aainstitutetext: Indian Institute of Science Education and Research, Homi
Bhabha Road, Pashan, Pune 411008, India.

# Resurgence in the Scalar Quantum Electrodynamics Euler-Heisenberg Lagrangian

Drishti Gupta a and Arun M. Thalapillil
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We explore the ideas of resurgence and Padé-Borel resummation in the Euler-
Heisenberg Lagrangian of scalar quantum electrodynamics, which has remained
largely unexamined in these contexts. We thereby extend the related seminal
works in spinor quantum electrodynamics, while contrasting the similarities
and differences in the two cases. We investigate in detail the efficacy of
resurgent extrapolations starting from just a finite number of terms in the
weak-field expansions of the 1-loop and 2-loop scalar quantum electrodynamics
Euler-Heisenberg Lagrangian. While we re-derive some of the well-known 1-loop
and 2-loop contributions in representations suitable for Padé-Borel analyses,
other contributions have been derived for the first time. For instance, we
find a closed analytic form for the one-particle reducible contribution at
2-loop, which until recently was thought to be unimportant. It is pointed out
that there could be an interesting interplay between the one-particle
irreducible and one-particle reducible terms in the strong-field limit. For
the 1-loop contribution, the resurgent analysis may be mapped effectively into
two copies of the spinor quantum electrodynamics weak-field expansions,
differing only in their prefactors and masses. This simple mapping is no
longer true at 2-loops. The singularity structures in the Padé-Borel
transforms at 1-loop and 2-loop are examined in some detail. Analytic
continuation to the electric field case and the generation of an imaginary
part is also studied. We compare the Padé-Borel reconstructions to closed
analytic forms or to numerically computed values in the full theory. It is
seen that Padé-Borel resummations based on even a small number of terms from
the weak-field expansions can accurately approximate the strong-field
behaviour.

##  1 Introduction

Understanding the diverse facets of quantum field theories and their
extensions is an ongoing endeavour (See, for instance, Polyakov:1987ez ;
Deligne:1999qp ; Witten:1997fz ; Witten:2018zxz  and related references).
These aspirations include both gaining a better understanding of the various
mathematical structures that arise in the variegated manifestations of quantum
field theories as well as developing effective methodologies to apply
appropriate quantum field theories to physical scenarios of interest.

The calculation of quantities in interacting theories is often a non-trivial
task. In many such situations, one resorts to perturbative methods. These
perturbative series for an observable of interest ($\mathcal{O}$) involving a
small parameter ($\epsilon\rightarrow 0$)—for instance, a weak coupling
constant, low temperature, or small field value—are in general a power series
in the small parameter

$\mathcal{O}(\epsilon)\overset{?}{=}\sum_{k\geq
0}\,c^{(0)}_{k}\,\epsilon^{k}\;.$ (1)

Interestingly, these perturbative series are more often than not asymptotic
series with vanishing radii of convergence (for a nice pedagogical discussion
of some of these aspects, please see, for instance, Bender1999 ; marino_2015
). Dyson was among the first to realize that, in many cases, the lack of
convergence in a perturbative series is related to the presence of
instabilities in other sectors of the theory dyson .

A question then arises about correctly interpreting the oft-written equality
in eq. (1) and broadly the perturbative series itself. In this context, a
related consideration is if there is a way to incorporate more information
from the asymptotic series towards the reconstruction of the full observable
$\mathcal{O}$ than the usual optimal truncation of the said asymptotic series.

Resummation methods and, more broadly, the mathematical frameworks of
resurgence theory and trans-series ecalle1985fonctions ;
10.1093/oso/9780198535850.002.0003 ; Sternin:1996 ; Costin:2008  address these
pertinent questions, and attempt to reconstruct the complete function from
just the perturbative expansion (see also, for instance, DORIGONI2019167914 ;
Aniceto:2018bis ). The observable $\mathcal{O}$ may be extended to incorporate
non-perturbative aspects, occasionally with ambiguities, from the associated
small-parameter perturbative series by applying resurgent analysis and trans-
series constructions. The latter is a natural extension of the perturbative
series in eq. (1) around $\epsilon=0$, and in its simplest construction may be
of a form ecalle1985fonctions ; DORIGONI2019167914 ; Aniceto:2018bis

$\mathcal{O}(\epsilon)=\sum_{k\geq 0}\,c^{(0)}_{k}\,\epsilon^{k}+\sum_{I\geq
1}\exp\left[-\frac{C_{I}}{\epsilon}\right]\sum_{l\geq
0}\,c^{(I)}_{l}\,\epsilon^{l}\;.$ (2)

Terms of the form $\exp\left[-C_{I}/\epsilon\right]$ have an essential
singularity at $\epsilon=0$ and are invisible at any order in the small-
parameter perturbative series. If the original perturbative series is
interpreted as an expansion about the free theory, the other terms are
heuristically like an expansion about the other non-trivial saddle points that
may be present. The framework of resurgence and Borel resummation provides a
method to connect the various parts of this trans-series—for instance,
relating the large-order behavior of $c_{k}^{(0)}$ to
$\exp\left[-C_{1}/\epsilon\right]$. Remarkably, even when only a finite number
of terms are accessible to us in the perturbative series, one may, in many
cases, via the method of Padé approximants (see baker1975essentials ;
baker1996pade , for instance), perform resummations to yield accurate
reconstructions of the full function.

Since its inception, resurgent analysis has been applied to various cases—for
example, see Voros1983 ; Jentschura:2004jg ; Dunne:2012ae ; Argyres:2012ka ;
Dunne:2012zk ; Marino:2012zq ; Aniceto:2014hoa ; Dunne:2016nmc . The study of
resurgence has continued to be a very active area of research as well in the
past few years, for instance, to give a random sampling of recent studies—in
the study of the large-order behaviour in quantum chromodynamics
MiravitllasMas:2019neb , investigating resurgent extrapolation with the
Painleve I equation as a prototype Costin:2019xql , study of superconductivity
Marino:2019wra , the study of renormalisation group equations Bersini:2019axn
, applications to the one-dimensional Hubbard model Marino:2020dgc , study of
quantum knot invariants Garoufalidis:2020nut , the study of renormalons in
six-dimensional $\phi^{3}$ theories Borinsky:2022knn , investigations of the
$O(3)$ and $O(4)$ sigma models Bajnok:2021zjm ; Bajnok:2021dri , analysis of
integrable field theories DiPietro:2021yxb , large-charge expansions in
$O(2N)$ models Dondi:2021buw , analysis of $SU(2)$ Chern-Simons partition
function Wu:2020dhl , study of $sl(2,\mathbb{C})$ Chern-Simons state integral
models Duan:2022ryd , the explicit construction of four-dimensional de Sitter
space via nodal diagrams emerging from Glauber-Sudarshan states Brahma:2022wdl
, in the context of matrix-models and D-branes Schiappa:2023ned , in the
context of conformal blocks and the large central-charge limit
Benjamin:2023uib , study of supersymmetric $\mathcal{N}=1$ Jackiw-Teitelboim
gravity Griguolo:2023jyy , studies at the intersection of quantum topology and
mathematical physics Costin:2023kla , to mention a few.

On the other hand, the study of quantum field theories in the strong-field
regime has an extensive history and is of much theoretical and experimental
current interest (for recent discussions, see, for instance, Altarelli:2019zea
; Hattori:2023egw ; Fedotov_2023  and related references). Understanding
strong-field effects is also of relevance in astrophysical and cosmological
contexts Ruffini:2009hg ; Kaspi:2017fwg ; Kim:2019joy ; Grasso:2000wj ;
Durrer:2013pga , where such terrestrially-inaccessible fields and conditions
may abound.

Apart from the theoretical and observational motivations, recent advancements
in generating ultra-intense background fields terrestrially have sparked
renewed efforts in the study of the applications of Quantum Electrodynamics
(QED) in the strong-field regime Fedotov_2023 . The E-144 experiment at SLAC
was the first of its kind to observe non-linear scattering of light in intense
background fields SLAC-E144-Non-Linear-QED ; SLAC-E144-Compton-Scattering ;
SLAC-E144-Positron-Production . Since then, multiple experimental setups such
as ELI-NP ELI-NP  and LUXE LUXE  have been proposed to achieve even higher
intensities of laser fields. The physics at ultra-high intensities of
background fields, in addition to non-linear scattering effects, is also
influenced by non-perturbative effects that cannot be calculated in the
paradigm of usual perturbation theory. One such effect is “Schwinger pair
production”, which is the decay of the QED vacuum to produce pairs of particle
and anti-particle in the presence of intense electric fields Sauter:1931zz ;
Heisenberg:1936nmg ; Schwinger_1951 . The Schwinger pair-production rate is
exponentially suppressed by a factor of $e^{-\pi E/E_{c}}$, where $E$ is the
electric field, $E_{c}=m^{2}/q$ (we adopt natural units, with $c=\hbar=1$, in
the study) is the critical value of the electric field, $m$ is the mass of the
pair-produced particles and $q$ are their charges. There are a few open and
interesting problems in strong-field (spinor) QED and scalar QED (sQED) that
have come to the fore in recent years (for example, see Ruffini:2009hg ;
Medina:2015qzc ; Draper:2018lyw ; Gould:2017fve ; Gould:2018ovk ;
Korwar:2018euc ; Torgrimsson:2019sjn ; Gould:2018efv ).

The Euler-Heisenberg Lagrangian (EHL) Heisenberg:1936nmg ; Weisskopf:1936hya
is the most commonly used framework to study the dynamics of constant external
background fields. The EHL describes non-linear corrections to the classical
Maxwellian Lagrangian in the case of constant electromagnetic backgrounds due
to matter field vacuum fluctuations. The EHL can be organized into several
“loop corrections” to the classical electromagnetic action, each of which is
an order $\alpha\equiv q^{2}/4\pi$ suppressed relative to the previous term
(see, for instance, DUNNE_2005 ; HUET_2012  for a review). The complete EHL
may be written formally as

$\mathcal{L}_{\text{EH}}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\sum_{l=1}^{\infty}\mathcal{L}^{(l)}\;.$
(3)

The EHL, its extensions, applications and related ideas in broader quantum
field theoretic contexts have all been fertile areas of focused research
DUNNE_2005 ; HUET_2012 ; Dunne:2012vv ; Gies:2016yaa ; Dumlu_2010 .

In the presence of constant magnetic field backgrounds, say, note that each
loop correction $\mathcal{L}^{(l)}$ is a function of two independent
quantities— $\alpha$ and $qB/m^{2}$, where $B=\lvert\vec{B}\rvert$ is the
magnitude of the applied constant magnetic field. Each of the corrections
appearing in the EHL may, in the case of weak fields, be expanded as a power
series in $qB/m^{2}$

$\mathcal{L}^{(l)}\propto
m^{4}\alpha^{l-1}\sum_{n=0}^{\infty}a_{n}^{(l)}\left(\frac{qB}{m^{2}}\right)^{2n}\
,\ \frac{qB}{m^{2}}\to 0\;.$ (4)

The weak-field expansion diverges for all values of $qB/m^{2}$, both in the
cases of (spinor) QED and sQED (see, for instance, Huet:2017ydx ). The lack of
convergence of the weak-field expansion signals the presence of tunnelling
instabilities in the theory–for instance, Schwinger pair-production in the
large electric field regimes. Generally, such tunnelling instabilities are
exponentially suppressed contributions that cannot be directly captured by
perturbative expansions of the kind in eq. (4) and require resurgent
extrapolations into trans-series of the form eq. (2).

Ideas from resurgence and Borel resummation have recently been brought to bear
on analysing spinor QED florio ; dunne_higher-loop_2021 ; non-linear-trident-
resummation ; Torgrimsson:2022ndq ; Dunne:2022esi ; Torgrimsson:2021wcj ;
Torgrimsson:2021zob . In the present work we hope to extend the resurgent
extrapolation studies to the scalar QED EHL, which has not been studied in
detail so far in the literature. There have only been a few preliminary
comments in the literature pertaining to these topics Huet:2017ydx . Since
sQED is a relatively well-understood quantum field theory, it is an ideal
framework to investigate the characteristics of resurgent extrapolation from
asymptotic data in the form of weak-field expansions. Analogues of sQED, for
instance, involving a dark-sector $U(1)$ gauge field, may also be potential
prototypes for models in the dark matter sector HOLDOM1986196 ;
doi:10.1146/annurev.nucl.012809.104433 . These make the investigations here
very germane.

We explore various theoretical aspects and subtleties of the Padé-Borel
resummation as applied to sQED EHL weak-field expansions at 1-loop and 2-loop
order. We will also contrast the differences and similarities in the sQED EHL
analysis with the QED case florio ; dunne_higher-loop_2021 . In our analysis
of the 2-loop sQED EHL, we will also include the recently discovered
contribution of one-particle-reducible (1-PR) diagrams Gies:2016yaa . These
have been relatively lesser studied in recent works on the subject. We derive
a closed form for these and compare them to the Padé-Borel reconstructed
functions. Additionally, we will also re-calculate, in modern notation, the
formal expressions for the 1-loop and 2-loop particle-irreducible (1-PI)
diagrams well-known in the literature v_i_ritus_connection_1977 , and study
the resurgent structures appearing in their weak-field expansions. It was
recently shown that the EHL for QED loses its meromorphic properties and has a
non-trivial trans-series structure at the order of 2-loops dunne_higher-
loop_2021 . We will use the Padé-Borel resummation method to demonstrate a
similar loss of meromorphic properties in both the 1-PI and 1-PR contributions
to the 2-loop EHL of sQED. Additionally, we will demonstrate that the loss of
meromorphic properties in the 1-PR contributions has surprising implications
in the non-perturbative behaviour of the 2-loop EHL.

Our work is organized as follows– In Sec. 2, we briefly review the basic ideas
and definitions related to EHL, while also fixing notations and conventions.
We follow this by deriving the formal expressions for the 1-loop and 2-loop
EHL for sQED. We give an overview of the derivation of 2-loop 1-PI
contribution to the sQED EHL in Appendix. B. Then, starting from the weak-
field expansions, we use the methods of Padé-Borel resummation to reconstruct
and theoretically investigate the 1-loop and 2-loop 1-PI and 2-loop 1-PR
functions in Sec. 3. Specifically, we compare the reconstructed expressions to
closed analytic forms in the case of 1-loop and 2-loop 1-PR contributions. We
compare the resurgent extrapolations to results from numerical integration for
the 2-loop 1-PI contributions since the formal expressions are significantly
more intricate and cannot be computed exactly. We derive the closed form for
the 1-loop and 2-loop 1-PR EHL for sQED in Appendix. A. In particular, we will
also investigate the trans-series structure at the one and two-loop orders and
study the implications of the loss of meromorphic properties in the 2-loop EHL
for sQED. We summarise the main results and conclude in Sec. 4.

##  2 Euler-Heisenberg Lagrangian of scalar quantum electrodynamics

In this section, we will introduce and review some of the pertinent concepts
related to the study. We will also explicitly state all the conventions and
definitions used to derive the 1-loop and 2-loop EHLs and related expressions
to study resurgence in sQED. While the 1-loop EHL for sQED is relatively well
understood from a modern field theoretic perspective, the derivation of the
2-loop contributions remains relatively obscure with a few subtleties.
Therefore, we will rederive the expressions while pointing out some of the
subtleties, thereby consolidating the pertinent expressions in our adopted
conventions. In addition, we will derive novel representations for the two-
loop 1-PR EHL for sQED, which have escaped attention until very recently, both
as a Schwinger proper time functional integral as well as in a closed analytic
form.

In the work, henceforth, $\mathcal{L}_{\text{sQED}}$ will refer to the sQED
EHL, and $\mathcal{L}_{\text{QED}}$ will refer to the QED EHL (i.e., with
spinors). Whenever the subscript is sometimes omitted in an expression, for
brevity, it is then always to be assumed that we are talking about sQED, the
main case of interest to us in the paper. We adopt a convention where the flat
spacetime metric $\eta_{\alpha\beta}\equiv(1,-1,-1,-1)$.

The EHL describes the dynamics of a constant electromagnetic background field
generated by an external source. The action corresponding to the EHL is most
commonly expressed as a classical part with “loop corrections” due to vacuum
fluctuations of the scalar field progressively added to it. This may be
formally expressed as

$\Gamma_{\text{HE}}[A]=\int d^{4}x\
\left(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right)+W[A]\;,$ (5)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. The loop
action $W[A]$ is the vacuum-persistence amplitude which, in the path integral
formalism, can be expressed as

$\exp(iW[A])=\int\mathcal{D}\widetilde{A}\ \exp\left[i\int d^{4}x\
\left(-\frac{1}{4}\widetilde{F}_{\mu\nu}^{2}+\mathcal{L}_{\phi}[A+\widetilde{A}]\right)\right]\;.$
(6)

where the integration variable $\widetilde{A}$ denotes fluctuations of the
photon field and
$\widetilde{F}^{\mu\nu}=\partial^{\mu}\widetilde{A}^{\nu}-\partial^{\nu}\widetilde{A}^{\mu}$.
Also, $\mathcal{L}_{\phi}$ is the Wilsonian effective Lagrangian, with the
scalar field integrated out, defined as

$\exp\left(i\int d^{4}x\
\mathcal{L}_{\phi}[A]\right)=\int\mathcal{D}\phi\mathcal{D}\phi^{*}\
\exp\left[i\int d^{4}x\
\bigg{(}(D_{\mu}\phi)^{*}(D^{\mu}\phi)-m^{2}\phi^{*}\phi\bigg{)}\right]\;.$
(7)

Here, $D_{\mu}=\partial_{\mu}+iqA_{\mu}$ is the covariant derivative.

