and action.)

For the magnetic case, upon momentum and proper-time integrals of (102) in the
same way with the electric case, we get

$\displaystyle\Gamma_{\mathrm{B},0}^{\pm}(2\gamma^{2}_{\mathrm{B}})$
$\displaystyle=\frac{e{\mathcal{B}}}{\sqrt{2}\omega^{2}}\int^{2\gamma^{2}_{\mathrm{B}}}{\mathrm{d}}z\,G_{\mathrm{B},0}^{\pm}(z),$
(111)

where

$\displaystyle G_{\mathrm{B},0}^{\pm}$
$\displaystyle=\pm\lim_{\varepsilon\rightarrow
0}\int_{C_{\mathrm{B}}^{\pm}}{\mathrm{d}}x\,\int^{\infty}_{0}\frac{{\mathrm{d}}t}{\sqrt{\pm
2\pi it}}\,e^{-t\left[\mp i\left(z-\sin^{2}x\right)\right]}$ (112)
$\displaystyle=\pm\frac{1}{\sqrt{2}}\int_{C_{\mathrm{B}}^{\pm}}\frac{{\mathrm{d}}x}{\sqrt{z-\sin^{2}x}}.$
(113)

This is the same expression with $G_{\mathrm{E},0}^{\pm}$ up to the sign
difference. Therefore, $C_{\mathrm{B}}^{\pm}$ lies in the classically allowed
region of the inverted periodic potential and the WKB dynamics is encoded by
the corresponding $A$-cycle.

With the above discussion, we reduced the phase space integrals (101) and
(102) to WKB integrals and associated each case with the WKB cycles in the
classical allowed region. However, for the electric case this corresponds to
$m^{2}<0$; therefore, it is an unphysical case. It is possible to obtain the
WKB actions for $\kappa>1$ by analytical continuation from the results of
$\kappa<1$ region. To see the connection to the WKB cycles for
$m^{2}>0\,(\kappa>1)$, on the other hand, we recall that
$\Delta\Gamma_{\mathrm{E},0}$ is associated to $A$-cycle which consists of a
path going from $-x_{\mathrm{E}}$ to $x_{\mathrm{E}}$ in the $1^{\mathrm{st}}$
Riemann sheet and coming back in the $2^{\mathrm{nd}}$ sheet. In $\kappa>1$,
however, the branch cuts are placed between complex turning points and complex
infinities. Therefore, any WKB cycles connecting neighbouring turning points
can contribute to $\Delta\Gamma_{\mathrm{E},0}$. This suggests that a linear
combination of $A^{\prime}$ and $B^{\prime}$ cycles in Fig. 4 should
contribute to $\Delta\Gamma_{\mathrm{E},0}$. In this section, we will compute
this contribution through the analytical continuation from the unphysical
region. In Section 4.3, using the EWKB approach, we will associated
$\Delta\Gamma_{\mathrm{E},0}$ to the WKB cycles in scattering region which
also yields a precise relationship between WKB cycles and WLI for the specific
problem we are discussing.

Now finally, we compute $\Delta\Gamma_{\mathrm{E},0}$ and
$\Delta\Gamma_{\mathrm{B},0}$. We start from the period integrals in (109) and
(113) using the relation ramanujan_part3

$\int_{0}^{\sin^{-1}\sqrt{u}}\frac{{\mathrm{d}}x}{\sqrt{u-\sin^{2}x}}=\frac{\pi}{2}\,{}_{2}F_{1}\Big{(}\frac{1}{2},\frac{1}{2};1,u\Big{)}$
(114)

we get

$\displaystyle\Delta\Gamma_{\mathrm{B},0}(2\gamma_{\mathrm{B}}^{2})$
$\displaystyle=\left(\frac{e\mathcal{E}}{\sqrt{2}\omega^{2}}\right)2\sqrt{2}\int^{2\gamma_{\mathrm{B}}^{2}}{\mathrm{d}}z\,{}_{2}F_{1}\Big{(}\frac{1}{2},\frac{1}{2};1,z\Big{)}$
(115) $\displaystyle\Delta\Gamma_{\mathrm{E},0}(2\gamma_{\mathrm{E}}^{2})$
$\displaystyle=-\left(\frac{e\mathcal{E}}{\sqrt{2}\omega^{2}}\right)2\sqrt{2}\int^{1+2\gamma_{\mathrm{E}}^{2}}{\mathrm{d}}\kappa\,{}_{2}F_{1}\Big{(}\frac{1}{2},\frac{1}{2};1,\kappa\Big{)}$
(116)

