††institutetext: aSchool of Natural Sciences, Institute for Advanced Study,
Princeton, NJ 08540, USA††institutetext: bJoseph Henry Laboratories, Princeton
University, Princeton, NJ 08544, USA††institutetext: cSimons Center for
Geometry and Physics, Stony Brook University, Stony Brook, NY,
11794††institutetext: dSISSA, Trieste, 34136, Italy††institutetext: eINFN,
Sezione di Trieste, Via Valerio 2, 34127 Trieste, Italy††institutetext:
fDepartment of Physics, Technion, Haifa 32000, Israel

#  Polyakov’s confinement mechanism for
generalized Maxwell theory

Matthew Heydeman,a,b Christian B. Jepsen,c Ziming Ji,d,e Amos Yaromf
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study fractional-derivative Maxwell theory, as appears in effective
descriptions of, for example, large $N_{f}$ QED3, graphene, and some types of
surface defects. We argue that when the theory is UV completed on a lattice,
monopole condensation leads to a confining phase via the Polyakov confinement
mechanism.

††preprint: PUPT- 2636

##  1 Introduction and summary

In the study of infrared effective field theories, it is often useful to
integrate out massive degrees of freedom. Contrariwise, it is not common to
integrate out massless degrees of freedom, mainly because integrating them out
generates non-local terms in the effective action. These obstacles
notwithstanding, there are theories for which this practice has proved
fruitful. One example is QED3 with a large number $N_{f}$ of massless fields,
which when integrated out produce a non-local photon propagator
Appelquist:1981vg ; Appelquist:1988sr  (see also Anselmi:2000fr ;
Borokhov:2002ib ; Witten:2003ya ; Giombi:2015haa ; Chester:2016ref ;
Giombi:2016fct  for a handful of related modern discussions). Another example
involves integrating out bulk fields in the semiclassical limit of the AdS/CFT
correspondence Maldacena:1997re ; Gubser:1998bc ; Witten:1998qj , which leads
to boundary generalized free field theories. References Vasiliev:1990en ;
Klebanov:2002ja ; Witten:2003ya ; Giombi:2012ms ; Giombi:2013yva  studied
$N_{f}$ QED3 and similar vector models in this holographic setup. Related
examples include theories on manifolds with a boundary where bulk field are
effectively integrated out, e.g., free bulk scalar fields which interact on
the boundary Paulos:2015jfa ; Herzog:2017xha ; Giombi:2019enr , quantum
systems interacting with an environment PhysRevLett.46.211 ; Callan:1989mm ;
Callan:1994ub , and the theory of four-dimensional photons that couple to
electrons restricted to live on a three-dimensional surface—a system which
admits a purely three-dimensional effective description involving a non-local
Maxwell term Marino:1992xi ; Teber:2012de ; Teber:2014ita ; Herzog:2017xha ;
Karch:2018uft ; Dudal:2018pta ; DiPietro:2019hqe ; Herzog:2020bqw ;
Pedrelli:2020zlw ; Heydeman:2020ijz ; Herzog:2022jqv , and which has been
usefully applied as a model of graphene Semenoff:2011jf ; Teber:2014ita .

Motivated by the above considerations, consider the generalized free Maxwell
theory in three dimensions,111We refer the reader to Ref.
frasca2021confinement  for a recent study of the RG flow and confinement of
non-local gauge theory in four dimensions, although there non-locality arises
from an exponentiated, rather than a fractional, Laplacian. whose action is
given by

$\displaystyle\begin{split}S&=\frac{1}{4e^{2}}\int_{\mathbb{R}^{3}}d^{3}x\,F_{\mu\nu}D^{s-2}F^{\mu\nu}\\\
&=\frac{1}{4e^{2}}\int_{\mathbb{R}^{3}}\frac{d^{3}k}{(2\pi)^{3}}\,\widetilde{F}_{\mu\nu}(-k)|k|^{s-2}\widetilde{F}^{\mu\nu}(k)\,,\end{split}$
(1)

where tilde’d quantities are Fourier transformed quantities,

$\widetilde{\phi}(k)=\int d^{d}x\,\phi(x)\,e^{ikx}\,,$ (2)

$D^{s-2}=(-\nabla^{2})^{(s-2)/2}$ is the fractional derivative operator (which
is defined through the second equality in (1)), $s$ is a real number between
zero and two, and $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is
the field strength associated to a $U(1)$ gauge field. We restrict ourselves
to flat space throughout this paper but refer the reader to
basteiro2022fractional  for recent work on fractional-derivative theories in
curved space.

In what follows, we will focus primarily on the case $s=1$, where (1)
corresponds precisely to the non-local kinetic term appearing in large $N_{f}$
QED and the effective theory of graphene mentioned above. Thus, when $s=1$ we
may think of (1) as an effective action of an ultimately local theory. As
such, we may take the gauge field to satisfy the Dirac quantization condition.
If we set $s=2$, we recover standard three-dimensional Maxwell theory with a
compact gauge group.

When $s\neq 1$ and $0<s\leq 2$ the theory cannot be obtained by integrating
out modes of a local theory in an integer number of dimensions. Nevertheless,
theories with a non-local kinetic term identical, or similar to that in (1)
appear throughout the literature. For instance, the continuum limit of the
long range Ising model possesses a non-local kinetic term, $D^{s}$,
PhysRevLett.29.917 ; Sak ; Honkonen:1988fq ; Honkonen:1990mr ; Koffel:2012cu ;
Behan:2017dwr ; Behan:2017emf ; Benedetti:2020rrq  and can be related to
integrating out a free bulk scalar field for appropriate values of $s$
CStheorem ; Paulos:2015jfa . Similarly, certain aspects of turbulent flow have
been tied to field theories with non-local kinetic terms in Oz:2017ihc ;
Levy:2018xpu ; Levy:2019tjl ; Kislev:2022emm . More recently, theories of this
type have been shown to posses a symmetry-broken phase, persistent at high
temperatures Chai:2021djc ; Chai:2021tpt . The theory in (1) for arbitrary $s$
has been related to models of superconductivity in LaNave:2019mwv  and has
been studied extensively in Kleinert:2002qk ; Herbut:2003bs . More general
properties of field theories with a non-local kinetic term have been
considered in doAmaral:1992td ; Marino:2014oba ; Basa:2019ywr ;
calcagni2021quantum ; heredia2022nonlocal ; calcagni2022ultraviolet .

Theories with non-local kinetic terms, of the type exemplified by (1), raise
many questions regarding the viability of the associated effective theory. For
instance, one may wonder whether it maintains unitarity or causality, or, what
type of initial value problem classical solutions satisfy. The theory in (1)
is free and unitary, but once interactions are introduced the resulting theory
is expected to break unitarity Heydeman:2020ijz . This general expectation
relies on a violation of the optical theorem in an effective boundary theory
generated by integrating out bulk modes.

The question of causality was addressed in Ref. doAmaral:1992td , which
studied the quantization and causality of generalized Maxwell theory and found
that causality is respected in the sense that the classical Green’s function
vanishes outside the light cone. The approach to canonical quantization
adopted in this reference consisted in introducing an auxiliary mass term
$\chi$ and performing an expansion of a fractional derivative $D^{\sigma}$:

$\displaystyle(-\nabla^{2})^{\frac{s-2}{2}}\rightarrow(-\nabla^{2}+\chi)^{\frac{s-2}{2}}=\chi^{\frac{s-2}{2}}-\frac{s-2}{2}\chi^{\frac{s-2}{2}-1}\nabla^{2}+\frac{(s-2)(s-4)}{8}\chi^{\frac{s-2}{2}-2}(\nabla^{2})^{2}+...\,,$
(3)

which results in infinitely many derivatives. Since each term is local,
Hamilton’s formalism applies, but there are no independent momenta.

In general, whether an infinite-derivative theory is a well-behaved or not
depends on whether the coefficients of the higher-derivative terms decay
sufficiently fast with the number of derivatives. The best kind of scenario is
illustrated by Ref. kleppe1992nonlocal , which studied a modified Yang-Mills
theory where the propagator was dressed with factors of
$\exp(-\nabla^{2}/\Lambda)$ and showed that on-shell tree-amplitudes are
unaffected by this regularization procedure. A related result was reported in
Ref. li2011holography , wherein it is showed that for theories with an
exponentially decaying tower of derivatives, the zero-temperature entanglement
entropy exhibits the usual area law scaling; in Ref. Basa:2019ywr  it was
further demonstrated that such theories can be me mapped onto purely local
theories. In contrast, however, for fractional-derivative theories like
generalized free Maxwell theory, Refs li2011holography ; Basa:2019ywr
explicitly find that this local behaviour is not present.

But while fractional-derivative theories cannot immediately be lumped together
with local theories, there are also issues with formally treating them as
infinite derivative theories, especially as far as the classical equations of
motion are concerned. The problem with an infinite-derivative theory of a
field $\phi(x)$ is that it has an infinite-derivative equation of motion. So
for the initial conditions, one needs to stipulate not just $\phi(0)$ and
$\phi^{\prime}(0)$, but all the higher derivatives as well,
$\phi^{\prime\prime}(0)$, $\phi^{\prime\prime\prime}(0)$, etc. But if all the
derivatives of a function at a given point are specified, then, up to issues
of non-analyticity, the entire function is determined.

Ultimately, fractional-derivative theories are not the same kind of beasts as
infinite-derivative theories. We can see this by studying the equations of
motion. In Fourier space the local ($s=2$) equations of motion resulting from
(1) read

$\displaystyle 0=k^{\mu}\widetilde{F}_{\mu\nu}(k)\,,$ (4)

while the non-local equations of motion read

$\displaystyle 0=\frac{k^{\mu}\widetilde{F}_{\mu\nu}(k)}{|k|^{2-s}}\,.$ (5)

It may be tempting to conclude that these two equations are equivalent, but
that would be too hasty. For example, Laplace’s equation $\nabla^{2}\phi=0$ in
$\mathbb{R}^{d}$ Fourier transforms to $0=k^{2}\widetilde{\phi}$, but this
does not imply that $0=\widetilde{\phi}$. There are solutions where
$\widetilde{\phi}$ is a generalized function with support only at $k=0$.
Similarly, there are solutions to (5) that are not solutions to (4), namely
solutions for which $k_{\mu}\widetilde{F}_{\mu\nu}(k)$ is a generalized
function with support only at $k=\infty$. Once we discard such pathological
solutions as not belonging to the space of functions over which the path
integral for generalized free Maxwell theory should be performed, then the
local and non-local equations of motion equivalent.

While its kinematics are non-local, the field strength $F$ is gauge invariant.
Due to the Bianchi identity, the current
$J^{\mu}=\epsilon^{\mu\nu\rho}F_{\nu\rho}$ is automatically conserved,
implying that the family of non-local free theories possess, in the language
of Gaiotto:2014kfa , a generalized global magnetic $0$-form symmetry. In
Heydeman:2020ijz , it was shown that for $s=1$, (1) is conformally invariant
and $J$ is a conserved current in this CFT. The existence of this topological
current corresponds to the presence of magnetically charged operators, the
monopole operators. In radial quantization on the cylinder $\mathbb{R}\times
S^{2}$, a monopole operator corresponds to a state on $S^{2}$ with non-
vanishing magnetic flux,

$\int_{S^{2}}F=2\pi q\,,$ (6)

where $q\in\mathbb{Z}$. The importance of magnetic sources in three dimensions
was first emphasized by Polyakov:1975rs , and later work established these
monopoles as local operators in conformal field theory (see Borokhov:2002ib ;
Pufu:2013vpa ; Dyer:2013fja ; Chester:2016wrc .) As it stands, (1) is a free
theory with no magnetically charged sources, so there is no dynamical
mechanism to study the effects of monopoles. We will remedy this by embedding
(1) in a bigger UV theory which does have monopole solutions which contribute
to the path integral as instantons.