It can be shown that the EHL action in eq. (6) may be rewritten as
dittrich_effective_1985

$\exp(iW[A])=\exp\left(\frac{1}{2}\frac{\delta}{\delta
J}D_{+}\frac{\delta}{\delta
J}\right)\exp\left(i\overline{W}^{(1)}[A+J]\right)\Bigg{|}_{J=0}\;,$ (8)

$\overline{W}^{(1)}$ being the Wilsonian action defined as

$\overline{W}^{(1)}[A]=\int d^{4}x\ \mathcal{L}_{\phi}[A]\;,$ (9)

$D_{+}^{\mu\nu}=g^{\mu\nu}D_{+}(x-y)$ is the photon propagator, and the
operator shorthand is defined as

$\frac{1}{2}\frac{\delta}{\delta J}D_{+}\frac{\delta}{\delta J}=\int
d^{4}x\,d^{4}y\ \frac{\delta}{\delta J(y)}D_{+}(x-y)\frac{\delta}{\delta
J(x)}\;.$ (10)

Expanding the right-hand side in eq. (8) gives successive loop corrections to
the Euler-Heisenberg action. Let us comment briefly on a few aspects. The
functional dependence of $\overline{W}^{(1)}$ on $A$ is expressed through the
combination $qA$. As a consequence, every functional derivative of
$\overline{W}^{(1)}[A+J]$ with respect to $J$ gives a multiplicative factor of
$q$. Consequently, each successive loop correction is an order $q^{2}$ more
than the previous one. Furthermore, as $W[A]$ is represented as the logarithm
of the quantity in the right-hand side of eq. (8), all the loop corrections
only have connected diagrams. Nevertheless, interestingly, this does not rule
out the existence of connected 1-PR diagrams at higher orders. While particle-
reducible diagrams are not present at the 1-loop order, it has recently been
demonstrated Gies:2016yaa  that the contribution of 1-PR diagrams at the two-
loop order is non-zero and must be accounted for properly to obtain the
complete answer. The work further intimates that the 1-PR contribution, in
fact, overtakes the contribution from the 1-PI counterpart in the strong-field
limit and, hence, cannot be overlooked in this limit of pertinent interest.
With this motivation, we derive explicit representations for the two-loop 1-PR
EHL in sQED for the first time.

In Secs. 2.1 and 2.2, we will re-express the expressions for the one and two-
loop contributions to the EHL in terms of the dressed scalar propagator
$G_{A}(x-y)$ in the background electromagnetic field. The dressed propagator
in the proper time representation Schwinger_1951  may be utilised to find the
integral representations for the one and two-loop EHLs.

In Sec. 3, for studying resurgent structures in sQED, we will then arrive at
functional approximations to these EHL corrections at the one and two-loop
orders. These will be obtained leveraging Padé-Borel resummations of their
weak-field expansions having just a finite number of terms. In this fashion,
we may contrast the resummed function constructed from finite-term weak-field
expansions to the full integral expression. Specifically, for instance, the
closed forms for the 1-loop and 1-PR EHLs may be analytically calculated,
making it possible to cross-check the validity of the Padé-Borel functional
approximations in both the weak and the strong field limit analytically. While
a closed analytic form for the 1-PI effective action is not yet known, the
function’s behaviour in the strong field limit has been analytically
calculated in the literature v_i_ritus_connection_1977 . In this case, we will
use numerical computations along with the strong field limit expression to
check the robustness of the Padé-Borel reconstructions.

###  2.1 1-Loop sQED EHL

Let us start by broadly reviewing the well-known 1-loop contribution to the
sQED EHL from a perspective that will be amenable to our resurgence analyses
later. The 1-loop correction corresponds to the first term in the expansion of
eq. (8). Analyzing eq. (8), it becomes evident that the 1-loop Euler-
Heisenberg action is represented just by $\overline{W}^{(1)}$; as already
augured by the notation adopted. Hence, at the level of 1-loop, the EHL is
equivalent to the Wilsonian effective action. This, for instance, facilitates
the use of the 1-loop EHL to determine light-by-light scattering amplitudes
Martin:2003gb ; Edwards:2018vjd . In practice, strictly speaking, a closed
analytic form for the 1-loop EHL has only been obtained for constant
electromagnetic backgrounds. This makes the utilisation of this particular
result for the calculation of scattering amplitudes viable only in the low-
frequency limit Dicus_1998 . The 1-loop contribution is conventionally
represented by a single loop of dressed propagator, as shown in figs. 1 and
2(a).

Figure 1: A depiction of the 1-loop EHL in the Schwinger proper time framework
in terms of the dressed scalar propagator. In terms of explicit Feynman
diagrams, it may be considered to include all contributions with one closed
scalar loop and any number of background photon insertions.

(a)

(b)

(c)

Figure 2: Diagrammatic representations of the three contributions to the
1-loop and 2-loop EHL in sQED. Fig. 2(a) is the only contribution to the
1-loop sQED EHL, whereas figs. 2(b) and 2(c) are respectively the 1-PI and
1-PR contributions to the 2-loop sQED EHL. The double dashed lines, as before,
denote dressing from the background photon interactions.

One may attempt to find an expression formally for $\mathcal{L}_{\phi}$ in the
background electromagnetic field by taking a functional derivative of eq. (7).
This gives us

$\partialderivative{\mathcal{L}_{\phi}}{A_{\mu}}\
(x)=-iq\bra{A}\phi^{*}(x)(D^{\mu}\phi(x))-\phi(x)(D^{\mu}\phi(x))^{*}\ket{A}\;.$
(11)

The right-hand side above is just the Noether current corresponding to the
$U(1)$ symmetry evaluated in the background electromagnetic field.
Particularly, note that the expectation value of the Noether current is being
taken, therefore, not with respect to the vacuum $\ket{\Omega}$, but with
respect to a state with a background electromagnetic field $A$ present. We
have demarcated this state by $\ket{A}$.

The Noether current in sQED is notably distinct compared to its counterpart in
spinor QED. This, interestingly, makes for a more intricate calculation of the
expectation value in eq. (11), compared to spinor QED. This intricacy is
especially pronounced when carrying out the calculation within the framework
of the Schwinger proper time formalism (see, for instance, schwartz2014quantum
for a pedagogical discussion) which often leads to this calculation being
carried out via functional determinants. However, we will explicitly carry out
this calculation in the Schwinger proper time formalism.

We begin the calculation by expressing eq. (11) in terms of scalar field
Green’s functions. By using translational invariance in eq. (11), or
equivalently from partial integration of the action in eq. (7), we arrive at
the form

$\partialderivative{\mathcal{L}_{\phi}}{A_{\mu}}\ (x)=-2iq\lim_{x\to
y}D^{\mu}_{x}G_{A}(x-y)\;,$ (12)

where $G_{A}(x-y)$ is the scalar field Green’s function in the background
electromagnetic field. This Green’s function, in the Schwinger proper time
formalism, may be expressed as

$G_{A}(x-y)=\bra{A}\widehat{T}\left[\phi(x)\phi^{*}(y)\right]\ket{A}=\int_{0}^{\infty}ds\
e^{-s\epsilon}e^{-ism^{2}}\bra{x}e^{-i\widehat{H}s}\ket{y}\;.$ (13)

$\widehat{T}$ denotes time ordering, and
$\widehat{H}=-(\widehat{p}-q\widehat{A})^{2}$, where $\widehat{p}^{\mu}$ is
the momentum operator, and $\widehat{A}^{\mu}=A^{\mu}(\widehat{x})$.

Operating both sides of eq. (13) by $D^{\mu}_{x}$ gives rise to a total
partial derivative on the right-hand side such that

$D^{\mu}_{x}G_{A}(x-y)=\frac{1}{2q}\frac{\partial}{\partial
A_{\mu}}\int_{0}^{\infty}\frac{ds}{s}\
e^{-s\epsilon}e^{-ism^{2}}\bra{x}e^{-i\widehat{H}s}\ket{x}\;.$ (14)

Substituting this result in eq. (12) gives the well-known 1-loop result for
sQED in the Schwinger proper time formalism

$\mathcal{L_{\phi}}(x)=-i\ \int_{0}^{\infty}\frac{ds}{s}\
e^{-s\epsilon}e^{-ism^{2}}\bra{x}e^{-i\widehat{H}s}\ket{x}\;.$ (15)

This is illustrated in fig. 1. Furry’s theorem or charge conjugation
invariance in sQED will render contributions with an odd number of external
photon legs vanishing.

As we shall see, the above integral expression for the 1-loop sQED EHL has a
much more tractable analytic form as compared to the corresponding 2-loop
contributions. In fact, one can even derive a closed analytic form for the
1-loop sQED contribution in terms of the Hurwitz-Zeta functions
$\zeta_{H}(\nu,\alpha)$ titchmarsh1986theory ; apostol1998introduction . This
gives (see Appendix A for details)

$\mathcal{L}^{(1)}(qB,m^{2})=-\frac{m^{4}}{16\pi^{2}}\left(\frac{2qB}{m^{2}}\right)^{2}\left[\zeta^{\prime}_{H}\left(-1,\frac{1}{2}+\frac{m^{2}}{2qB}\right)+\frac{3}{4}\left(\frac{m^{2}}{2qB}\right)^{2}+\right.\\\
\left.\left(1+\log\left(\frac{m^{2}}{2qB}\right)\right)\zeta_{H}\left(-1,\frac{1}{2}+\frac{m^{2}}{2qB}\right)\right]\;.$
(16)

The Hurwitz-zeta functions $\zeta_{H}(\nu,\alpha)$ above are defined as

$\zeta_{H}(\nu,\alpha)=\sum_{n=0}^{\infty}\,\frac{1}{(n+\alpha)^{\nu}}\;,$
(17)

with $\real(s)>1$ and $\alpha\in+\mathbb{Z}$. $\zeta^{\prime}_{H}(\nu,\alpha)$
denotes differentiation with respect to $\nu$.

###  2.2 2-Loop sQED EHL

Let us now discuss the corresponding 2-loop contributions broadly. The 2-loop
sQED EHL can be found by expanding the exponentials in eq. (8) up to the
second order. This gives

$1+iW^{(1)}[A]+iW^{(2)}[A]+\ldots=\\\ \left[1+\frac{1}{2}\frac{\delta}{\delta
J}D_{+}\frac{\delta}{\delta
J}\right]\left[1+i\overline{W}^{(1)}[A]+iJ\frac{\delta\overline{W}^{(1)}[A]}{\delta
A}+\frac{iJ^{2}}{2}\frac{\delta^{2}\overline{W}^{(1)}[A]}{\delta
A^{2}}-\frac{J^{2}}{2}\left(\frac{\delta\overline{W}^{(1)}[A]}{\delta
A}\right)^{2}\right]\Bigg{|}_{J=0}+\mathcal{O}(\alpha^{2})\;,$ (18)

where we have used the suggestive shorthand

$J\frac{\delta\overline{W}^{(1)}[A]}{\delta A}=\int d^{4}x\
J(x)\frac{\delta\overline{W}^{(1)}[A]}{\delta A(x)}\;,$ (19)

and so on.

Only two terms at the second order survive in this expansion. They are
illustrated in figs. 2(b) and 2(c). One of them corresponds to a one-particle-
irreducible (1-PI) contribution, and the other to a one-particle-reducible
(1-PR) contribution. Specifically, these contributions may be expressed as

$W^{(2)}_{\text{1-PI}}[A]=\frac{1}{2}\int d^{4}xd^{4}y\
\frac{\delta^{2}\overline{W}^{(1)}}{\delta A^{\mu}(y)\delta
A^{\nu}(x)}D^{\mu\nu}_{+}(x-y)\;,$ (20)

and

$W^{(2)}_{\text{1-PR}}[A]=\frac{i}{2}\int d^{4}xd^{4}y\
\frac{\delta\overline{W}^{(1)}}{\delta
A^{\mu}(x)}\frac{\delta\overline{W}^{(1)}}{\delta
A^{\nu}(y)}D^{\mu\nu}_{+}(x-y)\;.$ (21)

The 1-PI effective action contribution at 2-loop order for QED and sQED was
first calculated by Ritus v_i_ritus_connection_1977  and has since been
recalculated in the worldline formalism Schubert_2001 . To clarify discussions
pertinent to the study of resurgence in sQED, we rederive, in our conventions
and notations, the relevant expression in the Schwinger proper time formalism
(Please see Appendix B)). The 2-loop 1-PI sQED contribution is given formally
by the integral

$W^{(2)}_{\text{1-PI}}[A]=-iq^{2}\int
d^{4}xd^{4}y\bigg{[}4i\delta^{4}(x-y)\bra{x}\widehat{G}_{A}\ket{y}-\bra{x}\widehat{\Pi}^{\mu}\widehat{G}_{A}\ket{y}\bra{y}\widehat{\Pi}_{\mu}\widehat{G}_{A}\ket{x}-\\\
\bra{x}\widehat{\Pi}^{\mu}\widehat{G}_{A}\widehat{\Pi}_{\mu}\ket{y}\bra{y}\widehat{G}_{A}\ket{x}\bigg{]}D_{+}(x-y)\;,$
(22)

where the operators $\widehat{G}_{A}$ and $\widehat{\Pi}^{\mu}$ are defined as

$\displaystyle\bra{x}\widehat{G}_{A}\ket{y}=G_{A}(x-y)\;,$ (23a)
$\displaystyle\widehat{\Pi}^{\mu}=\widehat{p}^{\mu}-q\widehat{A}^{\mu}\;.$
(23b)

While the 1-PI effective action has been extensively studied in various
formalisms, the study of the 1-PR contribution has been largely overlooked. It
was previously assumed erroneously that this contribution to the QED and sQED
EHL was vanishing. It was only recently Gies:2016yaa  that it was demonstrated
that the 1-PR contribution in QED is, in fact, non-zero. This realization led
to subsequent studies being done in calculating the integral forms for the
1-PR contributions to the propagators and 2-loop EHLs both for QED and sQED
Ahmadiniaz_2019 ; EDWARDS2017339 . In fact, it has been shown Evans_2023  that
the 1-PR contribution to the EHL is necessary to account for the everlasting
polarization caused by the constant electromagnetic field in space. The closed
form for the 1-PR contributions to the 2-loop EHL in QED have been previously
calculated Gies:2016yaa , but a similar calculation has been lacking for the
case of sQED.

For sQED, the 2-loop 1-PR contribution may be formally expressed as

$W^{(2)}_{\text{1-PR}}[A]=\int d^{4}x\
\partialderivative{\mathcal{L}^{(1)}}{F^{\mu\nu}}\partialderivative{\mathcal{L}^{(1)}}{F_{\mu\nu}}\;.$
(24)

The formal expressions for the 1-PI and 1-PR diagrams can be simplified and
renormalized to get their expressions as proper time integrals. The proper
time integrals for 1-PI diagrams in sQED can be found in a representation
similar to their counterparts in QED v_i_ritus_connection_1977 . The 1-PI
diagrams in the EHL of QED for a pure magnetic field background have been
written as a double integral Lebedev-Ritus-Virial-rep ; dunne_higher-loop_2021
. We found a similar representation for sQED in the case of a pure magnetic
field background, and it is given by the intricate double integral (see
Appendix B for a sketch of the derivation)–

$\mathcal{L}^{(2)}_{\text{1-PI}}=\frac{q^{2}m^{4}}{256\pi^{4}}\left(\frac{qB}{m^{2}}\right)^{2}\int_{0}^{\infty}\frac{dt}{t^{3}}\
e^{-tm^{2}/qB}(T_{1}+T_{2}+T_{3})\;.$ (25)

Here, $T_{1}$, $T_{2}$ and $T_{3}$ are given by

$\displaystyle
T_{1}=-\frac{m^{2}t}{qB}\int_{0}^{1}\frac{du}{u(1-u)}\left[\frac{1}{a_{s}-b_{s}}\log\left(\frac{a_{s}}{b_{s}}\right)-\frac{t^{2}}{6}u(1-u)-\frac{t}{\sinh
t}\right]\;,$ (26a) $\displaystyle
T_{2}=\int_{0}^{1}\frac{du}{u(1-u)}\left[\frac{a_{s}-c_{s}}{(a_{s}-b_{s})^{2}}\log\left(\frac{a_{s}}{b_{s}}\right)-\frac{b_{s}-c_{s}}{b_{s}(a_{s}-b_{s})}+\frac{11t^{2}}{6}u(1-u)-\frac{t(1+t\coth
t)}{2\sinh t}\right]\;,$ (26b) $\displaystyle
T_{3}=\left[1-\frac{3tm^{2}}{qB}\left(\log\left(\frac{\gamma
tm^{2}}{qB}\right)-\frac{7}{6}\right)\right]\left[\frac{t}{\sinh
t}-1+\frac{t^{2}}{6}\right]\;,$ (26c)

with

$\displaystyle a_{s}$
$\displaystyle=\frac{\sinh(tu)\sinh(t(1-u))}{t^{2}u(1-u)}\;,$ (27a)
$\displaystyle b_{s}$ $\displaystyle=\frac{\sinh t}{t}\;,$ (27b)
$\displaystyle c_{s}$
$\displaystyle=\cosh(tu)\cosh(t(1-u))-3\sinh(tu)\sinh(t(1-u))\;.$ (27c)

$\log\gamma$ is the Euler’s constant; $\log\gamma\sim 0.577$
abramowitz1965handbook . Here, we have added the subscript ‘s’ to the
variables $a_{s}$, $b_{s}$, and $c_{s}$ to differentiate them from the values
of $a$, $b$, $c$ typically defined in the 1-PI EHL for QED. Note that while
the overall structure of the 1-PI integrals for QED and sQED is the same, the
functional form of $T_{1}$, $T_{2}$, and $T_{3}$ are very different from their
counterparts in QED. This is very much unlike the case of the 1-loop EHL,
where we can relate the QED and sQED EHLs with a simple equation as in eq.
(42).

