/tikzfeynman/momentum/arrow shorten = 0.3 /tikzfeynman/warn luatex = false
aainstitutetext: Department of Physics, University of Toronto, Toronto, ON M5S
1A7, Canada

# On the perturbative aspects of deformed Yang-Mills theory

J. Lai<EMAIL_ADDRESS>
###### Abstract

Centre-stabilised $SU(N)$ Yang-Mills theories on $\mathbb{R}^{3}\times S^{1}$
are QCD-like theories that can be engineered to remain weakly-coupled at all
energy scales by taking the $S^{1}$ circle length $L$ to be sufficiently
small. In this regime, these theories admit effective long-distance
descriptions as Abelian $U(1)^{N-1}$ gauge theories on $\mathbb{R}^{3}$, and
semiclassics can be reliably employed to study non-perturbative phenomena such
as colour confinement and the generation of mass gaps in an analytical
setting. At the perturbative tree level, the long-distance effective theory
contains $(N-1)$ free photons with identical gauge couplings $g^{2}_{3}\equiv
g^{2}/L$. Vacuum polarisation effects, from integrating out heavy charged
fields, lift this degeneracy to give $\lfloor\frac{N}{2}\rfloor$ distinct
values: $g^{2}(\frac{2}{L})\lesssim g_{3,\ell}^{2}L\lesssim
g^{2}(\frac{2\pi}{NL})$. In this work, we calculate these corrections to one-
loop order in theories where the centre-symmetric vacuum is stabilised by
$2\leq n_{f}\leq 5$ massive adjoint Weyl fermions with masses of order
$m_{\lambda}\sim\frac{2\pi}{NL}$, (also known as "deformed Yang-Mills,") and
show that our results agree with those found in previous studies in the
$m_{\lambda}\to 0$ limit. Then, we show that our result has an intuitive
interpretation as the running of the coupling in a "lattice momentum" in the
context of the non-perturbative "emergent latticised fourth dimension" in the
$N\to\infty$, fixed-$NL$ limit.

##  1 Introduction

Analytical methods to study the long-distance properties of four-dimensional
asymptotically free non-Abelian gauge theories are few and far between;
broadly speaking, it is a difficult problem to handle because the flow to
strong coupling causes theoretical control over the system to be lost at lo-
energy scales. While there are known models that are well-behaved enough to be
studied analytically, (e.g., Seiberg-Witten theory seibergwitten ,) these
typically require special structures such as supersymmetry, or otherwise make
use of gauge-gravity duality arguments and string-inspired tools (such as in
Ref. adscft ).

Over the past years, studies performed on "centre-stabilised" gauge theories
on $\mathbb{R}^{3}\times S^{1}$ have been remarkably fruitful for providing
insight into the non-perturbative dynamics of four-dimensional gauge theories.
These models are distinguished from the few known analytically-calculable
models in four dimensions by the fact that they can be engineered to remain
weakly-coupled at all energy scales, so that a semiclassical expansion in
terms of objects defined in the UV theory is reliable and self-consistent.

The basic idea behind these models is as follows: by compactifying
$\mathbb{R}^{4}$ to $\mathbb{R}^{3}\times S^{1}$, and "deforming" the pure
Yang-Mills (YM) theory by adding a non-local and non-renormalisable potential
to the Lagrangian, the well-known deconfining phase transition (cf. thermal
Yang-Mills gpy ,) at small circle lengths $L$ can be circumvented, and the
theory remains in the colour-confining phase for all values of $L$. Adiabatic
continuity to the full $\mathbb{R}^{4}$ theory of ultimate interest can
therefore be argued on grounds that the theories share identical (non-
spacetime) global symmetries for all $L\in[0,\infty]$. That is, they belong in
the same "universality class" unsalyaffe08 ; unsalyaffe2010 .

To be certain, the non-renormalisable "deformed" theory that we are describing
can be viewed as a lattice theory with a fixed finite lattice spacing unsal07
. On the other hand, it is also possible to define a UV-complete continuum
theory with the same desired properties by introducing $n_{f}$
$S^{1}$-periodic adjoint-representation fermion fields to the pure YM
Lagrangian: The desired deformation potential is realised as the fermionic
contribution to the dynamically-generated Gross-Pisarski-Yaffe (GPY) effective
potential at energy scales below $\sim\frac{1}{L}$ unsalyaffe08 ;
unsalshifman08 ; unsal09 ; unsalyaffe2010 . If the fermions are massless,111It
should be noted that QCD(adj) with $n_{f}$ massless fermions in its spectrum
has a global chiral symmetry not shared by the $\mathbb{R}^{4}$ pure Yang-
Mills and is therefore not covered by the aforementioned ”universality class
argument.” this class of theories is referred to as QCD(adj) if $2\leq
n_{f}\leq 5$, and super Yang-Mills (SYM) if $n_{f}=1$. It is called "deformed
Yang-Mills" (dYM), when the $2\leq n_{f}\leq 5$ fermions are massive, or if
the deformation potential is added "by hand," as in the lattice formulation.

From the theorist’s perspective, one of the most alluring features of these
admittedly artificial setups is that they admit a "weak-coupling regime" at
$\Lambda NL\ll 2\pi$, (where $\Lambda$ is the strong-coupling scale,) in which
the gauge coupling $g^{2}$ (and more pertinently, $g^{2}N$,) remains small at
all energy scales. Thus, in this regime, the semiclassical expansion over
high-energy monopole-instanton configurations is trustworthy, and can be
reliably employed to study the effects of the non-perturbative physics on the
low energy theory. The result is a theoretical laboratory in which a wide
variety of non-perturbative low-energy phenomena can be studied analytically:
For example, colour confinement, the generation of a non-perturbative mass gap
unsalyaffe08 ; unsal09 , a deconfining phase transition continuity ;
symallgroups , and certain aspects of chiral symmetry breaking unsal07 . For
this reason, $\Lambda NL\ll 2\pi$ is sometimes also called the "calculable
regime" in the context of centre-stabilised $\mathbb{R}^{3}\times S^{1}$
theories. For comprehensive reviews, see Refs. poppitzreview ; newmethods ;
semiclassical2012 .

At leading perturbative order and finite $N$, the IR effective theory of
$SU(N)$ dYM and QCD(adj) in the calculable regime is sometimes described as
being "rather boring" poppitzreview , because the gauge sector describes
$(N-1)$ free, massless photons in $\mathbb{R}^{3}$. When vacuum polarisation
effects are accounted for, the photons acquire $\lfloor\frac{N}{2}\rfloor$.
These corrections have been calculated to one-loop order by various methods
when the fermions are assumed to be massless: for $N=2,\,3$ QCD(adj) in Ref.
renormalons , for SYM (i.e., $n_{f}=1$) with arbitrary $N$ in Ref.
symallgroups , and in QCD(adj) with arbitrary $N$ in Ref. pelizzani .

In this study, we derive a more general expression for these corrections that
in particular covers the massive fermion case, for generic masses
$m_{1},...\,m_{n_{f}}$ such that the theory remains in the centre-symmetric
and weak-coupling regime. The final result is contained in Equation (13), and
the bulk of our exposition explains how we arrive at this result. Our
motivating aim is to confirm that the perturbative corrections to finite-$N$
$SU(N)$ dYM theory yield no unpleasant surprises even when the stabiliser
fermions are assumed to be heavy. This is a very reasonable assumption to
make, since ultimately we are interested in obtaining insight on pure Yang-
Mills on $\mathbb{R}^{4}$, and a continuum QCD-like theory that continues
smoothly to pure YM should not contain light adjoint fermions in its IR
spectrum.

Nevertheless, our results are not entirely devoid of novelty: Ref.
emergentdimensions  showed that in the $N\to\infty$, $L\to 0$, fixed-$NL$
limit of SYM, an emergent latticised fourth dimension appears, emerging out of
the space of fields — even though we should expect that taking $L\to 0$ ought
to result in a 3d theory. In particular, this emergent dimension exhibits
$z=2$ Lifschitz scaling invariance in SYM. In other words, the action is
quartic, rather than quadratic, in the momentum:
$\sim|\partial_{y}^{2}\Phi|^{2}$, where $\partial_{y}$ is the partial
derivative in the emergent latticised dimension. Simply put, this is because
in SYM, there is a discrete $\mathbb{Z}_{N}$ chiral symmetry (not to be
confused with the $\mathbb{Z}_{N}$ centre symmetry,) that forbids monopole-
instantons from contributing a bosonic potential of the form
$\sim|\partial_{y}\Phi|^{2}$ in the semiclassical expansion. Such a potential
is permitted, however, when the chiral symmetry is explicitly broken by a non-
zero fermion mass, as is in the case we study here. We find a satisfying and
intuitive interpretation of our massive correction in this emergent dimension
as the flow towards strong coupling for large values of the "lattice
momentum."

The rest of this paper is structured as follows: Section 2 contains an
overview of the essentials of dYM theory in an effort to make this paper more
self-contained. For the benefit of the impatient reader, we have placed our
main result, Equation (13) and its accompanying discussion, in Section 2.1.1.
A discussion of this result in the context of the emergent latticised
dimension of Ref. emergentdimensions  is contained in Section 2.2.1.

Section 3 covers the derivation of Equation (13) in detail, starting from the
very beginning with the UV dYM Lagrangian. Since this calculation is fairly
long and convoluted, we briefly summarise what we have done at the end of
subsections 3.1 and 3.2 to help the reader keep track of our progress. The
main "meat" of the calculation, and therefore of this paper, is mostly
contained in Section 3.3, especially Section 3.3.2.

In our calculation, we use the Mellin transform to rewrite certain infinite
sums in a form that allows their asymptotic behaviour to be more easily seen.
The details of this manipulation, which is mostly just complex analysis in one
variable, is given in Appendix A.

##  2 Background, results and discussion

###  2.1 Review of dYM: I. Perturbative aspects

Consider pure $SU(N)$ Yang-Mills theory on compactified $\mathbb{R}^{3}\times
S^{1}$, where the $S^{1}$ is a circle of circumference $L$:

$S_{\text{YM}}[\mathcal{A},\mathcal{F}]=\int_{\mathbb{R}^{3}\times
S^{1}}\frac{1}{2g^{2}}\mathrm{tr}(\mathcal{F}^{2}).$ (1)

This theory enjoys a global $\mathbb{Z}_{N}=Z(SU(N))$ centre symmetry, as it
only contains fields transforming in the adjoint representation of the gauge
group. The action of this symmetry may be thought of as a "gauge"222Though, of
course, the $g(x^{\mu},x^{4})$ so defined is not a true gauge transformation
by any means. That is, it is not a transition function between local
trivialisations of the principle bundle. transformation
$g(x^{\mu},x^{4}):\mathbb{R}^{3}\times S^{1}\to SU(N)$ that is periodic over
the $S^{1}$ modulo a $\mathbb{Z}_{N}$ factor:

$g(x^{\mu},0)=\omega g(x^{\mu},L)\,,\qquad\omega\equiv e^{i2\pi/N}.$ (2)

This acts on the fundamental representation Polyakov loop $\Omega$, the gauge
holonomy along the $S^{1}$,

$\Omega\equiv\mathcal{P}\exp\,i\int_{0}^{L}\\!dx^{4}\mathcal{A}_{4},$ (3)

as333According to the modern viewpoint, this $\mathbb{Z}_{N}$ belongs to a
class of ”generalised” global symmetries, which act on operators with non-
trivial spatial extent. In this context, Equation (4) defines the symmetry
gensym .

$\mathbb{Z}_{N}:\mathrm{tr}\Omega\to\omega\,\mathrm{tr}\Omega,$ (4)

where $\mathcal{A}_{4}$ is the $S^{1}$ part of the gauge field $\mathcal{A}$.

At large $L$, the centre symmetry is unbroken. That is to say,
$\langle\mathrm{tr}\Omega^{n}\rangle=0$ in the ground state for all $n\neq
0\mod N$. On the other hand, it is a well-known fact gpy  that in the
small-$L$ limit, the theory undergoes a deconfining phase transition
associated with the breaking of centre symmetry: In this regime, where
perturbative analyses can be trusted because of asymptotic freedom, Ref. gpy
showed that the theory produces a (GPY) effective potential,
$V_{\text{pert.}}[\Omega]$:

$V_{\text{pert.}}[\Omega]=-\frac{2}{\pi^{2}L^{4}}\sum_{n=1}^{\infty}\frac{|\mathrm{tr}\Omega^{n}|^{2}}{n^{4}}(1+O(g^{2})).$
(5)

This result can be found by integrating out the Kaluza-Klein modes at one-loop
order. This potential is minimised by $\Omega$ of the form
$\Omega=\omega^{k}\,1_{N}$ for any integer $k$, suggesting that the theory has
$N$ degenerate vacua related by the $\mathbb{Z}_{N}$ symmetry and describes a
gluon plasma phase.

The basic idea behind $\mathbb{R}^{3}\times S^{1}$ theories such as dYM is to
re-enforce the stability of the $\mathbb{Z}_{N}$ at small $L$ by "flipping"
the shape of the GPY potential, so to speak. This can be done in the most
direct way by simply adding a "double-trace" term to the YM action:

$\mathcal{L}_{\text{dYM}}=\mathcal{L}_{\text{YM}}+V_{\text{deformed}}[\Omega],$
(6a) where $V_{\text{deformed}}[\Omega]=\frac{1}{L^{4}}\sum_{n=1}^{\lfloor
N/2\rfloor}a_{n}|\mathrm{tr}\Omega^{n}|^{2}.$ (6b)

But such a term is manifestly non-local, being defined in terms of a non-local
operator. It is also non-renormalisable, as it contains infinitely many
irrelevant operators which blow up uncontrollably in the UV. As such, such a
deformation of the theory may be considered problematic to those with a
philosophical preference for continuum theories. We will return to address
this objection later, and focus on the effects of the double-trace deformation
potential on the IR theory for now.

The $\mathbb{Z}_{N}$ symmetry is said to be preserved if and only if the
vacuum state of the theory satisfies $\langle\mathrm{tr}\Omega^{n}\rangle=0$
for all $n\neq 0\mod N$, so the coefficients $a_{n}>0$ in Equation (6b) must
each be chosen so as to dominate the dynamically generated
$V_{\text{pert.}}[\Omega]$. With the centre symmetry stabilised, we can remove
the gauge redundancy of $\Omega$ by choosing a diagonal representative from
the class of physically equivalent minima:

$\Omega=\omega^{(1-N)/2}\textrm{diag}(1,\omega,...,\omega^{N-1}).$ (7)

This choice is in fact unique, up to permutations of the coefficients
corresponding to Weyl reflections that can also be gauged-fixed away by
working in the (affine) Weyl chamber. This allows us to write
$\langle\Omega\rangle$ as a physically meaningful expectation value despite
its uncontracted matrix indices.444To be certain, the action of the centre
$\mathbb{Z}_{N}$ in this gauge is, with $\Omega_{i}$ denoting the $i$’th
diagonal of $\Omega$ semiclassical2012 ,
$\mathbb{Z}_{N}:\Omega_{i}\to\begin{cases}\omega\,\Omega_{i-1}&i\neq 1,\\\
\omega\,\Omega_{N}&i=1.\end{cases}$ (8) The cyclic permutation is necessitated
by gauge-fixing to the Weyl chamber.

