††institutetext: Saha Institute of Nuclear Physics, Block-AF, Sector-1, Salt
Lake.
Kolkata 700064, India and Homi Bhabha National Institute, Training School
Complex, Anushakti Nagar.
Mumbai 400094, India

# Schwinger type pair production in Non-SuSy AdS/CFT

Udit Narayan Chowdhury<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study pair production of particles in presence of an external electric
field in a large N Non-supersymmetric Yang Mills theory using the holographic
duality. The dual geometry we consider is asymptotically AdS and is
effectively parametrized by two parameters, $u_{0}$ and
$-\sqrt{\frac{5}{2}}<\delta\leqslant 0$ both of which can be related to the
effective mass of quark/antiquark for Non-supersymmetric theories. We
numerically calculate the inter-quark potential profile and the effective
potential to study pair production and analytically find out the threshold
electric field beyond which one gets catastrophic pair creation by studying
rectangular Wilson loops using the holographic method. We also find out the
critical electric field from DBI analysis of a probe brane. Our initial
investigations reveal, that the critical electric field necessary for
spontaneous pair production increases or decreases w.r.t. its Non-
Supersymmetric value depending on the parameter $\delta$. Ultimately we find
out the pair production rate of particles in presence of an external electric
field by evaluating circular Wilson loops using perturbative methods. From the
later investigation we note the resemblance with our earlier prediction.
However, we also see that for and below another certain value of the parameter
$\delta$ the pair production rate of particle/antiparticle pairs blows up as
the external electric field is taken to zero. We thus infer that the vacuum of
the Non-SuSy gauge theory is unstable for a range of Non-supersymmetric
parameter $\delta$ and that the geometry/Non-SUSY field theory under
consideration has quite different characteristics than earlier reported.

##  1 Introduction

For the last few decades the AdS/CFT correspondence Maldacena:1997re
Witten:1998qj Aharony:1999ti Ammon:2015wua  (relating $\mathcal{N}=4$ Super-
Conformal Yang-Mills in 4 spacetime dimensions to quantum gravity in
asymptotic $AdS_{5}\otimes S^{5}$ spaces) and some of its modifications is one
of the exemplary ideas in theoretical physics. AdS/CFT chiefly comes from
black hole thermodynamicsTownsend:1997ku  and type IIB string theory
Blumenhagen:2013fgp  is thus inherently supersymmetric in nature. This is a
strong-weak duality meaning, strong coupling in the field theory side
corresponds to weak coupling in the Quantum Gravity side and vice-versa.
However even after so many years, no trace of supersymmetry has been found by
experiments and again conformal symmetry is not found quite much in nature.
Thus it is necessary to formulate a modification of AdS/CFT without
supersymmetry and conformal symmetry yet respecting its string
theory/supergravity origins. Such a solution is obtained in Nayek:2016hsi
Constable:1999ch . This solution for D3 branes has two parameters $\delta$ and
$u_{0}$ (i.e. the field theory dual is a two parameter deformation over usual
$\mathcal{N}=4$ Super Yang Mills)and has certain features which makes its an
attractive dual for large N QCD studies via holography Nayek:2018aei . Too the
best of our knowledge, an explicit interpretation of these two parameters in
terms of dual field theory measurables lacks till date. It had been previously
reported that Nayek:2016hsi Constable:1999ch Nayek:2018aei  those include,
running coupling + confinement in the infrared and absence of both SUSY and
conformal symmetry (thus the two parameter deformation commented on above
violates both SUSY and conformal symmetry). Part of what makes AdS/CFT
alluring is that when the field theory coupling is high the corresponding
coupling in the quantum gravity side is low and thus we are left with
classical gravity which is easily computable.

The coupling constant in field theory is usually used as a perturbative
parameter and observables are expressed in a series w.r.t. this parameter,
this is called perturbative field theory. However there are quite some effects
in quantum field theory which cannot be explained as such i.e. non-
perturbative effects. Amongst them the Schwinger Effect stands its ground. The
vacuum of QED or any gauge theory interacting with charged matter is full of
virtual particles and antiparticles (henceforth $q\bar{q}$). In presence of an
external electric field/gauge field this particles get the required energy and
become real particles. There is no magic involved in this. In realistic
situations the energy of the the real $q\bar{q}$ pairs is obtained from the
electric field. Schwinger calculated PhysRev.82.664  the pair production rate
for this process in U(1) gauge theory and obtained,

$\displaystyle\Gamma=\frac{(eE)^{3}}{(2\pi)^{3}}e^{\frac{-\pi m^{2}}{eE}}$ (1)

The exponential suppression hints that pair productions can be modeled as a
tunneling process. Assuming that the virtual $q\bar{q}$ pair has a separation
x the potential on a virtual quark in presence on an external electric field
is given as

$\displaystyle\mathcal{V}_{eff}=-\frac{\alpha}{x}-eEx+2m$ (2)

Imagine this to be the potential barrier though which the $q\bar{q}$ pairs
tunnel out in the opposite direction and become real. For
E$<\frac{m^{2}}{e\alpha}$ there exists two zero points of the potential and
$\mathcal{V}_{eff}$ is positive for intermediate values of x. That means there
is a potential barrier and quarks have to tunnel out through them justifying
the exponential factor stated above. However for E$>\frac{m^{2}}{e\alpha}$ the
potential becomes negative all along and stops putting up a potential barrier,
indicating a catastrophic instability of vacuum where the $q\bar{q}$ are
produced spontaneously. The value of electric field for which the potential
stops putting up a tunneling barrier is called "critical/threshold electric
field" $E_{c}$.

The Schwinger effect in holographic setting was first calculated in
Semenoff:2011ng  (see Gorsky:2001up  for an even earlier work) wherein the
pair modified pair production rate was found to be

$\displaystyle\Gamma\sim\text{exp}\Big{[}-\frac{\sqrt{\lambda}}{2}\Big{(}\sqrt{\frac{E_{c}}{E}}-\sqrt{\frac{E}{E_{c}}}\Big{)}^{2}\Big{]}\;\;\;\;\;;\;\;\;\;\;E_{c}=\frac{2\pi
m^{2}}{\sqrt{\lambda}}$ (3)

This formula matches with the one above for low electric field (much lower
than $E_{c}$). For field much higher than $E_{c}$ we don’t see a exponential
suppression anymore hinting at catastrophic decay. The chief idea of this work
was to place the probe brane at a finite position unlike what is done usually
(placing the probe brane at the conformal boundary of AdS) and then to
calculate the circular Wilson loop. Another approach was pioneered in
Sato:2013iua  which calculated the rectangular Wilson loop for virtual
$q\bar{q}$ pair and relate it to inter-quark potential and then find the
critical electric field from the same. Holographic Schwinger effect for
confining gauge theories have also been studied in literatureSato:2013dwa
Kawai:2015mha , and the confinement manifest itself in the presence of another
"threshold" electric field, below which pair production doesn’t happen at all.
In this work we want to study the Schwinger effect for Non-Supersymmetric
gauge theories via holographic methods using both of this methods, our chief
interest being twofold. On one hand we like to see the theoretical effect
absence of supersymmetry yields on the value of critical electric field at
least for large N Yang Mills theories(and if such a relation can be re-framed
to be an indirect experimental evidence towards presence or absence of
supersymmetry in real world nature). We also like to demonstrate the effect of
confinement (as reported earlier for large N non-supersymmetric YM theories
via holography) towards holographic Schwinger particle decay and look into
exotic results if any. For our purpose the virtual $q\bar{q}$ pairs are
imagined to be endpoint of a string in the boundary. We calculate the
rectangular Wilson loop in space-time direction to find out the inter-quark
potential. To account for an external electric field we add an extra term. We
analytically find out the critical electric field from the same. We also plot
figures to illustrate the tunneling phenomenon. Next we move on to finding out
the critical electric field from analysis of the DBI action using the fact
that the action should be real valued. Then we move on to finding out the
circular Wilson loop. It is impossible to do so without any simplification. We
thus expand the expressions to first order of the Non-SuSy deformation
parameter $(u_{0})^{4}$. Doing so we explicitly find out the profile of
circular Wilson Loop upto first order of $(u_{0})^{4}$ from which we find out
the pair production rate.

This paper is organized as follows. In section 2 we recap Non-SuSy D3 branes
and their decoupling limit from supergravity. We also show that the Non-SuSy
solution goes over to usual AdS when appropriate limits are taken. In section
3 we show the derivation of pair production in theory with U(1) gauge field
coupled to charged matter. Relevant expression for large N gauge theory is
also given. In section 4 we carry on Potential Analysis of virtual $q\bar{q}$
pairs from which the critical electric field is derived both by analytical and
numerical means. In section 5 we use the DBI action and find out the critical
electric field using the fact that the action should be real valued. In
section 6 we use perturbative analysis to find out the profile for circular
Wilson loop when the string ends at a finite position ($u_{b}$). Using this we
find the critical electric field and pair production rate and make some
comments about the later. We close this paper with conclusions in section 7.

##  2 Non SuSy Dp Branes and their Decoupling Limit

In this section we will take a brief recap of non-supersymmetric Dp brane
solutions Lu:2004ms  and show how to recover the BPS Dp brane solutions from
them. Then we will state the decoupling limit of Non-SuSy D3 branes by analogy
with the BPS case and make sure the the decoupling goes over to the BPS brane
decoupling limit when SuSy is restoredNayek:2016hsi . In addition we also show
by taking suitable co-ordinate transformation that the decoupled throat
geometry is actually identical with two parameter solution obtained previously
by Constable and Myres in which supersymmetry and conformal symmetry are both
broken Constable:1999ch . We start with the action for ten dimensional type II
supergravity which in addition to the string frame metric $g_{\mu\nu}$, has a
dilation $\phi$ field and a $(8-p)$ RR from gauge field $F_{[8-p]}$.

