#  UV completion of extradimensional Yang-Mills theory for Gauge-Higgs
unification

Álvaro Pastor-Gutiérrez<EMAIL_ADDRESS>Max-Planck-Institut für
Kernphysik P.O. Box 103980, D 69029, Heidelberg, Germany  Institut für
Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120
Heidelberg, Germany  Masatoshi Yamada<EMAIL_ADDRESS>Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16,
69120 Heidelberg, Germany

###### Abstract

The $SU(N)$ Yang-Mills theory in $\mathbb{R}^{4}\times S^{1}$ spacetime is
studied as a simple toy model of Gauge-Higgs unification. The theory is
perturbatively nonrenormalizable but could be formulated as an asymptotically
safe theory, namely a nonperturbatively renormalizable theory. We study the
fixed point structure of the Yang-Mills theory in $\mathbb{R}^{4}\times S^{1}$
by using the functional renormalization group in the background field
approximation. We derive the functional flow equations for the gauge coupling
and the background gauge-field potential. There exists a nontrivial fixed
point for both couplings at finite compactification radii. At the fixed point,
gauge coupling and vacuum energy are both relevant. The renormalization group
flow of the gauge coupling describes the smooth transition between the
ultraviolet asymptotically safe regime and the strong interacting infrared
limit.

##  I Introduction

Understanding the dynamics of gauge theories is a challenging problem in
theoretical physics. In particular, elucidating their phase structure is a big
issue. Gauge theories in four dimensional spacetime ($D=4$), including quantum
chromodynamics (QCD) and similar theories, have been both intensively and
extensively studied by various methods, such as Monte-Carlo simulations based
on lattice gauge theory and functional methods.

On the other hand, gauge theories in higher dimensional spacetime have not
been studied as well as those in $D=4$. One of typical reasons is that the
universe is well-described by four dimensional theories at least up to TeV
scales, and there is no experimental evidence for the existence of extra
dimensions so far. In addition, gauge theories in $D=4$ are perturbatively
renormalizable and asymptotically free. In terms of the renormalization group,
this means that the gauge coupling is relevant at the Gaussian (trivial) fixed
point. This fact is crucial for taking a continuum limit and removing lattice
artifacts in lattice gauge field theories. In contrast, gauge theories in
higher dimensions, e.g. in $D=5$, are perturbatively nonrenormalizable since
the mass dimension of the gauge coupling is negative and the viability of a
perturbative expansion over small coupling is lost. Consequently, those are
generally regarded as effective field theories with a certain ultraviolet (UV)
cutoff and hence predictable only at sufficiently lower energy than the UV
cutoff. Nevertheless, gauge theories in higher dimensional spacetime are
attractive as both theoretical and phenomenological models in quantum field
theory.

From the theoretical side, the notion of renormalizability in quantum field
theory can be generalized to nonperturbative theories. In general, most
theories in higher dimensional spacetime are _perturbatively
nonrenormalizable_ , while gauge theories $D>4$ could be _nonperturbatively
renormalizable_. Namely, there could exist a nontrivial fixed point providing
a UV-completeness of the theory. This is essential for the asymptotically safe
scenario Weinberg (1979) (see also Ref. Nagy (2014) as a review) where a non-
trivial fixed point exists in the theory. At such a fixed point, one can
classify couplings by looking at their energy scalings characterized by the
critical exponents. A coupling with a positive critical exponent is amplified
in the infrared (IR) limit and is called “relevant”. An “irrelevant” coupling
has a negative critical exponent and converges to the fixed point value in the
IR limit. An important fact is that relevant couplings are free parameters,
while irrelevant ones are predicted parameters. In general, the number of
relevant couplings is finite and thus the theory is predictive for the low
energy dynamics. Consequently, the theory becomes nonperturbatively
renormalizable in the sense that a nontrivial fixed point is generally found
by nonperturbative methods.

The existence of a nontrivial UV fixed point in high dimensional gauge
theories has been discussed by means of perturbative methods in Refs. Peskin
(1980); Morris (2005); Irges and Koutroulis (2018, 2021); De Cesare  _et al._
(2021):111 See also Refs. Antoniadis (1990); Dienes  _et al._ (1998, 1999);
Kobayashi  _et al._ (1999); Kubo  _et al._ (2000); Gies (2003) for the
renormalization group analysis on higher dimensional gauge theories.  The
$\epsilon$ expansion was applied for a $SU(N)$ Yang-Mills theory in
$4+\epsilon$ dimensional spacetime in order to derive the beta function for
the (squared) gauge coupling at the two-loop level as

$\displaystyle\beta(\tilde{g})=\epsilon\tilde{g}^{2}-\frac{22N}{3}\frac{\tilde{g}^{4}}{16\pi^{2}}-\frac{68N^{2}}{3}\frac{\tilde{g}^{6}}{(16\pi^{2})^{2}}\,,$
(1)

where $\tilde{g}^{2}=\mu^{\epsilon}g^{2}$ is the dimensionless gauge coupling
and $\mu$ is the renormalization scale. In addition to the Gaussian (trivial)
fixed point $\tilde{g}_{*}^{2}=0$, characterizing the perturbative theory, the
beta function (1) admits a nontrivial UV fixed point,

$\displaystyle\tilde{g}_{*}^{2}=\frac{16\pi^{2}}{N}\left(\frac{3}{22}\epsilon-\frac{153}{2662}\epsilon^{2}+\cdots\right).$
(2)

The critical exponent at this fixed point is found to be

$\displaystyle\nu^{-1}=-\frac{d\beta(\tilde{g})}{d\tilde{g}^{2}}\bigg{|}_{\tilde{g}^{2}=\tilde{g}_{*}^{2}}=\epsilon+\frac{51}{121}\epsilon^{2}+\cdots\,.$
(3)

For $\epsilon>0$, we observe $\nu>0$, so that the gauge coupling $\tilde{g}$
becomes a relevant coupling. In the limit $\epsilon\to 0$, corresponding to
$D=4$, the nontrivial fixed point (2) merges to the trivial fixed point at
which the gauge coupling becomes a marginally relevant coupling to be
asymptotically free.

However, the $\epsilon$ expansion is a kind of asymptotic expansion where
results for $\epsilon\gtrsim 1$ break down and thus may not be reliable.
Hence, in order to establish the asymptotic safety scenario for gauge theories
in higher dimensional spacetimes $D>4$, one requires methods which rely
neither on the expansion of spacetimes nor couplings.

On the phenomenological side, higher dimensional gauge theories are the
foundation for gauge-Higgs unification models Fairlie (1979); Manton (1979);
Forgacs and Manton (1980); Hosotani (1983a, b, 1989); Davies and McLachlan
(1988, 1989); Antoniadis and Benakli (1994); Antoniadis  _et al._ (2001) which
is one of the attractive scenarios for an extension of the Standard Model. The
simplest toy model may be a $SU(N)$ Yang-Mills theory in five dimensional
spacetime. The extra dimensional component of the gauge field is identified
with a scalar field, concretely the Higgs boson. For understanding the
dynamics of Gauge-Higgs unification models, the elucidation of the phase
structure of gauge theories in higher dimensional spacetime is crucial
especially when the extra dimensional direction is compactified to be a circle
$S^{1}$ and an extra gauge field has a nontrivial background configuration. In
this case, the gauge symmetry is broken into a subgroup. This noteworthy
phenomenon in gauge theories is associated to the Aharonov-Bohm (AB) effect in
quantum mechanics or its equivalent, the Hosotani mechanism, in non-Abelian
gauge theories Hosotani (1983a, b, 1989). Moreover, depending on fermionc
degrees of freedom and their representations of the gauge group, the gauge
theory tends to have a rich phase structure. See e.g. Hosotani (1989); Kashiwa
and Misumi (2013); Cossu  _et al._ (2014).

As aforementioned, due to the perturbative nonrenormalizablity of higher
dimensional gauge theories, physical quantities in the low energy scale such
as the vacuum expectation value and the mass of the the Higgs boson are
generally UV cutoff dependent. Nevertheless, it is conjectured that the
potential determining the vacuum configuration of the extra background gauge
field may be free from cutoff dependencies Hosotani (2005, 2006). It is indeed
shown in Maru and Yamashita (2006); Hosotani  _et al._ (2007); Hisano  _et
al._ (2020) that this fact holds at least up to the two-loop level. However,
it has been argued recently, in Ref. Yamada (2021), that the potential is UV
sensitive at the four-loop level. See also Ref. Freitas and Wiegand (2018) for
the study on the UV sensitivity in a universal extra dimension model. This
means that the vacuum structure is sensitive to higher dimensional operators
in perturbative higher loop levels and thus the Gauge-Higgs unification theory
fails to predict the low energy physics. Therefore, the UV completion of
higher dimensional gauge theories is necessary.

In this paper, we study the $SU(N)$ Yang-Mills theory in $\mathbb{R}^{4}\times
S^{1}$ spacetime and its UV completeness in terms of asymptotic safety towards
UV complete models of Gauge-Higgs unification. To study the fixed point
structure of the $SU(3)$ Yang-Mills theory in $\mathbb{R}^{4}\times S^{1}$
spacetime, we utilize the functional renormalization group Wetterich (1993);
Morris (1994); Reuter and Wetterich (1994a); Ellwanger (1994). See also
reviews Morris (1998); Berges  _et al._ (2002); Aoki (2000); Bagnuls and
Bervillier (2001); Polonyi (2003); Pawlowski (2007); Gies (2012); Delamotte
(2012); Rosten (2012); Kopietz  _et al._ (2010); Braun (2012); Dupuis  _et
al._ (2021). We derive the (nonperturbative) beta functions for the potential
for the background field of an extra gauge field in terms of the AB phases,
and for the gauge coupling. It will turn out that the gauge coupling has a
nontrivial UV fixed point depending on the compactification radius and then
converges to a finite value in the $k\to\infty$ limit, presenting an
asymptotically safe theory. Besides, the vacuum energy (the potential
independent of AB phases) has a nontrivial UV fixed point as well, so that the
potential is free from the UV divergence. These couplings are relevant at the
nontrivial UV fixed point, i.e. free parameters in the system.222 Strictly
speaking, the vacuum energy does not affect the dynamics of the gauge fields.
Solving the flow equation for the gauge coupling, it will be clearly seen that
the gauge coupling smoothly switches from an asymptotically free behavior to
an asymptotically safe one at the energy scale of order of the inverse
compactification radius. Then, it converges to a finite value in the UV
regime, while in the low energy regime, the gauge coupling becomes a strong
coupling which would induce the phase transition to the confinement phase in
the $SU(3)$ gauge theory.

However, we cannot address the confinement problem within the current setup.
Instead, we demonstrate a mechanism of the confinement by making a simple
modification to the beta function of the potential for AB phases. More
specifically, we take a gap mass of the gauge fields into account at a certain
energy (confinement) scale by hand in order to suppress loop effects of the
gauge fields. We will see that the contributions from the ghost fields
dominate below the confinement scale and yield nontrivial vacua corresponding
to the confinement phase.

This paper is organized as follows: In Section II we set up the $SU(N)$ Yang-
Mills theory in $\mathbb{R}^{4}\times S^{1}$ spacetime and briefly review the
Hosotani mechanism. We introduce the functional renormalization group and
explain the idea of asymptotic safety in Section III. We derive the flow
equation for the potential of AB phases in Section IV. The flow equation of
the gauge coupling is derived in Section V. We analyze these flow equations in
Section VI. We discuss in Section VII how a confinement mechanism could be
realized by a simple extension. Section VIII is devoted to summarizing this
work and to discussing prospects. In Appendix A, the heat kernel method, used
to derive the flow equation for the gauge coupling, is explained. Lastly, in
Appendix B, the detail of the derivation of the flow equation for the gauge
coupling is shown.

##  II Yang-Mills theory on $\mathbb{R}^{4}\times S^{1}$ spacetime

This section is devoted to a review on the $SU(N)$ Yang-Mills theory in
compactified spacetime. We start by defining a Yang-Mills theory in five
dimensional spacetime together with a summary of our conventions. Then, we
turn to the theory on the compactified spacetime denoted by
$\mathbb{R}^{4}\times S^{1}$. For understanding the quantum dynamics of the
Yang-Mills theory on $\mathbb{R}^{4}\times S^{1}$, we introduce the Wilson
line along with $S^{1}$ and the Polyakov loop which is defined as the Wilson
line traced in color space. The former is a key quantity especially for
spontaneous symmetry breaking of gauge symmetry, while the latter is for
confinement.

###  II.1 Action

We consider a pure $SU(N)$ Yang-Mills gauge theory in 5 dimensional Euclidean
spacetime, where a flat spacetime metric is given by
$\eta_{MN}=\text{diag}(1,1,1,1,1)$.

Here and hereafter, we denote 5 dimensional indices by capital letters e.g.
$M,\,N,\,\cdots=0,\ldots,3,\,5$, while greek letters are used to stand for 4
dimensional indices, i.e. $\mu,\,\nu,\cdots=0,\ldots,3$. The Yang-Mills gauge
field in 5 dimensional spacetime is then written as
$A^{a}_{M}=(A^{a}_{\mu},A^{a}_{5})$, where $a,\,b,\cdots$, denote indices for
the adjoint representation of $SU(N)$ whose generators are denoted by $T^{a}$
and are normalized as ${\rm tr}(T^{a}T^{b})=\frac{1}{2}\delta^{ab}$.

The classical action for the Yang-Mills theory in $d=5$ reads

$\displaystyle S_{\text{YM}}$
$\displaystyle=\frac{1}{2g^{2}}\int\text{d}^{5}x\,{\rm
tr}\left[F_{MN}F^{MN}\right]\,.$ (4)

Here, the field strength of $A_{M}^{a}$ and the covariant derivative are given
respectively by

$\displaystyle F_{MN}^{a}$
$\displaystyle=\partial_{M}A_{N}^{a}-\partial_{N}A_{M}^{a}+f^{abc}A_{M}^{b}A_{N}^{c}\,,$
(5) $\displaystyle D_{M}^{ab}$
$\displaystyle=\delta^{ab}\partial_{M}-f^{abc}A_{M}^{c}\,,$ (6)

with $f^{abc}$ the structure constants associated to $SU(N)$ and $g$ the gauge
coupling constant. Note that in 5 dimensional spacetime, the squared gauge
coupling $g^{2}$ has mass dimension $-1$.

###  II.2 Compactification and Kaluza-Klein modes

In five dimensional Euclidean spacetime $\mathbb{R}^{5}$, spacetime
coordinates $x_{M}$ are noncompact, i.e. $-\infty\leq x_{M}\leq\infty$ for
$M=0,\ldots,3,\,5$. The gauge transformation for $A_{M}=A_{M}^{a}T^{a}$ is
given by

$\displaystyle A_{M}\to
A_{M}^{\prime}=UA_{M}U^{-1}+\frac{i}{g}U\partial_{M}U^{-1}\,,$ (7)

where $U=\exp(i\alpha^{a}(x)T^{a})$ are elements of $SU(N)$.

We now define a system in which the $x_{5}$-direction is compactified so as to
be $|x_{5}|\leq R$, where $R$ is a compactification radius. For finite $R$,
the system is in $\mathbb{R}^{4}\times S^{1}$ spacetime. Thus, $R$ is a
parameter continuously connecting between $\mathbb{R}^{5}$ ($R\to\infty$) and
$\mathbb{R}^{4}$ ($R\to 0$) spacetimes.

We can generally consider the following boundary condition for the gauge field
on $\mathbb{R}^{4}\times S^{1}$:

$\displaystyle A_{M}(x,x_{5}+2\pi R)=VA_{M}(x,x_{5})V^{-1}\,,$ (8)

where $V\in SU(N)$. This boundary condition guarantees the Lagrangian on
$S^{1}$ to be single-valued. The boundary condition for the gauge field
transformed by Eq. (7) reads

$\displaystyle A^{\prime}_{M}(x,x_{5}+2\pi
R)=V^{\prime}A^{\prime}_{M}(x,x_{5})V^{\prime}{}^{-1}+iV^{\prime}\partial_{M}V^{\prime}{}^{-1}\,,$
(9)

with

$\displaystyle V^{\prime}=U(x,x_{5}+2\pi R)VU(x,x_{5})^{-1}\,.$ (10)

In particular, for the gauge transformation by gauge group elements
$U_{\text{res}}(x,x_{5})\in U(x,x_{5})$ satisfying $V^{\prime}=V$ in Eq. (10),
namely

$\displaystyle V=U_{\text{res}}(x,x_{5}+2\pi
R)VU_{\text{res}}(x,x_{5})^{-1}\,,$ (11)

the system on $\mathbb{R}^{4}\times S^{1}$ becomes gauge invariant. In this
sense, such a gauge transformation given by $U_{\text{res}}(x,x_{5})$
characterizes the “residual” gauge invariance of the system. The choice
$V=1_{3}$ (where $1_{3}$ is the $3\times 3$ unity matrix) does not spoil
physical consequences at the quantum level Hosotani (1989), so that hereafter
we employ this choice. Then, the gauge field satisfies the periodic condition
as can be seen from Eq. (8).

From Eq. (11), one finds the residual gauge transformation to be

$\displaystyle U_{\text{res}}(x,x_{5})={\rm
diag}\,\left(e^{in_{1}x_{5}/R},\,\cdots,\,e^{in_{N}x_{5}/R}\right)\,,$ (12)

where $\sum_{j}^{N}n_{j}=0$. This nonperiodic gauge transformation associates
to $Z_{N}$ center symmetry involved in $SU(N)$.

For the gauge field $A_{M}(x,x_{5})$ on $\mathbb{R}^{4}\times S^{1}$
spacetime, one expands the compactified direction as

$\displaystyle A_{M}(x,x_{5})=\frac{1}{\sqrt{2\pi
R}}\sum_{n=-\infty}^{\infty}A_{\mu}^{(n)}(x)e^{i\Omega_{n}x_{5}}\,,$ (13)

where the direction of the extra dimension in momentum space, $p_{5}$, is
given as discretized modes $\Omega_{n}=n/R$. The summation in Eq. (13)
corresponds to the Kaluza-Klein (KK) expansion, representing $n$ the different
KK modes. Further, performing the Fourier transformation for $A_{M}^{(n)}(x)$
in order to move into the 4 dimensional momentum space $p_{\mu}$, namely

$\displaystyle
A_{M}^{(n)}(x)=\int\frac{\text{d}^{4}p}{(2\pi)^{4}}A_{M}^{(n)}(p)e^{ip\cdot
x}\,,$ (14)

one has

$\displaystyle A_{M}(x,x_{5})=\frac{1}{\sqrt{2\pi
R}}\sum_{n=-\infty}^{\infty}\int\frac{\text{d}^{4}p}{(2\pi)^{4}}A_{M}^{(n)}(p)e^{ip\cdot
x+i\Omega_{n}x_{5}}\,.$ (15)

For $R\to 0$, the KK frequencies $\Omega_{n}$ except for $n=0$ become
infinite, so that only lowest mode $n=0$ contributes to the dynamics. Hence,
only $A_{M}^{(0)}$ are effective degrees of freedom at $R=0$. On the other
hand, the limit $R\to\infty$ yields $\Omega_{n}\to 0$ for all (finite) KK
modes and recovers the continuous momentum $p_{5}$. Only by performing the
full summation all KK modes $A_{M}^{(n)}$ are taken into account. This fact
will be developed in Section VI.

