The free and safe fate of symmetry non-restoration

Borut<EMAIL_ADDRESS>Adrián<EMAIL_ADDRESS>and
Francesco<EMAIL_ADDRESS>
a J. Stefan Institute, 1000 Ljubljana, Slovenia

b Instituto de Física de La Plata-CONICET, and Departamento de Física,

Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Argentina

c CP3-Origins & the Danish IAS, University of Southern Denmark, Denmark

d Dipartimento di Fisica, E. Pancini, Univ. di Napoli, Federico II and INFN
sezione di Napoli.

Complesso Universitario di Monte S. Angelo Edificio 6, via Cintia, 80126
Napoli, Italy

###### Abstract

We investigate the high temperature fate of four dimensional gauge-Yukawa
theories featuring short distance conformality of either interacting or non-
interacting nature. The latter is known as complete asymptotic freedom and, as
templates, we consider non-abelian gauge theories featuring either two singlet
scalars coupled to gauged fermions via Yukawa interactions or two gauged
scalars with(out) fermions. For theories with interacting fixed points at
short distance, known as asymptotically safe, we consider two calculable
examples. Exploring the landscape of safe and free theories above we discover
a class of complete asymptotically free theories for which symmetry breaks at
arbitrary high temperatures. In its minimal form this class is constituted by
a theory with two fundamental gauged scalars each gauged under an independent
group.

###### Contents

  1. 1 Introduction
  2. 2 Complete asymptotically free theories at high temperature
    1. 2.1 Symmetry restoration with singlet scalars
      1. 2.1.1  SU($N_{c}$) with two singlet scalars and fundamental fermions
      2. 2.1.2 More general result for singlet scalars
    2. 2.2 Exploring symmetry non-restoration with gauged scalars
      1. 2.2.1 $SU(N_{c})$ with $N_{s}$ fundamental scalars
      2. 2.2.2 $SU(N_{c})$ with two fundamental scalars
      3. 2.2.3 $SU(N_{c_{1}})\times SU(N_{c_{2}})$ with fundamental scalars:  Symmetry breaks at high temperatures
      4. 2.2.4 The IR story of $SU(N_{c_{1}})\times SU(N_{c_{2}})$ at nonzero temperature
      5. 2.2.5 Another example of symmetry breaking at high $T$: two adjoints in $SU(N_{c1})\times SU(N_{c2})$
  3. 3 Asymptotic safety at high temperature
    1. 3.1 Explicit examples of asymptotic safety
      1. 3.1.1 The Litim-Sannino (LS) model
    2. 3.2 A gauged scalar variant of the LS model
  4. 4 Conclusions
  5. A  The 1-loop RG equations
  6. B SU($N_{c}$) with two singlet scalars and fundamental fermions
  7. C Gauged scalars
    1. C.1  $SU(2)$ with two scalar triplets
    2. C.2  $SU(2)$ with one scalar singlet and one scalar triplet
    3. C.3 $SU(N_{c})$ with two scalar fundamentals
    4. C.4  $SU(N_{c_{1}})\times SU(N_{c_{2}})$ with two scalar fundamentals
    5. C.5  $SU(N_{c_{1}})\times SU(N_{c_{2}})$ with two scalar adjoints

##  1 Introduction

The phenomenon of symmetry non-restoration (for a review see for example [1,
2]) has been first noticed by Weinberg [3] and then studied in detail by
Mohapatra and Senjanović [4, 5] who were the first ones to successfully apply
the mechanism to phenomenology. Since then it has been employed in cosmology
to address various issues like the monopole [6, 7, 8], the domain wall [9] and
false vacuum problems. The phenomenon has also been invoked for other
phenomena including baryogenesis [10, 11, 12, 13, 14, 15] and inflation [16].

Symmetry non-restoration at high energy can occur also due to the concomitance
of other mechanisms such as the presence of large charges that can induce
either Bose-Einstein condensation or superconductivity. This mechanism has
been used in the literature [17, 18, 19, 20, 21, 22, 23, 24]. For example a
large charge can still be realistically related to the yet to be
experimentally determined neutrino lepton number [25, 26, 27, 28, 29, 30].

Symmetry non-restoration at high temperature cannot occur in supersymmetry
[31, 32, 33] unless we have flat directions [34, 35] and/or nonzero fixed
charge.

For non supersymmetric quantum field theories symmetry non-restoration has
been tested via different methods in [36, 37, 38, 39, 40, 41] for global
symmetries and non-restoration for local symmetries have been investigated in
[42]. The results seem to support the existence of symmetry non-restoration
although these claims have been challenged in [43, 44, 45, 46, 47].

Analyses including generalisation to different space-time dimensions including
$\epsilon$ dimensions away from four are summarised in Ref. [48, 49, 50, 51].
More precisely: symmetry non-restoration at high temperature is possible also
in lower [48, 50, 51] and non-integer dimensions [49].

A common feature of all the theories studied so far for symmetry non-
restoration at high temperature is that these can be viewed as effective
theories without a well defined ultraviolet completion. This fact implies that
the arbitrary large temperature limit cannot be taken.

In this work we go one step beyond with respect to what has been done so far
by analysing Weinberg’s symmetry non-restoration hypothesis within models that
are well defined at short distance. These are, according to Wilson [52, 53]
and Weinberg [54] classification of well defined theories, of either
asymptotically free or safe nature. Within these theories it is consistent to
consider the infinite temperature limit. It is worth recalling that for these
theories short scale conformality guarantees the existence of a well defined
theory at high energy making them UV complete. Asymptotic safety for gauge-
Yukawa theories was discovered in [55] with the corrections to the quantum
potential presented in [56]. Interestingly, once asymptotic freedom is lost in
the gauge-fermion sector, within perturbation theory, the fundamentality of
the theory can only be reinstated via Yukawa interactions. This implies that
elementary scalars are needed, for the first time, to tame the high energy
behaviour of the theory. The discovery of asymptotic safe quantum field
theories [55] has led to an ongoing number of theoretical [57, 58, 61, 59, 60]
and phenomenological investigations [62, 63, 65, 66, 67, 68, 64, 69],
including the recent discovery of safe non-supersymmetric grand unified
theories of [70] which naturally integrates and complements the supersymmetric
story of [58].