Hypergeometric function is analytic expect on the line $[1,\infty)$.
Therefore, it is straightforward to compute the integral (115):

$\Delta\Gamma_{\mathrm{B},0}(2\gamma_{\mathrm{B}}^{2})=\left(\frac{e\mathcal{E}}{\sqrt{2}\omega^{2}}\right)4\sqrt{2}\Big{[}\mathbb{E}(2\gamma^{2}_{\mathrm{E}})-(1-2\gamma^{2}_{\mathrm{E}})\,\mathbb{K}(2\gamma^{2}_{\mathrm{E}})\Big{]}$
(117)

which is a manifestly real function for $0<\gamma_{\mathrm{E}}<1$. For the
electric case, on the other hand, since we are interested in region
$\kappa>1$, we need an analytical continuation (NIST:DLMF, , Eqn. 15.2.3):

${}_{2}F_{1}\Big{(}\frac{1}{2},\frac{1}{2};1,u\Big{)}\rightarrow{}_{2}F_{1}\Big{(}\frac{1}{2},\frac{1}{2};1,u\Big{)}-i{}_{2}F_{1}\Big{(}\frac{1}{2},\frac{1}{2};1,1-u\Big{)}$
(118)

where we pick the sign which will be compatible with the definition

$\Delta G(\kappa)=\lim_{\varepsilon\rightarrow
0}\left(G(\kappa-i\varepsilon)-G(\kappa+i\varepsilon)\right).$

Finally, taking $\kappa$ integral of the analytical continued expression, we
get666Note that here, we defined the elliptic integral as
$\mathbb{K}(m)=\int_{0}^{\pi/2}\frac{{\mathrm{d}}x}{\sqrt{1-m\sin^{2}x}}$
where $m$ is the elliptic parameter rather than the modulus $k^{2}=m$. With
this definition it is related to the hypergeometric function as
$\mathbb{K}(m)=\frac{\pi}{2}{}_{2}F_{1}\Big{(}\frac{1}{2},\frac{1}{2};1,u\Big{)}$.

$\Delta\Gamma_{\mathrm{E},0}(2\gamma_{\mathrm{E}}^{2})=\left(\frac{e\mathcal{E}}{\sqrt{2}\omega^{2}}\right)\left[S_{\mathrm{R}}+S_{\mathrm{I}}\right],$
(119)

where

$\displaystyle S_{\mathrm{R}}$
$\displaystyle=-4\sqrt{2}\Big{[}\mathbb{E}(1+2\gamma^{2}_{\mathrm{E}})-2\gamma^{2}_{\mathrm{E}}\mathbb{K}(1+2\gamma^{2}_{\mathrm{E}})\Big{]},$
(120) $\displaystyle S_{\mathrm{I}}$
$\displaystyle=-i4\sqrt{2}\Big{[}\mathbb{E}(-2\gamma^{2}_{\mathrm{E}})-\left(1+2\gamma^{2}_{\mathrm{E}}\right)\mathbb{K}(-2\gamma^{2}_{\mathrm{E}})\Big{]}.$
(121)

As we expected, upon an appropriate analytical continuation,
$\Delta\Gamma_{\mathrm{E},0}$ is expressed as a linear combination of two WKB
cycles and its imaginary part

$\displaystyle\mathrm{Im}\Delta\Gamma_{\mathrm{E},0}(2\gamma^{2}_{\mathrm{E}})$
$\displaystyle=-\left(\frac{e\mathcal{E}}{\sqrt{2}\omega^{2}}\right)4\sqrt{2}\Big{[}\mathbb{E}(-2\gamma^{2}_{\mathrm{E}})+\left(1+2\gamma^{2}_{\mathrm{E}}\right)\mathbb{K}(-2\gamma^{2}_{\mathrm{E}})\Big{]}$
(122)

is equal to the exponent of the pair production probability given by the WLI
instanton in (89).

###  4.2 Direct Computations via Lefschetz Thimbles

In this subsection, we discuss another way to compute both
$\Delta\Gamma_{\mathrm{B},0}$ and $\Delta\Gamma_{\mathrm{E},0}$ by taking the
integrals in (101) and (102) without changing their order and we evaluate the
spatial integral along the associated Lefschetz thimbles. On the one hand, we
will obtain the same $\Delta\Gamma_{\mathrm{E},0}$ in (119) and
$\Delta\Gamma_{\mathrm{B},0}$ in (117) without a direct reference to WKB
formulation. Therefore, we can see that the discussion in this subsection
provide a Lefschetz thimble perspective to the WKB method. In addition to that
as we will see in the following, in this approach the analytical continuation
we imposed for the electric case in the previous subsection appears naturally.
Thus, in the following, no manipulation for $z$-parameter will be needed.