The rather complicated double integral of $\mathcal{L}^{(2)}_{\text{1-PI}}$
has no closed form. The 1-PR contribution, on the other hand, is relatively
straightforward. It may be evaluated starting from eq. (24), adapting the
methodologies detailed in Gies:2016yaa . Proceeding along these lines, for the
2-loop sQED EHL 1-PR contribution, one gets the double-integral expression
(for a constant magnetic field background)–

$\mathcal{L}^{(2)}_{\text{1-PR}}=\frac{q^{4}B^{2}}{512\pi^{4}}I_{2}^{2}\;.$
(28)

Here, $I_{2}$ denotes the derivative of the 1-loop integral and is given by

$I_{2}=\int_{0}^{\infty}\frac{ds}{s^{3}}\ e^{-m^{2}s/qB}\left[\frac{s}{\sinh
s}-\frac{s^{2}}{\sinh s\tanh s}+\frac{s^{2}}{3}\right]\;.$ (29)

In fact, we were able to derive a closed form for $I_{2}$ using methods
detailed in Appendix A–

$I_{2}=-4\zeta_{H}\left(-1,\frac{1+m^{2}/qB}{2}\right)-8\zeta_{H}^{\prime}\left(-1,\frac{1+m^{2}/qB}{2}\right)+2\left(\frac{m^{2}}{qB}\right)\zeta_{H}^{\prime}\left(0,\frac{1+m^{2}/qB}{2}\right)\\\
-\frac{1}{3}\log\left(\frac{m^{2}}{2qB}\right)\;.$ (30)

In the following section, we will now use these integral forms for the 1-loop,
2-loop 1-PI, and 2-loop 1-PR contributions to the sQED EHLs to investigate the
resurgent characteristics in sQED. For instance, we would like to understand
how well one may reconstruct the respective sQED EHL contributions, given
their weak-field asymptotic expansions up to some finite number of terms.

##  3 Padé-Borel analysis of the sQED EHL

In many physical situations, one is faced with perturbative series that are
occasionally factorially divergent and, therefore, don’t converge anywhere on
the complex plane. Dyson dyson  first noted the significance of such
divergences and argued heuristically that a divergent series may be an
indication of non-perturbative behaviour in some sectors of the theory—for
instance, signifying non-perturbative instabilities manifested by instantons.
That the nature of the non-perturbative sectors may already be gleaned from
the perturbative sectors is among the lessons from the broad program of
resurgence (see, for example, Aniceto:2018bis ; Costin:2020hwg , and
references therein). Particularly, in many cases, the methods of Padé-Borel
analysis (see, for example, Costin:2019xql ; pade-borel-review ; marino_2015 )
may help one re-construct the almost complete functional form from just a few
terms of the small-coupling or weak-field perturbative expansions.

In this section, we explore resurgent extrapolation in sQED as pertaining to
its large-field and weak-field behavior. The weak-field expansions of the sQED
EHL are factorially divergent at both one and two-loop orders, as will be
shown in Secs. 3.1 and 3.2. We investigate in detail how efficiently we can
reconstruct large-field characteristics of the theory from just a finite
number of terms of the weak-field expansions. In this context, let us start by
discussing some of the concepts in resurgent extrapolation we require for our
analyses.

Consider a factorially divergent asymptotic series $\mathcal{S}$ with the
coefficients $a_{n}$,

$\mathcal{S}(z)=\sum_{n=0}^{\infty}a_{n}z^{2n}\;,$ (31)

where the factorial divergence is of the form

$a_{n}\sim\sum_{\mathcal{P}=1}^{\infty}C_{\mathcal{P}}\
A_{\mathcal{P}}^{-2n-k_{\mathcal{P}}-1}\
\Gamma(2n+k_{\mathcal{P}}+1)+\text{corr.}\ ,\ n\to\infty\;.$ (32)

Here, $k_{\mathcal{P}}$ is a fixed constant. Every successive term in the sum
of eq. (32) is exponentially smaller than the next, since
$A_{1}<A_{2}<A_{3}\dots$ and we refer to them as “power-law contributions”
(see, for instance, dunne_higher-loop_2021 ; marino_2015 ). The corrections in
eq. (32) refer to sub-leading corrections to each power-law contribution. As
was mentioned previously, depending on the quantity under consideration,
factorial divergences may be manifestations of instabilities in some sectors
of the theory—they may be related to sub-dominant saddle points of the action
of the theory. As we shall see, our weak-field expansions will have features
similar to the asymptotic series $\mathcal{S}(z)$.

When the function underlying $\mathcal{S}$ is reconstructed using methods of
Borel summation, these instabilities are manifested as exponentially
suppressed imaginary contributions to the function and are therefore missed by
a naive analysis of the perturbation series in eq.(31). The idea of resurgence
is to relate these exponentially suppressed imaginary contributions to the
large order behaviour of $a_{n}$– in fact, in the particular case, it dictates
that each power-law contribution gives an exponentially suppressed
contribution proportional to $e^{-|A_{P}/z|}$. The study of these and other
features in trans-series is an active area of mathematical research trans-
series-research1 ; trans-series-research2 ; trans-series-research-3 ; trans-
series-research-4 . In the physics context, exponentially suppressed terms of
this form are associated with instanton contributions— for example, the
leading order contribution to the imaginary part $e^{-|A_{1}/z|}$ is
associated with the first instanton contribution.

In the context of QED and sQED, the values of $\\{A_{\mathcal{P}}\\}$ are
often imaginary, indicating instanton contributions to the imaginary part of
$\mathcal{S}$ for imaginary values of $z$. The Borel dispersion relations,
which have been expounded upon in Appendix C, may then be used to quantify the
above notion and explicitly compute the instanton contributions. These
instanton contributions combine to form a trans-series for the imaginary part
of $\mathcal{S}$ such that, (see, for instance, trans-series-ref1 ; trans-
series-ref2  and references therein),

$\text{Im}\ \mathcal{S}(it)=\sum_{\mathcal{P}=1}^{\infty}\frac{i\pi
C_{\mathcal{P}}}{2(it)^{k_{\mathcal{P}}+1}}\
e^{-A_{\mathcal{P}}/t}+\text{corr.}\;,$ (33)

The index $\mathcal{P}$ in the sum represents the $\mathcal{P}^{\text{th}}$
instanton contribution. Corrections to the power law causes each instanton
contribution to be an independent asymptotic series; this is a characteristic
of a trans-series.

In the following sub-sections, we will denote the $\mathcal{P}^{\text{th}}$
instanton contribution by $\mathcal{I}_{\mathcal{P}}$. Therefore, in this
notation, we can also rewrite eq. (33) as

$\text{Im}\
\mathcal{S}(it)=\sum_{\mathcal{P}=1}^{\infty}\mathcal{I}_{\mathcal{P}}^{(\mathcal{S})}(t)\;.$
(34)

The process of Padé-Borel reconstruction begins by defining the Borel
transform. We will define a general Borel transform
$\widehat{\mathcal{S}}_{p,N^{*}}(z)$ of the asymptotic series $\mathcal{S}(z)$
as

$\widehat{\mathcal{S}}_{p,N^{*}}(z)=\sum_{n=0}^{N^{*}}\frac{a_{n}}{(2n+p)!}\
z^{2n+p}\ ,\ p\geq 0\;.$ (35)

In the limit $N^{*}\to\infty$, the series $\widehat{\mathcal{S}}$ converges
within the radius of convergence $A_{1}$.

For finite $N^{*}$, the next step in the analysis is to find an analytical
continuation for the Borel transform such that it reproduces the properties of
$\widehat{\mathcal{S}}$ inside and on the radius of convergence while also
furnishing a satisfactory analytical continuation outside the radius of
convergence. The analytical continuation generally used for the single-
variable Borel transforms are the Padé approximants (see, for instance,
baker1975essentials ; baker1996pade ). These are rational functions denoted by
$P^{N}_{M}(x)$, with the parameters $N$ and $M$ representing the degrees of
the polynomials in the numerator and the denominator.

In our specific case, the Padé approximants will be constructed to have the
same weak-field expansion inside the radius of convergence as the weak-field
Borel transform. Specifically, in terms of $\widehat{\mathcal{S}}$, we equate

$P^{N}_{M}\big{[}\widehat{\mathcal{S}}\big{]}(z)\equiv\frac{U_{0}+U_{1}z+\dots+U_{N}z^{N}}{1+D_{1}z+\dots
D_{M}z^{M}}\sim\sum_{n=0}^{N^{*}}\frac{a_{n}}{(2n+p)!}z^{2n+p}\equiv\widehat{\mathcal{S}}_{p,N^{*}}(z)\,,~{}\
z\to 0\;,$ (36)

where $N^{*}$ denotes the number of coefficients required to construct
$P^{N}_{M}$ ($N^{*}$ will depend on $N$, $M$ and $p$). The diagonal and off-
diagonal Padé approximants, corresponding to $M=N$ and $M=N+1$, are the most
frequently used approximants in literature dunne_higher-loop_2021 ; florio .
We will denote the Padé approximants in use by $P^{N}_{N+f}$, where the
parameter $f$ is a measure of how far the Padé approximant is from the
diagonal Padé approximant.

The Borel sum of the formal series $\mathcal{S}$ is defined as a Laplace like
transform of $\widehat{S}$ such that (see, for instance, guillou1990large  and
references therein)

$\text{B}_{p}\big{[}\mathcal{S}\big{]}(z)=z^{-p}\lim_{N^{*}\to\infty}\int_{0}^{\infty}dt\
e^{-t}\ \widehat{\mathcal{S}}_{p,N^{*}}(zt)\;.$ (37)

It follows from a simple use of the definition of the Gamma function,
$\Gamma(n+1)=\int_{0}^{\infty}dt\ e^{-t}\ t^{n}$ abramowitz1965handbook , that
we get

$\mathcal{S}(z)\sim\text{B}_{p}\big{[}\mathcal{S}\big{]}(z)\ ,\ z\to 0\;.$
(38)

We can make the method of Borel sum better by constructing Padé approximants
of the Borel transform $\widehat{\mathcal{S}}$ out of only finite terms
$N^{*}$ and taking a Laplace-like transform to get the “reconstructed
function” $\text{PB}_{N,f,p}\big{[}\mathcal{S}\big{]}$. The Padé-Borel
reconstruction is defined as

$\text{PB}_{N,f,p}\big{[}\mathcal{S}\big{]}(z)=z^{-p}\int_{0}^{\infty}dt\
e^{-t}\ P^{N}_{N+f}\big{[}\widehat{\mathcal{S}}_{p,N^{*}}\big{]}(zt)\;.$ (39)

The parameters $f$ and $p$ are often omitted in conventional definitions of
Padé-Borel reconstructions. Nonetheless, it is important to note that varying
these parameters can change the properties of the reconstructed function in
meaningful ways. The apriori unknown optimal values of $f$ and $p$ can be
explored by looking at the asymptotic behaviour of the function to be
reconstructed in the limit of strong fields, as we will see in Secs. 3.1 and
3.2.

The perturbative information obtained from the asymptotic series in eq. (31)
is insufficient to uniquely reconstruct the underlying function $\mathcal{S}$
without additional information. Given just a perturbative series as in eq.
(4), it is possible to construct a range of functions that are equally
effective for small values of $z$ but exhibit different behaviour for larger
values of $z$. This relates to the concept of “non-perturbative ambiguity”,
which is typically quantified by the non-perturbative error $e^{-|A_{1}/z|}$
(see, for instance, marino_2015 ). This error represents the extent to which
$\mathcal{S}$ can be approximated. In Secs. 3.1 and 3.2, we will use different
schemes of $f$ and $p$ and explicitly demonstrate the non-perturbative
ambiguity that accompanies the reconstruction of asymptotic series using the
methods of Padé-Borel reconstruction.

In some cases like QED and sQED, obtaining some information or data points
regarding the behaviour of the function underlying $\mathcal{S}$ might be
possible for large values of $z$ by doing some non-perturbative analysis. A
unique functional approximation for $\mathcal{S}$ can be fixed in such cases.

In the following section, we will consider a finite number of terms in the
weak-field expansion of the sQED EHL at 1-loop and 2-loop orders and carefully
analyze the resurgent extrapolation in each case. Additionally, we will show
how a unique functional approximation can be fixed by using the large-field
behaviour of the 1-loop and 2-loop EHLs, which can be derived by comparing the
trace anomalies of the EHL with the full theory at $\mathcal{O}(\alpha^{2})$.

###  3.1 At the 1-loop order

Let us first focus on the 1-loop sQED EHL expression and the ensuing weak-
field expansion in the presence of a magnetic field. While resurgent
properties of the 1-loop EHL for QED, which is given by Heisenberg:1936nmg

$\mathcal{L}^{(1)}_{\text{QED}}(qB,m^{2})=-\frac{q^{2}B^{2}}{8\pi^{2}}\int_{0}^{\infty}\frac{dt}{t^{2}}\
e^{-m^{2}t/qB}\left[\coth t-\frac{1}{t}-\frac{t}{3}\right]\;,$ (40)

is relatively well studied DUNNE_2005 ; florio , similar studies for the sQED
EHL remain unexplored. We will start the Padé-Borel analysis by looking at the
integral form of the sQED EHL at 1-loop order in the presence of a constant
magnetic field background, given by (see, for instance, pady ; DUNNE_2005 )

$\mathcal{L}^{(1)}_{\text{sQED}}(qB,m^{2})=\frac{q^{2}B^{2}}{16\pi^{2}}\int_{0}^{\infty}\frac{dt}{t^{2}}\
e^{-m^{2}t/qB}\left[\frac{1}{\sinh t}-\frac{1}{t}+\frac{t}{6}\right]\;.$ (41)

We note, by comparing to the spinor QED case above, that eq. (41) can be
represented in terms of the 1-loop EHL of spinor QED, using the identity
$1/\sinh t=\coth\left(t/2\right)-\coth(t)$. Specifically, the correspondence
one obtains is

$\mathcal{L}^{(1)}_{\text{sQED}}(qB,m^{2})=\frac{1}{2}\mathcal{L}^{(1)}_{\text{QED}}(qB,m^{2})-\frac{1}{4}\mathcal{L}^{(1)}_{\text{QED}}(qB,2m^{2})\;.$
(42)

This relation between the 1-loop EHLs of QED and sQED will be helpful later
and will help in contrasting the resurgent properties in QED and sQED.

One may obtain a weak field expansion from the sQED 1-loop EHL in eq. (41).
This is of the form

$\mathcal{L}^{(1)}_{\text{sQED}}(qB,m^{2})\sim\frac{m^{4}}{4\pi^{2}}\left(\frac{qB}{m^{2}}\right)^{4}\sum_{n=0}^{\infty}a_{n}^{(1)}\left(\frac{qB}{m^{2}}\right)^{2n}\equiv\frac{m^{4}}{4\pi^{2}}\left(\frac{qB}{m^{2}}\right)^{4}\mathcal{S}^{(1)}_{\text{sQED}}(qB/m^{2})\
,\ ~{}\frac{qB}{m^{2}}\to 0\;,$ (43)

where the coefficients $a_{n}^{(1)}$ may be found exactly, and have the form

$a_{n}^{(1)}=(-1)^{n}\ \frac{2^{2n+3}-1}{(2\pi)^{2n+4}}\ \zeta(2n+4)\
(2n+1)!\;,$ (44)

and the zeta function is defined as $\zeta(\nu)=\sum_{n=1}^{\infty}1/n^{\nu}$.
The relative simplicity of sQED, along with the factorial divergence of the
coefficients of the weak-field expansion of the sQED EHL, makes it among the
ideal theoretical cases to explore the efficacy of Padé-Borel reconstructions.
We will omit the subscripts from $\mathcal{L}_{\text{sQED}}$ and
$\mathcal{S}_{\text{sQED}}$, as we will only talk about the EHL and weak-field
expansions for sQED.

We define the general Borel transform of $\mathcal{S}^{(1)}$ appearing in the
1-loop sQED EHL as

$\widehat{\mathcal{S}}_{p,N^{*}}^{(1)}(g_{B})=\sum_{n=0}^{N^{*}}\frac{a_{n}^{(1)}}{(2n+p)!}\
g_{B}^{2n+p}\ ,\ p\geq 0\;,$ (45)

where we have defined $g_{B}=qB/m^{2}$, a dimensionless measure of the
strength of the external magnetic field. Henceforth, we will occasionally use
this notation for brevity of expressions wherever required.

The reconstruction is done by approximating the Borel series
$\widehat{\mathcal{S}}_{p,N^{*}}^{(1)}$ by Padé approximants and then
performing a Laplace-like transform, as described in eq. (39). This furnishes
a functional approximation for $\mathcal{L}^{(1)}$.

$\text{PB}_{N,f,p}\big{[}\mathcal{S}^{(1)}\big{]}(g_{B})=g_{B}^{-p}\int_{0}^{\infty}dt\
e^{-t}\
P^{N}_{N+f}\big{[}\widehat{\mathcal{S}}^{(1)}_{p,N^{*}}\big{]}(g_{B}t)\;.$
(46)

To investigate the effectiveness of the Padé-Borel resummation, we compare the
reconstructed function $\text{PB}_{p,N,f}\mathcal{S}^{(1)}$ to the closed
form, as given in eq. (16), of $\mathcal{L}^{(1)}$ for arbitrary field values

$\mathcal{L}^{(1)}(qB,m^{2})\xLeftrightarrow{~{}~{}?~{}~{}}\frac{m^{4}}{4\pi^{2}}\left(\frac{qB}{m^{2}}\right)^{4}\text{PB}_{p,N,f}\big{[}\mathcal{S}^{(1)}\big{]}(g_{B})\;.$
(47)

Figure 3: A comparison of different parameter values $p$ and $f$ and their
effectiveness at replicating the original function. It is apparent that the
parameter combinations $p=1$, $f=1$ provide the best functional approximation.