For the issue of symmetry non-restoration at arbitrary high temperatures we
consider, at first, the landscape of complete asymptotically free non-abelian
gauge theories that feature either two singlet scalars coupled to gauged
fermions via Yukawa interactions or two gauged scalars without Yukawa
interaction.

The first model we encounter of complete asymptotically free theories for
which symmetry breaks at arbitrary high temperatures is constituted by two
gauged scalars transforming according to the fundamental representation of two
distinct gauge groups with fermions also transforming in the fundamental
representation and without Yukawa interactions.

To investigate the high-temperature fate of global symmetries for
asymptotically safe theories we consider the Litim-Sannino model of [55] and
one of its variations that has been used for perturbative safe extensions of
the standard model [71]. We show that for these examples the safe quantum
global symmetries are restored at high temperatures.

##  2 Complete asymptotically free theories at high temperature

Before embarking in our main quest, which is to investigate the symmetry
(non)restoration phenomenon for complete asymptotically free quantum field
theories we briefly summarise Weinberg’s (non-free) model mechanics. In its
most minimal form the model features two scalars with the following quartic
potential

$V=\frac{\lambda_{1}}{4}\phi_{1}^{4}+\frac{\lambda_{2}}{4}\phi_{2}^{4}-\frac{\lambda}{2}\phi_{1}^{2}\phi_{2}^{2}$
(2.1)

with discrete $Z_{2}\times Z_{2}$ symmetry $\phi_{1}\to-\phi_{1}$ and
$\phi_{2}\to-\phi_{2}$. For

$\lambda_{1,2}>0\;\;\;,\;\;\;\lambda^{2}<\lambda_{1}\lambda_{2}$ (2.2)

the model is bounded from below. At high temperature the following correction
arises: [3]:

$\Delta
V_{T}=\frac{T^{2}}{24}\left((3\lambda_{1}-\lambda)\phi_{1}^{2}+(3\lambda_{2}-\lambda)\phi_{2}^{2}\right)\
.$ (2.3)

With $\lambda>3\lambda_{2}$ (but with $\lambda_{1}$ satisfying (2.2)) the
field $\phi_{2}$ acquires a negative thermal mass squared at high temperature
which yields a non-zero vev $\langle\phi_{2}\rangle\neq 0$. Therefore in this
case the second $Z_{2}$ breaks at sufficiently high temperatures. This theory
is, however, not UV complete since the scalar couplings increase with the
energy. Assuming a physical cutoff, for temperatures below this cutoff one
therefore observes the phenomenon of symmetry non-restoration.

Because the theory is limited by a physical cutoff we cannot ask the relevant
question of whether the symmetry remains broken at arbitrary high
temperatures. This is exactly what our work wishes to achieve, i.e. what is
the ultimate fate of the symmetry in a truly UV complete theory (up to
gravity) at arbitrary large temperatures.

Here we analyse complete asymptotically free theories that are natural UV
completions of the Weinberg’s model. These require the presence of gauge
fields and the gauge sector to be asymptotically free given that it is this
sector the one responsible to drive the Yukawa and scalar couplings to be
asymptotically free as well.

We divide our theories in whether they feature gauge singlets or gauged
scalars.

###  2.1 Symmetry restoration with singlet scalars

To start with we consider an $SU(N_{c})$ gauge group with
$N_{f}=N_{f_{1}}+N_{f_{2}}$ Dirac fermions in the fundamental representation
coupled to the scalars $\phi_{1,2}$ via the following $Z_{2}\times Z_{2}$
symmetry ($\phi_{k}\to-\phi_{k}$, $\psi_{k}\to i\gamma_{5}\psi_{k}$)
preserving Yukawa terms:

${\cal
L}_{Y}=\phi_{1}\sum_{i=1}^{N_{f_{1}}}y_{1i}\overline{\psi}_{1i}\psi_{1i}+\phi_{2}\sum_{i=1}^{N_{f_{2}}}y_{2i}\overline{\psi}_{2i}\psi_{2i}\
.$ (2.4)

Because we are searching for asymptotically free solutions we must have that
$\alpha_{g}\propto 1/t$ for large $t=\log{(\mu/\mu_{0})}$ with $\mu$ the
renormalisation scale and $\mu_{0}$ a reference scale. Complete asymptotic
freedom requires that all couplings must vanish at infinity at least as fast
as $\alpha_{g}$ and therefore their scaling must be proportional to $1/t^{a}$
with $a\geq 1$. Additionally the requirement of a negative thermal mass given
in (A.19), necessary for symmetry breaking, implies that at least some scalar
quartic couplings cannot decrease faster than the gauge coupling, i.e. they
must approach zero as $1/t^{b}$ with $b\leq 1$. Therefore, for the purpose of
our work, it is sufficient to investigate the fixed flow solution according to
which all couplings vanish at infinity as $1/t$ [72]. This observation greatly
simplifies the following analyses by transforming a set of non-linear and
coupled first order ordinary differential equations into a system of non-
linear and coupled polynomial equations. In practice, by defining ($g$ is the
gauge coupling)

$\alpha_{g}=\frac{g^{2}}{(4\pi)^{2}}\;\;\;,\;\;\;\alpha_{y_{i}}=\frac{y_{i}^{2}}{(4\pi)^{2}}\;\;\;,\;\;\;\alpha_{\lambda_{i}}=\frac{\lambda_{i}}{(4\pi)^{2}}\;\;\;,\;\;\;\alpha_{\lambda}=\frac{\lambda}{(4\pi)^{2}}$
(2.5)

($i=1,2$) we will search for solutions of the asymptotic form

$\alpha_{a}=\frac{\tilde{\alpha}_{a}}{t}\;\;\;,\;\;\;a=g,y_{1},y_{2},\lambda_{1},\lambda_{2},\lambda,\ldots$
(2.6)

with constant $\tilde{\alpha}_{a}$.