Indeed, in his famous paper Polyakov:1975rs , Polyakov argued the $s=2$ theory
exhibits the phenomenon of confinement once monopole solutions are allowed to
proliferate in the path integral. One way to achieve this is to show that 3$d$
Maxwell theory arises as a continuum limit of a lattice gauge theory which
does have monopole solutions.222A more familiar situation in which dynamical
monopoles arise are 3$d$ non-abelian gauge theories in which the gauge
symmetry is broken to a subgroup containing $U(1)$. Charged matter fields can
then act as a localized source of magnetic flux. While we believe some of our
results should also apply to this case, it is more difficult to construct non-
abelian non-local gauge theories. We therefore only consider the lattice UV
completion which leads to a relatively simple non-local theory. When doing the
functional integral over $A_{\mu}$, one also must also sum over the
monopole/anti-monopole configurations. The monopole gas gives rise to a (non-
perturbative) effective screening potential which in some cases leads to a
massive photon and a linear potential between charges.

Let us briefly review this mechanism in broad strokes. When $s=2$, 3$d$
electric-magnetic duality allows us to recast Maxwell theory as a theory of a
compact scalar $\phi$ whose action is given by

$S=-\frac{e^{2}}{8\pi^{2}}\int d^{3}x\,\phi\nabla^{2}\phi\,.$ (7)

By UV completing Maxwell theory on a lattice, Polyakov carries out a
resummation of monopole solutions to obtain an effective cosine potential,
$V=\lambda\cos(\phi)$, where $\lambda$ is related to $e$ and the lattice
cutoff. In the infrared the cosine term dominates the dynamics and Wilson
lines can be shown to possess an area law indicating confinement.

It is natural to ask whether a similar confinement mechanism comes into play
in the effective description of the boundary dynamics and large $N_{f}$ QED of
a compact $U(1)$ field given by the action (1). The current work provides an
extended analysis of the resulting dynamics and closely follows
Polyakov:1975rs . In section 2 we discuss the non-local action (1) and its
associated dual photon. In section 3 we UV complete (1) on a lattice and sum
up monopole solutions to obtain a non-local (generalized) version of the sine-
Gordon theory in three dimensions generalizing (7).

In section 4 we perform a perturbative RG calculation of this generalized
sine-Gordon theory to third-order in the coupling constant of the cosine
potential, $\lambda$. Refs. Kleinert:2002qk ; Herbut:2003bs  previously
studied the RG flow of this model at orders $\lambda$ and $\lambda^{2}$
respectively. The first-order beta function for $\lambda$ reveals that the
theory exhibits a phase transition with the cosine potential changing from
being an irrelevant to a relevant deformation of the generalized free theory
as the value of the coupling $e$ is increased, a fact previously noted by
Kleinert:2002qk  in the context of an Abelian Higgs model. But, as observed in
Herbut:2003bs , second-order perturbation theory reveals that generically the
RG flow generates a relevant local kinetic term steering the theory towards
local, confining Maxwell theory. At third-order in $\lambda$, we find that as
the result of a delicate cancellation of terms, the theory remains
renormalizable, despite the non-locality of the model invalidating standard
proofs of renormalizability.333Ref. alves2022supersymmetric  recently provided
a proof of all-loop renormalizability of the supersymmetric version of
fractional-derivative Maxwell theory (1) with $s=1$.

When the cosine potential is relevant, the compact scalar acquires a mass, and
electric charges become confined. The arguments attesting to these facts in
the local case carry over to generalized Maxwell theory, subject to some
technical obstacles, which we tackle and resolve in section 5 where we compute
the expectation value of a Wilson loop. In particular, through a numerical
study of the generalized sine-Gordon equation, we demonstrate the existence of
a field configuration in which the compact scalar jumps when crossing the
plane bounded by a large loop. This allows us to use the method of steepest
descent to evaluate the expectation value of the loop and finally observe area
law behavior.

Section 6 wraps up the paper with a discussion of our results and the
implications of the RG flow of the compact scalar theory for generalized
Maxwell theory in the original gauge field formulation. A possible connection
to the ‘string’ mechanism of confinement is briefly discussed.

##  2 Non-locality and the dual photon

Our goal is to study the infrared physics of a non-local version of the
Polyakov model, which we will show is essentially the generalized free Maxwell
action in equation (1) deformed by relevant interactions coming from non-
perturbative effects. Because the free theory is expressed in terms of a
fractional-derivative operator $D^{\sigma}=(-\nabla^{2})^{\sigma/2}$ with
$\sigma\in\mathbb{R}$, we will establish our definition and conventions for
this operator. It is particularly simple in momentum space, where it amounts
to multiplication by a (possibly fractional) power of the Fourier momentum.

More concretely, let $\mathcal{F}$ denote the operator that implements the
Fourier transform on a function, $\mathcal{F}\phi=\tilde{\phi}$. If we let
$\pi_{\sigma}$ be an operator that acts on a function $f(x)$ by multiplying it
with a power of the norm of its argument, $\pi_{\sigma}f(x)=|x|^{\sigma}f(x)$,
then, as in Huang:2020aao , the fractional-derivative operator can be
expressed as

$\displaystyle D^{\sigma}=\mathcal{F}^{-1}\pi_{\sigma}\mathcal{F}\,.$

This representation makes it clear that the fractional derivative satisfies
the composition property,
$D^{\sigma_{1}+\sigma_{2}}=D^{\sigma_{1}}D^{\sigma_{2}}$. For general values
of $\sigma$, the fractional derivative in position space is an integro-
differential operator which acts as gubser2019non

$\displaystyle
D^{\sigma}f(x)=(2\pi)^{\sigma}\frac{\Gamma(\frac{d+\sigma}{2})}{\pi^{\sigma+d/2}\Gamma(-\frac{\sigma}{2})}\int\frac{d^{d}y}{|x-y|^{d+\sigma}}\bigg{[}f(y)-\sum_{r=0}^{\lfloor\sigma/2\rfloor}y^{2r}b_{r}(\nabla^{2})^{r}f(x)\bigg{]}\,,$
(8)

where the second term is essentially a subtraction to regulate the expression
and the coefficients are given by

$\displaystyle b_{r}=\frac{\Gamma(d/2)}{4^{r}\Gamma(r+d/2)\Gamma(r+1)}\,.$ (9)

This position space representation makes it clear that the action (1) is non-
local,

$\displaystyle
S=\frac{2^{s}\Gamma(\frac{1+s}{2})}{16\pi^{3/2}\Gamma(\frac{2-s}{2})e^{2}}\int
d^{3}x\,d^{3}y\frac{F_{\mu\nu}(x)F^{\mu\nu}(y)}{|x-y|^{1+s}}\,.$ (10)

In the limit $s\rightarrow 2$, this action tends to the usual local
Lagrangian, as can be verified by writing the action as an integral over
momentum space.

As mentioned in the introduction, local $U(1)$ Maxwell theory in three
dimensions may be written in terms of a dual photon. This is an example of an
electric-magnetic duality and is an essential step in incorporating the
effects of magnetic monopoles in the full Polyakov model in terms of local
field variables. To generalize this construction to the non-local theory (10),
we start with the path integral on $\mathbb{R}^{3}$ in terms of the electric
gauge field:

$\displaystyle Z$ $\displaystyle=\int DA\exp\left(-\frac{1}{4e^{2}}\int
d^{3}x\,F_{\mu\nu}D^{s-2}F^{\mu\nu}\right)\,.$ (11)

We take the fluxes $F$ to be in the integer cohomology, $F\in
H^{2}(\mathbb{R}^{3};\mathbb{Z})$. $A$ is a connection on this principle
$U(1)$ bundle, and, in general may be singular due to the presence of monopole
sources. In addition to the $U(1)$ gauge symmetry $A\rightarrow A+d\alpha$,
this action also has a global symmetry with conserved current $j=\ast F$ via
the Bianchi identity. In modern language, this topological current corresponds
to a generalized global $0$-form symmetry Gaiotto:2014kfa . Because this
theory lacks explicit $A_{\mu}$ dependence, there is also an additional
$1$-form symmetry

$A_{\mu}\rightarrow A_{\mu}+\Lambda_{\mu}\,,$ (12)

where $\Lambda$ is an arbitrary flat connection. Physically, local operators
such as monopole operators are charged under $0$-form symmetries, while non-
local operators (such as Wilson lines) are charged under $1$-form symmetries.
In addition to non-local observables of this kind, theories with fractional
kinetic terms might also have non-local operators given by $D^{\sigma}$ acting
on a local operator.444Natural examples of this kind of non-local operator
include conserved currents of non-local field theories Phillips:2019qkc ;
Heydeman:2020ijz . While we do not study these non-local operators directly,
in principle they have a straightforward definition in the path integral
approach similar to the treatment of topological operators.

The electric-magnetic duality transformation is implemented by integrating in
(and then out) certain degrees of freedom, as reviewed, for example, in
wittenlecture . Abstractly, one takes the action with a global symmetry and
couples it to a dynamical gauge field along with a topological term. The
intermediate gauge field is then integrated out. We can do this for the 1-form
symmetry by first introducing a 2-form gauge field $B_{\mu\nu}$. In addition
to the symmetry (12), we also have

$\displaystyle B_{\mu\nu}\rightarrow
B_{\mu\nu}+\partial_{[\mu}\Lambda_{\nu]}\,.$ (13)

To work with an arbitrary gauge parameter $\Lambda$ and not just a flat
1-form, we must covariantize the field strengths with respect to this larger
symmetry. An appropriate choice is

$\mathcal{F}=F-B\,.$ (14)

When $B$ is a flat higher form connection satisfying $dB=0$, this field
strength reduces to the usual one by gauging away $B$ (emphasizing that the
non-local kinetic term does not modify the local definition of the field
strength). By introducing an additional Lagrange multiplier $\phi$ whose
equation of motion enforces the flatness condition, we obtain the path
integral

$Z=\int\\!DA\,DB\,D\phi\,\exp\left(-\int\\!\\!d^{3}x\,\,\frac{1}{4e^{2}}\mathcal{F}_{\mu\nu}D^{s-2}\mathcal{F}^{\mu\nu}-\frac{i}{4\pi}\int
d^{3}x\,\epsilon^{\mu\nu\rho}\partial_{\mu}\phi B_{\nu\rho}\right)\,.$ (15)

We have normalized the Lagrange multiplier such that, as a consequence of
standard Dirac quantization, the field $\phi$ is a compact field with a
periodicity of $2\pi$.