Let us start with the computations of the spatial integrals. In both magnetic
and electric cases, the space integrals are in the same form:

$I_{x}^{\pm}=\int{\mathrm{d}}x\,e^{\mp it\sin^{2}x}\quad,\quad\arg t=0.$ (123)

While, the exponent in (123) does not contain any small/large parameter, it is
always possible to re-write the exponent as
$\frac{it}{\gamma^{2}_{\mathrm{E}}}\sin^{2}(\omega x)$. Therefore, in
$\gamma_{\mathrm{E}}\ll 1$ limit, it is reasonable to use the saddle point
approximation for (123). The proper way to compute the saddle point
approximation is allowing $x$ to be a complex variable and choosing “good
paths” appropriately such that the integrand decays exponentially at
infinities on those paths, i.e. $e^{-t\sin^{2}x}$ as $|x|\rightarrow\infty$.
These contours are called the Lefschetz thimbles and in the following we will
label them with $\mathcal{J}$. There is also another set of contours which
diverge exponentially at infinities. We would call them anti-thimbles and
label with $\mathcal{K}$. Note that both thimbles and anti-thimbles are the
most dominant paths in the saddle point approximation.

An important point is that the set of thimbles form a (relative homology)
basis such that all other possible well-behaved paths can be written as a
linear combination of them. (See Witten:2010cx ; Witten:2010zr  for a precise
construction, Aniceto:2018bis  for a review and Cherman:2014ofa  for a simple
example very similar to our case.) In this way, it is possible to reconstruct
any reasonable path using the set of thimbles.

Since (123) is only a one-dimensional integral, the saddle point analysis
simply leads to the stationary phase contours which are given by777Note that
in a generic $D$ dimensional setting, the line (124) would be generalized to a
$D$ dimensional surface corresponding to a solution of a flow equation
Witten:2010cx ; Witten:2010zr ; Aniceto:2018bis . However, we would not need
it in this example.

$\mathrm{Re}\sin^{2}x=\mathrm{const}.$ (124)

The good paths, on the other hand, determined by the imaginary part of
$\sin^{2}x$ and they should behave differently in $I_{x}^{+}$ and $I_{x}^{-}$
as the real time limit is taken from different directions. As
$|x|\rightarrow\infty$, the thimbles $\mathcal{J}^{+}$ obeys

$\mathrm{Im}[t\sin^{2}x]<0,$ (125)

while $\mathcal{J}^{-}$ obeys

$\mathrm{Im}[t\sin^{2}x]>0.$ (126)

The anti-thimbles $\mathcal{K}^{\pm}$ obey the opposite conditions to
$\mathcal{J}^{\pm}$ in their respective cases. Therefore, the thimbles and
anti-thimbles for $I_{x}^{+}$ and $I_{x}^{-}$ swap.

(a) Thimble and anti-thimble for forward time evolution.

(b) Thimble and anti-thimble for backward time evolution.

Figure 5: Lefschetz (anti-)thimbles for the spatial integrals of forward
(left) and backward (right) time evolutions for both magnetic and the electric
cases. In Lighter regions the integrals converge while they diverge in darker
regions. Paths for thimbles, i.e. $\color[rgb]{0.55,0.0,0.0}\mathcal{J}\pm$,
are plotted in red while paths for anti-thhimbles , i.e.
$\color[rgb]{0.0,0.0,0.55}\mathcal{K}^{\pm}$, are plotted in blue.

Due to the periodic nature of the exponent, we can restrict ourselves to the
region $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$. The thimbles for this
region are given in Fig. 5. Then, the effective path to the spatial integral
can be obtain by a simple decomposition of $\mathcal{J}^{\pm}$ as

$\displaystyle\mathcal{J}^{\pm}$
$\displaystyle=\left[-\frac{\pi}{2},\frac{\pi}{2}\right]+\left[-\frac{\pi}{4},-\frac{\pi}{4}\pm
i\infty\right]+\left[-\frac{\pi}{4}\pm i\infty,-\frac{\pi}{4}\right]$ (127)
$\displaystyle\quad+\left[\frac{\pi}{4},\frac{\pi}{4}\mp
i\infty\right]+\left[\frac{\pi}{4}\mp i\infty,\frac{\pi}{4}\right].$ (129)