$\displaystyle S=\frac{1}{16\pi G_{10}}\int
d^{10}x\sqrt{-\text{det}g_{\mu\nu}}\Big{[}R-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2(8-p)!}F_{[8-p]}^{2}\Big{]}$
(4)

We will be looking for solutions of the above using the ansatz,

$\displaystyle
ds_{str}^{2}=e^{2A(r)}\Big{(}-dt^{2}+dx_{1}^{2}+...+dx_{p}^{2}\Big{)}+e^{2B(r)}\Big{(}dr^{2}+r^{2}d\Omega_{8-p}^{2}\Big{)}$
(5) $\displaystyle F_{[8-p]}=Q\text{Vol}\big{(}\Omega_{8-p}\big{)}$ (6)

In the above the metric has an ISO(p,1) $\times$ SO(9-p) isometry and
represents a magnetically charged p brane in 10 dimensions with magnetic
charge $Q$. It can be shown that the above solution conserves super-symmetry
i.e. saturates the BPS bound if Duff:1994an ,

$\displaystyle(p+1)B(r)+(7-p)A(r)=0$ (7)

Solution of equations of (4) compatible with (5)-(7) leads to usual BPS p
branes. We will be looking for solutions with break the condition (7), and
thus breaks spacetime supersymmetries. In the rest of the paper we will be
concerned with non-supersymmetric D3 brane solution and thus will consider the
case where $p=3$. The non-supersymmetric D3 brane solution is given as,

$\displaystyle
ds^{2}=\tilde{F}(\rho)^{-\frac{1}{2}}G(\rho)^{\frac{\delta}{4}}\bigg{[}-dt^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\bigg{]}+\tilde{F}(\rho)^{\frac{1}{2}}G(\rho)^{\frac{1+\delta}{4}}\bigg{[}\frac{d\rho^{2}}{G(\rho)}+\rho^{2}d\Omega_{5}^{2}\bigg{]}$
$\displaystyle
e^{2\phi}=g_{s}^{2}G(\rho)^{\delta}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;F_{[5]}=\frac{1}{\sqrt{2}}(1+\star)\text{
Q Vol}(\Omega_{5})$ (8)

In the above the functions $\tilde{F}(\rho)$ and $G(\rho)$ are given as,

$\displaystyle\tilde{F}(\rho)=G(\rho)^{\frac{\alpha}{2}}\cosh^{2}\theta-G(\rho)^{-\frac{\beta}{2}}\sinh^{2}\theta$
$\displaystyle G(\rho)=1+\frac{\rho_{0}^{4}}{\rho^{4}}$ (9)

It can be shown that the non-SuSy solution (2) violates the condition (7) and
thus breaks spacetime supersymmetries. In the above $e^{2\phi}$ is the
effective string coupling constant and the solution is characterized by six
parameters i.e, $\alpha,\beta,\delta,\theta,\rho_{0},\text{Q}$, of which
$\rho_{0}$ has the dimensions of length, Q has dimensions of four volume and
others are dimensionless. One should further note from (2) that the solution
given above has a naked singularity at $\rho=0$ and the physical region is
given by $\rho>0$. In the context of string theory one hopes that quantum
fluctuations modify the behavior of the solution near the singularity point.
As $e^{2\phi}$ is the effective string coupling, for the supergravity
description to remain valid one needs the parameter $\delta$ to be less or
equal to zero so as to make the string coupling small. The parameters of the
solutions are not all independent but satisfy some consistency relations like,

$\displaystyle\alpha=\beta\;\;\;\;\;\;\;\;\;,\;\;\;\;\;\;\;\;\;\text{Q}=2\alpha\rho_{0}^{4}\sinh
2\theta\;\;\;\;\;\;\;\;\;,\;\;\;\;\;\;\;\;\;\alpha^{2}+\delta^{2}=\frac{5}{2}$
(10)

In arbitrary dimensions the solutions and the constraints are a bit
complicated and is given in Nayek:2015tta . Just like the BPS D3 brane
solution, the non-SuSy solution too is asymptotically flat. One can recover
the BPS solution from the non-SuSy solution given above by considering the
limits $\rho_{0}\rightarrow 0$ and $\theta\rightarrow\infty$ keeping
$\frac{\alpha}{2}\rho_{0}^{4}(\cosh^{2}\theta+\sinh^{2}\theta)\rightarrow
R^{4}=$fixed. Under this scaling one has $G(\rho)\rightarrow 1$ and
$\tilde{F}(\rho)\rightarrow 1+\frac{R^{4}}{\rho^{4}}$ and $\text{Q}\rightarrow
4R^{4}$ under which the standard BPS solution is regained.

The decoupling limit is a low energy limit in which interactions between the
bulk theory and theory living on the brane vanishes. To work out the
decoupling limit and henceforth the throat geometry one needs to make a change
of variables in analogy with the BPS D3 brane.

$\displaystyle\rho=\alpha^{\prime}u\;\;\;\;\;\;,\;\;\;\;\;\;\rho_{0}=\alpha^{\prime}u_{0}\;\;\;\;\;\;,\;\;\;\;\;\;\alpha\cosh^{2}\theta=\frac{\lambda}{{\alpha{\prime}}^{2}u_{0}^{4}}\;\;\;\;\;\;,\;\;\;\;\;\;\alpha^{\prime}\rightarrow
0$ (11)

In the above $u$ and $u_{0}$ have the dimensions of energy and are kept fixed.
From (10) and (11) it can be shown that $\frac{Q}{{\alpha^{\prime}}^{2}}\gg 1$
implying that the curvature of spacetime in string units must be very small
for the supergravity description to be valid. A justification of the above
decoupling limit is given explicitly in Nayek:2016hsi  and Nayek:2015tta .
Under the above said limit,

$\displaystyle G(\rho)\rightarrow G(u)=1+\frac{u_{0}^{4}}{u^{4}}=\text{fixed}$
(12)
$\displaystyle\tilde{F}(\rho)\rightarrow\tilde{F}(u)=\frac{\lambda}{{\alpha^{\prime}}^{2}}F(u)$
(13)

In the above $F(u)=\frac{1}{\alpha
u_{0}^{4}}\big{(}G(u)^{\frac{\alpha}{2}}-G(u)^{-\frac{\alpha}{2}}\big{)}$ and
the non-SuSy D3 brane throat geometry in the decoupling limit mentioned above
becomes

$\displaystyle
ds^{2}=\alpha^{\prime}\sqrt{\lambda}\Big{[}F(u)^{-\frac{1}{2}}G(u)^{\frac{\delta}{4}}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+F(u)^{\frac{1}{2}}G(u)^{\frac{1+\delta}{4}}\big{(}\frac{du^{2}}{G(u)}+u^{2}d\Omega_{5}^{2}\big{)}\Big{]}$
$\displaystyle e^{2\phi}=g_{s}^{2}G(u)^{\delta}$ (14)

In the above the spacetime co-ordinates has been rescaled as
$(t,x^{i})\rightarrow\sqrt{\lambda}(t,x^{i})$ where $\lambda$ is the ’t hooft
coupling. In the limit $u_{0}\rightarrow 0$ one has $G(u)\rightarrow 1$ and
$F(u)=\frac{1}{\alpha u_{0}^{4}}\big{[}\frac{\alpha
u_{0}^{4}}{u^{4}}+\mathcal{O}(\frac{u_{0}^{8}}{u^{8}})\big{]}\approx u^{4}$.
In this limit the non-SuSy throat geometry (2) goes over to the known
AdS${}_{5}\times$S5, and the effective string coupling becomes constant. To
check the relation of solution (2) with that of the previously known one by
Constable and Myres Constable:1999ch  which was conjectured to be dual to some
non-supersymmetric field theory, one has to re-write the solution in the
Einstein frame,

$\displaystyle
ds^{2}_{E}=\alpha^{\prime}\sqrt{\lambda}\Big{[}H(u)^{-\frac{1}{2}}G(u)^{\frac{\alpha}{4}}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+H(u)^{\frac{1}{2}}G(u)^{\frac{1-\alpha}{4}}\Big{(}\frac{du^{2}}{G(u)}+u^{2}d\Omega_{5}^{2}\Big{)}\Big{]}$
$\displaystyle e^{2\phi}=g_{s}^{2}G(u)^{\delta}$ (15)

In the above the function $H(u)$ is defined by
$H(u)=G(u)^{\frac{\alpha}{2}}F(u)=G(u)^{\alpha}-1$. Now one has to make a co-
ordinate transformation like
$u=r\Big{(}1+\frac{\omega^{4}}{r^{4}}\Big{)}^{-\frac{1}{4}}$ where
$\omega^{4}=\frac{u_{0}^{4}}{4}$. Under this transformation,
$G(u)\rightarrow\Big{(}1+2\frac{\omega^{4}}{r^{4}}\Big{)}^{2}$ and
$H(u)\rightarrow\Big{(}1+2\frac{\omega^{4}}{r^{4}}\Big{)}^{2\alpha}-1$. From
these relations and (2) one can exactly produce the two parameter family of
solutions as found in Constable:1999ch  in which both supersymmetry and
conformal symmetry is broken. The solution in Constable:1999ch  also exhibits
QCD like behavior like running gauge coupling and confinement in the infrared.
The geometry (2) exhibits a naked singularity at $u=0$, and thus should be
corrected by stringy corrections which should become dominant at low length
scales. Moreover the proper distance (spatial) from the exterior (say
$u=u_{b}$) to the interior is finite (which says that stringy corrections are
a must). In holography the proper distance is identified with mass of the
string hanging from the boundary to the interior Kinar:1998vq . To find the
same we have to choose a gauge of the form : $x_{0}=t$, $u=s$, $\text{all
others}=\text{constant}$. With this gauge the mass is given by

$\displaystyle
m=\frac{\sqrt{\lambda}}{2\pi}\int_{0}^{u_{b}}\;du\sqrt{\bigg{(}1+\frac{u_{0}^{4}}{u^{4}}\bigg{)}^{\frac{2\delta-3}{4}}}=\text{finite
and positive for all allowed values of}\;\delta.$ (16)

The integral can indeed be done in closed form. However the result is very
complicated (hypergeometric functions involved), and it is very difficult to
invert $u_{b}$ in terms of $m$. Thus we express our results in this work with
formula for mass($m_{0}$) of $\mathcal{N}=4$ SYM.