This vev precipitates a simulacrum of the Higgs mechanism in which the
$\mathcal{A}_{4}$ field plays the role of an adjoint Higgs field. The gauge
group generators left unbroken by $\langle\Omega\rangle$ form a Cartan
subalgebra $\mathfrak{t}\subset su(N)$, generating the maximal torus
$U(1)^{N-1}\subset SU(N)$. The Higgs mechanism endows fields in
$\mathfrak{t}^{\perp}$ (i.e., fields that carry charge under the $U(1)^{N-1}$)
with an effective mass $\geq\frac{2\pi}{NL}\equiv m_{W}$, the so-called
Abelianisation scale.

We can now perform the path integral around the centre symmetric vacuum:
Working perturbatively, (the treatment of the non-perturbative physics is left
to Section 2.2,) weak-coupling ensures that the $\mathcal{A}_{4}$ fluctuations
around $\langle\Omega\rangle$, of mass $\gtrsim\sqrt{g^{2}N}\,m_{W}$, can only
effect small corrections to the effective action. Weak-coupling, in turn, is
guaranteed by the weak-coupling assumption $m_{W}\gg\Lambda$ — meaning that
all dynamic charged fields can be safely integrated out before the onset of
strong coupling as we carry the theory towards the infrared. When the dust
settles, we are left with a weakly-coupled $U(1)^{N-1}$ gauge theory
containing no light charged fields in its spectrum. In fewer words: everything
works out fine.

In settings where it is desirable to have a theory that respects both locality
and UV-completeness, and yet preserve centre symmetry at all scales, we can
opt to have $V_{\text{deformed}}[\Omega]$ generated dynamically as well, by
adding sufficiently light, or massless, $S^{1}$-periodic adjoint fermions
$\lambda_{I}$ to the theory unsal09 , rather than inserting the deformation
potential "by hand." In such a setting, the periodicity requirement
$\lambda_{I}(x^{\mu},x^{4})=+\lambda_{I}(x^{\mu},x^{4}+L)$ prohibits a thermal
interpretation for the $S^{1}$, which must therefore be taken to be a spatial
circle.

For the theory with $1\leq n_{f}\leq 5$ Weyl fields555The theory loses
asymptotic freedom for $n_{f}>5$. of masses $m_{I}$ indexed by $1\leq I\leq
n_{f}$, the dynamically generated (GPY) effective potential is symallgroups ;
unsalyaffe2010 ; misumi2014

$V[\Omega]=-\frac{1}{\pi^{2}L^{4}}\sum_{n=1}^{\infty}\frac{1}{n^{4}}\Big{[}2-\sum_{I=1}^{n_{f}}(nLm_{I})^{2}K_{2}(nLm_{I})\Big{]}|\mathrm{tr}\Omega^{n}|^{2},$
(9)

where $K_{2}$ is the modified Bessel function of order 2. It is not hard to
find constraints on the $m_{I}$ for each value $n_{f}$ that stabilise the
centre-symmetric $\Omega$ in Equation (7); see e.g., Ref. misumi2014 . We also
note in passing that the $n_{f}=1$ potential vanishes (in fact, to all
perturbative orders,) in the massless case, and is centre-unstable otherwise.
This particular case is known as super Yang-Mills (SYM), in which the UV
theory enjoys an exact $\mathcal{N}=1$ supersymmetry, which allows many
aspects of its rich non-perturbative physics to be calculated exactly. But SYM
is outside of the scope of this study, along with the massless QCD(adj), and
we henceforth only consider $2\leq n_{f}\leq 5$ and $m_{I}>0$.

Assuming the fermion masses to be roughly equal, it turns out that centre
stability requires $m_{I}\lesssim m_{W}$. In particular, this means that we
can assume that the $m_{I}$ are $O(m_{W})$ so that the fermions disappear from
the low-energy theory, and the effective action can be written on
$\mathbb{R}^{3}$ as:

$S_{3d}=\int_{\mathbb{R}^{3}}\sum_{a,b=1}^{N}\kappa_{ab}F_{\mu\nu}^{a}F_{\mu\nu}^{b}+(\text{$A_{4}$
and higher order terms})$ (10a) for Abelian field strengths $F_{\mu\nu}^{a}$
and $\mathbb{R}^{3}$ indices $\mu,\nu\in\\{1,2,3\\}$ and Lie algebra indices
$a,b\in\\{1\,...\,N\\}$. There is also a a neutral scalar field $A_{4}^{a}$ in
the IR theory, which descends from the Abelian part of $\mathcal{A}_{4}$ and
corresponds to the oscillations of the eigenvalues of $\Omega$ around the
centre-symmetric vev. But this field receives a (mass)2 $\sim g^{2}Nm_{W}^{2}$
correction from the GPY potential and can be integrated out by moving the
theory to still lower energies.

The quantity $\kappa_{ab}$ in Equation (10a) is the quantum-corrected photon
coupling matrix:

$\kappa_{ab}=\frac{m_{W}^{-1}}{16\pi}\Big{(}\frac{8\pi^{2}}{Ng^{2}}\delta_{ab}+\,O(1)\,\Big{)}\,,\qquad
g^{2}\equiv g^{2}(4\pi/L),$ (10b)

where in Equation (10b), the gauge coupling is normalised with respect to the
$L\to\infty$, $m_{I}\to 0$ limit:

$\Lambda^{b_{0}}=\mu^{b_{0}}\exp\Big{(}-\frac{8\pi^{2}}{g^{2}(\mu)N}\Big{)}\,,\qquad
b_{0}\equiv\frac{11-2n_{f}}{3},$ (11)

and $b_{0}$ is the one-loop coefficient of the beta function of
$(g^{2}N)^{-1}$ in that limit. As stated before, these corrections have been
calculated in previous studies for arbitrary $N$ and $1\leq n_{f}\leq 5$ in
the limit $m_{I}=0$. Our calculation generalises to the massive case, and is a
new result. We also believe it to be a non-trivial problem in terms of
significance (as we will argue in this Section,) as well difficulty (which we
will demonstrate in the next).

####  2.1.1 The one-loop corrections to $\kappa_{ab}$

Compared to the writing out the matrix entries of $\kappa_{ab}$ explicitly in
the Cartan-Weyl basis, (given in Equation (76),) it is more enlightening to
present its eigenvalues, $\kappa_{\ell}$:

$\sum_{a,b=1}^{N}\kappa_{ab}F_{\mu\nu}^{a}F^{b\mu\nu}=\sum_{\ell=1}^{N}\kappa_{\ell}\tilde{F}_{\mu\nu}^{\ell}\tilde{F}^{\ell\mu\nu},$
(12a) where
$\tilde{F}_{\mu\nu}^{\ell}\equiv\frac{1}{\sqrt{N}}\sum_{a=1}^{N}\omega^{-\ell
a}F_{\mu\nu}^{a}\,,\qquad\kappa_{\ell}\equiv\frac{1}{N}\sum_{a,b=1}^{N}\omega^{\ell(a-b)}\kappa_{ab},$
(12b)

where again $\omega=e^{i\frac{2\pi}{N}}$, so Equation (12b) are really just
discrete Fourier transforms in the indices $a,b$. Then, assuming centre-
stabilising fermion masses $m_{I}$,666Please note that the $\ell=N$ mode must
be excluded from the spectrum as it corresponds to the trace of
$F^{a}_{\mu\nu}$, (as can be seen from the definition,) which is unphysical in
our theory; it would have been a physical mode if we had instead chosen our
gauge group to be $U(N)$.

$\begin{split}\kappa_{\ell}&\equiv\frac{1}{4}g_{3,\ell}^{-2}\\\
&=\frac{m_{W}^{-1}}{16\pi}\Bigg{[}\frac{8\pi^{2}}{Ng^{2}(m_{W}e^{-\gamma})}+\Big{(}\frac{11-2n_{f}}{3}\Big{)}\log\frac{1}{\sin\pi\frac{\ell}{N}}+\frac{2}{3}\sum_{I=1}^{n_{f}}W_{\ell}\Big{(}\frac{m_{I}}{m_{W}}\Big{)}\Bigg{]}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{for}\,\,1\leq\ell\leq
N-1.\end{split}$ (13)

$W_{\ell}=W_{N-\ell}$ is an $O(1)$ pure function of the masses $m_{I}$ in
units of $m_{W}$, which enjoys the following properties:

$\displaystyle W_{\ell}(0)=0$ for all integers $\ell$, (14a) $\displaystyle
W_{\ell}\Big{(}\frac{m}{m_{W}}\Big{)}\to\log\Big{(}\frac{m_{W}e^{-\gamma}}{m\sin\pi\frac{\ell}{N}}\Big{)}$
$\displaystyle\qquad\text{monotonically, as}\,\,\,m/m_{W}\to\infty,$ (14b)
$\displaystyle W_{\ell}\Big{(}\frac{m}{m_{W}}\Big{)}\to 0$
$\displaystyle\qquad\text{as}\,\,\,\frac{\ell}{N}\to 0\,,\,1\,,$ (14c)
$\displaystyle W_{\ell}(\tau^{\prime})<W_{\ell}(\tau)<0$
$\displaystyle\qquad\text{for}\,\,\,0<\tau<\tau^{\prime},$ (14d)
$\displaystyle W_{\lfloor
N/2\rfloor}(\tau)<W_{\ell^{\prime}}(\tau)<W_{\ell}(\tau)\leq 0$
$\displaystyle\qquad\text{for}\,\,\,\ell<\ell^{\prime}<\lfloor
N/2\rfloor\,,\quad\tau>0.$ (14e)

A plot of $W_{\ell}$ as a function of $\frac{\ell}{N}$ for a few select values
of $m$ is given in Figure 1.

Equation (13) is written so that all the information about the one-loop
corrections due to the fermion masses is encoded in the pure function
$W_{\ell}$. In particular, Equation (14d) implies that the $\ell=1$ mode
receives a vanishingly small mass correction in the $N\to\infty$ limit;
conversely, Equation (14e) implies the $\lfloor N/2\rfloor$ mode receives the
largest mass correction. Equations (14a) and (14b) together imply that
$\kappa_{1}\geq\kappa_{\ell}\geq\kappa_{\lfloor N/2\rfloor}$ for all $m_{I}$
and $\ell$. Note that in order for our results to make sense, we must require
$\kappa_{\lfloor N/2\rfloor}>0$; we will discuss the conditions that fulfill
this requirement later.

Figure 1: A numerical plot of $W_{\ell}$ as a function of $\frac{\ell}{N}$ for
some select values of $m$, in units of $m_{W}$. The graphs were plotted with
Equation (15b) for $m=1.0m_{W}$ and $m=1.2m_{W}$, and with (15a) otherwise.

We present two analytic expressions for $W_{\ell}$ with different convergence
properties: one expression holds for $m<m_{W}$, and the other, for $m\gtrsim
m_{W}$. The former of these is:

$\begin{split}W_{\ell}\Big{(}\frac{m}{m_{W}}\Big{)}&=\sum^{\infty}_{n=1}\frac{(-1)^{n}(2n)!}{2^{2n}(n!)^{2}}\Big{(}\frac{m}{m_{W}}\Big{)}^{2n}\Big{[}\zeta(2n+1)-Re\Big{(}\mathrm{Li}_{2n+1}(e^{2\pi
i\frac{\ell}{N}})\Big{)}\Big{]}\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{for}\,\,(m/m_{W})<1,\end{split}$
(15a) where $\zeta$ is the Riemann zeta function, and $\mathrm{Li}_{s}$ is the
polylogarithm function of order $s$. This expression fails to converge when
$m>m_{W}$777Observing that the terms in the square brackets of (15a) are
absolutely bounded for all $n\geq 1$, and that
$\displaystyle\frac{(2n)!}{(n!)^{2}2^{2n}}=\frac{1}{\sqrt{\pi}}\frac{\Gamma(n+\frac{1}{2})}{\Gamma(n)},$
the root test gives a radius of convergence of $(m/m_{W})<1$.; it is in this
regime where our second expression is more useful:
$\begin{split}W_{\ell}\Big{(}\frac{m}{m_{W}}\Big{)}=&\log\Big{[}\frac{m_{W}e^{-\gamma}}{m\sin\pi\frac{\ell}{N}}\Big{)}\\\
&\quad+\sum_{p=1}^{\infty}\Big{\\{}2K_{0}\Big{(}2\pi
p\frac{m}{m_{W}}\Big{)}-K_{0}\Big{(}2\pi\Big{(}p-1+\frac{\ell}{N}\Big{)}\frac{m}{m_{W}}\Big{)}\\\
&\qquad\qquad-
K_{0}\Big{(}2\pi\Big{(}p-\frac{\ell}{N}\Big{)}\frac{m}{m_{W}}\Big{]}\Big{\\}}.\end{split}$
(15b)

$K_{0}$ is the modified Bessel function of order $0$. Equation (15b) is one-
loop exact for all $m$, but it is more useful at large $m\gtrsim m_{W}$, where
it may be very well approximated by the first term of the series, as
$K_{0}(t)\sim e^{-t}$ at large $t$. Conversely, Equation (15b) is less useful
at small $m/m_{W}$ as $K_{0}(t)\sim\log t$ at small $t$.

Since $W_{\ell}\leq 0$, the massive correction competes against the massless-
limit corrections encoded in the log-sine term. Indeed, by taking $m_{I}\gg
m_{W}$, the $I$’th fermion decouples from the theory888Assuming, of course,
that the theory still remains in the centre-symmetric regime.,
$n_{f}\to(n_{f}-1)$, up to an overall renormalisation of $g^{2}N$, or,
equivalently, a re-definition of the strong-coupling scale $\Lambda$.

We can also use Equation (11) to define a "lattice-renormalised" ’t Hooft
coupling $\lambda_{\ell}$:

$\begin{split}\frac{1}{\lambda_{\ell}}\equiv
b_{0}\log\Big{(}\frac{m_{W}e^{-\gamma}}{\Lambda\sin\pi\frac{\ell}{N}}\Big{)}\,\quad\text{for}\,\,1\leq\ell\leq
N-1.\end{split}$ (16a) Then assuming for convenience equal fermion masses
$m_{I}=m_{\lambda}$ for all $I=1\,...\,n_{f}$, and abbreviating
$W_{\ell}(\frac{m_{\lambda}}{m_{W}})=W_{\ell}$, Equation (13) can be written
in a neater form:
$\kappa_{\ell}=\frac{m_{W}^{-1}}{16\pi}\Big{(}\frac{1}{\lambda_{\ell}}+\frac{2n_{f}}{3}W_{\ell}\Big{)}\quad\text{for}\,\,1\leq\ell\leq
N-1.$ (16b)

Figure 2: A numerical plot of $\kappa_{\ell}$ in units of
$\frac{m_{W}^{-1}}{16\pi}$, as a function of $\frac{\ell}{N}$ for some select
values of $m_{\lambda}$, for $n_{f}=4$ and $m_{W}=e^{3}\Lambda$. The plot
diverges at $\ell/N\to 0,1$ (not depicted) due to the log-sine running in
$\frac{1}{\lambda_{\ell}}$.