It is obvious that the lines in the imaginary direction cancel each other.
Then, after the cancellations, we get

$\displaystyle I_{x}^{\pm}(t)$
$\displaystyle=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}{\mathrm{d}}x\,e^{\mp
it\sin^{2}x}$ (130)
$\displaystyle=e^{\mp\frac{it}{2}}\int_{0}^{\pi}{\mathrm{d}}x\,e^{\mp\frac{it}{2}\cos
x}=\pi e^{\mp\frac{it}{2}}J_{0}\left(\frac{t}{2}\right)$ (131)

and re-write the effective action in (102) as

$\displaystyle\Gamma_{\mathrm{E},0}^{\pm}(2\gamma_{\mathrm{E}}^{2})$
$\displaystyle=\frac{e\mathcal{E}}{\sqrt{2}\omega^{2}}\int^{2\gamma_{\mathrm{E}}^{2}}{\mathrm{d}}z\,\left[\mp\sqrt{\frac{\pi}{2}}\int_{0}^{\infty}\frac{{\mathrm{d}}t\,e^{\mp
izt}}{\sqrt{\mp it}}{}_{1}F_{1}\left(\frac{1}{2},1;\mp it\right)\right],$
(132)

where ${}_{1}F_{1}\left(a,b;y\right)$ is the confluent hypergeometric function
and we used (NIST:DLMF, , Eqn. 10.16.5)

$e^{\mp\frac{it}{2}}J_{0}\left(\frac{t}{2}\right)={}_{1}F_{1}\left(\frac{1}{2},1;\mp
it\right)$ (133)

to obtain the final form of (132). Note that the part in the square brackets
is the trace of resolvent $G_{\mathrm{E},0}^{\pm}(z)$ in (106). Taking the
time integral (NIST:DLMF, , Eqn. 13.10.3), we get

$\displaystyle\Gamma_{\mathrm{E},0}^{\pm}(2\gamma_{\mathrm{E}}^{2})$
$\displaystyle=\mp\left(\frac{e\mathcal{E}}{\sqrt{2}\omega^{2}}\right)\frac{\pi}{\sqrt{2}}\int^{2\gamma_{\mathrm{E}}^{2}}\frac{{\mathrm{d}}z}{\sqrt{z}}\,{}_{2}F_{1}\left(\frac{1}{2},\frac{1}{2},1;-\frac{1}{z}\right)$
(134)

Although, the integral can be directly taken since
${}_{2}F_{1}\left(\frac{1}{2},\frac{1}{2},1;-\frac{1}{z}\right)$ is analytic
for $z>0$, it is instructive to re-write it as bateman1953higher ; Fettis1970

$\Gamma_{\mathrm{E},0}^{\pm}(2\gamma_{\mathrm{E}}^{2})=\mp\left(\frac{e\mathcal{E}}{\sqrt{2}\omega^{2}}\right)\frac{\pi}{\sqrt{2}}\int^{2\gamma_{\mathrm{E}}^{2}}{\mathrm{d}}z\,\left[{}_{2}F_{1}\left(\frac{1}{2},\frac{1}{2},1;1+z\right)\pm
i\,{}_{2}F_{1}\left(\frac{1}{2},\frac{1}{2},1;-z\right)\right]$ (135)

In (135), the term in the bracket together with $\mp\frac{\pi}{\sqrt{2}}$ term
is $G^{\pm}_{\mathrm{E},0}$. Using the behaviour of
${}_{2}F_{1}\left(\frac{1}{2},\frac{1}{2},1;1+z\right)$ on its branch cut
(NIST:DLMF, , Eqn. 15.2.3), we observe
$G_{\mathrm{E},0}^{+}=-G_{\mathrm{E},0}^{-}$. Then, we write
$\Delta\Gamma_{\mathrm{E},0}$ as

$\displaystyle\Delta\Gamma_{\mathrm{E},0}$
$\displaystyle=-\left(\frac{e\mathcal{E}}{\sqrt{2}\omega^{2}}\right)\sqrt{2}\pi\int^{2\gamma^{2}_{\mathrm{E}}}{\mathrm{d}}z\,\left[{}_{2}F_{1}\left(\frac{1}{2},\frac{1}{2},1;1+z\right)-i{}_{2}F_{1}\left(\frac{1}{2},\frac{1}{2},1;-z\right)\right].$
(136)

In this way, we recovered (116) with the analytical continuation given in
(118). Finally, integrating the expression in the bracket, we recover (119)

$\Delta\Gamma_{\mathrm{E},0}(2\gamma_{\mathrm{E}}^{2})=\left(\frac{e\mathcal{E}}{\sqrt{2}\omega^{2}}\right)\left[S_{\mathrm{R}}+S_{\mathrm{I}}\right],$
(137)

where $S_{\mathrm{R}}$ and $S_{\mathrm{I}}$ are given in (120) and (121).