We are now ready to investigate the first relevant examples with singlets
scalars and then we will generalise the results to a wider class of theories.

####  2.1.1  SU($N_{c}$) with two singlet scalars and fundamental fermions

In this model, described in detail in Appendix B, we consider two singlet
scalars coupled through Yukawa interactions to $N_{f_{1}}$ ($N_{f_{2}}$) Dirac
fermions in the fundamental representation of $SU(N_{c})$. We further allow
for $N_{f_{0}}$ Dirac fermions in the fundamental representation of the gauge
group that are inert with respect to the scalars, i.e. do not possess Yukawa
couplings.

We now provide an elegant proof that at high temperature this theory, if
complete asymptotically free, cannot break any symmetry. Let us start with the
thermal masses for the scalars that at one loop read (B.18):

$m_{i}^{2}(T)=(4\pi)^{2}\frac{T^{2}}{12\log{T}}\left(3\tilde{\alpha}_{\lambda_{i}}-\tilde{\alpha}_{\lambda}+2N_{c}N_{f_{i}}\tilde{\alpha}_{y_{i}}\right)\
,$ (2.7)

written in terms of (2.6) couplings. It is sufficient to consider one of the
two scalar masses to be negative. Here we choose to be $m^{2}_{1}$ that
requires

$\tilde{\alpha}_{\lambda}-2N_{c}N_{f_{1}}\tilde{\alpha}_{y_{1}}>3\tilde{\alpha}_{\lambda_{1}}>0\
.$ (2.8)

Under the assumption that there is a complete asymptotically free solution we
have (B.14)

$-\tilde{\alpha}_{\lambda_{1}}=18\tilde{\alpha}_{\lambda_{1}}^{2}+2\tilde{\alpha}_{\lambda}^{2}-8N_{c}N_{f_{1}}\tilde{\alpha}_{y_{1}}^{2}+8N_{c}N_{f_{1}}\tilde{\alpha}_{y_{1}}\tilde{\alpha}_{\lambda_{1}}\
,$ (2.9)

for the relevant scalar coupling as function of the other couplings. The
general form of the RGE equations can be found in the Appendix B.

Rewriting (2.9) as

$2\left(\tilde{\alpha}_{\lambda}^{2}-4N_{c}N_{f_{1}}\tilde{\alpha}_{y_{1}}^{2}\right)+\tilde{\alpha}_{\lambda_{1}}+18\tilde{\alpha}_{\lambda_{1}}^{2}+8N_{c}N_{f_{1}}\tilde{\alpha}_{y_{1}}\tilde{\alpha}_{\lambda_{1}}=0\
,$ (2.10)

we notice that every term except the first one is positive. This means that to
satisfy this equation, the first term must be negative for the fixed flow
solution to be possible. However, since the first term can be rewritten as

$\tilde{\alpha}_{\lambda}^{2}-4N_{c}N_{f_{1}}\tilde{\alpha}_{y_{1}}^{2}=\left(\tilde{\alpha}_{\lambda}-2N_{c}N_{f_{1}}\tilde{\alpha}_{y_{1}}\right)\left(\tilde{\alpha}_{\lambda}+2N_{c}N_{f_{1}}\tilde{\alpha}_{y_{1}}\right)+4N_{c}N_{f_{1}}\left(N_{c}N_{f_{1}}-1\right)\tilde{\alpha}_{y_{1}}^{2}\
,$ (2.11)

the simultaneous requirement of the presence of a negative mass squared term
implies that also the first term is positive due to (2.8).

We have therefore shown that (2.10) cannot have a solution and that the
symmetry must be restored for this model at high temperature once complete
asymptotic freedom is enforced.

####  2.1.2 More general result for singlet scalars

Let us consider the more general scalar potential

$V=\frac{\lambda}{4}\left(\phi^{T}\phi\right)^{2}-\frac{1}{2}\left(\phi^{T}\phi\right)\eta_{ij}\chi^{i}\chi^{j}+V(\chi)$
(2.12)

where $\phi$ is a real vector with $d_{\phi}$ components. The global symmetry
at the potential level over $\phi$ is $O(d_{\phi})$. Under this group $\phi$
transforms with a $d_{\phi}\times d_{\phi}$ orthogonal matrix $O$ of
$O(d_{\phi})$ as:

$\phi^{\prime}=O\phi\ .$ (2.13)

We further consider an arbitrary gauge group with Weyl fermions transforming
according to an arbitrary gauge representation compatible with asymptotic
freedom [73]. The Yukawa terms written directly in terms of the Weyl fermions
read:

${\cal L}_{Yukawa}=\frac{1}{2}\phi^{a}\psi_{i}Y^{a}_{ij}\psi_{j}+h.c.+{\cal
L}_{Yukawa}(\chi,\psi^{\prime})\ .$ (2.14)

Under the assumption that

$\psi^{\prime}=U\psi\quad,\quad\widetilde{O}^{a}_{b}U_{ki}Y^{b}_{kl}U_{lj}=Y^{a}_{ij}$
(2.15)

with $\widetilde{O}$ a rotation matrix that is part of a subgroup of
$O(d_{\phi})$ and $U$ a unitary transformation with $i,j,k=1,\cdots,N_{f}$
with $N_{f}$ the number of Weyl matter fields, the Yukawa terms preserves the
resulting subgroup of $O(d_{\phi})$. The information on which fermions couple
to $\phi$ is clearly hidden in the Yukawa matrix. The last unspecified Yukawa
terms in (2.14) contain interactions of the $\chi$ scalar fields with the Weyl
fermions $\psi^{\prime}$ that are not coupled to $\phi$. We now show that the
thermal mass of $\phi$ cannot be negative at high temperatures when the theory
is required to be asymptotically free also in all couplings.