###  II.3 Wilson line and Aharonov-Bohm phase

We introduce the Wilson line which is a key quantity for understanding the
gauge dynamics in Yang-Mills theory on $\mathbb{R}^{4}\times S^{1}$. The
Wilson line along $S^{1}$ is defined as

$\displaystyle W(x)=\mathcal{P}\exp\left\\{-i\int_{0}^{2\pi
R}\text{d}x_{5}\,A_{5}(x,x_{5})\right\\}\,,$ (16)

where $\mathcal{P}$ is the path ordering and $A_{5}=A_{5}^{a}T^{a}$. The
residual gauge transformation (12) gives

$\displaystyle W(x)$ $\displaystyle\to
U_{\text{res}}(x,0)W(x)U_{\text{res}}(x,2\pi R)^{-1}$
$\displaystyle=U_{\text{res}}(x,0)W(x)U_{\text{res}}(x,0)^{-1}\,.$ (17)

Hence, the eigenvalues of $W$ are gauge invariant quantities. We here denote a
set of their eigenvalues by $\theta_{H}\equiv\\{\theta_{i}\\}$
($i=1,\ldots,N$). From the fact that the determinant of gauge group elements
is unity, the phases $\theta_{i}$ have to satisfy $\sum_{i=1}^{N}\theta_{i}=0$
(mod $2\pi$). These phases correspond to AB phases in $SU(N)$ gauge theory on
$\mathbb{R}^{4}\times S^{1}$. In the next subsection, we see that the AB
phases play an important role for the breaking of gauge symmetry.

Another important quantity is the Polyakov loop (in the fundamental
representation) which is defined by

$\displaystyle P=\frac{1}{N}{\rm tr}\,W\,,$ (18)

where the trace acts on the fundamental representation space of $SU(N)$. This
is transformed under the residual gauge transformation (12) as $P\to
P^{\prime}=e^{2\pi in_{i}/(NR)}P$. The Polyakov loop is a gauge invariant
object, so that it plays a role of an exact order parameter for spontaneous
symmetry breaking of the $Z_{N}$ center symmetry in pure Yang-Mills theories.
Moreover, the expectation value of $P$ is written as $\langle
P\rangle=\exp(-RF_{q})$ where $F_{q}$ is the free energy of a static massive
quark (spectator) at a spacial position. In the confinement phase, $F_{q}$ has
to be infinite and thus we observe $\langle P\rangle=0$. On the other hand, a
finite value of $F_{q}$ entails $\langle P\rangle\neq 0$ in the deconfinement
phase. Hence, the spontaneous symmetry breaking of the $Z_{N}$ center symmetry
corresponds to deconfinement transition.

###  II.4 Hosotani mechanism

Here, we discuss a consequence of a certain nontrivial configuration of
$\theta_{i}$. In this subsection, we see that the phase configurations being
$\theta_{i}\neq\theta_{j}$ could break the $SU(N)$ gauge symmetry. This
mechanism is known as the Hosotani mechanism Hosotani (1983a, b, 1989), see
also Refs. Davies and McLachlan (1988, 1989).

We start by assuming that a vacuum state of the system takes $\langle
A_{\mu}\rangle=0$, $\langle A_{5}\rangle\neq 0$. To evaluate quantum
fluctuations around such a vacuum, it is convenient to expand the gauge field
into the background $\bar{A}_{M}$ and the fluctuation field $a_{M}$, i.e.

$\displaystyle A_{M}=A_{M}^{a}T^{a}=\bar{A}_{M}+a_{M}\,.$ (19)

We assume that the background field takes the following form:

$\displaystyle\bar{A}_{\mu}=0\qquad(M=\mu=0,\ldots,3)\,,$
$\displaystyle\bar{A}_{5}=\frac{1}{2\pi R}\begin{pmatrix}\theta_{1}&&\\\
&\ddots&\\\
&&\theta_{N}\end{pmatrix}\,,\quad\sum_{i=1}^{N}\theta_{i}=0\quad(M=5)\,.$ (20)

For constant $\theta_{i}$, the field strength of $\bar{A}_{M}$ vanishes, i.e.
$\bar{F}_{MN}=0$ and then one has

$\displaystyle\frac{1}{2}{\rm tr}\,\bar{F}_{MN}\bar{F}^{MN}=0\,.$ (21)

Consider here the Wilson line for the background (20),

$\displaystyle\bar{W}=\mathcal{P}\exp\left\\{-i\int_{0}^{2\pi
R}\text{d}x_{5}\,\bar{A}_{5}(x,x_{5})\right\\}\,.$ (22)

For a certain configuration of the AB phases, we observe $\bar{W}\neq 1_{3}$
or correspondingly nontrivial configurations of AB phases. Hence, the $SU(N)$
symmetry is broken into a subgroup $\mathcal{H}_{\text{sym}}$ with generators
satisfying $[T^{a},\bar{W}]=0$, while some generators satisfying
$[T^{a},\bar{W}]\neq 0$ represent the broken symmetry.

As an example, let us consider a $SU(3)$ gauge theory. There are three
constant angles [$\theta_{1}$, $\theta_{2}$, $\theta_{3}$ (with
$\sum_{i=1}^{3}\theta_{i}=0$ mod $2\pi$)] and then the Wilson line and the
Polyakov loop read

$\displaystyle\bar{W}$
$\displaystyle=\text{diag}\,\left(e^{i\theta_{1}},e^{i\theta_{2}},e^{i\theta_{3}}\right)\,,$
(23) $\displaystyle P$
$\displaystyle=\frac{1}{3}(e^{i\theta_{1}}+e^{i\theta_{2}}+e^{i\theta_{3}})\,.$
(24)

One can in general write down three different configurations of AB phases: (i)
$\theta_{1}=\theta_{2}=\theta_{3}=0,\,\pm\frac{2}{3}\pi$, for such
configurations, $W$ is proportional to $I$, so that for all Gell-Mann matrices
$T^{a}$, one has $[T^{a},\bar{W}]=0$, i.e. $SU(3)$ symmetry is not broken.
(ii) $\theta_{1}=\theta_{2}=\alpha$ and $\theta_{3}=-2\alpha$ ($\alpha\neq
0$). In this case, we observe that $[T^{a},\bar{W}]=0$ only for
$a=1,\,2,\,3,\,8$ which indicates $SU(3)\to SU(2)\times U(1)$. (iii)
$\theta_{1}=\beta_{1}$, $\theta_{2}=\beta_{2}$ and
$\theta_{3}=-\beta_{1}-\beta_{2}$ ($\beta_{1}\neq\beta_{2}$). This
configuration shows that only $T^{3}$ and $T^{8}$ commute with $\bar{W}$,
namely $SU(3)\to U(1)\times U(1)$. These possible symmetry breaking patterns
are summarized in Table 1.

Studies of the Hosotani mechanism have been done by means of lattice Monte-
Carlo simulations of gauge theories in $\mathbb{R}^{3}\times S^{1}$ in which
the continuum limit is manifest Cossu  _et al._ (2014). The terminologies of
the phases and the configurations were given in Ref. Cossu and D’Elia (2009).
In particular, AB phases in the confined phase called by $X$ obey the Haar
measure and give $P=1$. Their distribution is similar to the reconfined phase
denoted by $C$ in Table 1. AB phases in $X$ would be uniformly and randomly
distributed when normalized by the distribution of $\theta_{i}$ generated by
the Haar measure.

$\mathcal{H}_{\text{sym}}/\text{Phase}$ | config. | $\theta_{1}$ | $\theta_{2}$ | $\theta_{3}$
---|---|---|---|---
$SU(3)/\text{confined}$ | $X$ | — | — | —
| $A_{1}$ | $0$ | $0$ | $0$
$SU(3)/\text{deconfined}$ | $A_{2}$ | $\frac{2}{3}\pi$ | $\frac{2}{3}\pi$ | $\frac{2}{3}\pi$
| $A_{3}$ | $-\frac{2}{3}\pi$ | $-\frac{2}{3}\pi$ | $-\frac{2}{3}\pi$
| $B_{1}$ | $\alpha$ | $\alpha$ | $-2\alpha$
$SU(2)\times U(1)/\text{split}$ | $B_{2}$ | $\alpha$ | $-2\alpha$ | $\alpha$
| $B_{3}$ | $-2\alpha$ | $\alpha$ | $\alpha$
| $C_{1}$ | $\beta_{1}$ | $\beta_{2}$ | $-\beta_{1}-\beta_{2}$
$U(1)\times U(1)/\text{reconfined}$ | $C_{2}$ | $\beta_{1}$ | $-\beta_{1}-\beta_{2}$ | $\beta_{2}$
| $C_{3}$ | $-\beta_{1}-\beta_{2}$ | $\beta_{1}$ | $\beta_{2}$
Table 1:  Configurations of AB phases and symmetry breaking patters in $SU(3)$
due to the Hosotani mechanism Cossu and D’Elia (2009); Cossu  _et al._ (2014),
i.e. $SU(3)\to\mathcal{H}_{\rm sym}$, where $\alpha\neq 0$,
$\beta_{1}\neq\beta_{2},\,-\frac{1}{2}\beta_{2},\,-2\beta_{2}$. Phases in the
confined configuration $X$ obey the Haar-measure distribution which is similar
to that in the reconfined phase such that $P=0$. Hence, if the distribution of
AB phases in $X$ is normalized by that of the Haar-measure, AB phases in $X$
would be uniformly and randomly distributed. (See Ref. Cossu  _et al._
(2014).)

As mentioned in subsection II.3, the confinement-deconfiment phase can be
judged by the Polyakov loop. By setting $\alpha=\pi/3$, $\beta_{1}=0$ and
$\beta_{2}=2\pi/3$, we plot the distribution of the Polyakov loop (24) in Fig.
1 in accordance with the configurations listed in Table 1. The orange lines
are obtained by setting $\theta_{1}=\theta_{2}=\theta_{3}=\varphi$ and varying
$\varphi$ from $0$ to $2\pi$.

Figure 1: Configurations of the Polyakov loop. We set $\alpha=\pi/3$,
$\beta_{1}=0$ and $\beta_{2}=2\pi/3$. The horizontal and vertical axes
correspond respectively to the real and imaginary parts of the Polyakov loop.
The configurations names are listed in Table 1.

In pure $SU(N)$ Yang-Mills theories, it has been shown by perturbation theory
Hosotani (1989) and lattice simulations in $\mathbb{R}^{3}\times S^{1}$
spacetime Cossu and D’Elia (2009); Cossu  _et al._ (2014) that in the small
gauge coupling regime, the $SU(N)$ symmetry is not spontaneously broken, i.e.
the symmetric vacua $\theta_{1}=\cdots=\theta_{N}=0$ or $\pm 2\pi/N$ are
observed as global minima of the effective potential for AB phases. The
present functional renormalization group approach will lead to the same result
in the $N=3$ case as of those earlier works and will be presented in Section
IV.4.

##  III Functional flow for Yang-Mills theory

In this section, we set up the functional renormalization group framework for
extradimensional Yang-Mills theories. The main objects to be derived are the
full propagators for gauge, scalar and ghost fields with an $R_{\xi}$-gauge
fixing action. We perform our computations with the background field method.

To close this section, we briefly review the basic idea of asymptotic safety.
In particular, we focus on the important notion of “relevant” coupling.

###  III.1 Functional renormalization group

A central object in quantum field theory is the one-particle irreducible
effective action $\Gamma$. The functional renormalization group provides a
tool to obtain $\Gamma$ as the coarse-graining of quantum fluctuations. This
can be realized by integrating out quantum fluctuations with higher momentum
modes $|p|>k$ and defines the scale-dependent effective (average) action
$\Gamma_{k}$. Then the coarse-graining procedure, i.e. the scale dependence of
$\Gamma_{k}$ is described by a functional differential equation. One of forms
is the Wetterich equation Wetterich (1993)

$\displaystyle\partial_{t}\Gamma_{k}=\frac{1}{2}{\rm
Tr}\,\left[\left(\Gamma_{k}^{(2)}+\mathcal{R}_{k}\right)^{-1}\partial_{t}\mathcal{R}_{k}\right]\,.$
(25)

Here, $\partial_{t}=k\,\partial_{k}$ is the dimensionless scale derivative,
$\Gamma_{k}^{(2)}$ is the full two-point correlation function obtained by
taking the second order functional derivative for $\Gamma_{k}$ and
$\mathcal{R}_{k}$ is the regulator function. Tr denotes the functional trace
acting on all internal spaces where the fields are defined.

In this work, we employ the functional renormalization group equation (25) to
analyze the effective action

$\displaystyle\Gamma_{k}$ $\displaystyle=\int\text{d}^{5}x\,V_{\rm
5D}(\theta_{H})+\frac{Z_{k}}{2g^{2}}\,\int\text{d}^{5}x\,{\rm
tr}\,F_{MN}F^{MN}$ $\displaystyle\qquad+S_{\text{gf}}+S_{\text{gh}}\,,$ (26)

where $V(\theta_{H})$ is the potential for the background field $\bar{A}_{5}$,
i.e. is a function of AB phases. Note that $Z_{k}$ is the dimensionless field
renormalization factor for $A_{M}$. Here and hereafter the spacetime integral
is

$\displaystyle\int\text{d}^{5}x=\int\text{d}^{4}x\int^{2\pi
R}_{0}\text{d}x_{5}\,,$ (27)

from which if the background field satisfies ${\rm tr}\,F_{MN}F^{MN}={\rm
tr}\,F_{\mu\nu}F^{\mu\nu}$, the potential and the gauge coupling in $D=4$ are
defined as

$\displaystyle V(\theta_{H})=(2\pi R)V_{\rm 5D}(\theta_{H})\,,$ $\displaystyle
g^{2}_{4D}=\frac{g^{2}}{2\pi R}\,.$ (28)

We employ the gauge fixing action $S_{\text{gf}}$ and the ghost action
$S_{\text{gh}}$, respectively as

$\displaystyle S_{\text{gf}}$ $\displaystyle=\frac{Z_{k}}{\xi
g^{2}}\int\text{d}^{5}x\,{\rm Tr}\,[\mathcal{F}(a)]^{2}\,,$ (29)
$\displaystyle S_{\text{gh}}$
$\displaystyle=Z_{\text{gh}}\int\text{d}^{5}x\,{\rm
Tr}\,\left[\bar{c}\,\frac{\delta\mathcal{F}(a^{\alpha})}{\delta\alpha}c\right]\,,$
(30)

with $\xi$ the gauge fixing parameter and $Z_{\text{gh}}$ the ghost field
renormalization factor. We denote the ghost and anti-ghost fields by $c$ and
$\bar{c}$, respectively. For $\mathcal{F}(a)$, we give the $R_{\xi}$-gauge
fixing function

$\displaystyle\mathcal{F}(a)=\delta^{\mu\nu}\bar{D}_{\mu}a_{\nu}+\xi\bar{D}_{5}a_{5}\,,$
(31)

where $\bar{D}_{\mu}a_{\nu}=\partial_{\mu}a_{\nu}-i[\bar{A}_{\mu},a_{\nu}]$ is
the covariant derivative with the background field. From Eq. (31) and
$a_{\mu}^{\alpha}=a_{\mu}+D_{\mu}\alpha$, one has

$\displaystyle\frac{\delta\mathcal{F}(a^{\alpha})}{\delta\alpha}=\delta^{\mu\nu}\bar{D}_{\mu}D_{\nu}+\xi\bar{D}_{5}D_{5}\,.$
(32)

For the effective action (26), the Wetterich equation is expressed
diagrammatically as

$\displaystyle\partial_{t}\Gamma_{k}=\frac{1}{2}\vbox{\hbox{\includegraphics[width=42.67912pt]{one_loop_gluon}}}+\frac{1}{2}\vbox{\hbox{\includegraphics[width=42.67912pt]{one_loop_gluon_5}}}-\vbox{\hbox{\includegraphics[width=39.83385pt]{one_loop_ghost}}}\,,$
(33)

where curly, dashed and dotted lines represent the 4 dimensional gauge
($M=0,\ldots,3$), the extra gauge ($M=5$) and ghost fields, respectively, and
a crossed circle denotes the cutoff insertion, i.e.
$\partial_{t}\mathcal{R}_{k}$.

###  III.2 Asymptotic safety

Before we start to calculate the flow equation, we briefly review the basic
idea of asymptotic safety. Let us suppose a $d$ dimensional system whose
theory space $\Gamma_{k}$ is spanned by effective operators $\mathcal{O}_{i}$
with couplings $g_{i}$:

$\displaystyle\Gamma_{k}=\int\text{d}^{d}x\sum_{i}g_{i}{\mathcal{O}}_{i}\,.$
(34)

Here we denote the mass dimension of ${\mathcal{O}}_{i}$ by $d_{i}$. For the
couplings $g_{i}$, one can compute the renormalization group equations by
using Eq. (25) as

$\displaystyle\partial_{t}\tilde{g}_{i}=\beta_{i}(\\{\tilde{g}_{j}\\})\,,$
(35)

where $\tilde{g}_{i}=k^{d-d_{i}}g_{i}$ are dimensionless couplings and
$\\{\tilde{g}_{j}\\}$ denotes a set of dimensionless couplings.