Before, finishing this section, let us also compute
$\Delta\Gamma_{\mathrm{B},0}$ in the same fashion. Since the only difference
between the phase space part of (101) and (102) is the sign in the exponent of
the momentum integrand, it is straight forward to get

$\displaystyle\Gamma_{\mathrm{B},0}^{\pm}$
$\displaystyle=\pm\frac{e\mathcal{B}}{\sqrt{2}\omega^{2}}\int^{2\gamma_{\mathrm{B}}^{2}}{\mathrm{d}}z\,\sqrt{\frac{\pi}{2}}\int_{0}^{\infty}\frac{{\mathrm{d}}t\,e^{\pm
izt}}{\sqrt{\pm it}}{}_{1}F_{1}\left(\frac{1}{2},1;\mp it\right)$ (138)
$\displaystyle=\pm\left(\frac{e{\mathcal{B}}}{\sqrt{2}\omega^{2}}\right)\frac{\pi}{\sqrt{2}}\int^{2\gamma_{\mathrm{B}}^{2}}\frac{{\mathrm{d}}z}{\sqrt{z}}\,\,{}_{2}F_{1}\left(\frac{1}{2},\frac{1}{2},1;\frac{1}{z}\right).$
(139)

Then, re-writing the integrand using

$\frac{1}{\sqrt{z}}{}_{2}F_{1}\left(\frac{1}{2},\frac{1}{2},1;\frac{1}{z}\right)={}_{2}F_{1}\left(\frac{1}{2},\frac{1}{2},1;z\right)\mp
i{}_{2}F_{1}\left(\frac{1}{2},\frac{1}{2},1;1-z\right)$ (140)

we get

$\displaystyle\Delta\Gamma_{\mathrm{B},0}$
$\displaystyle=\left(\frac{e{\mathcal{B}}}{\sqrt{2}\omega^{2}}\right)\frac{\pi}{\sqrt{2}}\int^{2\gamma_{\mathrm{B}}^{2}}{\mathrm{d}}z\,{}_{2}F_{1}\left(\frac{1}{2},\frac{1}{2},1;z\right)$
(141)
$\displaystyle=\left(\frac{e{\mathcal{B}}}{\sqrt{2}\omega^{2}}\right)4\sqrt{2}\left[\mathbb{E}(2\gamma_{\mathrm{B}}^{2})-(1-2\gamma_{\mathrm{B}}^{2})\mathbb{K}(2\gamma_{\mathrm{B}}^{2})\right].$
(142)

and recover (117).

###  4.3 Connection to Exact WKB

So far in this section, we computed the WKB action for both electric and
magnetic backgrounds in two different ways: By reducing the classical limit of
the effective action to WKB integrals and by direct computation along
Lefschetz thimbles. Our findings are in accordance with the WLI computations
Dunne:2005sx ; Dunne:2006st  for the electric field and the lack of particle
creation for the magnetic background. Now, we discuss how these results fits
in the EWKB approach, which uses the Klein-Gordon equation rather than 1-loop
effective action in path integral description.

On the time dependent electric field side, the EWKB approach was recently
discussed in Taya:2020dco  where the particle creation was associated to the
connection matrices between different Stokes regions. In the setting presented
in Taya:2020dco , the connection problem is equivalent to the Bogoliubov
transformation between the mode functions in asymptotic regions which satisfy
the Klein-Gordon equation. However, the approach we presented throughout this
paper is based on 1-loop effective action. Therefore, instead of connecting
asymptotic regions, we will focus on the connection of successive turning
points which is associated to the leading order WLI as described in
Taya:2020dco

For our choice of the gauge potential in (94) and $\hbar=1$, the Klein-Gordon
equation is written as