Let us consider the thermal mass

$m_{\phi}^{2}(T)=(4\pi)^{2}\frac{T^{2}}{12\log{T}}\left((d_{\phi}+2)\tilde{\lambda}-\tilde{\eta}_{kk}+Tr\left(\tilde{Y}_{1}^{\dagger}\tilde{Y}_{1}\right)\right)\
,$ (2.16)

where we used

$Tr\left(Y^{a\dagger}Y^{b}\right)=\delta^{ab}Tr\left(Y_{1}^{\dagger}Y_{1}\right)\
.$ (2.17)

and defined as usual

$\lambda=(4\pi)^{2}\frac{\tilde{\lambda}}{t}\quad,\quad\eta_{ij}=(4\pi)^{2}\frac{\tilde{\eta}_{ij}}{t}\quad,\quad
Y^{a}=4\pi\frac{\tilde{Y}^{a}}{t^{1/2}}$ (2.18)

with $\tilde{\lambda}$, $\tilde{\eta}_{ij}$, $\tilde{Y}^{a}$ constants.

In order not to restore the symmetry carried by the potential term and the
Yukawa relative to $\phi$ the thermal mass (2.16) must be negative. This
implies:

$\tilde{\eta}_{kk}-Tr\left(\tilde{Y}_{1}^{\dagger}\tilde{Y}_{1}\right)>\left(d_{\phi}+2\right)\tilde{\lambda}>0\
.$ (2.19)

Let us now compute the RGE for $\tilde{\lambda}$ relative to achieving the
fixed flow solution:

$2\left(\tilde{\eta}_{ij}\tilde{\eta}_{ij}-2Tr\left(\tilde{Y}_{1}^{\dagger}\tilde{Y}_{1}\tilde{Y}_{1}^{\dagger}\tilde{Y}_{1}\right)\right)+2(d_{\phi}+8)\tilde{\lambda}^{2}+\tilde{\lambda}+4\tilde{\lambda}Tr\left(\tilde{Y}_{1}^{\dagger}\tilde{Y}_{1}\right)=0\
,$ (2.20)

where we used

$Tr\left(Y^{a\dagger}Y^{b}Y^{c\dagger}Y^{d}\right)=A\delta_{ab}\delta_{cd}+B\delta_{ac}\delta_{bd}+C\delta_{ad}\delta_{bc}\
,$ (2.21)

which follows from the symmetry properties of the Yukawa matrices (2.15).

To obtain a solution, the first term must be negative (all the others are
positive). However, we have

$\displaystyle\tilde{\eta}_{ij}\tilde{\eta}_{ij}-2Tr\left(\tilde{Y}_{1}^{\dagger}\tilde{Y}_{1}\tilde{Y}_{1}^{\dagger}\tilde{Y}_{1}\right)$
$\displaystyle=$
$\displaystyle\left(\tilde{\eta}_{kk}-Tr\left(\tilde{Y}_{1}^{\dagger}\tilde{Y}_{1}\right)\right)\left(\tilde{\eta}_{kk}+Tr\left(\tilde{Y}_{1}^{\dagger}\tilde{Y}_{1}\right)\right)$
$\displaystyle+$ $\displaystyle
2\sum_{i<j}\tilde{\eta}_{ij}\tilde{\eta}_{ij}+\left(\left(Tr\left(\tilde{Y}_{1}^{\dagger}\tilde{Y}_{1}\right)\right)^{2}-2Tr\left(\tilde{Y}_{1}^{\dagger}\tilde{Y}_{1}\right)^{2}\right)\
.$

The first term on the right-hand-side is positive due to the assumption of the
occurrence of a negative thermal mass squared (2.19), therefore the only
possible negative term could be the last one. Since the above traces are
invariant under unitary rotations of the Hermitian matrix
$\tilde{Y}_{1}^{\dagger}\tilde{Y}_{1}$, we are free to consider the basis with
diagonal

$\left(\tilde{Y}_{1}^{\dagger}\tilde{Y}_{1}\right)_{ij}=\tilde{y}_{1i}^{2}\delta_{ij}$
(2.23)

so that (2.1.2) becomes

$\displaystyle\left(Tr\left(\tilde{Y}_{1}^{\dagger}\tilde{Y}_{1}\right)\right)^{2}-2Tr\left(\tilde{Y}_{1}^{\dagger}\tilde{Y}_{1}\right)^{2}$
$\displaystyle=$
$\displaystyle\sum_{\mu}dim\left(R_{\mu}^{1}\right)\left(dim\left(R_{\mu}^{1}\right)-2\right)\tilde{y}_{1\mu}^{4}$
(2.24) $\displaystyle+$ $\displaystyle
2\sum_{\mu<\mu^{\prime}}dim\left(R_{\mu}^{1}\right)dim\left(R_{\mu^{\prime}}^{1}\right)\tilde{y}_{1\mu}^{2}\tilde{y}_{1\mu^{\prime}}^{2}$

with $\mu$ and $\mu^{\prime}$ running over the fermion representations. For
non gauge singlet fermions we have $dim\left(R_{\mu^{\prime}}\right)\geq 2$
and therefore the right hand side is positive. For gauge singlet fermions the
only solution compatible with a UV well defined theory is the one for which
the Yukawa coupling vanishes identically and therefore the previous equation
does not apply.