Integrating out $\phi$ returns us to the original theory.555We have ignored
the possibility that $B$ could have nontrivial holonomies on more general
manifolds, but a more careful argument shows $B$ reduces to the trivial flat
connection. Alternately, we can try to eliminate $A$ and then do the path
integral over B. To do this, we use the $1$-form gauge symmetry to set $A$ to
be zero everywhere.666While we do not elaborate on it here, the quantization
of non-local abelian gauge theories in the Faddeev-Popov or BRST formalism
will typically require non-local ghosts. This is the case for a covariant
gauge choice such as $\partial_{\mu}A^{\mu}=0$ or the Feynman $\xi$ gauge in
the original gauge theory. In the present case, the integral over the ghosts
and intermediate gauge fields produces a ratio of determinants which appears
as an overall multiplicative factor we do not include explicitly. The path
integral over $A$ is straightforward, and we are left with

$Z=\int\\!DB\,D\phi\,\exp\left(-\int\\!\\!d^{3}x\,\,\frac{1}{4e^{2}}B_{\mu\nu}D^{s-2}B^{\mu\nu}-\frac{i}{4\pi}\int
d^{3}x\,\epsilon^{\mu\nu\rho}\partial_{\mu}\phi B_{\nu\rho}\right)\,.$ (16)

This action is quadratic in $B$, but in contrast to the local ($s=2$) case, it
contains a non-local differential operator acting on $B$. Nevertheless, we can
integrate out $B$ by completing the square and solving the classical equation
of motion. Using the composition law
$D^{\sigma_{1}+\sigma_{2}}=D^{\sigma_{1}}D^{\sigma_{2}}$ allows us to invert
the non-local derivative, leading to the stationary phase configurations

$B_{\mu\nu}=-\frac{ie^{2}}{2\pi}\epsilon_{\mu\nu\rho}D^{2-s}\partial^{\rho}\phi\,.$
(17)

In the local ($s=2$) theory, this equation may be used to eliminate $B$ in
favor of $\phi$, leading to the dual action (7), and we assume the same is
true for the non-local equation of motion. The final result after integrating
out the auxiliary potential is:

$Z=\int D\phi\exp\left(-\frac{e^{2}}{8\pi^{2}}\int d^{3}x\,\partial_{\rho}\phi
D^{2-s}\partial^{\rho}\phi\right)\,.$ (18)

In the ordinary local case and here, the gauge field has been exchanged with a
compact boson and the coupling $e$ has been inverted. The new feature of the
non-local case is that the non-local exponent $s$ has also been inverted, in
the sense that a higher (lower) derivative gauge theory is exchanged with a
lower (higher) derivative scalar theory, respectively.

Local and non-local operator insertions may also be mapped across the duality
transformation, and of particular interest are Wilson lines (discussed in
section 5) and monopole operators which create field configurations satisfying
(6). For $s=1$, the dual action has a logarithmic propagator, and the operator
$e^{i\phi}$ becomes a candidate monopole operator. A related observation was
made previously in kapustin1999mirror , which pointed out that in the large
$N_{f}$ limit of $\mathcal{N}=2$ and $\mathcal{N}=4$ SQED, the exponentiated
generalized free field has the correct scaling dimension and vortex charge for
a monopole operator but does not, however, carry the appropriate $R$-charge.
In the large $N_{f}$ limit of regular QED3 without supersymmetry, as pointed
out in Borokhov:2002ib , the scaling dimension of the exponentiated
generalized free field no longer matches the monopole operator, a mismatch
Borokhov:2002ib  ascribed to generalized free theory not taking into account
fermionic zero modes.777The dimension of the exponentiated generalized free
field scales as $N_{f}/4$ at large $N_{f}$, whereas Ref. Borokhov:2002ib
determined the monopole scaling dimension, $\Delta_{\text{monopole}}$, to be
given to leading order in $N_{f}$ by
$\displaystyle\frac{\Delta_{\text{monopole}}}{N_{f}}=\frac{1}{4}+\frac{3}{8\,\Gamma(\frac{5}{2})}\sum_{n=2}^{\infty}\frac{(-1)^{n}(n-1)\Gamma(\frac{1}{2}+n)}{\Gamma(n+3)}\zeta(n)=0.2650955...\,.$
For monopole operators with higher vorticity, the mismatch gets increasingly
worse. This issue is not present for the purely bosonic action (1), nor the
issue of higher order interaction terms generated by integrating out fermions,
so that in our setting we can confidently equate $e^{i\phi}$ with the monopole
operator. We corroborate this identification in the next section by showing
how a sum over instanton configurations gives rise to the proliferation of
this operator.

##  3 Monopole solutions and proliferation

So far, we have reviewed the non-local Maxwell action and the equivalence with
a compact scalar in three dimensions. At the level of the field theory, we do
not yet have a mechanism to supply monopole solutions. This can be
incorporated by adding additional UV degrees of freedom which flow to the
Maxwell theory (or the non-local variant) in the infrared. Instead of
considering for instance a non-abelian gauge theory at a point in the moduli
space with an unbroken $U(1)$, we will instead follow Polyakov’s original
proposal of UV completion on a lattice. Suppose that the action (10) is the
low energy limit of an action $S_{UV}$ which satisfies

$S_{UV}(F_{\mu\nu})=S_{UV}(F_{\mu\nu}+2\pi N_{\mu\nu})$ (19)

where $N_{\mu\nu}$ is a lattice-valued tensor field. It is possible to find a
lattice action that satisfies (19) and flows to (10) in the infrared. Indeed,
consider a three-dimensional cubic lattice with lattice spacing $a$, link
variables

$U_{\mu}(x)=e^{-iaA_{\mu}(x)}\,,$ (20)

and plaquette variables

$\displaystyle\begin{split}W_{\mu,\nu}(x)&=U_{\mu}(x)U_{\nu}(x+\hat{\mu})U_{\mu}^{\dagger}(x+\hat{\nu})U_{\nu}^{\dagger}(x)\\\
&=e^{-ia^{2}F_{\mu\nu}}\end{split}$ (21)

(with $F$ the field strength associated with $A_{\mu}$). The non-local action
(10) is given by the continuum limit of

$S_{UV}=-\frac{2^{s}\,\Gamma(\frac{1+s}{2})a^{2}}{16\pi^{3/2}\,e^{2}\,\Gamma(\frac{2-s}{2})}\sum_{\mu\neq\nu}\sum_{x,y}W_{\mu,\nu}(x)\frac{W_{\mu,\nu}(y)-2}{|x-y|^{1+s}}\,,$
(22)

up to a constant term.

Monopole solutions located at $x^{\mu}_{i}$ with charges $q_{i}$, attached to
a Dirac string oriented in, for instance, the $x^{3}$ direction,

$\frac{1}{2}\epsilon^{\mu\nu\rho}F_{\nu\rho}=\sum_{i}\frac{q_{i}}{2}\frac{(x-x_{i})^{\mu}}{|x-x_{i}|^{3}}-\sum_{i}2\pi
q_{i}\delta^{\mu}_{3}\theta(x^{3}-x_{i}^{3})\delta(x^{1}-x_{i}^{1})\delta(x^{2}-x_{i}^{2})$
(23)

are solutions to the Bianchi identity,

$\epsilon^{\mu\nu\rho}\partial_{\mu}F_{\nu\rho}=0\,,$ (24)

and also solutions to the free, local, Maxwell equations (in a distributional
sense). We will refer to the first term on the right hand side of (23) as the
monopole contribution to the field configuration and to the second term on the
right hand side of (23) as the string contribution to it. Due to (19), the
string contribution will not contribute to the action $S_{UV}$.

Any solution to the local version of the Maxwell equations is also a solution
to the non-local equations of motion, c.f., the discussion surrounding (5).
Solutions to (5) which are not solutions to the local Maxwell equations are
distributions (in momentum space) with support at large $k^{\mu}$. Their
Fourier transform would correspond to a highly oscillatory function which
would vanish when integrated over a reasonable test function from the
stationary phase approximation. Since the monopole contribution to the field
strength solves the local Maxwell equation, we must take it into account when
evaluating a saddle point related to the infrared theory associated with (22);
this was one of the insights in Polyakov:1975rs .

Inserting the monopole contribution to the field configuration (23) into the
non-local action (10) we find, by making repeated use of

$\int\frac{d^{d}x}{|x-a|^{\alpha}|x-b|^{\beta}}=\pi^{d/2}\frac{\Gamma(\frac{d-\alpha}{2})\Gamma(\frac{d-\beta}{2})\Gamma(\frac{\alpha+\beta-d}{2})}{\,\Gamma(\frac{\alpha}{2})\Gamma(\frac{\beta}{2})\Gamma(\frac{2d-\alpha-\beta}{2})}|a-b|^{d-\alpha-\beta}\,,$
(25)

that, in the $d\to 3$ limit,

$S=\frac{K}{e^{2}}\sum_{i\neq
j}\frac{q_{i}q_{j}}{|x_{i}-x_{j}|^{s-1}}+\frac{\epsilon}{e^{2}}\sum_{i}q_{i}^{2}\,,$
(26)

where

$K=\frac{2^{s}\pi^{\frac{1}{2}}\Gamma(\frac{s-1}{2})}{8\,\Gamma(\frac{4-s}{2})}\,,\hskip
42.67912pt\epsilon=\frac{K^{\prime}}{a^{s-1}}\,,$ (27)

and $K^{\prime}$ is a scheme dependent dimensionless number. When $s=1$,
$\epsilon$ becomes dimensionless and (26) takes the logarithmic form of a
Coulomb gas.

A saddle point approximation of the monopole contributions to the Euclidean
generating function is given by

$Z\sim\sum_{N=0}^{\infty}\frac{1}{N!}\left[\prod_{i=1}^{N}\,\,\sum_{q_{i}=\pm
1,2,\dots}\\!\\!a^{-3}\int d^{3}x_{i}\right]e^{-S}\,,$ (28)

with $S$ given in (26). Using

$\int
D\varphi\exp\left(-\frac{K}{e^{2}}\frac{16\pi^{3/2}\Gamma(2-\frac{s}{2})}{2^{s}\Gamma(\frac{s-1}{2})}\frac{1}{C^{2}}\int
d^{3}x\left(\varphi D^{4-s}\varphi+2iC\sum_{i}\varphi
q_{i}\delta(x-x_{i})\right)\right)\\\
\propto\exp\left(-\frac{K}{e^{2}}\sum_{i,j}\int
d^{3}xd^{3}y\frac{q_{i}q_{j}\delta(x-x_{i})\delta(y-x_{j})}{|x-y|^{s-1}}\right)\,,$
(29)

with $C$ an arbitrary constant, we find

$\displaystyle\begin{split}Z&\sim\int
D\varphi\,e^{-\frac{4\pi^{2}}{e^{2}C^{2}}\int d^{3}x\,\frac{1}{2}\varphi
D^{4-s}\varphi}\sum_{N=0}^{\infty}\frac{1}{N!}\prod_{i=1}^{N}\frac{1}{a^{3}}\int
d^{3}x_{i}\,\sum_{n=1}^{\infty}e^{-\frac{n^{2}\epsilon}{e^{2}}}2\cos\left(n\frac{4\pi^{2}\varphi}{e^{2}C}\right)\\\
&=\int D\varphi\,e^{-S_{\varphi}}\,,\end{split}$ (30)

where

$S_{\varphi}=\frac{4\pi^{2}}{e^{2}C^{2}}\int
d^{3}x\,\bigg{(}\frac{1}{2}\varphi
D^{4-s}\varphi-\frac{e^{2}C^{2}}{2\pi^{2}a^{3}}\sum_{n=1}^{\infty}e^{-\frac{n^{2}\epsilon}{e^{2}}}\cos\Big{(}n\frac{4\pi^{2}\varphi}{e^{2}C}\Big{)}\bigg{)}\,.$
(31)

After a field redefinition to a new field $\phi=4\pi^{2}\varphi/(e^{2}C)$, we
may write (31) in the form

$S=\int d^{3}x\bigg{(}\frac{1}{2g}\phi
D^{\sigma}\phi-\sum_{n=1}^{\infty}\lambda_{n}\cos(n\phi)\bigg{)},$ (32)

with $\sigma=4-s$, $g=4\pi^{2}/e^{2}$, and
$\lambda_{n}=\frac{2}{a^{3}}e^{-\frac{\epsilon n^{2}}{e^{2}}}$. We observe
that the kinetic term for $\phi$ that we have now obtained through an
instanton summation exactly agrees with the dual photon action (18) that was
derived from a duality transformation of the free gauge theory. Having in each
case normalized $\phi$ to be $2\pi$-periodic, the coefficients of the kinetic
terms match.