$\displaystyle
0=\left[\partial_{0}^{2}+\frac{1}{2}\left(p_{3}-\frac{e\mathcal{E}}{\omega}\sin(\omega
x_{0})\right)^{2}+\frac{p_{\perp}^{2}}{2}+m^{2}\right]\phi\;\Longrightarrow\;\left(-\hbar^{2}\partial_{0}^{2}+\sin^{2}x_{0}\right)\phi=\kappa\phi,$
(143)

where $\hbar=\frac{\sqrt{2}\omega^{2}}{e\mathcal{E}}$,
$\kappa=1+2\gamma_{\mathrm{E}}^{2}$ and shifted $x_{0}$ by $\frac{\pi}{2}$. We
also set ${\mathbf{p}}=0$ to simplify the problem as before. Then,
$e\mathcal{E}\gg\omega$ limit becomes the semi-classical limit, i.e $\hbar\ll
1$ of the Schrödinger equation on the right which formulates a scattering
problem for periodic potential since $\kappa=1+2\gamma_{\mathrm{E}}^{2}>1$. In
the following, we will discuss EWKB formalism very briefly with a focus on its
connection to the semi-classical picture we illustrated in this section. The
detailed introductory discussions on EWKB method can be found in
kawai2005algebraic ; Sueishi:2020rug ; Enomoto:2020xlf ; Taya:2020dco .

In the EWKB perspective, the (formal) solutions of (143) are given as

$\phi_{\pm}(x_{0})=\frac{1}{\sigma_{\mathrm{even}}}\,\mathrm{exp}\left\\{\pm\int_{\tau}^{x_{0}}{\mathrm{d}}x_{0}^{\prime}\,\sigma_{\mathrm{even}}\right\\},$
(144)

where $\tau$ is the reference point which is chosen to one of the turning
points during computations and $S_{\mathrm{even}}$ is an infinite series in
even powers of $\hbar$. Its leading order term is

$\sigma_{0}=\sqrt{\sin^{2}x_{0}-\kappa}$ (145)

and the rest of the series can be found recursively. These solutions have
discontinuities, which are called Stokes lines and they are given by

$\mathrm{Im}\int_{\tau}^{x_{0}}{\mathrm{d}}x_{0}^{\prime}\,\sqrt{\sin^{2}x-\kappa}=0,$
(146)

where $\tau$ is chosen as one of the turning points.

When $\kappa>1$, the turning points associated to the equation (143) in region
$\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$ are
$x_{\pm}=\pm\frac{\pi}{2}+i\sin^{-1}\sqrt{\kappa}$ and their complex
conjugates. The corresponding Stokes diagrams are illustrated in Fig. 6. Note
that for real parameters, the Stokes lines are degenerate, i.e. a Stokes line
originated from a turning point hits another turning point. In order the EWKB
procedure works, these Stokes lines should be de-degenerated by an analytical
continuation.

Let us consider the choice $\kappa\rightarrow\kappa-i\varepsilon$ which
corresponds to the forward time evolution in the time dependent case. In this
case, the connection of the solutions $\phi_{\pm}\left(-\frac{\pi}{2}\right)$
to $\phi_{\pm}\left(\frac{\pi}{2}\right)$ is given by the matrix

$\displaystyle T^{(+)}$
$\displaystyle=\begin{pmatrix}e^{\Sigma_{x_{+}^{*}x_{+}}}&0\\\
0&e^{-\Sigma_{x_{+}^{*}x_{+}}}\end{pmatrix}\;\begin{pmatrix}1\;\;&0\\\
i\;\;&1\end{pmatrix}\;\begin{pmatrix}e^{\Sigma_{x_{-}x_{+}^{*}}}&0\\\
0&e^{-\Sigma_{x_{-}x_{+}^{*}}}\end{pmatrix}$ (147)
$\displaystyle=\begin{pmatrix}e^{\Sigma_{x_{-}x_{+}^{*}}+\Sigma_{x_{+}^{*}x_{+}}}\;\;\;&0\\\
i\,e^{\Sigma_{x_{-}x_{+}^{*}}-\Sigma_{x_{+}^{*}x_{+}}}\;\;\;&e^{-\Sigma_{x_{-}x_{+}^{*}}-\Sigma_{x_{+}^{*}x_{+}}}\end{pmatrix},$
(149)

where $\Sigma_{ab}$ is called the Voros coefficient and at the leading order
semi-classical approximation it is given by

$\Sigma_{ab}=\frac{1}{\hbar}\int_{a}^{b}{\mathrm{d}}x\sqrt{\sin^{2}x-\kappa}.$
(150)

Figure 6: Stokes diagrams of $V(x)=\sin^{2}x$ over the barrier region. Solid
lines are Stokes lines emanating from the turning points and wavy lines are
the branch cuts. Left: Degenerate case. Right: De-degenerate case with
analytical continuation $\kappa\rightarrow\kappa-i\varepsilon$ which is linked
to the forward time evoluation. The cycles associated to the exact WKB
procedure are denoted as  D’ and  B’. Note that this is different than the
conventional picture in Fig. 4 as
${\color[rgb]{1,0,0}D^{\prime}}={\color[rgb]{1,0,0}A^{\prime}}+{\color[rgb]{0,0,1}B^{\prime}}$
and the information about the pair production process arises from
${\color[rgb]{1,0,0}D^{\prime}}-2{\color[rgb]{0,0,1}B^{\prime}}$.