To get a sense of what the optimal values for the apriori unknown parameters
$f$, $p$ may be, let us consider the leading large-field behaviour of
$\mathcal{L}^{(1)}$. This may be found by comparing the trace anomalies of the
EHL DUNNE_2005  with the full theory, at first order in $\alpha$. For sQED,
this gives

$\mathcal{L}_{\text{sQED}}^{(1)}\sim\frac{q^{2}B^{2}}{96\pi^{2}}\log\left(\frac{qB}{m^{2}}\right)\
,~{}\frac{qB}{m^{2}}\rightarrow\infty\;.$ (48)

In passing, we note that the corresponding expression for QED, which, although
has the same logarithmic growth, has a different pre-factor due to differences
in the value of the coefficients of their respective beta functions DUNNE_2005
. It is given by

$\mathcal{L}^{(1)}_{\text{QED}}\sim\frac{q^{2}B^{2}}{24\pi^{2}}\log\left(\frac{qB}{m^{2}}\right)\
,\ \frac{qB}{m^{2}}\to\infty\;.$ (49)

We may compare the large-field behaviour of $\mathcal{L}_{\text{sQED}}$ with
that of the Padé-Borel reconstructed Lagrangian, which is given by

$\frac{m^{4}}{4\pi^{2}}\left(\frac{qB}{m^{2}}\right)^{4}\text{PB}_{N,f,p}\big{[}\mathcal{S}^{(1)}\big{]}(g_{B})\sim\frac{m^{4}}{4\pi^{2}}\left(\frac{qB}{m^{2}}\right)^{4-p}\int_{1/g_{B}}^{\infty}dt\
e^{-t}\ \frac{U_{N}/D_{N+f}}{(g_{B}t)^{f}}\ ,~{}\ g_{B}\to\infty\;,$ (50)

where $U_{N}$ and $D_{N+f}$ are the coefficients of the $N^{\text{th}}$ and
$(N+f)^{\text{th}}$ degree terms in the numerator and denominator of the Padé-
approximant as defined in eq. (36). The infrared region of the integral
$[0,1/g_{B})$ may be ignored in the strong-field limit as it will be
$\mathcal{O}(1/g_{B})$. The integral in eq. (50) is the incomplete Gamma
function $\Gamma(-f+1,x)$ with $x=1/g_{B}$, which is defined as
abramowitz1965handbook

$\Gamma(a,x)=\int_{x}^{\infty}dt\ e^{-t}\ t^{a-1}\;.$ (51)

The derivative of the incomplete Gamma function,
$\partial_{x}\Gamma(-f+1,x)=-\frac{e^{-x}}{x^{f}}$, can be expanded in the
limit $x\to 0$ to deduce that, up to a multiplicative constant, we would have

$\frac{m^{4}}{4\pi^{2}}\left(\frac{qB}{m^{2}}\right)^{4}\text{PB}_{N,f,p}\big{[}\mathcal{S}^{(1)}\big{]}\sim\frac{m^{4}}{4\pi^{2}}\left(\frac{qB}{m^{2}}\right)^{4-p}\begin{cases}\frac{g_{B}^{f-1}}{g_{B}^{f}},&\text{if
}f>1\\\ \frac{\log g_{B}}{g_{B}}&\text{if }f=1\end{cases}~{}\
,~{}~{}~{}\frac{qB}{m^{2}}\to\infty\;.$ (52)

From the above discussions, we deduce that the optimal parameter choices would
be $p=1$ and $f=1$.

Figure 4: A comparison of the effectiveness of the weak-field asymptotic
expansion and the Padé-Borel reconstruction at approximating
$\mathcal{L}^{(1)}$. The dashed lines represent the asymptotic expansion
truncated at different values of $N$. The coloured lines represent the Padé-
Borel reconstructions with Padé approximants of the form $P^{N}_{N+1}$ for
different values of $N$, and the black line represents the closed form. The
generalized Borel sum is used here with the parameter values $p=1$, $f=1$.

We see a similar trend in fig. 3– the superior approximation is given by
$p=1$, $f=1$. We were able to predict that these parameter values would give
the optimum approximation because we had some information about the behaviour
of $\mathcal{L}^{(1)}$ in the specific case of sQED for large $g_{B}$ due to
non-perturbative arguments (a similar thing also holds for QED; please refer
to dunne_higher-loop_2021 ). However, we would not know how to choose the
optimal parameter values without this information. Without non-perturbative
information, we obtain a set of functional approximations corresponding to
different parameter values $p$ and $f$, as in fig. (3). While all of these
approximations work equally well in the perturbative sector, they differ for
values of $g_{B}\sim 10$, indicative of a non-perturbative ambiguity.

In fig. 4, we compare the Padé-Borel reconstruction of the 1-loop sQED finite-
termed weak-field expansion with the full 1-loop sQED EHL. We have taken $f=1$
and $p=1$ here. Even with just two terms ($N=2$) from the truncated weak field
expansion, we see that the Padé-Borel reconstruction is still able to well-
approximate the full 1-loop sQED EHL in the strong-field regime; up to
$g_{B}\sim 10^{3}$.

Now, the above analysis was performed for a constant magnetic field
background. The analytical continuation $B\to\pm iE$ can extend this analysis
to constant electric field backgrounds. The motivation for such a rotation
stems from the Lorentz invariant quantity $B^{2}-E^{2}$, on which the EHL
should always depend. Our analysis will always choose the analytical
continuation $B\to-iE$. We provide a heuristic argument for our reason for
doing so; other arguments have also been presented in the literature
jentschura-gies .

Consider, for example, the 1-loop EHL for sQED. Its most preliminary
functional form is given by pady

$\mathcal{L}^{(1)}_{\text{sQED}}=-\int_{0}^{\infty}\frac{ds}{s^{3}}e^{-i(m^{2}-i\epsilon)s}F(qBs)\;,$
(53)

where the function $F$ is given by

$F(x)=\frac{x}{\sin x}-1-\frac{x^{2}}{6}\;.$ (54)

The $\epsilon$ prescription is necessary to ensure convergence of the
integral. This prescription causes the integration contour to lie slightly
below the real axis, as shown in fig. 5. We note here that because the
functional dependence of $s$ always comes in the form of $qBs$, this
coordinate is a more natural choice to work in. To compute this integral in
the presence of magnetic fields, the integration variable $s$ is traditionally
continued to the negative imaginary axis, i.e., $s\to-is$. Here, the
continuation from $s$ to $is$ is not viable due to the existence of poles
along the real axis. But because the functional dependence of $s$ comes in the
form of $qBs$, the analytic continuation $B\to iE$ should not be viable
either.

Figure 5: Valid analytical continuations in the $qBs$ plane. The continuations
$s\to-is$ and $B\to-iE$ are equivalent.

However, when the analytic continuation $g_{B}\to-ig_{E}$ is implemented on
the Padé-Borel reconstructed function in eq. (39), it is found that the Padé
approximant $P^{N}_{N+f}$ develops a series of poles on the imaginary axis, as
is shown in fig. 6. In this case, the singularities of the Padé approximant of
the Borel transform are simple poles at integral multiples of $i\pi$. This is
also expected because of the $1/\sinh$ term in the integrand in the exact
expression eq. (41).

(a)

(b)

Figure 6: Poles of the Padé approximants of the 1-loop EHL constructed in two
different ways. A naive construction of Padé approximants leads to arcs of
spurious poles as seen in fig. 6(b), whereas Padé-like approximants
constructed from the spinor 1-loop EHL do not give rise to such spurious
poles, as is seen in fig. 6(a).

In fig. 6, we have constructed the pole structure of the Padé-approximants
using two strategies. In the first strategy, we directly construct the Padé
approximant $P^{N}_{N+1}$ from $\widehat{\mathcal{S}}^{(1)}$. In the second
strategy, we instead compute separately the Padé-approximants for the Borel
transform series of the 1-loop EHLs of spinor QED, as they appear in eq. (42).
From this, we then construct Padé-like approximants for the 1-loop EHL of
sQED.

While both strategies show simple poles on the imaginary axis, we see
curiously, by comparing to the full theory, that the first strategy gives rise
to unwanted arcs of spurious poles. This may be related to the so-called Szegő
curves. Loosely speaking, Szegő curves are curves onto which the zeros of a
normalized partial sum of $e^{z}$, denoted as $s_{n}(nz)$ converge (see, for
instance, Szego-1 ; szego-2 ). It has been demonstrated that analogous curves
also manifest in the distribution of zeros of the normalized partial sums of
$\sin(z)$ and $\cos(z)$ functions varga_zeros_2000 . In our context, for the
first strategy, we similarly observe characteristic Szegő curves in the
distribution of the poles of the normalized Padé approximants. This is
depicted in fig. 7.

Such spurious poles are interestingly absent in the Padé-like approximants
constructed starting from the spinor Borel transform series, as is seen in
fig. 6(a). Therefore, the second strategy detailed above seems to be able to
circumvent the issue with spurious poles. Thus, comparing with the Padé
approximants derived from the integral in eq. (41), this methodology seems to
give a better resurgent extrapolation. Note that the answers obtained from the
two strategies should agree in the limit $N\to\infty$, as the Szegő curves are
artifacts from partial sums. In this limit, the arc of spurious poles is
pushed to infinity. This trend is also seen explicitly in fig. 6(b), where the
arc of spurious poles is pushed farther and farther away from the origin for
larger and larger $N$.

Figure 7: The poles of the normalized Padé approximant $P^{60}_{61}(120z)$
illustrating the Szegő curves. The plot shows the characteristic curve of the
spurious poles of a normalized partial sum $s_{n}(nz)$, as discussed in the
text.

The presence of poles in the Padé approximants seemingly introduces
singularities in the integral in eq. (39) because of the presence of poles in
the term $P^{N}_{N+f}(zt)$ for $z=-ig_{E}$. However, note that if this
integral is performed with the analytic continuation $B\to-iE+\epsilon$, where
$\epsilon\to 0$, we can circumvent these poles in the integral. This manoeuvre
is equivalent to doing the variable change $t\to it/g_{E}$ (where
$g_{E}=qE/m^{2}$) and a contour rotation to get that

$\text{PB}_{N,f,p}\big{[}\mathcal{S}^{(1)}\big{]}(g_{E})=\frac{1}{(-ig_{E})^{p+1}}\int_{0}^{\infty}dt\
e^{-it/g_{E}}\
P^{N}_{N+f}\big{[}\widehat{\mathcal{S}}^{(1)}_{N^{*},p}\big{]}\;.$ (55)

This integral will have both a real and an imaginary part. Therefore, an
analytical continuation to electric fields gives rise to an imaginary
contribution to the EHL. This imaginary contribution is responsible for the
false vacuum decay that leads to Schwinger pair production Schwinger_1951 .
The instability of the vacuum in the presence of strong electric fields leads
to the production of charged particle and anti-particle pairs and hence causes
vacuum polarization. This effect is entirely non-perturbative and is
suppressed by a factor of $e^{-\pi m^{2}/qE}$. Therefore, effects such as
these cannot be probed by conventional perturbative analysis methods and
require methodologies developed in the resurgence program.

Note that because the poles of $P^{N}_{N+f}$ only lie on the imaginary axis,
we do not encounter singularities while reconstructing the EHL in the integral
in eq. (39) in the case of a pure magnetic field background. Therefore, the
EHL doesn’t develop an imaginary part for a pure magnetic field background.

We will use the Padé-Borel reconstructions to reconstruct the imaginary part
of the EHL and show that it gives remarkably close approximations to the full
theory, even though the information used to construct it is perturbative. The
results from the scheme of Padé-Borel reconstruction as was described in eq.
(55) are compared in fig. 9. It is apparent that the Padé-Borel reconstruction
method is not only successful in approximating the EHL for magnetic field
backgrounds but can also effectively provide very good functional
approximations in the electric field regimes. The parameter values used here
are again $p=1$, $f=1$.

Figure 8: Padé-Borel reconstruction of the real part of the 1-loop EHL
analytically continued to electric fields, compared with the closed form. The
Padé-Borel resummations successfully approximate the closed form, even at the
kink at $g_{E}\sim 10$, which marks a transition in the function’s sign. The
parameter values used here are $p=1$, $f=1$ Figure 9: Padé-Borel
reconstruction of the imaginary of the 1-loop EHL analytically continued to
electric fields, compared with the closed form. The parameter values used here
are $p=1$, $f=1$

As was mentioned at the beginning of Sec. 3, the program of resurgence,
especially the Borel dispersion relations, can be used to relate the large-
order behaviour of weak-field coefficients to the imaginary part of the
function underlying the weak-field expansion. If the formal structure of the
weak-field expansion is known, we may determine the large order behaviour of
the coefficients $a_{n}^{(1)}$ up to all orders. From eq. (44), we deduce that
at large orders

$a_{n}^{(1)}\sim(-1)^{n}\ \frac{2^{2n+3}-1}{(2\pi)^{2n+4}}\
\Gamma(2n+2)\left[1+\frac{1}{2^{2n+4}}+\frac{1}{3^{2n+4}}+\dots\right]\ ,\
n\to\infty\;.$ (56)

Cognizant of the large order behaviour, we may now compute the exact
contribution of each instanton. This can be done by deducing the singularity
structure of the Borel transform $\widehat{S}$ using the large-order behaviour
of $a_{n}^{(1)}$ and then using this singularity structure to find its
contribution to the imaginary part of the Borel sum $\text{B}[\mathcal{S}]$.
The result of this procedure is known as the Borel dispersion relations, which
are detailed in Appendix C.

It is apparent from the structure of the large-n expansion of $a_{n}$ that the
Borel transform $\widehat{\mathcal{S}}^{(1)}$ must have multiple simple pole
singularities at integer multiples of $\pi$. This attribute is also realized
in the pole structure of the corresponding Padé approximants. Already, we see
evidence of non-perturbative behaviour lurking in the large order behaviour of
$a_{n}^{(1)}$. From the Borel dispersion relations in Appendix C, and the
large order behavior in eq. (56), we calculate the instanton contributions to
the 1-loop EHL to be of the form

$\text{Im}\
\mathcal{L}^{(1)}_{\text{sQED}}=\frac{m^{4}}{16\pi^{3}}\left(\frac{qE}{m^{2}}\right)^{2}\left[e^{-m^{2}\pi/qE}-\frac{1}{4}e^{-2m^{2}\pi/qE}+\frac{1}{9}e^{-3m^{2}\pi/qE}+\dots\right]\;.$
(57)

We may compare this with the imaginary part of the spinor QED 1-loop EHL,
reconstructed in a similar fashion florio ; dunne_higher-loop_2021

$\text{Im}\
\mathcal{L}^{(1)}_{\text{QED}}=\frac{m^{4}}{8\pi^{3}}\left(\frac{qE}{m^{2}}\right)^{2}\left[e^{-m^{2}\pi/qE}+\frac{1}{4}e^{-2m^{2}\pi/qE}+\frac{1}{9}e^{-3m^{2}\pi/qE}+\dots\right]\;.$
(58)

This reproduces the well-known result Lebedev-Ritus-Virial-rep ; DUNNE_2005 ;
Kim:2003qp  that the vacuum decay rate in the case of sQED, in the regime of
strong electric fields, is slightly suppressed as compared to that of charged
spinors.

###  3.2 At the 2-loop order

Let us now turn our focus on the 2-loop contributions to the sQED EHL and a
study of the resurgent extrapolation of the corresponding weak-field
expansions. At the 2-loop order, eq. (8) suggests, as discussed earlier, that
there may be two distinct contributions to the sQED EHL—the one-particle
irreducible contribution and a one-particle reducible contribution. It is
found that both these contributions are physically significant, with
interesting inter-plays between them as the magnitude of the background
electromagnetic field increases. Unlike the 1-loop expressions, the 2-loop
expressions are relatively less tractable analytically.

The above formal expression for 1-PI in eq. (25) has no known simple closed
analytic form, and we will evaluate these numerically for future comparisons.
In contrast, the expression for the 1-PR contribution in eq. (29) is
relatively straightforward.

We begin the Padé-Borel analysis of the 2-loop sQED EHL contributions by
performing a weak-field expansion for the 1-PI and 1-PR expressions for a
constant magnetic field background. We will denote the coefficients for the
1-PI EHL by $a_{n}^{(2I)}$, and that of the 1-PR EHL by $a_{n}^{(2R)}$. The
corresponding weak-field expansions have the form

$\displaystyle\mathcal{L}_{\text{1-PI}}^{(2)}$
$\displaystyle\sim\frac{q^{2}m^{4}}{256\pi^{4}}\left(\frac{qB}{m^{2}}\right)^{4}\sum_{n=0}^{\infty}a_{n}^{(2I)}\left(\frac{qB}{m^{2}}\right)^{2n}\equiv\frac{q^{2}m^{4}}{256\pi^{4}}\left(\frac{qB}{m^{2}}\right)^{4}\mathcal{S}^{(2I)}(qB/m^{2})\
,\ \frac{qB}{m^{2}}\to 0\;,$ (59a)
$\displaystyle\mathcal{L}_{\text{1-PR}}^{(2)}$
$\displaystyle\sim\frac{q^{2}m^{4}}{512\pi^{4}}\left(\frac{qB}{m^{2}}\right)^{6}\sum_{n=0}^{\infty}a_{n}^{(2R)}\left(\frac{qB}{m^{2}}\right)^{2n}\equiv\frac{q^{2}m^{4}}{512\pi^{4}}\left(\frac{qB}{m^{2}}\right)^{6}\mathcal{S}^{(2R)}(qB/m^{2})\
,\ \frac{qB}{m^{2}}\to 0\;.$ (59b)

Due to the more tractable analytic expression as compared to its 1-PI
counterpart, it is feasible to numerically compute the exact weak-field
coefficients of the 1-PR integral up to the desired orders relatively fast.
Conversely, the weak-field expansion of the 1-PI integral is significantly
more intricate, leading to quickly escalating computation times. We computed
the first 27 coefficients of the weak field expansion of the 1-PI integral and
have noted the first 15 of these values in Appendix. D for reference. The
first few of these coefficients in the 2-loop sQED EHL weak-field expansions
may be compared to the existing values in literature Ahmadiniaz_2019 ;
kors_effective_1999 , which in our convention may be represented as

$\displaystyle\mathcal{L}^{(2)}_{\text{1-PI}}\sim\frac{q^{2}m^{4}}{256\pi^{4}}\left[\frac{275}{648}\left(\frac{qB}{m^{2}}\right)^{4}-\frac{5159}{16200}\left(\frac{qB}{m^{2}}\right)^{6}+\frac{751673}{1058400}\left(\frac{qB}{m^{2}}\right)^{8}+\dots\right]\
,\ \frac{qB}{m^{2}}\to 0\;,$ (60a)
$\displaystyle\mathcal{L}^{(2)}_{\text{1-PR}}\sim\frac{q^{2}m^{4}}{512\pi^{4}}\left[\frac{49}{8100}\left(\frac{qB}{m^{2}}\right)^{6}-\frac{31}{2700}\left(\frac{qB}{m^{2}}\right)^{8}+\frac{58433}{1587600}\left(\frac{qB}{m^{2}}\right)^{10}+\dots\right]\
,\ \frac{qB}{m^{2}}\to 0\;.$ (60b)

The computed values in Appendix. D are seen to match these values in the
existing literature very well.