Therefore there is no solution to the RGE for $\tilde{\lambda}$. Or, in other
words, if a fixed flow solution exists, it cannot have a negative thermal
mass. The previous example with a $Z_{2}$ symmetry is included here by
assuming the original symmetry to be simply a $Z_{2}$ for $d_{\phi}=1$.

###  2.2 Exploring symmetry non-restoration with gauged scalars

So far we have shown that a great deal of gauge theories with scalar gauge
singlets do not support symmetry non-restoration at arbitrary high
temperatures. Does this phenomenon persists when considering gauged scalar
fields? This is the question we will answer in this section. We will find an
example with the opposite behaviour, i.e. we will explicitly present a theory
featuring two different gauge groups displaying simultaneously complete
asymptotic freedom and symmetry non-restoration.

To motivate the introduction of a second gauge group we will first show that
with a single gauge group symmetries will restore at arbitrary high
temperatures with(out) fermionic matter fields.

Although the models in this section share some features with the ones
investigated in [74] the main difference resides in the fact that we are
interested in symmetry non-restoration at arbitrary high temperatures. This
means that we investigate theories near their UV fixed point, while in [74]
the authors concentrate on symmetry non-restoration occurring near interacting
IR-fixed points.

####  2.2.1 $SU(N_{c})$ with $N_{s}$ fundamental scalars

The $SU(N_{c})\times SU(N_{f})\times SU(N_{s})$ symmetric Lagrangian is444If
$N_{c}=N_{s}=4$ one can add to the potential a new invariant
$w\,\mbox{det}\,{S}+w^{*}\,\mbox{det}\,{S^{\dagger}}$.

${\cal
L}=-\frac{1}{2}TrF_{\mu\nu}F^{\mu\nu}+Tr\left(\bar{Q}i\not{D}Q\right)+Tr\left(D^{\mu}SD_{\mu}S^{\dagger}\right)-v\left(TrSS^{\dagger}\right)^{2}-uTr\left(SS^{\dagger}\right)^{2}$
(2.25)

with the fields transforming as

$Q\sim\left(N_{c},N_{f},1\right)\quad,\quad S\sim\left(N_{c},1,N_{s}\right)$
(2.26)

The scalar thermal mass at high temperature is

$m_{S}^{2}(T)=(4\pi)^{2}\frac{T^{2}}{24\log{T}}\left(4(N_{s}N_{c}+1)\tilde{\lambda}_{1}+4(N_{s}+N_{c})\tilde{\lambda}_{2}+3\frac{N_{c}^{2}-1}{N_{c}}\tilde{\alpha}\right)$
(2.27)

where we introduced, following [75],

$v=(4\pi)^{2}\frac{\tilde{\lambda}_{1}}{t}\quad,\quad
u=(4\pi)^{2}\frac{\tilde{\lambda}_{2}}{t}\quad,\quad
g^{2}=(4\pi)^{2}\frac{\tilde{\alpha}}{t}$ (2.28)

with constant $\tilde{\lambda}_{1,2}$, $\tilde{\alpha}$.

The positivity of (2.27) follows from boundedness arguments. In fact The $T=0$
potential is bounded from below iff [75]

$\displaystyle\tilde{\lambda}_{2}\geq 0$ $\displaystyle:$ $\displaystyle
N_{s}\tilde{\lambda}_{1}+\tilde{\lambda}_{2}\geq 0$ (2.29)
$\displaystyle\tilde{\lambda}_{2}\leq 0$ $\displaystyle:$
$\displaystyle\tilde{\lambda}_{1}+\tilde{\lambda}_{2}\leq 0$ (2.30)

Since

  1. 1.

$\tilde{\lambda}_{2}\geq 0$:

$(N_{s}N_{c}+1)\tilde{\lambda}_{1}+(N_{s}+N_{c})\tilde{\lambda}_{2}=\frac{1}{N_{s}}\left(N_{s}N_{c}+1\right)\left(N_{s}\tilde{\lambda}_{2}+\tilde{\lambda}_{2}\right)+\left(N_{2}-\frac{1}{N_{s}}\right)\tilde{\lambda}_{2}\geq
0$ (2.31)

  2. 2.

$\tilde{\lambda}_{2}\geq 0$:

$(N_{s}N_{c}+1)\tilde{\lambda}_{1}+(N_{s}+N_{c})\tilde{\lambda}_{2}=(N_{s}N_{c}+1)\left(\tilde{\lambda}_{1}+\tilde{\lambda}_{2}\right)+\left(N_{s}-1\right)\left(N_{c}-1\right)\left|\tilde{\lambda}_{2}\right|\geq
0$ (2.32)

we can now conclude that the thermal mass is always positive

$m_{S}^{2}(T)>0$ (2.33)

i.e. the symmetry is restored at high temperature.

####  2.2.2 $SU(N_{c})$ with two fundamental scalars

One of the problems of the previous model was that there was too much symmetry
in the scalar potential. We now take the case of two scalars, $N_{s}=2$, but
instead of scalar SU(2) the symmetry of the potential will be just a discrete
symmetry. We can take either a single $Z_{2}$ for even $N_{c}=2n$ or
$Z_{2}\times Z_{2}$ for odd $N_{c}=2n+1$.

In fact:

  * •

$Z_{2}\subset Z_{2n}$ and for even $N_{c}=2n$ the center of $SU(N_{c})$ is
$Z_{2n}$ and so $Z_{2}\subset SU(N_{c})$. In other words, a common
$\vec{\varphi}_{i}\to-\vec{\varphi}_{i}$ is already present. So in this case
there is only one extra $Z_{2}$ possible, say
$\vec{\varphi}_{1}\to-\vec{\varphi}_{1}$.