As explained in Taya:2020dco , the pair production probability is related to
$|T_{21}|^{2}$ whose exponent in (149) can be written as

$\displaystyle\Sigma_{x_{-}x_{+}^{*}}-\Sigma_{x_{+}^{*}x_{+}}$
$\displaystyle=+\frac{i}{\hbar}\left(\int_{x_{-}}^{x_{+}}\sqrt{\kappa-\sin^{2}x}-2\int_{x_{+}^{*}}^{x_{+}}\sqrt{\kappa-\sin^{2}x}\right)$
(151)

where the choice of $(+)$ sign is compatible with
$\kappa\rightarrow\kappa-i\varepsilon$ choice:

$\sqrt{\sin^{2}x-(\kappa-i\varepsilon)}=\sqrt{e^{+i\pi}\left(\kappa-i\varepsilon-\sin^{2}x\right)}=+i\sqrt{\kappa-i\varepsilon-\sin^{2}x}.$

The integrals in (151) are well-known Connor84 :

$\displaystyle\int_{x_{-}}^{x_{+}}\sqrt{\kappa-\sin^{2}x}$
$\displaystyle=2\sqrt{2}\bigg{[}\Big{(}\mathbb{E}(\kappa)-(1-\kappa)\mathbb{K}(\kappa)\Big{)}-i\Big{(}\mathbb{E}(1-\kappa)-\kappa\mathbb{K}(1-\kappa)\Big{)}\Big{]}$
(152) $\displaystyle\int_{x_{+}^{*}}^{x_{+}}\sqrt{\kappa-\sin^{2}x}$
$\displaystyle=-i2\sqrt{2}\Big{(}\mathbb{E}(1-\kappa)-\kappa\,\mathbb{K}(1-\kappa)\Big{)},$
(153)

where again the sign of the imaginary part in (152) is chosen accordingly to
$\kappa\rightarrow\kappa-i\varepsilon$. Then, putting everything together and
setting $\kappa=1+2\gamma_{\mathrm{E}}^{2}$ and
$\hbar=\frac{\sqrt{2}\omega^{2}}{e\mathcal{E}}$, we get

$\Sigma_{x_{-}x_{+}^{\star}}-\Sigma_{x_{+}^{\star}x_{+}}=-\frac{i}{2}\left(S_{\mathrm{R}}+S_{\mathrm{I}}\right)=-\frac{i}{2}\Delta\Gamma_{\mathrm{E},0}.$
(154)

where $S_{\mathrm{R}}$ and $S_{\mathrm{I}}$ are given in (120) and (121)
respectively. Note that the right hand side of (154) is the contribution of
half of the cycles $B^{\prime}$ and $D^{\prime}$ in the $1^{\mathrm{st}}$
Riemann sheet while $\Delta\Gamma_{\mathrm{E},0}$ is the contribution of the
whole cycles. With this given knowledge, we can get the leading order non-
perturbative pair production probability as

$\mathcal{P}=|T^{(+)}_{21}|^{2}\sim
e^{-\mathrm{Im}\Delta\Gamma_{\mathrm{E},0}},$ (155)

which compatible with our previous discussion in Sections 4.1 and 4.2. This
also indicates that the contribution at 1 instanton level is equal to the sum
of integration over $B^{\prime}$ and $D^{\prime}$ cycles in Fig. 6 which can
also be seen as a linear combination of $A^{\prime}$ and $B^{\prime}$ cycles.
This is in complete agreement with the observation on WLI in Taya:2020dco .