$\displaystyle m_{0}=\frac{\sqrt{\lambda}}{2\pi}u_{b}$ (17)

##  3 Pair Production in presence of External Fields

In this section we will revisit the concept of pair production in presence of
external electric fields i.e the "Schwinger Effect". We will demonstrate the
effect using euclidean version of the electromagnetic action Affleck:1981bma
and generalize to large N gauge theories. The euclideanized version of U(1)
gauge theory coupled to a massive complex scalar field is given by

$\displaystyle S=\int
d^{4}x\Big{[}\frac{1}{4}F_{\mu\nu}^{2}+|(\partial_{\mu}+ieA_{\mu}+iea^{ex}_{\mu})\phi|^{2}+m^{2}|\phi|^{2}\Big{]}$
(18)

In the above $A_{\mu}$ refers to the dynamical U(1) gauge field and
$a_{\mu}^{ex}$ refers to the external value of (constant) electromagnetic
field. The pair production rate, $\Gamma$ can be written as Itzykson:1980rh

$\displaystyle V\Gamma$ $\displaystyle=-2\;\text{Im
ln}\int\mathcal{D}A\mathcal{D}\phi\;e^{-S}$ $\displaystyle=-2\;\text{Im
ln}\int\mathcal{D}Ae^{-S_{eff}}$ (19)

Where, $S_{eff}=\frac{1}{4}\int d^{4}xF_{\mu\nu}^{2}+\text{tr
ln}\big{[}-\big{(}\partial_{\mu}+ieA_{\mu}+iea^{ex}_{\mu}\big{)}^{2}+m^{2}\big{]}$.
For leading order calculations one can ignore the coupling of the dynamical
gauge field with the scalar field. Thus the expression above reduces to

$\displaystyle V\Gamma=-2\;\text{Im tr
ln}\Big{[}-(\partial_{\mu}+iea^{ex}_{\mu})^{2}+m^{2}\Big{]}$ (20)

Using the relation, $\text{tr
ln}(A)=-\int_{0}^{\infty}\;\frac{dT}{T}\;\text{tr}e^{-AT}$ and evaluating the
trace in position basis, one can rewrite the above expression to

$\displaystyle
V\Gamma=\text{Im}\int_{0}^{\infty}\;\frac{dT}{T}\;e^{-\frac{m^{2}T}{2}}\int
d^{4}x\;\langle
x|\exp\bigg{[}-T\Big{\\{}-(\partial_{\mu}+iea^{ex}_{\mu})^{2}\Big{\\}}\bigg{]}|x\rangle$
(21)

Note that the integrand under $d^{4}x$ is synonymous to the path integral of a
non-relativistic particle under the influence of the Hamiltonian
$H=\frac{1}{2}\Big{[}P_{\mu}+ea^{ex}_{\mu}\Big{]}^{2}$. Using quantum
mechanical path integral representation Schubert:2001he , one can write

$\displaystyle V\Gamma=$
$\displaystyle\text{Im}\int_{0}^{\infty}\;\frac{dT}{T}\;e^{-\frac{m^{2}T}{2}}\int_{x(0)=x(T)}\mathcal{D}x\exp\Big{[}-\frac{1}{2}\int_{0}^{T}d\tau\dot{x}^{2}+ie\oint
a^{ex}_{\mu}dx_{\mu}\Big{]}$ $\displaystyle=$
$\displaystyle\text{Im}\int_{0}^{\infty}\;\frac{dT}{T}\int_{x(0)=x(1)}\mathcal{D}x\exp\Big{[}-\frac{1}{2T}\int_{0}^{1}d\tau\dot{x}^{2}-\frac{m^{2}T}{2}+ie\oint
a^{ex}_{\mu}dx_{\mu}\Big{]}$ (22)

Where in the last line we have rescaled $\tau\rightarrow\frac{1}{T}\tau$. We
assume $m^{2}\int_{0}^{1}d\tau\dot{x}^{2}\gg 1$ (a condition signifying heavy
mass) and note that the integration over $T$ has the form of a modified Bessel
function
$K_{0}(x)=\int_{0}^{\infty}\frac{dt}{t}\exp\Big{(}-t-\frac{x^{2}}{4t}\Big{)}$
with the asymptotic behavior, $K_{0}(x)\simeq\sqrt{\frac{\pi}{2x}}e^{-x}$, for
large $x$. Thus the above integral becomes

$\displaystyle
V\Gamma=\text{Im}\int\mathcal{D}x\exp\big{[}-S_{p}\big{]}\frac{1}{m}\sqrt{\frac{2\pi}{T_{0}}}$
(23)

In the above , $T_{0}=\frac{1}{m}\sqrt{\int d\tau\dot{x}^{2}}$ and
$S_{p}=m\sqrt{\int d\tau\dot{x}^{2}}-ie\oint a^{ex}_{\mu}dx_{\mu}$ and
$a_{1}^{ex}=-iEx_{0}$ (signifying constant electric field of value E in
$x_{1}$ direction, iota comes in due to euclidean signature). We like to
evaluate the above integral by the method of steepest descent. The argument
within the exponential is the action for a relativistic particle executing a
periodic motion under influence of $a_{\mu}^{ex}$. The equation of motion for
it is given by

$\displaystyle\frac{1}{\sqrt{\int\dot{x}^{2}}}m\ddot{x}_{\mu}=eF^{ex}_{\mu\nu}\dot{x}_{\nu}$
(24)

Keeping in mind the periodic boundary conditions $x_{\mu}(0)=x_{\mu}(1)$,
$F^{ex}_{01}=E$ one has the following classical solution

$\displaystyle x_{\mu}^{cl}=R(0,0,\cos 2\pi\tau,\sin
2\pi\tau)\,\,\,\,\,,\,\,\,\,\,\text{R}=\frac{m}{eE}\,\,\,\,\,,\,\,\,\,\,S_{p}^{cl}=\frac{\pi
m^{2}}{eE}$ (25)

Using the above values one has
$\frac{1}{m}\sqrt{\frac{2\pi}{T_{0}}}=\frac{\sqrt{eE}}{m}$. Thus decay rate
can be approximated as

$\displaystyle V\Gamma\approx\frac{\sqrt{eE}}{m}e^{-\frac{\pi m^{2}}{eE}}$
(26)

Ideally one should go around calculating the one loop prefactor and complete
the steepest descent process Affleck:1981bma ,Dunne:2006st , the calculation
of which is indeed complicated. The modified prefactor is given by
$\frac{(eE)^{2}}{(2\pi)^{3}}$. Thus we see that the pair production rate does
to zero if the external electric field is switched off. In arbitrary coupling
one can no longer neglect the effect of the dynamical fields and one has to
include contribution from Wilson loops

$\displaystyle
V\Gamma=-2\text{Im}\int_{0}^{\infty}\frac{dT}{T}e^{-\frac{m^{2}T}{2}}\int\mathcal{D}x\exp\Big{[}-\frac{1}{2T}\int_{0}^{1}d\tau\dot{x}^{2}+ie\oint
a^{ex}_{\mu}dx_{\mu}\Big{]}\langle\exp\Big{(}ie\oint
A_{\mu}dx_{\mu}\Big{)}\rangle$ (27)

The pair production rate gets modified to Affleck:1981bma Kawai:2015mha

$\displaystyle\Gamma=\frac{(eE)^{2}}{(2\pi)^{3}}\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{2}}\exp\Big{(}-n\big{(}\frac{\pi
m^{2}}{eE}-\frac{e^{2}}{4}\big{)}\Big{)}$ (28)

From the above one can work out that the pair production rate is not
exponentially suppressed once the value of electric field exceeds the so
called critical value $E_{c}=\frac{4\pi m^{2}}{e^{3}}$, beyond which the
vacuum becomes unstable.

To implement this argument for AdS/CFT like theories one faces a number of
problems. Firstly the field theory in those circumstances is a conformal one
and one cannot get a mass term a priori. Moreover in the dual gauge theory
matter fields exists in the adjoint representation of SU(N) gauge group. To
evade these issues one uses the Higgs mechanism to break the symmetry group
from SU(N+1)$\rightarrow$SU(N) $\otimes$U(1). Because of this splitting one
has $5$ massive W bosons transforming in fundamental representation of SU(N)
and interacting with the background Yang Mills theory. Now the pair production
rate in presence on an external electric field is given by Kawai:2015mha
Chowdhury:2019mqi

$\displaystyle\Gamma\sim$
$\displaystyle-5\text{N}\;\int\mathcal{D}x\;\text{exp}\Big{(}-m\int_{0}^{1}d\tau\;\sqrt{\dot{x}^{2}}+i\int_{0}^{1}d\tau\;a^{(E)}_{\mu}\dot{x}_{\mu}\Big{)}\langle
W[x]\rangle$ (29)

Where W[x] is the SU(N) Wilson loop and can be calculated by holographic
means.