  * •

for odd $N_{c}=2n+1$ there is no $Z_{2}$ subgroup of $SU(N_{c})$. In fact
using the Levi-Civita tensor the invariant out of $N_{c}=2n+1$ fundamentals is
possible. Here it is thus possible to have an extra $Z_{2}\times Z_{2}$
symmetry for two fundamentals.

One way or another this means that each term of the potential can have only an
even number of fundamentals $\vec{\varphi}_{1}$ and anti-fundamentals
$\vec{\varphi}_{1}^{*}$ and an even number of fundamentals $\vec{\varphi}_{2}$
and anti-fundamentals $\vec{\varphi}_{2}^{*}$:

$\displaystyle V$ $\displaystyle=$
$\displaystyle\frac{\lambda_{1}}{2}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)^{2}+\frac{\lambda_{2}}{2}\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)^{2}+\lambda_{3}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)$
(2.34) $\displaystyle+$
$\displaystyle\lambda_{4}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{2}\right)\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{1}\right)+\frac{\lambda_{5}}{2}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{2}\right)^{2}+\frac{\lambda_{5}^{*}}{2}\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{1}\right)^{2}$

with $\lambda_{1,2,3,4}$ real and in general $\lambda_{5}$ complex.

By taking

$g^{2}=\frac{16\pi^{2}\tilde{\alpha}}{N_{c}t}\quad,\quad\lambda_{i}=\frac{16\pi^{2}\tilde{\lambda}_{i}}{N_{c}t}$
(2.35)

with constant $\tilde{\alpha}$, $\tilde{\lambda}_{i}$, and

$\tilde{\lambda}_{\pm}=\frac{1}{2}\left(\tilde{\lambda}_{1}\pm\tilde{\lambda}_{2}\right)$
(2.36)

the solutions to the RGE (see the appendix C.3) are

  1. 1.

$\tilde{\lambda}_{+}=\frac{6\tilde{\alpha}-1}{4}\quad,\quad\tilde{\lambda}_{3}^{2}+\tilde{\lambda}_{-}^{2}=\frac{24\tilde{\alpha}^{2}-12\tilde{\alpha}+1}{16}$
(2.37)

  2. 2.

$\tilde{\lambda}_{+}=\frac{6\tilde{\alpha}-1+a_{+}\sqrt{24\tilde{\alpha}^{2}-12\tilde{\alpha}+1}}{4}\quad,\quad
a_{+}^{2}=1\quad,\quad\tilde{\lambda}_{3}=\tilde{\lambda}_{-}=0$ (2.38)

acceptable only for $\tilde{\alpha}\geq(3+\sqrt{3})/12$.

The thermal potential at large $N_{c}$ is

$\Delta
V_{T}=(4\pi)^{2}\frac{T^{2}}{24\log{T}}\left(\left(2\left(\tilde{\lambda}_{1}+\tilde{\lambda}_{3}\right)+3\tilde{\alpha}\right)\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)+\left(2\left(\tilde{\lambda}_{2}+\tilde{\lambda}_{3}\right)+3\tilde{\alpha}\right)\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)\right)$
(2.39)

The masses are quite symmetric and the search for symmetry restoration boils
down to look for negative $\tilde{\lambda}_{3}=-|\tilde{\lambda}_{3}|$ which
leads to a negative mass square for $\vec{\varphi}_{1}$, i.e. a negative
combination

$\frac{12\tilde{\alpha}-1}{2}-2\left(\sqrt{\frac{24\tilde{\alpha}^{2}-12\tilde{\alpha}+1}{16}-|\tilde{\lambda}_{3}|^{2}}+|\tilde{\lambda}_{3}|\right)$
(2.40)

for

$\tilde{\alpha}\geq\frac{3+\sqrt{3}}{12}\quad,\quad
0\leq|\tilde{\lambda}_{3}|\leq\sqrt{\frac{24\tilde{\alpha}^{2}-a2\tilde{\alpha}+1}{16}}$
(2.41)

The function (2.40) is minimised for

$|\tilde{\lambda}_{3}|=\frac{1}{\sqrt{2}}\sqrt{\frac{24\tilde{\alpha}^{2}-12\tilde{\alpha}+1}{16}}$
(2.42)

which is however not enough for a negative mass square.

####  2.2.3 $SU(N_{c_{1}})\times SU(N_{c_{2}})$ with fundamental scalars:
Symmetry breaks at high temperatures

The model we will study now is similar to the previous one, but now we have
two simple groups, $SU(N_{c_{1}})\times SU(N_{c_{2}})$, so that each
$\varphi_{i}$ is in a fundamentals representation of its $SU(N_{ci})$ and a
singlet under the other one. The most general potential is

$V=\frac{\lambda_{1}}{2}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)^{2}+\frac{\lambda_{2}}{2}\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)^{2}-\lambda\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)$
(2.43)

Defining first

$\displaystyle i=1,2$ $\displaystyle:$ $\displaystyle
g_{i}^{2}=\frac{16\pi^{2}\tilde{\alpha}_{i}}{N_{ci}t}\quad,\quad\lambda_{i}=\frac{16\pi^{2}\tilde{\lambda}_{i}}{N_{ci}t}$
(2.44)
$\displaystyle\lambda=\frac{16\pi^{2}\tilde{\lambda}}{\sqrt{N_{c1}N_{c2}}t}$
(2.45)

with constant $\tilde{\alpha}_{i}$, $\tilde{\lambda}_{i}$, $\tilde{\lambda}$,
the thermal effective potential becomes at large $N_{ci}$ reads

$\displaystyle\Delta V_{T}=(4\pi)^{2}\frac{T^{2}}{24\log{T}}$
$\displaystyle\left(\left(2\left(\tilde{\lambda}_{1}-\sqrt{\frac{N_{c2}}{N_{c1}}}\tilde{\lambda}\right)+3\tilde{\alpha}_{1}\right)\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)\right.$
$\displaystyle\left.+\left(2\left(\tilde{\lambda}_{2}-\sqrt{\frac{N_{c1}}{N_{c2}}}\tilde{\lambda}\right)+3\tilde{\alpha}_{2}\right)\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)\right)$
(2.46)