##  4 RG flow of non-local sine-Gordon theory

Equation (32) is a non-local version of the sine-Gordon action with higher
harmonics present. Because the bare values of $\lambda_{n}$ decay
exponentially with $n^{2}$, and because $\cos(n\phi)$ will turn out to be
irrelevant for $g>12\pi^{2}/n^{2}$, we will retain only the leading harmonic
in the following and rename $\lambda_{1}$ as simply $\lambda$. Since fields
associated with non-local kinetic terms of the type appearing in (32) (for $s$
not even) do not receive wavefunction renormalization, (see e.g.,
Heydeman:2020ijz  for a gauge theory version of this non-renormalization
theorem), one might expect $\lambda$ to be the only running parameter in the
theory once we have discarded the $\lambda_{n}$ with $n>1$. However, if $s$ is
sufficiently large, a local kinetic term will become relevant and might
dominate the dynamics. This was not an issue in the free theory, and also in
the original Maxwell theory description in which the local term was
irrelevant. In general however, to understand the renormalization group flow
associated with (76), we consider the action

$S=\int d^{3}x\bigg{(}\frac{1}{2g}\phi
D^{\sigma}\phi-\frac{h}{2}\phi\nabla^{2}\phi-\lambda\cos(\phi)\bigg{)}\,,$
(33)

where we choose $h$ to be positive. Expanding the cosine shows that $\lambda$
controls the mass of the dual photon, so the flow of this parameter controls
whether or not the theory becomes confining in the infrared due to monopole
proliferation.

As stated earlier, the beta function for the non-local kinetic term,
$\beta_{g}$ will not receive quantum corrections for non even $s$ and will
vanish for $\sigma=3$ ($s=1$), which is the case we will focus on. To compute
the beta functions for $h$ and $\lambda$, $\beta_{h}$ and $\beta_{\lambda}$,
we will compute the one-particle irreducible (1-PI) 2-point vertex,
$\Sigma_{2}$, and renormalize the couplings, $h$ and $\lambda$, so that the
full propagator $G$, given by

$G^{-1}=G_{f}^{-1}-\Sigma_{2}$ (34)

with $G_{f}$ the propagator in the free theory, is finite. We will use a hard
IR cutoff $\mu$ in momentum space, and a hard UV cutoff $a$ in real space, the
latter in line with our lattice construction from the previous section. On a
practical level, we introduce dimensionless bare couplings by setting

$g=g_{0}\,,\qquad\lambda=\frac{\lambda_{0}}{a^{3}}\,,\qquad
h=\frac{h_{0}}{a}\,.$ (35)

In the following we will express $h_{0}$ and $\lambda_{0}$ in terms of
renormalized couplings $h_{r}$ and $\lambda_{r}$, chosen such as to ensure
that $G$ does not diverge in the $a\to 0$ limit AGG . We work perturbatively
in $\lambda_{0}$. Note that since $\lambda\to-\lambda$ together with
$\phi\to\phi+\pi$ is a symmetry of the action, $h$ can get perturbatively
renormalized only at even orders in $\lambda$, and $\lambda$ itself can only
get renormalized at odd powers.

Evaluating certain loop integrals below requires the free propagator in
position space, given by

$\displaystyle G_{f}(x)$ $\displaystyle=\int_{\mu}^{\infty}k^{2}dk\int
d\Omega_{k}\frac{e^{2\pi ik\cdot x}}{\frac{1}{g}(2\pi
k)^{3}+\frac{h_{0}}{a}(2\pi k)^{2}}$ (36)
$\displaystyle=\frac{g}{2\pi^{2}}\int_{2\pi
x\mu}^{\infty}\frac{\sin(\rho)}{\rho^{2}+\frac{gh_{0}x}{a}\rho}d\rho$
$\displaystyle=\frac{g}{4\pi^{2}\chi}\Big{(}2\cos(\chi)\text{Si}(2\pi
x\mu+\chi)-2(\sin(\chi)\text{Ci}(2\pi x\mu+\chi)+\text{Si}(2\pi
x\mu))-\pi\cos(\chi)+\pi\Big{)}\Bigg{|}_{\chi=\frac{gh_{0}x}{a}}\,,$

where Si and Ci are the sine integral and cosine integral functions
respectively,

$\hbox{Si}(y)=\int_{0}^{y}\frac{\sin(z)}{z}dz\qquad\hbox{Ci}(y)=-\int_{y}^{\infty}\frac{\cos(z)}{z}dz\,.$
(37)

Expanding $G_{f}$ around small values of $x$ we find

$G_{f}(x)=-\frac{g}{2\pi^{2}}\ln\left(\left(\frac{gh_{0}}{a}+2\pi\mu\right)cx\right)+\frac{g^{2}h_{0}x}{8\pi
a}+\mathcal{O}(x^{2})\,,$ (38)

where $c=e^{\gamma-1}$ with $\gamma$ Euler’s constant. Note that if $h_{0}=0$,
then the IR cutoff is needed to make $G_{f}(x)$ well-defined. Since we are
assuming a local term has been generated, we may set $\mu$ to 0 in almost all
of our computations. We will comment on the role of the IR regulator when
relevant.

In order for $G_{f}$ to be finite at tree-level, $h_{0}$ and $h_{r}$ must be
related via

$h_{0}=\kappa ah_{r}+\mathcal{O}(\lambda_{r})\,,$ (39)

where $\kappa$ is the RG scale. To compute the 1-PI 2 point vertex we work
perturbatively in $\lambda_{0}$, decomposing $\Sigma_{2}$ as

$\Sigma_{2}=\sum_{n=1}\Sigma_{2}^{(n)}\,,$ (40)

where $\Sigma_{2}^{(n)}$ is the term of order $(\lambda_{0})^{n}$ in the
expansion of $\Sigma_{2}$ with respect to $\lambda_{0}$. In the remainder of
this section we will compute the $\Sigma_{2}^{(n)}$ up to order $n=3$ and use
this result to compute the $\beta$ functions associated with $h_{r}$ and
$\lambda_{r}$. Our results extend those in Kleinert:2002qk ; Herbut:2003bs ,
first found in a somewhat different context.

###  4.1 Order $\lambda_{0}$

The leading order in $\lambda_{0}$ correction to the propagator comes from the
cosine term. The vertex involves a sum of bubble diagrams, and we find:

$\displaystyle\begin{split}\\\\[-40.0pt] \Sigma_{2}^{(1)}&=\hskip
8.53581pt\leavevmode\hbox to2.4pt{\vbox
to2.4pt{\pgfpicture\makeatletter\hbox{\hskip 1.2pt\lower-1.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(ll) at (-0.7,0)
{$p$}; \vertex(r) at (0.3,0);
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{tikz@color}{rgb}{0,0,0}\definecolor[named]{.}{rgb}{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.0pt}{0.0pt}\pgfsys@curveto{1.0pt}{0.55229pt}{0.55229pt}{1.0pt}{0.0pt}{1.0pt}\pgfsys@curveto{-0.55229pt}{1.0pt}{-1.0pt}{0.55229pt}{-1.0pt}{0.0pt}\pgfsys@curveto{-1.0pt}{-0.55229pt}{-0.55229pt}{-1.0pt}{0.0pt}{-1.0pt}\pgfsys@curveto{0.55229pt}{-1.0pt}{1.0pt}{-0.55229pt}{1.0pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\vertex(l) at (-0.3,0);
\diagram* { (l) -- (r) , }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\hskip
8.53581pt+\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(ll) at (-0.7,0)
{$p$}; \vertex(c) at (0,0); \vertex(r) at (0.4,0); \vertex(l) at (-0.4,0);
\diagram* { (l) -- (c) -- (r) , (c) -- [out=135, in=45, loop, min distance=2
cm] c, }; \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,+\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(ll) at (-0.85,0)
{$p$}; \vertex(c) at (0,0); \vertex(r) at (0.5,0); \vertex(l) at (-0.5,0);
\diagram* { (l) -- (c) -- (r) , (c) -- [out=160, in=90, loop, min distance=2
cm] c, (c) -- [out=90, in=20, loop, min distance=2 cm] c, };
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,+\hskip
14.22636pt\ldots\\\\[8.0pt]
&=\frac{\lambda_{0}}{a^{3}}\sum_{n=0}^{\infty}(-1)^{n}\frac{(2n-1)!!}{(2n)!}G_{f}(0)^{n}\\\
&=\frac{\lambda_{0}}{a^{3}}e^{-\frac{1}{2}G_{f}(0)}\,.\end{split}$ (41)

Unsurprisingly, the expression for $\Sigma_{2}^{(1)}$ in terms of $G_{f}$ is
identical to the one obtained in the local sine-Gordon theory, c.f., AGG . The
difference between the expression for $\Sigma_{2}^{(1)}$ in the local and non-
local theories is encoded in the kinetic term $G_{f}$.

Using (36) and ‘$a$’ as a hard UV cutoff, we find

$\Sigma_{2}^{(1)}=\lambda_{0}\,a^{\frac{g}{4\pi^{2}}-3}(cgh_{r}\kappa)^{\frac{g}{4\pi^{2}}}\left(1-\frac{g^{2}h_{r}}{16\pi}\kappa
a+\mathcal{O}(a^{2})\right)\,.$ (42)

When $g>12\pi^{2}$ the expression for $G^{-1}_{f}-\Sigma_{2}^{(1)}$ is finite.
For $g\leq 12\pi^{2}$ we must have

$\lambda_{0}=\lambda_{r}(\kappa
a)^{3-\frac{g}{4\pi^{2}}}+\mathcal{O}(\lambda_{r}^{3})\,$ (43)

in addition to (39) in order for the propagator to be divergenceless. Note
that, similar to the local sine-Gordon theory, the tree level scaling
dimension of $\lambda$ is $3-g/(4\pi)^{2}$ due to the non perturbative
contributions coming from $g$. For $g>12\pi^{2}$, we see that the cosine
interaction is irrelevant. For this reason, we focus on the regime $g\leq
12\pi^{2}$ in the remainder of this section.

###  4.2 Order $\lambda_{0}^{2}$

We now compute the next order of corrections via the diagrams:

$\displaystyle\begin{split}\leavevmode\hbox to1.98pt{\vbox
to1.98pt{\pgfpicture\makeatletter\hbox{\hskip 0.98999pt\lower-0.98999pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(r) at (1,0);
\vertex(l) at (-1,0); \vertex(fl) at (-1.5,0) {$p$}; \vertex(fr) at (1.3,0);
\vertex(d1) at (0,0.3) {}; \vertex(d2) at (0,0.15) {}; \vertex(d3) at (0,0.0)
{}; \diagram* { (fl) [particle=$p$]-- (l), (r) -- (fr), (l) --[out=90, in=90,
loop, min distance=1.7 cm, edge label=$k_{1}$] (r), (l) --[out=70, in=110,
loop, min distance=1 cm, edge label=$k_{2}$] (r), (l) --[out=270, in=270,
loop, min distance=1.7 cm, edge label=$k_{n}$] (r), (l) --[out=290, in=250,
loop, min distance=1 cm, edge label=$k_{n-1}$] (r), };
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@moveto{0.98999pt}{0.0pt}\pgfsys@curveto{0.98999pt}{0.54675pt}{0.54675pt}{0.98999pt}{0.0pt}{0.98999pt}\pgfsys@curveto{-0.54675pt}{0.98999pt}{-0.98999pt}{0.54675pt}{-0.98999pt}{0.0pt}\pgfsys@curveto{-0.98999pt}{-0.54675pt}{-0.54675pt}{-0.98999pt}{0.0pt}{-0.98999pt}\pgfsys@curveto{0.54675pt}{-0.98999pt}{0.98999pt}{-0.54675pt}{0.98999pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@moveto{0.98999pt}{0.0pt}\pgfsys@curveto{0.98999pt}{0.54675pt}{0.54675pt}{0.98999pt}{0.0pt}{0.98999pt}\pgfsys@curveto{-0.54675pt}{0.98999pt}{-0.98999pt}{0.54675pt}{-0.98999pt}{0.0pt}\pgfsys@curveto{-0.98999pt}{-0.54675pt}{-0.54675pt}{-0.98999pt}{0.0pt}{-0.98999pt}\pgfsys@curveto{0.54675pt}{-0.98999pt}{0.98999pt}{-0.54675pt}{0.98999pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@moveto{0.98999pt}{0.0pt}\pgfsys@curveto{0.98999pt}{0.54675pt}{0.54675pt}{0.98999pt}{0.0pt}{0.98999pt}\pgfsys@curveto{-0.54675pt}{0.98999pt}{-0.98999pt}{0.54675pt}{-0.98999pt}{0.0pt}\pgfsys@curveto{-0.98999pt}{-0.54675pt}{-0.54675pt}{-0.98999pt}{0.0pt}{-0.98999pt}\pgfsys@curveto{0.54675pt}{-0.98999pt}{0.98999pt}{-0.54675pt}{0.98999pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\quad&=\frac{(n+1)^{2}n!}{((n+1)!)^{2}}\left(\prod_{i=1}^{n}\int
d^{3}k_{i}\,\hat{G}(k_{i})\right)\delta\left(\sum_{i=1}^{n}k_{i}+p\right)\\\
&=\frac{1}{n!}\int d^{3}xe^{2\pi x\cdot p}G(x)^{n}\,,\end{split}$ (44)

and

$\displaystyle\begin{split}\leavevmode\hbox to1.98pt{\vbox
to1.98pt{\pgfpicture\makeatletter\hbox{\hskip 0.98999pt\lower-0.98999pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(r) at (0.5,0);
\vertex(l) at (-0.7,0) {$p$}; \vertex(c) at (0,0) ; \vertex(t) at (0,2);
\vertex(d1) at (-0.45,1) {}; \vertex(d2) at (-0.3,1) {}; \vertex(d3) at
(-0.15,1) {}; \diagram* { (l) -- (r), (c) --[out=160, in=200, loop, min
distance=2.5 cm, edge label=$k_{1}$] (t), (c) --[out=140, in=220, loop, min
distance=1.7 cm, edge label=$k_{2}$] (t), (c) --[out=40, in=-40, loop, min
distance=1.7 cm, edge label=$k_{n-1}$] (t), (c) --[out=20, in=-20, loop, min
distance=2.5 cm, edge label=$k_{n}$] (t), };
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@moveto{0.98999pt}{0.0pt}\pgfsys@curveto{0.98999pt}{0.54675pt}{0.54675pt}{0.98999pt}{0.0pt}{0.98999pt}\pgfsys@curveto{-0.54675pt}{0.98999pt}{-0.98999pt}{0.54675pt}{-0.98999pt}{0.0pt}\pgfsys@curveto{-0.98999pt}{-0.54675pt}{-0.54675pt}{-0.98999pt}{0.0pt}{-0.98999pt}\pgfsys@curveto{0.54675pt}{-0.98999pt}{0.98999pt}{-0.54675pt}{0.98999pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@moveto{0.98999pt}{0.0pt}\pgfsys@curveto{0.98999pt}{0.54675pt}{0.54675pt}{0.98999pt}{0.0pt}{0.98999pt}\pgfsys@curveto{-0.54675pt}{0.98999pt}{-0.98999pt}{0.54675pt}{-0.98999pt}{0.0pt}\pgfsys@curveto{-0.98999pt}{-0.54675pt}{-0.54675pt}{-0.98999pt}{0.0pt}{-0.98999pt}\pgfsys@curveto{0.54675pt}{-0.98999pt}{0.98999pt}{-0.54675pt}{0.98999pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@moveto{0.98999pt}{0.0pt}\pgfsys@curveto{0.98999pt}{0.54675pt}{0.54675pt}{0.98999pt}{0.0pt}{0.98999pt}\pgfsys@curveto{-0.54675pt}{0.98999pt}{-0.98999pt}{0.54675pt}{-0.98999pt}{0.0pt}\pgfsys@curveto{-0.98999pt}{-0.54675pt}{-0.54675pt}{-0.98999pt}{0.0pt}{-0.98999pt}\pgfsys@curveto{0.54675pt}{-0.98999pt}{0.98999pt}{-0.54675pt}{0.98999pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\quad&=-\frac{(n+2)!}{n!(n+2)!}\left(\prod_{i=1}^{n}\int
d^{3}k_{i}\,\hat{G}(k_{i})\right)\delta\left(\sum_{i=1}^{n}k_{i}\right)\\\
&=-\frac{1}{n!}\int d^{3}xG(x)^{n}\,.\end{split}$ (45)

We find

$\displaystyle\begin{split}\Sigma_{2}^{(2)}(p)&=\left(\Sigma_{2}^{(1)}\right)^{2}\left(\sum_{n=1}^{\infty}\leavevmode\hbox
to1.98pt{\vbox to1.98pt{\pgfpicture\makeatletter\hbox{\hskip
0.98999pt\lower-0.98999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(r) at (1,0);
\vertex(l) at (-1,0); \vertex(fl) at (-1.5,0) {}; \vertex(fr) at (1.3,0);
\vertex(d1) at (0,0.3) {}; \vertex(d2) at (0,0.15) {}; \vertex(d3) at (0,0.0)
{}; \diagram* { (fl) [particle=$p$]-- (l), (r) -- (fr), (l) --[out=90, in=90,
loop, min distance=1.7 cm, edge label=$k_{1}$] (r), (l) --[out=70, in=110,
loop, min distance=1 cm, edge label=$k_{2}$] (r), (l) --[out=270, in=270,
loop, min distance=1.7 cm, edge label=$k_{2n+1}$] (r), (l) --[out=290, in=250,
loop, min distance=1 cm, edge label=$k_{2n}$] (r), };
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@moveto{0.98999pt}{0.0pt}\pgfsys@curveto{0.98999pt}{0.54675pt}{0.54675pt}{0.98999pt}{0.0pt}{0.98999pt}\pgfsys@curveto{-0.54675pt}{0.98999pt}{-0.98999pt}{0.54675pt}{-0.98999pt}{0.0pt}\pgfsys@curveto{-0.98999pt}{-0.54675pt}{-0.54675pt}{-0.98999pt}{0.0pt}{-0.98999pt}\pgfsys@curveto{0.54675pt}{-0.98999pt}{0.98999pt}{-0.54675pt}{0.98999pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@moveto{0.98999pt}{0.0pt}\pgfsys@curveto{0.98999pt}{0.54675pt}{0.54675pt}{0.98999pt}{0.0pt}{0.98999pt}\pgfsys@curveto{-0.54675pt}{0.98999pt}{-0.98999pt}{0.54675pt}{-0.98999pt}{0.0pt}\pgfsys@curveto{-0.98999pt}{-0.54675pt}{-0.54675pt}{-0.98999pt}{0.0pt}{-0.98999pt}\pgfsys@curveto{0.54675pt}{-0.98999pt}{0.98999pt}{-0.54675pt}{0.98999pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@moveto{0.98999pt}{0.0pt}\pgfsys@curveto{0.98999pt}{0.54675pt}{0.54675pt}{0.98999pt}{0.0pt}{0.98999pt}\pgfsys@curveto{-0.54675pt}{0.98999pt}{-0.98999pt}{0.54675pt}{-0.98999pt}{0.0pt}\pgfsys@curveto{-0.98999pt}{-0.54675pt}{-0.54675pt}{-0.98999pt}{0.0pt}{-0.98999pt}\pgfsys@curveto{0.54675pt}{-0.98999pt}{0.98999pt}{-0.54675pt}{0.98999pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}+\sum_{n=1}^{\infty}\leavevmode\hbox
to1.98pt{\vbox to1.98pt{\pgfpicture\makeatletter\hbox{\hskip
0.98999pt\lower-0.98999pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } \feynman \vertex(r) at (0.5,0);
\vertex(l) at (-0.7,0) {$p$}; \vertex(c) at (0,0) ; \vertex(t) at (0,2);
\vertex(d1) at (-0.6,1) {}; \vertex(d2) at (-0.45,1) {}; \vertex(d3) at
(-0.3,1) {}; \diagram* { (l) -- (r), (c) --[out=160, in=200, loop, min
distance=2.5 cm, edge label=$k_{1}$] (t), (c) --[out=140, in=220, loop, min
distance=1.7 cm, edge label=$k_{2}$] (t), (c) --[out=40, in=-40, loop, min
distance=1.7 cm, edge label=$k_{2n-1}$] (t), (c) --[out=20, in=-20, loop, min
distance=2.5 cm, edge label=$k_{2n}$] (t), };
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@moveto{0.98999pt}{0.0pt}\pgfsys@curveto{0.98999pt}{0.54675pt}{0.54675pt}{0.98999pt}{0.0pt}{0.98999pt}\pgfsys@curveto{-0.54675pt}{0.98999pt}{-0.98999pt}{0.54675pt}{-0.98999pt}{0.0pt}\pgfsys@curveto{-0.98999pt}{-0.54675pt}{-0.54675pt}{-0.98999pt}{0.0pt}{-0.98999pt}\pgfsys@curveto{0.54675pt}{-0.98999pt}{0.98999pt}{-0.54675pt}{0.98999pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@moveto{0.98999pt}{0.0pt}\pgfsys@curveto{0.98999pt}{0.54675pt}{0.54675pt}{0.98999pt}{0.0pt}{0.98999pt}\pgfsys@curveto{-0.54675pt}{0.98999pt}{-0.98999pt}{0.54675pt}{-0.98999pt}{0.0pt}\pgfsys@curveto{-0.98999pt}{-0.54675pt}{-0.54675pt}{-0.98999pt}{0.0pt}{-0.98999pt}\pgfsys@curveto{0.54675pt}{-0.98999pt}{0.98999pt}{-0.54675pt}{0.98999pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@moveto{0.98999pt}{0.0pt}\pgfsys@curveto{0.98999pt}{0.54675pt}{0.54675pt}{0.98999pt}{0.0pt}{0.98999pt}\pgfsys@curveto{-0.54675pt}{0.98999pt}{-0.98999pt}{0.54675pt}{-0.98999pt}{0.0pt}\pgfsys@curveto{-0.98999pt}{-0.54675pt}{-0.54675pt}{-0.98999pt}{0.0pt}{-0.98999pt}\pgfsys@curveto{0.54675pt}{-0.98999pt}{0.98999pt}{-0.54675pt}{0.98999pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)\\\
&=4\pi(\Sigma_{2}^{(1)})^{2}\int_{a}^{\infty}dx\,x^{2}\bigg{(}\frac{\sin(2\pi
px)}{2\pi
px}\Big{(}\sinh\big{(}G_{f}(x)\big{)}-G_{f}(x)\Big{)}+1-\cosh\big{(}G_{f}(x)\big{)}\bigg{)}\,.\end{split}$
(46)

Once again, this expression is identical in form to the order
$\lambda_{0}^{2}$ correction of the local sine-Gordon theory which can be
found in, e.g., AGG .

We are interested in terms in $\Sigma_{2}^{(2)}$ which are divergent in the
limit $a\to 0$. Such terms will arise from the lower end of the integral.
Expanding the sine in powers of $p$, we find, to leading order

$\Sigma_{2}^{(2)}(0)=4\pi(\Sigma_{2}^{(1)})^{2}\int_{a}^{\infty}dx\,x^{2}\left(-e^{-G_{f}(x)}-G_{f}(x)+1\right)\,.$
(47)

Given the expansion in the last line of (36) we see that the integrand is such
that the integral in (47) remains finite in the $a\to 0$ limit.