##  4 Pair Production in Non-supersymmetric Theories via Holography

The ideal way to argue Schwinger effectSemenoff:2011ng Bolognesi:2012gr  is to
calculate the expectation value of circular Wilson loops and relate it to the
decay rate. However one can alternatively view the vacuum to be made of
virtual $q\bar{q}$ pairs in presence of an attractive potential and study the
influence of an external electric field Sato:2013iua . This basically amounts
to calculating the inter-quark potential which one does by considering the
rectangular Wilson loop. In doing so one has to make some additional
approximations. One considers that the time scale associated with the Wilson
loop is much lesser than the length scale. Intuitively, one thinks that the
quark anti-quark pairs are separated in the far past and unite in the far
future. In holography the Wilson loop is given by following formula
Maldacena:1998im  Drukker:1999zq

$\displaystyle\langle{W[\mathcal{C}]}\rangle=\frac{1}{\text{Vol}}\int_{\partial
X=\mathcal{C}}\mathcal{D}X\mathcal{D}h_{ab}\;e^{-S[X,h]}$ (30)

$S[X,h]$ is the Wick rotated action of the fundamental string
Blumenhagen:2013fgp  with endpoints ending at contour $\mathcal{C}$ situated
on the probe brane. In the classical limit ($\alpha^{\prime}\rightarrow 0$)
the extremal value of the string action dominates and thus the Wilson loop is
the extremal area of string world-sheet ending on the contour. To study the
rectangular Wilson loop we take the quark anti-quark dipole to be aligned in
the $x_{3}$ direction. The string action whose on-shell value we are
interested with is the Nambu-Goto action
$\mathcal{S}_{NG}=\frac{1}{2\pi\alpha^{\prime}}\int
dtds\sqrt{det\;G^{(in)}_{ab}}$ with $G^{(in)}_{ab}\equiv
G_{\mu\nu}\frac{\partial x^{\mu}}{\partial s^{a}}\frac{\partial
x^{\nu}}{\partial s^{b}}$ which has two diffeomorphism symmetries. We exploit
those to choose the following gauge

$\displaystyle
x^{0}(s,t)=t\;\;;\;\;x^{3}(s,t)=s\;\;;\;\;u(s,t)=u(s)\;\;;\;\;x^{1,2}=0\;\;;\;\;\Theta^{i}(s,t)=\text{constant}$
(31)

Figure 1: This figure illustrates the setup used. The probe brane is placed at
a finite position ($u_{b}$) on the holographic direction as in (1.2). On the
probe brane the placement of the Wilson loop is shown in (1.1), arrows
indicating the contour of the loop (not the propagation of the string). For
adiabatic interactions one can neglect the effects of the dotted lines and the
string profile becomes static.

For present purposes $x_{3}\equiv s$ is assumed to to range between $[-L,L]$
and temporal direction $x_{0}\equiv t$ is ranged between
$[-\mathcal{T},\mathcal{T}]$ with the assumption that $\mathcal{T}\gg L$. $2L$
indicates the inter-quark separation on the probe brane with the boundary
condition $u(\pm L)=u_{b}$ where $u_{b}$ indicates the position of the probe
brane along the holographic direction, see Figure1. Finally another word about
the configuration, it is possible to consider the $q\bar{q}$ pairs at a
velocity in the $x_{2}$ direction. However in the present case where the
virtual particles in vacuum are modeled as $q\bar{q}$ dipoles, such a
configuration seems hardly sensible. The induced metric as per the above gauge
choice reads (2)(31),

$\displaystyle\frac{1}{\alpha^{\prime}\sqrt{\lambda}}G^{(in)}_{ab}ds^{a}ds^{b}=-F(u(s))^{-\frac{1}{2}}$
$\displaystyle
G(u(s))^{\frac{\delta}{4}}dt^{2}+ds^{2}\bigg{[}F(u(s))^{-\frac{1}{2}}G(u(s))^{\frac{\delta}{4}}$
$\displaystyle+$ $\displaystyle
F(u(s))^{\frac{1}{2}}G(u(s))^{\frac{1+\delta}{4}}\frac{1}{G(u(s))}\left(\frac{du}{ds}\right)^{2}\bigg{]}$
(32)

From the above we have the determinant of the induced metric to be

$\displaystyle-\text{det}G^{(in)}_{ab}=(\alpha^{\prime}\sqrt{\lambda})^{2}\bigg{[}G(u(s))^{\frac{2\delta-3}{4}}\bigg{(}\left(\frac{du}{ds}\right)^{2}+G(u(s))^{\frac{3}{4}}F(u(s))^{-1}\bigg{)}\bigg{]}$
(33)

It is not possible to carry on analysis without some simplification. We
therefore assume that $\left(\frac{u_{0}}{u}\right)^{4}\ll 1$ and with the
mentioned, simplify the area i.e. on shell Nambu-Goto action to

$\displaystyle\mathcal{S}_{ng}$
$\displaystyle=\frac{1}{2\pi\alpha^{\prime}}\int_{-\mathcal{T}/2}^{\mathcal{T}/2}dt\int_{-L}^{L}ds\;\sqrt{-\text{det}G^{(in)}_{ab}}$
$\displaystyle=\frac{\sqrt{\lambda}}{2\pi}\mathcal{T}\int_{-L}^{L}ds\;\sqrt{\left(\frac{du}{ds}\right)^{2}\bigg{(}1+A\frac{u_{0}^{4}}{u^{4}}\bigg{)}+u^{4}\bigg{(}1+B\frac{u_{0}^{4}}{u^{4}}\bigg{)}}$
(34)

Wherein

$\displaystyle
B=\frac{\delta+1}{2}\;\;\;\;\;;\;\;\;\;\;A=\frac{2\delta-3}{4}\;\;\;\;\;;\;\;\;\;\;A+\frac{5}{4}=B$
(35)

Crudely speaking, this can be seen as treating the Non-Susy theory as
perturbation over the $\mathcal{N}$=4 supersymmetric Yang Mills. Since the
expression (4) doesn’t explicitly depend on the parameter $s$, the
corresponding "Hamiltonian", $Q$ is conserved.

$\displaystyle
Q=-\frac{du}{ds}\frac{dL_{ng}}{d\left(\frac{du}{ds}\right)}+L_{ng}=\frac{u^{4}+Bu_{0}^{4}}{\sqrt{\left(\frac{du}{ds}\right)^{2}\bigg{(}1+A\frac{u_{0}^{4}}{u^{4}}\bigg{)}+u^{4}\bigg{(}1+B\frac{u_{0}^{4}}{u^{4}}\bigg{)}}}$
(36)

A indicated in Rey:1998ik  the fundamental string is assumed to carry charges
at two of its endpoints and is otherwise symmetric about its origin. From the
above expression we see that $\frac{du}{ds}$ has both positive and negative
sign. Appealing to its symmetric nature there exists a point, namely turning
point ( with string parameter $s_{t}$ ) such that

$\displaystyle\bigg{(}\frac{du}{ds}\bigg{)}(u_{t})=0$ (37)

Using the above expression in (36) the value of the conserved Hamiltonian is
found in terms of the turning point

$\displaystyle Q=\sqrt{u_{t}^{4}+Bu_{0}^{4}}$ (38)

Putting the above value in (36) we get

$\displaystyle\frac{du}{ds}=u^{2}\sqrt{\frac{(u^{4}-u_{t}^{4})(u^{4}+Bu_{0}^{4})}{(u_{t}^{4}+Bu_{0}^{4})(u^{4}+Au_{0}^{4})}}$
(39)

The length of the (virtual) dipole can be calculated to be (see Figure 1)

$\displaystyle
L=\int_{-L/2}^{L/2}\;dx_{3}=\sqrt{u_{t}^{4}+Bu_{0}^{4}}\int_{u_{t}}^{u_{b}}du\frac{\sqrt{u^{4}+Au_{0}^{4}}}{u^{2}\sqrt{(u^{4}-u_{t}^{4})(u^{4}+Bu_{0}^{4})}}$
(40)

From, (39) and (4) we can find the on-shell value of inter-quark potential ,

$\displaystyle\mathcal{U}_{PE+SE}=\frac{\mathcal{S}_{ng}}{\mathcal{T}}=\frac{\sqrt{\lambda}}{2\pi}\int_{u_{t}}^{u_{b}}du\;\sqrt{(u^{4}+Au_{0}^{4})(u^{4}+Bu_{0}^{4})}\frac{1}{u^{2}\sqrt{u^{4}-u_{t}^{4}}}$
(41)

Notice from (40) that when $u_{t}\rightarrow u_{b}$, the value of the inter-
quark separation becomes small. But as said earlier we are in an approximation
where $\left(\frac{u_{0}}{u}\right)^{4}\ll 1$. Thus the calculations in this
section are trustable for large inter-quark separation. Now the expression in
(41) (see Figure 3 for the plot) doesn’t take the presence of an external
electric field into account. Thus we define an effective potential as

$\displaystyle\mathcal{V}_{eff}=\mathcal{U}_{PE+SE}-E.L=(1-r)E_{c}.L+G(u_{t}(L))$
(42)