A ratio test on the weak-field coefficients of the 1-PI integral shows that
they scale as

$a_{n}^{(2I)}\sim 8.0\ (-1)^{n}\ \frac{\Gamma(2n+2)}{\pi^{2n+2}}\;.$ (61)

The weak-field coefficients of the 1-PR integral reveal that, unlike
$a_{n}^{(2I)}$, $a_{n}^{(2R)}$ scale as

$a_{n}^{(2R)}\sim 0.1\ (-1)^{n}\ \frac{\Gamma(2n+3)}{\pi^{2n+3}}\;.$ (62)

The results of the ratio tests on $a_{n}^{(2I)}$ and $a_{n}^{(2R)}$ are
tabulated in tables. 1 and 2. Clearly, the growth is factorial, supporting the
asymptotic analysis above.

$n$ | $an^{2}+bn+c$
---|---
$5-10$ | $0.404191n^{2}+0.243097n+0.128455$
$10-15$ | $0.404741n^{2}+0.232291n+0.181699$
$15-20$ | $0.404984n^{2}+0.225143n+0.234264$
$20-25$ | $0.405094n^{2}+0.220784n+0.277354$
Table 1: A tabulation of $a_{n}^{(2I)}/a_{n-1}^{(2I)}$ fits to a quadratic polynomial $an^{2}+bn+c$ for different ranges of $n$. The left column depicts the ranges of $n$ for which the ratio $a_{n}^{(2I)}/a_{n-1}^{(2I)}$ was calculated and fitted, and the right column depicts the fit obtained from these ranges of $n$. The values of the fit-parameters $a$ and $b$ are seen to converge and indicate the asymptotic factorial growth of the 1-PI coefficients $a_{n}^{(2I)}\sim\Gamma(2n+2)/\pi^{2n}$. $n$ | $an^{2}+bn+c$
---|---
$20-30$ | $0.405136n^{2}+0.618621n-0.449305$
$30-40$ | $0.405238n^{2}+0.612768n-0.364672$
$35-45$ | $0.405254n^{2}+0.611527n-0.341554$
$40-50$ | $0.405264n^{2}+0.610714n-0.324364$
Table 2: $a_{n}^{(2R)}/a_{n-1}^{(2R)}$ ratio fits to the quadratic polynomial
$an^{2}+bn+c$. The values of the fit parameters $a$ and $b$ are again seen to
converge, indicating the asymptotic factorial growth of the 1-PR coefficients
$a_{n}^{(2R)}\sim\Gamma(2n+3)/\pi^{2n}$.

As in the 1-loop case, we perform a Borel transform and construct a Padé
approximant to the truncated Borel transform series. In this case, we will
have two Borel transforms pertaining to the 1-PI and 1-PR diagrams–

$\displaystyle\widehat{S}^{(2I)}_{p,N^{*}}(g_{B})=\sum_{n=0}^{N^{*}}\frac{a_{n}^{(2I)}}{(2n+p)!}g_{B}^{2n+p}\;,$
(63a)
$\displaystyle\widehat{S}^{(2R)}_{p,N^{*}}(g_{B})=\sum_{n=0}^{N^{*}}\frac{a_{n}^{(2R)}}{(2n+p)!}g_{B}^{2n+p}\;.$
(63b)

The Padé approximants are then used in a Laplace-like transform as in eq. (39)
to obtain the resurgent extrapolation of the 2-loop 1-PI and 1-PR sQED weak-
field expansions. We will then investigate the efficacy of different schemes
of Padé-Borel resummations in reconstructing the 1-PI and 1-PR contributions
to the 2-loop EHL of sQED.

$\displaystyle\mathcal{L}^{(2)}_{\text{1-PI}}(qB,m^{2})\xLeftrightarrow{~{}~{}?~{}~{}}\frac{q^{2}m^{4}}{256\pi^{4}}\left(\frac{qB}{m^{2}}\right)^{4}\text{PB}_{N,f,p}\big{[}\mathcal{S}^{(2I)}\big{]}(g_{B})\;,$
(64a)
$\displaystyle\mathcal{L}^{(2)}_{\text{1-PR}}(qB,m^{2})\xLeftrightarrow{~{}~{}?~{}~{}}\frac{q^{2}m^{4}}{512\pi^{4}}\left(\frac{qB}{m^{2}}\right)^{6}\text{PB}_{N,f,p}\big{[}\mathcal{S}^{(2R)}\big{]}(g_{B})\;.$
(64b)

Again, we will use different values of $p$ and different schemes of the Padé
approximants $P_{N}^{N+f}$ to construct Padé-Borel resummations from the weak
field coefficients $a_{n}^{(2I)}$ and $a_{n}^{(2I)}$, and further optimize the
functional approximation. To get a sense of what the optimal parameter values
$p$ and $f$ might be, as defined in eqs. (35) and (39), we may again compare
the strong-field behaviours, as we did in the previous sub-section. The strong
field behaviours for the 1-PI and 1-PR diagrams are known to be DUNNE_2005 ;
karbstein-all-loop —

$\displaystyle\mathcal{L}_{\text{1-PI}}^{(2)}\sim\frac{q^{4}B^{2}}{128\pi^{4}}\log\left(\frac{qB}{m^{2}}\right)\
,~{}\frac{qB}{m^{2}}\rightarrow\infty\;,$ (65a)
$\displaystyle\mathcal{L}^{(2)}_{\text{1-PR}}\sim\frac{1}{9}\frac{q^{4}B^{2}}{512\pi^{4}}\left[\log\left(\frac{qB}{m^{2}}\right)\right]^{2}\
,~{}\frac{qB}{m^{2}}\rightarrow\infty\;.$ (65b)

On the other hand, the large-order behaviour of the Padé-Borel reconstructed
2-loop Lagrangians will look like

$\frac{q^{2}m^{4}}{256\pi^{4}}\left(\frac{qB}{m^{2}}\right)^{4}\text{PB}_{N,f,p}\big{[}\mathcal{S}^{(2I)}\big{]}\sim\frac{q^{2}m^{4}}{256\pi^{4}}\left(\frac{qB}{m^{2}}\right)^{4-p}\begin{cases}\frac{g_{B}^{f-1}}{g_{B}^{f}},&\text{if
}f>1\\\ \frac{\log g_{B}}{g_{B}}&\text{if }f=1\end{cases}\;,$ (66)

and,

$\frac{q^{2}m^{4}}{512\pi^{4}}\left(\frac{qB}{m^{2}}\right)^{6}\text{PB}_{N,f,p}\big{[}\mathcal{S}^{(2R)}\big{]}\sim\frac{q^{2}m^{4}}{512\pi^{4}}\left(\frac{qB}{m^{2}}\right)^{6-p}\begin{cases}\frac{g_{B}^{f-1}}{g_{B}^{f}},&\text{if
}f>1\\\ \frac{\log g_{B}}{g_{B}}&\text{if }f=1\end{cases}\;.$ (67)

Comparing the two large-order behaviours, it is apparent that the best
approximation would correspond to the parameter values $p=1$, $f=1$ for the
1-PI Lagrangian and $p=3$, $f=1$ for the 1-PR Lagrangian.

To ascertain that these parameter values indeed give the best functional
approximations, we compare the efficacy of various Padé-Borel
reconstructions—computed using different parameter values of $p$ and $f$.
These comparisons are shown in figs. 10 and 11. Given that we lack a closed-
form expression for the 1-PI EHL, we have compared the Padé-Borel
reconstructed Lagrangian from $\text{PB}\big{[}\mathcal{S}^{(2I)}\big{]}$ with
results from the numerical integration of eq. (25). A comparison is also shown
with the analytic large-field behaviour of $\mathcal{L}^{(2)}_{\text{1-PI}}$.
In contrast, we can find a closed analytic form for the sQED 2-loop 1-PR
contribution (see Appendix A)–

$\mathcal{L}_{\text{1-PR}}^{(2)}=\frac{q^{4}B^{2}}{512\pi^{4}}\left[-4\zeta_{H}\left(-1,\frac{1+m^{2}/qB}{2}\right)-8\zeta^{\prime}_{H}\left(-1,\frac{1+m^{2}/qB}{2}\right)\right.\\\
\left.+2\left(\frac{m^{2}}{qB}\right)\zeta^{\prime}_{H}\left(0,\frac{1+m^{2}/qB}{2}\right)-\frac{1}{3}\log\left(\frac{m^{2}}{2qB}\right)\right]^{2}\;.$
(68)

Figs. 10 and 11 illustrate that optimal and robust functional approximations
for the 1-PI and 1-PR contributions are achieved when utilizing parameter
values of $p=1$, $f=1$ and $p=3$, $f=1$, respectively. For the optimal
parameter values, the 2-loop Padé-Borel reconstructed functions are seen to
give remarkably accurate functional approximations, even with a truncation at
$N=6$.

Figure 10: A comparison of the effectiveness of various schemes of Padé-Borel
approximations, parameterized by different values of $p$ and $f$, in
reconstructing $\mathcal{L}^{(2)}_{\text{1-PI}}$ for $N=6$. Clearly, the
parameter values $p=1$ and $f=1$ provide the closest approximation to the
numerically-computed closed form and the large-order behaviour of
$\mathcal{L}^{(2)}_{\text{1-PI}}$. Figure 11: A comparison of the
effectiveness of various schemes of Padé-Borel approximations, parameterized
by different values of $m$ and $p$, in reconstructing
$\mathcal{L}^{(2)}_{\text{1-PR}}$ for $N=6$

As was done in the 1-loop case, we can again analytically continue the
magnetic field case to the electric field case via the prescription $B\to-iE$.
We may then compare the imaginary part of the analytically continued full 1-PI
and 1-PR contribution to the instanton contribution computed using eqs. (61)
and (118) from the weak-field expansions.

The Borel dispersion relations, in this case, must be used with care, owing to
the different large-order behaviours of the 1-PI and 1-PR weak-field
coefficients. The dispersion relations, as is shown in Appendix C, are a
technique to find instanton contributions to the imaginary part of the EHL
using the large-order behaviour for weak-field expansions. Given a set of weak
field coefficients $a_{n}$ with the large order behaviour

$a_{n}\sim\ \sum_{\mathcal{P}=1}^{\infty}C_{\mathcal{P}}\
(iA_{\mathcal{P}})^{-2n-k_{\mathcal{P}}-1}\ \Gamma(2n+k_{\mathcal{P}}+1)\ ,\
n\to\infty\;,$ (69)

the Borel dispersion relations dictate that the imaginary contribution to the
EHL be given by

$\text{Im}\big{[}\text{PB}\big{[}\mathcal{S}\big{]}\big{]}(g_{E})=\frac{i\pi}{2}\sum_{\mathcal{P}=1}^{\infty}\frac{C_{\mathcal{P}}}{(ig_{E})^{k_{\mathcal{P}}+1}}e^{-A_{\mathcal{P}}/g_{E}}$
(70)

Making use of these relations for the 1-PI and 1-PR diagrams with $A_{1}=\pi$,
$C_{1}=-8.0$, $k_{1}=1$, and $A_{1}=\pi$, $C_{1}=-0.1i$, $k_{1}=2$
respectively, we get the instanton contributions $\mathcal{I}^{(2I)}_{1}$ and
$\mathcal{I}^{(2R)}_{1}$–

$\displaystyle\mathcal{I}^{(2I)}_{1}=\frac{q^{2}m^{4}}{256\pi^{4}}\
(-ig_{E})^{4}\ \frac{-8.0\ i\pi}{2(ig_{E})^{2}}\ e^{-\pi/g_{E}}$ (71a)
$\displaystyle\mathcal{I}^{(2R)}_{1}=\frac{q^{2}m^{4}}{512\pi^{4}}\
(-ig_{E})^{6}\ \frac{0.1\pi}{2(ig_{E})^{3}}\ e^{-\pi/g_{E}}$ (71b)

The comparison for the 1-PI case is presented in fig. 12, revealing that the
first instanton contribution computed from the weak-field expansion via the
Borel dispersion relation provides a sufficiently accurate approximation to
the behaviour of the imaginary part of the full 1-PI action.

Figure 12: Comparison of the Padé-Borel reconstruction of the imaginary part
of the 1-PI Lagrangian in the regime of strong electric fields with the first
instanton contribution.

Interestingly, this ceases to be the case for the 1-PR action. The kink
observed in the imaginary part of the 1-PR effective action, which signifies a
change in the function’s sign, is not effectively replicated by the first
instanton contribution constructed from the weak-field expansions. However,
such behaviour may possibly be accommodated by sub-leading corrections to the
large-order behaviour of $a_{n}^{(2R)}$. Let us model the sub-leading
corrections to be of the form

$a_{n}^{(2R)}\sim 0.1(-1)^{n}\ \frac{\Gamma(2n+3)}{\pi^{2n+3}}+(-1)^{n}\ C\
\frac{\Gamma(2n+k)}{\pi^{2n+k}}~{}\ ,~{}~{}n\to\infty\;,$ (72)

assuming $k<3$. This sub-leading correction to the large order behaviour of
the weak-field coefficients will consequently lead to a correction to the
instanton of the form (taking $k_{\mathcal{P}}=k-1$ in eq. (69))

$\mathcal{I}^{(2R)}_{1}=\frac{q^{2}m^{4}}{512\pi^{4}}\
g_{E}^{6}\left[-\frac{0.1i\pi e^{-\pi/g_{E}}}{2g_{E}^{3}}+C\frac{i\pi
e^{-\pi/g_{E}}}{2g_{E}^{k}}\right]\;.$ (73)

Figure 13: Padé-Borel reconstruction of the real part of the analytically
continued 1-PR EHL compared with its closed form. The behaviour of the 1-PR
EHL in the electric field regime is quite different from that of its 1-loop
and two-loop 1-PI counterparts in that it changes its sign twice. The Padé-
Borel resummation provides an excellent approximation to the closed form,
effectively reproducing both of its kinks. The parameter values used for the
reconstruction are $f=1$, and $p=3$ Figure 14: Comparison of the Padé-Borel
reconstruction of the imaginary part of the analytically continued 1-PR
Lagrangian with the closed form. Unlike its 1-loop and two-loop 1-PI
counterparts, the imaginary contribution of the 1-PR EHL in the regime of
electric fields flips sign, as is signified by the kink in the graph. While
the naive estimate of the first instanton contribution (illustrated in grey)
is unable to reproduce this kink, a sub-leading correction to the first
instanton contribution (illustrated in golden) can explain this behaviour. The
parameter values for the Padé-Borel approximations are $f=1$, $p=3$

This instanton correction, while suppressed in the regime of weak fields, can
rapidly overtake the leading instanton contribution for sufficiently strong
fields and can, in fact, reverse the sign of $\mathcal{I}^{(2R)}_{1}$.

We note purely empirically that a fitted correction to first instanton
contribution with $k=1.5$ and $C=0.027$ (plotted in fig. 14) is a much better
approximation to the non-perturbative behaviour of
$\mathcal{L}^{(2)}_{\text{1-PR}}$ than the uncorrected instanton contribution.

One of the consequences of the Borel dispersion relations (as is shown at the
end of Appendix C) is that a fractional fit for $k$ implies the presence of
algebraic singularities that manifest as branch cuts in the pole structure of
the 2-loop sQED 1-PR Borel transforms. In the case of spinor QED 1-PI
reconstructions, a similar feature has also been commented on recently
dunne_higher-loop_2021 . Indeed, the presence of branch cuts are suggested
when the singularities of the Padé approximants of
$\widehat{\mathcal{S}}_{N^{*}}^{(2I)}$ and
$\widehat{\mathcal{S}}_{N^{*}}^{(2R)}$ are plotted on the complex plane in
fig. 15. The presence of accumulated simple poles of the Padé-approximants,
seen near integer multiples of $\pi$, is an attempt at mimicking the
properties of the multi-valued function despite being single-valued
themselves. In addition to poles on the imaginary axis, spurious poles
surround the poles accumulated on the imaginary axis. As in the case of the
1-loop sQED EHL analysis, these spurious poles are pushed to infinity in the
limit $N\to\infty$.

(a)

(b)

Figure 15: Poles of the Padé approximants of the Borel sum for the 1-PI (fig.
15(a)) and 1-PR (fig. 15(b)) contributions. The presence of branch cuts is
suggestive in both singularity structures.

The weak field expansions in eq. (59) indicate that for weak fields, the 1-PI
contributions dominate over their 1-PR counterparts, and one could potentially
ignore the latter contribution in this limit. However, it has recently been
pointed out Gies:2016yaa  that the 1-PR diagrams can compete with the
contributions of the 1-PI diagrams in the strong-field regime. This is indeed
also seen when we compare the values attained by the two contributions in
figs. 10 and 11. We observe that for $g_{E}\sim\mathcal{O}(10^{3})$, the 1-PI
and 1-PR contributions to the EHL for a magnetic field background are of the
same order. Already at the order of $g_{E}\sim 10^{3}$, we realize that the
1-PR contributions cannot be ignored.

Another curious property of the 1-PR diagrams in sQED is the kink in its
imaginary contribution to the 2-loop EHL, as demonstrated in fig. 14. This
kink signifies a change of sign for $g_{E}\sim 2$. Also, the instanton
contribution in eq. (73) suggests that while the value of the imaginary part
of $\mathcal{L}^{(2)}_{\text{1-PR}}$ is negative for small value of $g_{E}$,
the imaginary part changes sign for a sufficiently large $g_{E}$. Physically,
this implies that while 1-PR diagrams hinder the vacuum decay rate in the weak
field regime by contributing negatively to the imaginary part of the EHL, they
start facilitating the pair production rate after a critical electric field
value.

