aainstitutetext: Racah Institute, The Hebrew University. Jerusalem 9190401,
Israel

# Symmetry breaking at high temperatures in large N gauge theories

Soumyadeep Chaudhuri a , Eliezer Rabinovici<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Considering marginally relevant and relevant deformations of the weakly
coupled $(3+1)$-dimensional large $N$ conformal gauge theories introduced in
Chaudhuri:2020xxb , we study the patterns of phase transitions in these
systems that lead to a symmetry-broken phase in the high temperature limit.
These deformations involve only the scalar fields in the models. The
marginally relevant deformations are obtained by varying certain double trace
quartic couplings between the scalar fields. The relevant deformations, on the
other hand, are obtained by adding masses to the scalar fields while keeping
all the couplings frozen at their fixed point values. At the
$N\rightarrow\infty$ limit, the RG flows triggered by these deformations
approach the aforementioned weakly coupled CFTs in the UV regime. These UV
fixed points lie on a conformal manifold with the shape of a circle in the
space of couplings. As shown in Chaudhuri:2020xxb , in certain parameter
regimes a subset of points on this manifold exhibits thermal order
characterized by the spontaneous breaking of a global $\mathbb{Z}_{2}$ or
$U(1)$ symmetry and Higgsing of a subset of gauge bosons at all nonzero
temperatures. We show that the RG flows triggered by the marginally relevant
deformations lead to a weakly coupled IR fixed point which lacks the thermal
order. Thus, the systems defined by these RG flows undergo a transition from a
disordered phase at low temperatures to an ordered phase at high temperatures.
This provides examples of both inverse symmetry breaking and symmetry
nonrestoration. For the relevant deformations, we demonstrate that a variety
of phase transitions are possible depending on the signs and magnitudes of the
squares of the masses added to the scalar fields. Using thermal perturbation
theory, we derive the approximate values of the critical temperatures for all
these phase transitions. All the results are obtained at the
$N\rightarrow\infty$ limit. Most of them are found in a reliable weak coupling
regime and for others we present qualitative arguments.

##  1 Introduction

Spontaneous breaking of symmetries plays an important role in distinguishing
the different phases of matter. Usually symmetries that are broken at low
temperatures are eventually restored as the temperature becomes sufficiently
high.111Here, as well as in the rest of the paper, we are referring to the
spontaneous breaking of ordinary (0-form) global symmetries. There are known
instances of spontaneous breaking of higher form symmetries at high
temperatures. One familiar example of this is the spontaneous breaking of the
1-form $\mathbb{Z}_{N}$ center symmetry in pure Yang-Mills theory with $SU(N)$
gauge group.  This is true also for systems in which a symmetry that is
unbroken at low temperatures is spontaneously broken at a higher critical
temperature PhysRevE.72.046107  \- a phenomenon called inverse symmetry
breaking. In each such instance, the broken symmetry is restored at an even
higher temperature. The ubiquity of such symmetry restoration in nature raised
the question of whether there can be models where some symmetry remains broken
up to arbitrarily high temperatures.

This question has been explored for several decades in varied contexts ; it
has important implications for several areas of physics. We refer the reader
to Weinberg:1974hy ; Orloff:1996yn ; Bimonte:1999tw ; Pinto:1999pg ;
Mohapatra:1979qt ; Langacker:1980kd ; Dodelson:1989ii ; Dodelson:1991iv ;
Dvali:1995cj ; Meade:2018saz ; Bai:2021hfb ; Carena:2021onl  for a sample of
the literature. The main difficulty of coming up with a conclusive answer lay
in the fact that the theories considered were UV-incomplete, thereby putting
an upper cutoff on the temperatures that can be explored.222 Examples of
symmetry nonrestoration have been found in certain models with imaginary or
random chemical potentials Komargodski:2017dmc ; Aitken:2017ayq ;
Tanizaki:2017qhf ; Dunne:2018hog ; Wan:2019oax ; Hong:2000rk . The dynamics of
these models suffer from the lack of unitarity. Some holographic models were
also explored as candidates for theories with symmetry nonrestoration
Buchel:2009ge ; Donos:2011ut ; Gursoy:2018umf ; Buchel:2018bzp ;
Buchel:2020thm ; Buchel:2020xdk ; Buchel:2020jfs . Unlike the phases discussed
in this paper, the symmetry-broken phases in these models do not correspond to
stable vacua in thermal states.  To overcome this problem certain Wilson-
Fisher-like conformal field theories (CFTs) with global $O(N)$ symmetries were
studied for both infinite and finite $N$ in fractional dimensions Chai:2020zgq
; Chai:2020onq .333There may be issues with unitarity of such Wilson-Fisher-
like fixed points at finite $N$ due to the potential existence of operators
with complex scaling dimensions Hogervorst:2015akt . The advantage of studying
CFTs is that the absence of any intrinsic scale in the theory leads the system
to be in the same phase (whatever it may be) at all nonzero temperatures. A
symmetry-broken phase was indeed found to occur in the models considered in
Chai:2020zgq ; Chai:2020onq . This was followed by the construction of
(3+1)-dimensional large $N$ conformal gauge theories in Chaudhuri:2020xxb
where certain global $\mathbb{Z}_{2}$ or $U(1)$ symmetries were shown to be
broken at all nonzero temperatures. It was also shown that this symmetry
breaking is accompanied by the Higgsing of a subset of gauge bosons which
leads the system to be in a persistent Brout-Englert-Higgs (BEH) phase. In
this context, we refer the reader to Bajc:2021ope  for examples of similar
persistent Higgsing at high temperatures in some asymptotically free large N
gauge theories. Certain asymptotically safe theories Litim:2014uca  were also
considered in Bajc:2021ope , but they failed to show a symmetry-broken phase
in the high temperature limit.

The study of the above-mentioned CFTs has demonstrated the existence of
persistent symmetry breaking in the large $N$ limit. However, this still
leaves open the question of how the persistent symmetry breaking
characteristics change in the presence of an extra scale. In this work we
address this question by considering relevant and marginally relevant
deformations of the large $N$ CFTs introduced in Chaudhuri:2020xxb . These
CFTs are weakly coupled. They lie on a circle in the space of couplings. In
appropriate parameter regimes, a subset of points on this fixed circle
demonstrates spontaneous breaking of some global $\mathbb{Z}_{2}$ or $U(1)$
symmetries at any nonzero temperature. We consider two kinds of deformations
of these CFTs with thermal order. As we will discuss shortly, these
deformations involve only the scalar fields in the model. At the
$N\rightarrow\infty$ limit, the RG flows triggered by these deformations end
up at the aforementioned CFTs in the UV regime. So, in this limit these
systems provide examples of symmetry nonrestoration in theories with
nontrivial UV fixed points. Exploring the IR regimes of these flows, we find
interesting patterns of phase transitions at nonzero temperatures. Let us
briefly describe these deformations and the corresponding phase transitions in
the following two paragraphs.

The first class of deformations that we consider consists of marginally
relevant ones involving variations of certain quartic couplings between the
scalar fields. Such marginally relevant deformations may exist in four
dimensional theories only in the presence of non-Abelian gauge interactions
Zee:1973gn . This is the case here. We show that the RG flows corresponding to
these deformations lead to a non-Gaussian fixed point in the IR. The theory
remains weakly coupled throughout the flow allowing the use of perturbation
theory to study it. As mentioned earlier, under certain conditions, some of
the UV fixed points of these flows demonstrate persistent thermal order. On
the other hand, the IR fixed point lacks thermal order. From this we conclude
that there must be an inverse symmetry breaking phase transition at an
intermediate temperature. By a perturbative analysis, we determine the
critical temperature corresponding to this phase transition. Above the
critical temperature, the system persistently remains in a symmetry-broken
phase.

The second class of deformations involves adding masses to the scalar fields
while keeping all the couplings frozen at the fixed points exhibiting thermal
order. We show that at high temperatures, the effects of these masses are
insignificant and the system exists in the same phase as the UV fixed point
where a $\mathbb{Z}_{2}$ (or $U(1)$) global symmetry remains persistently
broken. As the temperature is decreased the effects of the renormalized masses
of the scalar fields become increasingly pronounced. At sufficiently low
temperatures, such renormalized masses can induce phase transitions in the
system. We show that in certain cases such phase transitions can be studied
using thermal perturbation theory. We derive estimates of the critical
temperatures corresponding to these phase transitions. We show that a variety
of phases can exist slightly below such critical temperatures. The nature of
these phases crucially depend on the magnitudes and signs of the squares of
the renormalized masses. Far below the critical temperatures, the perturbative
analysis breaks down. So we cannot say anything definite about the phases in
this regime.

We emphasize that all the results mentioned above are derived in the
$N\rightarrow\infty$ limit. Whether the symmetry nonrestoration in these
models persists for finite $N$ remains unresolved.444In this context, we refer
the reader to the recent work 1869270  where symmetry nonrestoration was found
to occur even at finite $N$ in some $d$-dimensional nonlocal CFTs with
$1<d<4$. In the conclusion of the paper we comment on the potential problems
that may arise for symmetry nonrestoration at finite $N$.

##### Organization of the paper:

In section 2, we review the models that were introduced in Chaudhuri:2020xxb .
We discuss the features of the planar beta functions of the couplings in these
models and the corresponding fixed points with special emphasis on the fixed
points with thermal order.

In section 3, we demonstrate the existence of marginally relevant deformations
of the fixed points exhibiting thermal order. We show that the systems defined
by these deformations undergo inverse symmetry breaking phase transitions at
nonzero critical temperatures. We derive estimates of these critical
temperatures.

In section 4, we add masses to the scalar fields in the models, and study the
phase transitions induced by these masses at nonzero critical temperatures. We
show that there are distinct patterns of phase transitions for the different
signs and magnitudes of the squares of these masses. We provide estimates of
the critical temperatures corresponding to these phase transitions.

In section 5, we conclude by summarizing our results and commenting on how
finite $N$ corrections can alter the features found in the
$N\rightarrow\infty$ limit.

In appendix A we show that the tree-level quartic terms in the effective
potential of the scalar fields are positive-definite throughout the RG flows
discussed in section 3.

In appendix B we derive the 1-loop beta functions of the masses of the scalar
fields that are introduced in section 4.

##  2 Review of the double bifundamental models

In this section we will describe the models that were originally introduced in
Chaudhuri:2020xxb  and which will be the objects of interest in this work.
These models have gauge groups of the form

$\displaystyle G=\prod\limits_{i=1}^{2}G_{i}\times G_{i}$ (1)

where $G_{i}$ can be either $SO(N_{ci})$ or $SU(N_{ci})$.555The ranks $N_{c1}$
and $N_{c2}$ can be different.  The models where the $G_{i}$’s are
$SO(N_{ci})$ and $SU(N_{ci})$ were called the real double bifundamental model
and the complex double bifundamental model respectively in Chaudhuri:2020xxb .
These names were coined keeping in mind the representations in which certain
scalar fields in the model transform under the gauge group. Henceforth, we
will use the abbreviations RDB and CDB for these models. The two sectors of
the gauge group are labeled by the index $i$. The matter fields in each sector
transform only under the gauge group $(G_{i}\times G_{i})$ in that sector and
they are invariant under the gauge transformations in the other sector. The
fields in the $i^{\text{th}}$ sector include two sets of massless fermions
($\psi_{i}$ and $\chi_{i}$), each of which has $N_{fi}$ flavors and transforms
in the fundamental representation of one of the $G_{i}$’s while being
invariant under the other $G_{i}$. These fermions are Majorana spinors in the
RDB model and Dirac spinors in CDB model. In addition to these fermions, there
is an $N_{ci}\times N_{ci}$ matrix of massless scalar fields which transform
in the bifundamental representation of $G_{i}\times G_{i}$. These scalars are
real in the RDB model and complex in the CDB model. The scalar fields in each
sector interact via both single trace and double trace quartic couplings.
There is an additional double trace interaction coupling the scalar fields in
the two sectors. In figure 1 we provide a schematic diagram indicating the
representations in which the different fields transform in these models.

$N_{c1}$$N_{c1}$$N_{c2}$$N_{c2}$$N_{f1}$$N_{f1}$$N_{f2}$$N_{f2}$$N_{f2}$$\Phi_{1}$$\Phi_{2}$$\psi_{1}$$\chi_{1}$$\psi_{2}$$\chi_{2}$

Figure 1: A schematic diagram indicating the representations in which the
different fields transform in the double bifundamental models: The two sectors
are represented by the two subdiagrams which are connected by a dashed line.
This dashed line represents a double trace quartic interaction between the
scalar fields in the two sectors. The two green nodes in each sector represent
the two $G_{i}$’s in the gauge group $G_{i}\times G_{i}$ of that sector. Each
of these $G_{i}$’s is $SO(N_{ci})$ for the RDB model and $SU(N_{ci})$ for the
CDB model. The line connecting the two green nodes in each sector represents
the scalar fields in that sector which transform in the bifundamental
representation of $G_{i}\times G_{i}$. These scalar fields interact via both
single trace and double trace quartic couplings. The two yellow nodes in each
sector represent the $N_{fi}$ flavors of the two fermions $\psi_{i}$ and
$\chi_{i}$ in that sector. Each line connecting a green node and a yellow node
indicates that the respective fermion is an $(N_{ci}\times N_{fi})$ matrix
which transforms in the fundamental representation of the corresponding
$G_{i}$.

The renormalized Lagrangians of these models have the following common
form666In case of the RDB model, $\Phi_{i}^{\dagger}=\Phi_{i}^{T}$ since the
scalar fields are real.:

$\begin{split}\mathcal{L}=&-\frac{1}{2}\sum_{i,\gamma=1}^{2}\text{Tr}\Big{[}(F_{i\gamma})_{\mu\nu}(F_{i\gamma})^{\mu\nu}\Big{]}+i\kappa\sum_{i=1}^{2}\text{Tr}\Big{[}\overline{\psi}_{i}\not{D}\psi_{i}\Big{]}+i\kappa\sum_{i=1}^{2}\text{Tr}\Big{[}\overline{\chi}_{i}\not{D}\chi_{i}\Big{]}\\\
&+\kappa\sum_{i=1}^{2}\text{Tr}\Big{[}\Big{(}D_{\mu}\Phi_{i}\Big{)}^{\dagger}D^{\mu}\Phi_{i}\Big{]}-\sum_{i=1}^{2}\widetilde{h}_{i}\text{Tr}\Big{[}(\Phi_{i}^{\dagger}\Phi_{i})^{2}\Big{]}-\sum_{i=1}^{2}\widetilde{f}_{i}\Big{(}\text{Tr}\Big{[}\Phi_{i}^{\dagger}\Phi_{i}\Big{]}\Big{)}^{2}\\\
&-2\widetilde{\zeta}\text{Tr}\Big{[}\Phi_{1}^{\dagger}\Phi_{1}\Big{]}\text{Tr}\Big{[}\Phi_{2}^{\dagger}\Phi_{2}\Big{]},\end{split}$
(2)

where $\kappa=\frac{1}{2}$ for the RDB model, and $\kappa=1$ for the CDB
model. The index $\gamma$ distinguishes the two $G_{i}$’s in the
$i^{\text{th}}$ sector. The fermionic fields $\psi_{i}$ and $\chi_{i}$ are
$(N_{ci}\times N_{fi})$ matrices.

In Chaudhuri:2020xxb  the fixed points of these models were studied at the
Veneziano limit Veneziano:1976wm  where $N_{ci},N_{fi}\rightarrow\infty$ while
$r\equiv\frac{N_{c2}}{N_{c1}}$ and $x_{fi}\equiv\frac{N_{fi}}{N_{ci}}$ are
kept finite, and the different couplings scale as

$g_{i}^{2}=\frac{16\pi^{2}\lambda_{i}}{N_{ci}},\
\widetilde{h}_{i}=\frac{16\pi^{2}h_{i}}{N_{ci}},\
\widetilde{f}_{i}=\frac{16\pi^{2}f_{i}}{N_{ci}^{2}},\
\widetilde{\zeta}=\frac{16\pi^{2}\zeta}{N_{c1}N_{c2}}$ (3)

with $g_{i}$ being the gauge coupling in the $i^{\text{th}}$ sector 777The
gauge couplings for the two $G_{i}$’s in the $i^{\text{th}}$ sector are taken
to be equal.. We will continue to work in this limit. In this limit, there is
an orbifold equivalence Bershadsky:1998cb ; Schmaltz:1998bg ; Dymarsky:2005nc
; Dymarsky:2005uh ; Pomoni:2008de ; Cherman:2010jj ; Hanada:2011ju ;
Dunne:2016nmc ; Aitken:2019shs ; Jepsen:2020czw  between the RDB model and the
CDB model Chaudhuri:2020xxb  with the couplings in the two dual theories
related by

$\begin{split}\lambda_{i}^{C}=\frac{\lambda_{i}^{R}}{2},\
h_{i}^{C}=2h_{i}^{R},\ f_{i}^{C}=2f_{i}^{R},\
\zeta^{C}=2\zeta^{R}.\end{split}$ (4)

Here the superscript ‘R’ or ‘C’ indicates whether the coupling belongs to the
RDB or the CDB model. This planar equivalence between the two models will
allow us to restrict our attention to the RDB model. All results that will be
derived in this work will have their counterparts in the CDB model.