An important notion is the fixed point at which all beta functions (the right-
hand side of Eq. (35)) vanish, namely

$\displaystyle\beta_{i}(\\{\tilde{g}_{j*}\\})=0\qquad\text{for all $i$}\,.$
(36)

If such a fixed point $\\{\tilde{g}_{i*}\\}$ is found, one linearizes Eq. (35)
around it as

$\displaystyle\partial_{t}\tilde{g}_{i}\simeq\sum_{j}\frac{\partial\beta_{i}}{\partial\tilde{g}_{j}}\bigg{|}_{\\{\tilde{g}_{i}\\}=\\{\tilde{g}_{i*}\\}}(\tilde{g}_{j}-\tilde{g}_{j})\,.$
(37)

By diagonalizing the stability matrix
$\frac{\partial\beta_{i}}{\partial\tilde{g}_{j}}|_{\\{\tilde{g}_{i}\\}=\\{\tilde{g}_{i*}\\}}$
these equations can be solved such that

$\displaystyle\tilde{g}_{i}=\tilde{g}_{i*}+\sum_{j}C^{j}_{i}\left(\frac{k}{k_{0}}\right)^{-\lambda_{j}}\,,$
(38)

where $k_{0}$ is a reference scale and $C_{i}^{j}$ is a constant matrix. Here
$\lambda_{j}$ are eigenvalues of the stability matrix
$-\frac{\partial\beta_{i}}{\partial\tilde{g}_{j}}|_{\\{\tilde{g}_{i}\\}=\\{\tilde{g}_{i*}\\}}$
called “critical exponents” and characterize the energy scaling of
$\tilde{g}_{i}$ around the fixed point. Associated couplings with positive
$\lambda_{i}$s are amplified for $k\to 0$ and therefore called “relevant”,
whereas the “irrelevant” couplings have negative values of $\lambda_{i}$. The
couplings with vanishing critical exponents are “marginal” couplings. In this
case, we need to expand the beta functions up to the next order and evaluate
whether the next order contribution gives positive contributions to the
critical exponents or not. When mixing effects between different operators are
negligible, i.e. $C_{i}^{j}\approx c_{i}\delta_{i}^{j}$ with constant $c_{i}$,
the critical exponent $\lambda_{i}$ for a fixed $i$ characterizes the energy
scaling for $\tilde{g}_{i}$. Relevant couplings are free parameters at the
fixed point. Then, the finite number of relevant couplings means the theory to
be renormalizable. In particular, at the Gaussian fixed point
$\tilde{g}_{*}=0$, the critical exponents are equivalent to the canonical
scaling of the couplings, i.e. $\lambda_{i}=d-d_{i}$. The gauge coupling of
the Yang-Mills theory in $D=4$ is a marginally relevant parameter which leads
to asymptotic freedom.

In this work, we are especially interested in the renormalization group
equation for the dimensionless gauge coupling $\tilde{g}^{2}$. Its
renormalization group equation is schematically given by

$\displaystyle\partial_{t}\tilde{g}^{2}=\beta_{\tilde{g}^{2}}(\tilde{g}^{2};\bar{R})\,,$
(39)

where $\bar{R}=k\,R$ is the dimensionless radius. The explicit definition of
$\tilde{g}^{2}$ and beta function are given in Eq. (73) in Section V. We look
for fixed points by solving $\beta_{\tilde{g}^{2}}(\tilde{g}^{2};\bar{R})=0$
as a function of $\bar{R}$. For fixed, finite $\bar{R}$, we will find two
fixed points: One is the trivial fixed point $\tilde{g}^{2}_{*}=0$ for which
the canonical scaling term dominates and then the corresponding critical
exponent becomes $\lambda_{\tilde{g}^{2}}=-1$ for $R\to\infty$, while for
$R\to 0$ being $\lambda_{\tilde{g}^{2}}=0$. The second is the nontrivial fixed
point $\tilde{g}^{2}_{*}\neq 0$ at which the critical exponent reads

$\displaystyle\nu_{\tilde{g}^{2}}^{-1}=\lambda_{\tilde{g}^{2}}=-\frac{\partial\beta_{\tilde{g}^{2}}}{\partial\tilde{g}^{2}}\bigg{|}_{\tilde{g}^{2}=\tilde{g}^{2}_{*}}\,.$
(40)

We will present the fixed point value and the critical exponent for the gauge
coupling as functions of $\bar{R}$ in Section VI. In the asymptotic safety
scenario, the vacuum density, i.e. the $\theta_{H}$-independent part of the
potential, should have a finite fixed point too. This is indeed realized in
the current study. We will argue this point together with the gauge coupling
in Section VI.

##  IV Renormalization group for effective potential

In this section, we derive the flow equation for AB phases $\theta_{i}$ within
the background field approximation Reuter and Wetterich (1994a, 1993a, 1993b,
b); Wetterich (2018a). See also Refs. Reuter (1996); Gies (2002, 2012). After
computing the Hessians for the effective action (26), we derive the flow
equation for $V(\theta_{H})$. We then show that the vacuum energy $V(0)$
contains a UV divergence, while $V(\theta_{H})-V(0)$ becomes finite. We solve
the flow equation for $V(\theta_{H})-V(0)$ to find its minima at the IR limit
($k\to 0$).

###  IV.1 Hessians

To derive the flow equations, we need to compute the Hessian (the two-point
function) by performing the second-order functional derivative of
$\Gamma_{k}$, i.e.
$\Gamma_{k}^{(2)}=\frac{\delta^{2}\Gamma_{k}}{\delta\varphi\delta\varphi}$
where $\varphi=(a_{\mu},a_{5},c)$. The Hessians for the effective action (26)
are obtained as

$\displaystyle(\Gamma_{k}^{(2)})_{a_{\mu}a_{\nu}}$
$\displaystyle=\frac{Z_{k}}{g^{2}}\left\\{\delta^{\mu\nu}\left(-\partial^{2}-\bar{D}_{5}^{2}\right)+\left(1-\frac{1}{\xi}\right)\partial^{\mu}\partial^{\nu}\right\\}\,,$
$\displaystyle(\Gamma_{k}^{(2)})_{a_{5}a_{5}}$
$\displaystyle=\frac{Z_{k}}{g^{2}}\left(-\partial^{2}-\xi\bar{D}_{5}^{2}\right)\,,$
$\displaystyle(\Gamma_{k}^{(2)})_{\bar{c}c}$
$\displaystyle=Z_{\text{gh}}\left(-\partial^{2}-\xi\bar{D}_{5}^{2}\right)\,.$
(41)

We note here that $V(\theta_{H})$ in Eq. (26) does not contribute to the
Hessian since it is a function of the background field.

We perform the KK expansion (13) for $\varphi=(a_{\mu},a_{5},c)$:

$\displaystyle a_{M}(x,x_{5})$
$\displaystyle=\sum_{n=-\infty}^{\infty}a_{M}^{(n)}(x)\frac{e^{inx_{5}/R}}{\sqrt{2\pi
R}}\,,$ (42) $\displaystyle c(x,x_{5})$
$\displaystyle=\sum_{n=-\infty}^{\infty}c^{(n)}(x)\frac{e^{inx_{5}/R}}{\sqrt{2\pi
R}}\,,$ (43) $\displaystyle\bar{c}(x,x_{5})$
$\displaystyle=\sum_{n=-\infty}^{\infty}\bar{c}^{(n)}(x)\frac{e^{inx_{5}/R}}{\sqrt{2\pi
R}}\,.$ (44)

Inserting these KK expansions, the covariant derivative $\bar{D}_{5}^{2}$
acting on $\varphi$ becomes

$\displaystyle(-\bar{D}_{5}^{2}\varphi)_{ij}=\frac{1}{R^{2}}\left(n-\frac{\theta_{i}-\theta_{j}}{2\pi}\right)^{2}\varphi_{ij}\equiv
M_{ij,n}^{2}\varphi_{ij}\,,$ (45)

where $i$, $j$ are indices for the fundamental representation of $SU(N)$.
Hence, the covariant derivative of the extra dimension acts as a mass term for
the gauge field. The propagators for $a_{\mu}^{(n)}(p)$, $a_{5}^{(n)}(p)$ and
the ghost fields in momentum space are given respectively by

$\displaystyle\Pi^{(n)}_{\mu\nu,ij}(p^{2})=\frac{g^{2}}{Z_{k}}\bigg{[}\frac{\delta_{\mu\nu}}{p^{2}+M_{ij,n}^{2}}$
$\displaystyle\qquad+\frac{p_{\mu}p_{\nu}}{M^{2}_{ij,n}}\bigg{(}\frac{1}{p^{2}+M_{ij,n}^{2}}-\frac{1}{p^{2}+\xi
M_{ij,n}^{2}}\bigg{)}\bigg{]}\,,$ (46)
$\displaystyle\Pi^{(n)}_{55,ij}(p^{2})=\frac{g^{2}}{Z_{k}}\frac{1}{p^{2}+\xi
M_{ij,n}^{2}}\,,$ (47)
$\displaystyle\Pi_{\text{gh}}^{(n)}{}_{ij}(p^{2})=\frac{1}{Z_{\rm
gh}}\frac{1}{p^{2}+\xi M_{ij,n}^{2}}\,.$ (48)

At this point, we can understand the symmetry breaking patterns from the mass
spectra as discussed in Section II.4. For $\theta_{i}=0$ or
$\pm\frac{2}{3}\pi$, $M_{ij,n}^{2}$ has contributions from all KK modes and
not from the non-trivial background.

###  IV.2 Derivation of flow equation

Let us derive the flow equation of the potential by using the flow equation
(25). We first introduce the regulator $\mathcal{R}_{k}$ such that
$-\partial^{2}$ (or $p^{2}$) in Eq. (41) is replaced to
$P_{k}(p^{2})=p^{2}+R_{k}(p^{2})$ by the loop integration, i.e. more
specifically, we use the following regulators

$\displaystyle(\mathcal{R}_{k}(p^{2}))^{\mu\nu}=\delta^{\mu\nu}\frac{Z_{k}}{g^{2}}R_{k}(p^{2})$
$\displaystyle\text{for $a^{(n)}_{\mu}$}\,,$ (49)
$\displaystyle\mathcal{R}_{k}(p^{2})=\frac{Z_{k}}{g^{2}}R_{k}(p^{2})$
$\displaystyle\text{for $a^{(n)}_{5}$}\,,$ (50)
$\displaystyle\mathcal{R}_{k}(p^{2})=Z_{\text{gh}}R_{k}(p^{2})$
$\displaystyle\text{for $\bar{c}^{(n)}$, $c^{(n)}$}\,,$ (51)

with a Litim-type cutoff function Litim (2001)

$\displaystyle R_{k}(p^{2})=(k^{2}-p^{2})\theta(k^{2}-p^{2})\,.$ (52)

The flow equation (33) in the system reads

$\displaystyle\partial_{t}\Gamma_{k}$ $\displaystyle=\frac{1}{2}{\rm
Tr}\,\left[\hat{\Pi}^{(n)}_{\mu\alpha,ij}(P_{k})(\partial_{t}{\mathcal{R}}_{k})^{\alpha\nu}\right]$
$\displaystyle\quad+\frac{1}{2}{\rm
Tr}\,\left[\hat{\Pi}^{(n)}_{55,ij}(P_{k})\partial_{t}{\mathcal{R}}_{k}\right]$
$\displaystyle\qquad-{\rm
Tr}\,\left[\hat{\Pi}_{\text{gh}}^{(n)}{}_{ij}(P_{k})\partial_{t}{\mathcal{R}}_{k}\right]\,,$
(53)

where $\hat{\Pi}$s denote the regulated propagators in Eqs. (49) – (51). We
define the potential of AB phases as

$\displaystyle\frac{V(\theta_{H})}{2\pi R}=\frac{1}{2\pi
R{\mathcal{V}}_{4}}\Gamma_{k}[\bar{A}_{5}]\,,$ (54)

where ${\mathcal{V}}_{4}=\int\text{d}^{4}x$ is the volume in 4 dimensional
spacetime and then $2\pi R{\mathcal{V}}_{4}=\int\text{d}^{5}x$. The functional
trace in Eq. (53) involves summations for gauge group and Lorentz indices and
the momenta integration, i.e. for a flow kernel
$W(p^{2})=W_{ij;n;\mu\nu}(p^{2})=\hat{\Pi}^{(n)}_{\mu\alpha,ij}(P_{k})(\partial_{t}{\mathcal{R}}_{k})^{\alpha\nu}$,

$\displaystyle{\rm Tr}\,[W(p^{2})]$
$\displaystyle={\mathcal{V}}_{4}\sum_{n=-\infty}^{\infty}\int\frac{\text{d}^{4}p}{(2\pi)^{4}}\sum_{i,j=1}^{N}\sum_{\mu=0}^{3}W_{ij;n;\mu}{}^{\mu}(p^{2})\,.$
(55)

We show the flow equation in the Feynman gauge $\xi=1$ by performing the
momentum integral and the KK mode summation,

$\displaystyle\partial_{t}V(\theta_{H})$
$\displaystyle=\frac{5k^{4}}{2(4\pi)^{2}}\left(1-\frac{\eta_{g}}{6}\right){\mathcal{I}}_{N}(\bar{R};\theta_{H})$
$\displaystyle\quad-\frac{k^{4}}{(4\pi)^{2}}\left(1-\frac{\eta_{\text{gh}}}{6}\right){\mathcal{I}}_{N}(\bar{R};\theta_{H})\,,$
(56)

where $\bar{R}=kR$ is the dimensionless compactification radius, and
$\eta_{g}=-\partial_{t}\log Z_{g}$ and $\eta_{\text{gh}}=-\partial_{t}\log
Z_{\text{gh}}$ are the anomalous dimensions of the gauge and ghost fields,
respectively. We have defined the threshold function as

$\displaystyle{\mathcal{I}}_{N}(\bar{R};\theta_{H})$
$\displaystyle=\sum_{i,j=1}^{N}\sum_{n=-\infty}^{\infty}\frac{1}{1+\bar{M}_{ij,n}^{2}}$
$\displaystyle=\pi\bar{R}\sum_{i,j=1}^{N}\frac{\sinh(2\pi\bar{R})}{\cosh(2\pi\bar{R})-\cos(\theta_{i}-\theta_{j})}\,,$
(57)

where $\bar{M}_{ij,n}^{2}=M_{ij,n}^{2}/k^{2}$. The computation of $\eta_{g}$
within the background field approximation is given in Section V.

###  IV.3 Independence from UV cutoff

In general, a solution to the differential equation (56) requires a UV cutoff
$\Lambda$. However, by subtracting the $\theta_{H}$-independent part
(corresponding to the vacuum energy) from the potential, namely evaluating
$V(\theta_{H})-V(0)$, the result is free from the UV cutoff. To see this, we
consider

$\displaystyle V(\theta_{H})-V(0)$
$\displaystyle=\frac{3}{2(4\pi)^{2}}\int_{0}^{\Lambda}\text{d}k\,k^{3}({\mathcal{I}}_{N}(\bar{R};\theta_{H})-{\mathcal{I}}_{N}(\bar{R};0))$
$\displaystyle=\frac{-3\pi\bar{R}}{2(4\pi)^{2}}\sum_{i,j=1}^{N}\int_{0}^{\Lambda}\text{d}k\,k^{4}\frac{\coth(\pi\bar{R})\sin^{2}\left(\frac{\theta_{i}-\theta_{j}}{2}\right)}{\cosh(2\pi\bar{R})-\cos(\theta_{i}-\theta_{j})}$
$\displaystyle=\frac{-3}{2(4\pi)^{2}}\sum_{i,j=1}^{N}\sin^{2}\left(\frac{\theta_{i}-\theta_{j}}{2}\right)$
$\displaystyle\quad\times\frac{1}{(\pi R)^{4}}\int_{0}^{\pi
R\Lambda}\text{d}x\frac{x^{4}\coth(x)}{\cosh(2x)-\cos(\theta_{i}-\theta_{j})}\,,$
(58)

where $x=\pi Rk$ and we set $\eta_{g}=0$ for simplicity. Hence, the problem is
whether the limit $\Lambda\to\infty$ can be taken or not. In other words, we
study the convergence of the integral

$\displaystyle\int^{\infty}_{0}\text{d}x\frac{x^{4}\coth(x)}{\cosh(2x)-a}\qquad(-1\leq
a\leq 1)\,.$ (59)

Since its integrand increases monotonically for increasing $a$, we see that

$\displaystyle\int^{\infty}_{0}\text{d}x\frac{x^{4}\coth(x)}{\cosh(2x)+1}=\frac{93}{64}\zeta(5)\leq\int^{\infty}_{0}\text{d}x\frac{x^{4}\coth(x)}{\cosh(2x)-a}$
$\displaystyle\qquad\qquad\leq\int^{\infty}_{0}\text{d}x\frac{x^{4}\coth(x)}{\cosh(2x)-1}=\frac{3}{2}\zeta(3)\,,$
(60)

where $\zeta(3)\simeq 1.202$ is the Riemann zeta function of 3. We conclude
now that Eq. (58) is finite for $\Lambda\to\infty$. Note that for particular
choices of $a$, one has

$\displaystyle\int^{\infty}_{0}\text{d}x\frac{x^{4}\coth(x)}{\cosh(2x)-a}=\begin{cases}\frac{3}{2}\zeta(3)&(a=1)\,,\\\\[8.61108pt]
\frac{1581}{1024}\zeta(5)&(a=0)\,,\\\\[8.61108pt]
\frac{93}{64}\zeta(5)&(a=-1)\,.\end{cases}$ (61)

The determination of configurations of AB phases relies on the difference
$V(\theta_{H})-V(0)$. The vacuum part (the cosmological constant) $V(0)$ is
generally divergent. In perturbation theory, the divergence of the
$\theta_{H}$-independent part $V(0)$ in higher dimensional spacetime cannot be
subtracted by the standard renormalization, while in the asymptotic safety
scenario $V(0)$ could have a nontrivial fixed point, so that it could be an
asymptotically safe coupling. Although the vacuum energy does not affect the
dynamics of the gauge fields, it is important to see its finiteness as an
asymptotically safe feature. We define the dimensionless quantity

$\displaystyle V(0)$
$\displaystyle=k^{4}u(0)=k^{4}\mathcal{I}_{N}(\bar{R};0)\tilde{u}(0)\,.$ (62)

We reparametrize the potential in such a way that the fifth dimension scales
with the dimensionless radius. Taking the $t$-derivative on the both sides of
Eq. (62), the flow of $\tilde{u}(0)$ then reads

$\displaystyle\partial_{t}\tilde{u}(0)=-d_{\text{eff}}(\bar{R};0)\,\tilde{u}(0)+\frac{1}{k^{4}\,\mathcal{I}_{N}\left(\bar{R}\right)}\,\partial_{t}V(0)\,,$
(63)

where the last term in the right-hand side is given by Eq. (56) with
$\theta_{H}=0$. Here we introduce the effective dimension Kubo  _et al._
(2000) as

$\displaystyle
d_{\text{eff}}(\bar{R};\theta_{H})=4+\frac{\text{d}\log{\mathcal{I}}_{N}(\bar{R};\theta_{H})}{\text{d}\log\bar{R}}\,,$
(64)

in which we used $\partial_{t}=\partial/\partial\log
k=\partial/\partial\log\bar{R}$. We will investigate the fixed point structure
of Eq. (63) in Section VI.