In the above we have assumed the presence on an critical electric field
$E_{c}$, above which the effective inter-quark force becomes repulsive for all
values of the inter-quark separation. The quantity $G(u_{t})$ is

$\displaystyle
G(u_{t})=\mathcal{U}_{PE+SE}-E_{c}L=\int_{u_{t}}^{u_{b}}du\frac{\sqrt{u^{4}+Au_{0}^{4}}}{u^{2}\sqrt{u^{4}-u_{t}^{4}}}\Big{[}\frac{\sqrt{\lambda}}{2\pi}\sqrt{u^{4}+Bu_{0}^{4}}-E_{c}\frac{\sqrt{u_{t}^{4}+Bu_{0}^{4}}}{\sqrt{u^{4}+Bu_{0}^{4}}}\Big{]}$
(43)

The parameter r is the ratio of applied electric field to its critical value.
The slope of the effective potential is given as

$\displaystyle\frac{d\mathcal{V}_{eff}}{dL}=(1-r)E_{c}+\frac{du_{t}}{dL}\frac{dG(u_{t})}{du_{t}}$
(44)

We now proceed to find the value of the critical electric field. Note, at
$u_{t}=u_{b}$ the inter quark separation (40) and the inter-quark potential
(41) vanishes, see Figure 2. At the critical value of the electric field
$r=1$, the $1^{st}$ term of (44) ceases to contribute, and the behavior of the
inter-quark force will be completely governed by the second term of (44).
Criticality demands that the potential ceases to put up a tunneling barrier
for all values of inter-quark separation, see red line in Figure 4. Given that
$G(u_{t}(L))$ vanishes at $L=0$ we need to show that $G(u_{t}(L))$ is a
monotonically decreasing function with respect to L whose slope vanishes at
$L=0$. (This is because critical electric field is the least one for which
pair production happens spontaneously). From (40) we have

$\displaystyle\frac{dL}{du_{t}}=-\frac{\sqrt{u_{t}^{4}+Au_{0}^{4}}}{u_{t}^{2}\sqrt{\big{(}u_{t}^{4}+\epsilon\big{)}^{4}-u_{t}^{4}}}+2\int_{u_{t}+\epsilon}^{u_{b}}du\;\frac{u_{t}^{3}}{u^{2}}\frac{\sqrt{\big{(}u^{4}+Au_{0}^{4}\big{)}\big{(}u^{4}+Bu_{0}^{4}\big{)}}}{\sqrt{\big{(}u^{4}-u_{t}^{4}\big{)}^{3}\big{(}u_{t}^{4}+Bu_{0}^{4}\big{)}}}$
(45)

Similarly we have from (43)

$\displaystyle\frac{dG}{du_{t}}(u_{t})=-$
$\displaystyle\frac{\sqrt{u_{t}^{4}+Au_{0}^{4}}}{u_{t}^{2}\sqrt{(u_{t}+\epsilon)^{4}-u_{t}^{4}}}\Big{[}\frac{\sqrt{\lambda}}{2\pi}\sqrt{u_{t}^{4}+Bu_{0}^{4}}-E_{c}\Big{]}$
$\displaystyle+2u_{t}^{3}\int_{u_{t}+\epsilon}^{u_{b}}du\;\frac{\sqrt{(u^{4}+Au_{0}^{4})(u^{4}+Bu_{0}^{4})}}{u^{2}\big{(}\sqrt{u^{4}-u_{t}^{4}}\big{)}^{3}}\Big{[}\frac{\sqrt{\lambda}}{2\pi}-\frac{E_{c}}{\sqrt{u_{t}^{4}+Bu_{0}^{4}}}\Big{]}$
$\displaystyle=$
$\displaystyle\Big{[}\frac{\sqrt{\lambda}}{2\pi}\sqrt{u_{t}^{4}+Bu_{0}^{4}}-E_{c}\Big{]}\frac{dL}{du_{t}}$
(46)

Thus we get

$\displaystyle\frac{d\mathcal{V}_{eff}}{dL}=(1-r)E_{c}+\Big{[}\frac{\sqrt{\lambda}}{2\pi}\sqrt{u_{t}^{4}+Bu_{0}^{4}}-E_{c}\Big{]}$
(47)

At threshold condition the slope of the potential should be zero at when
inter-quark separation vanishes i.e. $u_{t}=u_{b}$. Implementing the same in
(47) we get

$\displaystyle E_{c}=$
$\displaystyle\frac{\sqrt{\lambda}}{2\pi}u_{b}^{2}\sqrt{1+\frac{\delta+1}{2}\frac{u_{0}^{4}}{u_{b}^{4}}}$
$\displaystyle=$
$\displaystyle\frac{2\pi}{\sqrt{\lambda}}m_{0}^{2}\sqrt{1+\frac{\lambda^{2}}{32\pi^{4}}\big{(}\delta+1\big{)}\frac{u_{0}^{4}}{m_{0}^{4}}}$
(48)

We thus have ,

$\displaystyle\frac{d\mathcal{V}_{eff}}{dL}=(1-r)E_{c}+\Big{[}\frac{\sqrt{\lambda}}{2\pi}\sqrt{u_{t}^{4}+Bu_{0}^{4}}-\frac{\sqrt{\lambda}}{2\pi}\sqrt{u_{b}^{4}+Bu_{0}^{4}}\Big{]}$
(49)

Figure 2: This is the graph of L v/s $u_{t}$. Note that the function is an
isomorphism. The values used are $\delta=-0.75$ and $\frac{u_{0}}{u_{b}}=0.01$
Figure 3: This is the graph of $U_{PE}$ v/s L. The rest mass has been duly
subtracted. Note that for small values of L the graph is approximately linear
and for large L coulombic behavior is mimicked. Deviation from usual coulombic
behavior is evident.The values used are $\delta=-0.75$,
$\frac{u_{0}}{u_{b}}=0.01$ and $\lambda=4\pi^{2}$.  Figure 4: The plot
indicates the effective potential (in presence of external electric field) v/s
the inter-quark separation. Imagine this to be then potential through which
q$\bar{q}$ tunnels out. The green line indicates r=0.25, blue line for r=0.75.
The parameter r is the ratio of the applied field to it’s threshold value. The
red line which exhibits the threshold behavior i.e. no potential barrier
stands for r=1.0 and cyan for r=1.75 shows catastrophic decay of vacuum. Note
that at the threshold/ critical value, the slope of the potential vanishes at
$L=0$ and is negative for nonzero value of L which is precisely the conditions
we have used to analytically find out the value of $E_{c}$. Also note that
none of the above plots exhibit confining behavior as earlier reported in
literature! The values used are $\delta=-0.75$, $\frac{u_{0}}{u_{b}}=0.01$ and
$\lambda=4\pi^{2}$.

From Figure 2 we see that L increases as $u_{t}$ decreases using which we can
say from (49) that $\frac{d\mathcal{V}_{eff}}{dL}$ is a monotonically
decreasing function of L at $r=1$. It can be easily understood that from $r>1$
the effective potential in totally repulsive. Thus we establish the existence
of a critical electric field with value given by (4). We see that as $\delta$
switches over $-1$, the critical electric field increases and decreases
respectively compared to the supersymmetric value. Not even that, just at
$\delta=-1$, the critical field has the same value as that of the
supersymmetric theory. Will this kind of behavior remain when one considers
higher orders? How much of the calculation in this section should be trusted
for small values of inter-quark separation? The answer to this question will
be found in the next section.

It so happens that analytical solutions to (40),(41),(42) cannot be found out
in a closed from via Mathematica. Thus we resort to numerical methods. Some
plots to illustrate the situation are given.

##  5 DBI Analysis of Critical Electric Field

In this section we look to find out the critical electric field from analysis
of the DBI action of the probe brane in presence of an external electric
field. In due course we will also answer the question raised in section 4.

As earlier we imagine the probe brane situated at $u=u_{b}$ (see Figure 1) in
the holographic dual with an electric field switched on at the brane position.
The DBI action is given as

$\displaystyle\mathcal{S}_{\text{DBI}}=\frac{1}{(2\pi)^{3}g_{s}\alpha^{\prime}}\int_{u=u_{b}}d^{4}x\sqrt{-\text{det}(P[g]_{\mu\nu}+B_{\mu\nu}+2\pi\alpha^{\prime}F_{\mu\nu})}$
(50)

In the above$P[g]_{\mu\nu}$ is the pullback of the curved metric on the probe
brane, $B_{\mu\nu}$ is the NS 2-form which is zero in the present case.
$F_{\mu\nu}$ is the Faraday tensor which we set to the value $F_{03}=E$, to
indicate the presence of an external electric field. Evaluating the above from
(2) we have

$\displaystyle P[g]_{\mu\nu}+2\pi\alpha^{\prime}F_{\mu\nu}=$
$\displaystyle\begin{pmatrix}-\alpha^{\prime}\sqrt{\lambda}F(u_{b})^{-\frac{1}{2}}G(u_{b})^{\frac{\delta}{4}}&0&0&2\pi\alpha^{\prime}E\\\
0&\alpha^{\prime}\sqrt{\lambda}F(u_{b})^{-\frac{1}{2}}G(u_{b})^{\frac{\delta}{4}}&0&0\\\
0&0&\alpha^{\prime}\sqrt{\lambda}F(u_{b})^{-\frac{1}{2}}G(u_{b})^{\frac{\delta}{4}}&0\\\
-2\pi\alpha^{\prime}E&0&0&\alpha^{\prime}\sqrt{\lambda}F(u_{b})^{-\frac{1}{2}}G(u_{b})^{\frac{\delta}{4}}\end{pmatrix}$
(51)

Thus the DBI action becomes

$\displaystyle\mathcal{S}_{\text{DBI}}=\frac{\alpha^{\prime}\lambda}{(2\pi)^{3}g_{s}}\int_{u=u_{b}}d^{4}x\;F(u_{b})^{-\frac{1}{2}}G(u_{b})^{\frac{\delta}{4}}\sqrt{F(u_{b})^{-1}G(u_{b})^{\frac{\delta}{4}}-\left(\frac{2\pi
E}{\sqrt{\lambda}}\right)^{2}}$ (52)

Figure 5: This is the plot of critical electric field $E_{c}$ v/s the non-
supersymmetric parameter $\delta$. In this figure we have set
$\frac{\sqrt{\lambda}}{2\pi}=1$, $\frac{u_{0}}{u_{b}}=\frac{1}{0.5}$.
According to this values the supersymmetric critical field would haven been
0.25.