Now, the planar beta functions (in the $\overline{\text{MS}}$ scheme) of the
different couplings in the RDB model have the following forms:

$\begin{split}&\beta_{\lambda_{i}}=-\Big{(}\frac{21-4x_{fi}}{6}\Big{)}\lambda_{i}^{2}+\Big{(}\frac{-27+13x_{fi}}{6}\Big{)}\lambda_{i}^{3}+\cdots\
,\\\
&\beta_{h_{i}}=16h_{i}^{2}-6h_{i}\lambda_{i}+\frac{3}{16}\lambda_{i}^{2}+\cdots\
,\\\
&\beta_{f_{i}}=8f_{i}^{2}+32f_{i}h_{i}-6f_{i}\lambda_{i}+24h_{i}^{2}+\frac{9}{16}\lambda_{i}^{2}+8\zeta^{2}+\cdots\
,\\\
&\beta_{\zeta}=\zeta\Bigg{[}8f_{1}+8f_{2}+16h_{1}+16h_{2}-3\lambda_{1}-3\lambda_{2}\Bigg{]}+\cdots\
.\end{split}$ (5)

The dots indicate higher order corrections. There are several features of
these beta functions which will be important for our analysis in this work. We
would like to mention two of them here. Firstly, note that these beta
functions are independent of the ratio $r=\frac{N_{c2}}{N_{c1}}$. This means
that the RG flows that we will study in section 3 will be independent of this
ratio. Secondly, the beta functions of the single trace couplings
($\lambda_{i}$ and $h_{i}$) are independent of the double trace couplings
($f_{i}$ and $\zeta$).888This is a general feature of planar beta functions of
single trace couplings in large N gauge theories Dymarsky:2005uh . These
properties of the beta functions were shown to survive up to all orders at the
Veneziano limit in Chaudhuri:2020xxb .

Based on the forms of the above beta functions, one can show that there are
unitary perturbative fixed points999These fixed points are akin to the Banks-
Zaks-Caswell fixed points Belavin:1974gu ; PhysRevLett.33.244 ; Banks:1981nn
in more familiar QCDs. which are of the following two kinds.

  1. 1.

There is a discrete set of fixed points where the two sectors are decoupled
and the couplings have the following values at leading order:

$\displaystyle\lambda_{i}={21-4x_{fi}\over 13x_{fi}-27},\
h_{i}={3-\sqrt{6}\over 16}\lambda_{i},\
f_{i}={2\sqrt{6}+\sigma_{i}\sqrt{18\sqrt{6}-39}\over 16}\lambda_{i},\ \zeta=0$
(6)

with $\sigma_{i}=\pm 1$. Note that to get unitary fixed points in a
perturbative regime, one must set $x_{fi}=\frac{21}{4}-\epsilon_{i}$ with
$0<\epsilon_{i}\ll 1$.

  2. 2.

The second class of fixed points consists of theories where the two sectors
are coupled. These fixed points exist only when $x_{f1}=x_{f2}\equiv x_{f}$.
They form a conformal manifold which has the shape of a circle in the space of
couplings101010A couple of points on this circle where $\zeta=0$ have already
been included in the first class.. The values of the different couplings on
this fixed circle at leading order are as follows:

$\displaystyle\lambda_{1}=\lambda_{2}\equiv\lambda={21-4x_{f}\over
13x_{f}-27},\ h_{1}=h_{2}\equiv h={3-\sqrt{6}\over 16}\lambda,$ (7)
$\displaystyle f_{p}=\frac{\sqrt{6}}{8}\lambda,\
f_{m}^{2}+\zeta^{2}=\Big{(}{18\sqrt{6}-39\over 256}\Big{)}\lambda^{2}.$

where $f_{p}\equiv\frac{f_{1}+f_{2}}{2}$ and
$f_{m}\equiv\frac{f_{1}-f_{2}}{2}$.

All these fixed points survive under higher loop corrections at the planar
limit and have stable effective potentials (at least up to leading order in
perturbation theory)Chaudhuri:2020xxb . Moreover, in appropriate regimes of
the ratio $r$ certain points on the fixed circle exhibit thermal order, i.e.,
a $\mathbb{Z}_{2}$ symmetry in the sector with the smaller rank is
spontaneously broken at all nonzero temperatures. The relevant
$\mathbb{Z}_{2}$ symmetry in the $i^{\text{th}}$ sector transforms the
different fields in that sector as follows:

$\begin{split}&\Phi_{i}\rightarrow\mathcal{T}_{i}\Phi_{i},\ \
\psi_{i}\rightarrow\mathcal{T}_{i}\psi_{i},\
(V_{i1})_{\mu}\rightarrow\mathcal{T}_{i}(V_{i1})_{\mu}\mathcal{T}_{i}^{-1},\end{split}$
(8)

where $\mathcal{T}_{i}$ is the following $N_{ci}\times N_{ci}$ diagonal
matrix:

$\displaystyle\mathcal{T}_{i}\equiv\text{diag}\\{-1,1,\cdots,1\\}.$ (9)

It leaves the fields in the other sector unchanged. These symmetry
transformations map one class of gauge-equivalent field configurations to
another.111111See appendix A of Chaudhuri:2020xxb  for a proof of this. These
global symmetries were called baryon symmetries in Chaudhuri:2020xxb  because
the corresponding order parameters are the expectation values of the
determinants of the scalar fields.

When $N_{c2}<N_{c1}$, the baryon symmetry in the first sector remains unbroken
at all temperatures. On the other hand, the baryon symmetry in the second
sector can be spontaneously broken in a thermal state for a subset of points
on the conformal manifold when
$r<r_{\text{max}}\equiv\sqrt{\frac{6\sqrt{6}-13}{61-6\sqrt{6}}}\approx 0.191$.
These points can be expressed simply by defining the following polar
coordinates in the $f_{m}$-$\zeta$ plane:

$\displaystyle f_{m}=R\sin\theta,\ \zeta=R\cos\theta,$ (10)

where $\theta\in[0,2\pi)$. For each $r<r_{\text{max}}$, the fixed points
exhibiting thermal order lie in the angular interval
$\theta\in(\theta_{1},\theta_{2})$ where121212One can check that the angular
interval given here is equivalent to the intervals of $f_{m}$ and $\zeta$
given in Chaudhuri:2020xxb .

$\displaystyle\theta_{1}=\cos^{-1}(\nu_{1}),\
\theta_{2}=\pi-\text{sgn}(r-r_{0})\Big{(}\pi-\cos^{-1}(\nu_{2})\Big{)}$ (11)

with $r_{0}\equiv\frac{\sqrt{18\sqrt{6}-39}}{12}$ and

$\displaystyle\nu_{1}\equiv
r\Bigg{[}\frac{-12+\sqrt{\Big{(}18\sqrt{6}-183\Big{)}r^{2}+\Big{(}18\sqrt{6}-39\Big{)}}}{\Big{(}1+r^{2}\Big{)}\sqrt{18\sqrt{16}-39}}\Bigg{]}\
,$ (12) $\displaystyle\nu_{2}\equiv
r\Bigg{[}\frac{-12-\sqrt{\Big{(}18\sqrt{6}-183\Big{)}r^{2}+\Big{(}18\sqrt{6}-39\Big{)}}}{\Big{(}1+r^{2}\Big{)}\sqrt{18\sqrt{16}-39}}\Bigg{]}\
.$

Here the range of the function $\cos^{-1}$ is the interval $[0,\pi]$. The
plots of $\theta_{1}$ and $\theta_{2}$ as functions of $r$ are shown in figure
2. From these plots, one can see that for all values of
$r\in(0,r_{\text{max}})$ the end points of the above angular interval satisfy
$\frac{\pi}{2}<\theta_{1}<\theta_{2}<\frac{3\pi}{2}$ with
$\theta_{1}\rightarrow\frac{\pi}{2}$ and $\theta_{2}\rightarrow\frac{3\pi}{2}$
as $r\rightarrow 0$, and
$\theta_{1,2}\rightarrow\cos^{-1}\Big{(}-\frac{\sqrt{61-6\sqrt{6}}}{4\sqrt{3}}\Big{)}\approx
2.952$ as $r\rightarrow r_{\text{max}}$. Thus, $\cos\theta$ (or equivalently,
$\zeta=R\cos\theta$) is always negative for the fixed points demonstrating
thermal order.

When $N_{c1}<N_{c2}$, one would get similar fixed points with spontaneous
breaking of the baryon symmetry in the first sector at all nonzero
temperatures. These fixed points can be obtained from those demonstrating
thermal order in the second sector by the transformations
$r\rightarrow\frac{1}{r}$ and $\theta\rightarrow 2\pi-\theta$.

Figure 2: Graphs of $\theta_{1}$ and $\theta_{2}$ against $r$: For each value
of $r<r_{\text{max}}\equiv\sqrt{\frac{6\sqrt{6}-13}{61-6\sqrt{6}}}$,
$\theta_{1}$ and $\theta_{2}$ are the endpoints of the angular interval on the
fixed circle which exhibits thermal order in the second sector. They always
lie in the regime $\frac{\pi}{2}<\theta_{1}<\theta_{2}<\frac{3\pi}{2}$. As
$r\rightarrow 0$, $\theta_{1}\rightarrow\frac{\pi}{2}$ and
$\theta_{2}\rightarrow\frac{3\pi}{2}$. As $r\rightarrow r_{\text{max}}$,
$\theta_{1,2}\rightarrow\cos^{-1}\Big{(}-\frac{\sqrt{61-6\sqrt{6}}}{4\sqrt{3}}\Big{)}\approx
2.952$.

It was shown in Chaudhuri:2020xxb  that for all the fixed points where the
baryon symmetry in a sector is spontanously broken at nonzero temperatures,
this phenomenon is accompanied by the Higgsing of half of the gauge bosons in
the same sector. This leads the system to be in a persistent Brout-Englert-
Higgs (BEH) phase at all nonzero temperatures.

A similar spontaneous breaking of a baryon symmetry at nonzero temperatures
and a persistent BEH phase were also found at the corresponding fixed points
of the dual CDB model. In this case, the relevant baryon symmetry in each
sector is a $U(1)$ symmetry which can be obtained by replacing
$\mathcal{T}_{i}$ by the $N_{ci}\times N_{ci}$ diagonal matrix defined below:

$\displaystyle(\mathcal{T}_{i})_{\phi}\equiv\text{diag}\\{e^{i\phi},1,\cdots,1\\}.$
(13)

with $\phi\in[0,2\pi)$.

An important result which was derived in Chaudhuri:2020xxb  is that the above-
mentioned baryon symmetries are not broken in a thermal state for the fixed
points where the two sectors are decoupled. We will find this result to be
consequential as one of these fixed points is the IR limit of the RG flows
corresponding to the marginally relevant deformations discussed in section 3.

##  3 Phase transitions for marginally relevant deformations

In this section we will consider certain marginally relevant deformations of
the points on the fixed circle in the RDB model by varying the quartic
couplings between the scalars fields. In subsection 3.1 we will demonstrate
the existence of such a marginally relevant deformation for each point on the
fixed circle. Later in subsection 3.2 we will study the RG flows triggered by
these deformations. We will show that in the IR limit these RG flows take the
corresponding systems to a unique fixed point that lacks thermal order.
Therefore, at low temperatures all these systems are in a disordered phase. On
the other hand, in the UV limit some of these systems flow to the points on
the fixed circle which exhibit thermal order. This means that at high
temperatures these systems are in an ordered phase. Therefore, each of these
systems must undergo an inverse symmetry breaking phase transition at some
critical temperature. Above this critical temperature, the system remains
persistently in a symmetry-broken phase. In subsection 3.3, we will provide an
estimate of this critical temperature.

###  3.1 Deformations of the double trace quartic couplings

Let us now turn our attention to the deformations of the quartic couplings
away from their values on the fixed circle. The deformations that are of
interest to us involve only the double trace couplings ($f_{i}$ and $\zeta$).
Since the beta functions of the single trace couplings ($\lambda_{i}$ and
$h_{i}$) are independent of the double trace ones at the Veneziano limit,
these couplings remain frozen at their fixed point values throughout the RG
flows triggered by such deformations. Therefore, all such RG flows take place
only in a subspace where $\lambda_{1}=\lambda_{2}=\lambda$ and $h_{1}=h_{2}=h$
with $\lambda$ and $h$ being the values given in (7) at leading order in
perturbation theory. In this subspace the full planar beta functions of the
double trace couplings have the following forms 131313We refer the reader to
section 5 of Chaudhuri:2020xxb  for a derivation of these forms of the beta
functions of the double trace couplings up to all orders in perturbation
theory at the planar limit.:

$\begin{split}&\beta_{f_{p}}=F_{1}(f_{p},\sqrt{f_{m}^{2}+\zeta^{2}}),\
\beta_{f_{m}}=f_{m}\widetilde{F}_{2}(f_{p},\sqrt{f_{m}^{2}+\zeta^{2}}),\
\beta_{\zeta}=\zeta\widetilde{F}_{2}(f_{p},\sqrt{f_{m}^{2}+\zeta^{2}}),\end{split}$
(14)

where $F_{1}$ and $\widetilde{F}_{2}$ are two functions of $f_{p}$ and
$\sqrt{f_{m}^{2}+\zeta^{2}}$. Switching to the polar coordinates introduced in
(10), the beta functions can be re-expressed as

$\begin{split}&\beta_{f_{p}}=F_{1}(f_{p},R),\ \beta_{R}=F_{2}(f_{p},R),\
R\beta_{\theta}=0,\end{split}$ (15)

where $F_{2}(f_{p},R)\equiv R\widetilde{F}_{2}(f_{p},R)$. The expressions of
$F_{1}(f_{p},R)$ and $F_{2}(f_{p},R)$ at leading order are

$\begin{split}&F_{1}(f_{p},R)=(8f_{p}+32h-6\lambda)f_{p}+8R^{2}+24h^{2}+\frac{9}{16}\lambda^{2}\
,\\\ &F_{2}(f_{p},R)=R(16f_{p}+32h-6\lambda)\ .\\\ \end{split}$ (16)

We can see that $\beta_{\theta}=0$ for all points in the space of the double
trace couplings where $R\neq 0$. Moreover, the beta functions of $f_{p}$ and
$R$ are independent of the angular coordinate $\theta$. Therefore, for a fixed
point of these beta functions where $R\neq 0$, a shift in the value of
$\theta$ spans the conformal manifold. Another important consequence of the
vanishing of $\beta_{\theta}$ is that the projections of all RG flows on the
$f_{m}$-$\zeta$ plane are radially directed. Furthermore, the fact that
$\beta_{f_{p}}$ and $\beta_{R}$ are independent of $\theta$ ensures that such
flows are identical for all values of $\theta$.

Now, suppose $f_{0p}$ and $R_{0}$ are the values of $f_{p}$ and $R$ for the
points on the conformal manifold. The leading order expressions of $f_{0p}$
and $R_{0}$ are (see (7))

$\displaystyle f_{0p}=\frac{\sqrt{6}}{8}\lambda,\
R_{0}=\Big{(}\frac{\sqrt{18\sqrt{6}-39}}{16}\Big{)}\lambda.$ (17)

Consider the following small deformations of $f_{p}$ and $R$ away from their
values on the fixed circle:

$\displaystyle f_{p}=f_{0p}+\delta f_{p},\ R=R_{0}+\delta R.$ (18)

The beta functions of these deformations in the values of $f_{p}$ and $R$ (at
linear order) are

$\begin{split}&\beta_{\delta f_{p}}=\frac{\partial F_{1}}{\partial
f_{p}}(f_{0p},R_{0})\delta f_{p}+\frac{\partial F_{1}}{\partial
R}(f_{0p},R_{0})\delta R,\\\ &\beta_{\delta R}=\frac{\partial F_{2}}{\partial
f_{p}}(f_{0p},R_{0})\delta f_{p}+\frac{\partial F_{2}}{\partial
R}(f_{0p},R_{0})\delta R.\end{split}$ (19)

The coefficient matrix multiplying the deformations in the above equations is

$\displaystyle\mathcal{M}=\begin{pmatrix}\frac{\partial F_{1}}{\partial
f_{p}}(f_{0p},R_{0})&\frac{\partial F_{1}}{\partial R}(f_{0p},R_{0})\\\
\frac{\partial F_{2}}{\partial f_{p}}(f_{0p},R_{0})&\frac{\partial
F_{2}}{\partial R}(f_{0p},R_{0})\end{pmatrix}=16R_{0}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}.$ (20)

The second equality in the above equation is obtained by substituting $F_{1}$,
$F_{2}$ and the couplings $h$, $f_{0p}$ and $R_{0}$ by their leading order
expressions. The eigenvalues of this matrix and the corresponding eigenvectors
are as follows:

$\begin{split}&v_{1}=-16R_{0},\ v_{2}=16R_{0},\\\ &e_{1}=\begin{pmatrix}1\\\
-1\end{pmatrix},\ e_{2}=\begin{pmatrix}1\\\ 1\end{pmatrix}.\end{split}$ (21)

Therefore, the deformations along $e_{1}$ and $e_{2}$ are marginally relevant
and marginally irrelevant respectively. The existence of a marginally relevant
deformation clearly demonstrates that all points on the conformal manifold are
UV fixed points of certain RG flows. Next, let us study the features of these
RG flows.