Introducing the new variables

$\tilde{\lambda}_{\pm}=\frac{1}{2}\left(\tilde{\lambda}_{1}\pm\tilde{\lambda}_{2}\right)\quad,\quad\tilde{\alpha}_{\pm}=\frac{1}{2}\left(\tilde{\alpha}_{1}\pm\tilde{\alpha}_{2}\right)$
(2.47)

one finds the following solution555The other possible solution
$\tilde{\alpha}_{+}=1/4$, $\tilde{\lambda}_{+}=\frac{1}{8}$,
$\left(\tilde{\lambda}_{-}-\frac{3}{2}\tilde{\alpha}_{-}\right)^{2}+\tilde{\lambda}^{2}=\frac{1}{32}\left(48\tilde{\alpha}_{-}^{2}-1\right)$
describes a $T=0$ potential which is unbounded from below. of the RGE (see the
appendix C.4):

$\displaystyle\tilde{\alpha}_{-}$ $\displaystyle=$ $\displaystyle 0$ (2.48)
$\displaystyle\tilde{\lambda}_{+}$ $\displaystyle=$
$\displaystyle\frac{6\tilde{\alpha}_{+}-1}{4}$ (2.49)
$\displaystyle\tilde{\lambda}_{-}^{2}+\tilde{\lambda}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\left(24\tilde{\alpha}_{+}^{2}-12\tilde{\alpha}_{+}+1\right)$
(2.50)

valid for

$\tilde{\alpha}_{+}\geq(3+\sqrt{3})/12$ (2.51)

We will now prove that this solution supports symmetry non-restoration at
arbitrary high temperatures.

Denoting by $\mu^{2}_{i}$ the coefficient in front of
$\left(\vec{\varphi}_{i}^{*}\cdot\vec{\varphi}_{i}\right)$ in the parenthesis
on the right-hand-side of (2.2.3) we have

$\displaystyle\mu_{1}^{2}$ $\displaystyle=$
$\displaystyle\frac{12\tilde{\alpha}_{+}-1}{2}+2\tilde{\lambda}_{-}-2\sqrt{\frac{N_{c2}}{N_{c1}}}\tilde{\lambda}$
(2.52) $\displaystyle\mu_{2}^{2}$ $\displaystyle=$
$\displaystyle\frac{12\tilde{\alpha}_{+}-1}{2}-2\tilde{\lambda}_{-}-2\sqrt{\frac{N_{c1}}{N_{c2}}}\tilde{\lambda}$
(2.53)

We are searching for positive

$\tilde{\lambda}=\left|\tilde{\lambda}\right|$ (2.54)

and, up to redefinitions of what is 1 and what is 2, we can take

$\tilde{\lambda}_{-}=-\sqrt{\frac{24\tilde{\alpha}_{+}^{2}-12\tilde{\alpha}_{+}+1}{16}-\left|\tilde{\lambda}\right|^{2}}$
(2.55)

so that

$\displaystyle\mu_{1}^{2}$ $\displaystyle=$
$\displaystyle\frac{12\tilde{\alpha}_{+}-1}{2}-2\left(\sqrt{\frac{24\tilde{\alpha}_{+}^{2}-12\tilde{\alpha}_{+}+1}{16}-\left|\tilde{\lambda}\right|^{2}}+\sqrt{\frac{N_{c2}}{N_{c1}}}\left|\tilde{\lambda}\right|\right)$
(2.56) $\displaystyle\mu_{2}^{2}$ $\displaystyle=$
$\displaystyle\frac{12\tilde{\alpha}_{+}-1}{2}+2\left(\sqrt{\frac{24\tilde{\alpha}_{+}^{2}-12\tilde{\alpha}_{+}+1}{16}-\left|\tilde{\lambda}\right|^{2}}-\sqrt{\frac{N_{c1}}{N_{c2}}}\left|\tilde{\lambda}\right|\right)$
(2.57)

Minimising the expression for $\mu_{1}^{2}$ we obtain:

$\left|\tilde{\lambda}\right|^{2}=\frac{1}{1+N_{c1}/N_{c2}}\frac{24\tilde{\alpha}_{+}^{2}-12\tilde{\alpha}_{+}+1}{16}$
(2.58)

The minimised mass parameter

$\mu_{1}^{2}=\frac{12\tilde{\alpha}_{+}-1}{2}-2\sqrt{\frac{24\tilde{\alpha}_{+}^{2}-12\tilde{\alpha}_{+}+1}{16}}\sqrt{1+\frac{N_{c2}}{N_{c1}}}$
(2.59)

can now be negative by a suitable choice of number of colours.

Let us now demonstrate that the previous solution leads to a bounded
potential. The latter occurs if

$\lambda_{1}\lambda_{2}-\lambda^{2}>0$ (2.60)

which can be rewritten first as

$\tilde{\lambda}_{+}^{2}-\tilde{\lambda}_{-}^{2}-\tilde{\lambda}^{2}>0$ (2.61)

and then as

$\frac{(6\tilde{\alpha}_{+}-1)^{2}}{16}-\frac{24\tilde{\alpha}_{+}^{2}-12\tilde{\alpha}_{+}+1}{16}=\frac{12\tilde{\alpha}_{+}^{2}}{16}$
(2.62)

which is indeed positive.