The dependence of $\kappa_{\ell}$ on $\frac{\ell}{N}$ is illustrated in Fig. 2
for a few sample values of $m_{\lambda}$, for fixed $nf=4$ and
$m_{W}=e^{3}\Lambda$. Given a fixed value of
$m_{W}/\Lambda=\frac{2\pi}{\Lambda NL}\gg 1$, it is a straightforward exercise
in numerical analysis to find constraints on $m_{\lambda}$ so that
$\kappa_{\lfloor N/2\rfloor}>0$ in order for our result to make sense.
Conversely, $\mathbb{Z}_{N}$-stability requires that e.g.
$m_{\lambda}\lessapprox 1.08m_{W}$ for $n_{f}=4$, and this bound can in
particular be saturated by taking $m_{W}/\Lambda\gtrapprox e^{3}$. On the
other hand, $\mathbb{Z}_{N}$ stability requires $m_{\lambda}\lessapprox
1.2m_{W}$ for $n_{f}=5$, and saturation of that bound would require
$m_{W}/\Lambda\gtrapprox e^{9}$, which is substantially larger.

Equations (13), (15a), and (15b) comprise the main results of this paper; they
are derived in detail in Section 3, with reference to some results from
Appendix A. The rest of this Section discusses how to interpret these results
in the context of the non-perturbative physics of dYM theory, particularly
with regards to the "emergent latticised dimension" of Ref. emergentdimensions
.

###  2.2 Review of dYM: II. Non-perturbative aspects

Let us now very quickly summarise the derivation of the low-energy effective
Lagrangian in dYM theory at leading order in the semiclassical expansion. The
basic idea is essentially the same as Polyakov’s version of confinement in the
Georgi-Glashow (GG) model in $2+1$ dimensions shifman2012advanced , although
there are crucial differences due to the intrinsically four-dimensional nature
of dYM theory. The reader interested in a more detailed exposition is referred
to Refs. unsalyaffe08 ; stringtensions ; smallqcd .

The contribution of the non-perturbative physics to the path integral in a
weakly-coupled Euclidean QFT can be approximated to first exponential order by
summing over classical field configurations that are inundated by a "gas" of
weakly interacting minimal-action instantons. This is the so-called dilute
instanton gas approximation, and it is applicable in dYM because weak coupling
can be reliably assumed to hold at all scales provided that $NL\Lambda\ll
2\pi$.

In addition to a topological charge
$Q\sim\int\mathrm{tr}\,\mathcal{F}\wedge\mathcal{F}=\frac{1}{N}$, the
instantons of dYM theory carry a magnetic charge ( $\sim\int F$) under the
$U(1)^{N-1}$ — they are essentially ’t Hooft-Polyakov monopoles, with
$\mathcal{A}_{4}$ again standing in for the adjoint Higgs field. In
particular, we call them monopole-instantons.

Among these, there are $(N-1)$ "BPS"999This is a common abuse of terminology:
outside of the supersymmetric case, the BPS bound cannot be saturated because
the "Higgs" potential $V[\Omega]$ cannot be set to zero. So strictly speaking
we are expanding around "almost-BPS" configurations.monopoles, each carrying a
magnetic charge corresponding to a simple root $\alpha_{i}$ of the gauge
group. In distinction to the 3-dimensional Polyakov model, in
$\mathbb{R}^{3}\times S^{1}$ theories there is also an $N$’th "twisted," or
Kaluza-Klein, (KK) monopole, which carries charge
$\sum_{i=1}^{N-1}(-\alpha_{i})\equiv\alpha_{N}$, the affine root. In addition
to these, there are also the antiparticles carrying charge $-\alpha_{i}$. In a
sense, the ($N-1$) BPS+KK monopole-instantons can be thought of as the
"dissociation" of the BPST instanton in 4-dimensional $SU(N)$ Yang-Mills into
$N$ sub-constituents lee1997 ; kraan1998 .

We unfortunately do not have exact expressions for the monopole-instantons
outside of the supersymmetric $n_{f}=1,m=0$ case. But as it turns out, they
will not be required as far as our presentation is concerned: we need only
know that these charged objects interact with a long-range Coulombic
interaction, and have a non-linear "$A_{4}$/Higgs condensate" core of size
$\sim m_{W}^{-1}$. In addition, there is also a "medium-range" ($\sim
1/g\sqrt{N}m_{W}$) Yukawa interaction arising from $A_{4}$/Higgs exchange.

Every insertion of a monopole-instanton in the path integral comes with three
translation zero modes and a Boltzmann suppression factor
$e^{-S_{0}}\sim(NL\Lambda)^{b_{0}}\ll 1$, where
$S_{0}\approx\frac{8\pi^{2}}{Ng^{2}(m_{W})}$ is the one-monopole action. This
means the typical monopole-instanton separation $d\sim e^{S_{0}/3}$ is much
greater than the monopole diameter $\sim m_{W}^{-1}$, allowing us to ignore
the contribution from paths with overlapping monopole-instanton cores. It also
means that we can ignore the effects of $A_{4}$ exchange. The proliferation of
magnetic charges in the vacuum gives rise to a potential for the photon. This
potential which is most conveniently described in terms of the dual photon
$\sigma^{a}$, defined as

$\frac{1}{2}\varepsilon_{\mu\nu\rho}\kappa_{ab}F^{a}_{\mu\nu}=\frac{1}{16\pi}\partial_{\rho}\sigma^{b}.$
(17)

Written in terms of $\sigma^{a}$, the IR behaviour of dYM theory is described
to first order in the semiclassical expansion, by the 3d Lagrangian
$\mathcal{L}_{3d,\text{dual}}$:

$\mathcal{L}_{3d,\text{dual}}=\frac{1}{2(8\pi)^{2}}\kappa^{-1}_{ab}\partial_{\mu}\sigma^{a}\partial_{\mu}\sigma^{b}+\zeta\sum_{k=1}^{N}\Big{[}1-\cos(\sigma^{k+1}-\sigma^{k})\Big{]},$
(18)

where $\sigma^{N+1}\equiv\sigma^{1}$, and $\kappa^{-1}_{ab}$ is the
inverse101010Actually, it should be the pseudoinverse since the eigenvalue
corresponding to the $\ell=N$ mode diverges, but the difference is immaterial
since the $\ell=N$ mode is unphysical. of $\kappa_{ab}$, and $\zeta$ is the
monopole fugacity:

$\zeta\equiv Am_{W}^{3}(g^{2}N)^{-2}e^{-8\pi^{2}/Ng^{2}(m_{W})}.$ (19)

$A(\\{m_{I}\\},n_{f})$ is an $O(1)$ pre-exponential factor.111111 As an aside,
let us note that calculating the pre-factor $A(\\{m_{I}\\},n_{f})$ is a highly
non-trivial open calculation, and has only been performed in the SYM case,
first in davies00 , and later corrected in continuity ; symallgroups . This is
because it involves matrix determinants in a monopole-instanton background,
for which we do not even have an exact analytic expression, as mentioned. We
make no attempt to calculate $A$ here.  In the Fourier basis, the 3d dual
photon Lagrangian is, to quadratic accuracy in the fields,

$\mathcal{L}_{3d,\text{dual}}=\sum_{\ell=1}^{N-1}\Bigg{[}\frac{\kappa_{\ell}^{-1}}{(8\pi)^{2}}|\partial_{\mu}\tilde{\sigma}^{\ell}|^{2}+\zeta\sin^{2}\Big{(}\pi\frac{\ell}{N}\Big{)}|\tilde{\sigma}^{\ell}|^{2}\Bigg{]}+O(\tilde{\sigma}^{4}),$
(20)

where $\tilde{\sigma}^{\ell}$ is the discrete Fourier transform of
$\sigma^{a}$:

$\tilde{\sigma}^{\ell}\equiv\frac{1}{\sqrt{N}}\sum_{a=1}^{N}\omega^{-\ell
a}\sigma^{a}.$ (21)

From this expression we can read off the dual photon masses-squared:

$m^{2}_{\sigma,\,\ell}\sim\zeta\sin^{2}\Big{(}\pi\frac{\ell}{N}\Big{)}\,\kappa_{\ell}.$
(22)

Let us take $\Lambda NL$ to be sufficiently small so that
$\frac{2}{3}n_{f}\lambda_{\ell}W_{\ell}\ll 1$ can be treated as a small
correction for all $\ell$. In that case, we can write a mass-corrected
expression for the scaling behaviour of the $k$-wall thicknesses. Recalling
(16a),

$\frac{m_{\sigma,k}}{m_{\sigma,1}}\approx\frac{\sin\pi\frac{k}{N}}{\sin\frac{\pi}{N}}\left(\frac{\lambda_{1}}{\lambda_{k}}\right)^{1/2}\left[1+\frac{n_{f}}{3}\left(\lambda_{k}W_{k}-\lambda_{1}W_{1}\right)\right].$
(23)

The multiplicative sine factor is the expected tree-level scaling behaviour;
the factor of $(\lambda_{1}/\lambda_{k})^{1/2}$ is due to the one-loop
corrections in the massless limit, and the factor in the square brackets gives
the massive correction. The dependence of $m^{2}_{\sigma,\ell}$ on $\ell$ in
units of $m^{2}_{\sigma,N/2}$ is graphically depicted in Fig. 3.

Figure 3: A numerical plot of $m_{\sigma,\ell}^{2}/m_{\sigma,N/2}^{2}$ as a
function of $\frac{\ell}{N}$, for $n_{f}=4$ and $m_{W}=e^{4}\Lambda$.

####  2.2.1 Emergent dimension at large $N$: a 4d interpretation of the mass
correction

Let us now consider the large-$N$ limit. To do this, we simultaneously take
$N\to\infty$ and $L\to 0$ whilst keeping $NL$ constant so as to stay inside
the weak-coupling regime.121212The ’t Hooft coupling $g^{2}N$ is under control
in this limit, as can be seen from Equation (11). This is known as the
"Abelian large-$N$ limit" poppitzreview . In this setup, we can treat
$\frac{\ell}{N}\in[0,1]$ as though it were on a continuum, and the potential
in Equation (20) has an interpretation as the kinetic energy on a latticised
and compact fourth dimension, with a quadratic (as opposed to quartic, as is
the case in SYM,) dependence on a lattice momentum $p_{y}$. But what is the
scale of this momentum? Since the mass gap for the dual photon
$m^{2}_{\sigma,\,1}$ vanishes in the large-$N$ limit, the only remaining mass
scale to characterise the low-energy theory is $m^{2}_{\sigma,\,N/2}\equiv
m_{N/2}$, the (Debye) mass of the heaviest dual photon:

$\begin{split}m_{N/2}\sim m_{W}\lambda^{-3/2}e^{-1/2\lambda},\end{split}$
(24a) where $\begin{split}\lambda&\equiv\frac{Ng^{2}(m_{W})}{8\pi^{2}}\\\
&\approx\lambda_{N/2},\end{split}$ (24b) up to small corrections. This allows
us to define $p_{y,\ell}$ as an honest-to-goodness lattice momentum:
$p_{y,\ell}\equiv m_{N/2}\sin\Big{(}\pi\frac{\ell}{N}\Big{)}.$ (25)

We can also read off the two-point function directly from (20): Defining
$x^{M}\equiv(\vec{x},y)$, $p_{M}\equiv(\vec{p},p_{y})$, and momentarily
disregarding the massive correction,

$\int
d^{4}xe^{ip_{M}x^{M}}\langle\sigma(x_{M})\sigma(0)\rangle\sim(\lambda_{\ell}\,\vec{p}\,{}^{2}\\!+\lambda
p_{y}^{2})^{-1}.$ (26)

We observe that there is a restored Lorentz symmetry which is broken by an
anomalous scaling dimension $\Delta=b_{0}\lambda$ as $\lambda_{\ell}\sim
p_{y}^{b_{0}\lambda}$.

Put another way, the dual photon coupling $\lambda_{\ell}$ exhibits
logarithmic running in the lattice momentum
$p^{y}\sim\sin(\pi\frac{\ell}{N})$:

$p_{y}\frac{d}{dp_{y}}\Bigg{(}\frac{1}{\lambda_{\ell}}\Bigg{)}=p_{y}\frac{d}{dp_{y}}\Big{(}b_{0}\log\frac{1}{p_{y}}+(\text{const.})\Big{)}=-b_{0}.$
(27)

In particular, the scaling behaviour is opposite to that of the
$\mathbb{R}^{4}$ theory (cf. Equation (11)):

$\mu\frac{d}{d\mu}\Bigg{(}\frac{1}{\lambda(\mu)}\Bigg{)}=+b_{0}.$ (28)

We can also show how this analogy can be extended to encompass the mass-
correction terms $\sim W_{\ell}$: The one-loop correction to the coupling due
to a single adjoint fermion with mass $m$ in an $SU(N)$ theory on
$\mathbb{R}^{4}$, renormalised at some scale $\mu$ in the $\overline{MS}$
scheme is, (cf. Equation (68)):

$\begin{split}\mathcal{M}_{\mathbb{R}^{4}}^{\text{fermion}}(P)&=-2\int^{1}_{0}\\!dx\,x(1-x)\log\Big{(}\frac{P^{2}x(1-x)+m^{2}}{\mu^{2}}\Big{)}\\\
&\sim\begin{cases}-\frac{2}{3}\log(\frac{P}{\mu})&P^{2}\gg m^{2},\\\
-\frac{2}{3}\log(\frac{m}{\mu})&P^{2}\ll m^{2},\end{cases}\end{split}$ (29)

where $x$ is a Feynman parameter. We can compare this with our result of the
contribution in the $\mathbb{R}^{3}\times S^{1}$ theory, which can be read off
from (13):

$\begin{split}\mathcal{M}_{\mathbb{R}^{3}\times
S^{1},\,\ell}^{\text{fermion}}&=\frac{2}{3}\Big{[}\log\Big{(}\sin\pi\frac{\ell}{N}\Big{)}+W_{\ell}\Big{(}\frac{m}{m_{W}}\Big{)}\Big{]}\\\
&\sim\begin{cases}+\frac{2}{3}\log\sin(\pi\frac{\ell}{N})&m_{W}\gg m,\\\
-\frac{2}{3}\log(\frac{m}{m_{W}})&m_{W}\ll m.\end{cases}\end{split}$ (30)

This result is consistent with our interpretation of $p_{y}$ as a momentum,
with the mass correction behaving as we should expect in the $\mathbb{R}^{4}$
theory, albeit with opposite momentum-scaling behaviour in $p_{y}$.

##  3 Perturbative analysis: theory and practice

The remainder of this paper mainly focuses on deriving and calculating loop
integrals and Matsubara sums. Our approach is extremely straightforward —
essentially identical to the analysis of a thermal gauge theory at
temperatures $T=1/L$, but for the fact that our $S^{1}$ is spacelike rather
than timelike. This means, in particular for the fermions, that the $S^{1}$
momenta $\omega_{n}$ assume integer values $\omega_{n}=\frac{2\pi n}{L}$,
rather than half-integer $\omega_{n}=\frac{2\pi}{L}(n+\frac{1}{2})$. As the
calculation is rather involved, our presentation will try to go into as much
detail as we can without being overly cumbersome. For the convenience of the
reader, we will summarise the contents of Sections 3.1 and 3.2 at the end of
their respective Sections.

Let us start by defining our notation. We will use $M,N\in\\{1,2,3,4\\}$ for
Euclidean indices on $\mathbb{R}^{3}\times S^{1}$, (with $x^{4}$ the
coordinate on $S^{1}$) and $\mu,\nu\in\\{1,2,3\\}$ for indices on the
$\mathbb{R}^{3}$. We will use $a,b,c,i,j,k\in\\{1,\dots N\\}$ to denote Lie
algebra indices.