Next we have

$\displaystyle\begin{split}\Sigma_{2}^{(2)}(p)-\Sigma_{2}^{(2)}(0)&=\sum_{n=1}^{\infty}(-1)^{n}\left(\Sigma_{2}^{(1)}\right)^{2}p^{2n}\frac{\pi^{2n+1}2^{2n+2}}{(2n+1)!}\int_{a}^{\infty}dx\,x^{2n+2}\Big{(}\sinh(G_{f})-G_{f}\Big{)}\\\
&=\sum_{n=1}^{\infty}(-1)^{n}\left(\Sigma_{2}^{(1)}\right)^{2}p^{2n}\frac{\pi^{2n+1}2^{2n+1}}{(2n+1)!}\left(\int_{a}^{\infty}dx\,x^{2n+2}e^{G_{f}(x)}+\mathcal{O}(a^{2n+3})\right).\end{split}$
(48)

(where, in the first equality, we assumed that we can swap the summation over
momentum modes with an integral over $x$).

It is convenient to define

$I_{k}(a)=\int_{a}^{\infty}dx\,x^{2k+2}e^{G_{f}(x)}\,,$ (49)

The dependence of $I_{1}$ on $a$ in the small $a$ limit will provide us with
the information we need on the renormalization of $h_{0}$ at this order in
perturbation theory. If $I_{k}$ diverges for $k>1$, then this implies that we
need to add higher order local kinetic terms to the action. As is the case
when studying local theories, these higher order kinetic terms are irrelevant
so we may safely ignore them. Omitting constant terms, we find

$I_{k}(a)=\left(g\kappa h_{r}c\right)^{-\frac{g}{2\pi^{2}}}\\\
\times\begin{cases}a^{2k+3-\frac{g}{2\pi^{2}}}\left(\frac{1}{\frac{g}{2\pi^{2}}-\left(2k+3\right)}+\frac{g^{2}h_{r}}{8\pi}\frac{\kappa
a}{\frac{g}{2\pi^{2}}-\left(2k+4\right)}+\mathcal{O}(a^{2})\right)&\begin{subarray}{c}g\neq(2k+n)2\pi^{2}\\\
n=3,\,4,\,5,\,...\end{subarray}\\\ -\ln(a)-\frac{g^{2}h_{r}}{8\pi}\kappa
a+\mathcal{O}(a^{2})&g=(2k+3)2\pi^{2}\\\
\frac{1}{a}\left(1-\frac{g^{2}h_{r}}{8\pi}\kappa
a\ln(a)+\mathcal{O}(a^{2})\right)&g=(2k+4)2\pi^{2}\\\
\ldots&\ldots\end{cases}$ (50)

Inserting (42) and (50) into (48), we find

$\displaystyle\begin{split}\frac{\partial}{\partial
p^{2}}\Sigma_{2}^{(2)}\Big{|}_{p=0}=&\lambda_{r}^{2}\begin{cases}-\frac{2\pi^{5}}{3}(a\kappa)^{5-\frac{g}{2\pi^{2}}}\kappa\left(\frac{4}{g-10\pi^{2}}+\frac{g^{2}h_{r}\pi}{(g-10\pi^{2})(g-12\pi^{2})}(a\kappa)+\mathcal{O}(a^{2})\right)&\begin{subarray}{c}g\neq
2n\pi^{2}\\\ n=5,\,6,\,\ldots\end{subarray}\\\
\frac{4}{3}\pi^{3}\kappa\ln(a)+\frac{50h_{r}\pi^{6}\kappa}{3}\kappa
a\left(1-\ln(a)\right)+\mathcal{O}(a^{2})&g=10\pi^{2}\\\
-\frac{4\pi^{3}}{3a}+24\pi^{6}g^{2}h_{r}\kappa\left(1+\ln
a\right)+\mathcal{O}(a)&g=12\pi^{2}\\\ \ldots&\ldots\end{cases}\end{split}$
(51)

For $g<10\pi^{2}$ we see that $\frac{\partial}{\partial
p^{2}}\Sigma_{2}^{(2)}$ is finite in the $a\to 0$ limit. For $g=2n\pi^{2}$
with $n=5,\,6,\,\ldots$, $\frac{\partial}{\partial
p^{2}}\Sigma_{2}^{(2)}\Big{|}_{p=0}$ will possess logarithmic divergences in
the $a\to 0$ limit. As $g$ increases beyond $g=10\pi^{2}$, more terms in the
expansion of (51) will contribute to the ultraviolet divergences of
$\frac{\partial}{\partial p^{2}}\Sigma_{2}^{(2)}$.

The divergences in the Green’s function (34) arising from the
$\mathcal{O}(p^{2})$ terms in $\Sigma_{2}^{(2)}$ may be removed by an order
$\lambda_{r}^{2}$ correction to $h_{r}$. More explicitly, in order for the
order $p^{2}$ term in $\frac{h_{0}}{a}(2\pi p)^{2}-\Sigma_{2}^{(2)}(p)$ to
remain finite in the $a\to 0$ limit we may use a prescription of the form

$\frac{h_{0}}{a\kappa}=h_{r}+\lambda_{r}^{2}\begin{cases}0&0<g<10\pi^{2}\\\
\frac{\pi}{3}\ln(\kappa a)+C_{3}h_{r}&g=10\pi^{2}\\\
-\frac{2\pi^{3}}{3(g-10\pi^{2})}(a\kappa)^{5-\frac{g}{2\pi^{2}}}+C_{1}(g)+C_{2}(g)h_{r}&10\pi^{2}<g<12\pi^{2}\\\
-\frac{\pi}{3a\kappa}+6\pi^{4}h_{r}\ln(\kappa a)+C_{4}&g=12\pi^{2}\\\
\ldots&\ldots\\\ \end{cases}\,,$ (52)

up to order $\lambda_{r}^{4}$ corrections. Here $C_{1}(g)$, $C_{2}(g)$,
$C_{3}$ and $C_{4}$ are prescription dependent finite counterterms which we
will choose later. Other counterterms are also allowed but we have not
included them since, as we will see later, they will not affect the beta
function for $h_{r}$.

###  4.3 Order $\lambda_{0}^{3}$

At order $\lambda_{0}^{3}$ there are eight classes of diagrams that contribute
to $\Sigma_{2}^{(3)}$. As explained earlier, since $\lambda\to-\lambda$ and
$\phi\to\phi+\pi$ is a symmetry, $h_{r}$ will not be renormalized at order
$\lambda_{0}^{3}$. So we may evaluate the diagrams at zero momentum. Since
this computation is identical to that of the local theory, we will not write
out the diagrams explicitly, but quote the result of AGG ,

$\Sigma_{2}^{(3)}=\left(\Sigma_{2}^{(1)}\right)^{3}\frac{1}{8}\int_{a}^{\infty}d^{3}x\int_{a}^{\infty}d^{3}y\,\bigg{(}e^{G_{f}(x)+G_{f}(y)-G_{f}(x-y)}-e^{G_{f}(x)}-e^{G_{f}(y)}\bigg{)}\,.$
(53)

Since the dependence of $\Sigma_{2}^{(3)}$ on the UV cutoff is tied to the
lower end of the integration region in (53), we separate the integration into
four regions:

$\displaystyle\begin{split}\bf{I}_{1}&=\\{x,y\,|\,a<|x|<\Delta,\,a<|y|<\Delta\\}\\\
\bf{I}_{2}&=\\{x,y\,|\,a<|x|<\Delta,\,\Delta<|y|<\infty\\}\\\
\bf{I}_{3}&=\\{x,y\,|\,\Delta<|x|<\infty,\,a<|y|<\Delta\\}\\\
\bf{I}_{4}&=\\{x,y\,|\,\Delta<|x|<\infty,\,\Delta<|y|<\infty\\}\,,\end{split}$
(54)

where $\Delta$ is a length scale much bigger than $a$. Because we are only
interested in the part of $\Sigma_{2}^{(3)}$ that diverges in the
$a\rightarrow 0$ limit, we can discard $\bf{I}_{4}$. The integrals over
$\bf{I}_{2}$ and $\bf{I}_{3}$ are equal:

$\Sigma_{2}^{(3)}\Big{|}_{\bf{I}_{2}}=\Sigma_{2}^{(3)}\Big{|}_{\bf{I}_{3}}=\left(\Sigma_{2}^{(1)}\right)^{3}\frac{1}{8}\int_{a}^{\Delta}d^{3}x\,e^{G_{f}(x)}\int_{\Delta}^{\infty}d^{3}y\,\bigg{(}e^{G_{f}(y)-G_{f}(x-y)}-1\bigg{)}+\mathcal{O}(a^{0})\,.$
(55)

Each divergent piece in $\bf{I}_{2}$, $\bf{I}_{3}$, and $\bf{I}_{4}$ admits of
an expansion in powers of $\kappa\Delta$, with logarithmic factors also
allowed. Since $\Sigma_{2}^{(3)}$ does not depend on $\Delta$, the $\Delta$
dependencies of $\bf{I}_{1}$ to $\bf{I}_{4}$ cancel among each other, meaning
that only $(\kappa\Delta)^{0}$ terms in the divergent pieces contribute to
$\Sigma_{2}^{(3)}$. We can single out these pieces by taking the limit
$\kappa\Delta\rightarrow 0$ (while still demanding that $\Delta>>a$). In this
limit the integral over $\bf{I}_{1}$ disappears,888 For comparison with the RG
computation of 2$d$ sine-Gordon theory, we note that this vanishing does not
happen in $2d$, where the integral over $\bf{I}_{1}$ contains a divergent
piece proportional to $(\kappa\Delta)^{0}\log(a/\Delta)$. and we find that

$\left(\Sigma_{2}^{(1)}\right)^{-3}\Sigma_{2}^{(3)}=\frac{1}{4}\int_{a}^{\infty}d^{3}x\,e^{G_{f}(x)}\int_{a}^{\infty}d^{3}y\,\bigg{(}e^{G_{f}(y)-G_{f}(x-y)}-1\bigg{)}+\mathcal{O}(a^{0})\,.$
(56)

To evaluate the integral on the right hand side of (56) we expand the
integrand over the $y$ coordinate in powers of $x$ and integrate over the
angular coordinates. We find

$\displaystyle\begin{split}\frac{1}{4}\int_{a}^{\infty}&{d^{3}x}\,e^{G_{f}(x)}\int_{a}^{\infty}d^{3}y\left(e^{G_{f}(y)-G_{f}(x-y)}-1\right)\\\
&=\frac{2\pi^{2}}{3}I_{1}(a)\int_{a}^{\infty}dy\,\Bigg{(}y^{2}\left(\frac{\partial
G_{f}}{\partial y}\right)^{2}-\frac{\partial}{\partial
y}\left(y^{2}\frac{\partial G_{f}}{\partial
y}\right)\Bigg{)}+\mathcal{O}(a^{0})\,.\end{split}$ (57)

The total derivative on the right hand side of (57) can be evaluated using
(36) Reinserting the infrared cutoff $\mu$, we find

$\displaystyle\int_{a}^{\infty}dy\frac{\partial}{\partial
y}\left(y^{2}\frac{\partial G_{f}}{\partial y}\right)$
$\displaystyle=y^{2}\frac{\partial G_{f}}{\partial
y}\Big{|}_{y=\infty}-y^{2}\frac{\partial G_{f}}{\partial y}\Big{|}_{y=a}$ (58)
$\displaystyle=-\frac{g}{2\pi^{2}}\frac{\sin(2\pi
y\mu)}{gh_{r}\kappa+2\pi\mu}\left(1+\mathcal{O}(y^{-1})\right)\Big{|}_{y=\infty}+\mathcal{O}\left(a\right)$

In the $\mu\to 0$ limit this expression is linear in $a$ and, as we will see
shortly, will not contribute to the divergences of $\Sigma_{2}^{(2)}$ for the
values of $g$ we are interested in. The same type of behavior is expected if
we use a mass regulator for the infrared behavior.999 With a positive mass
regulator $\Sigma_{0}$, the inverse propagator is $\frac{1}{g}(2\pi
k)^{3}+h(2\pi k)^{2}+\Sigma_{0}$ and the position space propagator $G_{f}(y)$
is obtained by the Fourier transform with no cutoffs. In this case the large
$y$ asymptotic of $G_{f}(y)$ is $e^{-y\sqrt{\frac{\Sigma_{0}}{h}}}/(4\pi hy)$.
Thus, the limit $\lim_{y\to\infty}y^{2}G_{f}^{\prime}(y)=0$.  If we set the
infrared cutoff to zero from the start we obtain a constant contribution to
the right-hand side of (58).