##  5 Summary

In this paper, we investigated the pair production problems using the real
proper time approach. In the first part, we focused on keeping the final
result unitary. Investigating electric fields with a general time and space
dependence of the form ${\mathbf{E}}=\mathcal{E}\frac{{\mathrm{d}}H(\omega
x_{0})}{{\mathrm{d}}x_{0}}$ and ${\mathbf{E}}=\mathcal{E}\nabla
H_{0}(\omega{\mathbf{x}})$, we computed the first two orders of the non-
perturbative pair production probability in the locally constant field limit,
i.e. $\omega\ll 1$. Our method relies only on the perturbative expansion of
the real proper time propagator $U^{\pm}=e^{\mp iHt}$ which we generated by a
simple recursion relation. After summing this series via Pade approximation,
we associated the pair production probability as a difference between the
discontinuities of the effective actions of forward and backward time
directions, i.e. $\Delta\Gamma_{\mathrm{E}}=\Gamma^{+}_{\mathrm{E}}$ \-
$\Gamma^{-}_{\mathrm{E}}$. The discontinuities arises due to a Borel like
integration over a singularity of ${\mathrm{Tr}}U^{\pm}(t)$ at a complex non-
zero point in $t$-plane. In this sense, we showed that the singularity of
${\mathrm{Tr}}U^{\pm}$ acts as the source for the pair production process.

In addition to computing the unitary preserved pair production probability, we
also managed to understand how this real time approach is linked to the well
known semi-classical approaches, namely the Euclidean worldline instantons and
WKB methods. First, we found an exact agreement between our calculations and
the Euclidean time worldline instanton calculations in Dunne:2005sx ;
Dunne:2006st  which is also a verification of a conjecture between action of
the classical motion and the singularities of the propagator as conjectured by
André Voros voros94 .

Later, focusing on one dimensional periodic electric background potential,
i.e. $\sin^{2}x$, we showed how the WKB integrals appear in the classical
limit of the real time dependent formulation and linked the WKB cycles of the
potential $V(x)$ to the discontinuities of the electric and magnetic cases in
$\hbar\rightarrow 0$ limit, which we denoted as $\Delta\Gamma_{\mathrm{E},0}$
and $\Delta\Gamma_{\mathrm{B},0}$ respectively. In this way, we showed that
the electric case is linked to a linear combination of WKB cycles in the
scattering region of the potential while the magnetic case is represented by
the perturbative WKB cycle inside the well of $\sin^{2}x$. We computed both
$\Delta\Gamma_{\mathrm{E}}$ and $\Delta\Gamma_{\mathrm{B}}$ by integrating the
WKB integrals we obtained as well as taking the phase-space integrals directly
along the Lefschetz thimbles without any direct reference to the WKB cycles.
For the electric case, $\mathrm{Im}\Delta\Gamma_{\mathrm{E},0}$ matches with
WLI action while we obtained real $\Delta\Gamma_{\mathrm{B},0}$ which
indicates the lack of particle creation as expected. Note also that the
thimble approach presents a new way to obtain the WKB actions which can be
utilized in higher dimensional problems as well.

We completed our discussion with a connection to EWKB method and showed which
specific WKB cycles are responsible with the particle creation. In complete
harmony with the time dependent approach, we found that it is a linear
combination of the two linearly independent cycles over the barrier region.
Since the integration cycles in EWKB approach are linked to WLI Taya:2020dco ,
this also provided a precise example on the connection between WLI and the
classical action $\Delta\Gamma_{\mathrm{E},0}$.

Note that the pair production probability
$e^{-\mathrm{Im}\Delta\Gamma_{\mathrm{E},0}}$ we get in EWKB approach offers
an exponentiation of $\Delta\Gamma_{\mathrm{E},0}$ by including higher order
terms. It would be interesting to explore possible methods to reconstruct the
exponential instanton contribution to the semi-classical limit of
$\Delta\Gamma_{\mathrm{E}}$. Apart from very hard analytical explorations in
this direction, it might also be desirable to rely on the perturbative
expansions and making use of the conformal Pade methods Costin:2019xql ;
Costin:2020hwg ; Costin:2020pcj ; Florio:2019hzn  in an $\hbar$ expansion
rather than a LCFA limit. On the one side, it would strengthen the connection
between WKB formulation, WLI and singularity structure of
${\mathrm{Tr}}U^{\pm}$, it would enable to probe the singularity not only in
$\gamma^{2}_{\mathrm{E}}\gg 1$ limit but for all values of
$\gamma_{\mathrm{E}}^{2}$. (See Dunne:2022esi  for an example.)

## Acknowledgment

The author thanks Dieter Van den Bleeken, Can Kozçaz, İlmar Gahramanov and
Mithat Ünsal for various discussions in the earlier stages of this research.
He also thanks to Reiko Toriumi and Rudrajit Banerjee for discussions on the
exact WKB method.

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