We also define the (over-complete) Cartan-Weyl basis on $su(N)$:

$(H_{i})_{ab}=\delta_{ia}\delta_{ib}=\text{diag}(0,\,...\,,\,\overbrace{1}^{i\text{'th}}\,,\,...\,,\,0)\qquad
1\leq i\leq N,$ (31a) which span the Cartan subalgebra $\mathfrak{t}$. These
are accompanied by the raising and lowering operators spanning
$\mathfrak{t}^{\perp}$, the orthogonal complement of $\mathfrak{t}$,
$(E_{\beta_{ij}})_{ab}=\delta_{ai}\delta_{bj},$ (31b) for $N$-component
vectors $\beta_{ij}$ in the root lattice of $su(N)$, which in our basis are
written $\begin{split}\beta^{a}_{ij}&\equiv\delta^{a}_{i}-\delta^{a}_{j}\\\
&=(0,...\overbrace{1}^{i\text{'th}},...,\overbrace{-1}^{j\text{'th}},...,0)\,,\qquad
1\leq i\leq j\leq N.\end{split}$ (31c) Perhaps a bit idiosyncratically, we say
that the subscripts on $\beta_{ij}$ are a set of antisymmetric indices
labelling the roots of $su(N)$: $\beta_{ij}=-\beta_{ji}$, and the superscript
$a$ denotes its $a$’th vector component.

$E_{\beta_{ij}},E_{-\beta_{ij}}$ are respectively raising and lowering
operators for the $su(2)$ subalgebra associated with the root $\beta_{ij}$:

$\displaystyle[H_{i},\,H_{j}]=0\,,\quad[H_{k},\,E_{\beta_{ij}}]=\beta^{k}_{ij}E_{\beta_{ij}}\,,\quad[E_{\beta_{ij}},\,E_{-\beta_{ij}}]=\sum_{k}\beta_{ij}^{k}H_{k},$
(32a) (no sums over $i,j$). We also have $\displaystyle
E_{\beta_{ij}}^{\dagger}=E_{-\beta_{ij}}=E_{\beta_{ji}}\,,\quad
H_{k}^{\dagger}=H_{k}.$ (32b) These generators are normalised as:
$\displaystyle\mathrm{tr}[H_{i}H_{j}]=\delta_{ij}\,,\quad\mathrm{tr}[E_{\beta}E_{-\beta^{\prime}}]=\delta_{\beta\beta^{\prime}}.$
(32c)

In the interest of brevity, we will frequently abuse notation and treat
$\beta$ as though it were the index on the root space and omit the subscripts
$ij$, as we have just done above. To avoid confusion, there will be no
implicit sum over $su(N)$ indices unless otherwise specified,.

As a matter of convenience, we normalise the components of $su(N)$-valued
fields $\psi$ as:

$\psi(x^{\mu},x^{4})=\frac{1}{2}\sum_{k}\psi^{k}(x^{\mu},x^{4})H_{k}+\frac{1}{\sqrt{2}}\sum_{\beta}\psi^{\beta}(x^{\mu},x^{4})E_{\beta},$
(33a) obeying hermiticity conditions:
$(\psi^{k})^{*}=\psi^{k}\,,\qquad(\psi^{\beta})^{*}=\psi^{-\beta},$ (33b) and
constrained by a trace-free condition: $\sum_{k}\psi^{k}(x^{\mu},x^{4})=0,$
(33c)

so that the expansion (33a) is unique although it is written in terms of an
over-complete basis.

###  3.1 Formal setup: beginnings

Let us start with a 4-dimensional Euclidean $SU(N)$ gauge theory with non-
Abelian field strength $\mathcal{F}_{MN}$ and $n_{f}$ two-component massive
adjoint fermions $\lambda_{I}$. As we are performing a perturbative
calculation, the vacuum angle is "invisible" to us, so we might as well set
the fermion masses to be real and the topological angle $\theta=0$:

$\begin{split}\mathcal{L}_{4d}&=\mathrm{tr}\Big{[}\frac{1}{2g^{2}}(\mathcal{F}_{MN})^{2}\\\
&+2i\sum_{I=1}^{n_{f}}\Big{(}\bar{\lambda}_{I}{}_{\dot{\alpha}}\bar{\sigma}^{M\dot{\alpha}\alpha}(\nabla_{M}\lambda_{I})_{\alpha}+\frac{m_{I}}{2}\lambda_{I}{}_{\alpha}\lambda_{I}{}_{\beta}\varepsilon^{\alpha\beta}+\frac{m_{I}}{2}\bar{\lambda}_{I}{}^{\dot{\alpha}}\bar{\lambda}_{I}{}^{\dot{\beta}}\varepsilon_{\dot{\alpha}\dot{\beta}}\Big{)}\Big{]}.\end{split}$
(34)

$\nabla_{M}$ is the covariant derivative on adjoint-representation fields:

$\nabla_{M}\equiv\partial_{M}+i[\mathcal{A}_{M},\,\cdot\,\,],$ (35)

and $\bar{\sigma}^{M}=(\,i\vec{\sigma}\,,\,1_{2}\,)$ are the Euclidean sigma
matrices.

Formally integrating out the high-energy ( $\gtrsim m_{W}$) degrees of freedom
around the centre-symmetric $\Omega$ gives us the effective 3d Lagrangian,
(10a).

We want to explicitly integrate out the high-energy ( $\gtrsim m_{W}$) degrees
of freedom to obtain the effective 3d Lagrangian, (10a) to find the one-loop
corrections to $\kappa_{ab}$, the photon coupling matrix. The methods we use
can also be applied almost verbatim to find $\rho_{ab}$, the corrected scalar
couplings, as well as $M_{ab}$ the scalar masses. Since these are not as
interesting to us, we simply quote their Fourier-transformed results in
Equations(85) and (86b).

Following Abbott’s approach, (e.g., Ref. abbott ,) we use an adapted
background field gauge method to calculate vacuum polarisation. This is fairly
standard textbook material, but to review: first, we treat the $su(N)$-valued
gauge field $A_{M}$ as the sum of a "classical" background field and a
"quantum" high-frequency field:

$\mathcal{A}_{M}=\underbrace{A_{M}}_{\text{classical}}+\underbrace{g\,a_{M}}_{\text{quantum}}.$
(36)

The normalisation is for convenience. We say that these fields have two
complementary expressions of gauge symmetry: for
$U,\,\tilde{U}:\mathbb{R}^{3}\times S^{1}\to SU(N)$:

$A_{M}\rightarrow U(A_{M}+i\partial_{M})U^{-1}\,,\quad a_{M}\rightarrow
Ua_{M}U^{-1}\quad\text{(gauge transformation under $U$) },$ (37a)
$a_{M}\rightarrow\tilde{U}(a_{M}+i\partial_{M})\tilde{U}^{-1}\,,\quad
A_{M}\rightarrow\tilde{U}A_{M}\tilde{U}^{-1}\quad\text{(gauge transformation
under $\tilde{U}$) }.$ (37b)

Anticipating a 3d and Abelian theory, we take $A_{M}$ to be Abelian and
trivial over $x^{4}$, and call its field strength $F_{MN}$:

$\partial_{4}A_{M}=0\,,\qquad[A_{M},A_{N}]=0,$ (38a)
$F_{MN}\equiv\partial_{M}A_{N}-\partial_{N}A_{M}.$ (38b)

We want to fix the gauge under $\tilde{U}$ in order to integrate out $a_{M}$,
which, as we will see, are basically $W$-bosons. To do this, we would impose
the condition

$D^{M}a_{M}+ig[a^{M},a_{M}]=0,$ (39a) where $D_{M}$ is the "covariant
derivative with connection $A_{M}$" :
$D_{M}\equiv\partial_{M}+i[A_{M},\,\cdot\,\,],$ (39b)

which can be done by adding a Gaussian term to the Lagrangian,

$\Delta\mathcal{L}_{a}=\mathrm{tr}\,(\nabla^{M}a_{M})^{2},$ (40)

and a Lagrangian $\mathcal{L}_{c}$ for scalar-yet-Grassmannian $su(N)$-valued
ghost fields $c$, $\bar{c}$.

Since $A_{4}$ has a non-zero vev, we must write

$A_{4}\equiv\frac{\phi}{L}+A_{4}^{0},$ (41a) where $\phi$ is the (constant in
$x^{\mu}$) vev, $\phi\equiv-iL\log\Omega,$ (41b)

and $A_{4}^{0}$ represents the fluctuations around the vev. But we can mostly
ignore $A_{4}^{0}$ as it is $U(1)^{N-1}$-neutral and therefore not involved in
the corrections to $\kappa_{ab}$ at one-loop. Gauge-fixing the centre-
symmetric $\Omega$ as in Equation (7), $\phi$ has vector components:

$\begin{split}\phi^{k}&=\frac{\pi}{N}\sum_{\beta>0}\beta^{k}\\\
&=\frac{2\pi}{N}\left(\frac{N+1}{2}-k\right),\end{split}$ (42)

where the sum in Equation (42) is over the positive roots. In particular, this
means

$\phi\\!\cdot\\!\beta_{ij}=\frac{2\pi}{N}(j-i).$ (43)

All together, the gauge-fixed Lagrangian has the form:

$\begin{split}\mathcal{L}&=\underset{\textrm{classical
fields}}{\underbrace{\mathcal{L}_{cl.}}}+\underset{\textrm{$W$-bosons}}{\underbrace{\mathcal{L}_{a}+\Delta\mathcal{L}_{a}}}\\\
&\qquad+\underset{\textrm{ghosts}}{\underbrace{\mathcal{L}_{c}}}+\underset{\textrm{fermions}}{\underbrace{\mathcal{L}_{\lambda}}}+O(\hbar^{3}),\end{split}$
(44)

where $\mathcal{L}_{a}$ contains the $\sim Aaa$, $AAaa$ terms upon expanding
$\mathcal{L}_{4d}$ in terms of $A_{M}$ and $a_{M}$, and similarly for
$\mathcal{L}_{c}$ and $\mathcal{L}_{\lambda}$. We observe that by choosing
$A_{M}$ to be Abelian, the Abelian parts of each of the quantum fields
$a,\lambda,c,\bar{c}$ cannot contribute to $\kappa_{ab}$ at one-loop order, so
we may forget about them altogether for the rest of this analysis.

Let us take any $su(N)$-valued field $\psi$ and simultaneously expand in the
KK modes and the Cartan-Weyl basis, recalling our convention as in Equation
(33a),

$\displaystyle\psi(x^{\mu},x^{4})$
$\displaystyle=\frac{1}{2}\sum_{k,z}e^{i\frac{2\pi
z}{L}x^{4}}\psi^{k,z}(x^{\mu})H_{k}+\frac{1}{\sqrt{2}}\sum_{\beta,z}e^{i\frac{2\pi
z}{L}x^{4}}\psi^{\beta,k}(x^{\mu})E_{\beta},$

so that

$iD_{4}\psi=\frac{1}{\sqrt{2}}\sum_{z,\beta}e^{i\frac{2\pi
z}{L}x^{4}}\Big{(}\frac{2\pi
z+\phi\cdot\beta}{L}\Big{)}\psi^{\beta}E_{\beta}+\text{(Abelian and
$O(\hbar^{2})$ parts)},$ (45)

so asymptotically, the derivative operator $iD_{4}$ diagonalises with
eigenvalues

$iD_{4}\to\frac{2\pi z+\phi\\!\cdot\\!\beta_{ij}}{L}\,,\quad
z\,\,\text{integer},$ (46)

so that fields in $\mathfrak{t}^{\perp}$ with charge $\beta$ and circle
momentum $2\pi z/L$ of a field with mass $m$ obtains an effective 3d (mass)2:

$m^{2}+m_{W}^{2}\Big{[}Nz+(i-j)\Big{]}^{2}\geq m_{W}^{2}.$ (47)

Thus only $U(1)^{N-1}$-neutral and $x^{4}$-trivial fields survive in the IR
theory at scales $\ll m_{W}$, consistent with our hypotheses on $A_{M}$.

#### Summary of Section 3.1:

We outlined the background field approach to perturbation theory: with an eye
toward the infrared theory, we set the background $A_{M}$ to be
$x^{4}$-trivial and Abelian, and showed that only "quantum fields"
proportional to the broken gauge generators may contribute to the corrections
of $\kappa_{ab}$ at one-loop order. We further showed that this assumption is
self-consistent, because all fields with non-vanishing $x^{4}$ momentum or
carrying charge under the $U(1)^{N-1}$ acquire an effective mass $\geq m_{W}$
through the Higgs mechanism.

###  3.2 The 1-loop Wilsonian action

We can write an expression for the Wilsonian effective action $\Gamma[A]$ by
formally integrating out the quantum fields under the path integral sign.
Setting the vacuum energy to zero,

$\begin{split}\Gamma[A]&=-\log\Big{[}\int
DaD\bar{c}Dc\prod_{I}^{n_{f}}(D\bar{\lambda}_{I}D\lambda_{I})\,e^{-\int\mathcal{L}[A,c,\lambda_{I},a]}\Big{]}\\\
&=\int_{\mathbb{R}^{3}}\Big{[}\frac{L}{4g^{2}}(F_{\mu\nu}^{k})^{2}+\frac{L}{2g^{2}}(\partial_{\mu}A_{4}^{k})^{2}\Big{]}\\\
&\quad+\sum_{s=0,\frac{1}{2},1}\sum_{f_{s}}\chi(s)\mathrm{Tr}\log(-D_{(s)}^{2}+m_{f_{s},s}^{2})\\\
&\quad+\text{(higher loop contributions)}.\end{split}$ (48)

There is a lot of notation to define in Equation (48), but it will make life
easier by formatting the problem so that the entire non-trivial part of the
calculation is contained in the single expression
"$\mathrm{Tr}\log(-D_{(s)}^{2}+m_{f_{s},s}^{2})$," which we will only have to
evaluate once to cover all the relevant cases, rather than having to work with
massive or massless, spinor, scalar, and vector integrals separately.