The remaining term in (57) requires more work.

$\displaystyle\begin{split}\int_{a}^{\infty}dy\,y^{2}\left(\frac{\partial
G_{f}}{\partial y}\right)^{2}&=\frac{g^{4}h_{r}^{2}\kappa^{2}}{4\pi^{4}}\int
d\rho\,d\sigma\,dx\frac{\rho\sigma\sin(\rho)\sin(\sigma)}{(gh_{r}\kappa
x\rho+\rho^{2})^{2}(gh_{r}\kappa x\sigma+\sigma^{2})^{2}}\\\
&=\frac{g}{4\pi^{4}h_{r}\kappa}\int
d\rho\,d\sigma\frac{\sin(\rho)\sin(\sigma)(\rho^{2}-\sigma^{2}+2\rho\sigma\ln(\sigma/\rho))}{\rho\sigma(\rho-\sigma)^{3}}+\mathcal{O}(a)\\\
&=\frac{g}{4\pi^{4}h_{r}\kappa}\int
d\beta\,d\sigma\frac{\sin(\sigma)\sin(\beta\sigma)(\beta^{2}-1-2\beta\ln(\beta))}{\beta\sigma^{2}(-1+\beta)^{3}}+\mathcal{O}(a)\\\
&=\frac{g}{8\pi^{3}h_{r}\kappa}+\mathcal{O}(a)\,.\end{split}$ (59)

Thus,

$\left(\Sigma_{2}^{(1)}\right)^{-3}\Sigma_{2}^{(3)}=\frac{g}{12\pi
h_{r}\kappa}\left(I_{1}(a)-I_{1}(\Delta)\right)\,.$ (60)

It is now straightforward to compute $\Sigma_{2}^{(3)}$ explicitly. One finds
that it contains a logarithmic divergence for $g^{2}=10\pi^{2}$, power law
divergences for $g/2\pi^{2}<5$ and non-integer, and a mix of logarithmic and
power law divergences for integer $g/2\pi^{2}>5$, reminiscent of (51).
Somewhat magically, these divergent contributions are exactly cancelled by the
renormalized coupling $h_{r}$ in (52) when evaluating
$\Sigma_{2}^{(1)}(0)+\Sigma_{2}^{(3)}(0)$. Thus,

$\lambda_{0}=\lambda_{r}(\kappa
a)^{3-\frac{g}{4\pi^{2}}}+\mathcal{O}(\lambda_{r}^{5})\hskip 28.45274ptg\leq
12\pi^{2}\,.$ (61)

The order $\lambda_{r}^{3}$ divergences in $\Sigma_{2}^{(1)}(0)$ and
$\Sigma_{2}^{(3)}(0)$, which delicately cancel one another, are of order
$\frac{1}{h_{r}}$. Had these terms not cancelled, the $h_{r}\rightarrow 0$
limit would have been singular, and it would have been unclear whether the
purely non-local theory ($h_{r}=0$) was well-defined. If one undertakes a
perturbative analysis of purely non-local sine-Gordon theory, it becomes
necessary to introduce an IR regulator $\mu$ as mentioned in the beginning of
4. It then turns out that order $\lambda_{0}^{3}$ divergences are
$\mu$-dependent and can only be eliminated if $\lambda_{r}$ is a function, not
just of $\lambda_{0}$, $\kappa$, and $a$, but also of $\mu$. In other words,
the cancellation of $\lambda_{0}^{3}$ divergences cures the purely non-local
theory of pathological UV-IR mixing.

###  4.4 The $\beta$ functions

We are now in a position to compute the beta function for the renormalized
couplings $\lambda_{r}$, $h_{r}$, (and $g_{r}$). We find:

$\displaystyle\begin{split}\beta_{g}=&\,0\,,\\\
\beta_{h}=&\left.-h_{r}+\lambda_{r}^{2}\times\begin{cases}-\frac{g-10\pi^{2}}{2\pi^{2}}C_{1}(g)-\frac{g-12\pi^{2}}{2\pi^{2}}C_{2}(g)h_{r}&\begin{subarray}{c}\displaystyle
g\neq 2n\pi^{2}\\\ n=5,\,6,\,\ldots\end{subarray}\\\
-\frac{\pi}{3}+C_{3}h_{r}&g=10\pi^{2}\\\ -C_{4}-6\pi^{4}h_{r}&g=12\pi^{2}\\\
\end{cases}\hskip 2.84526pt\right\\}+\mathcal{O}(\lambda_{r}^{4})\,,\\\
\beta_{\lambda}=&\left(\frac{g}{4\pi^{2}}-3\right)\lambda_{r}+\mathcal{O}(\lambda_{r}^{5})\qquad
0<g\leq 12\pi^{2}\,.\end{split}$ (62)

Some comments are in order. We note that one can choose the $C_{i}$’s such
that the $\beta$ functions are not continuous functions of $g$. This might
seem somewhat alarming at first, but we remind the reader that $g$ is an
exactly marginal direction in the space of couplings, so there is no flow in
the $g$ direction.

In addition, one may always generate a smooth beta function by an appropriate
prescription for finite counterterms. In particular, one can choose $C_{1}(g)$
and $C_{2}(g)$ to be functions that are smooth for $g\neq 10\pi^{2}$ and
$g\neq 12\pi^{2}$ and which satisfy

$\displaystyle\begin{split}&C_{1}(g)\hskip
14.22636pt\rightarrow\frac{2\pi^{2}}{g-10\pi^{2}}\times\frac{\pi}{3}\text{ as
}g\rightarrow 10\pi^{2}\\\
&C_{1}(12\pi^{2})=\frac{2\pi^{2}}{g-10\pi^{2}}C_{4}\\\
&C_{2}(10\pi^{2})=-\frac{2\pi^{2}}{g-12\pi^{2}}C_{3}\\\ &C_{2}(g)\hskip
14.22636pt\rightarrow\frac{2\pi^{2}}{g-12\pi^{2}}\times 6\pi^{4}\text{ as
}g\rightarrow 12\pi^{2}\end{split}$ (63)

In particular, one can choose
$C_{1}(g)=\frac{2\pi^{2}}{g-10\pi^{2}}\times\frac{\pi}{3}$,
$C_{2}(g)=\frac{2\pi^{2}}{g-12\pi^{2}}\times 6\pi^{4}$, $C_{3}=-6\pi^{4}$,
$C_{4}=\frac{\pi}{3}$ in which case,

$\beta_{h}=-h_{r}+\left(-\frac{\pi}{3}-6\pi^{4}h_{r}\right)\lambda_{r}^{2}$
(64)

at least for $g\leq 12\pi^{2}$. Alternately, one may always choose $C_{1}$ and
$C_{2}$ so that $\beta_{h}=-h_{r}+\mathcal{O}(\lambda_{r}^{4})$ for $g$
sufficiently smaller than $10\pi^{2}$.

Finally, one may inquire whether it is possible to modify the $-\pi/3$ term in
the beta function for $h_{r}$ at $g=10\pi^{2}$ or the $-6\pi^{4}h_{r}$ term at
$g=12\pi^{2}$. We claim that no such counterterms are possible. Our argument
relies on the Poincare-Dulac theorem (see, e.g., n61 ); both of the
aforementioned terms are associated with resonant monomials.

Thus, it seems that $\lambda_{r}$ is marginal for $g=12\pi^{2}$ at least up to
order $\lambda_{r}^{5}$. For $g<12\pi^{2}$ (and sufficiently small starting
value of $\lambda_{r}$), the sine potential is relevant in the infrared. The
coupling to the local kinetic term, $h_{r}$ cannot be tuned away at
$g=10\pi^{2}$. Since $h_{r}$ is relevant, if it is not tuned to zero, it will
dominate the dynamics in the infrared, and the theory will flow towards the
three-dimensional theory studied in Polyakov:1975rs . We claim that even when
$h_{r}=0$, the infrared theory is confining for $g<12\pi^{2}$. This is the
content of the next section.

##  5 Confinement via an area law

To assess whether the original gauge theory is in a confining phase once a gas
of monopoles is included, we may evaluate the Wilson loop. This non-local
operator depends on a contour $C$ and will exhibit an area law growth for a
large loop in the confining phase. In terms of the holonomy, we write,

$F(C)=\left<\exp(-W(C))\right>=\int
DA_{\mu}\,e^{i\oint_{C}A_{\mu}dx^{\mu}}e^{-S},$ (65)

where $C$ is a planar contour orthogonal to the $x^{3}$ axis. Recall that in
the absence of Wilson loops we have, via (30),

$\int DA_{\mu}\,e^{-S}\sim\int D\phi\,e^{-S_{\phi}}\,,$ (66)

with

$S_{\phi}=\frac{1}{2g}\int d^{3}x\bigg{(}\phi
D^{\sigma}\phi-2g\lambda\cos\phi\bigg{)}\,,$ (67)

where we omitted the higher harmonics.

To obtain a manageable expression for the Wilson loop $W(C)$ we write

$\int DA_{\mu}\,e^{-S}\sim\int D\phi\exp\left(-\frac{1}{2g}\int d^{3}x\,\phi
D^{\sigma}\phi\right)\sum_{N=0}^{\infty}\sum_{\\{q_{i}=\pm
1\\}}\frac{\lambda^{N}}{N!}\int\prod_{i=1}^{N}d^{3}x_{i}\,e^{iq_{i}\phi(x_{i})}$
(68)

and notice that evaluated on monopole configurations with charges $q_{i}$ we
have

$e^{i\int_{C}A_{\mu}dx^{\mu}}=e^{i\int_{B}F_{3}dS^{3}}=e^{i\sum_{i,q_{i}}q_{i}\eta(x_{i})}$
(69)

where

$\eta(x)=\ \int_{B}\frac{(x^{\mu}-y^{\mu})}{2|x-y|^{3}}\delta^{3}_{\mu}d^{2}y$
(70)

where the integration region is in the interior of $B$ ($\partial B=C$). Thus,
after a change of variables $\phi\to\phi+\eta$ we find

$F(C)=\int D\phi\exp\left(-\frac{1}{2g}\int
d^{3}x\Big{(}(\phi-\eta)D^{\sigma}(\phi-\eta)-2g\lambda\cos\phi\Big{)}\right)\,.$
(71)

The Wilson loop was simple to write in the original variables, but there it is
difficult to describe the monopoles. In passing to the dual description, the
Wilson loop is the recipe that $\phi$ sees a background connection with a
specified monodromy around $C$. In the semi-classical approximation, we can
evaluate $W(C)$ by saddle point with these boundary conditions. In practice,
for a large area contour we look for field configurations which ‘jump’ between
two vacua of $\phi$ in a neighborhood of $B$. This means solving the non-local
equation of motion for $\phi$,