Thus we see that (52) is not real for all values of the external electric
field and there is an upper limit of the same. This limiting value is nothing
but the critical electric field

$\displaystyle
E_{c}=\frac{\sqrt{\lambda}}{2\pi}F(u_{b})^{-\frac{1}{2}}G(u_{b})^{\frac{\delta}{4}}$
(53)

The functions $F(u)$ and $G(u)$ has been defined before (2) . One can check
that upto $\mathcal{O}\left(\frac{u_{0}}{u_{b}}\right)^{4}$, (53) reduces to
(4). However in finding (53) we have refrained from using perturbations of any
sort and thus (53) is the exact value. Let us check the behavior of it with
respect to the parameter $\delta$.

We see from Figure 5 that the critical electric field is same as that of its
supersymmetric cousin somewhere around $\delta=-0.85$, which matches more or
less with our perturbative analysis is the last section. For values of
$\delta>0.85$, the critical electric field is greater than the supersymmetric
counterpart, and for $\delta<0.85$, the critical field is lesser. Thus the
question raised in the last section is answered in the affirmative. The
calculation in section 4 won’t be affected drastically for small values of
inter-quark separation. (This is because it is the small separation behavior
that decides the critical value.)

##  6 Holographic Pair Production Rate for Non-supersymmetric Theories

In this section we calculate the pair production rate by using the method of
circular Wilson loops. As indicated earlier in (29) and (30), to find the pair
production rate, we need to find the on-shell value of the Nambu Goto action
with string endpoints ending on a circular contour at the probe brane
($u=u_{b}$). For pure AdS the calculation of the same has been presented in
Semenoff:2011ng Kawai:2015mha Sato:2013pxa . However, it is not possible to
find exact solutions to the relevant equation of motions for the present case
(2). Thus we will resort to perturbative treatments like that of
Allahbakhshi:2013rda  to calculate the circular Wilson loop and hence the
decay rate to first order of the non-susy deformation parameter ($u_{0}^{4}$).
Since the metric (2) enjoys circular symmetry we start by making an ansatz

$\displaystyle
x^{0}=r(\sigma)\cos\tau\;\;\;\;\;\;;\;\;\;\;\;x^{3}=r(\sigma)\sin\tau\;\;\;\;\;;\;\;\;\;\;\;u=u(\sigma)$
(54)

In the above all other co-ordinates have been put to be constants as circular
symmetry would imply. The parameter $\tau$ ranges from $(0,2\pi)$ while the
parameter $\sigma$ is still arbitrary. Their exists a diffeomorphism
invariance of the Nambu-Goto action with which we can set $u=u(\sigma)$ to a
function of our choosing. Putting the ansatz (54) in (2) we have the induced
metric to be

$\displaystyle ds^{2}=\alpha^{\prime}\sqrt{\lambda}\Bigg{[}$
$\displaystyle\bigg{(}F(u)^{-\frac{1}{2}}G(u)^{\frac{\delta}{4}}\left(\frac{dr}{d\sigma}\right)^{2}+F(u)^{\frac{1}{2}}G(u)^{\frac{\delta-3}{4}}\left(\frac{du}{d\sigma}\right)^{2}\bigg{)}d\sigma^{2}+$
$\displaystyle
r^{2}F(u)^{-\frac{1}{2}}G(u)^{\frac{\delta}{4}}d\tau^{2}\Bigg{]}$ (55)

From the above one can get the Nambu-Goto action to be of the form

$\displaystyle\mathcal{S}_{ng}=\frac{(\alpha^{\prime}\sqrt{\lambda})}{2\pi\alpha^{\prime}}\int_{0}^{2\pi}d\tau\int_{0}^{\sigma_{b}}d\sigma\;\sqrt{r^{2}G(u)^{\frac{\delta}{2}}\bigg{(}F(u)^{-1}\left(r^{\prime}\right)^{2}+G(u)^{-\frac{3}{4}}\left(u^{\prime}\right)^{2}\bigg{)}}$
(56)

For purposes of calculation we expand the function $F(u)$ and $G(u)$ in their
leading order to the non-supersymmetric deformation parameter and we have,

$\displaystyle\mathcal{S}_{ng}$
$\displaystyle=\sqrt{\lambda}\int_{0}^{\sigma_{b}}d\sigma\;\sqrt{r^{2}\Big{(}1+\frac{\delta}{2}\frac{u_{0}^{4}}{u^{4}}\Big{)}\Bigg{(}\left(r^{\prime}\right)^{2}u^{4}\Big{(}1+\frac{1}{2}\frac{u_{0}^{4}}{u^{4}}\Big{)}+\left(u^{\prime}\right)^{2}\Big{(}1-\frac{3}{4}\frac{u_{0}^{4}}{u^{4}}\Big{)}\Bigg{)}+\mathcal{O}\Big{(}\frac{u_{o}^{8}}{u^{8}}\Big{)}}$
$\displaystyle\approx\sqrt{\lambda}\int_{0}^{\sigma_{b}}d\sigma\;\sqrt{r^{2}\Bigg{(}\left(r^{\prime}\right)^{2}u^{4}\Big{(}1+A\frac{u_{0}^{4}}{u^{4}}\Big{)}+\left(u^{\prime}\right)^{2}\Big{(}1+B\frac{u_{0}^{4}}{u^{4}}\Big{)}\Bigg{)}}$
(57)

Wherein

$\displaystyle
B=\frac{\delta+1}{2}\;\;\;\;\;;\;\;\;\;\;A=\frac{2\delta-3}{4}\;\;\;\;\;;\;\;\;\;\;A+\frac{5}{4}=B$
(58)

The above binomial expansion and all the others that follow is simply treating
the Non-SuSy theory as a perturbation over the regular $\mathcal{N}=4$ SYM. In
this paper we limit ourselves to first order perturbations. Recall that we
still had one diffeomorphism invariance left as mentioned before, with the
help of which we set $\frac{du(\sigma)}{d\sigma}=1$. Thus (6) is simplified to

$\displaystyle\mathcal{S}_{ng}=\sqrt{\lambda}\int_{u_{t}}^{u_{b}}du\;\sqrt{r^{2}\bigg{(}\frac{dr}{du}\bigg{)}^{2}\Big{(}u^{4}+Au_{0}^{4}\Big{)}+\frac{r^{2}}{u^{4}}\Big{(}u^{4}+Bu_{0}^{4}\Big{)}}$
(59)

We would like to find out the function $r=r(u)$ which extremizes (59).
Extremizing the same one has to encounter the equation

$\displaystyle u^{4}$
$\displaystyle\big{(}u^{4}+Au_{0}^{4}\big{)}\bigg{(}\frac{d\rho}{du}\bigg{)}^{2}\bigg{(}2\big{(}u^{4}+Bu_{0}^{4}\big{)}-u^{7}\frac{d\rho}{du}\bigg{)}-4\rho\bigg{(}u^{3}\frac{d\rho}{du}\big{(}ABu_{0}^{8}+3Bu^{4}u_{0}^{4}+2u^{8}\big{)}$
$\displaystyle-\big{(}u^{4}+Bu_{0}^{4}\big{)}^{2}\bigg{)}-2u^{4}\big{(}u^{4}+Au_{0}^{4}\big{)}\big{(}u^{4}+Bu_{0}^{4}\big{)}\rho\frac{d^{2}\rho}{du^{2}}=0$
(60)

Where $\rho=r^{2}$. The above equation is very hard to solve in closed form.
Thus we adopt perturbative techniques like that of Chowdhury:2019mqi . To do
so we decompose the solution to (6) as $\rho=\rho_{0}+u_{0}^{4}\rho_{1}$ in
which $\rho_{0}=-\frac{1}{u^{2}}$, and $\rho_{1}$ indicates the perturbation.
From (6) the equation for $\rho_{1}$ to the leading order of $u_{0}^{4}$ is

$\displaystyle
2u^{2}\bigg{(}6(B-A)+2u^{7}\frac{d\rho_{1}}{du}+u^{8}\frac{d^{2}\rho_{1}}{du^{2}}\bigg{)}=0$
(61)

One can check that the above is solved by

$\displaystyle\rho_{1}(u)=\frac{A-B}{5u^{6}}+K^{\prime}$ (62)

Thus the full solution is

$\displaystyle r^{2}(u)=\rho(u)=$ $\displaystyle
u_{0}^{4}K^{\prime}-\frac{1}{u^{2}}+\frac{A-B}{5u^{6}}\frac{u_{0}^{4}}{u^{6}}$
$\displaystyle\equiv$ $\displaystyle
K-\frac{1}{u^{2}}+\frac{A-B}{5}\frac{u_{0}^{4}}{u^{6}}$ $\displaystyle=$
$\displaystyle K-\frac{1}{u^{2}}-\frac{1}{4}\frac{u_{0}^{4}}{u^{6}}$ (63)