###  3.2 RG flows triggered by the marginally relevant deformations

To analyze the RG flows triggered by the marginally relevant deformations, we
will rely on the 1-loop expressions of the beta functions. This approximation
is justified near the UV fixed points on the conformal manifold as these fixed
points are weakly coupled. However, the RG flows triggered by the marginally
relevant deformations can take the system to a strongly coupled regime. We
will show that this is not the case when we choose the marginally relevant
deformation such that it has a radially inward component in the
$f_{m}$-$\zeta$ plane, i.e., $\delta R<0$. Therefore, in this case one can
trust the results based on the 1-loop beta functions. If the deformation is
chosen to be in the opposite direction, i.e., $\delta R>0$, then the theory
indeed flows to a strongly coupled regime. This can be verified by noticing
that the RG flows obtained from the 1-loop beta functions lead to divergences
of the couplings at finite energy scales.

With the above comments in mind, we choose the marginally relevant deformation
to be

$\displaystyle\delta f_{p}=-\delta R=\frac{R_{0}}{2}\widetilde{\epsilon}$ (22)

at a reference energy scale $\Lambda$. We take $0<\widetilde{\epsilon}<1$. The
RG flow generated by the 1-loop beta functions leads to the following values
of the couplings at a scale $\mu$:

$\begin{split}&f_{p}=f_{0p}+\frac{R_{0}}{2}\Bigg{[}\frac{(1-k)\Big{(}1-\tanh(8R_{0}t)\Big{)}}{1+k\tanh(8R_{0}t)}\Bigg{]}\
,\
R=\frac{R_{0}}{2}\Bigg{[}\frac{(1+k)\Big{(}1+\tanh(8R_{0}t)\Big{)}}{1+k\tanh(8R_{0}t)}\Bigg{]}\
,\\\ \end{split}$ (23)

where $t\equiv\ln(\mu/\Lambda)$ and $k\equiv 1-\widetilde{\epsilon}$. In
figure 3, a graphical plot of this RG flow is given for $\lambda=0.001,\
\widetilde{\epsilon}=0.1$.

Figure 3: RG flow for $\lambda=0.001,\ \widetilde{\epsilon}=0.1$: In the UV
limit ($t\rightarrow\infty$) the couplings flow to a point on the fixed
circle. In the IR limit ($t\rightarrow-\infty$) they flow to a fixed point
where the two sectors are decoupled.

Note that the theory remains weakly coupled throughout the RG flow. In the
deep UV, it flows to a point on the fixed circle as expected. In the deep IR,
it flows to a fixed point where the two sectors are decoupled and
$f_{1}=f_{2}={2\sqrt{6}+\sqrt{18\sqrt{6}-39}\over 16}\lambda$. Due to the
rotational symmetry in the $f_{m}$-$\zeta$ plane, this behavior of the RG flow
is identical for marginally relevant deformations of all points on the fixed
circle. In particular, when $r<r_{\text{max}}$, it holds for the deformations
of the fixed points demonstrating thermal order in the second sector. The RG
flows triggered by these deformations lead to the IR fixed point where the two
sectors are decoupled. As we have mentioned earlier, the baryon symmetries in
both the sectors are unbroken at any nonzero temperature for this IR fixed
point. Therefore, for the systems that flow from the UV fixed points with
thermal order to this IR fixed point without thermal order, there must be a
transition from a disordered phase to an ordered phase as the temperature is
increased. We will now determine the critical temperatures at which this phase
transition takes place in these systems.

###  3.3 Estimates of the critical temperatures

To determine the critical temperatures, we will consider the thermal effective
potential of the scalar fields. At leading order in perturbation theory, this
potential has both quadratic and quartic terms. The quartic terms are as
follows:

$\displaystyle
V_{\text{quartic}}=16\pi^{2}\Bigg{[}\sum_{i=1}^{2}\frac{h_{i}}{N_{ci}}\text{Tr}[(\Phi_{i}^{T}\Phi_{i})^{2}]+\sum_{i=1}^{2}\frac{f_{i}}{N_{ci}^{2}}\Big{(}\text{Tr}[\Phi_{i}^{T}\Phi_{i}]\Big{)}^{2}+\frac{2\zeta}{N_{c1}N_{c2}}\text{Tr}[\Phi_{1}^{T}\Phi_{1}]\text{Tr}[\Phi_{2}^{T}\Phi_{2}]\Bigg{]}.$
(24)

In appendix A we have shown that these quartic terms are positive-definite
throughout the RG flows that we have discussed in the previous subsection.

The quadratic terms in the thermal effective potential are generated due to
integration over the nonzero Matsubara modes of the different fields in the
theory. These terms have the following structure

$\displaystyle
V_{\text{quadratic}}=\frac{1}{2}\sum_{i=1}^{2}m_{\text{th},i}^{2}\text{Tr}[\Phi_{i}^{T}\Phi_{i}]$
(25)

where $m_{\text{th},1}^{2}$ and $m_{\text{th},2}^{2}$ are the thermal masses
(squared) of the scalar fields $\Phi_{1}$ and $\Phi_{2}$ respectively. The
contributions of 1-loop diagrams to such thermal masses (squared) were
computed for a general 4-dimensional gauge theory in Weinberg:1974hy . Using
these general results, the 1-loop expressions of $m_{\text{th},1}^{2}$ and
$m_{\text{th},2}^{2}$ were derived in Chaudhuri:2020xxb . These expressions
reduce to the following forms at the planar limit:

$\begin{split}&m_{\text{th},1}^{2}=\frac{16\pi^{2}T^{2}}{3}\Bigg{[}2h_{1}+f_{1}+r\zeta+\frac{3}{8}\lambda_{1}\Bigg{]},\\\
&m_{\text{th},2}^{2}=\frac{16\pi^{2}T^{2}}{3}\Bigg{[}2h_{2}+f_{2}+\frac{\zeta}{r}+\frac{3}{8}\lambda_{2}\Bigg{]},\\\
\end{split}$ (26)

where $T$ is the temperature. These thermal masses (squared) quantify the
behavior of the effective potential of the scalar fields in the neighborhood
of the point $\Phi_{1}=\Phi_{2}=0$, viz., the origin of the field space. Their
signs determine the fates of the baryon symmetries at any given temperature.
If either $m_{\text{th},1}^{2}$ or $m_{\text{th},2}^{2}$ is negative then the
minimum of the effective potential cannot be at the origin of the field space.
In fact, the positivity of the tree-level quartic terms in the potential
ensures the existence of a minimum of the potential away from the origin. This
would imply the spontaneous breaking of at least one of the baryon symmetries.
In particular, when $m_{\text{th},i}^{2}<0$, the baryon symmetry in the
$i^{\text{th}}$ sector would be spontaneously broken Chaudhuri:2020xxb .

To track the effective potential at different temperatures we will set the
renormalization scale $\mu=T$. Let us explain this choice of the
renormalization scale. Our aim is to check perturbatively whether the thermal
effective potential of the scalar fields has a minimum away from the origin.
When both the thermal masses (squared) are positive, the minimum of the
potential is at the origin. In this case, as long as we are sufficiently away
from the critical temperature, we will have
$m_{\text{th},1}^{2},m_{\text{th},2}^{2}\sim\frac{16\pi^{2}}{3}\lambda
T^{2}$.141414This is due to the fact that the different couplings are at most
$O(\lambda)$ throughout the RG flows discussed in the previous subsection.
From the expressions given in (26), one can see that if $r$ is too small or
too large then the magnitude of one of the thermal masses (squared) can become
much larger than $\frac{16\pi^{2}}{3}\lambda T^{2}$. In this paper, we will
work in a regime where this is not the case. At higher orders in the
perturbative expansion of the potential near the origin, there are terms which
contain the logarithm of the ratio of $m_{\text{th},i}^{2}$ and $\mu^{2}$. The
lowest order term of this kind is the Coleman-Weinberg term Coleman:1973jx ;
Lee:1974fj ; Gould:2021oba . By setting $\mu=T$ we ensure that such
logarithmic terms are suppressed by at least a factor of $\lambda\ln\lambda\ll
1$ compared to the leading order terms that we are retaining.151515This
suppression persists even when $T$ is very small. A similar argument can be
given when either of the thermal masses (squared) is negative and the minimum
of the potential is away from the origin. In this case, again as long as we
are sufficiently away from the critical point, we will have
$|m_{\text{th},i}^{2}|\sim\frac{16\pi^{2}}{3}\lambda T^{2}$ irrespective of
the sign of $m_{\text{th},i}^{2}$.161616In fact, this behavior of the thermal
masses (squared) would persist in the high temperature limit. Comparing this
behavior of the thermal masses (squared) with the quartic terms given in (24),
one can check that if there is symmetry-breaking in the $i^{\text{th}}$
sector, then the value of $\Phi_{i}$ (appropriately normalized by $N_{ci}$) at
the minimum would be of the same order as the temperature scale $T$. To be
more precise, if $(\Phi_{i})_{0}$ is the value of $\Phi_{i}$ at the minimum,
then $\frac{\sqrt{\text{Tr}[(\Phi_{i})_{0}^{T}(\Phi_{i})_{0}]}}{N_{ci}}\sim
T$.171717See Chaudhuri:2020xxb  for the explicit forms of such thermal
expectation values of the scalar fields in the UV CFTs. Similar expressions
would hold in the models that we are are studying presently. We want the
perturbative expansion of the potential to be valid at this scale. At higher
orders in this expansion, there are terms which contain the logarithm of the
ratio of $\lambda$ times the square of this scale and $\mu^{2}$. By taking
$\mu=T$, we again ensure that such logarithmic corrections are small. So we
can rely on the perturbative analysis sufficiently away from the critical
temperature on both sides of this temperature. To come up with an estimate of
the critical temperature, we will interpolate the thermal masses (squared) in
the intermediate interval with the expressions given in (26) and search for
the point at which one of the thermal masses (squared) vanishes.

One may be able to improve the above-mentioned perturbative analysis by
choosing $\mu$ to be some $O(1)$ numerical coefficient times $T$. Such a
numerical factor can be absorbed in the reference energy scale $\Lambda$ by
rescaling it appropriately while doing thermal perturbation theory. Due to the
ambiguity in the value of this numerical factor, there would be theoretical
uncertainties in the critical temperatures of the phase transitions that we
will study.181818Such theoretical uncertainties were discussed in
Croon:2020cgk  for first order phase transitions in some models. Henceforth,
we will ignore this subtlety as our aim is to find the qualitative nature of
the phases in different temperature regimes and to determine the critical
temperatures for the phase transitions in terms of the unspecified reference
energy scale $\Lambda$.

Having provided the rationale for choosing $\mu=T$ in the effective potential,
let us now look at the behavior of the thermal masses at different temperature
scales. For this it is convenient to define the dimensionless quantity

$\displaystyle\widetilde{m}_{\text{th},i}^{2}\equiv\frac{3}{16\pi^{2}T^{2}}m_{\text{th},i}^{2}.$
(27)

Substituting the values of the the different couplings along the RG flow
triggered by the marginally relevant deformation, we get

$\begin{split}\widetilde{m}_{\text{th},1}^{2}=&\frac{R_{0}}{2}(1+k)\Big{(}r\cos\theta+\sin\theta-1\Big{)}C(t)+\Big{(}\frac{3}{4}\lambda+R_{0}\Big{)},\\\
\widetilde{m}_{\text{th},2}^{2}=&\frac{R_{0}}{2}(1+k)\Big{(}\frac{\cos\theta}{r}-\sin\theta-1\Big{)}C(t)+\Big{(}\frac{3}{4}\lambda+R_{0}\Big{)},\\\
\end{split}$ (28)

where $\theta$ is the angular location of the UV fixed point on the conformal
manifold and $C(t)$ is the following function of
$t\equiv\ln(\frac{T}{\Lambda})$:

$\displaystyle C(t)\equiv\frac{1+\tanh(8R_{0}t)}{1+k\tanh(8R_{0}t)}.$ (29)

Note that the parameter $k=1-\widetilde{\epsilon}$ controls the behavior of
the thermal masses at different temperatures. As
$\widetilde{\epsilon}\rightarrow 0$, $C(t)\rightarrow 1$ and the rescaled
thermal masses (squared), $\widetilde{m}_{\text{th},1}^{2}$ and
$\widetilde{m}_{\text{th},2}^{2}$, stop changing with the temperature. In this
limit, they just reduce to the rescaled thermal masses (squared) of the points
on the conformal manifold which were analyzed in Chaudhuri:2020xxb .

To study the variations of the thermal masses (squared) with the temperature
concretely, we choose the UV fixed point to be the point where $\theta=\pi$,
or equivalently, $f_{m}=0,\ \zeta=-R_{0}$. As can be verified from (11), this
fixed point has thermal order in the second sector only when $r<r_{0}\approx
0.188$. So to stay in this regime, we choose $r=0.1$. To keep the values of
the couplings and the deformations small we choose $\lambda=0.001$ and
$\widetilde{\epsilon}=0.1$ as before. For this set of parameters, the plots of
the rescaled thermal masses (squared) at different temperature scales are
given in figure 4.

Figure 4: Plots of the rescaled thermal masses (squared) against
$t=\ln(T/\Lambda)$ for $r=0.1$, $\theta=\pi$, $\lambda=0.001$,
$\widetilde{\epsilon}=0.1$: In the low temperature regime, both
$\widetilde{m}_{\text{th},1}^{2}$ and $\widetilde{m}_{\text{th},2}^{2}$
saturate at positive values. This indicates that the baryon symmetries in both
the sectors are unbroken in this regime. In the high temperature limit,
$\widetilde{m}_{\text{th},1}^{2}$ saturates at a positive value, while
$\widetilde{m}_{\text{th},2}^{2}$ saturates at a negative value. This means
that the baryon symmetry in the second sector remains persistently broken
above a critical temperature. This critical temperature ($T_{c}$) is given by
$t_{c}\equiv\ln(T_{c}/\Lambda)\approx-1172.1$.

As one can see form these plots, both $\widetilde{m}_{\text{th},1}^{2}$ and
$\widetilde{m}_{\text{th},2}^{2}$ decrease monotonically with increase in the
temperature. However, $\widetilde{m}_{\text{th},1}^{2}$ remains positive at
all temperatures indicating the absence of thermal order in the first sector.
On the other hand, $\widetilde{m}_{\text{th},2}^{2}$ starts off from a
positive value at low temperatures, but eventually becomes negative at high
temperatures indicating a transition to an ordered phase. The critical
temperature at which this transition happens is where
$\widetilde{m}_{\text{th},2}^{2}=0$. This behavior is similar for all the
systems where the UV fixed point exhibits thermal order in the second sector.
One can obtain a general expression of the critical temperature
($T_{c}\equiv\Lambda e^{t_{c}}$) for all these systems by solving the equation
$\widetilde{m}_{\text{th},2}^{2}=0$. We provide this expression below:

$\displaystyle t_{c}=\frac{1}{8R_{0}}\tanh^{-1}(\rho),$ (30)

where

$\displaystyle\rho\equiv\frac{-(1+k)\frac{R_{0}}{2r}\Big{(}-\cos\theta+r(1+\sin\theta)\Big{)}+(\frac{3\lambda}{4}+R_{0})}{(1+k)\frac{R_{0}}{2r}\Big{(}-\cos\theta+r(1+\sin\theta)\Big{)}-k(\frac{3\lambda}{4}+R_{0})}.$
(31)

From the above expression, we can see that $t_{c}$ has a real value only when
$\rho\in(-1,1)$. This puts a restriction on the UV fixed points for which
there can be a phase transition. The allowed fixed points are precisely the
ones where $\theta\in(\theta_{1},\theta_{2})$ with $\theta_{1}$ and
$\theta_{2}$ being the values given in (11). To study the variation of $t_{c}$
in this domain, we choose $r=0.1$, $\lambda=0.001$, $\widetilde{\epsilon}=0.1$
and plot the values of $t_{c}$ for different values of $\theta$ in figure 5.