"$\mathrm{Tr}$" refers to the trace over the respective Hilbert spaces, and
$-D^{2}_{(s)}$ is a differential operator defined in Equation (51). The terms
on the third row of Equation (48) are due to the $W$-bosons $a$, ($s=1$,) the
gauge ghosts $c,\,\bar{c}$, ($s=0$,) and the fermions $\lambda_{I}$ ($s=1/2$).
The $s=1/2$ term is obtained by doubling then halving the trace-log of the
massive Weyl operator:

$\begin{split}\sum_{I=1}^{n_{f}}\frac{1}{2}\mathrm{Tr}\log(i\bar{\sigma}\\!\cdot\\!D+im_{I})\equiv\sum_{I=1}^{n_{f}}\frac{1}{4}\mathrm{Tr}\log\Big{(}-D_{(1/2)}^{2}+m_{I}^{2}\Big{)}.\end{split}$
(49)

$\sum_{f_{s}}$ is a sum over flavour indices $I$ when $s=1/2$;
$m_{f_{s},s}^{2}=0$ for $s\neq 1/2$. $\chi(s)$ is a pre-factor determined by
the statistics of each field:

$\begin{split}\chi(s)\equiv\begin{cases}-1&(s=0),\\\ -1/4&(s=1/2),\\\
+1/2&(s=1).\\\ \end{cases}\end{split}$ (50)

To define $-D_{(s)}^{2}$, let $A,B$ denote indices in the spin-$s$ irrep of
the (Euclidean) Lorentz group, then

$\begin{split}(-D_{s}^{2})_{AB}&=-D_{M}D^{M}\delta_{AB}+F^{k}_{MN}(H^{\text{adj.}}_{k})(\sigma^{(s)}_{MN})_{AB}\\\
&=\Big{[}(i\partial_{\mu})^{2}+\Big{(}i\partial_{3}+\frac{\phi^{k}}{L}(H_{k}^{\text{adj.}})\Big{)}^{2}\,\Big{]}(\delta^{(s)})_{AB}\\\
&\quad+(\Sigma_{1}^{(s)})_{AB}+(\Sigma_{2}^{(s)})_{AB}+(\Sigma_{F}^{(s)})_{AB},\end{split}$
(51)

(with an implicit sum over $k$,) where
$(H_{k}^{\text{adj.}})\equiv[H_{k},\,\cdot\,\,]$, and

$\displaystyle(\Sigma_{F}^{(s)})_{AB}$ $\displaystyle\equiv
F^{k}_{MN}(H_{k}^{\text{adj.}})(\sigma^{(s)}_{MN})_{AB},$ (52a)
$\displaystyle(\Sigma_{1}^{(s)})_{AB}$ $\displaystyle\equiv-
iA^{k}_{M}\overleftrightarrow{\partial}^{M}(H_{k}^{\text{adj.}})(\delta^{(s)})_{AB},$
(52b) $\displaystyle(\Sigma_{2}^{(s)})_{AB}$
$\displaystyle\equiv[A^{k}_{M}(H_{k}^{\text{adj.}})]^{2}(\delta^{(s)})_{AB},$
(52c)

(again, with an implicit sum over $k$). $\Sigma_{1}^{(s)}$ and
$\Sigma_{2}^{(s)}$ are respectively the 3- and 4- point interactions of a
charged adjoint field, and $\Sigma_{F}^{(s)}$ is the spin-field coupling term
responsible for asymptotic freedom in non-Abelian theories. $\delta^{(s)}$ and
$\sigma_{MN}^{(s)}$ are respectively the identity matrix and the generators of
rotations in the spin-$s$ representation. Explicitly, (and abusing notation
slightly by mixing indices,)

$(\sigma_{MN}^{(s)})_{AB}=\begin{cases}\,0&(s=0),\\\
\,\frac{i}{4}(\bar{\sigma}_{[M}\sigma_{N]})_{AB}&(s=1/2),\\\
\,-i(\delta_{AM}\delta_{NB}\\!\\!-\\!\\!\delta_{AN}\delta_{MB})\quad&(s=1).\end{cases}$
(53)

so the one-loop correction to the Wilsonian can be written, to quadratic order
in $A_{M}$:

$\begin{split}\mathrm{Tr}\log\Big{(}\frac{-D^{2}_{s}+m_{s}^{2}}{-\partial^{2}+m_{s}^{2}}\Big{)}=&\mathrm{Tr}\Big{(}\frac{\Sigma^{(s)}_{2}}{-\partial^{2}+m_{s}^{2}}\Big{)}\\\
-\frac{1}{2}&\mathrm{Tr}\Big{[}\Big{(}\frac{\Sigma^{(s)}_{1}}{-\partial^{2}+m_{s}^{2}}\Big{)}^{2}+\Big{(}\frac{\Sigma^{(s)}_{F}}{-\partial^{2}+m_{s}^{2}}\Big{)}^{2}\Big{]}+O(A^{3}).\end{split}$
(54)

There is no $\sim\Sigma_{F}\Sigma_{1}$ cross-term because the trace of
$\sigma^{(s)}_{MN}$ vanishes. Expanding in a Fourier basis to quadratic order
in the fields, Equation (48) becomes

$\begin{split}\Gamma[A_{M}^{k};\mu]=2\sum_{a,b}\int\\!\frac{d^{3}p}{(2\pi)^{3}}\Big{[}&A^{\mu
a}\kappa_{ab}(p^{2}\delta_{\mu\nu}-p_{\mu}p_{\nu})A^{\nu b}\\\
+&A_{4}^{a}\Big{(}p^{2}\rho_{ab}+\frac{L}{2g^{2}}M^{2}_{ab}\Big{)}A_{4}^{b}\Big{]}+O(A^{3}).\end{split}$
(55)

We note in passing that the GPY potential $V[\Omega]$ still appears in
Equation (55) through $M^{2}_{ab}$, its second derivative.

Now we are ready to draw some Feynman diagrams. Let $p^{M}=(p^{\mu},0)$ denote
the external momentum of $A_{M}$, and for convenience, define an effective
loop momentum $K^{M}_{(\beta,z)}$.

$K^{M}\equiv K^{M}_{(\beta,z)}\equiv\Big{(}k^{\mu},\frac{2\pi
z+\phi\cdot\beta}{L}\Big{)}.$ (56)

Using Equations (52a), (52b), (52c), and (54) and reading off from (55), we
can write down the corrections to the $\sim A_{M}^{a}A_{N}^{b}$ term in the
action:

$\displaystyle(\Pi_{2}^{(s)})_{MN}^{ab}$
$\displaystyle\equiv\frac{1}{2}\frac{\delta^{2}}{\delta A^{a}_{M}\delta
A^{b}_{N}}\mathrm{Tr}\Big{(}\frac{\Sigma^{(s)}_{2}}{-\partial^{2}+m_{s}^{2}}\Big{)}$
$\displaystyle=\frac{1}{L}\sum_{z,\beta}\int\\!\frac{d^{3}\\!k}{(2\pi)^{3}}d(s)\beta^{a}\beta^{b}\delta_{MN}\Bigg{[}\frac{1}{(K^{2}+\Delta_{s}^{2})}+\frac{(1-2x)^{2}p^{2}}{2(K^{2}+\Delta_{s}^{2})^{2}}\Bigg{]},$
(57a) $\displaystyle(\Pi_{1}^{(s)})_{MN}^{ab}$
$\displaystyle\equiv\frac{1}{2}\frac{\delta^{2}}{\delta A^{a}_{M}\delta
A^{b}_{N}}\Bigg{[}-\frac{1}{2}\mathrm{Tr}\Big{(}\frac{\Sigma^{(s)}_{1}}{-\partial^{2}+m_{s}^{2}}\Big{)}^{2}\Bigg{]}$
$\displaystyle=-\frac{1}{L}\sum_{z,\beta}\int\\!\frac{d^{3}\\!k}{(2\pi)^{3}}\frac{1}{2}d(s)\beta^{a}\beta^{b}\int^{1}_{0}\\!dx\frac{4K_{M}K_{N}+(1-2x)^{2}p_{M}p_{N}}{(K^{2}+\Delta_{s})^{2}},$
(57b) $\displaystyle(\Pi_{F}^{(s)})_{MN}^{ab}$
$\displaystyle\equiv\frac{1}{2}\frac{\delta^{2}}{\delta A^{a}_{M}\delta
A^{b}_{N}}\Bigg{[}-\frac{1}{2}\mathrm{Tr}\Big{(}\frac{\Sigma^{(s)}_{F}}{-\partial^{2}+m_{s}^{2}}\Big{)}^{2}\Bigg{]}$
$\displaystyle=-\frac{1}{L}\sum_{z,\beta}\int\\!\frac{d^{3}\\!k}{(2\pi)^{3}}c(s)\beta^{a}\beta^{b}\int^{1}_{0}\\!dx\frac{2(p^{2}\delta_{MN}-p_{M}p_{N})}{(K^{2}+\Delta_{s})^{2}}.$
(57c)

(a) ($\Pi_{2}^{(s)})_{MN}^{ab}$

(b) ($\Pi_{1}^{(s)})_{MN}^{ab}$

(c) ($\Pi_{F}^{(s)})_{MN}^{ab}$

Figure 4: Representations of the loop integrals in Equation (57) in terms of
Feynman diagrams. The $\Sigma_{F}$ vertex is distinguished from the
$\Sigma_{1}$ vertex with a (red) dot.

These integrals are pictorially represented by the Feynman diagrams in Fig. 4.
In each of the integrals above we have employed a Feynman parameter $x$, and
shifted our loop momentum $K^{M}\to K^{M}-xp^{M}$. We have also defined an
effective (mass)2, $\Delta_{s}$ (not to be confused with the 3d effective
(mass)2 in Equation (47),)

$\Delta_{s}\equiv m_{s}^{2}+p^{2}x(1-x),$ (58)

We have further defined $d(s)$, the number of spin states in the spin-$s$
representation, and $c(s)$, the spin-field coupling coefficient:131313Note
that $d(1/2)=4$ for us, because we doubled the number of polarisations in
Equation (49); this is already compensated for by an additional factor of
$1/2$ in front of the fermion determinant in Equation (48).

$d(s)\equiv\mathrm{tr}(\delta^{(s)})=\begin{cases}1&(s=0),\\\ 4&(s=1/2),\\\
4&(s=1),\\\ \end{cases}\,\qquad
c(s)\equiv\mathrm{tr}(\sigma^{(s)}_{MN}\sigma^{(s)}{}^{MN})=\begin{cases}0&(s=0),\\\
1&(s=1/2),\\\ 2&(s=1),\\\ \end{cases}$ (59)

where the traces are over the spin indices, which we have omitted. The rest of
our report will be largely concerned with evaluating these three integrals.

#### Summary of Section 3.2:

We introduced some formal notation to write down the one-loop effective action
in a more compact form, Equation (48). This allowed us to write the integrals
of each of $a,\lambda,c$ in terms of the loop integrals $\Pi^{(s)}_{2}$,
(Equation (57a),) $\Pi^{(s)}_{1}$, (Equation (57b),) and $\Pi^{(s)}_{F}$
(Equation (57c)). As we will see, the evaluation of these integrals are by no
means a trivial task, but we will make them much more tractable with a handful
of clever manipulations.

###  3.3 Outline of the calculation

We have written the integrals in (57) to superficially respect the Euclidean
Lorentz group $SO(4)$. But to evaluate them, we must rewrite (57) to reflect
the broken rotational symmetry $SO(4)\to SO(3)$. Symmetry considerations tell
us that averaging $K_{M}K_{N}$ must give:

$(\overline{K_{M}K_{N}})_{(\beta,z)}=\frac{k^{2}}{3}\delta_{\mu\nu}\mathcal{P}^{\mu\nu}_{MN}+\Big{(}\frac{2\pi
z+\phi\cdot\beta}{L}\Big{)}^{2}\mathcal{P}^{44}_{MN},$ (60a) where
$\mathcal{P}^{\mu\nu}_{MN}$ and $\mathcal{P}^{44}_{MN}$ are projection
operators to $\mathbb{R}^{3}$ and $S^{1}$ respectively:
$\displaystyle\mathcal{P}^{\mu\nu}_{MN}$
$\displaystyle\equiv\delta_{M}^{\mu}\delta_{N}^{\nu},$ (60b)
$\displaystyle\mathcal{P}^{44}_{MN}$
$\displaystyle\equiv\delta_{M}^{4}\delta_{N}^{4},$ (60c)

Integrating over the angular coordinates and summing the three graphs in
Equation (57), we get

$\begin{split}(\Pi^{(s)})_{\mu\nu}^{ab}&\equiv\sum_{\mathcal{I}=2,1,F}(\Pi_{\mathcal{I}}^{(s)})_{\mu\nu}^{ab}\\\
&=\sum_{z,\beta}\beta^{a}\beta^{b}\\!\int^{1}_{0}\\!dx\int^{\infty}_{0}\frac{d(kL)}{2\pi^{2}}(kL)^{2}\Bigg{\\{}\Big{[}\frac{(1-2x)^{2}}{2}d(s)-2c(s)\Big{]}(p^{2}\delta_{\mu\nu}-p_{\mu}p_{\nu})S_{1}(b,\omega_{s}L)\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{d(s)}{L^{2}}\delta_{\mu\nu}\Big{[}S_{0}(b,\omega_{s}L)-\frac{2}{3}(kL)^{2}S_{1}(b,\omega_{s}L)\Big{]}\Bigg{\\}}.\end{split}$
(61a) We also write out the $(44)$ part, which are needed to renormalise:
$\begin{split}(\Pi^{(s)})_{44}^{ab}&\equiv\sum_{\mathcal{I}}(\Pi_{\mathcal{I}}^{(s)})_{44}^{ab}\\\
&=\sum_{z,\beta}\beta^{a}\beta^{b}\\!\int^{1}_{0}\\!dx\int^{\infty}_{0}\frac{d(kL)}{2\pi^{2}}(kL)^{2}\Bigg{\\{}\Big{[}\frac{(1-2x)^{2}}{2}d(s)-2c(s)\Big{]}p^{2}\,S_{1}(b,\omega_{s}L)\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{d(s)}{L^{2}}\Big{[}S_{0}(b,\omega_{s}L)-2S_{2}(b,\omega_{s}L)\Big{]}\Bigg{\\}}.\end{split}$
(61b) where $(\Pi^{(s)})_{\mu\nu}^{ab}$ and $(\Pi^{(s)})_{44}^{ab}$ are
defined in the obvious way:
$(\Pi^{(s)})_{MN}^{ab}\equiv(\Pi^{(s)})_{\mu\nu}^{ab}\mathcal{P}^{\mu\nu}_{MN}+(\Pi^{(s)})_{44}^{ab}\mathcal{P}^{44}_{MN},$
(61c) and we have also defined:
$\omega_{s}\equiv\sqrt{k^{2}+\Delta_{s}}\,,\qquad b\equiv\phi\cdot\beta,$
(61d) and dimensionless sums over the KK modes, $S_{0,1,2}$: $S_{1}(b,\omega
L)\equiv\sum_{n\in\mathbb{Z}}\frac{1}{[(2\pi n+b)^{2}+(\omega
L)^{2}]^{2}}\,,\quad S_{2}(b,\omega L)\equiv\sum_{n\in\mathbb{Z}}\frac{(2\pi
n+b)^{2}}{[(2\pi n+b)^{2}+(\omega L)^{2}]^{2}}.$ (61e) The third sum, $S_{0}$,
is a standard result. It can be evaluated exactly by e.g., Matsubara
summation: $\begin{split}S_{0}(b,\omega
L)&\equiv\sum_{n\in\mathbb{Z}}\frac{1}{(2\pi n+b)^{2}+(\omega L)^{2}}\\\
&=\frac{1}{2\omega L}+\frac{1}{2\omega
L}Re\Big{(}\frac{1}{e^{L\omega+ib}-1}\Big{)}\\\ &\equiv
I_{0}^{\text{vac.}}(\omega L)+\delta I_{0}(b,\omega L).\end{split}$ (61f)

Where we have defined a function $I^{\text{vac.}}_{0}\equiv\frac{1}{2\omega
L}$ that falls off as a negative power in $\omega L$, and another, $\delta
I_{0}\equiv\frac{1}{2\omega L}Re(\frac{1}{e^{L\omega+ib}-1})$, that falls off
exponentially.