Where a redefinition of constant has been made. Now its time to relate the
constant K to physical parameters. At $u=u_{b}$ the value of $r$ is the radius
of the Wilson loop $R$. Thus we have,

$\displaystyle
K=R^{2}+\frac{1}{u_{b}^{2}}-\frac{A-B}{5}\frac{u_{0}^{4}}{u_{b}^{6}}$ (64)

From the above we can also find the value of the turning point $u_{t}$, since
at the turning point the radius $r(u_{t})=0$. Thus the equation which
determines the turning point is,

$\displaystyle
K=\frac{1}{u_{t}^{2}}\Big{(}1-\frac{A-B}{5}\frac{u_{0}^{4}}{u_{t}^{4}}\Big{)}$
(65)

Now we proceed to calculate the on shell value of the Nambu Goto action (59)
on the solution (6). We have,

$\displaystyle\mathcal{S}_{ng}=$
$\displaystyle\sqrt{\lambda}\int_{u_{t}}^{u_{b}}du\sqrt{\frac{1}{4}\bigg{(}\frac{d(r^{2})}{du}\bigg{)}^{2}\big{(}u^{4}+Au_{0}^{4}\big{)}+(r^{2})\Big{(}1+B\frac{u_{0}^{4}}{u^{4}}\Big{)}}$
$\displaystyle=$
$\displaystyle\sqrt{\lambda}\int_{u_{t}}^{u_{b}}du\sqrt{\frac{1}{4}\cdotp\frac{4}{u^{6}}\Big{(}1-\frac{3}{5}(A-B)\frac{u_{0}^{4}}{u^{4}}\Big{)}^{2}\big{(}u^{4}+Au_{0}^{4}\big{)}+\Big{(}1+B\frac{u_{0}^{4}}{u^{4}}\Big{)}\Big{(}K-\frac{1}{u^{2}}+\frac{A-B}{5}\frac{u_{0}^{4}}{u^{6}}\Big{)}}$
$\displaystyle=$
$\displaystyle\sqrt{\lambda}\int_{u_{t}}^{u_{b}}du\sqrt{K+KB\frac{u_{0}^{4}}{u^{4}}+\mathcal{O}\big{(}u_{0}^{8}\big{)}}$
(66)

We neglect the $\mathcal{O}\big{(}u_{0}^{8}\big{)}$ term in the above. The
integral of the remaining part cannot be done in closed form by using
Mathematica. So we resort to perturbative methods again, and write

$\displaystyle\mathcal{S}_{ng}=$
$\displaystyle\sqrt{\lambda}\int_{u_{t}}^{u_{b}}\;du\sqrt{K}\bigg{[}1+\frac{B}{2}\frac{u_{0}^{4}}{u^{4}}+\mathcal{O}\big{(}u_{0}^{8}\big{)}\bigg{]}$
$\displaystyle\approx$
$\displaystyle\sqrt{\lambda}\Bigg{(}\bigg{[}\sqrt{K}u\bigg{]}^{u_{b}}_{u_{t}}-\bigg{[}\frac{2B\sqrt{K}u_{0}^{4}}{u^{3}}\bigg{]}^{u_{b}}_{u_{t}}\Bigg{)}$
(67)

So far so good, however the reader may agree that working with (6) is still
daunting given that we now have to substitute the highly non-linear relations
(64),(65) into it. Happily there is a way out of this mess. Recall that our
theme has been to work in the leading order of $u_{0}^{4}$ and the last two
terms of (6) come with a $u_{0}^{4}$ of their own. Thus to leading order we
may substitute the usual AdS relations (relating K to $u_{t}$ and $u_{b}$) in
the last term of(6), but use the Non-Susy relations (64),(65) in the first
term of the same. Doing so we have

$\displaystyle\mathcal{S}_{ng}=\sqrt{\lambda}$
$\displaystyle\Bigg{(}\;\sqrt{\big{(}R^{2}u_{b}^{2}+1\big{)}-\frac{A-B}{5}\frac{u_{0}^{4}}{u_{b}^{4}}}-\sqrt{1-\frac{A-B}{5}\frac{u_{0}^{4}}{u_{t}^{4}}}$
$\displaystyle+u_{0}^{4}\bigg{[}\frac{2B}{u_{t}^{4}}-\frac{2B\sqrt{R^{2}u_{b}^{2}+1}}{u_{b}^{4}}\bigg{]}\;\Bigg{)}$
$\displaystyle\approx\sqrt{\lambda}$
$\displaystyle\Bigg{(}\;\sqrt{\big{(}R^{2}u_{b}^{2}+1\big{)}-\frac{A-B}{5}\frac{u_{0}^{4}}{u_{b}^{4}}}-\sqrt{1-\frac{A-B}{5}\frac{u_{0}^{4}}{u_{b}^{4}}\big{(}R^{2}u_{b}^{2}+1\big{)}^{2}}$
$\displaystyle+\frac{u_{0}^{4}}{u_{b}^{4}}\bigg{[}2B\big{(}R^{2}u_{b}^{2}+1\big{)}^{2}-2B\sqrt{R^{2}u_{b}^{2}+1}\bigg{]}\;\Bigg{)}$
$\displaystyle\approx\sqrt{\lambda}$
$\displaystyle\Bigg{(}\;\sqrt{\big{(}R^{2}u_{b}^{2}+1\big{)}}-1+\frac{u_{0}^{4}}{u_{b}^{4}}\bigg{[}\frac{A+19B}{10}\big{(}R^{2}u_{b}^{2}+1\big{)}^{2}$
$\displaystyle-\frac{A-B}{10\sqrt{R^{2}u_{b}^{2}+1}}-2B\sqrt{R^{2}u_{b}^{2}+1}\bigg{]}\;\Bigg{)}$
(68)

In the second line of the above we have used the usual AdS relations for the
term $\frac{1}{u_{t}^{4}}$ to leading order as it is accompanied by a
$u_{0}^{4}$. Again in the third line we have used a binomial expansion and
retained terms of leading order in $\frac{u_{0}^{4}}{u_{b}^{4}}$. Now, in
presence of an electric field the effective action of the string has an extra
piece, $S_{B}=T_{0}\int\;d\sigma d\tau
B_{\mu\nu}\partial_{\sigma}x^{\mu}\partial_{\tau}x^{\nu}$. Specializing to
constant electric field, the contribution of $S_{B}$ is a pure boundary term
with on-shell value $\pi R^{2}E$, where R is the radius of the Wilson loop,
$E=B_{01}$ and all other components of the $B_{\mu\nu}$ is set to zero. The
effective action is given by

$\displaystyle S_{eff}=$ $\displaystyle S_{ng}+S_{B}$ $\displaystyle=$
$\displaystyle\sqrt{\lambda}\Bigg{(}\sqrt{x}-1+\frac{u_{0}^{4}}{u_{b}^{4}}\bigg{[}\frac{A+19B}{10}x^{2}-\frac{A-B}{10\sqrt{x}}-2B\sqrt{x}\bigg{]}-\mathcal{E}x+\mathcal{E}\Bigg{)}$
(69)

In the above, $x=R^{2}u_{b}^{2}+1$ and
$E=\frac{\sqrt{\lambda}u_{b}^{2}}{\pi}\mathcal{E}$. Thus the radius R (x) is a
free parameter in the expression (6). Following Semenoff:2011ng
,Bolognesi:2012gr  the radius should be set to an an extremum of (6). Instead
of extremizing w.r.t. $R$, we extremize the action (6) w.r.t. the parameter
$y=\sqrt{x}$. Doing so we find,

$\displaystyle
0=\frac{dS_{eff}}{dy}=\sqrt{\lambda}\Bigg{(}1-2\mathcal{E}y+\frac{u_{0}^{4}}{u_{b}^{4}}\bigg{[}\frac{2(A+19B)}{5}y^{3}+\frac{A-B}{10y^{2}}-2B\bigg{]}\Bigg{)}$
(70)

The radius $R$ should be set to be the solution of (70), recall
$y=\sqrt{R^{2}u_{b}^{2}+1}$. Thus the value of $y$ in the above equation is
constrained and should always be greater than $1$. This is because the radius
of the Wilson loop should be a real number. A subtle point is that the range
of parameter y should be restricted to half of the real line, because the
radius R is nonnegative. The critical electric field $\mathcal{E}_{c}$ is the
one for which the radius $R=0$, i.e. $y=1$. Setting so in the above we see,

$\displaystyle
1-2\mathcal{E}_{c}+\frac{u_{0}^{4}}{u_{b}^{4}}\bigg{[}\frac{2(A+19B)}{5}+\frac{A-B}{10}-2B\bigg{]}=0$
(71)

Thus

$\displaystyle E_{c}=$
$\displaystyle\frac{\sqrt{\lambda}}{2\pi}u_{b}^{2}\bigg{[}1+\frac{1}{8}\frac{u_{0}^{4}}{u_{b}^{4}}\Big{(}23+24\delta\Big{)}\bigg{]}$
$\displaystyle=$
$\displaystyle\frac{2\pi}{\sqrt{\lambda}}m_{0}^{2}\bigg{[}1+\frac{\lambda^{2}}{128\pi^{4}}\Big{(}23+24\delta\Big{)}\frac{u_{0}^{4}}{m_{0}^{4}}\bigg{]}$
(72)

We see that like (4) and (53) the value of the critical electric field is
greater than he supersymmetric value for the value $\delta=-\frac{23}{24}$ and
less than the supersymmetric cousin otherwise. Our perturbative analysis has
even shown that the value of parameter $\delta$ for which this phase
transition occurs is slightly bigger than -1 as can be seen from the non-
perturbative DBI analysis. Now to find the expression of the pair production
rate we need to solve (70) for y. As can be seen, that is not analytically
possible. We thus resort to perturbative treatments again and write,

$\displaystyle y=y_{0}+\frac{u_{0}^{4}}{u_{b}^{4}}y_{1}$ (73)

$y_{0}$ being the usual AdS solution i.e. $u_{0}=0$ in (70). The value of
$y_{0}$ is $\frac{1}{2\mathcal{E}}$. We put the above relation in the equation
in (70) to get upto leading order in $\frac{u_{0}^{4}}{u_{b}^{4}}$.