###  IV.4 Minimum of the potential

Let us here solve the flow equation for the AB-phase potential in the $N=3$
case,

$\displaystyle\partial_{t}(V(\theta_{H})-V(0))$
$\displaystyle\quad=\frac{5k^{4}}{2(4\pi)^{2}}\left(1-\frac{\eta_{g}}{6}\right)\left({\mathcal{I}}_{N}(\bar{R};\theta_{H})-{\mathcal{I}}_{N}(\bar{R};0)\right)$
$\displaystyle\qquad-\frac{k^{4}}{(4\pi)^{2}}\left({\mathcal{I}}_{N}(\bar{R};\theta_{H})-{\mathcal{I}}_{N}(\bar{R};0)\right)\,,$
(65)

or equivalently analyze the potential (58). As will be seen in Section VI, the
anomalous dimension of the gauge field is bounded $-1<\eta_{g}<0$ and
suppressed by a factor $1/6$ and consequently it can be neglected in the flow
equation (65).

To solve the differential equation (65), we set $V(\theta_{H})-V(0)=0$ and
$\bar{R}=1$ at $k=\Lambda=10$ and then evolve the potential
$V(\theta_{H})-V(0)$ until the low energy scale $k\to 0$. In Fig. 2, the
renormalization group evolution of the potential at the $\theta_{2}=0$ plane
is presented. Starting from the constant potential at the UV scale, we can see
that the potential is deformed and its vacuum structure becomes clearer in
lower energy scales. The potential shows a minimum at $\theta_{1}=0$, so that
the vacuum is $SU(3)$ symmetric. In order to see this clearly, we plot the
potential on the $\theta_{1}$-$\theta_{2}$ plane. In Fig. 3, the potential in
the IR limit is shown by both 3 dimensional and contour plots. It can be seen
that there are vacua at $\theta_{1}=\theta_{2}=\theta_{3}=0$ and
$\theta_{1}=\theta_{2}=\theta_{3}=\pm\frac{2}{3}\pi$. As listed in Table 1,
these vacua are $SU(3)$ symmetric and are in a deconfinement phase.

The fact that the $SU(N)$ symmetry is not spontaneously broken in a pure Yang-
Mills theory has been actually seen by perturbation theory Hosotani (1989) and
lattice simulations of gauge theories on $\mathbb{R}^{3}\times S^{1}$ Cossu
and D’Elia (2009); Cossu  _et al._ (2014). We have now confirmed the same
conclusion from the functional renormalization group method.

Figure 2: The renormalization group evolution of the potential
$V(\theta_{H})-V(0)$ at the $\theta_{2}=0$ plane (with
$\theta_{3}=-\theta_{1}-\theta_{2}$) from the UV to IR scales. The energy
units of $k$ are arbitrary. Each dotted line corresponds to the potential at a
lowed scale by $\Delta k=1$.

Figure 3: 3 dimensional plot (left) and contour plot (right) of the potential
$V(\theta_{H})-V(0)$ at the IR scale ($k\to 0$). Note
$\theta_{3}=-\theta_{1}-\theta_{2}$.

##  V Flowing gauge coupling

In this section we proceed to derive the flow equation for the gauge coupling.
From earlier studies, e.g. Hosotani (1989); Kashiwa and Misumi (2013); Cossu
_et al._ (2014) and the previous section, it turns out that a pure Yang-Mills
theory entails the $SU(N)$ symmetric vacuum $\langle
A_{5}\rangle=\bar{A}_{5}=0$. Therefore, hereafter we assume a general constant
background $\bar{A}_{\mu}\neq 0$ and $\bar{A}_{5}=0$ so that

$\displaystyle\frac{Z_{k}}{2}\frac{1}{g^{2}}\int\text{d}^{5}x\,{\rm
tr}\left[\bar{F}_{MN}\bar{F}^{MN}\right]$
$\displaystyle\qquad\qquad=\frac{Z_{k}}{2}\frac{2\pi
R}{g^{2}}\int\text{d}^{4}x\,{\rm
tr}\left[\bar{F}_{\mu\nu}\bar{F}^{\mu\nu}\right]\,.$ (66)

In such a background, the Hessian is obtained by the replacement
$\partial_{\mu}\to\bar{D}_{\mu}$ and $\bar{D}_{5}\to\partial_{5}$ in Eq. (41)
for which we employ the regulator, i.e.
$P_{k}(\bar{\Delta})=\bar{\Delta}+R_{k}(\bar{\Delta})$ with
$\bar{\Delta}=-\bar{D}^{2}$. A useful method to obtain the flow equation is
the heat kernel technique. Its detailed description is given in Appendix A.
For a flow kernel $W(\bar{\Delta}_{i})$ with the Laplacian $\bar{\Delta}_{i}$
acting on spin-$i$ field, the heat kernel expansion results in

$\displaystyle{\rm Tr}\,_{(i)}[W(\bar{\Delta}_{i})]$
$\displaystyle=\frac{\mathcal{V}_{4}}{(4\pi)^{2}}\sum_{n=-\infty}^{\infty}\bigg{[}b_{0}^{(i)}Q_{2}[W]$
$\displaystyle\quad+b_{2}^{(i)}Q_{0}[W]{\rm
tr}\,\bar{F}_{\mu\nu}\bar{F}^{\mu\nu}+\cdots\bigg{]}\,.$ (67)

Here $Q_{n}[W]$ are the threshold functions given by

$\displaystyle Q_{2}[W]$ $\displaystyle=\int_{0}^{\infty}\text{d}z\,zW(z)\,,$
(68) $\displaystyle Q_{0}[W]$ $\displaystyle=W(0)\,,$ (69)

and $b_{j}^{(i)}$ are the corresponding heat kernel coefficients for the spin
representation $(i)$ of the field. The first term on the right-hand side in
Eq. (67) corresponds to the beta functions for $V(\theta_{H}=0)$. The flow
equation for $Z_{k}$ is thus obtained from

$\displaystyle\frac{\partial_{t}Z_{k}}{2}\frac{2\pi
R}{g^{2}}=\frac{1}{{\mathcal{V}}_{4}}\partial_{t}\Gamma_{k}\bigg{|}_{F_{\mu\nu}^{2}}\,.$
(70)

The evaluation of the right-hand side is presented in Appendix B. Here we show
the result obtained from the flow equation for the gauge coupling

$\displaystyle\frac{\partial_{t}Z_{k}}{2}\frac{2\pi R}{g^{2}}$
$\displaystyle=\frac{N}{2(4\pi)^{2}}\Bigg{[}\frac{20}{3}\left(1-\frac{\eta_{g}}{2}\right)-\frac{1}{3}\left(1-\frac{\eta_{g}}{2}\right)$
$\displaystyle\quad+\frac{2}{3}\left(1-\frac{\eta_{\rm
gh}}{2}\right)\Bigg{]}\frac{1}{N^{2}}{\mathcal{I}}_{N}(\bar{R};\theta_{H}=0)\,.$
(71)

Defining the dimensionless renormalized gauge coupling

$\displaystyle\tilde{g}^{2}=\frac{Z^{-1}_{k}g^{2}}{2\pi
R}\,\frac{\mathcal{I}_{N}(\bar{R};0)}{N^{2}}=Z^{-1}_{k}g^{2}_{\text{4D}}\,\frac{\mathcal{I}_{N}(\bar{R};0)}{N^{2}}\,,$
(72)

and taking the derivative of both sides in Eq. (72) with respect to $t$ which
acts on $\tilde{g}$, $Z_{k}^{-1}$ and $\bar{R}$, the flow equation for
$\tilde{g}$ can be written as

$\displaystyle\partial_{t}\tilde{g}^{2}=\left(d_{\text{eff}}(\bar{R};0)-4+\eta_{g}\right)\tilde{g}^{2}\,,$
(73)

where $d_{\text{eff}}(\bar{R};\theta_{H})$ has been defined in Eq. (64). Here,
the anomalous dimension reads

$\displaystyle\eta_{g}$ $\displaystyle=-\frac{\partial_{t}Z_{k}}{Z_{k}}$
$\displaystyle=-\frac{N}{(4\pi)^{2}}\left(7-\frac{19\eta_{g}}{6}+\frac{\eta_{\rm
gh}}{3}\right)\frac{Z_{k}^{-1}g^{2}}{2\pi
R}\frac{{\mathcal{I}}_{N}(\bar{R};0)}{N^{2}}\,.$ (74)

Setting $\eta_{g}=\eta_{\text{gh}}=0$ in the right-hand side of Eq. (74)
produces the one-loop contribution to the beta function

$\displaystyle\eta_{g}=-7N\frac{\tilde{g}^{2}}{(4\pi)^{2}}\,.$ (75)

This one-loop result agrees with that of Ref. Freitas and Wiegand (2018) in
which a Yang-Mills theory in universal extra dimensions is studied. Setting
only $\eta_{\text{gh}}=0$ and solving Eq. (74) to $\eta_{g}$ yield the
“resummed” anomalous dimension of the gauge field as

$\displaystyle\eta_{g}=-\frac{7N\,\frac{\tilde{g}^{2}}{(4\pi)^{2}}}{1-\frac{19N}{6}\,\frac{\tilde{g}^{2}}{(4\pi)^{2}}}\,.$
(76)

This corresponds to the loop resummation, i.e. the nonperturbative anomalous
dimension. Indeed, expanding it into the polynomial of $\tilde{g}^{2}$ gives

$\displaystyle\eta_{g}=-7N\frac{\tilde{g}^{2}}{(4\pi)^{2}}-\frac{133N^{2}}{6}\frac{\tilde{g}^{4}}{(4\pi)^{4}}+\mathcal{O}\left(\tilde{g}^{6}\right)\,.$
(77)

The first term agrees with the one-loop result given in Eq. (75). We should
here note that the one-loop coefficient ($-7$) is different from that
($-22/3$) of Eq. (1) because the present system contains an additional scalar
one-loop contribution from $A_{5}$. Subtracting its contribution from the
anomalous dimension by hand actually reproduces the same coefficient as of the
one-loop result in Yang-Mills. The two-loop coefficient also differs from
Yang-Mills: This is because the background field approximation Reuter and
Wetterich (1994a, 1993a, 1993b, b); Wetterich (2018a) misses some
contributions from two-loop effects besides the $A_{5}$ contribution.

##  VI Fixed point structure, critical exponent and renormalization group flow

Before studying the fixed point structure of $\tilde{u}(0)$ and
$\tilde{g}^{2}$, let us see the behavior of the effective dimension $d_{\rm
eff}(\theta_{H},\bar{R})$ defined in Eq. (64). The effective dimension is
depicted in Fig. 4 for $N=3$:

Figure 4: The effective dimension $d_{\rm eff}$ as a function of the
dimensionless compactification radius. Left: Dependence of $d_{\rm eff}$ on
truncation limits on the KK mode summations as $|n|=N_{\rm tr}$ in the
symmetric phase ($\theta_{1}=\theta_{2}=\theta_{3}=0$). Right: The effective
dimension with the full resummation of KK modes for the different AB phases
shown in Tabel 1. Note $\theta_{3}=-\theta_{1}-\theta_{2}$.

The left-hand side panel shows $d_{\rm eff}$ for different truncations for the
KK modes, i.e. the KK summation is truncated as $\sum_{n=-N_{\rm tr}}^{N_{\rm
tr}}$. We see that the effective dimension smoothly interpolates between four
and five dimensional spacetimes varying the compactification radius if all KK
modes are summed up ($N_{\rm tr}=\infty$):

$\displaystyle d_{\rm eff}(\bar{R};\theta_{H})=\begin{cases}4&(\bar{R}\to
0)\,,\\\\[8.61108pt] 5&(\bar{R}\to\infty)\,.\end{cases}$ (78)

For finite truncations ($N_{\rm tr}<\infty$), however, the full dimensional
transition is not achieved. This originates from the fact that all KK modes
$M_{n}=n/R$ in $R\to\infty$ become massless and thus contribute to the loop
integration. The effective dimension $d_{\rm eff}(\bar{R}:\theta_{H})$ is
defined from the threshold function (loop integral) (57), so that it describes
the dynamical change of the dimensionality in the system. In the right-hand
side panel of Fig. 4, the effective dimension is shown for the different
breaking patterns listed in Table 1. We see a nontrivial dependence of the
dimensionality on the background field configuration. Such a dependence is
observed especially around the compactification radii
$0.001\lesssim\bar{R}\lesssim 1$ for which the transition of dimensionality
takes place. Nevertheless, the behavior of Eq. (78) is independent of the
vacuum configurations.

Figure 5:  The nontrivial fixed point value $\tilde{g}_{*}^{2}$ (left) and the
critical exponent $\nu_{\tilde{g}^{2}}^{-1}$ (right) of the gauge coupling as
a function of the dimensionless compactification radius $\bar{R}$.

We now study the fixed point solutions in the flow equations:

$\displaystyle\partial_{t}\tilde{u}(0)$
$\displaystyle=-d_{\text{eff}}(\bar{R};0)\,\tilde{u}(0)+\frac{5}{2(4\pi)^{2}}\left(1-\frac{\eta_{g}}{6}\right)-\frac{1}{(4\pi)^{2}}\,,$
(79) $\displaystyle\partial_{t}\tilde{g}^{2}$
$\displaystyle=\left(d_{\text{eff}}(\bar{R};0)-4+\eta_{g}\right)\tilde{g}^{2}\equiv\beta_{\tilde{g}^{2}}\,,$
(80)

with the effective dimension (64) and the anomalous dimension of the gauge
field

$\displaystyle\eta_{g}=\begin{cases}\displaystyle-\frac{\frac{21}{(4\pi)^{2}}\tilde{g}^{2}}{1-\frac{57}{6(4\pi)^{2}}\,\tilde{g}^{2}}&(\text{resumed})\,,\\\\[21.52771pt]
\displaystyle-\frac{21}{(4\pi)^{2}}\tilde{g}^{2}&(\text{one-
loop})\,,\end{cases}$ (81)

where “one-loop” corresponds to the lowest order of the Taylor series in terms
of $\tilde{g}^{2}$ for the “resumed” anomalous dimension. We set $\eta_{\rm
gh}=0$ in this analysis. Let us first discuss the nontrivial UV fixed point of
the gauge coupling. We find it analytically as

$\displaystyle\tilde{g}_{*}^{2}=\begin{cases}\displaystyle\frac{32\pi^{2}(\sinh(2\pi\bar{R})-2\pi\bar{R})}{61\sinh(2\pi\bar{R})-38\pi\bar{R}}&\text{(resumed)}\,,\\\\[17.22217pt]
\displaystyle\frac{16\pi^{2}}{21}-\frac{16\pi^{2}}{21}\frac{2\pi\bar{R}}{\sinh(2\pi\bar{R})}&\text{(one-
loop)}\,.\end{cases}$ (82)

Fig. 5 exhibits this fixed point value and the critical exponent of the gauge
coupling $\tilde{g}^{2}$ as functions of $\bar{R}$ in the symmetric vacuum
$\theta_{1}=\theta_{2}=\theta_{3}=0$. For $\bar{R}\to\infty$ ($\mathbb{R}^{5}$
spacetime), the fixed point value $\tilde{g}_{*}^{2}$ is given by

$\displaystyle\tilde{g}_{*}^{2}\big{|}_{\bar{R}\to\infty}=\begin{cases}\displaystyle\frac{32\pi^{2}}{61}\simeq
5.18&\text{(resumed)}\,,\\\\[17.22217pt]
\displaystyle\frac{16\pi^{2}}{21}\simeq 7.52&\text{(one-loop)}\,,\end{cases}$
(83)

while for $\bar{R}\to 0$ corresponding to the system in $D=4$, the fixed point
value goes to zero, namely, the gauge coupling converges to the Gaussian fixed
point which entails asymptotic freedom of the gauge coupling in $D=4$. This
can be also seen from Fig. 6 in which the behavior of the beta function
$\beta_{\tilde{g}^{2}}$ in Eq. (80) as a function of $\tilde{g}^{2}$ is
depicted. As the compactification radius becomes smaller, the nontrivial fixed
point value comes closer to the Gaussian fixed point. For $\bar{R}\to 0$, the
nontrivial fixed point merges with the Gaussian fixed point. The critical
exponent has been defined in Eq. (40) which reads for Eq. (80) at the
nontrivial fixed point as

$\displaystyle\nu_{\tilde{g}^{2}}^{-1}=4-d_{\text{eff}}(\bar{R};0)-\frac{\partial}{\partial\tilde{g}^{2}}(\eta_{g}\tilde{g}^{2})\Big{|}_{\tilde{g}^{2}=\tilde{g}^{2}_{*}}\,.$
(84)

For a finite compactification radius, one has $d_{\rm eff}(\bar{R};0)>4$, so
that the first two terms in the right-hand side of Eq. (84) yield a negative
contribution to the critical exponent. Since in the Gaussian fixed point
$\tilde{g}_{*}^{2}=0$, the last term (anomalous dimension) in Eq. (84)
vanishes, the gauge coupling becomes irrelevant. This reflects the
perturbative nonrenormalizability of higher dimensional Yang-Mills theories.
On the other hand, at the nontrivial fixed point (83), a finite positive value
of the anomalous dimension is induced and consequently the total value of the
critical exponent could be positive. The right-hand side panel of Fig. 5 tells
us actually that the critical exponent of the gauge coupling is positive for
all compactification radii. Hence, the gauge coupling is a relevant coupling
and thus a free parameter. The value of the critical exponent
$\nu_{\tilde{g}^{2}}^{-1}$ especially for $\bar{R}\to\infty$ becomes

$\displaystyle\nu_{\tilde{g}^{2}}^{-1}=\begin{cases}\displaystyle\frac{61}{42}\simeq
1.45&\text{(resumed)}\,,\\\\[17.22217pt] \displaystyle 1&\text{(one-
loop)}\,.\end{cases}$ (85)

Figure 6: The one-loop beta function of $\tilde{g}^{2}$ as a function of
$\tilde{g}^{2}$. The purple points stand for the nontrivial fixed point, while
the red point indicates the Gaussian fixed point.