Since the summand of $S_{0}$ is monotone decreasing in $|n|$, differentiation
commutes with summation, so $S_{1,2}$ can be trivially evaluated by taking
derivatives of both sides of Equation (61f):

$\displaystyle S_{1}(b,\omega L)$
$\displaystyle=-\frac{\partial}{\partial{(\omega L)^{2}}}S_{0}(b,\omega L)$
$\displaystyle\equiv I^{\text{vac.}}_{1}(\omega L)+\delta I_{1}(b,\omega L),$
(62a) $\displaystyle S_{2}(b,\omega L)$
$\displaystyle=\frac{\partial}{\partial{(\omega L)^{2}}}\Big{[}(\omega
L)^{2}S_{0}(b,\omega L)\Big{]}$ $\displaystyle\equiv
I^{\text{vac.}}_{2}(\omega L)+\delta I_{2}(b,\omega L),$ (62b)

where $I^{\text{vac.}}_{1,2}$ and $\delta I_{1,2}$ are defined in terms of
derivatives of $I_{0}^{\text{vac.}}$ and $\delta I_{0}$ respectively, in the
obvious ways as suggested by the notation. The point is that we can split the
integrals in Equation (61a):

$(\Pi^{(s)})_{\mu\nu}^{ab}=(\Pi^{(s),\textrm{vac.}})_{\mu\nu}^{ab}+(\delta\Pi^{(s)})_{\mu\nu}^{ab},$
(63)

by collecting the $I^{\text{vac.}}_{0,1,2}$ terms into
$(\Pi^{(s),\textrm{vac.}})_{\mu\nu}^{ab}$, and the $\delta I_{0,1,2}$ terms
into $(\delta\Pi^{(s)})_{\mu\nu}^{ab}$, and similarly for
$(\Pi^{(s)})_{44}^{ab}$. We will call these the _vacuum integral_ and _pseudo-
thermal integral_ contributions respectively, and we consider them separately
in the following.

The basic idea is this: we can see by the asymptotics that the
$(\Pi^{(s),\text{vac.}})_{MN}^{ab}$ integrals remain unchanged in the
$L\to\infty$ limit. This means we can evaluate those integrals in terms of the
familiar loop integrals in $\mathbb{R}^{4}$, in a way we show explicitly.
Obviously these integrals are UV divergent, but they can be renormalised in
the $\overline{MS}$ scheme in the usual way. On the other hand, the
$SO(4)$-breaking, $L$-dependent parts of $(\Pi^{(s)})_{MN}^{ab}$ are contained
entirely within $(\delta\Pi^{(s)})_{MN}^{ab}$ : the exponential decay of the
$\delta I_{0,1,2}$ means that the integrands of
$(\delta\Pi^{(s)})_{\mu\nu}^{ab}$ are uniformly convergent in $kL$. Then we
may use the identities (62a), (62b) to integrate by parts in $kL$ and obtain a
much more tractable expression.

####  3.3.1 The vacuum integrals

We can evaluate the loop integrals in $\Pi^{(s),\textrm{vac.}}$ by "undoing"
an integral over an auxiliary continuous variable $k_{4}$. For example,
(defining $\omega\equiv\sqrt{k^{2}+\Delta}$ for positive $\Delta$,)

$\begin{split}\int\\!\frac{d^{3}k}{(2\pi)^{3}}LI^{\text{vac.}}_{0}&=\int\\!\frac{d^{3}k}{(2\pi)^{3}}\frac{1}{2\omega}\\\
&=\int\\!\frac{d^{3}k}{(2\pi)^{3}}\\!\\!\int^{\infty}_{-\infty}\frac{dk_{4}}{2\pi}\frac{1}{(k_{4})^{2}+k^{2}+\Delta},\end{split}$

thus mapping the integral over $k\in\mathbb{R}^{3}$ to one over
$\tilde{k}\in\mathbb{R}^{4}$. Then we regulate the expression by taking the
analytic continuation to $d\equiv 4-\varepsilon$ dimensions. In summary:

$\displaystyle\int\\!\frac{dk}{2\pi^{2}}k^{2}\,LI^{\text{vac.}}_{1}$
$\displaystyle\longrightarrow\mu^{-\varepsilon}\int\\!\frac{d^{d}\tilde{k}}{(2\pi)^{d}}\frac{1}{(\tilde{k}^{2}+\Delta)^{2}},$
(64a) $\displaystyle\int\\!\frac{dk}{2\pi^{2}}k^{2}\,L^{3}I^{\text{vac.}}_{2}$
$\displaystyle\longrightarrow\mu^{-\varepsilon}\int\\!\frac{d^{d}\tilde{k}}{(2\pi)^{d}}\frac{(k_{4})^{2}}{(\tilde{k}^{2}+\Delta)^{2}},$
(64b) $\displaystyle\int\\!\frac{dk}{2\pi^{2}}k^{2}LI^{\text{vac.}}_{0}$
$\displaystyle\longrightarrow\mu^{-\varepsilon}\int\\!\frac{d^{d}\tilde{k}}{(2\pi)^{d}}\frac{1}{(\tilde{k}^{2}+\Delta)}.$
(64c)

The expressions on the LHS are the relevant $\mathbb{R}^{3}$ integrals, and
$\mu$ is the $\overline{MS}$ scale of the theory. "$\longrightarrow$" means
"analytically continues to." On the other hand, we also have the following
series of relations under the integral sign:

$\begin{split}\frac{\tilde{k}^{2}}{d}\delta^{MN}|_{\mathbb{R}^{d}}\equiv&\,\overline{(\tilde{k}^{M}\tilde{k}^{N})}\,|_{\mathbb{R}^{d}}\\\
\longleftarrow&\,\,[\,\overline{(k^{\mu}k^{\nu})}\mathcal{P}_{MN}^{\mu\nu}+(k_{4})^{2}\mathcal{P}_{MN}^{44}]\,|_{\mathbb{R}^{3}\times
S^{1}}\\\
=&\,\Big{[}\,\frac{k^{2}}{3}\mathcal{P}_{MN}^{\mu\nu}\delta_{\mu\nu}+(k_{4})^{2}\mathcal{P}_{MN}^{44}\Big{]}\,|_{\mathbb{R}^{3}\times
S^{1}}.\end{split}$ (65)

where $\mathcal{P}_{MN}^{44},\mathcal{P}_{MN}^{\mu\nu}$ are the projectors
defined in (60b) and (60c). Combining these expressions, the vacuum integrals
can be re-written as integrals in $d=(4-\varepsilon)$ dimensions by restoring
the $SO(4)$ symmetry:

$\begin{split}(\Pi^{(s),\textrm{vac.}})_{MN}^{ab}&\equiv\sum_{\mathcal{I}}(\Pi_{\mathcal{I}}^{(s),\textrm{vac.}})_{MN}^{ab}\\\
&=\sum_{\beta}\beta^{a}\beta^{b}\int^{1}_{0}\\!dx\int\\!\frac{d^{d}\tilde{k}}{(2\pi)^{d}}\frac{\mu^{4-d}}{(\tilde{k}^{2}+\Delta_{s})^{2}}\Bigg{\\{}\Big{[}\frac{d-2}{2}\tilde{k}^{2}+\Delta_{s}\Big{]}d(s)\delta_{MN}\\\
&\qquad\qquad\qquad+\Big{[}\frac{(1-2x)^{2}}{2}d(s)-2c(s)\Big{]}\Big{[}(p^{2}\delta_{\mu\nu}-p_{\mu}p_{\nu})\mathcal{P}^{\mu\nu}_{MN}+p^{2}\,\mathcal{P}^{44}_{MN}\Big{]}\Bigg{\\}}.\end{split}$
(66)

Expanding in powers of $1\gg\varepsilon>0$, it is easy to regulate the
$\tilde{k}$ integral to get a convergent result. The (Abelian part of the) UV
counterterm $\delta Z_{s}\mathrm{tr}F_{MN}F^{MN}$ contributes,
diagrammatically,

$\begin{split}\Bigg{[}\feynmandiagram[horizontal=btoc,layeredlayout,inline=(b.base)]{a[particle={$A^{a}_{M}$}]--[boson]b[crosseddot]--[boson]c[particle={$A^{b}_{N}$}],};\Bigg{]}^{\textrm{CT},(s)}&=\delta
Z_{s}\sum_{\beta}\beta^{a}\beta^{b}\Big{[}(p^{2}\delta_{\mu\nu}-p_{\mu}p_{\nu})\mathcal{P}^{\mu\nu}_{MN}+p^{2}\,\mathcal{P}^{44}_{MN}\Big{]}.\end{split}$
(67a) So for each $s$, we choose $\displaystyle-\delta Z_{s}$
$\displaystyle\equiv\frac{1}{32\pi^{2}}\Big{(}\frac{d(s)}{3}-4c(s)\Big{)}\Big{(}\frac{2}{\varepsilon}-\gamma+\log
4\pi\Big{)},$ (67b)

and the sum of the three regulated vacuum integrals is therefore:

$\begin{split}(\tilde{\Pi}^{(s),\textrm{vac.}})_{MN}^{ab}&\equiv(\Pi^{(s),\textrm{vac.}})_{MN}^{ab}+(\textrm{counterterms})\\\
&=\sum_{\beta}\frac{\beta^{a}\beta^{b}}{32\pi^{2}}\int^{1}_{0}\\!dx\Big{[}d(s)(1-2x)^{2}-4c(s)\Big{]}\\\
&\qquad\qquad\qquad\qquad\times\Big{[}(p^{2}\delta_{\mu\nu}-p_{\mu}p_{\nu})\mathcal{P}^{\mu\nu}_{MN}+p^{2}\mathcal{P}^{44}_{MN}\Big{]}\log\Big{(}\frac{\mu^{2}}{\Delta_{s}}\Big{)}.\end{split}$
(68)

####  3.3.2 The pseudo-thermal integrals

Now we consider the pseudo-thermal integrals. Using Equations (62a), (62b), we
can simplify the loop integrals immensely by integrating by parts by changing
variables $\frac{\partial}{\partial(\omega
L)^{2}}=\frac{1}{2kL}\frac{\partial}{\partial(kL)}$. We find that all boundary
terms vanish, and the results are, in summary,

$\displaystyle\int^{\infty}_{0}\\!d(kL)\,(kL)^{2}\delta I_{1}(b,\omega L)=$
$\displaystyle\frac{1}{2}\int^{\infty}_{0}\\!d(kL)\,\delta I_{0}(b,\omega L),$
(69a) $\displaystyle\int^{\infty}_{0}\\!d(kL)\,(kL)^{4}\delta I_{1}(b,\omega
L)=$ $\displaystyle\frac{3}{2}\int^{\infty}_{0}\\!d(kL)\,(kL)^{2}\delta
I_{0}(b,\omega L),$ (69b)
$\displaystyle\int^{\infty}_{0}\\!d(kL)\,(kL)^{2}\delta I_{2}(b,\omega L)=-$
$\displaystyle\frac{1}{2}\int^{\infty}_{0}\\!d(kL)\,(\omega L)^{2}\delta
I_{0}(b,\omega L).$ (69c)

Plugging into Equations (61a), (61b), the pseudo-thermal integrals may be
written:

$(\delta\Pi^{(s)})_{\mu\nu}^{ab}=\sum_{\beta}\frac{\beta^{a}\beta^{b}}{8\pi^{2}}\int^{1}_{0}\\!dx\,\Big{[}d(s)(1-2x)^{2}-4c(s)\Big{]}(p^{2}\delta_{\mu\nu}-p_{\mu}p_{\nu})R^{b}_{0}(\sqrt{\Delta_{s}}L),$
(70)

Where we have defined141414The integral in the second line can be carried out
by expanding in series in $|e^{-\sqrt{(kL)^{2}+\Delta L^{2}}-ib}|<1$.

$\begin{split}R^{b}_{0}(\sqrt{\Delta}L)&\equiv\int^{\infty}_{0}\\!d(kL)\cdot\delta
I_{0}\Big{(}b,\sqrt{(kL)^{2}+\Delta L^{2}}\Big{)}\\\
&=\int^{\infty}_{0}\frac{d(kL)}{2\sqrt{(kL)^{2}+\Delta
L^{2}}}Re\Big{(}\frac{1}{e^{\sqrt{(kL)^{2}+\Delta L^{2}}+ib}-1}\Big{)}\\\
&=\sum_{n=1}^{\infty}K_{0}(n\sqrt{\Delta}L)\cos(n\phi\cdot\beta_{ij}),\end{split}$
(71)

where $K_{0}$ is the modified Bessel function of order $0$. This represents
the only remaining non-trivial sum, as far as the corrections to $\kappa_{ab}$
are concerned. We have not given an expression for
$(\delta\Pi^{(s)})_{44}^{ab}$ as it is not needed to find $\kappa_{ab}$.

Summing the result with the vacuum contribution,

$\begin{split}&(\Pi^{(s),\textrm{vac.}})_{\mu\nu}^{ab}+(\delta\Pi^{(s)})_{\mu\nu}^{ab}+\text{(counterterms)}\\\
&=\frac{(p^{2}\delta_{\mu\nu}-p_{\mu}p_{\nu})}{8\pi^{2}}\int^{1}_{0}dx\Big{[}d(s)(1-2x)^{2}-4c(s)\Big{]}\Big{(}\mathcal{R}_{0}^{ab}(\sqrt{\Delta_{s}}L)+\delta^{ab}N\log\frac{\mu}{\sqrt{\Delta_{s}}}\Big{)},\end{split}$
(72)

where we have defined

$\begin{split}\mathcal{R}_{0}^{ab}(\sqrt{\Delta}L)&\equiv\sum_{i,j}\beta^{a}_{ij}\beta^{b}_{ij}R_{0}^{\phi\cdot\beta_{ij}}(\sqrt{\Delta}L)\\\
&=\sum_{i,j}\beta^{a}_{ij}\beta^{b}_{ij}\sum_{n=1}^{\infty}K_{0}(n\sqrt{\Delta}L)\cos(n\phi\cdot\beta_{ij}).\end{split}$
(73)

In Appendix A.1 we explicitly show that

$\begin{split}R_{0}^{b}(\sqrt{\Delta}L)=\frac{1}{2}\log\frac{\sqrt{\Delta}L}{4\pi}+\tilde{R}_{0}^{b}(\sqrt{\Delta}L),\end{split}$
(74)

where $\tilde{R}_{0}^{b}(t)$ is a pure function that has a power series
expansion around $t=0$ for fixed $b\in(0,2\pi)$. We know that Equation (74)
must be true because the running of the coupling $g^{2}$ must freeze out at
scales below $m_{W}$. Equation (74) allows us to disregard the
$p^{2}$-dependence in $R_{0}^{b}(\sqrt{\Delta}L)$ as higher-derivative
corrections, and integrate over the Feynman parameter $x$ trivially. Recalling
Equations (59) and (50),

$\sum_{s=0,1}\chi(s)\Big{(}4\,c(s)-\frac{d(s)}{3}\Big{)}=\frac{11}{3},$ (75a)
and $\chi(1/2)\Big{(}4\,c(1/2)-\frac{d(1/2)}{3}\Big{)}=-\frac{2}{3}.$ (75b)

we have

$\kappa_{ab}=\frac{m_{W}^{-1}}{16\pi}\Big{[}\frac{8\pi^{2}}{Ng^{2}(\frac{4\pi}{L})}\delta_{ab}+\frac{1}{N}\sum_{i,j}\beta^{a}_{ij}\beta^{b}_{ij}\Big{(}\frac{11}{3}\tilde{R}^{\phi\cdot\beta_{ij}}_{0}(0)-\frac{2}{3}\sum_{I=1}^{n_{f}}\tilde{R}^{\phi\cdot\beta_{ij}}_{0}(m_{I}L)\Big{)}\Big{]}.$
(76)

An expression for $\tilde{R}_{0}^{\phi\cdot\beta_{ij}}(t)$ is derived in
Equation (103). All that remains now is to diagonalise Equation (76).