$\displaystyle
y_{1}=\frac{1}{2\mathcal{E}}\bigg{[}\frac{A+19B}{20}\frac{1}{\mathcal{E}^{3}}+\frac{2(A-B)}{5}\mathcal{E}^{2}-2B\bigg{]}$
(74)

Now we put (73) and (74) in (6), i.e. find out the on-shell action. Retaining
terms in leading order of $\frac{u_{0}^{4}}{u_{b}^{4}}$ leads us to.

$\displaystyle S_{eff}^{onshell}=$
$\displaystyle\frac{\sqrt{\lambda}}{2}\Bigg{(}\frac{1}{2\mathcal{E}}-2+2\mathcal{E}+\frac{u_{0}^{4}}{u_{b}^{4}}\bigg{[}\frac{A+19B}{80}\frac{1}{\mathcal{E}^{4}}+\frac{B-A}{10}\mathcal{E}-\frac{2B}{\mathcal{E}}\bigg{]}\Bigg{)}$
$\displaystyle=\frac{\sqrt{\lambda}}{2}\Bigg{(}\frac{1}{2\mathcal{E}}-2+2\mathcal{E}+\frac{u_{0}^{4}}{u_{b}^{4}}\bigg{[}\frac{40\delta+39}{320}\frac{1}{\mathcal{E}^{4}}+\frac{1}{40}\mathcal{E}-\frac{\delta+1}{\mathcal{E}}\bigg{]}\Bigg{)}$
(75)

The pair production rate of quark anti-quark pairs per unit volume per unit
time is given by the formula, $\Gamma\sim e^{-S_{eff}^{onshell}}$. Note that
we are using reduced parameter
$\mathcal{E}=\frac{\pi}{\sqrt{\lambda}u_{b}^{2}}E$, in terms of which the pure
AdS pair production rate(per unit volume per unit time) is given by
Semenoff:2011ng Bolognesi:2012gr Sato:2013pxa .

$\displaystyle\Gamma_{susy}\sim\text{exp}\Bigg{[}-\frac{\sqrt{\lambda}}{2}\bigg{(}\sqrt{\frac{\frac{1}{2}}{\mathcal{E}}}-\sqrt{\frac{\mathcal{E}}{\frac{1}{2}}}\bigg{)}^{2}\Bigg{]}$
(76)

For the pure AdS/Supersymmetric scenario, the critical electric field is
$\mathcal{E}_{c}=\frac{1}{2}$ i.e.
$E_{c}=\sqrt{\lambda}\frac{u_{b}^{2}}{2\pi}$. Now unlike the supersymmetric
case, the pair production rate cannot be brought in closed form. We will have
to resort to numerical calculations. We present the plots of pair production
rate. Computation of the fluctuation prefactor (i.e. the
$\frac{(eE)^{2}}{(2\pi)^{3}}$ term in (28)) is somewhat a open question in
holography, which is the reason we have plotted $e^{-S_{onshell}}$ instead of
$\Gamma$.

Figure 6: The figure on the left illustrates relation between the pair
production rate and applied electric field for pure $\mathcal{N}=4$ SYM (76).
In our units the critical electric field is at $\mathcal{E}_{c}=\frac{1}{2}$.
The right figure is for non-supersymmetric case (6) for the value
$\delta=-0.75$. Almost no change in the profile is found !

Figure 7: We plot the pair production rate for non-supersymmetric Yang Mills
for the value $\delta=-\sqrt{2}$. On the right we have
$\big{(}\frac{u_{0}}{u_{b}}\big{)}^{4}=\big{(}\frac{1}{5}\big{)}^{4}$ and on
the left
,$\big{(}\frac{u_{0}}{u_{b}}\big{)}^{4}=\big{(}\frac{1}{3.75}\big{)}^{4}$.

Physical interpretation of the plots : As commented in the caption, no stark
contrast is found between the susy and non-susy case in Figure 6. This is the
case when the parameter is greater than (somewhere around) $-0.975$. The
reason for this can be seen from (6). For low electric field the pair
production rate is dominated by the
$\frac{u_{0}^{4}}{u_{b}^{4}}\frac{40\delta+39}{320}\frac{1}{\mathcal{E}^{4}}$
term. Above $\delta=-0.975$ the effective correction at the low electric field
limit is positive. At the limit of high electric field limit the correction of
pair production rate due to the non-susy deformation parameter is always
positive. This is reason that the high electric field limit, the behavior of
non-susy pair production rate is same as its supersymmetric cousin for all
values of parameters. Startling effects happen when the parameter
$\delta<-0.975$, for which the plot is shown in Figure 7. Let us recall, the
prefactor of the pair production rate is given by field theoretic calculations
to be $\frac{(eE)^{2}}{(2\pi)^{3}}$, see (28). Although the holographic
calculation of the fluctuation prefactor is currently a mystery, it should
definitely match with field theoretic calculation for low electric field. For
small applied electric fields the production rate shoots up signaling in non-
perturbative instability of the vacuum. We say "non-perturbative" because
Schwinger effect is by itself a non-perturbative phenomenon. We see that the
limit $\delta=-0.975$ apprx. is for more interesting than earlier imagined.

Let us end by writing down the pair production rate per unit spatial volume
per unit time for non-susy Yang Mills.

$\displaystyle\Gamma_{\text{non-
susy}}\approx\text{exp}\Bigg{[}-\frac{\sqrt{\lambda}}{2}\Bigg{(}$
$\displaystyle\frac{2\pi
m_{0}^{2}}{\sqrt{\lambda}}\frac{1}{E}-2+\frac{\sqrt{\lambda}}{2\pi
m_{0}^{2}}E+\frac{\lambda^{2}u_{0}^{4}}{16\pi^{4}m_{0}^{4}}\bigg{\\{}\frac{4\pi^{4}m_{0}^{8}\big{(}40\delta+39\big{)}}{5\lambda^{2}}\frac{1}{E^{4}}$
$\displaystyle+\frac{\sqrt{\lambda}}{160\pi m_{0}^{2}}E-\frac{4\pi
m_{0}^{2}\big{(}\delta+1\big{)}}{\sqrt{\lambda}}\frac{1}{E}\bigg{\\}}\Bigg{)}\Bigg{]}$
(77)

##  7 Conclusion

In this paper we have studied pair production (Schwinger Effect) in presence
of external electric field for Non-susy AdS/CFT using three methods in the
literature. In section 4 we have done a potential analysis by calculating
rectangular Wilson loops and have analytically calculated the critical
electric field below which pair production happens via a tunneling phenomenon
(and above which the quark-antiquark potential ceases to put up a potential
barrier. We have seen that the critical electric field is higher / lower than
its supersymmetric counterpart depending on the value of the non-
supersymmetric parameter $\delta$(in that section we have used a metric
perturbation to ease up the calculation). We have also confirmed the same from
the DBI analysis of the critical electric field in section 5 where no such
approximation has been made. As can be seen from the main body of work, the
correction absence of supersymmetry yields on the critical electric field
(w.r.t. its supersymmetric value) is neither positive nor negative
definite(4)(53)(6). Thus such a relation cannot be conveniently used to be a
indirect evidence for presence or absence of supersymmetry since the modulus
of the correction is parameter dependent. Also note that from Figure 4 no
trace of confinement is seen contrary to earlier findings in literature (for
confinement some of the plot should have been positive with non negative slope
all along). Next in section 6 we have performed the analysis for pair
production rate for quark-antiquark pairs using circular Wilson loops. Since
the relevant equations are rather impossible to solve, we have resorted to
perturbative analysis, which can be thought of as perturbation over
$\mathcal{N}=4$ SYM by a supersymmetry breaking term with coupling constant
proportional to $\big{(}\frac{u_{0}}{u_{b}}\big{)}^{4}$ parametrized by
$\delta$. We have explicitly found out the profile for circular Wilson loop
for Non-susy AdS/CFT upto first order of $u_{0}^{4}$. To our knowledge this is
the first time such a solution has been obtained. We proceed to find the on-
shell value of the Nambu-Goto action on the profile found and relate it to
pair production rate. We see for a regime of allowed value of the parameter
$\delta$ the pair production rate shoots up as external electric field
decreases towards zero. In stark contrast to confinement (as earlier reported)
this signals that the vacuum of the dual Non-SuSy gauge theory is non-
perturbatively unstable for the regime of the parameter
$-\sqrt{\frac{5}{2}}\leqslant\delta<-\frac{39}{40}$ (apprx) (a confining
potential would show the exact opposite). Thus the field theory dual of the
Non-SUSY geometry considered here is not confining but is the exact opposite
i.e non-perturbatively unstable. A relevant question is to find this
instability from potential analysis and DBI analysis, something which eludes
us at this moment.

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