We next analyze the fixed point and the critical exponent for $\tilde{u}(0)$.
To this end, we note that the value of the anomalous dimension $\eta_{g}$ at
the UV fixed point is in $-1<\eta_{g}|_{\tilde{g}^{2}=\tilde{g}^{2}_{*}}<0$
for a finite compactification radius and therefore the effect of $\eta_{g}$ in
the vacuum energy (79) is negligible due to the suppression factor $1/6$. A
nontrivial UV fixed point for a certain $\bar{R}$ is obtained as

$\displaystyle\tilde{u}^{\rm
UV}_{*}(0)=\frac{23}{960\pi^{2}}+\frac{\bar{R}}{240\pi(2\pi\bar{R}-5\sinh(2\pi\bar{R}))}\,.$
(86)

In Fig. 7, this fixed point value of $\tilde{u}(0)$ is shown as a function of
$\bar{R}$. Its critical exponent is evaluated by using Eq. (79) as

$\displaystyle\lambda_{\tilde{u}(0)}=-\frac{\partial(\partial_{t}\tilde{u}(0))}{\partial\tilde{u}(0)}\Big{|}_{\tilde{u}(0)=\tilde{u}^{\rm
UV}_{*}(0)}=d_{\text{eff}}(\bar{R};0)\,.$ (87)

Thus, the value of the critical exponent can be read off from Fig. 4. It
becomes always positive for all compactification radii and thus relevant. In
addition to the nontrivial UV fixed point (86), a nontrivial IR fixed point is
found to be

$\displaystyle\tilde{u}^{\rm
IR}_{*}(0)=\frac{3}{32\pi^{2}\left(5-\frac{2\pi\bar{R}}{\sinh(2\pi\bar{R})}\right)}\,.$
(88)

Figure 7: The nontrivial UV (solid purple line) and IR (dashed red line) fixed
point values as a function of the dimensionless compactification radius
$\bar{R}$.

Finally, we show the renormalization group flow of the gauge coupling and the
vacuum energy by solving Eqs. (79) and (80) numerically. To do so, we use the
resumed anomalous dimension of the gauge field in Eq. (81). In the left-hand
side panel of Fig. 8, the renormalization group flow of the gauge coupling is
presented. We can see that for a fixed value of $\bar{R}$, the behavior of the
gauge coupling changes: In the energy region $k<1/R$, we observe the
asymptotically free behavior of $\tilde{g}^{2}$, namely the system can be
regarded as effectively a $D=4$ gauge theory. On the other hand, in the energy
region $k>1/R$ the gauge coupling turns to an increasing behavior and then
converges to the fixed point value (83). Hence, the gauge coupling is
asymptotically safe and we do not suffer from any UV divergence. In high
energy scales, the system behaves as a $D=5$ gauge theory. Such a dynamical
deformation of the dimension takes place thanks to the change of the effective
dimension $d_{\rm eff}(\bar{R};\theta_{H})$ which arises from the threshold
function (57).

Whereas the gauge coupling is finite in the deep UV regime, it diverges in the
IR regime. We expect from this that in such a strong interacting regime, the
gauge fields acquire a mass gap and we observe the confinement of the gauge
bosons. This implies that there exits a phase transition to the confinement
phase (denoted by $X$ in Table 1). The present setup however does not allow us
to address confinement phenomena. For those, more elaborate methods of the
functional renormalization group beyond the background approximation are
required. Instead, in this work, we infer from the finite temperature Yang-
Mills theory in $D=4$ how the functional renormalization group can access the
confinement phase and evaluate the gap mass of the gauge fields. We argue it
in the next section.

The renormalization group flow of $\tilde{u}(0)$ is exhibited in the right-
hand side panel of Fig. 8. In the deep UV regime, $\tilde{u}(0)$ converges to
the nontrivial UV fixed point (86) for $\bar{R}\to\infty$, while in the IR
regime, we observe the convergence of $\tilde{u}(0)$ to the nontrivial IR
fixed point (88) for $\bar{R}\to 0$. The transition between the nontrivial UV
and IR fixed points takes place around the energy scale $k\sim 1/R$. We
conclude now that the vacuum energy in pure Yang-Mills theories could be both
UV and IR finite.

Figure 8:  Flow of the gauge coupling $\tilde{g}^{2}$ (left) and the vacuum
energy (right) for several fixed values of the dimensionless compactification
radius $\bar{R}$. The energy unit of $k$ is arbitrary.

##  VII Mechanism of confinement

In the previous section, we have observed how the gauge coupling becomes
strong in the IR limit. Indeed, Monte-Carlo simulations for $SU(3)$ pure Yang-
Mills theory in $D=4$ at finite temperature (corresponding to
$\mathbb{R}^{3}\times S^{1}$) show the confinement-deconfinement phase
transition at a critical coupling Fukugita  _et al._ (1989). Within the
current setup, we cannot address the confinement dynamics as accounting for
the dynamical emergence of a longitudinal mass for the gauge fields would
require a more elaborate formulation of our setup within the FRG Fister and
Pawlowski (2013). Here, instead, we discuss a possible mechanism of the
confinement and make a small modification to the beta function of the
effective potential for AB phases.

In the renormalization group evolution of $V(\theta_{H})-V(0)$ which is shown
in Fig. 2, its value at the origin $\theta_{H}=0$ does not change from the
initial value, while the potential in nonvanishing $\theta_{H}$ region
evolutes towards larger values than the initial ones. This is because the beta
function of $V(\theta_{H})-V(0)$ receives positive contributions. Thus, in
order for the potential of AB phases (65) to produce the nontrivial
configuration $\theta_{H}\neq 0$ realizing the confinement phase, negative
contributions to its beta function are required. In the potential (65) the
gauge fields give positive contributions to the beta function, while the ghost
fields contributions are negative. Hence, we may here think of two potential
modifications to mimic confinement: first, enhancing the ghost-loop effects or
second, suppressing the gauge-loop effects. Naively thinking, both could be
realized by a large negative value of $\eta_{\rm gh}<-6$ and a large positive
value of $\eta_{g}>6$. However, this may be unlikely since $\eta_{g}$ and
$\eta_{\rm gh}$ are the second order of the derivative expansion for the
effective action; namely, such large values of the anomalous dimensions could
imply the breakdown of the derivative expansion and the higher derivative
operators cannot be ignored. This situation is thus not appropriate in the
current setup or its simple extensions.

A more realistic mechanism for the suppression of the gauge fields could be
considered: We could include a gap mass $m_{A}^{2}$ depending on the
compactification radius in the propagator of the gauge fields which suppresses
the loop effects of the gauge fields. More specifically, we modify the
propagator of the gauge fields so as to be

$\displaystyle\Pi(p^{2})=\frac{1}{p^{2}+M_{n,ij}^{2}+m_{A}^{2}(R)}\,.$ (89)

This modification is associated with the quantum corrections to the two-point
function $\Gamma_{k}^{(2)}(p)$. Specifically, from the vanishing point of the
full two-point function, one could read off a dressed mass $m_{A}$ such that
$\Gamma_{k}^{(2)}(p=m_{A})=0$. In the strong coupling regime, it is expected
that quantum corrections to vertices become significant. In this work,
however, we have used the classical propagators and vertices for the $n$-point
functions ($n\geq 2$) as the background field approximation, so that such a
dressed mass $m_{A}$ cannot be obtained. A systematic improvement for those
functions can be actually realized in the vertex expansion scheme Braun  _et
al._ (2016); Mitter  _et al._ (2015); Cyrol  _et al._ (2018a, b); Fischer  _et
al._ (2009); Fister and Pawlowski (2011, 2013); Herbst  _et al._ (2015); Cyrol
_et al._ (2016); Corell  _et al._ (2018); Eichmann  _et al._ (2021);
Napetschnig  _et al._ (2021); Horak  _et al._ (2022); Christiansen  _et al._
(2014, 2016, 2015); Meibohm and Pawlowski (2016); Christiansen (2016); Denz
_et al._ (2018); Christiansen  _et al._ (2018a, b); Eichhorn  _et al._ (2018a,
b); Pawlowski and Reichert (2020); Bonanno  _et al._ (2022); Fehre  _et al._
(2021) in the functional renormalization group,333 The Schwinger-Dyson
equation together with the functional renormalization group is also useful for
the evaluation of the full two-point function Gao and Pawlowski (2020, 2021a);
Gao  _et al._ (2021); Gao and Pawlowski (2021b).  but this is out of the scope
of the current work and then is left for future work.

From the functional renormalization group analysis for a Yang-Mills theory in
$D=4$ at finite temperature Cyrol  _et al._ (2018b), i.e. the gauge system in
$\mathbb{R}^{3}\times S^{1}$ where temperature $T$ corresponds to $1/(2\pi R)$
in the current setup, we emulate the behaviour of the mass gap as

$\displaystyle m_{A}^{2}(R)$ $\displaystyle=m^{2}_{D}(R)+m_{\rm
gap}^{2}\theta(k-k_{\rm conf})\,,$ (90)

where $\theta(x)$ is the step function: $\theta(x)\geq 1$ for $x>0$, while
$\theta(x)=0$ for $x<0$. Here, $k_{\rm conf}$ is the confinement scale and
$m_{D}$ is associated to the Debye screening mass which would naively take the
form

$\displaystyle m_{D}(R)$ $\displaystyle=\frac{c_{D}}{2\pi R}\,,$ (91)

as observed in finite temperature gauge theories. At the one-loop level, the
Debye screening mass is obtained by the evaluation of the (non-perturbative)
daisy diagram and then it turns out that the parameter $c_{D}$ is proportional
to the gauge coupling; $c_{D}\sim g$. There is a prescription taking into
account higher-order effects in QCD Arnold and Yaffe (1995) and it modifies
that the form of the Debye screening mass from Eq. (91). But, in this current
rough sketch of the gauge-field mass, we simply treat $c_{D}$ as a constant
parameter and set it unity: $c_{D}=1$. Then, there are two free parameters:
$m_{\rm gap}^{2}$ represents the magnitude of the gap mass and $k_{\rm conf}$
is the confinement scale at which the gauge fields are gapped.

Including the gap mass (90) into the propagators of the gauge fields, the flow
equation for the potential $V(\theta_{H})$ is given by

$\displaystyle\partial_{t}V(\theta_{H})$
$\displaystyle=\frac{5k^{4}}{2(4\pi)^{2}}\widehat{\mathcal{I}}_{N}(\bar{R};\theta_{H};\bar{m}^{2}_{A})-\frac{k^{4}}{(4\pi)^{2}}{\mathcal{I}}_{N}(\bar{R};\theta_{H})\,,$
(92)

Here the threshold function for the gauge fields are modified by including the
gap mass so as to be

$\displaystyle\widehat{\mathcal{I}}_{N}(\bar{R};\theta_{H};\bar{m}^{2}_{A})$
$\displaystyle=\sum_{i,j=1}^{N}\frac{i\pi\bar{R}}{2\sqrt{1+\bar{m}^{2}_{A}}}\bigg{[}\cot\left(\frac{1}{2}\bar{R}\left(\theta_{i}-\theta_{j}+2i\pi\sqrt{1+\bar{m}^{2}_{A}}\right)\right)$
$\displaystyle\quad+\cot\left(-\frac{1}{2}\bar{R}\left(\theta_{i}-\theta_{j}-2i\pi\sqrt{1+\bar{m}^{2}_{A}}\right)\right)\bigg{]}\,,$
(93)

with $\bar{m}^{2}_{A}(\bar{R})=m^{2}_{A}(R)/k^{2}$ the dimensionless gap mass.
Moreover, note that
$\widehat{\mathcal{I}}_{N}(\bar{R};\theta_{H};\bar{m}^{2}_{A}=0)={\mathcal{I}}_{N}(\bar{R};\theta_{H})$.

We solve the flow equation $\partial_{t}(V(\theta_{H})-V(0))$ with the initial
condition $V(\theta_{H})-V(0)=0$ and $\bar{R}=1$ at $k=\Lambda=10$. To
represent the evolution of the potential, we set $m_{\rm gap}^{2}=10^{2}$ and
$k_{\rm conf}=9.9$. In the left-hand side panel of Fig. 9, we display the
renormalization group evolution of the potential $V(\theta_{H})-V(0)$ with
$k_{\rm conf}=9.9$ at the $\theta_{2}=0$ plane. We see that the potential
evolves towards positive values until $k=k_{\rm conf}=9.9$ at which the gap
mass enters into the propagator of the gauge fields. After $k=k_{\rm conf}$,
the gauge-field contributions are suppressed and the ghost fields dominate. In
the IR scale $k\to 0$, the potential converges to the red line and has
nontrivial vacua at $\theta_{1}=\pm 2\pi/3$ and $\theta_{2}=0$. The right-hand
panel of Fig. 9 is the contour plot of $V(\theta_{H})-V(0)=0$ at $k=0$ in
$\theta_{1}$–$\theta_{2}$ plane. Minima of the potential are located at
$(\theta_{1},\,\theta_{2},\,\theta_{3})=(0,\,\frac{2\pi}{3},\,-\frac{2\pi}{3})$
and its permutations which entails $P=0$. Hence, this case demonstrates the
potential in the confinement phase denoted by $X$. The distribution of AB
phases in the confinement phase may be described by the Haar measure; see Ref.
Cossu  _et al._ (2014). Fig. 10 exhibits the potential at the IR scale $k\to
0$ for various values of $m_{\rm gap}^{2}$ and $k_{\rm conf}$. We see that for
larger $m_{\rm gap}^{2}$ and $k_{\rm conf}$, the potential tends to have
confinement vacua due to enough suppressions of loop effects of the gauge
fields.

Figure 9:  The potential with $m^{2}_{\rm gap}=10^{2}$ and $k_{\rm conf}=9.9$
in a confinement-emulated scenario. Left: The renormalization evolution of the
potential at the $\theta_{2}=0$ plane from $\Lambda=10$ to $k=0$. Right:
Contour plot of the potential at the IR scale $k\to 0$ in
$\theta_{1}$–$\theta_{2}$ plane.

Figure 10:  The IR potential at the $\theta_{2}=0$ plane in a confinement-
emulated scenario. Left: Various values of $m^{2}_{\rm gap}$ with $k_{\rm
conf}=9$ fixed. Right: Various values of $k_{\rm conf}$ with with $m^{2}_{\rm
gap}=10^{2}$ fixed.

##  VIII Summary and Discussions

In this work, we have studied the $SU(N)$ pure Yang-Mills theory in
$\mathbb{R}^{4}\times S^{1}$ by using the functional renormalization group. We
have derived the flow equations for the potential of the AB phases and the
gauge coupling. It has been shown that there exists a nontrivial UV fixed
point for the gauge coupling and the vacuum energy, which entails asymptotic
safety of the $SU(N)$ pure Yang-Mills theory in $\mathbb{R}^{4}\times S^{1}$.
At the nontrivial UV fixed point, those couplings are relevant parameters,
i.e. free parameters in the system. We have seen also the dynamical and smooth
transition of their renormalization group flows as in Fig. 8 around the energy
scale of order of the inverse compactification radius. Thus, this system is UV
complete and does not lose the predictivity to the low energy dynamics. These
facts can be addressed only by nonperturbative treatments.

We have investigated only the two lowest order operators, i.e. $V(\theta_{H})$
and $F_{MN}F^{MN}$ in the effective action; however in general an infinite
number of effective operators is induced by quantum effects. In order to
confirm the nonperturbative renormalizability of Yang-Mills theory in five
dimensions, higher dimensional operators such as $(F_{MN}F^{MN})^{2}$ must be
studied. Nevertheless, as discussed in Section VI, whether higher dimensional
operators are still irrelevant or not relies on the magnitude of the anomalous
dimension induced at the nontrivial fixed point. We expect that the magnitude
of $\eta_{g}$ does not drastically change from the current setup since impacts
of the higher dimensional operators on $\eta_{g}$ tend to be weaker than the
lower dimensional ones. It implies that only the gauge coupling $\tilde{g}$ is
a free parameter for the prediction to the low energy dynamics and all higher
dimensional operators are predicted to be the nontrivial UV fixed point value
in the $SU(N)$ pure Yang-Mills theory in $\mathbb{R}^{4}\times S^{1}$. Note
that the $\tilde{u}(0)$ does not play a role for the gauge dynamics in the
system since it does not couple to any fields. In a gravity-matter system,
however, $\tilde{u}(0)$ corresponds to the cosmological constant which couples
to the metric field.444 More specifically, the cosmological constant term
reads  $\displaystyle\int\text{d}^{D}x\sqrt{g}\,\tilde{u}(0)\,,$ (94) where
$\sqrt{g}$ is the determinant of the metric field.  Thus, the relevance of the
cosmological constant is significant for the prediction to the low energy
dynamics in a gravity-matter system. This is part of the asymptotically safe
scenario for quantum gravity Weinberg (1979); Reuter (1998); Souma (1999). See
Refs. Niedermaier and Reuter (2006); Niedermaier (2007); Percacci (2007);
Codello  _et al._ (2009); Reuter and Saueressig (2012); Percacci (2017);
Reuter and Saueressig (2019); Wetterich (2019); Pawlowski and Reichert (2020);
Bonanno  _et al._ (2020); Reichert (2020); Eichhorn (2018a, b, 2022) for
reviews. We found an IR fixed point of the cosmological constant in addition
to the UV one. In a gravity-matter system, such an IR fixed point may be a key
point to resolve the cosmological constant problem Wetterich (2017, 2018b). We
also note that the existence of a nontrivial fixed point in $SU(N)$ Yang-Mills
theory in $D>5$ has been discussed in Ref. Gies (2003) where a critical
dimension $D_{\rm cr}$ beyond which the nontrivial fixed point disappears was
found to be $5\lesssim D_{\rm cr}<6$.

As can be seen from the behavior of the gauge coupling in the low energy
regime (see the left-hand side panel of Fig. 8), the gauge coupling becomes
large. This may lead to confinement and induce a mass gap of the gauge field.
Within the current setup, unfortunately we cannot address the dynamics of the
confinement. To study confinement phenomena, we need improvements of the
approximation. This can be systematically done within the functional
renormalization group with the vertex expansion scheme Fischer  _et al._
(2009); Fister and Pawlowski (2011, 2013); Braun  _et al._ (2016); Mitter  _et
al._ (2015); Herbst  _et al._ (2015); Cyrol  _et al._ (2016, 2018a, 2018b);
Corell  _et al._ (2018, 2018); Eichmann  _et al._ (2021); Napetschnig  _et
al._ (2021); Horak  _et al._ (2022); Christiansen  _et al._ (2014, 2016,
2015); Meibohm and Pawlowski (2016); Christiansen (2016); Denz  _et al._
(2018); Christiansen  _et al._ (2018a, b); Eichhorn  _et al._ (2018a, b);
Pawlowski and Reichert (2020); Bonanno  _et al._ (2022); Fehre  _et al._
(2021). We can take into account vertex corrections of the gauge fields.
Indeed, confinement phenomena in $D=4$ QCD at zero and finite temperatures
have been observed by using the functional renormalization group Fischer  _et
al._ (2009); Fister and Pawlowski (2011, 2013); Cyrol  _et al._ (2016, 2018b);
Herbst  _et al._ (2015). See also Refs. Braun  _et al._ (2010a); Eichhorn  _et
al._ (2011); Braun  _et al._ (2010b); Haas  _et al._ (2013). In a future work,
the same technique will be applied for $SU(N)$ pure Yang-Mills theory in
$\mathbb{R}^{4}\times S^{1}$. Instead, in this work, we have demonstrated the
confinement potential by adding a gap mass $m_{\rm gap}$ of the gauge fields
at a certain scale $k_{\rm conf}$ to the propagator.