Figure 5: Plot of $t_{c}$ against $\theta$ for $r=0.1,\lambda=0.001,\
\widetilde{\epsilon}=0.1$: The critical temperature increases sharply near the
edges of the interval $(\theta_{1},\theta_{2})$, i.e.,
$t_{c}\rightarrow\infty$ as $\theta\rightarrow\theta_{1}$ or
$\theta\rightarrow\theta_{2}$.

As one can see, $t_{c}$ increases rapidly as one approaches the edges of the
interval $(\theta_{1},\theta_{2})$. As $\theta\rightarrow\theta_{1}$ or
$\theta\rightarrow\theta_{2}$, $t_{c}\rightarrow\infty$ thereby indicating the
absence of the phase transition at the end points. For each point in the
interval $\theta\in(\theta_{1},\theta_{2})$, there is a finite temperature at
which the system goes to an ordered phase characterized by the spontaneous
breaking of the baryon symmetry and Higgsing of half of the gauge bosons in
the second sector. In the Veneziano limit, the system remains in this phase at
all higher temperatures.

The line of critical points shown in figure 5 separates the following two
phases:

  * •

Phase 1: The high temperature phase where the baryon symmetry in the first
sector remains unbroken while the baryon symmetry in the second sector is
spontaneously broken.

  * •

Phase 2: The low temperature phase where the baryon symmetries in both the
sectors are unbroken.

In figure 6, we show the phase diagram of the systems with $r=0.1$,
$\lambda=0.001$, $\widetilde{\epsilon}=0.1$. The phase diagrams for other
values of $r$ in the domain $r<r_{\text{max}}$ would be similar to that shown
in this figure.

Figure 6: Phase diagram of the systems with $r=0.1$, $\lambda=0.001$,
$\widetilde{\epsilon}=0.1$: In Phase 1, the baryon symmetry in the first
sector is unbroken while the baryon symmetry in the second sector is
spontaneously broken. In Phase 2, the baryon symmetries in both the sectors
are unbroken. The red line in this diagram is the line of critical points
separating the two phases. It is the same as the curve given in figure 5.

Let us end this section with a remark on the zero temperature phase of the
systems that we have been considering. At $T=0$, the thermal masses (squared)
vanish and it is no longer possible to set the renormalization scale $\mu$ to
$T$. In this case, one may wonder whether the Coleman-Weinberg terms can lead
to a symmetry-broken phase. Such a symmetry-broken phase was indeed found in
Coleman:1973jx  for massless scalar QED at zero temperature. As shown in
Coleman:1973jx , the quartic coupling in that model is of the same order as
the fourth power of the gauge coupling at certain renormalization scales. This
leads to the Coleman-Weinberg term becoming comparable to the quartic term
slightly away from the origin which, in turn, allows for a perturbatively
reliable minimum in this domain. In our case, however, the quartic couplings
remain $O(\lambda)$ throughout the RG flow with $\lambda$ being related to the
gauge couplings $g_{i}$ by $\lambda=\frac{N_{ci}g_{i}^{2}}{16\pi^{2}}$. The
suppression of the Coleman-Weinberg terms compared to the quartic terms in
this case is analogous to the same in massless $\phi^{4}$ theory where one
does not get a perturbatively reliable minimum away from the origin at zero
temperature Coleman:1973jx . Therefore, in the systems that we are
considering, we expect the baryon symmetries in both the sectors to remain
unbroken at zero temperature. We would like to establish this more rigorously
in the future.

##  4 Adding masses to the scalar fields

In the previous section we studied marginally relevant deformations of the
points on the fixed circle in the RDB model obtained by varying the double
trace quartic couplings between the scalar fields. In this section we will
consider relevant deformations of the same fixed points obtained by adding
masses to the scalar fields.191919The planar equivalence between the RDB model
and the CDB model can be extended to the case where the scalar fields are
massive. These masses in the two dual theories are equal to each other. For
such deformations, the Lagrangian has the following terms in addition to those
given in (2):

$\displaystyle\mathcal{L}_{\text{mass}}=-\frac{1}{2}\sum_{i=1}^{2}m_{i}^{2}\text{Tr}[\Phi_{i}^{T}\Phi_{i}],$
(32)

where $m_{1}^{2}$ and $m_{2}^{2}$ are the renormalized masses (squared) of the
scalar fields in the two sectors. The 1-loop planar contributions to their RG
flows (in the $\overline{\text{MS}}$ scheme) are derived in appendix B. These
contributions are as follows:

$\begin{split}&\mu\frac{dm_{1}^{2}}{d\mu}=\Big{(}16h_{1}+8f_{1}-3\lambda_{1}\Big{)}m_{1}^{2}+8r\zeta
m_{2}^{2},\\\
&\mu\frac{dm_{2}^{2}}{d\mu}=\Big{(}16h_{2}+8f_{2}-3\lambda_{2}\Big{)}m_{2}^{2}+\frac{8\zeta}{r}m_{1}^{2}.\end{split}$
(33)

Since we are working in a mass-independent renormalization scheme for the RG
flows of the quartic couplings, we can freeze these couplings at their fixed
point values and study the flows of the masses. Substituting the couplings by
their leading order values on the fixed circle as given in (7) and switching
to the polar coordinates in the $f_{m}$-$\zeta$ plane (see (10)), we get

$\begin{split}&\mu\frac{dm_{1}^{2}}{d\mu}=8R_{0}\Big{(}\sin\theta\
m_{1}^{2}+r\cos\theta\ m_{2}^{2}\Big{)},\\\
&\mu\frac{dm_{2}^{2}}{d\mu}=8R_{0}\Big{(}\frac{\cos\theta}{r}\
m_{1}^{2}-\sin\theta\ m_{2}^{2}\Big{)},\end{split}$ (34)

where $R_{0}$ is the radius of the fixed circle whose value at leading order
in $\lambda$ is given by202020We remind the reader that
$\lambda_{1}=\lambda_{2}\equiv\lambda=\frac{21-4x_{f}}{13x_{f}-27}$ on the
fixed circle.

$\displaystyle R_{0}=\Big{(}\frac{\sqrt{18\sqrt{6}-39}}{16}\Big{)}\lambda.$
(35)

Solving the above equations, we get

$\begin{split}&m_{1}^{2}=\frac{1}{2}(1-\sin\theta)M_{1}^{2}\
e^{-8R_{0}t}+\frac{1}{2}(1+\sin\theta)M_{2}^{2}\ e^{8R_{0}t},\\\
&m_{2}^{2}=\frac{1}{2r}\Big{(}-M_{1}^{2}\ e^{-8R_{0}t}+M_{2}^{2}\
e^{8R_{0}t}\Big{)}\cos\theta,\end{split}$ (36)

where $t\equiv\ln(\mu/\Lambda)$ with $\Lambda$ being a reference energy scale.
$M_{1}^{2}$ and $M_{2}^{2}$ are integration constants having the dimension of
mass2. When both of them are nonzero, they determine the asymptotic behavior
of the renormalized masses (squared) in the IR and UV regimes respectively.

At any nonzero temperature $T$ there are additional contributions to the
effective masses (squared) of the scalar fields from the the thermal masses
(squared) which were discussed in the previous section. For the massive
theories that we are considering presently, the expressions of the 1-loop
thermal masses (squared) given in (26) are approximately valid when
$T^{2}\gg|m_{i}^{2}|$. These expressions are given below:

$\begin{split}&m_{\text{th},1}^{2}=\frac{16\pi^{2}T^{2}}{3}\Bigg{[}2h_{1}+f_{1}+r\zeta+\frac{3}{8}\lambda_{1}\Bigg{]},\\\
&m_{\text{th},2}^{2}=\frac{16\pi^{2}T^{2}}{3}\Bigg{[}2h_{2}+f_{2}+\frac{\zeta}{r}+\frac{3}{8}\lambda_{2}\Bigg{]}.\\\
\end{split}$ (37)

For the points on the fixed circle, these expressions reduce to

$\begin{split}m_{\text{th},1}^{2}=&\frac{16\pi^{2}T^{2}}{3}\Bigg{[}R_{0}\Big{(}r\cos\theta+\sin\theta\Big{)}+\frac{3}{4}\lambda\Bigg{]},\\\
m_{\text{th},2}^{2}=&\frac{16\pi^{2}T^{2}}{3}\Bigg{[}R_{0}\Big{(}\frac{\cos\theta}{r}-\sin\theta\Big{)}+\frac{3}{4}\lambda\Bigg{]}.\\\
\end{split}$ (38)

The overall effective masses (squared) of the scalar fields at the temperature
$T$ are

$\begin{split}&m_{\text{eff},1}^{2}=(m_{1}^{2}+m_{\text{th},1}^{2})|_{\mu=T},\
m_{\text{eff},2}^{2}=(m_{2}^{2}+m_{\text{th},2}^{2})|_{\mu=T}.\end{split}$
(39)

These effective masses (squared) determine the behavior of the effective
potential of the scalar fields in the neighborhood of the point
$\Phi_{1}=\Phi_{2}=0$. A negative value of $m_{\text{eff},i}^{2}$ indicates
that the baryon symmetry in the $i^{\text{th}}$ sector is spontaneously
broken.

Here, as in the previous section, we have set $\mu=T$ to avoid large
logarithms from appearing in the thermal perturbative expansion of the
effective potential of the scalar fields. However, now one must be careful as
the validity of this choice of the renormalization scale rests on the
magnitudes of effective masses (squared) being $O(\frac{16\pi^{2}}{3}\lambda
T^{2})$.212121The validity of the perturbation theory also requires
$|m_{i}^{2}|_{\mu=T}|\ll T^{2}.$ However, note that this is automatically
ensured when $|m_{\text{eff},i}^{2}|\sim\frac{16\pi^{2}}{3}\lambda T^{2}$
because
$\displaystyle|m_{i}^{2}|_{\mu=T}|=|m_{\text{eff},i}^{2}-m_{\text{th},i}^{2}|<|m_{\text{eff},i}^{2}|+|m_{\text{th},i}^{2}|\sim\frac{32\pi^{2}}{3}\lambda
T^{2}\ll T^{2}.$ (40) Conversely, when the magnitudes of the effective masses
(squared) become much larger than $O(\frac{16\pi^{2}}{3}\lambda T^{2})$, the
renormalized masses (squared) can become comparable to $T^{2}$. In that case,
the expressions of the thermal masses (squared) given in (38) are no longer
reliable. At sufficiently low temperatures, the contributions of the
renormalized masses (squared) will drive the effective masses (squared) out of
this regime. So, unlike the previous section, here we cannot study the phase
of the system down to arbitrarily low temperatures. However, we will show that
in some cases, the renormalized masses can induce a phase transition at scales
where the perturbative analysis is reliable, i.e., near the critical
temperatures corresponding to these phase transitions, the effective masses
(squared) remain $O(\frac{16\pi^{2}}{3}\lambda T^{2})$. In the rest of this
section, we will use the term ‘low temperatures’ to mean temperatures slightly
below such critical temperatures where the perturbative analysis is valid. We
hope the reader will not be misled by this laxity.

The above-mentioned phase transitions and the corresponding critical
temperatures depend on the magnitudes and the signs of two mass2 scales
$M_{1}^{2}$ and $M_{2}^{2}$ which were introduced in (36). To study these
phase transitions let us define the following two dimensionless quantities
which are obtained via dividing $M_{1}^{2}$ and $M_{2}^{2}$ by
$\frac{32\pi^{2}}{3}\Lambda^{2}$:

$\displaystyle c_{1}\equiv\frac{3}{32\pi^{2}}\frac{M_{1}^{2}}{\Lambda^{2}},\
c_{2}\equiv\frac{3}{32\pi^{2}}\frac{M_{2}^{2}}{\Lambda^{2}}.$ (41)

It is also convenient to switch to the following dimensionless quantities
obtained via dividing the effective masses (squared) by
$\frac{16\pi^{2}}{3}T^{2}$ :

$\widetilde{m}_{\text{eff},1}^{2}\equiv\frac{3}{16\pi^{2}}\frac{m_{\text{eff},1}^{2}}{T^{2}},\
\widetilde{m}_{\text{eff},2}^{2}\equiv\frac{3}{16\pi^{2}}\frac{m_{\text{eff},2}^{2}}{T^{2}}.$
(42)

In the regime where $m_{i}^{2}|_{\mu=T}\ll T^{2}$ and the expressions of the
thermal masses (squared) given in (38) are reliable, the rescaled effective
masses (squared) defined above have the following values:

$\begin{split}&\widetilde{m}_{\text{eff},1}^{2}=(1-\sin\theta)c_{1}\
e^{-(2+8R_{0})t}+(1+\sin\theta)c_{2}\
e^{-(2-8R_{0})t}+R_{0}\Big{(}r\cos\theta+\sin\theta\Big{)}+\frac{3}{4}\lambda,\\\
&\widetilde{m}_{\text{eff},2}^{2}=\frac{1}{r}\Big{(}-c_{1}\
e^{-(2+8R_{0})t}+c_{2}\
e^{-(2-8R_{0})t}\Big{)}\cos\theta+R_{0}\Big{(}\frac{\cos\theta}{r}-\sin\theta\Big{)}+\frac{3}{4}\lambda.\end{split}$
(43)

In the high temperature limit (i.e., large $t$), the rescaled effective masses
(squared) are dominated by the contributions of the thermal masses (squared)
and we have

$\begin{split}&\widetilde{m}_{\text{eff},1}^{2}\xrightarrow{t\rightarrow\infty}R_{0}\Big{(}r\cos\theta+\sin\theta\Big{)}+\frac{3}{4}\lambda,\\\
&\widetilde{m}_{\text{eff},2}^{2}\xrightarrow{t\rightarrow\infty}R_{0}\Big{(}\frac{\cos\theta}{r}-\sin\theta\Big{)}+\frac{3}{4}\lambda.\end{split}$
(44)

These high temperature behaviors of the effective masses (squared) are exactly
as they would be for the UV CFTs. Therefore, for the systems where
$r<r_{\text{max}}=\sqrt{\frac{6\sqrt{6}-13}{61-6\sqrt{6}}}$ and
$\theta\in(\theta_{1},\theta_{2})$ with $\theta_{1}$ and $\theta_{2}$ being
the values given in (11), the baryon symmetry in the first sector remains
unbroken in the high temperature regime, whereas the baryon symmetry in the
second sector is persistently broken in the same regime. In the rest of this
section we will restrict our attention to these systems which exhibit thermal
order in the second sector at the high temperature limit. As the temperature
is lowered in these systems, i.e., as $t$ decreases, the contributions of the
renormalized masses (squared) start becoming comparable to the thermal masses
(squared). This is what allows for phase transitions in the system. From the
expressions given in (43) one can see that the signs of $c_{1}$ and $c_{2}$
(or equivalently, the signs of the mass2 scales $M_{1}^{2}$ and $M_{2}^{2}$)
play the key role in determining these phase transitions. We will study these
phase transitions in the following 2 cases:

  * •

Case 1: $M_{1}^{2}\neq 0,M_{2}^{2}=0$,

  * •

Case 2: $M_{1}^{2}=0,M_{2}^{2}\neq 0$.

We will see that for each of these cases, there are distinct patterns of phase
transitions depending on the signs of the nonzero $M_{i}^{2}$. We will discuss
the low temperature phases for the different signs of the nonzero $M_{i}^{2}$
in each case as separate subcases. Later, in subsection 4.3 we will summarize
these results and comment on the possible phase transitions when both
$M_{1}^{2}$ and $M_{2}^{2}$ are nonzero.

###  4.1 Case 1: $M_{1}^{2}\neq 0,M_{2}^{2}=0$

In this case the expressions of the rescaled effective masses (squared) reduce
to the following forms:

$\begin{split}&\widetilde{m}_{\text{eff},1}^{2}=(1-\sin\theta)c_{1}\
e^{-(2+8R_{0})t}+R_{0}\Big{(}r\cos\theta+\sin\theta\Big{)}+\frac{3}{4}\lambda,\\\
&\widetilde{m}_{\text{eff},2}^{2}=-\frac{1}{r}\cos\theta\ c_{1}\
e^{-(2+8R_{0})t}+R_{0}\Big{(}\frac{\cos\theta}{r}-\sin\theta\Big{)}+\frac{3}{4}\lambda.\end{split}$
(45)

We remind the reader that we are restricting our attention to the systems
which exhibit thermal order in the second sector at high temperatures, i.e.,
the systems where $r<r_{\text{max}}$ and $\theta\in(\theta_{1},\theta_{2})$.
As we have already mentioned in section 2, $\cos\theta$ is always negative for
such systems. Moreover, the quantity $(1-\sin\theta)$ is positive for all
values of $\theta$ in this domain. Therefore, from the expressions given in
(45) we can see that in the low temperature regime where the renormalized
masses (squared) dominate over the thermal masses (squared), both the
effective masses (squared) have the same sign. This sign depends on whether
$c_{1}$ (or equivalently, $M_{1}^{2}$) is positive or negative.