###  3.4 The sums over $\beta$: linear algebra on the root lattice

Let us consider the sums over the root vectors $\beta$. It is not hard to show
by standard Fourier analysis that, for any integer $n$,

$\begin{split}C^{ab}_{n}:&=\sum_{i,\,j}\beta^{a}_{ij}\beta^{b}_{ij}\cos\Big{(}\frac{2\pi}{N}n(i-j)\Big{)}\\\
&=2(N\delta_{N\equiv
n}\delta^{ab}-1)\cos\Big{(}\frac{2\pi}{N}n(a-b)\Big{)},\end{split}$ (77)

where

$\delta_{n\equiv k}=\begin{cases}1&n\equiv k,\\\ 0&n\not\equiv k.\end{cases}$
(78)

The relation "$\equiv$" is to be understood here as equality in the mod $N$
sense (we instead use "$:=$" to denote "is defined to be" for this
subsection).

The matrix in Equation (77) is diagonalised by the (trace-free) eigenvectors
$u_{\ell}$ with vector components:

$(u_{\ell})^{b}:=e^{i\frac{2\pi}{N}\ell b},\qquad 1\leq\ell\leq N-1,$ (79)

and have eigenvalues indexed by $\ell$:

$\begin{split}\sum_{b=1}^{N}C^{ab}_{n}(u_{\ell})^{b}&=N(2\delta_{n\equiv
N}-\delta_{n\equiv\ell}-\delta_{n\equiv N-\ell})(u_{\ell})^{a}.\end{split}$
(80)

Plugging Equation (80) into Equation (73), and recalling
$m_{W}=\frac{2\pi}{NL},$ we can read off the eigenvalues $\mathcal{R}_{0\ell}$
of $\mathcal{R}_{0}^{ab}$:

$\begin{split}\mathcal{R}_{0\ell}(mL)&=N\sum_{p=1}^{\infty}\Big{\\{}2K_{0}\Big{(}2\pi
p\frac{m}{m_{W}}\Big{)}-K_{0}\Big{[}2\pi\Big{(}p-\frac{\ell}{N}\Big{)}\frac{m}{m_{W}}\Big{]}\\\
&\qquad-
K_{0}\Big{[}2\pi\Big{(}p-1+\frac{\ell}{N}\Big{)}\frac{m}{m_{W}}\Big{]}\Big{\\}}.\end{split}$
(81)

When $m\gtrsim m_{W}$, this series is very well-approximated by the $p=1$
term. However, some extra work is needed to extract information about the
$m\ll m_{W}$ case. In Appendix A.2, we perform the sum over $p$ by taking the
Mellin transform and find (cf. Equation (110)):

$\mathcal{R}_{0\ell}(mL)=N\Big{[}\gamma+\log\frac{m}{m_{W}}\sin\pi\frac{\ell}{N}+W_{\ell}\Big{(}\frac{m}{m_{W}}\Big{)}\Big{]},$
(82a) where, as mentioned before, $W_{\ell}$ is an $O(1)$ function such that
$W_{\ell}(0)=0$, and has a power series expansion for $\tau\equiv(m/m_{W})<1$:
$W_{\ell}(\tau)=\sum^{\infty}_{n=1}\frac{(2n)!}{(n!)^{2}}\Big{(}\frac{i\tau}{2}\Big{)}^{2n}\Big{[}\zeta(2n+1)-Re\Big{(}\mathrm{Li}_{2n+1}e^{2\pi
i\frac{\ell}{N}}\Big{)}\Big{]}\,,\qquad\tau<1.$ (82b)

This is exactly Equation (15a), and Equation (15b) follows directly from
Equations (81) and (82a). Putting everything together, we finally obtain
Equation (13):

$\begin{split}\kappa_{\ell}=\frac{m_{W}^{-1}}{16\pi}\Bigg{[}\frac{8\pi^{2}}{Ng^{2}(m_{W}e^{-\gamma})}+b_{0}\log\frac{1}{\sin\pi\frac{\ell}{N}}+\frac{2}{3}\sum_{I}^{n_{f}}W_{\ell}\Big{(}\frac{m_{I}}{m_{W}}\Big{)}\Bigg{]}\,,\quad
1\leq\ell\leq N-1.\end{split}$ (83)

Note that although heretofore the fermion masses only appeared in the
combination $mL$, Equations (82a), (83) suggest that they are in fact more
naturally measured in units of $m_{W}$, as we should expect.

On the other hand, the $44$ parts of the integrals also give us $M_{ab}^{2}$,
the scalar (mass)2 matrix. Omitting the intermediate steps,

$\begin{split}M^{2}_{ab}&=g^{2}\sum_{\beta}\sum_{n=1}^{\infty}\frac{\beta^{a}\beta^{b}}{4\pi^{2}L^{2}}\Big{[}\sum_{I}^{n_{f}}(m_{I}L)^{2}K_{2}(nm_{I}L)-\frac{2}{n^{2}}\Big{]}\cos(n\beta\cdot\phi),\end{split}$
(84)

where $K_{2}$ is the modified Bessel function of order 2; this matches the
result from taking the second derivative of the GPY potential, (9), which
serves as a "sanity check" on our calculations. For completeness, we present
$M^{2}_{\ell}$, the physical scalar (masses)2:

$\begin{split}M^{2}_{\ell}&=g^{2}Nm_{W}^{2}\Bigg{[}\sum_{I}^{n_{f}}F_{\ell}\Big{(}\frac{m_{I}}{m_{W}}\Big{)}-F_{\ell}(0)\Bigg{]},\end{split}$
(85a) where
$\begin{split}F_{\ell}(\tau)\equiv\frac{\tau^{2}}{4\pi^{2}}\sum_{p=1}^{\infty}\Big{\\{}&K_{2}\Big{[}2\pi\Big{(}p-1+\frac{\ell}{N}\Big{)}\tau\Big{]}+K_{2}\Big{[}2\pi\Big{(}p-\frac{\ell}{N}\Big{)}\tau\Big{]}-2K_{2}(2\pi
p\tau)\Big{\\}}.\end{split}$ (85b)

We also present $\rho_{\ell}$, the eigenvalues of $\rho_{ab}$,

$\rho_{\ell}=\kappa_{\ell}+\frac{m_{W}^{-1}}{96\pi}\Bigg{[}1-\sum_{I}^{n_{f}}X_{\ell}\Big{(}\frac{m_{I}}{m_{W}}\Big{)}\Bigg{]},$
(86a) where $X_{\ell}$ is an $O(1)$ function defined in terms of $W_{\ell}$:
$X_{\ell}(\tau)=1+4\tau^{2}\frac{d}{d\tau^{2}}W_{\ell}(\tau).$ (86b)

Equations (85) and (86b) are only presented for completeness, although they
may be found without too much difficulty using the methods described in this
paper.

##  4 Future directions

In this study we have derived an explicit one-loop expression for the
eigenvalues of $\kappa_{ab}$, the polarisation operator of the $SU(N)$ dYM
theory with massive fermions, and provisionally surveyed some properties of
the emergent fourth dimension. It would be interesting to numerically examine
the effect of these one-loop corrections on the $k$-string tensions, (as was
done for SYM in Ref. doublestring ,) but to do so would require us to compute
the matrix determinants in the monopole measure, $\zeta$ — a daunting task
(see the discussion in footnote (11)).

Additionally, the topological angle $\theta$-dependence in Yang-Mills theory
has been the subject of much attention unsal12 ; anber13 ; ttt17 ; bonati18 ;
bonati19 : we should also like to examine the dependence of the $k$-string
tensions on the topological angle $\theta$ as well as on the circle length
$L$, at the tree-order level, to compare against results on the lattice.

Finally, we would also like to further study the confining properties of dYM
outside of the calculable regime, $NL\Lambda\gg 1$ and its conjectured
continuity with the small $NL\Lambda$ regime, on the lattice.

##  Appendix A The Mellin transform, and some results

In this Appendix, we explicitly evaluate the sums over $p$ in (81) to obtain
an expression for $\kappa_{\ell}$ in terms of analytic functions. To this end,
we introduce the Mellin transform, an integral transform on real-valued
functions.

######  Definition A.1.

The Mellin transform $\mathcal{M}$ is an integral transform defined on the
space of real integrable functions $f:\mathbb{R}^{+}\to\mathbb{R}$ as:

$\begin{split}\varphi(s)&\equiv\mathcal{M}_{s}\,[f(t)]\\\
&\equiv\int^{\infty}_{0}dt\,t^{s-1}f(t).\end{split}$ (87a) In particular, for
each $\lambda>0$, $\mathcal{M}_{s}[f(\lambda
t)]=\lambda^{-s}\mathcal{M}_{s}[f(t)].$ (87b) The inverse transform
$\mathcal{M}^{-1}$ is, formally,
$\begin{split}f(t)&=\mathcal{M}_{t}^{-1}\,[\varphi(s)]\\\ &=\frac{1}{2\pi
i}\int^{c+i\infty}_{c-i\infty}ds\,t^{-s}\varphi(s),\end{split}$ (87c)

where $c$ is some real number chosen so that the integral in (87c) converges
(NIST:DLMF, , Sec. 2.5). Usually what this means is to take the sum over the
residues of the poles of $\varphi(s)$ on the real half-line,
$s\in(-\infty,c]$. To illustrate with a simple example, let us compute the
Mellin transform of $f(t)=e^{-t}$, and its inverse:

######  Example A.1.

Directly from the definition,

$\begin{split}\mathcal{M}_{s}[e^{-t}]&\equiv\int^{\infty}_{0}dt\,t^{s-1}e^{-t}\\\
&=\Gamma(s).\end{split}$ (88)

Now consider the inverse transform. Since $\Gamma(s)$ has poles at
$s=0,-1,-2...$, we evaluate the integral by limiting the integration contour
$c\to 0^{+}$ and closing the contour over the $Re(s)<0$ half-plane. The
integral over the arc goes to zero at large radius, so

$\begin{split}\lim_{c\to 0^{+}}\frac{1}{2\pi
i}\int^{c+i\infty}_{c-i\infty}ds\,t^{-s}\Gamma(s)&=\sum_{n=0}^{\infty}\underset{s}{\text{res}}\Big{(}t^{-s}\Gamma(s),-n\Big{)}\\\
&\\!\\!\overset{\ref{gamma_series}}{=}\frac{(-t)^{n}}{n!}\\\
&=e^{-t},\end{split}$ (89)

as expected, because near the poles of $\Gamma(s)$,

$\Gamma(s)=\frac{(-1)^{n}}{n!}\Big{[}(s+n)^{-1}+\psi^{(0)}(1+n)\Big{]}+O(s+n)\,,\qquad
n=0,1,2...$ (90)

where $\psi^{(0)}(z)\equiv\frac{d}{dz}\log(\Gamma(z))$ is the polygamma
function (of order $0$).

###  A.1 Proof of Equation (74)

We are now prepared to prove Equation (74) and derive a series expression for
$\tilde{R}_{0}^{b}$. The idea is to perform the sums over $n$ in "Mellin
space," then transform back to "mass space" to obtain a series expansion in
$t$. Like in Example A.1, the inverse transform involves evaluating the
residue of a chain of poles on the real axis.

To begin, we observe the Mellin transform of the modified Bessel function of
order $\nu$, $K_{\nu}$, is known to be (weisstein, , 03.04.22.0004.01):

$\mathcal{M}_{s}[K_{\nu}(t)]=2^{s-2}\Gamma\Big{(}\frac{s+\nu}{2}\Big{)}\Gamma\Big{(}\frac{s-\nu}{2}\Big{)}.$
(91)

Plugging this into Equation (71),

$\begin{split}\mathcal{M}_{s}[R^{b}_{0}(t)]&=\sum_{n=1}^{\infty}\mathcal{M}_{s}[K_{0}\big{(}nt\big{)}]\cos(nb)\\\
&\overset{\ref{eq:mellin_dilation}}{=}2^{s-3}\Gamma\Big{(}\frac{s}{2}\Big{)}^{2}\sum_{n=1}^{\infty}n^{-s}(e^{inb}+e^{-inb})\\\
&\overset{\ref{polylog_def}}{=}2^{s-3}\Gamma\Big{(}\frac{s}{2}\Big{)}^{2}\Big{(}\textrm{Li}_{s}e^{ib}+\textrm{Li}_{s}e^{-ib}\Big{)},\end{split}$
(92)

where $\mathrm{Li}_{s}$ is the polylogarithm function of order $s$:

$\sum_{k=1}^{\infty}\frac{e^{ikb}}{k^{s}}=\textrm{Li}_{s}e^{ib}\,,\quad\text{$b$
real, $s>0$}.$ (93)

Changing back to the original variable $t$,

$\begin{split}R^{b}_{0}(t)&=\mathcal{M}_{t}^{-1}\mathcal{M}_{s}[R^{b}_{0}(t^{\prime})]\\\
&=\frac{1}{2\pi
i}\int^{+i\infty+c}_{-i\infty+c}\\!ds\,t^{-s}2^{s-3}\Gamma\Big{(}\frac{s}{2}\Big{)}^{2}\Big{(}\textrm{Li}_{s}e^{ib}+\textrm{Li}_{s}e^{-ib}\Big{)}.\\\
\end{split}$ (94)

Note that the integral in Equation (94) is over the order $s$ of the
polylogarithm, rather than its argument. As in Example (A.1), we can evaluate
this integral by letting the integration contour approach the imaginary axis
from the right, $c\to 0^{+}$, and close the contour over the half plane
$Re(s)\leq 0$. The polylogarithm terms are regular for all $s$ for real
$0<b<2\pi$, so we are left with the residues from the chain of poles at
$s=0,-2,-4...$ where the gamma function diverges.

Unfortunately, the poles of $\Gamma(s/2)^{2}$ are of order 2, so evaluating
the residues with the integrand of (94), as is, would involve the expression
$\frac{d}{ds}\mathrm{Li}_{s}e^{ib}$, which produces a result that is even more
opaque than our original expression.