Monte Carlo simulations based on lattice gauge theory are important for the
elucidation of $SU(N)$ pure Yang-Mills theories. In order to remove artifacts
due to a finite lattice spacing, the continuum limit is necessary. However,
the existence of such continuum limit in $\mathbb{R}^{4}\times S^{1}$ gauge
theories is not obvious. Therefore, Monte-Carlo studies of the Hosotani
mechanism have been made in $\mathbb{R}^{3}\times S^{1}$ thanks to its
asymptotically free nature Cossu  _et al._ (2014), instead of
$\mathbb{R}^{4}\times S^{1}$. The present work indicates the existence of a
nontrivial UV fixed point to which the continuum limit can be taken. Indeed,
the existence of such continuum limit in $SU(N)$ pure Yang-Mills theories in
$\mathbb{R}^{4}\times S^{1}$ has been explored by Refs. Kawai  _et al._
(1992); Ejiri  _et al._ (2000, 2002); Kurkela  _et al._ (2009); de Forcrand
_et al._ (2010); Del Debbio  _et al._ (2012); Itou  _et al._ (2014); Knechtli
and Rinaldi (2016); Florio  _et al._ (2021). Complementary studies on higher
dimensional $SU(N)$ pure Yang-Mills theories will be more important in order
to elucidate the dynamics and phase structure of gauge theories.

Finally, we briefly mention the status on Gauge-Higgs unification models. A
simple $SU(N)$ Yang-Mills theory in $\mathbb{R}^{3}\times S^{1}$ is actually
not realistic as a model of Gauge-Higgs unification. One of the reasons is the
absence of chiral fermions in $\mathbb{R}^{3}\times S^{1}$ spacetime. To
introduce those in higher dimensional spacetime, one could consider the
orbifold Pomarol and Quiros (1998), i.e. $\mathbb{R}^{3}\times S^{1}/Z_{2}$.
For instance, $SU(3)$ symmetry is broken into $SU(2)_{L}\times U(1)_{Y}$ by a
boundary condition of the orbifold. Then, depending on the configuration of AB
phases, the $SU(2)_{L}\times U(1)_{Y}$ symmetry is further broken by the
Hosotani mechanism into its subgroups such as $U(1)_{\rm em}$ and $U(1)_{\rm
em}\times U(1)_{Z}$ Kubo  _et al._ (2002). Such a model however cannot
reproduce the experimental results Kubo  _et al._ (2002). A realistic model of
Gauge-Higgs unification has been built on the Randall-Sundrum warped spacetime
Randall and Sundrum (1999) and its gauge group is given as $SO(5)\times
U(1)_{X}$: See e.g. Agashe  _et al._ (2005); Medina  _et al._ (2007); Hosotani
_et al._ (2008). Hence, a Gauge-Higgs unification model for the standard model
is constructed as a $SU(3)_{c}\times SO(5)\times U(1)_{X}$ theory; see e.g.
Funatsu  _et al._ (2020a, b). Moreover, enlarged gauge groups involving
$SU(3)_{c}\times SO(5)\times U(1)_{X}$ as their subgroup have been discussed
as Gauge-Higgs grand unification theories Haba and Shimizu (2003); Lim and
Maru (2007); Kojima  _et al._ (2011); Hosotani and Yamatsu (2015); Yamatsu
(2016); Furui  _et al._ (2016); Angelescu  _et al._ (2022). A UV completion
for those models in the asymptotic safety scenario would be an interesting
theme.

### Acknowledgements

We thank Jan M. Pawlowski and Florian Goertz for many valuable discussions and
comments. M. Y. thanks also Felipe Attanasio and Kin-ya Oda for helpful
discussions. This work is supported by the DFG Collaborative Research Centre
“SFB 1225 (ISOQUANT)”, Germany’s Excellence Strategy EXC-2181/1-390900948 (the
Heidelberg Excellence Cluster STRUCTURES). The work of M. Y. is supported by
the Alexander von Humboldt Foundation.

##  Appendix A Heat kernel technique

In this appendix, we summarize the heat kernel expansion technique. For a
review on this topic, see Refs. Vassilevich (2003); Codello (2015). This
technique is used to derive the beta function for the gauge coupling in
Appendix B.

###  A.1 Heat kernel expansion

We consider the following quantity

$\displaystyle\zeta=\frac{1}{2}{\rm tr}\,_{(i)}W(\Delta_{i})\,,$ (95)

where $\Delta_{i}$ is the Laplacian acting on a spin-$i$ field and ${\rm
tr}\,_{(i)}$ denotes the trace for the Laplacian (sum of eigenvalues of
$\Delta_{i}$). We perform the Laplace transform

$\displaystyle{\rm
tr}\,_{(i)}W(\Delta_{i})=\int^{\infty}_{0}\text{d}s\,\widetilde{W}(s)\,{\rm
tr}\,_{(i)}\left[e^{-s\Delta_{i}}\right]\,.$ (96)

Let us here define

$\displaystyle K(s,x,y)=\langle x|e^{-s\Delta_{i}}|y\rangle\,,$ (97)
$\displaystyle K(s)=\int\text{d}^{d}x\,K(s,x,x)={\rm
tr}\,_{(i)}\left[e^{-s\Delta_{i}}\right]\,.$ (98)

The kernel $K(s,x,y)$ satisfies the heat diffusion equation

$\displaystyle\left(\frac{\partial}{\partial
s}+\Delta_{i}\right)K(s,x,y)=0\,,$ (99)

with $\displaystyle\lim_{s\to 0}K(s,x,y)=\delta^{d}(x-y)$. Therefore, $K(s)$
is called “heat kernel”. In particular, when the Laplacian is simply given as
$-\partial^{2}$, its solution is

$\displaystyle K(s,x,y)=\frac{\exp\left(-\frac{(x-y)^{2}}{4s}\right)}{(4\pi
s)^{d/2}}\,.$ (100)

We now consider the solution of Eq. (99) with a general covariant derivative.
Supposing $s$ to be small, we expand $K(s,x,y)$ into a polynomial of $s$ and
obtain

$\displaystyle K(s,x,y)=\frac{\exp\left(-\frac{(x-y)^{2}}{4s}\right)}{(4\pi
s)^{d/2}}\sum_{j=0}s^{j}{\mathcal{B}}^{(i)}_{j}(x,y)\,,$ (101)

from which we give

$\displaystyle K(s)=K(s,x,x)=\frac{1}{(4\pi
s)^{d/2}}\sum_{j=0}s^{j}{\mathcal{A}}^{(i)}_{j}\,.$ (102)

with

$\displaystyle{\mathcal{A}}^{(i)}_{j}=\int\text{d}^{d}x\,{\mathcal{B}}^{(i)}_{j}(x,x)\,.$
(103)

This expansion is the so-called “heat kernel expansion”. When
$\Delta=-\partial^{2}$, we see from Eq. (100) that
${\mathcal{B}}^{(i)}_{0}(x,y)=1$ and ${\mathcal{B}}_{j}^{(i)}=0$ for $j>0$,
and then ${\mathcal{A}}^{(i)}_{j}=\int\text{d}^{d}x\equiv{\mathcal{V}}_{d}$.
Inserting the heat kernel expansion (102) into Eq. (96), we obtain

$\displaystyle{\rm tr}\,_{(i)}W(\Delta_{i})$
$\displaystyle=\frac{1}{(4\pi)^{d/2}}\sum_{j=0}Q_{\frac{d}{2}-j}[W]{\rm
tr}\,_{(i)}{\mathcal{A}}^{(i)}_{j}\equiv\frac{1}{(4\pi)^{d/2}}\sum_{j=0}\int\text{d}^{d}x\,Q_{\frac{d}{2}-j}[W]b^{(i)}_{j}\,.$
(104)

Here, the threshold functions are defined by

$\displaystyle Q_{n}[W]$
$\displaystyle=\int^{\infty}_{0}\text{d}s\,s^{-n}\widetilde{W}(s)=\begin{cases}\displaystyle\frac{1}{\Gamma(n)}\int^{\infty}_{0}\text{d}z\,z^{n-1}W(z)&\text{for
$n\geq 1$}\\\\[17.22217pt]
\displaystyle(-1)^{-n}\frac{\partial^{-n}W(z)}{\partial
z^{-n}}\Big{|}_{z=0}&\text{for $n\leq 0$}\end{cases}\,,$ (105)

where the Mellin transformation is performed in the second equality. To
summarize, the heal kernel expansion for the flow generator (95) is given by

$\displaystyle\zeta=\frac{1}{2(4\pi)^{d/2}}\int\text{d}^{d}x\left[b_{0}^{(i)}Q_{\frac{d}{2}}[W]+b_{1}^{(i)}Q_{\frac{d}{2}-1}[W]+\cdots\right]\,.$
(106)

Here, $b_{n}^{(i)}$ are the heat kernel coefficients. For $\Delta=-D^{2}+U$,
those are evaluated by

$\displaystyle b_{0}^{(i)}$ $\displaystyle={\rm tr}\,[I]\,,$ (107)
$\displaystyle b_{1}^{(i)}$ $\displaystyle={\rm tr}\,[U]\,,$ (108)
$\displaystyle b_{2}^{(i)}$ $\displaystyle={\rm
tr}\,\left[\frac{1}{2}U^{2}+\frac{1}{6}D^{2}U+\frac{1}{12}[D_{\rho},D_{\sigma}][D^{\rho},D^{\sigma}]\right]\,,$
(109)

where $I$ is the unity matrix. Let us now evaluate these coefficients
explicitly in a Yang-Mills theory for which we take
$(U)^{ab}=2i(F_{\mu\nu})^{ab}=2f^{abc}F^{c}_{\mu\nu}$ in the adjoint
representation. The trace acts on Lorentz and adjoint color spaces. The heat
kernel coefficients for $\Delta$ acting on spin-1 vector field are computed by

$\displaystyle b^{(1)}_{0}$ $\displaystyle={\rm tr}\,_{\text{Lorentz}}[I]{\rm
tr}\,_{\text{adj}}[I]=[\delta^{\mu\nu}\delta_{\mu\nu}][\delta^{ab}\delta_{ab}]=d(N^{2}-1)\,,$
(110) $\displaystyle b_{1}^{(1)}$ $\displaystyle={\rm
tr}\,_{\text{Lorentz}}{\rm tr}\,_{\text{adj}}[2f^{abc}F^{a\mu\nu}]=0\,,$ (111)
$\displaystyle b_{2}^{(1)}$ $\displaystyle={\rm tr}\,_{\text{Lorentz}}{\rm
tr}\,_{\text{adj}}\left[\frac{1}{2}U^{2}+\frac{1}{12}+\frac{1}{12}[D_{\mu},D_{\nu}][D^{\mu},D^{\nu}]\right]$
$\displaystyle=\delta^{ab}\delta^{\mu\nu}\bigg{[}\frac{1}{2}(2f^{adc}F^{c}_{\mu\rho})(2f^{dbe}F^{e\rho}{}_{\nu})+\frac{\delta_{\mu\nu}}{12}(-f^{adc}F^{c}_{\rho\sigma})(-f^{dbe}F^{e\rho\sigma})\bigg{]}$
$\displaystyle=\frac{(24-d)N}{12}F^{a}_{\mu\nu}F^{a\mu\nu}\,,$ (112)

where we used

$\displaystyle{\rm tr}\,_{\rm
adj}(f^{abc}f^{bde})=f^{abc}f^{bae}=N\delta^{ce}\,,$ (113)

and $(T^{a})_{bc}=-if^{abc}$ and
$[D_{\mu},D_{\nu}]^{ab}=-f^{abc}F^{c}_{\mu\nu}$ from which one finds

$\displaystyle
U^{2}=(2iF_{\mu\rho})_{ab}(2iF^{\rho}{}_{\nu})_{bd}=4(F^{c}_{\mu\rho}T^{c})_{ab}(F^{e\rho}{}_{\nu}T^{e})_{bd}=f^{abc}f^{bde}F^{c}_{\mu\rho}F^{e\rho}{}_{\nu}\,,$
(114)
$\displaystyle[D_{\rho},D_{\sigma}]^{ac}[D^{\rho},D^{\sigma}]^{cb}=f^{abc}f^{bde}F^{c}_{\rho\sigma}F^{e\rho\sigma}\,.$
(115)

Note $b^{(1)}_{1}=0$ in Yang-Mills theories because there is no corresponding
Lorentz- and gauge-invariant operator in Yang-Mills gauge theories
($\delta^{\mu\nu}F_{\mu\nu}=0$), while in gravity the curvature scalar (Ricci
scalar) is an invariant operator with the finite value of $b^{(1)}_{1}$. In
the same manner, the heat kernel coefficients for $\Delta$ acting on spin-0
(scalar) field with $U=0$ are found so that

$\displaystyle b^{(0)}_{0}$ $\displaystyle=N^{2}-1\,,$ (116) $\displaystyle
b_{1}^{(0)}$ $\displaystyle=0\,,$ (117) $\displaystyle b_{2}^{(0)}$
$\displaystyle=\frac{N}{12}F^{a}_{\mu\nu}F^{a\mu\nu}\,.$ (118)

Here, we note an alternative introduction of the heat kernel Wetterich and
Yamada (2019): One starts from writing the flow kernel as

$\displaystyle\zeta=\frac{1}{2}{\rm
tr}\,_{(i)}W(\Delta_{i})=\frac{1}{2}\sum_{m}W(\lambda_{m}^{(i)})\,,$ (119)

where $\lambda_{m}^{(i)}$ are (positive) eigenvalues of the Laplacian. Using a
definition of the $\delta$-function

$\displaystyle\delta(z-\lambda_{m}^{(i)})=\int_{a-i\infty}^{a+i\infty}\frac{\text{d}s}{2\pi
i}e^{s(z-\lambda_{m}^{(i)})}\,,$ (120)

Eq. (119) can be rewritten as

$\displaystyle\zeta$
$\displaystyle=\frac{1}{2}\sum_{m}\int\text{d}z\,\delta(z-\lambda_{m}^{(i)})W(z)=\frac{1}{2}\sum_{m}\int\text{d}z\,W(z)\int_{a-i\infty}^{a+i\infty}\frac{\text{d}s}{2\pi
i}e^{sz}{\rm Tr}\,_{(i)}\left[e^{-s\Delta_{i}}\right]\,,$ (121)

where we used $\sum_{m}e^{-s\lambda_{m}^{(i)}}={\rm
tr}\,_{(i)}e^{-s\Delta_{i}}$. Inserting the heat kernel expansion (102), we
obtain again

$\displaystyle{\rm tr}\,_{(i)}W(\Delta_{i})$
$\displaystyle=\frac{1}{(4\pi)^{d/2}}\sum_{j=0}\int\text{d}^{d}x\,Q_{\frac{d}{2}-j}[W]b^{(i)}_{j}\,.$
(122)

Here the threshold function reads in this case

$\displaystyle Q_{n}[W]$
$\displaystyle=\int\text{d}z\,W(z)\int_{a-i\infty}^{a+i\infty}\frac{\text{d}s}{2\pi
i}e^{sz}s^{-n}=\frac{1}{\Gamma(n)}\int\text{d}z\,z^{n-1}W(z)\,,$ (123)

where we have used

$\displaystyle\int_{a-i\infty}^{a+i\infty}\frac{\text{d}s}{2\pi
i}e^{sz}s^{-n}$
$\displaystyle=z^{n-1}\frac{i}{2\pi}\int_{\tilde{a}-i\infty}^{\tilde{a}+i\infty}\text{d}\tilde{s}\,e^{-\tilde{s}}(-\tilde{s})^{-n}=\frac{z^{n-1}}{\Gamma(n)}\,,$
(124)

with $\tilde{s}=-sz$ and $\tilde{a}=-a/z$. This gives the same result as Eq.
(105).

##  Appendix B Derivation of flow equations

We show the details of the computations for the flow equations for the
potential and the gauge coupling using the heat kernel technique.

###  B.1 Hessian and background field approximation

We consider fluctuations of the gauge field around a nonvanishing background
$\bar{A}_{M}$, i.e.

$\displaystyle A_{M}=\bar{A}_{M}+a_{M}\,.$ (125)

Here and hereafter, the bar denotes the background field. Our starting
effective action is

$\displaystyle\Gamma_{k}$
$\displaystyle=\frac{Z_{k}}{2g^{2}}\,\int\text{d}^{5}x\,{\rm
tr}\,F_{MN}F^{MN}+\frac{Z_{k}}{\xi g^{2}}\int\text{d}^{5}x\,{\rm
Tr}\,[\mathcal{F}(a)]^{2}+Z_{\text{gh}}\int\text{d}^{5}x\,{\rm
Tr}\,\left[\bar{c}\mathcal{M}c\right]\,.$ (126)

with the $R_{\xi}$-gauge fixing function
$\mathcal{F}(a)=\delta^{\mu\nu}\bar{D}_{\mu}a_{\nu}+\xi\bar{D}_{5}a_{5}$ and
the ghost kinetic operator
$\mathcal{M}=\delta^{\mu\nu}\bar{D}_{\mu}D_{\nu}+\xi\bar{D}_{5}D_{5}$.