####  4.1.1 Subcase 1: $M_{1}^{2}>0$

When $M_{1}^{2}>0$, both the effective masses (squared) are positive in the
low temperature regime. This indicates that the baryon symmetries in both the
sectors are unbroken in this regime. Therefore, there should be a phase
transition at some critical temperature where the baryon symmetry in the
second sector is spontaneously broken.

At the critical temperature ($T_{c}$) corresponding to this phase transition
$m_{\text{eff},2}^{2}$ vanishes, i.e., the renormalized mass (squared)
$m_{2}^{2}|_{\mu=T_{c}}$ exactly cancels the thermal mass (squared)
$m_{\text{th},2}^{2}$ . Therefore, we have

$\displaystyle|m_{2}^{2}|_{\mu=T_{c}}|\sim\frac{16\pi^{2}}{3}\lambda
T_{c}^{2}\ll T_{c}^{2}.$ (46)

Moreover, note that the two renormalized masses (squared) are related as
follows

$\displaystyle m_{1}^{2}|_{\mu=T_{c}}=\alpha m_{2}^{2}|_{\mu=T_{c}}$ (47)

where $\alpha\equiv-\frac{r(1-\sin\theta)}{\cos\theta}$. Since we are
considering systems where $r\in(0,r_{\text{max}})$ and
$\theta\in(\theta_{1},\theta_{2})$ with $\theta_{1}$ and $\theta_{2}$
satisfying $\frac{\pi}{2}<\theta_{1}<\theta_{2}<\frac{3\pi}{2}$, both the
numerator and denominator in $\alpha$ are nonzero. In fact, the denominator
can become small only when $\theta$ is close to $\pi/2$ or $3\pi/2$. Such
values of $\theta$ can lie in the range $(\theta_{1},\theta_{2})$ only when
$r$ is close to zero. Therefore, in this case the factor of $r$ in the
numerator of the expression of $\alpha$ ensures that $\alpha$ remains small
despite the denominator having a small value. In figure 7 we show the values
of $\alpha$ for all values $(r,\theta)$ satisfying $r<r_{\text{max}}$ and
$\theta_{1}<\theta<\theta_{2}$. We can see that $\alpha<1$ at all points in
this domain.

Figure 7: Plot of $\alpha=-\frac{r(1-\sin\theta)}{\cos\theta}$ against $r$ and
$\theta$ in the domain $r<r_{\text{max}}$ and
$\theta\in(\theta_{1},\theta_{2})$.

Thus, from (46) and (47) we can conclude that

$\displaystyle|m_{1}^{2}|_{\mu=T_{c}}|\lesssim\frac{16\pi^{2}}{3}\lambda
T_{c}^{2}\ll T_{c}^{2}$ (48)

for all these systems. Note that $|m_{1,2}^{2}|_{\mu=T_{c}}|\ll T_{c}^{2}$
means that the expressions of the thermal masses (squared) given in (38) are
reliable at the critical temperature. Moreover, the fact that
$|m_{1}^{2}|_{\mu=T_{c}}|\lesssim\frac{16\pi^{2}}{3}\lambda T_{c}^{2}$ ensures
that at the critical temperature
$|m_{\text{eff},1}^{2}|\sim\frac{16\pi^{2}}{3}\lambda T_{c}^{2}$. This means
that the perturbation theory with $\mu=T$ can be trusted at temperatures near
$T_{c}$. Hence, one can study the phase transition using this renormalization
scale.

To study this phase transition with a concrete example, we plot the rescaled
effective masses (squared) against $t\equiv\ln(T/\Lambda)$ for the case where
$r=0.1,\ \theta=\pi$, $\lambda=0.001$, $c_{1}=0.1$ and $c_{2}=0$ in figure 8.
One can check that the choice of $r$ and $\theta$ is such that the UV fixed
point exhibits thermal order in the second sector.

Figure 8: Plots of the rescaled effective masses (squared) against
$t=\ln(T/\Lambda)$ for $r=0.1$, $\theta=\pi$, $\lambda=0.001$, $c_{1}=0.1$,
$c_{2}=0$: Both $\widetilde{m}_{\text{eff},1}^{2}$ and
$\widetilde{m}_{\text{eff},2}^{2}$ grow as the temperature decreases. In the
high temperature limit, $\widetilde{m}_{\text{eff},1}^{2}$ saturates at a
positive value while $\widetilde{m}_{\text{eff},2}^{2}$ saturates at a
negative value. This indicates a phase transition at an intermediate
temperature where the baryon symmetry in the second sector is broken. The
critical temperature ($T_{c}$) corresponding to this phase transition is given
by $t_{c}\equiv\ln(T_{c}/\Lambda)\approx 3.659$.

Note that $\widetilde{m}_{\text{eff},1}^{2}$ remains positive for all values
of $t$. It grows as one goes to lower temperatures. At high temperatures, it
saturates at a positive value. This means that the baryon symmetry in the
first sector remains unbroken at all temperatures where the perturbative
analysis is reliable. On the other hand, $\widetilde{m}_{\text{eff},2}^{2}$
also grows as the temperature is lowered, but at high temperatures it
saturates at a negative value. This means that the baryon symmetry in the
second sector is spontaneously broken at a critical temperature and then it
remains broken at all higher temperatures.

In order to determine this critical temperature, we can solve for the value of
$t$ at which $\widetilde{m}_{\text{eff},2}^{2}$ vanishes. Setting
$\widetilde{m}_{\text{eff},2}^{2}=0$ at $t=t_{c}=\ln(T_{c}/\Lambda)$, we get

$\begin{split}&t_{c}=-\frac{1}{2+8R_{0}}\ln\Bigg{[}\frac{r}{c_{1}\cos\theta}\Bigg{\\{}R_{0}\Big{(}\frac{\cos\theta}{r}-\sin\theta\Big{)}+\frac{3}{4}\lambda\Bigg{\\}}\Bigg{]}.\end{split}$
(49)

In figure 9, we provide the plot of $t_{c}$ corresponding to different values
of $\theta\in(\theta_{1},\theta_{2})$ for $\ r=0.1,\ \lambda=0.001,\
c_{1}=0.1,\ c_{2}=0$.

Figure 9: Plot of $t_{c}$ against $\theta$ for $r=0.1,\ \lambda=0.001,\
c_{1}=0.1,\ c_{2}=0$: The critical temperature increases sharply near the
edges of the interval $(\theta_{1},\theta_{2})$,i.e., $t_{c}\rightarrow\infty$
as $\theta\rightarrow\theta_{1,2}$.

Just like what we saw for the critical temperatures corresponding to the phase
transitions discussed in section 3, the critical temperature in this case
increases sharply as one approaches the edges of the domain
$(\theta_{1},\theta_{2})$, and $t_{c}\rightarrow\infty$ as
$\theta\rightarrow\theta_{1}$ or $\theta\rightarrow\theta_{2}$. This means
that at these edges, the system no longer undergoes the aforementioned phase
transition and the baryon symmetries in both the sectors remain unbroken at
all temperatures where the perturbative analysis can be trusted.

####  4.1.2 Subcase 2: $M_{1}^{2}<0$

When $M_{1}^{2}<0$, both the effective masses (squared) are negative in the
low temperature regime. This means that the baryon symmetries in both the
sectors are spontaneously broken in this regime. On the other hand, in the
high temperature regime, the baryon symmetry in the first sector is unbroken
while the baryon symmetry in the second sector is spontaneously broken.
Therefore, we expect a phase transition at some critical temperature where the
baryon symmetry in the first sector is restored.

At this critical temperature, the effective mass (squared)
$m_{\text{eff},1}^{2}$ should vanish, i.e., $m_{1}^{2}|_{\mu=T_{c}}$ should
cancel the contribution of the $m_{\text{th},1}^{2}$, and consequently

$\displaystyle|m_{1}^{2}|_{\mu=T_{c}}|\sim\frac{16\pi^{2}}{3}\lambda
T_{c}^{2}\ll T_{c}^{2}.$ (50)

Now, we can again use the relation (47) between $m_{1}^{2}$ and $m_{2}^{2}$ to
get

$\displaystyle m_{2}^{2}|_{\mu=T_{c}}=\frac{m_{1}^{2}|_{\mu=T_{c}}}{\alpha}$
(51)

where $\alpha\equiv-\frac{r(1-\sin\theta)}{\cos\theta}$. As we had noted
earlier, $\alpha<1$ for all the UV fixed points which we are considering,
i.e., the fixed points where $r<r_{\text{max}}$ and
$\theta\in(\theta_{1},\theta_{2})$. From the plot given in figure 7 we can see
that in certain regimes of $r$ and $\theta$, $\alpha$ can be much smaller than
1. In such regimes, the magnitude of $m_{2}^{2}|_{\mu=T_{c}}$ will be much
larger than $\frac{16\pi^{2}}{3}\lambda T_{c}^{2}$ making the perturbative
analysis with $\mu=T$ unreliable near the critical temperature.

In order to reliably employ the perturbative analysis to study the phase
transition, we choose a system where $r=0.03$ and $\theta=4.52$. One can check
that the fixed point to which this system flows in the UV lies in the domain
given in (11), i.e., it exhibits thermal order in the second sector. For this
point, we have $\alpha\approx 0.311$, and hence

$\displaystyle m_{2}^{2}|_{\mu=T_{c}}\approx 3.216\ m_{1}^{2}|_{\mu=T_{c}}.$
(52)

Therefore, near the critical temperature, we remain in a regime where
$|m_{\text{eff},2}^{2}|\sim\frac{16\pi^{2}}{3}\lambda T^{2}$, and the
perturbative analysis is reliable.

In figure 10, we plot $\widetilde{m}_{\text{eff},1}^{2}$ and
$\widetilde{m}_{\text{eff},2}^{2}$ against $t\equiv\ln(T/\Lambda)$ for the
case where $r=0.03$, $\theta=4.52$, $\lambda=0.001$ $c1=-0.1$ and $c_{2}=0$.

Figure 10: Plots of the rescaled effective masses (squared) against
$t=\ln(T/\Lambda)$ for $r=0.03$, $\theta=4.52$, $\lambda=0.001$, $c_{1}=-0.1$,
$c_{2}=0$: Both $\widetilde{m}_{\text{eff},1}^{2}$ and
$\widetilde{m}_{\text{eff},2}^{2}$ are negative in the low temperature regime,
and their magnitudes grow as the temperature is decreased. In the high
temperature limit, $\widetilde{m}_{\text{eff},1}^{2}$ saturates at a positive
value while $\widetilde{m}_{\text{eff},2}^{2}$ saturates at a negative value.
This indicates that the baryon symmetry in the second sector remains broken at
all temperatures where the perturbative analysis can be trusted, while the
baryon symmetry in the first sector is restored at a critical temperature.
This critical temperature ($T_{c}$) is given by
$t_{c}=\ln(T_{c}/\Lambda)\approx 2.889$.

Here $\widetilde{m}_{\text{eff},2}^{2}$ remains negative for all values of
$t$. Its magnitude grows as one goes to lower temperatures. At high
temperatures, it saturates at a negative value. This means that the baryon
symmetry in the second sector remains broken at all temperatures where the
perturbative analysis is valid. The behavior of
$\widetilde{m}_{\text{eff},1}^{2}$ is quite similar to that of
$\widetilde{m}_{\text{eff},2}^{2}$ in the low temperature regime, i.e., in
this regime $\widetilde{m}_{\text{eff},1}^{2}<0$ and
$|\widetilde{m}_{\text{eff},1}^{2}|$ grows as the temperature is lowered.
However, at high temperatures $\widetilde{m}_{\text{eff},1}^{2}$ saturates at
a positive value. This means that the baryon symmetry in the first sector is
restored at a critical temperature. The value of of this critical temperature
($T_{c}$) is given by $t_{c}=\ln(T_{c}/\Lambda)\approx 2.889$.

###  4.2 Case 2: $M_{1}^{2}=0,M_{2}^{2}\neq 0$

In this case the expressions of the rescaled effective masses (squared) are

$\begin{split}&\widetilde{m}_{\text{eff},1}^{2}=(1+\sin\theta)c_{2}\
e^{-(2-8R_{0})t}+R_{0}\Big{(}r\cos\theta+\sin\theta\Big{)}+\frac{3}{4}\lambda,\\\
&\widetilde{m}_{\text{eff},2}^{2}=\frac{1}{r}\cos\theta\ c_{2}\
e^{-(2-8R_{0})t}+R_{0}\Big{(}\frac{\cos\theta}{r}-\sin\theta\Big{)}+\frac{3}{4}\lambda.\end{split}$
(53)

As we have noted earlier, $\cos\theta<0$ for the systems that we are
considering, and $(1+\sin\theta)>0$ for all values of $\theta$ in this domain.
Therefore, at sufficiently low temperatures, when the renormalized masses
(squared) dominate over the thermal masses (squared),
$\widetilde{m}_{\text{eff},1}^{2}$ and $\widetilde{m}_{\text{eff},2}^{2}$ have
opposite signs. These signs depends on whether $c_{2}$ (or equivalently,
$M_{2}^{2}$) is positive or negative.

####  4.2.1 Subcase 1: $M_{2}^{2}>0$

When $M_{2}^{2}>0$, i.e., $c_{2}>0$, then $\widetilde{m}_{\text{eff},1}^{2}>0$
and $\widetilde{m}_{\text{eff},2}^{2}<0$ in the low temperature regime. This
means that in this regime the baryon symmetry in the first sector remains
unbroken and the baryon symmetry in the second sector is spontaneously broken.
This is identical to the phase in the high temperature limit. Therefore, we
expect the system to remain in the same phase at all temperatures in the
domain of validity of the perturbative analysis. To verify this we plot
$\widetilde{m}_{\text{eff},1}^{2}$ and $\widetilde{m}_{\text{eff},2}^{2}$ at
different temperatures for $r=0.1$, $\theta=\pi$, $\lambda=0.001$, $c_{1}=0$
and $c_{2}=0.1$ in figure 11. We can see that
$\widetilde{m}_{\text{eff},1}^{2}>0$ and $\widetilde{m}_{\text{eff},2}^{2}<0$
for all values of $t$. This indicates that the system remains in the same
phase at all temperatures where perturbation theory can be trusted.

Figure 11: Plots of the rescaled effective masses (squared) against
$t=\ln(T/\Lambda)$ for $r=0.1$, $\theta=\pi$, $\lambda=0.001$, $c_{1}=0$,
$c_{2}=0.1$: $\widetilde{m}_{\text{eff},1}^{2}>0$ and
$\widetilde{m}_{\text{eff},2}^{2}<0$ at all temperatures indicating the
absence of any phase transition in the domain of validity of the perturbative
analysis.

####  4.2.2 Subcase 2: $M_{2}^{2}<0$

When $M_{2}^{2}<0$, i.e., $c_{2}<0$, then $\widetilde{m}_{\text{eff},1}^{2}<0$
and $\widetilde{m}_{\text{eff},2}^{2}>0$ in the low temperature regime. This
means that in this regime the baryon symmetry in the first sector is
spontaneously broken and the baryon symmetry in the second sector is unbroken.
On the other hand, in the high temperature limit, as we have already seen, the
baryon symmetry in the first sector is unbroken and the baryon symmetry in the
second sector is spontaneously broken. Therefore, as one goes from the low
temperature regime to the high temperature regime, the baryon symmetry in the
first sector would be restored at some critical temperature $T_{c1}$ and the
baryon symmetry in the second sector would be broken at a possibly different
critical temperature $T_{c2}$. At these two critical temperatures,

$\displaystyle|m_{1}^{2}|_{\mu=T_{c1}}|\sim\frac{16\pi^{2}}{3}\lambda
T_{c1}^{2}\ll T_{c1}^{2},\
|m_{2}^{2}|_{\mu=T_{c2}}|\sim\frac{16\pi^{2}}{3}\lambda T_{c2}^{2}\ll
T_{c2}^{2}.$ (54)

From the relation between the two renormalized masses (squared), we get

$\displaystyle|m_{2}^{2}|_{\mu=T_{c1}}|=\frac{|m_{1}^{2}|_{\mu=T_{c1}}|}{\widetilde{\alpha}},\
|m_{1}^{2}|_{\mu=T_{c2}}|=\widetilde{\alpha}|m_{2}^{2}|_{\mu=T_{c2}}|$ (55)

where $\widetilde{\alpha}\equiv-\frac{r(1+\sin\theta)}{\cos\theta}$. Just like
$\alpha$ in the previous case, $\widetilde{\alpha}<1$ in the domain
$r<r_{\text{max}}$ and $\theta\in(\theta_{1},\theta_{2})$ as can be seen from
the plot given in figure 12. Therefore,

$\displaystyle|m_{1}^{2}|_{\mu=T_{c2}}|\lesssim\frac{16\pi^{2}}{3}\lambda
T_{c2}^{2}\ll T_{c2}^{2}.$ (56)

Figure 12: Plot of $\widetilde{\alpha}=-\frac{r(1+\sin\theta)}{\cos\theta}$
against $r$ and $\theta$ in the domain $r<r_{\text{max}}$ and
$\theta\in(\theta_{1},\theta_{2})$.