##  4 Summary and conclusions

The program of resurgent extrapolations, resummation methods, approximants and
trans-series ecalle1985fonctions ; 10.1093/oso/9780198535850.002.0003 ;
Sternin:1996 ; Costin:2008 ; DORIGONI2019167914 ; Aniceto:2018bis ;
baker1975essentials ; baker1996pade  have been subjects of intense research in
recent years. At the same time, the study of quantum field theories in strong-
field regimes has profound theoretical and experimental relevance, especially
with the advent of new high-intensity LASER facilities and observation of
astrophysical environments where such large fields are routinely present
Altarelli:2019zea ; Hattori:2023egw ; Fedotov_2023 ; Ruffini:2009hg ;
Kaspi:2017fwg ; Kim:2019joy ; Grasso:2000wj ; Durrer:2013pga . The present
work lies at the interface of these two broad topics.

In this work, we investigated resurgence and Padé-Borel resummation in the
theory of scalar quantum electrodynamics by looking at the resurgent
extrapolation of its weak-field expansions. There have already been a few
pioneering works analysing spinor QED florio ; dunne_higher-loop_2021 ; non-
linear-trident-resummation ; Torgrimsson:2022ndq ; Dunne:2022esi ;
Torgrimsson:2021wcj ; Torgrimsson:2021zob  in the context of resurgent
extrapolations. Remarkably, in contrast to spinor quantum electrodynamics,
scalar electrodynamics has not been studied in much detail in these settings
so far. There have only been a few preliminary comments in the existing
literature Huet:2017ydx . Apart from the fact that scalar quantum
electrodynamics is a relatively well-understood theory, thereby helping us
compare the Padé-Borel reconstructed functions to the full theory relatively
easily, it also may be a prototype for potential models in the dark matter
sector, like millicharged particles HOLDOM1986196 ;
doi:10.1146/annurev.nucl.012809.104433 .

We looked carefully at the 1-loop and 2-loop contributions to the scalar
quantum electrodynamics Euler-Heisenberg Lagrangian in the Schwinger proper
time framework and their corresponding weak-field expansions. In these
contexts, we re-derived a few well-known expressions in the literature in
forms amenable to resurgent analysis while also deriving for the first time
closed-form expressions for others, such as the one-particle reducible
contribution at two loops.

We then investigated in some detail Padé-Borel resummation methods to
reconstruct the 1-loop and 2-loop Euler-Heisenberg Lagrangian of scalar
quantum electrodynamics using only a finite number of terms from their weak-
field expansions. We tried different schemes of Padé-Borel reconstructions in
each case and commented on the optimal prescription. At 1-loop, we pointed out
the curious correspondence that for the Padé-Borel reconstruction, the scalar
quantum electrodynamics Euler-Heisenberg Lagrangian may be mapped to two
copies of the spinor quantum electrodynamics Euler-Heisenberg Lagrangian,
which only differ in their prefactor coefficients and masses.

Among the distinctive aspects of our analysis was the inclusion of one-
particle reducible diagrams in the 2-loop Euler-Heisenberg Lagrangian of
scalar quantum electrodynamics. These contributions were thought to be
irrelevant until recently. We compared our reconstructions of the 1-loop and
2-loop Euler-Heisenberg Lagrangian with the derived analytic closed forms and
results from numerical computations. Remarkably, it is found that even a few
terms from the weak-field expansions may be sufficient to reconstruct the full
function resurgently.

As in the case of spinor quantum electrodynamics, we found that the scalar
quantum electrodynamics Padé-Borel reconstructed Euler-Heisenberg Lagrangian
seems to lose its meromorphic properties at the 2-loop order. This occurs both
for the one-particle irreducible and one-particle reducible parts. For both
spinors and scalars, this is a distinction between the 1-loop and 2-loop Borel
transform counterparts. The Borel transform of the former only has simple
poles in both cases. We analyzed the trans-series structure of both these at
the order of the first instanton contribution.

Through analytic continuation to the electric field case, we studied the
appearance of instabilities in the theory, manifested as scalar particle
Schwinger pair production. By analyzing the large-order behaviour and through
the application of Borel dispersion relations, we were able to derive the
imaginary parts explicitly. Additionally, our findings revealed that the one-
particle reducible contributions to the vacuum decay rate have the surprising
behaviour of suppressing the decay rate in the weak field limit for
$qE/m^{2}\lesssim 2$ and enhancing the decay rate in the strong-field regime.
It would be interesting to explore the implications of this behaviour in
physical settings.

Apart from the germane discussions and expressions in Secs. 2 and 3, some of
our main results are encapsulated in figs. 3, 4, 9, 10, 11, 12, 13, and 14. In
summary, in this work, we studied in relative detail and successfully checked
the efficacy of Padé-Borel reconstructed Euler-Heisenberg Lagrangians in
scalar quantum electrodynamics.

There may be a few avenues for future explorations. Gaining a better
understanding of the singularity and trans-series structures from higher-loop
orders in scalar electrodynamics may be a fertile direction for future
research. Another interesting avenue will be exploring finite temperature and
matter density effects, which are of physical relevance in many scenarios of
interest.

###### Acknowledgements.

DG acknowledges support from an INSPIRE fellowship from the Department of
Science and Technology, Government of India.

##  Appendix A Closed form for integral expressions

It is well known that integrals like the ones in eqs. (41) and (29) can be
expressed in the closed form in terms of Hurwitz-Zeta functions
$\zeta_{H}(s,a)$ DUNNE_2005 ; titchmarsh1986theory . For completeness, we will
derive these forms in this appendix, in our conventions and notations,
following the ideas developed in Karbstein_2015 ; Karbstein_2013 .

Let us start with the integral in eq. (41), which we will denote by $I_{1}$:

$I_{1}=\int_{0}^{\infty}\frac{ds}{s^{2}}\ e^{-m^{2}s/qB}\left[\frac{1}{\sinh
s}-\frac{1}{s}+\frac{s}{6}\right]\;.$ (74)

There are some identities that we will use extensively in the coming
derivations (see the identities in (apostol1998introduction, , p. 251))

$\displaystyle\int_{0}^{\infty}ds\ s^{\nu-1}e^{-\beta s}=\beta^{-\nu}\
\Gamma(\nu)\;,$ (75a) $\displaystyle\int_{0}^{\infty}ds\
s^{\nu-1}\frac{1}{\sinh s}\ e^{-\beta s}=2^{1-\nu}\ \Gamma(\nu)\
\zeta_{H}\left(\nu,\frac{1+\beta}{2}\right)\;.$ (75b)

These identities are only valid for $\text{Re}\ \beta>0$ and $\text{Re}\
\nu>0$, and hence cannot be applied to eq. (74) directly. To perform an
appropriate analytic continuation, we introduce the parameter $\epsilon$ in
the integral:

$I_{1}=\lim_{\epsilon\to 0}\int_{0}^{\infty}\frac{ds}{s^{2-\epsilon}}\
e^{-m^{2}s/qB}\left[\frac{1}{\sinh s}-\frac{1}{s}+\frac{s}{6}\right]\;.$ (76)

The individual integrals in eq. (76) can now be performed and, in the leading
and first subleading order, be written as

$\int_{0}^{\infty}\frac{ds}{s^{2-\epsilon}}\ \frac{1}{\sinh s}\
e^{-m^{2}s/qB}=-\frac{4}{\epsilon}\
\zeta_{H}\left(-1,\frac{1+m^{2}/qB}{2}\right)+\zeta_{H}\left(-1,\frac{1+m^{2}/qB}{2}\right)(4\log
2-4+4\gamma)-\\\ 4\zeta_{H}^{\prime}\left(-1,\frac{1+m^{2}/qB}{2}\right)\;,$
(77)

where the derivative of the Hurwitz-zeta function is with respect to the first
argument and

$\displaystyle\int_{0}^{\infty}\frac{ds}{s^{3-\epsilon}}\
e^{-m^{2}s/qB}=\frac{1}{2}\left(\frac{m^{2}}{qB}\right)^{2}\left[\frac{1}{\epsilon}+\frac{3}{2}-\gamma-\log\left(\frac{m^{2}}{qB}\right)\right]\;,$
(78a) $\displaystyle\int_{0}^{\infty}\frac{ds}{s^{1-\epsilon}}\
e^{-m^{2}s/qB}=\frac{1}{\epsilon}-\log\left(\frac{m^{2}}{qB}\right)-\gamma\;.$
(78b)

Using the property of the zeta function,

$\zeta_{H}\left(-1,\frac{1+x}{2}\right)=\frac{1}{24}-\frac{x^{2}}{8}\;,$ (79)

we see that the divergent parts in $I_{1}$ cancel out, and the finite part
that remains is

$I_{1}=-4\left[\zeta_{H}^{\prime}\left(-1,\frac{1+m^{2}/qB}{2}\right)+\zeta_{H}\left(-1,\frac{1+m^{2}/qB}{2}\right)\left(\log\left(\frac{m^{2}}{2qB}\right)+1\right)+\frac{3}{16}\left(\frac{m^{2}}{qB}\right)^{2}\right]\;.$
(80)

Now, we will try to perform similar manipulations on the integral in eq. (29).
In this case, the following manipulation using integration by parts will come
in handy:

$\begin{split}&\int_{0}^{\infty}\frac{ds}{s^{3}}\
e^{-m^{2}s/qB}\left[\frac{s}{\sinh s}-\frac{s^{2}}{\sinh s\tanh
s}+\frac{s^{2}}{3}\right]\\\ &=\int_{0}^{\infty}\frac{ds}{s^{2}}\
e^{-m^{2}s/qB}\ \derivative{s}\left[\frac{s}{\sinh
s}+\frac{s^{2}}{6}-1\right]\\\
&=\frac{m^{2}}{qB}\int_{0}^{\infty}\frac{ds}{s^{2}}\
e^{-m^{2}s/qB}\left[\frac{s}{\sinh
s}+\frac{s^{2}}{6}-1\right]+2\int_{0}^{\infty}\frac{ds}{s^{3}}\
e^{-m^{2}s/qB}\left[\frac{s}{\sinh s}+\frac{s^{2}}{6}-1\right]\;.\end{split}$
(81)

We can now calculate the closed form $I_{2}$ using the same techniques we used
above to get that

$I_{2}=-4\zeta_{H}\left(-1,\frac{1+m^{2}/qB}{2}\right)-8\zeta_{H}^{\prime}\left(-1,\frac{1+m^{2}/qB}{2}\right)+2\left(\frac{m^{2}}{qB}\right)^{2}\zeta_{H}^{\prime}\left(0,\frac{1+m^{2}/qB}{2}\right)\\\
-\frac{1}{3}\log\left(\frac{m^{2}}{2qB}\right)\;.$ (82)

##  Appendix B Derivation of 1-PI effective action in Schwinger proper time

To simplify eq. (20), we need to evaluate the second functional derivative of
$\overline{W}^{(1)}$ with respect to the background gauge field. We already
know the first functional derivative from eq. (12),

$\frac{\delta\overline{W}^{(1)}}{\delta
A_{\mu}(x)}=\partialderivative{\mathcal{L}_{\phi}}{A_{\mu}}\
(x)=-2iq\lim_{x\to y}D^{\mu}_{x}G_{A}(x-y)\;.$ (83)

From this expression, it is apparent that we will need the functional
derivative of $G_{A}$ with respect to $A_{\mu}(x)$ to proceed further in the
calculations. To do this, we start with the Green’s function equation,

$(D_{z}^{2}+m^{2})G_{A}(z-y)=-i\delta^{4}(z-y)\;,$ (84)

where $D^{\mu}_{z}=\partial^{\mu}_{z}+iqA^{\mu}(z)$ is the covariant
derivative. Now, we multiply by the free scalar propagator $G_{F}(x-z)$ on
both sides and integrate over the $z$ space, expand the covariant derivative,
and use the identity

$G_{F}(x-z)\partial_{z}^{2}G_{A}(z-y)=G_{A}(z-y)\partial_{z}^{2}G_{F}(x-z)-\partial_{z}(G_{A}(z-y)\partial_{z}G_{F}(x-z))+\partial_{z}(G_{F}(x-z)\partial_{z}G_{A}(z-y))$
(85)

to get the relation

$-iG_{A}(x-y)-q^{2}\int d^{4}z\ A^{2}(z)G_{A}(z-y)G_{F}(x-z)+2iq\int d^{4}z\
A(z)G_{F}(x-z)\partial_{z}G_{A}(z-y)+\\\ iq\int d^{4}z\ \partial
A(z)G_{F}(x-z)G_{A}(z-y)=-iG^{F}(x-y)\;.$ (86)

This equation, in the operator notation, can be written as

$\widehat{G}_{A}=\widehat{G}_{F}+iq^{2}\widehat{G}_{F}\widehat{A}^{2}\widehat{G}_{A}-2iq\widehat{G}_{F}\widehat{A}\widehat{p}\widehat{G}_{A}+q\widehat{G}_{F}\partial\widehat{A}\widehat{G}_{A}\;,$
(87)

where $\widehat{p}^{\mu}$ is the momentum operator,
$\widehat{A}^{\mu}=A^{\mu}(\widehat{x})$, the operator $\widehat{G}_{A}$ is
defined as $\bra{x}\widehat{G}_{A}\ket{y}=G_{A}(x-y)$, and $\widehat{G}_{F}$
is defined in a similar manner. It will be useful to keep in mind that this
equation can be rearranged to give

$\widehat{G}_{A}=(1-iq^{2}\widehat{G}_{F}\widehat{A}^{2}+2iq\widehat{G}_{F}\widehat{A}\widehat{p}-q\widehat{G}_{F}\partial\widehat{A})^{-1}\widehat{G}_{F}\;.$
(88)

Now, we will take a functional derivative on both sides of eq. (86) to get
that

$\frac{\delta G_{A}(x-y)}{\delta A_{\mu}(w)}=\\\ iq^{2}\int d^{4}z\
\left[2G_{F}(x-z)A^{\mu}(z)\delta^{4}(z-w)G_{A}(z-y)+G_{F}(x-z)A^{2}(z)\frac{\delta
G_{A}(z-y)}{\delta A_{\mu}(w)}\right]\\\ +2q\int d^{4}z\
\left[G_{F}(x-z)\delta^{4}(z-w)\partial_{z}G_{A}(z-y)+G_{F}(x-z)A(z)\partial_{z}\frac{\delta
G_{A}(z-y)}{\delta A_{\mu}(w)}\right]\\\ +q\int d^{4}z\
\left[G_{F}(x-z)\partial^{\mu}_{z}(\delta^{4}(z-w))G_{A}(z-y)+G_{F}(x-z)\partial
A(z)\frac{\delta G_{A}(z-y)}{\delta A_{\mu}(w)}\right]\;.$ (89)

A couple of simplifications can be done to arrive at the final answer–

$\partial^{\mu}_{z}(\delta^{4}(z-w))G_{A}(x-z)G_{A}(z-y)=-\delta^{4}(z-w)\partial^{\mu}_{z}(G_{A}(x-z)G_{A}(z-y))\;,$
(90)

and

$\begin{split}\partial^{\mu}_{z}\bra{x}\widehat{G}_{A}\ket{z}&=(\partial^{\mu}_{z}\bra{z}\widehat{G}_{A}^{\dagger}\ket{x})^{*}\;,\\\
&=(-i\bra{z}\widehat{p}^{\mu}\widehat{G}_{A}^{\dagger}\ket{x})^{*}\;,\\\
&=i\bra{x}\widehat{G}_{A}\widehat{p}^{\mu}\ket{z}\;.\end{split}$ (91)

After this simplification, we will use eq. (88) to arrive at the first
functional derivative of $G_{A}$,

$\frac{\delta\widehat{G}_{A}}{\delta
A_{\mu}(w)}=-iq\bigg{[}\bigg{(}\widehat{G}_{A}\ket{w}\bra{w}\widehat{\Pi}^{\mu}\widehat{G}_{A}\bigg{)}+\bigg{(}\widehat{G}_{A}\widehat{\Pi}^{\mu}\ket{w}\bra{w}\widehat{G}_{A}\bigg{)}\bigg{]}\;,$
(92)

where $\widehat{\Pi}^{\mu}=\widehat{p}^{\mu}-\widehat{A}^{\mu}$. Substituting
this in eq. (20), we have that

$W^{(2)}_{\text{1-PI}}[A]=-iq^{2}\int
d^{4}xd^{4}y\bigg{[}4i\delta^{4}(x-y)\bra{x}\widehat{G}_{A}\ket{y}-\bra{x}\widehat{\Pi}^{\mu}\widehat{G}_{A}\ket{y}\bra{y}\widehat{\Pi}_{\mu}\widehat{G}_{A}\ket{x}-\\\
\bra{x}\widehat{\Pi}^{\mu}\widehat{G}_{A}\widehat{\Pi}_{\mu}\ket{y}\bra{y}\widehat{G}_{A}\ket{x}\bigg{]}D_{+}(x-y)\;.$
(93)

This expression can be computed by substituting the values of the photon and
dressed electron propagator in proper time form Schwinger_1951 –

$D_{+}(x-y)=\frac{i}{16\pi^{2}}\int_{0}^{\infty}\frac{ds}{s^{2}}\
\exp\left[\frac{-i(x-y)^{2}}{4s}-\varepsilon s\right]\;,$ (94)

and

$G_{A}(x-y)=-\frac{i}{16\pi^{2}}\Phi(x,y)\int_{0}^{\infty}\frac{ds}{s^{2}}\
e^{-s\epsilon}\exp\left[-\frac{i}{4}z^{\mu}\beta_{\mu\nu}(s)z^{\nu}-L(s)-ism^{2}\right]\;,$
(95)

where $\Phi(x,y)$ is the gauge-dependent phase the final answer is independent
of, $z=x-y$, and the matrix $\beta$ and scalar $L(s)$ is given by

$\displaystyle\beta=qF\coth(qFs)\;,$ (96a) $\displaystyle
L(s)=\frac{1}{2}\text{Tr}\log[(qFs)^{-1}\sinh(qFs)]\;.$ (96b)