A central object in the functional renormalization group is the full two-point
function $\Gamma_{k}^{(2)}$ (Hessian) which is obtained by the second
functional derivative of $\Gamma_{k}$ with respect to fields. To get it, we
evaluate the variations for the effective action

$\displaystyle\Gamma_{k}=\Gamma_{k}+\delta\Gamma_{k}+\frac{1}{2}\delta^{2}\Gamma_{k}\,.$
(127)

We need the second order variational term $\delta^{2}\Gamma_{k}$ to read off
$\Gamma_{k}^{(2)}$. The second order variation of $A_{M}$ yields

$\displaystyle\delta^{2}\Gamma_{k}$
$\displaystyle=\frac{Z_{k}}{g^{2}}\int\text{d}^{5}x\,a_{M}\left[({\mathcal{D}}_{T})^{MN}+D^{M}D^{N}\right]a_{M}$
$\displaystyle\qquad-\frac{Z_{k}}{\xi
g^{2}}\int\text{d}^{5}x\left[a_{\mu}\bar{D}^{\mu}\bar{D}^{\nu}a_{\nu}+\xi
a_{\mu}\bar{D}^{\mu}\bar{D}_{5}a_{5}+\xi
a_{5}\bar{D}_{5}\bar{D}^{\nu}a_{\nu}+\xi^{2}a_{5}(\bar{D}_{5}^{2})a_{5}\right]\,,$
(128)

where the derivative operator is defined as

$\displaystyle({\mathcal{D}}_{T})_{MN}$
$\displaystyle=(-D_{\mu}^{2}-D_{5}^{2})\delta_{MN}+2iF_{MN}\,.$ (129)

The background field approximation enforces then

$\displaystyle\Gamma_{k}^{(2)}[\bar{A},A]=\Gamma^{(2)}_{k}[\bar{A},A]\Big{|}_{\bar{A}=A}=\Gamma_{k}^{(2)}[\bar{A}]+S_{\text{gf}}^{(2)}[\bar{A}]\,.$
(130)

Here, we suppose a background gauge field $\bar{A}_{M}$ such that
$\bar{F}_{MN}=(1-\delta_{M5})(1-\delta_{N5})\bar{F}_{MN}\to\bar{F}_{\mu\nu}$
where $\bar{A}_{5}$ is given by Eq. (20). The mixing terms between $A_{\mu}$
and $a_{5}$ are eliminated thanks to the $R_{\xi}$-type gauge fixing. Hence,
the Hessians for $A_{\mu}$ and $A_{5}$ are given independently as

$\displaystyle(\Gamma_{k}^{(2)}[\bar{A}])^{\mu\nu}=\frac{Z_{k}}{g^{2}}\left\\{(\bar{\mathcal{D}}_{T})^{\mu\nu}-\delta^{\mu\nu}\bar{D}_{5}^{2}+\left(1-\frac{1}{\xi}\right)\bar{D}^{\mu}\bar{D}^{\nu}\right\\}\,,$
(131)
$\displaystyle\Gamma_{k}^{(2)}[\bar{A}]=\frac{Z_{k}}{g^{2}}\left(\bar{\Delta}-\xi\bar{D}_{5}^{2}\right)\,.$
(132)

with $\bar{\Delta}=-\bar{D}^{2}$. The Hessian for ghost fields reads from the
last term in Eq. (126) as

$\displaystyle
S^{(2)}_{\text{gh}}[\bar{A}]=Z_{\text{gh}}\left(\bar{\Delta}-\xi\bar{D}_{5}^{2}\right)\,.$
(133)

For the background $\bar{A}_{5}$ given in Eq. (20) and
$\varphi=\varphi^{a}T^{a}=(a_{\mu},a_{5},c,\bar{c})$ in $\mathbb{R}^{4}\times
S^{1}$, the covariant derivative $\bar{D}_{5}$ turns to the KK mass spectra as

$\displaystyle(\bar{D}_{5}\varphi)_{ij}=\partial_{5}\varphi_{ij}-[\bar{A}_{5},\varphi]_{ij}=i\left[\frac{n}{R}-\frac{1}{2\pi
R}(\theta_{i}-\theta_{j})\right]\varphi_{ij}=iM_{n,ij}\varphi_{ij}\,.$ (134)

The propagators, i.e. the inverse forms of Eqs. (131)–(133) are given in Eqs.
(46)–(48).

We employ the regulator $\mathcal{R}_{k}$ such that

$\displaystyle\Big{(}\Gamma_{k}^{(2)}+\mathcal{R}_{k}(\bar{\mathcal{D}}_{T})\Big{)}_{a_{\mu}a_{\nu}}^{\mu\nu}$
$\displaystyle=\frac{Z_{k}}{g^{2}}\left\\{(P_{k}(\bar{\mathcal{D}}_{T}))^{\mu\nu}+\delta^{\mu\nu}M_{n,ij}^{2}+\left(1-\frac{1}{\xi}\right)\bar{D}^{\mu}\bar{D}^{\nu}\right\\}\,,$
(135)
$\displaystyle\Big{(}\Gamma_{k}^{(2)}+\mathcal{R}_{k}(\bar{\Delta})\Big{)}_{a_{5}a_{5}}$
$\displaystyle=\frac{Z_{k}}{g^{2}}\left(P_{k}(\bar{\Delta})+\xi
M_{n,ij}^{2}\right)\,,$ (136)
$\displaystyle\Big{(}\Gamma_{k}^{(2)}+\mathcal{R}_{k}(\bar{\Delta})\Big{)}_{\bar{c}c}$
$\displaystyle=Z_{\text{gh}}\left(P_{k}(\bar{\Delta})+\xi
M_{n,ij}^{2}\right)\,.$ (137)

Their inverse forms correspond to the regulated propagators for $a_{\mu}$,
$a_{5}$ and the ghost fields. More specifically, we find

$\displaystyle\hat{\Pi}^{(n)}_{\mu\alpha,ij}(P_{k})$
$\displaystyle=\frac{g^{2}}{Z_{k}}\bigg{[}\frac{\delta_{\mu\nu}}{P_{k}+M_{ij,n}^{2}}+\frac{p_{\mu}p_{\nu}}{M^{2}_{ij,n}}\left(\frac{1}{p^{2}+M_{ij,n}^{2}}-\frac{1}{P_{k}+\xi
M_{ij,n}^{2}}\right)\bigg{]}\,,$ (138)
$\displaystyle\hat{\Pi}^{(n)}_{55,ij}(P_{k})$
$\displaystyle=\frac{g^{2}}{Z_{k}}\frac{1}{P_{k}+\xi M_{ij,n}^{2}}\,,$ (139)
$\displaystyle\hat{\Pi}_{\text{gh}}^{(n)}{}_{ij}(P_{k})$
$\displaystyle=\frac{1}{Z_{\rm gh}}\frac{1}{P_{k}+\xi M_{ij,n}^{2}}\,.$ (140)

###  B.2 Heat kernel expansion for Yang-Mills theory

We apply the heat kernel technique for the Yang-Mills theory in
$\mathbb{R}^{4}\times S^{1}$ spacetime. The flow equation for $\Gamma_{k}$ is
given by

$\displaystyle\partial_{t}\Gamma_{k}$ $\displaystyle=\frac{1}{2}{\rm
Tr}\,_{(1)}\left[\hat{\Pi}^{(n)}_{\mu\alpha,ij}(P_{k})(\partial_{t}{\mathcal{R}}_{k})^{\alpha\nu}\right]+\frac{1}{2}{\rm
Tr}\,_{(0)}\left[\hat{\Pi}^{(n)}_{55,ij}(P_{k})\partial_{t}{\mathcal{R}}_{k}\right]-{\rm
Tr}\,_{(0)}\left[\hat{\Pi}_{\text{gh}}^{(n)}{}_{ij}(P_{k})\partial_{t}{\mathcal{R}}_{k}\right]$
$\displaystyle=\zeta_{4}+\zeta_{5}+\zeta_{\rm gh}\,.$ (141)

We note that ${\rm Tr}\,_{(s)}$ stands for the functional trace acting on all
internal spaces in which a spin-$s$ field is defined. More specifically, in
the current system, the functional trace is decomposed as

$\displaystyle{\rm Tr}\,_{(s)}[\mathcal{O}]={\rm
tr}\,_{\text{Lorentz}}\otimes{\rm tr}\,_{\text{KK}}\otimes{\rm
tr}\,_{\text{color}}\otimes{\rm
tr}\,_{(s)}[\mathcal{O}(\bar{\Delta}_{(s)})]=\sum_{\mu=0}^{3}\delta^{\mu\nu}\sum_{n=-\infty}^{\infty}{\rm
tr}\,_{\text{color}}{\rm
tr}\,_{(s)}{\mathcal{O}}_{\mu\nu;ij;n}(\bar{\Delta}_{(s)})\,,$ (142)

where ${\rm tr}\,_{\text{color}}$ acts on the internal space of $SU(N)$.

Let us first evaluate the flow generator for $A_{\mu}$ with the Hessian (131).
To this end, we take the Feynman gauge $\xi=1$. As discussed in Section IV.4,
the dynamics of pure Yang-Mills theories entails the symmetric vacuum $\langle
A_{5}\rangle=0$. Therefore, we can set $\theta_{i}=0$, i.e. $M_{n,ij}^{2}=0$.

The flow generator for $a_{\mu}$ (the first term in Eq. (141)) reads

$\displaystyle\zeta_{4}$ $\displaystyle=\frac{1}{2}{\rm
Tr}\,_{(1)}W[\bar{\mathcal{D}}_{T}]=\frac{1}{2}{\rm
Tr}\,_{(1)}\frac{\partial_{t}R_{k}(\bar{\mathcal{D}}_{T})-\eta_{g}R_{k}(\bar{\mathcal{D}}_{T})}{P_{k}(\bar{\mathcal{D}}_{T})}=\frac{1}{2(4\pi)^{2}}\int\text{d}^{4}x\sum_{n=-\infty}^{\infty}\left[b^{(1)}_{0}Q_{2}[W]+b^{(1)}_{2}Q_{0}[W]\right]\,.$
(143)

Here, the heat kernel coefficients are given in Eqs. (110) and (112), and the
threshold functions reads

$\displaystyle Q_{2}[W]$
$\displaystyle=\frac{1}{\Gamma(2)}\int_{0}^{\infty}\text{d}z\,z\frac{\partial_{t}R_{k}(z)-\eta_{g}R_{k}(z)}{P_{k}(z)}\,,$
(144) $\displaystyle Q_{0}[W]$
$\displaystyle=\frac{\partial_{t}R_{k}(z)-\eta_{g}R_{k}(z)}{P_{k}(z)}\bigg{|}_{z\to
0}\,.$ (145)

Next, we evaluate the flow generator for $a_{5}$

$\displaystyle\zeta_{5}$ $\displaystyle=\frac{1}{2}{\rm
tr}\,_{(0)}W[\bar{\Delta}]=\frac{1}{2}{\rm
tr}\,_{(0)}\frac{\partial_{t}R_{k}(\bar{\Delta})-\eta_{g}R_{k}(\bar{\Delta})}{P_{k}(\bar{\Delta})}=\frac{1}{2(4\pi)^{2}}\int\text{d}^{4}x\sum_{n=-\infty}^{\infty}\left[b^{(0)}_{0}Q_{2}[W]+b_{2}^{(0)}Q_{0}[W]\right]\,.$
(146)

The heat kernel coefficients in this case are obtained in Eqs. (116) and (118)
and the threshold functions are the same as in Eqs. (144) and (145).

Finally, the ghost contribution gives

$\displaystyle\zeta_{\text{gh}}$ $\displaystyle=-{\rm
tr}\,_{(0)}W_{\text{gh}}=-{\rm
tr}\,_{(0)}\frac{\partial_{t}R_{k}(\bar{\Delta})-\eta_{\text{gh}}R_{k}(\bar{\Delta})}{P_{k}(\bar{\Delta})}=-\frac{1}{(4\pi)^{2}}\int\text{d}^{4}x\sum_{n=-\infty}^{\infty}\left[b^{(0)}_{0}Q_{2}[W_{\text{gh}}]+b^{(0)}Q_{0}[W_{\text{gh}}]\right]\,.$
(147)

The heat kernel coefficients and the threshold functions are the same as
$a_{5}$, while the threshold functions are given as in Eqs. (144) and (145)
with the replacement $\eta_{g}\to\eta_{\rm gh}$.

Let us read off the beta function for the gauge coupling which is defined by
the projection

$\displaystyle\frac{\partial_{t}Z_{k}}{2}\frac{2\pi R}{g^{2}}$
$\displaystyle=\frac{1}{\mathcal{V}_{4}}\frac{\partial^{2}}{\partial\bar{F}^{a}_{\mu\nu}\partial\bar{F}^{a\mu\nu}}\zeta\,.$
(148)

This yields

$\displaystyle\frac{1}{2}\frac{\partial_{t}Z_{k}}{2}\frac{2\pi R}{g^{2}}$
$\displaystyle=\frac{N}{2(4\pi)^{2}}\sum_{n=-\infty}^{\infty}\left[\frac{10}{3}Q_{0}[W]-\frac{1}{12}Q_{0}[W]+\frac{1}{6}Q_{0}[W_{\rm
gh}]\right]$
$\displaystyle=\frac{N}{2(4\pi)^{2}}\left[\frac{10}{3}\left(1-\frac{\eta_{g}}{2}\right)-\frac{1}{6}\left(1-\frac{\eta_{g}}{2}\right)+\frac{1}{3}\left(1-\frac{\eta_{\rm
gh}}{2}\right)\right]\sum_{n=-\infty}^{\infty}\frac{1}{1+\left(\frac{n}{\bar{R}}\right)^{2}}$
$\displaystyle=\frac{N}{2(4\pi)^{2}}\left[\frac{10}{3}\left(1-\frac{\eta_{g}}{2}\right)-\frac{1}{6}\left(1-\frac{\eta_{g}}{2}\right)+\frac{1}{3}\left(1-\frac{\eta_{\rm
gh}}{2}\right)\right]\frac{1}{N^{2}}{\mathcal{I}}_{N}(\bar{R};\theta_{H}=0)\,,$
(149)

where

$\displaystyle\frac{1}{N^{2}}{\mathcal{I}}_{N}(\bar{R};\theta_{H})=\frac{1}{N^{2}}\sum_{n=-\infty}^{\infty}\frac{1}{1+\left(\frac{n}{\bar{R}}\right)^{2}}=\frac{1}{N^{2}}\sum_{i,j=1}^{N}\sum_{n=-\infty}^{\infty}\frac{1}{1+\left(\frac{n}{\bar{R}}\right)^{2}}=\pi\bar{R}\coth(2\pi\bar{R})\,.$
(150)