Note that $|m_{i}^{2}|_{\mu=T_{c2}}|\ll T_{c2}^{2}$ means that the expressions
of the thermal masses (squared) given in (38) are reliable near the critical
temperature $T_{c2}$.
$|m_{1}^{2}|_{\mu=T_{c2}}|\lesssim\frac{16\pi^{2}}{3}\lambda T_{c2}^{2}$
further implies that near the critical temperature $T_{c2}$,
$m_{\text{eff},1}^{2}$ satisfies the condition

$\displaystyle|m_{\text{eff},1}^{2}|\sim\frac{16\pi^{2}}{3}\lambda T^{2}.$
(57)

This means that we can rely on the perturbative analysis with $\mu=T$ in this
regime.

Near the other critical temperature ($T_{c1}$) the perturbative analysis can
break down if $\widetilde{\alpha}$ is very small. To avoid this, we choose a
system where $r=0.1$ and $\theta=2.1$ for which $\widetilde{\alpha}\approx
0.369$. In this system, we have

$\displaystyle|m_{2}^{2}|_{\mu=T_{c1}}|\approx
2.71|m_{1}^{2}|_{\mu=T_{c1}}|\sim\frac{16\pi^{2}}{3}\lambda T_{c1}^{2}.$ (58)

Therefore, in this case we can rely on the perturbative analysis even near the
critical temperature $T_{c1}$. In figure 13 we provide the plots of the
rescaled effective masses (squared) against $t=\ln(T/\Lambda)$ for $r=0.1$,
$\theta=2.1$, $\lambda=0.001$, $c_{1}=0$ and $c_{2}=-0.1$.

Figure 13: Plots of the rescaled effective masses (squared) against
$t=\ln(T/\Lambda)$ for $r=0.1$, $\theta=2.1$, $\lambda=0.001$, $c_{1}=0$,
$c_{2}=-0.1$: $\widetilde{m}_{\text{eff},1}^{2}<0$ and
$\widetilde{m}_{\text{eff},2}^{2}>0$ in the low temperature regime. In the
high temperature limit, $\widetilde{m}_{\text{eff},1}^{2}$ saturates at a
positive value while $\widetilde{m}_{\text{eff},2}^{2}$ saturates at a
negative value. This indicates that as the temperature is increased, there are
two phase transitions. The first phase transition corresponds to the
restoration of the baryon symmetry in the first sector. At the second phase
transition, the baryon symmetry in the second sector is spontaneously broken.
The two critical temperatures $T_{c1}$ and $T_{c2}$ are given by
$t_{c1}=\ln(T_{c1}/\Lambda)\approx 2.688$ and
$t_{c2}=\ln(T_{c2}/\Lambda)\approx 4.355$.

As we can see from these plots, $\widetilde{m}_{\text{eff},1}^{2}$ and
$\widetilde{m}_{\text{eff},2}^{2}$ both flip their signs as the temperature is
increased. The critical temperatures ($T_{c1}$ and $T_{c2}$) corresponding to
these phase transitions are given by $t_{c1}=\ln(T_{c1}/\Lambda)\approx 2.688$
and $t_{c2}=\ln(T_{c2}/\Lambda)\approx 4.355$.

Since the perturbative analysis with $\mu=T$ is valid near the critical
temperature $T_{c2}$ for different values of $(r,\theta)$ in the domain
$r<r_{\text{max}}$, $\theta\in(\theta_{1},\theta_{2})$, we can provide a
general expression for $T_{c2}$. This critical temperature can be evaluated by
demanding that $\widetilde{m}_{\text{eff},2}^{2}=0$ at $T=T_{c2}$. This gives
us the following expression for $t_{c2}\equiv\ln(T_{c2}/\Lambda)$:

$\begin{split}&t_{c2}=-\frac{1}{2-8R_{0}}\ln\Bigg{[}-\frac{r}{c_{2}\cos\theta}\Bigg{\\{}R_{0}\Big{(}\frac{\cos\theta}{r}-\sin\theta\Big{)}+\frac{3}{4}\lambda\Bigg{\\}}\Bigg{]}.\end{split}$
(59)

In figure 14, we plot $t_{c2}$ against $\theta$ in the domain
$\theta\in(\theta_{1},\theta_{2})$ for $r=0.1$, $\lambda=0.001$, $c_{1}=0$ and
$c_{2}=-0.1$.

Figure 14: Plot of $t_{c2}$ against $\theta$ for $r=0.1,\ \lambda=0.001,\
c_{1}=0,\ c_{2}=-0.1$: $t_{c2}\rightarrow\infty$ as
$\theta\rightarrow\theta_{1}$ or $\theta\rightarrow\theta_{2}$.

From this plot, we can see that $t_{c2}$ increases sharply as one approaches
the edges of the domain $(\theta_{1},\theta_{2})$, and
$t_{c2}\rightarrow\infty$ as $\theta\rightarrow\theta_{1}$ or
$\theta\rightarrow\theta_{2}$. This means that at these edges, the baryon
symmetries in the second sector remains unbroken at all temperatures where the
perturbative analysis is valid.

###  4.3 Summary of the results and comments on the case $M_{1}^{2}\neq 0,\
M_{2}^{2}\neq 0$

In this section we studied the systems that are obtained by introducing masses
to the scalar fields in the RDB model. We restricted our attention to the
systems that flow to the UV fixed points exhibiting thermal order in the
second sector. Due to this behavior of the systems in the UV regime, these
systems remain in a persistently symmetry-broken phase in the high temperature
limit. We found that as the temperature is lowered, the effects of the
renormalized masses of the scalar fields start becoming significant. At
sufficiently low temperatures, these renormalized masses can lead to phase
transitions in the system. We studied such phase transitions for different
values of two mass2 scales $M_{1}^{2}$ and $M_{2}^{2}$ that appear in the
renormalized masses (squared). When both these scales are nonzero, they
determine the asymptotic behavior of the renormalized masses (squared) in the
IR and UV regimes respectively. However, to simplify the analysis, we
considered the following two cases where only one of these scales is nonzero:

  1. 1.

$M_{1}^{2}\neq 0,M_{2}^{2}=0$,

  2. 2.

$M_{1}^{2}=0,M_{2}^{2}\neq 0$.

We showed that in each of these cases, the sign of the nonzero $M_{i}^{2}$
determines the phase transitions occurring in the system. We showed that in
certain parameter regimes, one can study these phase transitions using
perturbation theory with the renormalization scale $\mu=T$. In these regimes,
the perturbation theory remains valid slightly below the critical temperatures
corresponding to the phase transitions. We called the phases at these scales
the ‘low temperature’ phases of the systems. These low temperature phases and
the associated phase transitions for the different cases are summarized in
table 1.

Table 1: Phase transitions for the systems in which the baryon symmetry in the first sector is unbroken and the baryon symmetry in the second sector is broken at the high temperature limit. | Fate of the baryon symmetry | Phase transitions as the
---|---|---
$M_{1}^{2},\ M_{2}^{2}$ | in the low temperature regime | temperature is increased
| First sector | Second sector |
|  |  | Baryon symmetry in the
$M_{1}^{2}>0,\ M_{2}^{2}=0$ | unbroken | unbroken | second sector is broken
|  |  | at a critical temperature.
|  |  | Baryon symmetry in the
$M_{1}^{2}<0,\ M_{2}^{2}=0$ | broken | broken | first sector is restored
|  |  | at a critical temperature.
$M_{1}^{2}=0,\ M_{2}^{2}>0$ | unbroken | broken | No phase transition.
|  |  | 2 phase transitions:
|  |  | At a critical temperature $T_{c1}$
|  |  | baryon symmetry in the
$M_{1}^{2}=0,\ M_{2}^{2}<0$ | broken | unbroken | first sector is restored.
|  |  | At another critical temperature
|  |  |  $T_{c2}$, baryon symmetry in the
|  |  | second sector is broken.

We expect similar phase transitions even outside the aforementioned parameter
regimes. However, to be sure of this, one would need to improve the
perturbative analysis which lies beyond the scope of this paper.

Finally, let us end this section by commenting on the possible phase
structures when both $M_{1}^{2}$ and $M_{2}^{2}$ are nonzero. In general, it
is hard to ascertain the regime of validity of the perturbative analysis near
the critical temperatures in these cases. So, the following discussion will be
mostly speculative. In the presence of both $M_{1}^{2}$ and $M_{2}^{2}$, we
expect the sign of $M_{1}^{2}$ to determine the phase in the low temperature
regime. This is due to the fact that as the temperature is lowered, the terms
with the coefficient $c_{1}\equiv\frac{3M_{1}^{2}}{32\pi^{2}\Lambda^{2}}$ in
the rescaled effective masses (squared) given in (43) grow faster than the
terms with the coefficient
$c_{2}\equiv\frac{3M_{2}^{2}}{32\pi^{2}\Lambda^{2}}$. However, at intermediate
temperatures the phase can depend on $M_{2}^{2}$ as well. The situations in
which $M_{2}^{2}$ can play an important role in determining the phase at such
intermediate temperature scales can be guessed from the low temperature phases
in the presence of a single $M_{i}^{2}$ that are given in table 1. If the
signs of $M_{1}^{2}$ and $M_{2}^{2}$ are such that individually both of them
lead to the same phase in a sector at low temperatures, then the phases in
that sector at different temperature regimes would be identical to those in
the presence of $M_{1}^{2}$ alone. On the other hand, if the signs of
$M_{1}^{2}$ and $M_{2}^{2}$ are such that individually they lead to distinct
phases in a sector at low temperatures, then there are the following two
possibilities. If there is a phase transition in that sector when only
$M_{1}^{2}$ is nonzero, we do not expect anything dramatic to happen in the
presence of $M_{2}^{2}$ apart from a modification of the critical temperature
corresponding to this phase transition. However, if there is no phase
transition in that sector when only $M_{1}^{2}$ is nonzero, then a pair of
phase transitions might be induced by the presence of $M_{2}^{2}$. Whether
such a pair of phase transitions occurs or not would depend on the relative
magnitudes of $M_{1}^{2}$ and $M_{2}^{2}$. In case such phase transitions
indeed occur in a particular sector, the phase in that sector between the two
critical temperatures would be the same as the low temperature phase in the
presence of $M_{2}^{2}$ alone. We do not undertake a detailed study of such
double phase transitions as the perturbation theory with $\mu=T$ may not be
appropriate for analyzing the phases near the two critical temperatures.

To illustrate the possibility of such a pair of phase transitions and the
associated problems with the perturbative analysis, let us consider a system
where $r=0.1$, $\theta=2.1$, $\lambda=0.001$ and $c_{1}=c_{2}=-0.2$. In figure
15 we plot the rescaled effective masses (squared) against $t=\ln(T/\Lambda)$
for this system.

Figure 15: Plots of the rescaled effective masses (squared) against
$t=\ln(T/\Lambda)$ for $r=0.1$, $\theta=2.1$, $\lambda=0.001$,
$c_{1}=c_{2}=-0.2$: A pair of phase transitions occurs in the second sector.

From these plots, we can see that at sufficiently low temperatures, both
$\widetilde{m}_{\text{eff},1}^{2}$ and $\widetilde{m}_{\text{eff},2}^{2}$ are
negative which indicates that the baryon symmetry in both the sectors are
broken in this regime. This behavior at lower temperatures is determined by
the negative sign of $M_{1}^{2}$ (see table 1). As the temperature is
increased, due to the presence of a negative $M_{2}^{2}$,222222See the low
temperature phase for $M_{2}^{2}<0$ in table 1.
$\widetilde{m}_{\text{eff},2}^{2}$ becomes positive at a critical temperature
indicating the restoration of the baryon symmetry in the second sector. When
the temperature is increased further, $\widetilde{m}_{\text{eff},2}^{2}$ again
becomes negative at a critical temperature and remains so at higher
temperatures. This second phase transition is driven by the thermal mass
(squared) $m_{\text{th},2}^{2}$ which begins to dominate over the renormalized
mass (squared) $m_{2}^{2}|_{\mu=T}$ in this regime.232323There would be a
third phase transition as well where the baryon symmetry in the first sector
is restored. At this second critical temperature, the system re-enters into a
phase where the baryon symmetry in the second sector is spontaneously broken.
However, note that $|\widetilde{m}_{\text{eff},1}^{2}|$ becomes much larger
than $\lambda$ at temperatures higher than both the critical temperatures
mentioned above, i.e., $|m_{\text{eff},1}^{2}|$ no longer remains
$O(\frac{16\pi^{2}}{3}\lambda T^{2})$ near these critical temperatures. This
indicates that the perturbative analysis with $\mu=T$ is unreliable in these
regimes. We hope to come up with a more reliable analysis of such double phase
transitions in the future.

##  5 Conclusion and discussion

In this paper we considered relevant and marginally relevant deformations of
the large $N$ CFTs introduced in Chaudhuri:2020xxb  and showed that in the
$N\rightarrow\infty$ limit, the resulting RG flows end at these CFTs in the UV
regime. The gauge groups in these theories have the product form given in (1).
This structure of the gauge group implies the existence of two distinct
sectors in the models. The ranks of the gauge groups in the two sectors can be
unequal. The matter fields in each sector transform in certain representations
of the gauge group in that sector while remaining invariant under the gauge
transformations in the other sector. In each of these sectors, there is a
global $\mathbb{Z}_{2}$ (or $U(1)$) baryon symmetry. It is the fate of these
symmetries at different temperatures that was of interest to us in this work.
We studied the RG flows of these theories to analyze whether the baryon
symmetries in the two sectors are spontaneously broken or not at different
temperature scales. As we mentioned above, the UV fixed points of these RG
flows are the large $N$ CFTs explored in Chaudhuri:2020xxb . These CFTs are
weakly coupled and they lie on a conformal manifold with the shape of a circle
in the space of couplings. It was shown in Chaudhuri:2020xxb  that when the
ratio of the ranks of the gauge groups in the two sectors is sufficiently away
from 1, an angular interval on this circle of fixed points demonstrates
thermal order characterized by the spontaneous breaking of the
$\mathbb{Z}_{2}$ (or $U(1)$) global baryon symmetry in the sector with the
smaller gauge group. It was also shown that this symmetry breaking is
accompanied by the Higgsing of half of the gauge bosons in the same sector at
all nonzero temperatures. From these facts, we concluded that the systems
described by the RG flows ending at these UV fixed points exhibit a symmetry-
broken phase in the high temperature regime. We studied the IR limits of the
RG flows corresponding to the different deformations and found interesting
patterns of phase transitions. Let us summarize the features of these phase
transitions for the different deformations below.

The first class of deformations that we considered involved variations of the
double trace quartic couplings of the scalar fields from their values on the
fixed circle. We showed that for each point on the fixed circle there is a
marginally relevant deformation belonging to this class. The RG flows
triggered by these marginally relevant deformations lead to a weakly coupled
fixed point in the IR where the two sectors are decoupled. In fact, the
theories remain weakly coupled throughout these RG flows. At the IR fixed
point of these flows, the baryon symmetries in both the sectors are unbroken
at any nonzero temperature. On the other hand, as mentioned earlier, the UV
fixed points for some of these flows demonstrate thermal order characterized
by spontaneous breaking of the baryon symmetry in the sector with the smaller
gauge group. We showed that for each of these flows, there is an inverse
symmetry breaking at a critical temperature where the baryon symmetry in the
sector with the smaller gauge group is spontaneously broken. We determined
this critical temperature by finding the temperature scale at which the
thermal mass of the scalar fields in the respective sector goes to zero.