($F$ without the tensor indices denotes the electromagnetic matrix that
follows the conventional rules of matrix multiplication). Now, we can easily
compute that

$\begin{split}\bra{x}\widehat{\Pi}^{\mu}G_{A}\ket{y}&=(i\partial_{\mu}-qA_{\mu})G_{A}(x-y)\;,\\\
&=-\frac{i}{16\pi^{2}}\Phi(x,y)\int_{0}^{\infty}\frac{ds}{s^{2}}\
e^{-s\epsilon}\frac{1}{2}\bigg{(}qF^{\mu\nu}z_{\mu}+\beta^{\mu\nu}(s)z_{\nu}\bigg{)}\exp\left[-\frac{i}{4}z\beta(s)z-L(s)-ism^{2}\right]\;,\end{split}$
(97)
$\begin{split}\bra{x}G_{A}\widehat{\Pi}^{\mu}\ket{y}&=\bigg{(}\bra{y}\widehat{\Pi}_{\mu}(\widehat{G}_{A})^{*}\ket{x}\bigg{)}^{*}\;,\\\
&=-\frac{i}{16\pi^{2}}\Phi(x,y)\int_{0}^{\infty}\frac{ds}{s^{2}}\
e^{-s\epsilon}\frac{1}{2}\bigg{(}-qFz+\beta(s)z\bigg{)}\exp\left[-\frac{i}{4}z\beta(s)z-L(s)-ism^{2}\right]\;.\end{split}$
(98)

To find $\bra{x}\widehat{\Pi}^{\mu}G_{A}\widehat{\Pi}_{\mu}\ket{y}$, we can
use a relation that is apparent from eqs. (97) and (98):

$\widehat{G}_{A}\widehat{\Pi}_{\mu}=\widehat{\Pi}_{\mu}\widehat{G}_{A}-qFz\widehat{G}_{A}\;.$
(99)

Multiplying both sides by $\widehat{\Pi}^{\mu}$, taking an expectation value,
and using Green’s equation for the evolution of $G_{A}$, we have the
simplified relation

$\bra{x}\widehat{\Pi}^{\mu}\widehat{G}_{A}\widehat{\Pi}_{\mu}\ket{y}=i\delta^{4}(x-y)+m^{2}G_{A}(x-y)-\frac{q^{2}}{2}(zFFz)G_{A}(x-y)\;.$
(100)

The multiplicative term $e^{-L(s)}$ can be simplified as

$\begin{split}\exp\left[-\frac{1}{2}\text{Tr}\log\left(\frac{\sinh(qFs)}{qFs}\right)\right]&=\text{det}\left[\exp\left(-\frac{1}{2}\log\left(\frac{\sinh(qFs)}{qFs}\right)\right)\right]\;,\\\
&=\text{det}\left[\frac{qFs}{\sinh(qFs)}\right]^{1/2}\;,\\\
&=\frac{q^{2}s^{2}\eta\epsilon}{\sin(q\eta s)\sinh(q\epsilon
s)}\;,\end{split}$ (101)

where $\eta$ and $\epsilon$ are the eigenvalues of $F^{\mu\nu}$ Schwinger_1951
. All the terms that contain the delta function factor $\delta^{4}(x-y)$ are
divergent and, in fact, renormalize the mass of the 1-loop EHL. Finally, there
are two types of terms left in the action:

$R_{1}=-m^{2}\int d^{4}zG_{A}(z)D(z)G_{A}(-z)\;,$ (102)

and

$R_{2}=\int d^{4}z\ \left(\frac{3q^{2}}{4}zFFz+\frac{1}{4}z\beta\beta
z\right)G_{A}(z)D(z)G_{A}(-z)\;.$ (103)

Using eq. (101) and straightforward Gaussian integrations, we arrive at the
unrenormalized result v_i_ritus_connection_1977

$\mathcal{L}^{(2)}_{\text{1-PI}}=\frac{iq^{2}}{256\pi^{4}}\int_{0}^{\infty}ds\int_{0}^{\infty}ds^{\prime}\
\frac{(q\eta)^{2}(q\epsilon)^{2}e^{-i(s+s^{\prime})(m^{2}-i\epsilon)}}{\sin(q\eta
s)\sin(q\eta s^{\prime})\sinh(q\epsilon s)\sinh(q\epsilon
s^{\prime})}\left(m^{2}I_{0}+\frac{iI}{2}\right)\;,$ (104)

where

$\displaystyle
I=\frac{(q-p)\log(b/a)}{(b-a)^{2}}-\frac{\frac{q}{b}-\frac{p}{a}}{b-a}\;,$
(105a) $\displaystyle I_{0}=\frac{1}{b-a}\log\left(\frac{b}{a}\right)\;,$
(105b)

with the definitions

$\displaystyle a=q\eta\cot(q\eta s)+q\eta\cot(q\eta s^{\prime})\;,$ (106a)
$\displaystyle b=q\epsilon\coth(q\epsilon s)+q\epsilon\coth(q\epsilon
s^{\prime})\;,$ (106b) $\displaystyle p=2(q\eta)^{2}(\cot(q\eta s)\cot(q\eta
s^{\prime})+3)\;,$ (106c) $\displaystyle q=2(q\epsilon)^{2}(\coth(q\epsilon
s)\coth(q\epsilon s^{\prime})-3)\;.$ (106d)

As in the case of the 1-loop EHL, the divergences in this integral are due to
the vacuum energy and a term that is proportional to the free Lagrangian.
These divergences can be removed by expanding the integrand in the limit
$\eta\to 0$, $\epsilon\to 0$ and subtracting the resultant divergences. After
a lengthy procedure of renormalization, contour rotation, and the change of
variables

$\displaystyle t$ $\displaystyle=s+s^{\prime}\;,$ (107a) $\displaystyle u$
$\displaystyle=\frac{s^{\prime}}{s+s^{\prime}}\;,$ (107b)

we get the final result stated in eq. (25).

##  Appendix C Borel Dispersion relations

The Borel dispersion relations establish a link between the asymptotic
behavior of weak-field coefficients and the non-perturbative dynamics of the
Borel sum observed in the strong-field regime. Consider the formal series in
eq. (31). For our purposes, we will state the modified result of the Borel
dispersion relation for weak-field coefficients that have large-order
asymptotics of the form

$a_{n}\sim\ \sum_{\mathcal{P}=1}^{\infty}C_{\mathcal{P}}\
(iA_{\mathcal{P}})^{-2n-k_{\mathcal{P}}-1}\ \Gamma(2n+k_{\mathcal{P}}+1)\ ,\
n\to\infty\;.$ (108)

The corresponding Borel transform,

$\widehat{\mathcal{S}}_{p,N^{*}}(z)=\sum_{n=0}^{N^{*}}\frac{a_{n}}{(2n+p)!}z^{2n+p}\;,$
(109)

will have singularities at $\pm iA_{\mathcal{P}}$ in the limit
$N^{*}\to\infty$. The singularity structure of $\hat{S}_{p,N^{*}}$ can be, for
example, algebraic, such that

$\widehat{\mathcal{S}}_{p}(\pm iA_{\mathcal{P}}\pm i\xi)\sim(\pm 1)^{p}\
\widetilde{C}_{\mathcal{P}}\ (-\xi)^{-b}\ ,\ \xi\to 0\;,$ (110)

or can have a simple pole such that

$\widehat{\mathcal{S}}_{p}(\pm iA_{\mathcal{P}}\pm i\xi)\sim-(\pm 1)^{p}\
\frac{\widetilde{C}_{\mathcal{P}}}{i\xi}\ ,\ \xi\to 0\;,$ (111)

where
$\widehat{\mathcal{S}}_{p}=\lim_{N^{*}\to\infty}\widehat{\mathcal{S}}_{p,N^{*}}$.
To find out the nature of the singularity, we consider a contour integral of
the function $\widehat{\mathcal{S}}_{p}$ around the origin, along the contour
shown in fig. 16(a). By the Cauchy integral theorem, we know that

$\frac{a_{n}}{\Gamma(2n+p+1)}=\frac{1}{2\pi i}\oint_{\mathcal{C}_{0}}d\zeta\
\frac{\widehat{\mathcal{S}}_{p}(\zeta)}{\zeta^{2n+p+1}}\;.$ (112)

(a)

(b)

Figure 16: The contours used for deriving the Borel dispersion relation in eq.
(118). The Cauchy contour integral in fig. 16(a) is deformed to surround the
poles of the Borel sum $\widehat{\mathcal{S}}(\zeta)$ in fig. 16(b). The poles
of $\widehat{\mathcal{S}}$ are denoted by filled circles on the imaginary
axis.

Now, this contour near the origin can be deformed in the way described in fig.
16 to obtain the so-called “Hankel contour” (see, for example,
krantz2012handbook ; marino_2015 ). This contour contains two parts– the
circular part $\mathcal{C}_{\infty}$, the integration over which vanishes for
a very large radius, and the Hankel contours $\mathcal{H}_{A}$ and
$\mathcal{H}_{-A}$ around the singularities of $\widehat{\mathcal{S}}$. The
integral around the Hankel contours $\mathcal{H}_{A}$ and $\mathcal{H}_{-A}$
have the same contribution because of the symmetry between the poles
$iA_{\mathcal{P}}$ and $-iA_{\mathcal{P}}$. Using all of these facts, we can
now easily do the contour integral over $\mathcal{H}_{A}$ using the
singularity structure of $\widehat{\mathcal{S}}$. For instance, when
$\widehat{\mathcal{S}}$ only has simple poles, one gets the result

$\frac{a_{n}}{\Gamma(2n+p+1)}\sim
2\sum_{\mathcal{P}=1}^{\infty}\widetilde{C}_{\mathcal{P}}(iA_{\mathcal{P}})^{-2n-p-1}\
,\ n\to\infty\;.$ (113)

Comparing this with the large-order behaviour that we started with in eq.
(108), we get that

$p=k_{\mathcal{P}}\;,$ (114)

and

$\widetilde{C_{\mathcal{P}}}=\frac{C_{\mathcal{P}}}{2}\;.$ (115)

Now, we can also use the singularity structure of $\widehat{\mathcal{S}}$ to
find out its contribution to the imaginary part of
$\text{B}_{p}[\mathcal{S}]$. To do this, consider the Borel transform in eq.
(37) with $z=-ig_{E}$ with the variable change $t\to t/z$,

$\text{B}_{p}[\mathcal{S}](g_{E})=\frac{1}{(-ig_{E})^{p+1}}\lim_{N^{*}\to\infty}\int_{0}^{-i\infty}dt\
e^{-it/g_{E}}\ \widehat{\mathcal{S}}_{p,N^{*}}(\zeta)\;.$ (116)

The imaginary part of this integral can be calculated by performing contour
integral around the singularities of $\widehat{\mathcal{S}}$ on the negative
imaginary axis to get that,

$2\ \text{Im}[\text{B}_{p}[\mathcal{S}]](g_{E})=\frac{2\pi
i}{(ig_{E})^{p+1}}\sum_{\mathcal{P}=0}^{\infty}\widetilde{C}_{\mathcal{P}}e^{-A_{\mathcal{P}}/g_{E}}\;.$
(117)

Using these relations, we get a neat relation that gives the imaginary part of
$\text{B}[\mathcal{S}]$ in terms of $C_{\mathcal{P}}$ and $k_{\mathcal{P}}$–

$\text{Im}[\text{B}_{p}[\mathcal{S}]](g_{E})=\frac{\pi
i}{2}\sum_{\mathcal{P}=1}^{\infty}\frac{C_{\mathcal{P}}}{(ig_{E})^{k_{\mathcal{P}}+1}}e^{-A_{\mathcal{P}}/g_{E}}\;.$
(118)

This is the Borel dispersion relation. It can be proved that the same relation
is obtained for the case of algebraic and logarithmic singularity as well. It
will be useful to note that when this procedure is adopted in the case of an
algebraic singularity, we get a relation between $b$, $k_{\mathcal{P}}$, and
$p$ akin to eq. (114)–

$k_{\mathcal{P}}=p+b-1\;.$ (119)

Note the significance of this relation– it says that if $k_{\mathcal{P}}$ is
fractional, $b$ must also be fractional. Therefore, fractional values of
$k_{\mathcal{P}}$ lead to algebraic singularities.

##  Appendix D Coefficient values for the weak field expansion of
$\mathcal{L}^{(2)}_{\text{1-PI}}$ and $\mathcal{L}^{(2)}_{\text{1-PR}}$ for
sQED

While the weak-field coefficients of the 1-loop EHL have a closed form and can
be computed up to arbitrary order, the weak-field expansions of the 2-loop EHL
don’t have any closed form. They are computationally challenging to compute,
even numerically. We list 15 of the numerically computed weak-field
coefficients of both 1-PI and 1-PR contributions of the 2-loop EHL for the
case of sQED.

$n$ | $a_{n}^{(2I)}$ | $a_{n}^{(2R)}$
---|---|---
0 | $\frac{275}{648}$ | $\frac{49}{8100}$
1 | $-\frac{5159}{16200}$ | $-\frac{31}{2700}$
2 | $\frac{751673}{1058400}$ | $\frac{58433}{1587600}$
3 | $-\frac{931061}{291600}$ | $-\frac{5161351}{26195400}$
4 | $\frac{139252117469}{5762988000}$ | $\frac{629915089}{378378000}$
5 | $-\frac{1749526192619}{6324318000}$ | $-\frac{35555913023}{1702701000}$
6 | $\frac{60143228485253}{13471920000}$ | $\frac{2806953752810761}{7641722088000}$
7 | $-\frac{342211558219703077}{3542980240800}$ | $-\frac{552956876118767}{64017954000}$
8 | $\frac{111676851457699685948647}{41379925106520000}$ | $\frac{15229529901011542321}{58256338140000}$
9 | $-\frac{7924897558024929739223}{83765030580000}$ | $-\frac{650336486466192684637}{65776701790800}$
10 | $\frac{121800315777228315482407336783}{29954709995530320000}$ | $\frac{428964436921204922260687}{939667168440000}$
11 | $-\frac{31118443383101537277412172791}{147997579029300000}$ | $-\frac{3012135397426829997023747}{119265448302000}$
12 | $\frac{1372366357724439120548020044517}{106558256901096000}$ | $\frac{6785551823434972770541748948323}{4115850620902020000}$
13 | $-\frac{12360815173481596296265109135631547}{13404026888392140000}$ | $-\frac{8462680145742528698979498502657}{67508660977758000}$
14 | $\frac{101430749192220049807898294140516375455307}{1328991702086778649920000}$ | $\frac{3876135337081567976230866367314231534199}{352968763887563791968000}$
15 | $-\frac{38745242595653082092152082819293043866847}{5358837508414430040000}$ | $-\frac{3885536678316440235723614259967162597697}{3539125306359262620000}$
Table 3: A tabulation of the first fifteen coefficients of the weak-field
expansion of the 1-PI and 1-PR contributions to the 2-loop EHL of sQED.