## References

  * Weinberg (1979) S. Weinberg, Chap. 16 in General Relativity ed. by Hawking, S.W. and Israel, W. (1979).
  * Nagy (2014) S. Nagy, Annals Phys. 350, 310 (2014), arXiv:1211.4151 [hep-th] .
  * Peskin (1980) M. E. Peskin, Phys. Lett. B 94, 161 (1980).
  * Morris (2005) T. R. Morris, JHEP 01, 002 (2005), arXiv:hep-ph/0410142 .
  * Irges and Koutroulis (2018) N. Irges and F. Koutroulis, Nucl. Phys. B 937, 135 (2018), arXiv:1804.06306 [hep-th] .
  * Irges and Koutroulis (2021) N. Irges and F. Koutroulis, Eur. Phys. J. C 81, 187 (2021), arXiv:2008.03981 [hep-th] .
  * De Cesare  _et al._ (2021) F. De Cesare, L. Di Pietro, and M. Serone, Phys. Rev. D 104, 105015 (2021), arXiv:2107.00342 [hep-th] .
  * Antoniadis (1990) I. Antoniadis, Phys. Lett. B 246, 377 (1990).
  * Dienes  _et al._ (1998) K. R. Dienes, E. Dudas, and T. Gherghetta, Phys. Lett. B 436, 55 (1998), arXiv:hep-ph/9803466 .
  * Dienes  _et al._ (1999) K. R. Dienes, E. Dudas, and T. Gherghetta, Nucl. Phys. B 537, 47 (1999), arXiv:hep-ph/9806292 .
  * Kobayashi  _et al._ (1999) T. Kobayashi, J. Kubo, M. Mondragon, and G. Zoupanos, Nucl. Phys. B 550, 99 (1999), arXiv:hep-ph/9812221 .
  * Kubo  _et al._ (2000) J. Kubo, H. Terao, and G. Zoupanos, Nucl. Phys. B 574, 495 (2000), arXiv:hep-ph/9910277 .
  * Gies (2003) H. Gies, Phys. Rev. D 68, 085015 (2003), arXiv:hep-th/0305208 .
  * Fairlie (1979) D. B. Fairlie, J. Phys. G 5, L55 (1979).
  * Manton (1979) N. S. Manton, Nucl. Phys. B 158, 141 (1979).
  * Forgacs and Manton (1980) P. Forgacs and N. S. Manton, Commun. Math. Phys. 72, 15 (1980).
  * Hosotani (1983a) Y. Hosotani, Phys. Lett. B 126, 309 (1983a).
  * Hosotani (1983b) Y. Hosotani, Phys. Lett. B 129, 193 (1983b).
  * Hosotani (1989) Y. Hosotani, Annals Phys. 190, 233 (1989).
  * Davies and McLachlan (1988) A. T. Davies and A. McLachlan, Phys. Lett. B 200, 305 (1988).
  * Davies and McLachlan (1989) A. T. Davies and A. McLachlan, Nucl. Phys. B 317, 237 (1989).
  * Antoniadis and Benakli (1994) I. Antoniadis and K. Benakli, Phys. Lett. B 326, 69 (1994), arXiv:hep-th/9310151 .
  * Antoniadis  _et al._ (2001) I. Antoniadis, K. Benakli, and M. Quiros, New J. Phys. 3, 20 (2001), arXiv:hep-th/0108005 .
  * Kashiwa and Misumi (2013) K. Kashiwa and T. Misumi, JHEP 05, 042 (2013), arXiv:1302.2196 [hep-ph] .
  * Cossu  _et al._ (2014) G. Cossu, H. Hatanaka, Y. Hosotani, and J.-I. Noaki, Phys. Rev. D 89, 094509 (2014), arXiv:1309.4198 [hep-lat] .
  * Hosotani (2005) Y. Hosotani, in  _2004 International Workshop on Dynamical Symmetry Breaking_ (2005) pp. 17–34, arXiv:hep-ph/0504272 .
  * Hosotani (2006) Y. Hosotani, (2006), arXiv:hep-ph/0607064 .
  * Maru and Yamashita (2006) N. Maru and T. Yamashita, Nucl. Phys. B 754, 127 (2006), arXiv:hep-ph/0603237 .
  * Hosotani  _et al._ (2007) Y. Hosotani, N. Maru, K. Takenaga, and T. Yamashita, Prog. Theor. Phys. 118, 1053 (2007), arXiv:0709.2844 [hep-ph] .
  * Hisano  _et al._ (2020) J. Hisano, Y. Shoji, and A. Yamada, JHEP 02, 193 (2020), arXiv:1908.09158 [hep-ph] .
  * Yamada (2021) A. Yamada, PTEP 2021, 093B01 (2021), arXiv:2103.06269 [hep-ph] .
  * Freitas and Wiegand (2018) A. Freitas and D. Wiegand, JHEP 08, 094 (2018), arXiv:1805.12142 [hep-ph] .
  * Wetterich (1993) C. Wetterich, Phys. Lett. B301, 90 (1993).
  * Morris (1994) T. R. Morris, Int. J. Mod. Phys. A9, 2411 (1994), arXiv:hep-ph/9308265 [hep-ph] .
  * Reuter and Wetterich (1994a) M. Reuter and C. Wetterich, Nucl. Phys. B 417, 181 (1994a).
  * Ellwanger (1994) U. Ellwanger, Z. Phys. C62, 503 (1994), arXiv:hep-ph/9308260 [hep-ph] .
  * Morris (1998) T. R. Morris,  _Nonperturbative QCD: Structure of the QCD vacuum: Proceedings, Yukawa International Seminar, YKIS’97, Kyoto, Japan, December 2-12, 1997_ , Prog. Theor. Phys. Suppl. 131, 395 (1998), [,395(1998)], arXiv:hep-th/9802039 [hep-th] .
  * Berges  _et al._ (2002) J. Berges, N. Tetradis, and C. Wetterich, Phys. Rept. 363, 223 (2002), arXiv:hep-ph/0005122 [hep-ph] .
  * Aoki (2000) K. Aoki, Int.J.Mod.Phys. B14, 1249 (2000).
  * Bagnuls and Bervillier (2001) C. Bagnuls and C. Bervillier, Phys. Rept. 348, 91 (2001), arXiv:hep-th/0002034 [hep-th] .
  * Polonyi (2003) J. Polonyi, Central Eur. J. Phys. 1, 1 (2003), arXiv:hep-th/0110026 [hep-th] .
  * Pawlowski (2007) J. M. Pawlowski, Annals Phys. 322, 2831 (2007), arXiv:hep-th/0512261 [hep-th] .
  * Gies (2012) H. Gies, Lect.Notes Phys. 852, 287 (2012), arXiv:hep-ph/0611146 [hep-ph] .
  * Delamotte (2012) B. Delamotte, Lect. Notes Phys. 852, 49 (2012), arXiv:cond-mat/0702365 [cond-mat.stat-mech] .
  * Rosten (2012) O. J. Rosten, Phys. Rept. 511, 177 (2012), arXiv:1003.1366 [hep-th] .
  * Kopietz  _et al._ (2010) P. Kopietz, L. Bartosch, and F. Schütz,  _Introduction to the functional renormalization group_, Vol. 798 (2010).
  * Braun (2012) J. Braun, J. Phys. G39, 033001 (2012), arXiv:1108.4449 [hep-ph] .
  * Dupuis  _et al._ (2021) N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, and N. Wschebor, Phys. Rept. 910, 1 (2021), arXiv:2006.04853 [cond-mat.stat-mech] .
  * Cossu and D’Elia (2009) G. Cossu and M. D’Elia, JHEP 07, 048 (2009), arXiv:0904.1353 [hep-lat] .
  * Reuter and Wetterich (1993a) M. Reuter and C. Wetterich, Nucl. Phys. B 391, 147 (1993a).
  * Reuter and Wetterich (1993b) M. Reuter and C. Wetterich, Nucl. Phys. B 408, 91 (1993b).
  * Reuter and Wetterich (1994b) M. Reuter and C. Wetterich, Nucl. Phys. B 427, 291 (1994b).
  * Wetterich (2018a) C. Wetterich, Nucl. Phys. B 934, 265 (2018a), arXiv:1710.02494 [hep-th] .
  * Reuter (1996) M. Reuter, in  _5th Hellenic School and Workshops on Elementary Particle Physics_ (1996) arXiv:hep-th/9602012 .
  * Gies (2002) H. Gies, Phys. Rev. D 66, 025006 (2002), arXiv:hep-th/0202207 .
  * Litim (2001) D. F. Litim, Phys. Rev. D64, 105007 (2001), arXiv:hep-th/0103195 [hep-th] .
  * Fukugita  _et al._ (1989) M. Fukugita, M. Okawa, and A. Ukawa, Phys. Rev. Lett. 63, 1768 (1989).
  * Fister and Pawlowski (2013) L. Fister and J. M. Pawlowski, Phys. Rev. D 88, 045010 (2013), arXiv:1301.4163 [hep-ph] .
  * Braun  _et al._ (2016) J. Braun, L. Fister, J. M. Pawlowski, and F. Rennecke, Phys. Rev. D 94, 034016 (2016), arXiv:1412.1045 [hep-ph] .
  * Mitter  _et al._ (2015) M. Mitter, J. M. Pawlowski, and N. Strodthoff, Phys. Rev. D 91, 054035 (2015), arXiv:1411.7978 [hep-ph] .
  * Cyrol  _et al._ (2018a) A. K. Cyrol, M. Mitter, J. M. Pawlowski, and N. Strodthoff, Phys. Rev. D 97, 054006 (2018a), arXiv:1706.06326 [hep-ph] .
  * Cyrol  _et al._ (2018b) A. K. Cyrol, M. Mitter, J. M. Pawlowski, and N. Strodthoff, Phys. Rev. D 97, 054015 (2018b), arXiv:1708.03482 [hep-ph] .
  * Fischer  _et al._ (2009) C. S. Fischer, A. Maas, and J. M. Pawlowski, Annals Phys. 324, 2408 (2009), arXiv:0810.1987 [hep-ph] .
  * Fister and Pawlowski (2011) L. Fister and J. M. Pawlowski, (2011), arXiv:1112.5440 [hep-ph] .
  * Herbst  _et al._ (2015) T. K. Herbst, J. Luecker, and J. M. Pawlowski, (2015), arXiv:1510.03830 [hep-ph] .
  * Cyrol  _et al._ (2016) A. K. Cyrol, L. Fister, M. Mitter, J. M. Pawlowski, and N. Strodthoff, Phys. Rev. D 94, 054005 (2016), arXiv:1605.01856 [hep-ph] .
  * Corell  _et al._ (2018) L. Corell, A. K. Cyrol, M. Mitter, J. M. Pawlowski, and N. Strodthoff, SciPost Phys. 5, 066 (2018), arXiv:1803.10092 [hep-ph] .
  * Eichmann  _et al._ (2021) G. Eichmann, J. M. Pawlowski, and J. a. M. Silva, Phys. Rev. D 104, 114016 (2021), arXiv:2107.05352 [hep-ph] .
  * Napetschnig  _et al._ (2021) M. Napetschnig, R. Alkofer, M. Q. Huber, and J. M. Pawlowski, Phys. Rev. D 104, 054003 (2021), arXiv:2106.12559 [hep-ph] .
  * Horak  _et al._ (2022) J. Horak, F. Ihssen, J. Papavassiliou, J. M. Pawlowski, A. Weber, and C. Wetterich, (2022), arXiv:2201.09747 [hep-ph] .
  * Christiansen  _et al._ (2014) N. Christiansen, D. F. Litim, J. M. Pawlowski, and A. Rodigast, Phys. Lett. B728, 114 (2014), arXiv:1209.4038 [hep-th] .
  * Christiansen  _et al._ (2016) N. Christiansen, B. Knorr, J. M. Pawlowski, and A. Rodigast, Phys. Rev. D93, 044036 (2016), arXiv:1403.1232 [hep-th] .
  * Christiansen  _et al._ (2015) N. Christiansen, B. Knorr, J. Meibohm, J. M. Pawlowski, and M. Reichert, Phys. Rev. D92, 121501 (2015), arXiv:1506.07016 [hep-th] .
  * Meibohm and Pawlowski (2016) J. Meibohm and J. M. Pawlowski, Eur. Phys. J. C76, 285 (2016), arXiv:1601.04597 [hep-th] .
  * Christiansen (2016) N. Christiansen, (2016), arXiv:1612.06223 [hep-th] .
  * Denz  _et al._ (2018) T. Denz, J. M. Pawlowski, and M. Reichert, Eur. Phys. J. C78, 336 (2018), arXiv:1612.07315 [hep-th] .
  * Christiansen  _et al._ (2018a) N. Christiansen, D. F. Litim, J. M. Pawlowski, and M. Reichert, Phys. Rev. D97, 106012 (2018a), arXiv:1710.04669 [hep-th] .
  * Christiansen  _et al._ (2018b) N. Christiansen, K. Falls, J. M. Pawlowski, and M. Reichert, Phys. Rev. D97, 046007 (2018b), arXiv:1711.09259 [hep-th] .
  * Eichhorn  _et al._ (2018a) A. Eichhorn, P. Labus, J. M. Pawlowski, and M. Reichert, SciPost Phys. 5, 031 (2018a), arXiv:1804.00012 [hep-th] .
  * Eichhorn  _et al._ (2018b) A. Eichhorn, Y. Hamada, J. Lumma, and M. Yamada, Phys. Rev. D97, 086004 (2018b), arXiv:1712.00319 [hep-th] .
  * Pawlowski and Reichert (2020) J. M. Pawlowski and M. Reichert, Front. Phys. 8, 551848 (2020), arXiv:2007.10353 [hep-th] .
  * Bonanno  _et al._ (2022) A. Bonanno, T. Denz, J. M. Pawlowski, and M. Reichert, SciPost Phys. 12, 001 (2022), arXiv:2102.02217 [hep-th] .
  * Fehre  _et al._ (2021) J. Fehre, D. F. Litim, J. M. Pawlowski, and M. Reichert, (2021), arXiv:2111.13232 [hep-th] .
  * Gao and Pawlowski (2020) F. Gao and J. M. Pawlowski, Phys. Rev. D 102, 034027 (2020), arXiv:2002.07500 [hep-ph] .
  * Gao and Pawlowski (2021a) F. Gao and J. M. Pawlowski, Phys. Lett. B 820, 136584 (2021a), arXiv:2010.13705 [hep-ph] .
  * Gao  _et al._ (2021) F. Gao, J. Papavassiliou, and J. M. Pawlowski, Phys. Rev. D 103, 094013 (2021), arXiv:2102.13053 [hep-ph] .
  * Gao and Pawlowski (2021b) F. Gao and J. M. Pawlowski, (2021b), arXiv:2112.01395 [hep-ph] .
  * Arnold and Yaffe (1995) P. B. Arnold and L. G. Yaffe, Phys. Rev. D 52, 7208 (1995), arXiv:hep-ph/9508280 .
  * Reuter (1998) M. Reuter, Phys. Rev. D57, 971 (1998), arXiv:hep-th/9605030 [hep-th] .
  * Souma (1999) W. Souma, Prog. Theor. Phys. 102, 181 (1999), arXiv:hep-th/9907027 [hep-th] .
  * Niedermaier and Reuter (2006) M. Niedermaier and M. Reuter, Living Rev. Rel. 9, 5 (2006).
  * Niedermaier (2007) M. Niedermaier, Class. Quant. Grav. 24, R171 (2007), arXiv:gr-qc/0610018 [gr-qc] .
  * Percacci (2007) R. Percacci, (2007), arXiv:0709.3851 [hep-th] .
  * Codello  _et al._ (2009) A. Codello, R. Percacci, and C. Rahmede, Annals Phys. 324, 414 (2009), arXiv:0805.2909 [hep-th] .
  * Reuter and Saueressig (2012) M. Reuter and F. Saueressig, New J. Phys. 14, 055022 (2012), arXiv:1202.2274 [hep-th] .
  * Percacci (2017) R. Percacci,  _An Introduction to Covariant Quantum Gravity and Asymptotic Safety_, 100 Years of General Relativity, Vol. 3 (World Scientific, 2017).
  * Reuter and Saueressig (2019) M. Reuter and F. Saueressig,  _Quantum Gravity and the Functional Renormalization Group_ (Cambridge University Press, 2019).
  * Wetterich (2019) C. Wetterich, (2019), arXiv:1901.04741 [hep-th] .
  * Bonanno  _et al._ (2020) A. Bonanno, A. Eichhorn, H. Gies, J. M. Pawlowski, R. Percacci, M. Reuter, F. Saueressig, and G. P. Vacca, Front. in Phys. 8, 269 (2020), arXiv:2004.06810 [gr-qc] .
  * Reichert (2020) M. Reichert, PoS Modave2019, 005 (2020).
  * Eichhorn (2018a) A. Eichhorn, Found. Phys. 48, 1407 (2018a), arXiv:1709.03696 [gr-qc] .
  * Eichhorn (2018b) A. Eichhorn, (2018b), arXiv:1810.07615 [hep-th] .
  * Eichhorn (2022) A. Eichhorn (2022) arXiv:2201.11543 [gr-qc] .
  * Wetterich (2017) C. Wetterich, Phys. Lett. B773, 6 (2017), arXiv:1704.08040 [gr-qc] .
  * Wetterich (2018b) C. Wetterich, Phys. Rev. D 98, 026028 (2018b), arXiv:1802.05947 [gr-qc] .
  * Braun  _et al._ (2010a) J. Braun, H. Gies, and J. M. Pawlowski, Phys. Lett. B 684, 262 (2010a), arXiv:0708.2413 [hep-th] .
  * Eichhorn  _et al._ (2011) A. Eichhorn, H. Gies, and J. M. Pawlowski, Phys. Rev. D 83, 045014 (2011), [Erratum: Phys.Rev.D 83, 069903 (2011)], arXiv:1010.2153 [hep-ph] .
  * Braun  _et al._ (2010b) J. Braun, A. Eichhorn, H. Gies, and J. M. Pawlowski, Eur. Phys. J. C 70, 689 (2010b), arXiv:1007.2619 [hep-ph] .
  * Haas  _et al._ (2013) L. M. Haas, R. Stiele, J. Braun, J. M. Pawlowski, and J. Schaffner-Bielich, Phys. Rev. D 87, 076004 (2013), arXiv:1302.1993 [hep-ph] .
  * Kawai  _et al._ (1992) H. Kawai, M. Nio, and Y. Okamoto, Prog. Theor. Phys. 88, 341 (1992).
  * Ejiri  _et al._ (2000) S. Ejiri, J. Kubo, and M. Murata, Phys. Rev. D 62, 105025 (2000), arXiv:hep-ph/0006217 .
  * Ejiri  _et al._ (2002) S. Ejiri, S. Fujimoto, and J. Kubo, Phys. Rev. D 66, 036002 (2002), arXiv:hep-lat/0204022 .
  * Kurkela  _et al._ (2009) A. Kurkela, P. de Forcrand, and M. Panero, PoS LAT2009, 050 (2009), arXiv:0911.3609 [hep-lat] .
  * de Forcrand  _et al._ (2010) P. de Forcrand, A. Kurkela, and M. Panero, JHEP 06, 050 (2010), arXiv:1003.4643 [hep-lat] .
  * Del Debbio  _et al._ (2012) L. Del Debbio, A. Hart, and E. Rinaldi, JHEP 07, 178 (2012), arXiv:1203.2116 [hep-lat] .
  * Itou  _et al._ (2014) E. Itou, K. Kashiwa, and N. Nakamoto, (2014), arXiv:1403.6277 [hep-lat] .
  * Knechtli and Rinaldi (2016) F. Knechtli and E. Rinaldi, Int. J. Mod. Phys. A 31, 1643002 (2016), arXiv:1605.04341 [hep-lat] .
  * Florio  _et al._ (2021) A. Florio, J. a. M. V. P. Lopes, J. Matos, and J. a. Penedones, JHEP 12, 076 (2021), arXiv:2103.15242 [hep-lat] .
  * Pomarol and Quiros (1998) A. Pomarol and M. Quiros, Phys. Lett. B 438, 255 (1998), arXiv:hep-ph/9806263 .
  * Kubo  _et al._ (2002) M. Kubo, C. S. Lim, and H. Yamashita, Mod. Phys. Lett. A 17, 2249 (2002), arXiv:hep-ph/0111327 .
  * Randall and Sundrum (1999) L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999), arXiv:hep-ph/9905221 .
  * Agashe  _et al._ (2005) K. Agashe, R. Contino, and A. Pomarol, Nucl. Phys. B 719, 165 (2005), arXiv:hep-ph/0412089 .
  * Medina  _et al._ (2007) A. D. Medina, N. R. Shah, and C. E. M. Wagner, Phys. Rev. D 76, 095010 (2007), arXiv:0706.1281 [hep-ph] .
  * Hosotani  _et al._ (2008) Y. Hosotani, K. Oda, T. Ohnuma, and Y. Sakamura, Phys. Rev. D 78, 096002 (2008), [Erratum: Phys.Rev.D 79, 079902 (2009)], arXiv:0806.0480 [hep-ph] .
  * Funatsu  _et al._ (2020a) S. Funatsu, H. Hatanaka, Y. Hosotani, Y. Orikasa, and N. Yamatsu, Phys. Rev. D 101, 055016 (2020a), arXiv:1909.00190 [hep-ph] .
  * Funatsu  _et al._ (2020b) S. Funatsu, H. Hatanaka, Y. Hosotani, Y. Orikasa, and N. Yamatsu, Phys. Rev. D 102, 015005 (2020b), arXiv:2002.09262 [hep-ph] .
  * Haba and Shimizu (2003) N. Haba and Y. Shimizu, Phys. Rev. D 67, 095001 (2003), [Erratum: Phys.Rev.D 69, 059902 (2004)], arXiv:hep-ph/0212166 .
  * Lim and Maru (2007) C. S. Lim and N. Maru, Phys. Lett. B 653, 320 (2007), arXiv:0706.1397 [hep-ph] .
  * Kojima  _et al._ (2011) K. Kojima, K. Takenaga, and T. Yamashita, Phys. Rev. D 84, 051701 (2011), arXiv:1103.1234 [hep-ph] .
  * Hosotani and Yamatsu (2015) Y. Hosotani and N. Yamatsu, PTEP 2015, 111B01 (2015), arXiv:1504.03817 [hep-ph] .
  * Yamatsu (2016) N. Yamatsu, PTEP 2016, 043B02 (2016), arXiv:1512.05559 [hep-ph] .
  * Furui  _et al._ (2016) A. Furui, Y. Hosotani, and N. Yamatsu, PTEP 2016, 093B01 (2016), arXiv:1606.07222 [hep-ph] .
  * Angelescu  _et al._ (2022) A. Angelescu, A. Bally, S. Blasi, and F. Goertz, Phys. Rev. D 105, 035026 (2022), arXiv:2104.07366 [hep-ph] .
  * Vassilevich (2003) D. V. Vassilevich, Phys. Rept. 388, 279 (2003), arXiv:hep-th/0306138 .
  * Codello (2015) A. Codello, Phys. Rev. D 91, 065032 (2015), arXiv:1304.2059 [hep-th] .
  * Wetterich and Yamada (2019) C. Wetterich and M. Yamada, Phys. Rev. D 100, 066017 (2019), arXiv:1906.01721 [hep-th] .