The second class of deformations involved adding masses to the scalar fields
while keeping all the couplings in the model frozen at a fixed point which
exhibits thermal order in the smaller sector. We studied the RG flows of these
masses. This analysis showed that there are two mass2 scales, $M_{1}^{2}$ and
$M_{2}^{2}$, which determine the asymptotic behavior of the renormalized
masses (squared) in the IR and the UV regimes respectively when both of them
are nonzero. We found that in the high temperature limit, the effects of both
these scales are suppressed compared to those of the thermal masses (squared)
of the scalar fields, and the system remains in the same phase as the UV fixed
point. Therefore, the baryon symmetry in the smaller sector remains
persistently broken in this regime. The baryon symmetry in the larger sector,
on the other hand, is unbroken in the same regime. As the temperature is
lowered, the effects of $M_{1}^{2}$ and $M_{2}^{2}$ start becoming
significant. At sufficiently low temperatures, they can induce phase
transitions in the system. The nature of these phase transitions depends on
the magnitudes and signs of $M_{1}^{2}$ and $M_{2}^{2}$. We did a detailed
analysis of these phase transitions for the cases where only one of the
$M_{i}^{2}$’s is nonzero. The results of this analysis for the systems where
the second sector has the smaller gauge group are summarized in table 1. We
also found that when both $M_{1}^{2}$ and $M_{2}^{2}$ are nonzero, in some
cases there may be a pair of phase transitions occurring in a sector at
different critical temperatures. We discussed the limitations of our
perturbative analysis in studying the detailed features of such double phase
transitions and estimating the corresponding critical temperatures.

We now comment on a few subtleties in the analysis done in this work. Firstly,
note that we relied on the 1-loop expressions of the beta functions and the
thermal masses to determine the phases at different temperatures. We expect
the higher loop corrections to refine the estimates of the critical
temperatures while not changing the qualitative features of the phases in the
different regimes. In this context, it is also important to remember that
thermal perturbation theory is riddled with infrared problems Linde:1980ts ;
Gross:1980br .242424We refer the reader to Laine:2016hma  for detailed
discussion on such IR problems. In the models that we considered there are
three mass scales at any temperature $T$ where we could employ the
perturbative analysis. These three mass scales are associated with the
following modes:

  1. 1.

The “hard” nonzero Matsubara modes of the different fields; these modes have
masses of the order of $\pi T$,

  2. 2.

The “soft” zero Matsubara modes of the scalar fields and the longitudinal
gauge bosons252525In a symmetry-broken phase, a subset of the transverse gauge
bosons also get Higgsed. The zero Matubara modes of these Higgsed transverse
gauge bosons also belong to the class of soft modes.; these modes have masses
of the order of $\sqrt{\frac{16\pi^{2}}{3}\lambda}T$,

  3. 3.

The “ultrasoft” zero Matsubara modes of the un-Higgsed transverse gauge
bosons; these modes have masses of the order of $\frac{16\pi^{2}}{3}\lambda
T$.

The different phases in our models are determined by the physics at the soft
scale while the IR problems arise from the ultrasoft modes. They appear in the
form of an infinite number of diagrams at a subleading order in the effective
potential. This leads to breakdown of perturbation theory at this order.
Usually this non-perturbative physics is studied by employing the
dimensionally reduced effective theory framework and the techniques of lattice
gauge theories Laine:2016hma ; Farakos:1994kx ; Kajantie:1995kf ;
Kajantie:1996qd ; Moore:2000jw ; Kainulainen:2019kyp ; Gould:2019qek ;
Niemi:2020hto . Such studies on a variety of models indicate that these non-
perturbative contributions typically do not alter the qualitative features of
the phases in different regimes.262626They may lead to corrections in the
critical temperatures corresponding to the phase transitions. We expect a
similar thing in the models considered in this paper. It may be useful to
check this explicitly in the future.

Another point to which we would like to draw the reader’s attention is that
the UV fixed points of the RG flows studied in this paper are like the Banks-
Zaks-Caswell fixed points in more familiar QFTs. Such fixed points are
typically considered to be IR fixed points as small variations of the gauge
couplings from their values at these fixed points usually lead to asymptotic
freedom. In our analysis, we avoided this by fixing the gauge couplings (as
well as the single trace quartic couplings) at their fixed point values. This
could be done in the massive theories that we considered in section 4 because
the RG flows of the masses do not backreact on the flows of the gauge
couplings (as well as the quartic couplings) in the $\overline{\text{MS}}$
scheme.272727Freezing the gauge couplings and the quartic couplings in the
$\overline{\text{MS}}$ scheme at their fixed point values while allowing the
masses to flow involves a fine-tuning which may take a different form in mass-
dependent renormalization schemes. In case of the massless theories considered
in section 3, the single trace couplings could be kept frozen at their fixed
point values as their beta functions are independent of the double trace
couplings at the planar limit Chaudhuri:2020xxb .

Finally, let us end with some comments on how finite $N$ corrections may alter
the features that we have presented at the planar limit. Currently, it is not
clear to us whether the UV fixed points lying on the conformal manifold
survive under such corrections. We refer the reader to the analysis in
appendix H of Chaudhuri:2020xxb  for some perturbative computations of $1/N$
corrections which suggest that these fixed points may survive under such
corrections. Another potential problem would arise at finite N due to the fact
that the double trace couplings start affecting the RG flows of the single
trace couplings. This can eventually drive these couplings away from their
planar fixed point values at sufficiently high temperatures where the
cumulative effects of $1/N$ corrections become significant. Resolving both the
issues mentioned above is complicated by the fact that to take into account
$1/N$ corrections while staying in a regime where $1/N\ll$ ’t Hooft couplings,
one would need to re-sum the perturbative expansions in the ’t Hooft
couplings. It would be interesting to explore whether the thermal order
persists under $1/N$ corrections after such re-summations. We would like to
return to these issues in the future.

###### Acknowledgements.

We thank Changha Choi for useful discussions. We are also grateful to Changha
Choi, Zohar Komargodski and Michael Smolkin for their comments on an initial
draft of the paper. This work is partially supported by the Israel Science
Foundation Center of Excellence (Grant No. 2289/ 18).

##  Appendix A Positivity of the quartic terms in the thermal effective
potential at leading order

In this appendix we will show that the tree-level quartic terms in the thermal
effective potential of the scalar fields remain positive throughout the RG
flows studied in section 3.2.282828Our analysis will closely follow the
derivation of the same result for the UV CFTs in Chaudhuri:2020xxb . These
quartic terms have the following form:

$\displaystyle
V_{\text{quartic}}=16\pi^{2}\Bigg{[}\sum_{i=1}^{2}\frac{h_{i}}{N_{ci}}\text{Tr}[\rho_{i}^{2}]+\sum_{i=1}^{2}\frac{f_{i}}{N_{ci}^{2}}\Big{(}\text{Tr}[\rho_{i}]\Big{)}^{2}+\frac{2\zeta}{N_{c1}N_{c2}}\text{Tr}[\rho_{1}]\text{Tr}[\rho_{2}]\Bigg{]},$
(60)

where $\rho_{i}\equiv\Phi_{i}^{T}\Phi_{i}$. Note that $\rho_{1}$ and
$\rho_{2}$ are positive-definite matrices. From this one can easily see that a
sufficient condition for the positivity of $V_{\text{quartic}}$ is

$\displaystyle h_{i}>0,\ f_{i}>0,\ f_{1}f_{2}-\zeta^{2}>0.$ (61)

Let us now check whether these inequalities are satisfied throughout the
above-mentioned RG flows. Firstly, let us note that $h_{1}$ and $h_{2}$ are
frozen at the following common value in these RG flows:

$\displaystyle h_{1}=h_{2}=h=\Big{(}\frac{3-\sqrt{6}}{16}\Big{)}\lambda.$ (62)

Since $\lambda$ and the coefficient $\Big{(}\frac{3-\sqrt{6}}{16}\Big{)}$ are
both positive, we can see that $h>0$. Now, let us look at the double trace
couplings $f_{1}=f_{p}+R\sin\theta$ and $f_{2}=f_{p}-R\sin\theta$. The values
of $f_{p}$ and $R$ along these RG flows are as follows.

$\begin{split}&f_{p}=f_{0p}+\frac{R_{0}}{2}\Bigg{[}\frac{(1-k)\Big{(}1-\tanh(8R_{0}t)\Big{)}}{1+k\tanh(8R_{0}t)}\Bigg{]}\
,\
R=\frac{R_{0}}{2}\Bigg{[}\frac{(1+k)\Big{(}1+\tanh(8R_{0}t)\Big{)}}{1+k\tanh(8R_{0}t)}\Bigg{]}\
.\end{split}$ (63)

From these values of the couplings along the RG flow, we get

$\begin{split}&f_{1,2}=f_{p}\pm R\sin\theta\geq
f_{p}-R=f_{0p}-R_{0}+(1-k)R_{0}\Bigg{[}\frac{1-\tanh(8R_{0}t)}{1+k\tanh(8R_{0}t)}\Bigg{]}.\end{split}$
(64)

Here the quantities $k$ and $\tanh(8R_{0}t)$ satisfy the following
inequalities:

$\displaystyle 0<k<1,\ -1<\tanh(8R_{0}t)<1.$ (65)

From this, we can infer that

$\begin{split}&(1-k)R_{0}\Bigg{[}\frac{1-\tanh(8R_{0}t)}{1+k\tanh(8R_{0}t)}\Bigg{]}>0.\end{split}$
(66)

This leads to the positivity of the couplings $f_{1}$ and $f_{2}$ as shown
below:

$\begin{split}&f_{1,2}>f_{0p}-R_{0}=\Bigg{[}\frac{\sqrt{6}}{8}-\frac{\sqrt{18\sqrt{6}-39}}{16}\Bigg{]}\lambda\approx
0.165\lambda>0.\end{split}$ (67)

Note that for $\theta=\frac{\pi}{2}$ the positivity of $f_{1}$ and $f_{2}$
implies

$\begin{split}&f_{p}\pm R>0.\end{split}$ (68)

This also leads to the positivity of the quantity $(f_{1}f_{2}-\zeta^{2})$ as
shown below:

$\begin{split}&f_{1}f_{2}-\zeta^{2}=f_{p}^{2}-R^{2}=(f_{p}+R)(f_{p}-R)>0.\end{split}$
(69)

Therefore, we have shown that all the conditions given in (61) are satisfied
by the couplings throughout the RG flows discussed in section 3.2. Therefore
the corresponding tree-level quartic terms in the thermal effective potential
of the scalar fields are positive-definite.

##  Appendix B RG flows of the masses of the scalar fields

In this appendix we will derive the RG flows of the masses of the scalar
fields that were introduced in section 4. For this we will rely on the
expressions of the 1-loop beta functions of such masses in a general gauge
theory given in Luo:2002ti . To be able to use these expressions, we need to
re-express the quadratic and quartic terms involving the scalar fields in the
Lagrangian of the RDB model as follows:

$\displaystyle\mathcal{L}_{\text{mass}}=-\frac{1}{2}\overline{m}^{2}_{a_{r}i_{r},b_{s}j_{s}}(\Phi_{r})_{a_{r}i_{r}}(\Phi_{s})_{b_{s}j_{s}},$
(70)
$\displaystyle\mathcal{L}_{\text{quartic}}=-\frac{1}{4!}\overline{\lambda}_{a_{r}i_{r},b_{s}j_{s},c_{t}k_{t},d_{u}l_{u}}(\Phi_{r})_{a_{r}i_{r}}(\Phi_{s})_{b_{s}j_{s}}(\Phi_{t})_{c_{t}k_{t}}(\Phi_{u})_{d_{u}l_{u}}$

where $\overline{m}^{2}_{a_{r}i_{r},b_{s}j_{s}}$ and
$\overline{\lambda}_{a_{r}i_{r},b_{s}j_{s},c_{t}k_{t},d_{u}l_{u}}$ are
symmetric under exchange of indices. By comparing the above forms with those
given in (32) and (2), we can see that

$\begin{split}&\overline{m}^{2}_{a_{1}i_{1},b_{1}j_{1}}\equiv
m_{1}^{2}\delta_{a_{1}b_{1}}\delta_{i_{1}j_{1}},\
\overline{m}^{2}_{a_{2}i_{2},b_{2}j_{2}}\equiv
m_{2}^{2}\delta_{a_{2}b_{2}}\delta_{i_{2}j_{2}},\
\overline{m}^{2}_{a_{1}i_{1},b_{2}j_{2}}=\overline{m}^{2}_{a_{2}i_{2},b_{1}j_{1}}\equiv
0,\end{split}$ (71)
$\begin{split}\overline{\lambda}_{a_{1}i_{1},b_{1}j_{1},c_{1}k_{1},d_{1}l_{1}}\equiv&4\widetilde{h}_{1}\Bigg{[}\delta_{i_{1}j_{1}}\delta_{k_{1}l_{1}}(\delta_{a_{1}c_{1}}\delta_{b_{1}d_{1}}+\delta_{a_{1}d_{1}}\delta_{b_{1}c_{1}})+\delta_{i_{1}k_{1}}\delta_{j_{1}l_{1}}(\delta_{a_{1}b_{1}}\delta_{c_{1}d_{1}}+\delta_{a_{1}d_{1}}\delta_{b_{1}c_{1}})\\\
&\quad+\delta_{i_{1}l_{1}}\delta_{j_{1}k_{1}}(\delta_{a_{1}b_{1}}\delta_{c_{1}d_{1}}+\delta_{a_{1}c_{1}}\delta_{b_{1}d_{1}})\Bigg{]}\\\
&+8\widetilde{f}_{1}\Bigg{[}\delta_{i_{1}j_{1}}\delta_{k_{1}l_{1}}\delta_{a_{1}b_{1}}\delta_{c_{1}d_{1}}+\delta_{i_{1}k_{1}}\delta_{j_{1}l_{1}}\delta_{a_{1}c_{1}}\delta_{b_{1}d_{1}}+\delta_{i_{1}l_{1}}\delta_{j_{1}k_{1}}\delta_{a_{1}d_{1}}\delta_{b_{1}c_{1}}\Bigg{]},\end{split}$
(72)
$\begin{split}\overline{\lambda}_{a_{2}i_{2},b_{2}j_{2},c_{2}k_{2},d_{2}l_{2}}\equiv&4\widetilde{h}_{2}\Bigg{[}\delta_{i_{2}j_{2}}\delta_{k_{2}l_{2}}(\delta_{a_{2}c_{2}}\delta_{b_{2}d_{2}}+\delta_{a_{2}d_{2}}\delta_{b_{2}c_{2}})+\delta_{i_{2}k_{2}}\delta_{j_{2}l_{2}}(\delta_{a_{2}b_{2}}\delta_{c_{2}d_{2}}+\delta_{a_{2}d_{2}}\delta_{b_{2}c_{2}})\\\
&\quad+\delta_{i_{2}l_{2}}\delta_{j_{2}k_{2}}(\delta_{a_{2}b_{2}}\delta_{c_{2}d_{2}}+\delta_{a_{2}c_{2}}\delta_{b_{2}d_{2}})\Bigg{]}\\\
&+8\widetilde{f}_{2}\Bigg{[}\delta_{i_{2}j_{2}}\delta_{k_{2}l_{2}}\delta_{a_{2}b_{2}}\delta_{c_{2}d_{2}}+\delta_{i_{2}k_{2}}\delta_{j_{2}l_{2}}\delta_{a_{2}c_{2}}\delta_{b_{2}d_{2}}+\delta_{i_{2}l_{2}}\delta_{j_{2}k_{2}}\delta_{a_{2}d_{2}}\delta_{b_{2}c_{2}}\Bigg{]},\end{split}$
(73)
$\begin{split}&\overline{\lambda}_{a_{1}i_{1},b_{1}j_{1},c_{2}k_{2},d_{2}l_{2}}=\overline{\lambda}_{a_{1}i_{1},c_{2}k_{2},b_{1}j_{1},d_{2}l_{2}}=\overline{\lambda}_{a_{1}i_{1},c_{2}k_{2},d_{2}l_{2},b_{1}j_{1}}\\\
&=\overline{\lambda}_{c_{2}k_{2},a_{1}i_{1},b_{1}j_{1},d_{2}l_{2}}=\overline{\lambda}_{c_{2}k_{2},a_{1}i_{1},d_{2}l_{2},b_{1}j_{1}}=\overline{\lambda}_{c_{2}k_{2},d_{2}l_{2},a_{1}i_{1},b_{1}j_{1}}\equiv
8\widetilde{\zeta}\delta_{a_{1}b_{1}}\delta_{i_{1}j_{1}}\delta_{c_{2}d_{2}}\delta_{k_{2}l_{2}}.\end{split}$
(74)

Now, we can use the expressions given in Luo:2002ti  to get the following
1-loop beta functions of $\overline{m}^{2}_{a_{r}i_{r},b_{s}j_{s}}$:

$\begin{split}&(4\pi)^{2}\mu\frac{d}{d\mu}(\overline{m}^{2}_{a_{1}i_{1},b_{1}j_{1}})=\overline{m}^{2}_{c_{1}k_{1},d_{1}l_{1}}\overline{\lambda}_{a_{1}i_{1},b_{1}j_{1},c_{1}k_{1},d_{1}l_{1}}+\overline{m}^{2}_{c_{2}k_{2},d_{2}l_{2}}\overline{\lambda}_{a_{1}i_{1},b_{1}j_{1},c_{2}k_{2},d_{2}l_{2}}\\\
&\qquad\qquad\qquad\qquad\qquad-3\sum_{r,\gamma=1}^{2}g_{r}^{2}\Lambda^{S,r\gamma}_{a_{1}i_{1},b_{1}j_{1}},\\\
&(4\pi)^{2}\mu\frac{d}{d\mu}(\overline{m}^{2}_{a_{2}i_{2},b_{2}j_{2}})=\overline{m}^{2}_{c_{2}k_{2},d_{2}l_{2}}\overline{\lambda}_{a_{2}i_{2},b_{2}j_{2},c_{2}k_{2},d_{2}l_{2}}+\overline{m}^{2}_{c_{1}k_{1},d_{1}l_{1}}\lambda_{a_{2}i_{2},b_{2}j_{2},c_{1}k_{1},d_{1}l_{1}}\\\
&\qquad\qquad\qquad\qquad\qquad-3\sum_{r,\gamma=1}^{2}g_{r}^{2}\Lambda^{S,r\gamma}_{a_{2}i_{2},b_{2}j_{2}}.\end{split}$
(75)

Here

$\displaystyle\Lambda^{S,r\gamma}_{a_{1}i_{1},b_{1}j_{1}}\equiv
2C_{2}^{r\gamma}(S_{1})\overline{m}^{2}_{a_{1}i_{1},b_{1}j_{1}},\
\Lambda^{S,r\gamma}_{a_{2}i_{2},b_{2}j_{2}}\equiv
2C_{2}^{r\gamma}(S_{2})\overline{m}^{2}_{a_{2}i_{2},b_{2}j_{2}},$ (76)

where $C_{2}^{r\gamma}(S_{i})$ is the quadratic Casimir of the representation
in which the scalar fields in the $i^{th}$ sector transform under the
$\gamma^{\text{th}}$ $G_{r}$ in the $r^{\text{th}}$ sector.292929We remind the
reader that $G_{r}=SO(N_{cr})$ in the RDB model. The values of these quadratic
Casimirs are as follows:

$\begin{split}C_{2}^{r\gamma}(S_{1})=\delta_{r1}\Big{(}\frac{N_{cr}-1}{4}\Big{)},\
C_{2}^{r\gamma}(S_{2})=\delta_{r2}\Big{(}\frac{N_{cr}-1}{4}\Big{)}.\end{split}$
(77)

Substituting these quadratic Casimirs as well as the parameters
$\overline{m}^{2}_{a_{r}i_{r},b_{s}j_{s}}$ and
$\overline{\lambda}_{a_{r}i_{r},b_{s}j_{s},c_{t}k_{t},d_{u}l_{u}}$ by their
values given above, and then switching to the ’t Hooft couplings introduced in
(3), we get

$\begin{split}&\mu\frac{dm_{1}^{2}}{d\mu}=8\Bigg{[}h_{1}\Bigg{(}2+\frac{1}{N_{c1}}\Bigg{)}+f_{1}\Bigg{(}1+\frac{2}{N_{c1}^{2}}\Bigg{)}\Bigg{]}m_{1}^{2}+8\zeta\frac{N_{c2}}{N_{c1}}m_{2}^{2}-3\lambda_{1}\Bigg{(}1-\frac{1}{N_{c1}}\Bigg{)}m_{1}^{2},\\\
&\mu\frac{dm_{2}^{2}}{d\mu}=8\Bigg{[}h_{2}\Bigg{(}2+\frac{1}{N_{c2}}\Bigg{)}+f_{2}\Bigg{(}1+\frac{2}{N_{c2}^{2}}\Bigg{)}\Bigg{]}m_{2}^{2}+8\zeta\frac{N_{c1}}{N_{c2}}m_{1}^{2}-3\lambda_{2}\Bigg{(}1-\frac{1}{N_{c2}}\Bigg{)}m_{2}^{2}.\end{split}$
(78)

At the planar limit ($N_{c1},N_{c2}\rightarrow\infty$), these beta functions
reduce to

$\begin{split}&\mu\frac{dm_{1}^{2}}{d\mu}=\Big{(}16h_{1}+8f_{1}-3\lambda_{1}\Big{)}m_{1}^{2}+8r\zeta
m_{2}^{2},\\\
&\mu\frac{dm_{2}^{2}}{d\mu}=\Big{(}16h_{2}+8f_{2}-3\lambda_{2}\Big{)}m_{2}^{2}+\frac{8\zeta}{r}m_{1}^{2},\end{split}$
(79)

where $r\equiv\frac{N_{c2}}{N_{c1}}$ in the same limit.

## References

  * (1) S. Chaudhuri, C. Choi and E. Rabinovici, _Thermal order in large N conformal gauge theories_ , _JHEP_ 04 (2021) 203 [2011.13981].
  * (2) N. Schupper and N. M. Shnerb, _Inverse melting and inverse freezing: A spin model_ , _Phys. Rev. E_ 72 (2005) 046107.
  * (3) S. Weinberg, _Gauge and Global Symmetries at High Temperature_ , _Phys. Rev._ D9 (1974) 3357.
  * (4) J. Orloff, _The UV price for symmetry nonrestoration_ , _Phys. Lett._ B403 (1997) 309 [hep-ph/9611398].
  * (5) G. Bimonte, D. Iniguez, A. Tarancon and C. L. Ullod, _Inverse symmetry breaking on the lattice: An Accurate MC study_ , _Nucl. Phys._ B559 (1999) 103 [hep-lat/9903027].
  * (6) M. B. Pinto and R. O. Ramos, _A Nonperturbative study of inverse symmetry breaking at high temperatures_ , _Phys. Rev._ D61 (2000) 125016 [hep-ph/9912273].
  * (7) R. N. Mohapatra and G. Senjanovic, _Soft CP Violation at High Temperature_ , _Phys. Rev. Lett._ 42 (1979) 1651.
  * (8) P. Langacker and S.-Y. Pi, _Magnetic Monopoles in Grand Unified Theories_ , _Phys. Rev. Lett._ 45 (1980) 1.
  * (9) S. Dodelson and L. M. Widrow, _BARYON SYMMETRIC BARYOGENESIS_ , _Phys. Rev. Lett._ 64 (1990) 340.
  * (10) S. Dodelson, B. R. Greene and L. M. Widrow, _Baryogenesis, dark matter and the width of the Z_ , _Nucl. Phys._ B372 (1992) 467.
  * (11) G. R. Dvali, A. Melfo and G. Senjanovic, _Is There a monopole problem?_ , _Phys. Rev. Lett._ 75 (1995) 4559 [hep-ph/9507230].
  * (12) P. Meade and H. Ramani, _Unrestored Electroweak Symmetry_ , _Phys. Rev. Lett._ 122 (2019) 041802 [1807.07578].
  * (13) Y. Bai, S. J. Lee, M. Son and F. Ye, _Global Electroweak Symmetric Vacuum_ , 2103.09819.
  * (14) M. Carena, C. Krause, Z. Liu and Y. Wang, _A New Approach to Electroweak Symmetry Non-Restoration_ , 2104.00638.
  * (15) Z. Komargodski, A. Sharon, R. Thorngren and X. Zhou, _Comments on Abelian Higgs Models and Persistent Order_ , _SciPost Phys._ 6 (2019) 003 [1705.04786].
  * (16) K. Aitken, A. Cherman, E. Poppitz and L. G. Yaffe, _QCD on a small circle_ , _Phys. Rev. D_ 96 (2017) 096022 [1707.08971].
  * (17) Y. Tanizaki, T. Misumi and N. Sakai, _Circle compactification and ’t Hooft anomaly_ , _JHEP_ 12 (2017) 056 [1710.08923].
  * (18) G. V. Dunne, Y. Tanizaki and M. Ünsal, _Quantum Distillation of Hilbert Spaces, Semi-classics and Anomaly Matching_ , _JHEP_ 08 (2018) 068 [1803.02430].
  * (19) Z. Wan and J. Wang, _Higher anomalies, higher symmetries, and cobordisms III: QCD matter phases anew_ , _Nucl. Phys. B_ 957 (2020) 115016 [1912.13514].
  * (20) S.-I. Hong and J. B. Kogut, _Symmetry nonrestoration in a Gross-Neveu model with random chemical potential_ , _Phys. Rev. D_ 63 (2001) 085014 [hep-th/0007216].
  * (21) A. Buchel and C. Pagnutti, _Exotic Hairy Black Holes_ , _Nucl. Phys. B_ 824 (2010) 85 [0904.1716].
  * (22) A. Donos and J. P. Gauntlett, _Superfluid black branes in AdS_4\times S^7_ , _JHEP_ 06 (2011) 053 [1104.4478].
  * (23) U. Gürsoy, E. Kiritsis, F. Nitti and L. Silva Pimenta, _Exotic holographic RG flows at finite temperature_ , _JHEP_ 10 (2018) 173 [1805.01769].
  * (24) A. Buchel, _Klebanov-Strassler black hole_ , _JHEP_ 01 (2019) 207 [1809.08484].
  * (25) A. Buchel, _Thermal order in holographic CFTs and no-hair theorem violation in black branes_ , _Nucl. Phys. B_ 967 (2021) 115425 [2005.07833].
  * (26) A. Buchel, _Holographic conformal order in supergravity_ , _Phys. Lett. B_ 814 (2021) 136111 [2007.09420].
  * (27) A. Buchel, _Fate of the conformal order_ , _Phys. Rev. D_ 103 (2021) 026008 [2011.11509].
  * (28) N. Chai, S. Chaudhuri, C. Choi, Z. Komargodski, E. Rabinovici and M. Smolkin, _Thermal Order in Conformal Theories_ , _Phys. Rev. D_ 102 (2020) 065014 [2005.03676].
  * (29) N. Chai, S. Chaudhuri, C. Choi, Z. Komargodski, E. Rabinovici and M. Smolkin, _Symmetry Breaking at All Temperatures_ , _Phys. Rev. Lett._ 125 (2020) 131603.
  * (30) M. Hogervorst, S. Rychkov and B. C. van Rees, _Unitarity violation at the Wilson-Fisher fixed point in 4- $\epsilon$ dimensions_, _Phys. Rev. D_ 93 (2016) 125025 [1512.00013].
  * (31) B. Bajc, A. Lugo and F. Sannino, _Asymptotically free and safe fate of symmetry nonrestoration_ , _Phys. Rev. D_ 103 (2021) 096014 [2012.08428].
  * (32) D. F. Litim and F. Sannino, _Asymptotic safety guaranteed_ , _JHEP_ 12 (2014) 178 [1406.2337].
  * (33) A. Zee, _Study of the renormalization group for small coupling constants_ , _Phys. Rev. D_ 7 (1973) 3630.
  * (34) N. Chai, A. Dymarsky and M. Smolkin, _A model of persistent breaking of discrete symmetry_ , 2106.09723.
  * (35) G. Veneziano, _Some Aspects of a Unified Approach to Gauge, Dual and Gribov Theories_ , _Nucl. Phys. B_ 117 (1976) 519.
  * (36) M. Bershadsky and A. Johansen, _Large N limit of orbifold field theories_ , _Nucl. Phys._ B536 (1998) 141 [hep-th/9803249].
  * (37) M. Schmaltz, _Duality of nonsupersymmetric large N gauge theories_ , _Phys. Rev. D_ 59 (1999) 105018 [hep-th/9805218].
  * (38) A. Dymarsky, I. Klebanov and R. Roiban, _Perturbative gauge theory and closed string tachyons_ , _JHEP_ 11 (2005) 038 [hep-th/0509132].
  * (39) A. Dymarsky, I. Klebanov and R. Roiban, _Perturbative search for fixed lines in large N gauge theories_ , _JHEP_ 08 (2005) 011 [hep-th/0505099].
  * (40) E. Pomoni and L. Rastelli, _Large N Field Theory and AdS Tachyons_ , _JHEP_ 04 (2009) 020 [0805.2261].
  * (41) A. Cherman, M. Hanada and D. Robles-Llana, _Orbifold equivalence and the sign problem at finite baryon density_ , _Phys. Rev. Lett._ 106 (2011) 091603 [1009.1623].
  * (42) M. Hanada and N. Yamamoto, _Universality of Phases in QCD and QCD-like Theories_ , _JHEP_ 02 (2012) 138 [1103.5480].
  * (43) G. V. Dunne and M. Ünsal, _New Nonperturbative Methods in Quantum Field Theory: From Large-N Orbifold Equivalence to Bions and Resurgence_ , _Ann. Rev. Nucl. Part. Sci._ 66 (2016) 245 [1601.03414].
  * (44) K. Aitken, C. Choi and A. Karch, _New and Old Fermionic Dualities from 3d Bosonization_ , _JHEP_ 01 (2020) 035 [1909.04036].
  * (45) C. B. Jepsen, I. R. Klebanov and F. K. Popov, _RG limit cycles and unconventional fixed points in perturbative QFT_ , _Phys. Rev. D_ 103 (2021) 046015 [2010.15133].
  * (46) A. Belavin and A. Migdal, _Calculation of anomalous dimensions in non-abelian gauge field theories_ , _Pisma Zh. Eksp. Teor. Fiz._ 19 (1974) 317.
  * (47) W. E. Caswell, _Asymptotic behavior of non-abelian gauge theories to two-loop order_ , _Phys. Rev. Lett._ 33 (1974) 244.
  * (48) T. Banks and A. Zaks, _On the Phase Structure of Vector-Like Gauge Theories with Massless Fermions_ , _Nucl. Phys._ B196 (1982) 189.
  * (49) S. R. Coleman and E. J. Weinberg, _Radiative Corrections as the Origin of Spontaneous Symmetry Breaking_ , _Phys. Rev. D_ 7 (1973) 1888.
  * (50) S. Y. Lee and A. M. Sciaccaluga, _Evaluation of Higher Order Effective Potentials with Dimensional Regularization_ , _Nucl. Phys. B_ 96 (1975) 435.
  * (51) O. Gould and T. V. I. Tenkanen, _On the perturbative expansion at high temperature and implications for cosmological phase transitions_ , _JHEP_ 06 (2021) 069 [2104.04399].
  * (52) D. Croon, O. Gould, P. Schicho, T. V. I. Tenkanen and G. White, _Theoretical uncertainties for cosmological first-order phase transitions_ , _JHEP_ 04 (2021) 055 [2009.10080].
  * (53) A. D. Linde, _Infrared Problem in Thermodynamics of the Yang-Mills Gas_ , _Phys. Lett. B_ 96 (1980) 289.
  * (54) D. J. Gross, R. D. Pisarski and L. G. Yaffe, _QCD and Instantons at Finite Temperature_ , _Rev. Mod. Phys._ 53 (1981) 43.
  * (55) M. Laine and A. Vuorinen, _Basics of Thermal Field Theory_ , vol. 925. Springer, 2016, 10.1007/978-3-319-31933-9, [1701.01554].
  * (56) K. Farakos, K. Kajantie, K. Rummukainen and M. E. Shaposhnikov, _3-D physics and the electroweak phase transition: Perturbation theory_ , _Nucl. Phys. B_ 425 (1994) 67 [hep-ph/9404201].
  * (57) K. Kajantie, M. Laine, K. Rummukainen and M. E. Shaposhnikov, _The Electroweak phase transition: A Nonperturbative analysis_ , _Nucl. Phys. B_ 466 (1996) 189 [hep-lat/9510020].
  * (58) K. Kajantie, M. Laine, K. Rummukainen and M. E. Shaposhnikov, _A Nonperturbative analysis of the finite T phase transition in SU(2) x U(1) electroweak theory_ , _Nucl. Phys. B_ 493 (1997) 413 [hep-lat/9612006].
  * (59) G. D. Moore and K. Rummukainen, _Electroweak bubble nucleation, nonperturbatively_ , _Phys. Rev. D_ 63 (2001) 045002 [hep-ph/0009132].
  * (60) K. Kainulainen, V. Keus, L. Niemi, K. Rummukainen, T. V. I. Tenkanen and V. Vaskonen, _On the validity of perturbative studies of the electroweak phase transition in the Two Higgs Doublet model_ , _JHEP_ 06 (2019) 075 [1904.01329].
  * (61) O. Gould, J. Kozaczuk, L. Niemi, M. J. Ramsey-Musolf, T. V. I. Tenkanen and D. J. Weir, _Nonperturbative analysis of the gravitational waves from a first-order electroweak phase transition_ , _Phys. Rev. D_ 100 (2019) 115024 [1903.11604].
  * (62) L. Niemi, M. J. Ramsey-Musolf, T. V. I. Tenkanen and D. J. Weir, _Thermodynamics of a two-step electroweak phase transition_ , _Phys. Rev. Lett._ 126 (2021) 171802 [2005.11332].
  * (63) M.-x. Luo, H.-w. Wang and Y. Xiao, _Two loop renormalization group equations in general gauge field theories_ , _Phys. Rev. D_ 67 (2003) 065019 [hep-ph/0211440].