## References

  * (1) A. M. Polyakov, _Gauge Fields and Strings_ , vol. 3. 1987.
  * (2) P. Deligne, P. Etingof, D. S. Freed, L. C. Jeffrey, D. Kazhdan, J. W. Morgan et al., eds., _Quantum fields and strings: A course for mathematicians. Vol. 1, 2_. 1999.
  * (3) E. Witten, _Duality, space-time and quantum mechanics_ , _Phys. Today_ 50N5 (1997) 28.
  * (4) E. Witten, _APS Medal for Exceptional Achievement in Research: Invited article on entanglement properties of quantum field theory_ , _Rev. Mod. Phys._ 90 (2018) 045003 [1803.04993].
  * (5) C. M. Bender and S. A. Orszag, _Summation of Series_ , in _Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory_ , (New York, NY), pp. 368–416, Springer New York, (1999), DOI.
  * (6) M. Mariño, _Instantons and Large N: An Introduction to Non-Perturbative Methods in Quantum Field Theory_. Cambridge University Press, 2015, 10.1017/CBO9781107705968.
  * (7) F. J. Dyson, _Divergence of perturbation theory in quantum electrodynamics_ , _Phys. Rev._ 85 (1952) 631.
  * (8) J. Ecalle, _Les fonctions resurgentes_ , Publications mathématiques d’Orsay. Université de Paris-Sud. Departement de Mathématique, 1985.
  * (9) B. Shawyer and B. Watson, _Preface_ , in _Borel’s Methods of Summability: Theory and Applications_ , Oxford University Press, (1994).
  * (10) S. B. Yu and S. V. E, _Borel-Laplace transform and asymptotic theory : introduction to resurgent analysis_. CRC, 1996.
  * (11) O. Costin, _Asymptotics and Borel Summability_. Chapman and Hall/CRC, 2008\.
  * (12) D. Dorigoni, _An introduction to resurgence, trans-series and alien calculus_ , _Annals of Physics_ 409 (2019) 167914.
  * (13) I. Aniceto, G. Basar and R. Schiappa, _A Primer on Resurgent Transseries and Their Asymptotics_ , _Phys. Rept._ 809 (2019) 1 [1802.10441].
  * (14) G. Baker, _Essentials of Padé Approximants_. Elsevier Science, 1975.
  * (15) G. Baker and P. Graves-Morris, _Pade Approximants: Encyclopedia of Mathematics and It’s Applications, Vol. 59 George A. Baker, Jr., Peter Graves-Morris_ , Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1996.
  * (16) A. Voros, _The return of the quartic oscillator. the complex wkb method_ , _Annales de l’I.H.P. Physique théorique_ 39 (1983) 211.
  * (17) U. D. Jentschura and J. Zinn-Justin, _Instantons in quantum mechanics and resurgent expansions_ , _Phys. Lett. B_ 596 (2004) 138 [hep-ph/0405279].
  * (18) G. V. Dunne and M. Unsal, _Resurgence and Trans-series in Quantum Field Theory: The CP(N-1) Model_ , _JHEP_ 11 (2012) 170 [1210.2423].
  * (19) P. C. Argyres and M. Unsal, _The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects_ , _JHEP_ 08 (2012) 063 [1206.1890].
  * (20) G. V. Dunne and M. Ünsal, _Continuity and Resurgence: towards a continuum definition of the $\mathbb{CP}$(N-1) model_, _Phys. Rev. D_ 87 (2013) 025015 [1210.3646].
  * (21) M. Mariño, _Lectures on non-perturbative effects in large $N$ gauge theories, matrix models and strings_, _Fortsch. Phys._ 62 (2014) 455 [1206.6272].
  * (22) I. Aniceto, J. G. Russo and R. Schiappa, _Resurgent Analysis of Localizable Observables in Supersymmetric Gauge Theories_ , _JHEP_ 03 (2015) 172 [1410.5834].
  * (23) G. V. Dunne and M. Ünsal, _New Nonperturbative Methods in Quantum Field Theory: From Large-N Orbifold Equivalence to Bions and Resurgence_ , _Ann. Rev. Nucl. Part. Sci._ 66 (2016) 245 [1601.03414].
  * (24) R. Miravitllas Mas, _Asymptotic expansions, resurgence and large order behaviour of quantum chromodynamics_ , Ph.D. thesis, Barcelona, Autonoma U., 2019\.
  * (25) O. Costin and G. V. Dunne, _Resurgent extrapolation: rebuilding a function from asymptotic data. Painlevé I_ , _J. Phys. A_ 52 (2019) 445205 [1904.11593].
  * (26) M. Mariño and T. Reis, _Resurgence for superconductors_ , 1905.09569.
  * (27) J. Bersini, A. Maiezza and J. C. Vasquez, _Resurgence of the renormalization group equation_ , _Annals Phys._ 415 (2020) 168126 [1910.14507].
  * (28) M. Marino and T. Reis, _Resurgence and renormalons in the one-dimensional Hubbard model_ , _SciPost Phys._ 13 (2022) 113 [2006.05131].
  * (29) S. Garoufalidis, J. Gu and M. Marino, _The Resurgent Structure of Quantum Knot Invariants_ , _Commun. Math. Phys._ 386 (2021) 469 [2007.10190].
  * (30) M. Borinsky and D. Broadhurst, _Resonant resurgent asymptotics from quantum field theory_ , _Nucl. Phys. B_ 981 (2022) 115861 [2202.01513].
  * (31) Z. Bajnok, J. Balog, A. Hegedus and I. Vona, _Instanton effects vs resurgence in the O(3) sigma model_ , _Phys. Lett. B_ 829 (2022) 137073 [2112.11741].
  * (32) Z. Bajnok, J. Balog and I. Vona, _Analytic resurgence in the O(4) model_ , _JHEP_ 04 (2022) 043 [2111.15390].
  * (33) L. Di Pietro, M. Mariño, G. Sberveglieri and M. Serone, _Resurgence and 1/N Expansion in Integrable Field Theories_ , _JHEP_ 10 (2021) 166 [2108.02647].
  * (34) N. Dondi, I. Kalogerakis, D. Orlando and S. Reffert, _Resurgence of the large-charge expansion_ , _JHEP_ 05 (2021) 035 [2102.12488].
  * (35) D. H. Wu, _Resurgent analysis of SU(2) Chern-Simons partition function on Brieskorn spheres $\Sigma(2,3,6n+5)$_, _JHEP_ 21 (2020) 008 [2010.13736].
  * (36) Z. Duan and J. Gu, _Resurgence in complex Chern-Simons theory at generic levels_ , _JHEP_ 05 (2023) 086 [2208.14188].
  * (37) S. Brahma, K. Dasgupta, M.-M. Faruk, B. Kulinich, V. Meruliya, B. Pym et al., _Resurgence of a de Sitter Glauber-Sudarshan State: Nodal Diagrams and Borel Resummation_ , _Fortsch. Phys._ 2023 (2022) 2300136 [2211.09181].
  * (38) R. Schiappa, M. Schwick and N. Tamarin, _All the D-Branes of Resurgence_ , 2301.05214.
  * (39) N. Benjamin, S. Collier, A. Maloney and V. Meruliya, _Resurgence, conformal blocks, and the sum over geometries in quantum gravity_ , _JHEP_ 05 (2023) 166 [2302.12851].
  * (40) L. Griguolo, J. Papalini, L. Russo and D. Seminara, _The resurgence of the plateau in supersymmetric ${\cal N}=1$ Jackiw-Teitelboim gravity_, 2310.06768.
  * (41) O. Costin, G. V. Dunne, A. Gruen and S. Gukov, _Going to the Other Side via the Resurgent Bridge_ , 2310.12317.
  * (42) M. Altarelli, R. Assmann, F. Burkart, B. Heinemann, T. Heinzl, T. Koffas et al., _Summary of strong-field QED Workshop_ , 4, 2019, 1905.00059.
  * (43) K. Hattori, K. Itakura and S. Ozaki, _Strong-field physics in QED and QCD: From fundamentals to applications_ , _Prog. Part. Nucl. Phys._ 133 (2023) 104068 [2305.03865].
  * (44) A. Fedotov, A. Ilderton, F. Karbstein, B. King, D. Seipt, H. Taya et al., _Advances in QED with intense background fields_ , _Physics Reports_ 1010 (2023) 1.
  * (45) R. Ruffini, G. Vereshchagin and S.-S. Xue, _Electron-positron pairs in physics and astrophysics: from heavy nuclei to black holes_ , _Phys. Rept._ 487 (2010) 1 [0910.0974].
  * (46) V. M. Kaspi and A. Beloborodov, _Magnetars_ , _Ann. Rev. Astron. Astrophys._ 55 (2017) 261 [1703.00068].
  * (47) S. P. Kim, _Astrophysics in Strong Electromagnetic Fields and Laboratory Astrophysics_ , in _15th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories_ , 5, 2019, 1905.13439.
  * (48) D. Grasso and H. R. Rubinstein, _Magnetic fields in the early universe_ , _Phys. Rept._ 348 (2001) 163 [astro-ph/0009061].
  * (49) R. Durrer and A. Neronov, _Cosmological Magnetic Fields: Their Generation, Evolution and Observation_ , _Astron. Astrophys. Rev._ 21 (2013) 62 [1303.7121].
  * (50) C. Bamber, S. J. Boege, T. Koffas, T. Kotseroglou, A. C. Melissinos, D. D. Meyerhofer et al., _Studies of nonlinear qed in collisions of 46.6 gev electrons with intense laser pulses_ , _Phys. Rev. D_ 60 (1999) 092004.
  * (51) C. Bula, K. T. McDonald, E. J. Prebys, C. Bamber, S. Boege, T. Kotseroglou et al., _Observation of nonlinear effects in compton scattering_ , _Phys. Rev. Lett._ 76 (1996) 3116.
  * (52) D. L. Burke, R. C. Field, G. Horton-Smith, J. E. Spencer, D. Walz, S. C. Berridge et al., _Positron production in multiphoton light-by-light scattering_ , _Phys. Rev. Lett._ 79 (1997) 1626.
  * (53) M. Pentia, C. R. Badita, D. Dumitriu, A. R. Ionescu and H. Petrascu, _The strong field qed approach of the vacuum interaction processes at eli-np_ , 2023\.
  * (54) H. Abramowicz, M. Altarelli, R. Aßmann, T. Behnke, Y. Benhammou, O. Borysov et al., _Letter of intent for the luxe experiment_ , 2019.
  * (55) F. Sauter, _Uber das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs_ , _Z. Phys._ 69 (1931) 742.
  * (56) W. Heisenberg and H. Euler, _Consequences of Dirac’s theory of positrons_ , _Z. Phys._ 98 (1936) 714 [physics/0605038].
  * (57) J. Schwinger, _On gauge invariance and vacuum polarization_ , _Phys. Rev._ 82 (1951) 664.
  * (58) L. Medina and M. C. Ogilvie, _Schwinger Pair Production at Finite Temperature_ , _Phys. Rev. D_ 95 (2017) 056006 [1511.09459].
  * (59) P. Draper, _Virtual and Thermal Schwinger Processes_ , _Phys. Rev. D_ 98 (2018) 125014 [1809.10768].
  * (60) O. Gould and A. Rajantie, _Thermal Schwinger pair production at arbitrary coupling_ , _Phys. Rev. D_ 96 (2017) 076002 [1704.04801].
  * (61) O. Gould, A. Rajantie and C. Xie, _Worldline sphaleron for thermal Schwinger pair production_ , _Phys. Rev. D_ 98 (2018) 056022 [1806.02665].
  * (62) M. Korwar and A. M. Thalapillil, _Finite temperature Schwinger pair production in coexistent electric and magnetic fields_ , _Phys. Rev. D_ 98 (2018) 076016 [1808.01295].
  * (63) G. Torgrimsson, _Thermally versus dynamically assisted Schwinger pair production_ , _Phys. Rev. D_ 99 (2019) 096007 [1902.07196].
  * (64) O. Gould, S. Mangles, A. Rajantie, S. Rose and C. Xie, _Observing Thermal Schwinger Pair Production_ , _Phys. Rev. A_ 99 (2019) 052120 [1812.04089].
  * (65) V. Weisskopf, _The electrodynamics of the vacuum based on the quantum theory of the electron_ , _Kong. Dan. Vid. Sel. Mat. Fys. Med._ 14N6 (1936) 1.
  * (66) G. V. DUNNE, _Heisenberg–Euler effective Lagrangians: basica and extensions_ , in _From Fields to Strings: Circumnavigating Theoretical Physics_ , pp. 445–522, WORLD SCIENTIFIC, (feb, 2005), DOI.
  * (67) I. HUET, M. R. D. TRAUBENBERG and C. SCHUBERT, _THE EULER-HEISENBERG LAGRANGIAN BEYOND ONE LOOP_ , _International Journal of Modern Physics: Conference Series_ 14 (2012) 383.
  * (68) G. V. Dunne, _The Heisenberg-Euler Effective Action: 75 years on_ , _Int. J. Mod. Phys. A_ 27 (2012) 1260004 [1202.1557].
  * (69) H. Gies and F. Karbstein, _An Addendum to the Heisenberg-Euler effective action beyond one loop_ , 1612.07251.
  * (70) C. K. Dumlu and G. V. Dunne, _Stokes phenomenon and schwinger vacuum pair production in time-dependent laser pulses_ , _Physical Review Letters_ 104 (2010) .
  * (71) I. Huet, M. Rausch de Traubenberg and C. Schubert, _Asymptotic Behaviour of the QED Perturbation Series_ , _Adv. High Energy Phys._ 2017 (2017) 6214341 [1707.07655].
  * (72) A. Florio, _Schwinger pair production from Padé-Borel reconstruction_ , _Phys. Rev. D_ 101 (2020) 013007 [1911.03489].
  * (73) G. V. Dunne and Z. Harris, _Higher-loop Euler-Heisenberg transseries structure_ , _Physical Review D_ 103 (2021) 065015.
  * (74) G. Torgrimsson, _Nonlinear trident in the high-energy limit: Nonlocality, coulomb field, and resummations_ , _Phys. Rev. D_ 102 (2020) 096008.
  * (75) G. Torgrimsson, _Resummation of the $\alpha$ expansion for nonlinear pair production by an electron in a strong electromagnetic field_, _Phys. Rev. D_ 107 (2023) 016019 [2207.05031].
  * (76) G. V. Dunne and Z. Harris, _Resurgence of the effective action in inhomogeneous fields_ , _Phys. Rev. D_ 107 (2023) 065003 [2212.04599].
  * (77) G. Torgrimsson, _Resummation of Quantum Radiation Reaction in Plane Waves_ , _Phys. Rev. Lett._ 127 (2021) 111602 [2102.11346].
  * (78) G. Torgrimsson, _Resummation of quantum radiation reaction and induced polarization_ , _Phys. Rev. D_ 104 (2021) 056016 [2105.02220].
  * (79) B. Holdom, _Two u(1)’s and $\epsilon$ charge shifts_, _Physics Letters B_ 166 (1986) 196.
  * (80) J. Jaeckel and A. Ringwald, _The low-energy frontier of particle physics_ , _Annual Review of Nuclear and Particle Science_ 60 (2010) 405 [https://doi.org/10.1146/annurev.nucl.012809.104433].
  * (81) V. I. Ritus, _Connection between strong-field quantum electrodynamics with short-distance quantum electrodynamics_ , _J. Exp. Theor. Phys._ 46 (1977) 423.
  * (82) W. Dittrich and M. Reuter, _Effective Lagrangians in quantum electrodynamics_ , no. 220 in Lecture notes in physics. Springer, Berlin, 1985\.
  * (83) L. C. Martin, C. Schubert and V. M. Villanueva Sandoval, _On the low-energy limit of the QED N photon amplitudes_ , _Nucl. Phys. B_ 668 (2003) 335 [hep-th/0301022].
  * (84) J. P. Edwards, A. Huet and C. Schubert, _On the low-energy limit of the QED N-photon amplitudes: part 2_ , _Nucl. Phys. B_ 935 (2018) 198 [1807.10697].
  * (85) D. A. Dicus, C. Kao and W. W. Repko, _Effective lagrangians and low energy photon-photon scattering_ , _Physical Review D_ 57 (1998) 2443.
  * (86) M. Schwartz, _Quantum Field Theory and the Standard Model_ , Quantum Field Theory and the Standard Model. Cambridge University Press, 2014.
  * (87) E. Titchmarsh and D. Heath-Brown, _The Theory of the Riemann Zeta-function_ , Oxford science publications. Clarendon Press, 1986.
  * (88) T. Apostol, _Introduction to Analytic Number Theory_ , Undergraduate Texts in Mathematics. Springer New York, 1998.
  * (89) C. Schubert, _Perturbative quantum field theory in the string-inspired formalism_ , _Physics Reports_ 355 (2001) 73.
  * (90) N. Ahmadiniaz, J. P. Edwards and A. Ilderton, _Reducible contributions to quantum electrodynamics in external fields_ , _Journal of High Energy Physics_ 2019 (2019) .
  * (91) J. P. Edwards and C. Schubert, _One-particle reducible contribution to the one-loop scalar propagator in a constant field_ , _Nuclear Physics B_ 923 (2017) 339.
  * (92) S. Evans and J. Rafelski, _Improving euler-heisenberg-schwinger effective action with dressed photons_ , _Acta Physica Polonica A_ 143 (2023) S13.
  * (93) S. L. Lebedev and V. I. Ritus, _Virial representation of the imaginary part of the Lagrange function of the electromagnetic field_ , _Sov. Phys. JETP_ 59 (1984) 237.
  * (94) M. Abramowitz and I. Stegun, _Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables_ , Applied mathematics series. Dover Publications, 1965.
  * (95) O. Costin and G. V. Dunne, _Physical Resurgent Extrapolation_ , _Phys. Lett. B_ 808 (2020) 135627 [2003.07451].
  * (96) J. Zinn-Justin, _Field theory: Perturbative expansion and summation methods_ , in _From Random Walks to Random Matrices_ , Oxford University Press, (06, 2019), https://academic.oup.com/book/0/chapter/369312315/chapter-pdf/50126330/oso-9780198787754-chapter-23.pdf, DOI.
  * (97) M. Marino, R. Miravitllas and T. Reis, _On the structure of trans-series in quantum field theory_ , 2302.08363.
  * (98) E. Laenen, C. Marinissen and M. Vonk, _Resurgence analysis of the Adler function at $\mathcal{O}$(1/${N}_{f}^{2}$)_, _JHEP_ 09 (2023) 103 [2302.13715].
  * (99) A. van Spaendonck and M. Vonk, _Painlevé I and exact WKB: Stokes phenomenon for two-parameter transseries_ , _J. Phys. A_ 55 (2022) 454003 [2204.09062].
  * (100) R. Schiappa, M. Schwick and N. Tamarin, _All the D-Branes of Resurgence_ , 2301.05214.
  * (101) L. van den Dries, _Transseries, model theory, and hardy fields_ , _Notices of the American Mathematical Society_ 69 (2022) 557.
  * (102) G. A. Edgar, _Transseries for beginners_ , 2009.
  * (103) J. Guillou and J. Zinn-Justin, _Large-order Behaviour of Perturbation Theory_ , Current Physics Series. North-Holland, 1990.
  * (104) T. Padmanabhan, _Quantum Field Theory: The Why, What and How_ , Graduate Texts in Physics. Springer, 2016, 10.1007/978-3-319-28173-5.
  * (105) U. D. Jentschura, H. Gies, S. R. Valluri, D. R. Lamm and E. J. Weniger, _Qed effective action revisited_ , _Canadian Journal of Physics_ 80 (2002) 267 [https://doi.org/10.1139/p01-139].
  * (106) I. E. Pritsker and R. S. Varga, _The szego curve, zero distribution and weighted approximation_ , _Transactions of the American Mathematical Society_ 349 (1997) 4085.
  * (107) A. CARPENTER, R. VARGA and J. WALDVOGEL, _Asymptotics for the zeros of the partial sums of e z . i_ , _The Rocky Mountain Journal of Mathematics_ 21 (1991) 99.
  * (108) R. S. Varga and A. J. Carpenter, _Zeros of the partial sums of cos(z) and sin(z)._ , _Numerical Algorithms_ 25 (2000) 363.
  * (109) S. P. Kim and D. N. Page, _Schwinger pair production in electric and magnetic fields_ , _Phys. Rev. D_ 73 (2006) 065020 [hep-th/0301132].
  * (110) B. Körs and M. Schmidt, _The effective two-loop Euler-Heisenberg action for scalar and spinor QED in a general constant background field_ , _The European Physical Journal C_ 6 (1999) 175.
  * (111) F. Karbstein, _All-loop result for the strong magnetic field limit of the heisenberg-euler effective lagrangian_ , _Phys. Rev. Lett._ 122 (2019) 211602.
  * (112) F. Karbstein and R. Shaisultanov, _Photon propagation in slowly varying inhomogeneous electromagnetic fields_ , _Physical Review D_ 91 (2015) .
  * (113) F. Karbstein, _Photon polarization tensor in a homogeneous magnetic or electric field_ , _Physical Review D_ 88 (2013) .
  * (114) S. Krantz, _Handbook of Complex Variables_. Birkhäuser Boston, 2012.