$D^{\sigma}\phi-D^{\sigma}\eta=-g\lambda\sin(\phi)\,,$ (72)

where for $\sigma>2$, we can safely write $D^{\sigma}$ as
$-D^{\sigma-2}\nabla^{2}$ and then we have

$D^{\sigma}\eta=-D^{\sigma-2}(2\pi\delta^{\prime}(z)\theta_{B}(x_{\bot}))$
(73)

and

$D^{\sigma}\phi+D^{\sigma-2}(2\pi\delta^{\prime}(z)\theta_{B}(x_{\bot}))=-g\lambda\sin(\phi)\,,$
(74)

with $\theta_{B}(x)$ equal to unity inside $B$ and zero elsewhere. The
boundary conditions on $\phi$ are that it vanishes at infinity. In particular,

$\lim_{z\to\infty}\phi=0\,.$ (75)

In order to solve (74) it is useful to consider its one-dimensional
homogeneous version

$D^{\sigma}\phi_{0}(z)=-g\lambda\sin(\phi_{0}(z))$ (76)

which possesses kink type solutions. While our main interest is $\sigma=3$, we
can also consider other cases. For $\sigma=2$, we have the standard domain
wall solution,

$\phi_{0}(z)=4\arctan(e^{-\sqrt{g\lambda}z})$ (77)

or its anti-kink version $2\pi-\phi_{0}(z)=\phi_{0}(-z)$. As it turns out, the
kink solution for $\sigma=1$ can be found explicitly and it takes the simple
analytic form

$\phi_{0}(z)=\pi-2\arctan(g\lambda z)\,,$ (78)

(with $g\lambda$ positive). For other values of $\sigma$ we need to resort to
numerics to solve (76). The dependence of $\phi_{0}$ on $g\lambda$ can be
obtained by a scaling argument. If $\phi_{0}(z)$ is a solution to (76) with
$g\lambda=1$ then $\phi_{0}(z(g\lambda)^{\frac{1}{\sigma}})$ is a solution to
(76) for general $g$ and $\lambda$. Thus, when looking for numerical solutions
to (76) we may focus on the $g\lambda=1$ case.

Our strategy for solving (76) numerically is to expand $\phi_{0}$ as a series
of sigmoid functions, $S(z)$ (such that $S(z)$ interpolates between $1$ and
$0$ and is centered around $z=0$),

$\phi_{0}(z)=\sum_{n=-N}^{N}a_{n}S(z-n\Delta)\,,$ (79)

with $\Delta$ a small positive constant, whose value is chosen such as to
optimize the fit. One can use, for example,

$S(z)=\frac{1}{2}\left(1-\tanh z\right).$ (80)

Other sigmoid functions we have used involve $\arctan(z)$, $1/(1+e^{-z})$ or
$\hbox{erf}(z)$. It is straightforward to evaluate the non-local derivative of
the sigmoid functions by evaluating them in Fourier space, e.g.,

$D^{\sigma}\left(\frac{1}{2}(1-\hbox{erf}(z))\right)=-\frac{2^{\sigma}}{\pi}\Gamma\left(\frac{1+\sigma}{2}\right){}_{1}F_{1}\left(\frac{\sigma+1}{2},\,\frac{3}{2},\,-z^{2}\right)z\,.$
(81)

With $D^{\sigma}S(x)$ at hand we can use standard Newton-Raphson methods to
search for $a_{n}$’s such that

$\sum_{n=-N}^{N}a_{n}D^{\sigma}S(z-n\Delta)+g\lambda\sin\left(\sum_{n=-N}^{N}a_{n}S(z-n\Delta)\right)=0\,,$
(82)

is satisfied on $2N+1$ sampling points $z=n\Delta$. We found that our
algorithm becomes more stable if we constrain the $a_{n}$’s to satisfy

$a_{n}=a_{-n}$ (83)

for $n\neq 0$, and $\phi_{0}(0)=\pi$ or equivalently

$a_{0}=2\Big{(}\pi-\sum_{n=1}^{N}a_{n}\Big{)}\,.$ (84)

A typical numerical kink solution can be found in figure 1.

Figure 1: Solutions to $D^{\sigma}\phi_{0}(z)=-\sin(\phi_{0}(z))$ with
$\sigma=1$, $\sigma=2$, and $\sigma=3$. The $\sigma=3$ solution is generated
for $N=50$, $\Delta=0.4$, and $S(x)=\frac{1}{2}\left(1-\tanh x\right)$.

With the homogeneous solution to (76) at hand we can solve (74) by gluing two
such solutions,

$\phi=\begin{cases}\phi_{0}(z)&z>0,\,\\{x,y\\}\in B\\\
\phi_{0}(z)-2\pi&z<0,\,\\{x,y\\}\in B\\\ 0&\\{x,y\\}\notin B\end{cases}$ (85)

valid away from the contour $C$. One can now evaluate (71) in the saddle point
approximation,

$F(C)=\left<\exp\left({-W(C)}\right)\right>=\exp\left(-\gamma\int_{B}d^{2}y\right)$
(86)

where

$\gamma=\frac{1}{2g}\int
dz\bigg{(}\phi_{0}D^{\sigma}\phi_{0}+2g\lambda\Big{(}1-\cos(\phi_{0})\Big{)}\bigg{)}\,.$
(87)

Thus the Wilson loop $W(C)$ in (86) satisfies an area law implying that the
theory is in a confining phase.

##  6 Discussion

In this paper we considered a generalized free Maxwell theory with Lagrangian
$\frac{1}{e^{2}}F_{\mu\nu}D^{-1}F^{\mu\nu}$ and used electric-magnetic duality
to recast the theory in terms of a compact boson with kinetic term
$\frac{1}{2g}\varphi D^{3}\varphi$. We argued that if the theory is defined on
a lattice, then instanton effects generate a cosine potential for the compact
scalar. When $g\geq 10\pi^{2}$ we observed that the non-local sine-Gordon
theory is renormalizable only if we introduce an additional (relevant) local
kinetic term to the theory. In the generic case when a local kinetic term is
present, we expect to be in a confining phase of the theory by the arguments
presented in Polyakov:1975rs ; Polyakov:1976fu . In the regime where the
kinetic term is not generated, or if we fine-tune the theory to set it to
zero, we obtain a purely non-local theory. In this theory, we have seen that
the cosine potential is an irrelevant deformation for $g>12\pi^{2}$, while for
$g<12\pi^{2}$ the cosine term is relevant and by the arguments of section 5,
we again expect to be in a confining phase.

It is, perhaps, useful to relate the above behavior of the non-local sine-
Gordon theory in three dimensions to the behavior of the local sine Gordon
theory in two dimensions by analytically continuing in dimensionality, $d$.
Indeed, consider the non-local theory of a dimensionless compact scalar,

$\displaystyle S=\int d^{d}x\bigg{(}\frac{1}{2g}\varphi
D^{d}\varphi-\lambda\cos(\varphi)\bigg{)}\,,$ (88)

where $2\leq d\leq 3$. In this more general non-local theory, one can show
that a local kinetic counterterm is required for
$g\geq\frac{d+2}{2}(4\pi)^{d/2}\Gamma(\frac{d}{2})$. When the local kinetic
term is tuned away, the cosine is relevant for
$g<d(4\pi)^{d/2}\Gamma(\frac{d}{2})$. When $d=2$, these two curves intersect
at the value $g=8\pi$, which marks the location of the Berezinskii-Kosterlitz-
Thouless phase transition. We plot the different regimes of the purely non-
local theory in figure 2, where we also show the region where the potential
term $\cos(2\varphi)$ is relevant,
$g<\frac{d+2}{8}(4\pi)^{d/2}\Gamma(\frac{d}{2})$, and have excluded the region
in which the monopole operator $e^{i\varphi}$ violates the unitarity bound,
$g<2\pi^{2}(d-2)$. In the figure we have not included the higher harmonics,
which are all irrelevant everywhere outside the excluded region in 3$d$.

Figure 2: Phase diagram of generalized sine-Gordon theory with action (88) for
$2<d\leq 3$ with the local kinetic deformation tuned to zero.

It may seem puzzling that while in gauge field language the non-local term
$F_{\mu\nu}D^{-1}F^{\mu\nu}$ dominates over the standard Maxwell term
$F_{\mu\nu}F^{\mu\nu}$ in the IR, in the dual language of compact bosons, the
local term $(\partial_{\mu}\varphi)(\partial^{\mu}\varphi)$ dominates over the
non-local term $\varphi D^{3}\varphi$. One may ask, in the original gauge
theory language, what is the relevant operator that deforms
$F_{\mu\nu}D^{-1}F^{\mu\nu}$ corresponding to the generation of the local
kinetic term in the non-local sine-Gordon theory? To answer this question, we
note that by electric-magnetic duality the compact scalar deformation

$\displaystyle\varphi
D^{3}\varphi\rightarrow\varphi\Big{(}D^{3}+hD^{2}\Big{)}\varphi$ (89)

corresponds to the following deformation of the generalized Maxwell term:

$\displaystyle F_{\mu\nu}D^{-1}F^{\mu\nu}\rightarrow
F_{\mu\nu}\frac{1}{D^{1}+h}F^{\mu\nu}\,.$ (90)

Taylor expanding the RHS for small $h$, we see that the generalized Maxwell
action is initially deformed by the relevant non-local term
$hF_{\mu\nu}D^{-2}F^{\mu\nu}$. In fact, retaining the higher order terms in
the Taylor expansion, we see that the monopole background induces deformations
by an infinite tower of terms $h^{n}F_{\mu\nu}D^{-1-n}F^{\mu\nu}$ with
$n=1,2,3$… It is a general concern for non-local theories that once locality
is abandoned as a principle to constrain the possible terms that can appear in
the action, the RG flow may generate uncontrollably many terms. Luckily, the
infinite tower of terms resums to the RHS of (90), as reflected in the fact
that the dual sine-Gordon theory turned out to be renormalizable.

Finally, we comment on a possible connection to the string picture of
confinement. Given the arguments in section 5, we have established that the
purely non-local theory is confining when the cosine term is relevant. The
area law of the Wilson loop comes from the semi-classical saddle in which the
dual scalar ‘jumps’ when crossing the loop. There has been long standing
analytical and numerical evidence that there exists a dual formulation of the
confinement mechanism due to effective strings which describe this interface
(see Karliner:1983ab ; Caselle:1995fh ; Teper:1998te ; Caselle:2005xy ;
Caselle:2016mqu ; Athenodorou:2018sab  for a partial list of references). It
would be interesting to find an effective string theory which describes this
fractional version of Maxwell theory, perhaps along the lines of
Gubser:2019uyf .

The evidence we have surveyed indicates that generalized free Maxwell theory
with its quantum deformations represents a benign theory that may fruitfully
be applied as an effective model of the type of four-dimensional systems with
photons coupling to a particles on a codimension-one surface that motivated
its study. It remains an intriguing possibility to find or engineer natural
systems of this kind which exhibit the confinement mechanism and fine-tuned
deconfining phase transition that we have analysed.

## Acknowledgements

The authors are grateful to Shai Chester, Mykola Dedushenko, Igor Klebanov,
Petr Kravchuk, Silviu Pufu, Samson Shatashvili, and Edward Witten for helpful
comments and illuminating discussions. M.H. is supported by the U.S.
Department of Energy Grant DE-SC0009988 and the Institute for Advanced Study.
Z.J. is supported by the ERC-COG grant NP-QFT No. 864583 “Non-perturbative
dynamics of quantum fields: from new deconfined phases of matter to quantum
black holes”, by the MIUR-SIR grant RBSI1471GJ, and by the INFN “Iniziativa
Specifica ST&FI”. AY is supported in part by an Israeli Science Foundation
excellence center grant 2289/18.

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