However, a known identity (NIST:DLMF, , Eq. 25.13.3) relates the
polylogarithms to the Hurwitz zeta function, $\zeta$:

$i^{-s}\mathrm{Li}_{s}(e^{ib})+i^{s}\mathrm{Li}_{s}(e^{-ib})=\frac{(2\pi)^{s}}{\Gamma(s)}\zeta\Big{(}1-s,\frac{b}{2\pi}\Big{)}\,,\qquad
0<b<2\pi.$ (95)

Where $\zeta(z,x)$ satisfies:

$\zeta(z,x)=\sum_{n=1}^{\infty}(n+x)^{-z}\,,\quad\text{$Re(s)>1$ and $x\neq
0,1,2...$}$ (96)

We can sum the expression in Equation (95) with $b\to 2\pi-b$ and divide by
$(i^{s}+i^{-s})$ to rewrite the integrand of (94) as:

$\begin{split}t^{-s}\mathcal{M}_{s}[R^{b}_{0}(t^{\prime})]&\overset{\ref{R0_integral}}{=}2^{-3}\,\Gamma\Big{(}\frac{s}{2}\Big{)}^{2}\Big{(}\mathrm{Li}_{s}(e^{ib})+\mathrm{Li}_{s}(e^{-ib})\Big{)}\Big{(}\frac{t}{2}\Big{)}^{-s}\\\
&\overset{\ref{hurwitz}}{=}\frac{2^{-4}}{\,(1+i^{-2s})}\frac{\Gamma(\frac{s}{2})^{2}}{\Gamma(s)}\Big{(}\frac{it}{4\pi}\Big{)}^{-s}\Big{[}\,\zeta\Big{(}1-s,\frac{b}{2\pi}\Big{)}+\zeta\Big{(}1-s,1-\frac{b}{2\pi}\Big{)}\,\Big{]}\end{split}$
(97)

This is helpful because the factor of $\Gamma(s)^{-1}$ in Equation (95)
reduces the order of the poles by one, and the zetas in the parentheses in
Equation (97) are regular except at $s=0$, so the poles at $s=-2,-4,-6...$ are
simple.

#### The residue at $s=0$:

Near $s=0$, the zeta terms diverge like $1/s$:

$\zeta(1-s,\frac{b}{2\pi})+\zeta\Big{(}1-s,1-\frac{b}{2\pi}\Big{)}=-\frac{2}{s}-\psi^{(0)}\\!\Big{(}\frac{b}{2\pi}\Big{)}-\psi^{(0)}\\!\Big{(}1-\frac{b}{2\pi}\Big{)}+O(s).$
(98)

So we must also look at the series expansion of the regular terms in (97) near
$s=0$:

$(1+i^{-2s})^{-1}\Big{(}\frac{it}{4\pi}\Big{)}^{-s}=\frac{1}{2}-\frac{1}{2}\log\frac{t}{4\pi}s+O(s^{2}).$
(99)

Combining Equations (98), (99) and (90), we find our famous logarithmic term:

$\underset{s}{\mathrm{res}}(t^{-s}\mathcal{M}_{s}[R^{b}_{0}]\,,\,0)=\frac{1}{2}\log\frac{t}{4\pi}-\frac{1}{4}\Big{[}\,\psi^{(0)}\\!\Big{(}\frac{b}{2\pi}\Big{)}+\psi^{(0)}\\!\Big{(}1-\frac{b}{2\pi}\Big{)}\,\Big{]}.$
(100)

#### The residues at $s=-2,\,-4,\,-6...$:

Since the poles at $s=-2,-4,-6...$ can only contribute terms $\sim t^{2n}$ for
$n=1,2,3...$, we have proven our claim in (74), so we are actually done, but
since we have already done most of the work,

$\begin{split}\underset{s}{\mathrm{res}}(\,t^{-s}\mathcal{M}_{s}[R^{b}_{0}]\,,\,-2n\,)=&\frac{1}{4}\frac{(2n)!}{(n!)^{2}}\Big{(}\frac{it}{4\pi}\Big{)}^{2n}\Big{[}\,\zeta\Big{(}1+2n,\frac{b}{2\pi}\Big{)}+\zeta\Big{(}1+2n,1-\frac{b}{2\pi}\Big{)}\,\Big{]}\\\
\overset{\ref{polygamma}}{=}&\frac{-1}{4(n!)^{2}}\Big{(}\frac{it}{4\pi}\Big{)}^{2n}\Big{[}\,\psi^{(2n)}\\!\Big{(}\frac{b}{2\pi}\Big{)}+\psi^{(2n)}\\!\Big{(}1-\frac{b}{2\pi}\Big{)}\,\Big{]}\,,\quad
n=1,2,3...\end{split}$ (101)

where $\psi^{(2n)}$, the polygamma function of order $2n$, is related to
$\zeta$ by (NIST:DLMF, , 25.11.12)

$\psi^{(2n)}(z)=-(2n)!\zeta(2n+1,z)\,,\qquad n=1,2,3...$ (102)

Putting our results together,

$\displaystyle\tilde{R}^{b}_{0}(t)$
$\displaystyle=-\frac{1}{4}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n!)^{2}}\Big{(}\frac{t}{4\pi}\Big{)}^{2n}\Big{[}\,\psi^{(2n)}\\!\Big{(}\frac{b}{2\pi}\Big{)}+\psi^{(2n)}\\!\Big{(}1-\frac{b}{2\pi}\Big{)}\,\Big{]},$
(103)

Plugging this result into Equation (76) and taking the massless limit, $t=0$,
the correction to the photon coupling matches that of the SYM result derived
in Ref. symallgroups .

###  A.2 Derivation of Equation (82b)

Now, consider our expression for $\mathcal{R}_{0\ell}$, the eigenvalues of
$\mathcal{R}_{0}^{ab}$, Equation (81). Let us define

$\displaystyle\nu_{\ell}(\tau)$
$\displaystyle\equiv\sum_{p=1}^{\infty}K_{0}\Big{[}\Big{(}\frac{\ell}{N}+p-1\Big{)}\tau\Big{]}+K_{0}\Big{[}\Big{(}-\frac{\ell}{N}+p\Big{)}\tau\Big{]},$
(104a) $\displaystyle\xi(\tau)$
$\displaystyle\equiv\sum_{p=1}^{\infty}2K_{0}(p\tau),$ (104b)
$\displaystyle\tau$ $\displaystyle\equiv 2\pi\frac{m}{m_{W}}=NLm,$ (104c)

Starting with $\nu_{\ell}$: the intermediate steps are largely the same as in
the preceding subsection,

$\displaystyle\allowdisplaybreaks\nu_{\ell}(\tau)\equiv$
$\displaystyle\,\mathcal{M}_{t}^{-1}\mathcal{M}_{s}\Bigg{\\{}\sum_{p=1}^{\infty}K_{0}\Big{[}\tau\Big{(}p-1+\frac{\ell}{N}\Big{)}\Big{]}+K_{0}\Big{[}\tau\Big{(}p-\frac{\ell}{N}\Big{)}\Big{]}\Bigg{\\}}$
$\displaystyle\overset{\ref{eq:mellin_dilation}}{=}$
$\displaystyle\,\mathcal{M}_{t}^{-1}\Big{[}\,\tau^{-s}2^{s-2}\Gamma\Big{(}\frac{s}{2}\Big{)}^{2}\sum_{p=1}^{\infty}\Big{[}\Big{(}p-1+\frac{\ell}{N}\Big{)}^{-s}+\Big{(}p-\frac{\ell}{N}\Big{)}^{-s}\Big{]}$
$\displaystyle\overset{\ref{hurwitz_def}}{=}$
$\displaystyle\,\sum_{\text{poles}}\underset{s}{\text{res}}\Big{[}\,\tau^{-s}2^{s-2}\Gamma\Big{(}\frac{s}{2}\Big{)}^{2}\Big{[}\zeta\Big{(}s,1-\frac{\ell}{N}\Big{)}+\zeta\Big{(}s,\frac{\ell}{N}\Big{)}\Big{]}\,,\,\\{\text{pole}\\}\,\Big{]}$
$\displaystyle=$
$\displaystyle\,\frac{\pi}{\tau}+\sum_{n=0}^{\infty}\frac{1}{(n!)^{2}}\Big{(}\frac{\tau}{2}\Big{)}^{2n}\Big{[}\zeta^{\prime}\Big{(}-2n,1-\frac{\ell}{N}\Big{)}+\zeta^{\prime}\Big{(}-2n,\frac{\ell}{N}\Big{)}$
$\displaystyle\qquad\qquad+\Big{[}\cancelto{0}{\zeta\Big{(}-2n,1-\frac{\ell}{N}\Big{)}+\zeta\Big{(}-2n,\frac{\ell}{N}\Big{)}}\Big{]}\Big{(}\psi(n+1)-\log\frac{\tau}{2}\Big{)}\Big{]}$
$\displaystyle\overset{\ref{bernoulli}}{=}$
$\displaystyle\,\frac{\pi}{\tau}+\sum_{n=0}^{\infty}\frac{1}{(n!)^{2}}\Big{(}\frac{\tau}{2}\Big{)}^{2n}\Big{[}\zeta^{\prime}\Big{(}-2n,1-\frac{\ell}{N}\Big{)}+\zeta^{\prime}\Big{(}-2n,\frac{\ell}{N}\Big{)}\Big{]}$
$\displaystyle\overset{\ref{nice_identity}}{=}$
$\displaystyle\,\frac{\pi}{\tau}+\frac{1}{2}\sum_{n=0}^{\infty}\frac{(2n)!}{(n!)^{2}}\Big{(}\frac{i\tau}{4\pi}\Big{)}^{2n}\Big{(}\mathrm{Li}_{2n+1}e^{2\pi
i\frac{\ell}{N}}+\mathrm{Li}_{2n+1}e^{-2\pi i\frac{\ell}{N}}\Big{)},$

The expression in the third line has order-$2$ poles at $s=0,-2,-4...$ (from
the gammas,) and a simple pole at $s=1$, (from the zetas,) and
$\zeta^{\prime}(s,x)\equiv\frac{d}{ds}\zeta(s,x)$.

The striked out term in the fourth line vanishes because (NIST:DLMF, ,
25.11.14)

$\zeta(-2n,x)=-\frac{B_{2n+1}(x)}{2n+1}=\frac{B_{2n+1}(-x)}{2n+1}\,,\qquad
x\,\,\text{real}\,,\,n=0,1,2\,...\\\ $ (105)

where $B_{k}(x)$ is the Bernoulli polynomial of order $k$, which has parity
$(-1)^{k}$ under $x\to 1-x$.

Lastly, the final line follows from (adamchik97, , Eq. 13):

$\begin{split}\zeta^{\prime}(-2n,1-x)+\zeta^{\prime}(-2n,x)&=\frac{(2n)!}{(2\pi
i)^{2n}}\Big{(}\mathrm{Li}_{2n+1}e^{2\pi ix}+\mathrm{Li}_{2n+1}e^{-2\pi
ix}\Big{)},\\\ &\qquad\qquad\qquad\qquad\qquad
x\,\,\text{real}\,,\,n=0,1,2\,...\\\ \end{split}$ (106)

and since $\mathrm{Li}_{1}(z)=-\log(1-z)$,

$\begin{split}\nu_{\ell}(\tau)=\frac{\pi}{\tau}&+\log\frac{1}{2\sin\pi\frac{\ell}{N}}\\\
&+\frac{1}{2}\sum_{n=1}^{\infty}\frac{(2n)!}{(n!)^{2}}\Big{(}\frac{i\tau}{4\pi}\Big{)}^{2n}\Big{(}\mathrm{Li}_{2n+1}e^{2\pi
i\frac{\ell}{N}}+\mathrm{Li}_{2n+1}e^{-2\pi
i\frac{\ell}{N}}\Big{)}.\end{split}$ (107)

To solve for $\xi$, we observe that, for the plain (Riemann) zetas
$\zeta(z,0)\equiv\zeta(z)$ (weisstein, , 10.01.20.0022.01),

$\zeta^{\prime}(-2n)=\frac{(2n)!}{(2\pi i)^{n}}\zeta(2n+1),\\\ $ (108)

and the sum over $p$ goes through almost verbatim. The result is:

$\begin{split}\xi(\tau)&=\,\sum_{\text{poles}}\underset{s}{\text{res}}\Big{[}\,\tau^{-s}2^{s-2}\Gamma\Big{(}\frac{s}{2}\Big{)}^{2}\zeta(s)\,,\,\\{\text{pole}\\}\,\Big{]}\\\
&=\frac{\pi}{\tau}+\gamma+\log\frac{\tau}{4\pi}+\sum^{\infty}_{n=1}\Big{(}\frac{i\tau}{4\pi}\Big{)}^{2n}\frac{(2n)!}{(n!)^{2}}\zeta(2n+1).\end{split}$
(109)

As before, the poles are located at $s=0,-2,-4....$ and $s=1$. So together,

$\begin{split}N^{-1}\mathcal{R}_{0\ell}&=\xi-\nu_{\ell}\\\
&=\gamma+\log\Big{(}\sin\pi\frac{\ell}{N}\tau\Big{)}+W_{\ell}(\tau),\end{split}$
(110)

where $W_{\ell}$ has a power series expansion in $\tau$:

$W_{\ell}(\tau)\equiv\sum^{\infty}_{n=1}\frac{(2n)!}{(n!)^{2}}\Big{(}\frac{i\tau}{2}\Big{)}^{2n}\Big{[}\zeta(2n+1)-\frac{1}{2}\Big{(}\mathrm{Li}_{2n+1}e^{2\pi
i\frac{\ell}{N}}+\mathrm{Li}_{2n+1}e^{-2\pi i\frac{\ell}{N}}\Big{)}\Big{]}.$
(111a) As mentioned in Footnote 7, the root test shows that this infinite
series diverges for $m\geq m_{W}$. In that case, $W_{\ell}$ can be
approximated by
$\begin{split}W_{\ell}(\tau)&=\gamma+\log\sin\pi\frac{\ell}{N}\tau+2K_{0}(2\pi\tau)\\\
&\quad-
K_{0}\Big{[}2\pi\Big{(}1-\frac{\ell}{N}\Big{)}\tau\Big{]}-K_{0}\Big{(}2\pi\frac{\ell}{N}\tau\Big{)}+O(e^{-2\pi\tau}).\end{split}$
(111b)

Manipulating the series expansions for $\zeta(2n+1)$ and $\mathrm{Li}_{2n+1}$,
Equations (93) and (96), we can also write $W_{\ell}(\tau)$ purely in terms of
elementary functions:

$W_{\ell}(\tau)=2\sum_{k=1}^{\infty}\Big{[}\Big{(}k^{2}+\tau^{2}\Big{)}^{-1/2}-k^{-1}\Big{]}\sin^{2}\Big{(}\pi\frac{\ell}{N}k\Big{)}.$
(111c)

We note that Equation (111c) converges much more slowly than the previous
ones, but on the other hand, it clearly shows that $W_{\ell}$ is strictly
negative, and goes to zero as $\frac{\ell}{N}\to 0$. Observing that

$\frac{d^{2}}{dx^{2}}\mathrm{Li}_{2n+1}(e^{ix})=-\mathrm{Li}_{2n-1}(e^{ix}),$
(112)

a similar argument shows that $W_{\ell}$ is strictly concave up in $\ell$ for
all $1\leq\ell\leq\lfloor N/2\rfloor$.

Finally, Equation (81) shows that $\mathcal{R}_{0\ell}\to 0$ rapidly as
$\tau\to\infty$, so (110) demands

$W_{\ell}(\tau)\to\log\frac{e^{-\gamma}}{\tau\sin\pi\frac{\ell}{N}}\,,\quad\tau\to\infty,$
(113)

which concludes the proofs for the statements we made about $W_{\ell}$ in
Equation (14).

## Acknowledgements

The author thanks E. Poppitz for many helpful discussions throughout the
production of this work. This work is funded by an NSERC Discovery Grant.

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