Finally, requiring equal gauge couplings in the large $N_{ci}$ limit

$\tilde{\alpha}_{1}=\tilde{\alpha}_{2}$ (2.63)

means that the original not rescaled couplings satisfy the relation

$N_{c1}g_{1}^{2}=N_{c2}g_{2}^{2}$ (2.64)

This is achieved by the following suitable choice of number of matter
fermions:

$\frac{N_{f1}}{N_{c1}}=\frac{N_{f2}}{N_{c2}}$ (2.65)

Because of (2.51), they must satisfy

$\frac{1}{2}\left(2+3\sqrt{3}\right)\leq\frac{N_{fi}}{N_{ci}}<\frac{11}{2}$
(2.66)

We arrive at the result, similar to Weinberg’s model, that only one thermal
mass is negative.

####  2.2.4 The IR story of $SU(N_{c_{1}})\times SU(N_{c_{2}})$ at nonzero
temperature

Interestingly the model of the previous subsection can feature also an IR
Banks-Zaks fixed point. This can be achieved by tuning the number of fermions
to maintain both gauge couplings equality and the occurrence o a perturbative
IR fixed point. Once this is achieved the remaining equations for the IR fixed
point values666N.B. These values should not to be confused with the tilded
$1/t$ coefficients used for the fixed flow solutions in the UV. are:

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
2\lambda_{1}^{2}+2\lambda^{2}-6\alpha_{1}\lambda_{1}+\frac{3}{2}\alpha_{1}^{2}$
(2.67) $\displaystyle 0$ $\displaystyle=$ $\displaystyle
2\lambda_{2}^{2}+2\lambda^{2}-6\alpha_{2}\lambda_{2}+\frac{3}{2}\alpha_{2}^{2}$
(2.68) $\displaystyle 0$ $\displaystyle=$ $\displaystyle
2\left(\lambda_{1}+\lambda_{2}\right)\lambda-3\left(\alpha_{1}+\alpha_{2}\right)\lambda$
(2.69)

The solutions are

$\displaystyle\lambda_{+}$ $\displaystyle=$
$\displaystyle\frac{3}{2}\alpha_{+}$ (2.70)
$\displaystyle\lambda_{-}^{2}+\lambda^{2}$ $\displaystyle=$
$\displaystyle\frac{3}{2}\alpha_{+}^{2}$ (2.71)

if and only if

$\alpha_{-}=0$ (2.72)

The thermal effective potential is

$\displaystyle\Delta V_{T}=\frac{T^{2}}{24}$ $\displaystyle\times$
$\displaystyle\left(\left(2\left(\lambda_{1}-\sqrt{\frac{N_{c2}}{N_{c1}}}\lambda\right)+3\alpha_{+}\right)\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)\right.$
(2.73)
$\displaystyle\left.+\left(2\left(\lambda_{2}-\sqrt{\frac{N_{c1}}{N_{c2}}}\lambda\right)+3\alpha_{+}\right)\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)\right)$

so that the thermal masses are proportional to

$\displaystyle\mu_{1}^{2}$ $\displaystyle=$ $\displaystyle
6\alpha_{+}+2\lambda_{-}-2\sqrt{\frac{N_{c2}}{N_{c1}}}\lambda$ (2.74)
$\displaystyle\mu_{2}^{2}$ $\displaystyle=$ $\displaystyle
6\alpha_{+}-2\lambda_{-}-2\sqrt{\frac{N_{c1}}{N_{c2}}}\lambda$ (2.75)

Searching again for the branch

$\lambda=\left|\lambda\right|\quad,\quad\lambda_{-}=-\sqrt{\frac{3}{2}\alpha_{+}^{2}-\left|\lambda\right|^{2}}$
(2.76)

we have first

$\displaystyle\mu_{1}^{2}$ $\displaystyle=$ $\displaystyle
6\alpha_{+}-2\left(\sqrt{\frac{3}{2}\alpha_{+}^{2}-\left|\lambda\right|^{2}}+\sqrt{\frac{N_{c2}}{N_{c1}}}\left|\lambda\right|\right)$
(2.77) $\displaystyle\mu_{2}^{2}$ $\displaystyle=$ $\displaystyle
6\alpha_{+}+2\left(\sqrt{\frac{3}{2}\alpha_{+}^{2}-\left|\lambda\right|^{2}}-\sqrt{\frac{N_{c1}}{N_{c2}}}\left|\lambda\right|\right)$
(2.78)

$\mu_{1}^{2}$ is minimised for

$\left|\lambda\right|^{2}=\frac{1}{1+N_{c1}/N_{c2}}\frac{3}{2}\alpha_{+}^{2}$
(2.79)

so that the thermal mass

$\mu_{1}^{2}=6\alpha_{+}-2\sqrt{\frac{3}{2}\alpha_{+}^{2}}\sqrt{1+\frac{N_{c2}}{N_{c1}}}$
(2.80)

is negative for (but still in the Veneziano limit $N_{ci}\to\infty$)

$\frac{N_{c2}}{N_{c1}}>5$ (2.81)

Since

$\lambda_{+}^{2}-\lambda_{-}^{2}-\lambda^{2}=\frac{3}{4}\alpha_{+}^{2}>0$
(2.82)

the parameter choice describes a $T=0$ potential which is bounded from below.

We have therefore found an example in which symmetry non-restoration occurs
near an IR fixed point which is more minimal than the one presented in [74].

####  2.2.5 Another example of symmetry breaking at high $T$: two adjoints in
$SU(N_{c1})\times SU(N_{c2})$

This model is similar to the previous one, except that adjoint scalars are
considered instead of fundamental scalars. The details are described in
appendix C.5. The most general quartic potential is

$V=\frac{\lambda_{1}^{\prime}}{4}Tr\Sigma_{1}^{4}+\frac{\lambda_{2}^{\prime}}{4}Tr\Sigma_{2}^{4}+\frac{\lambda_{1}}{4}\left(Tr\Sigma_{1}^{2}\right)^{2}+\frac{\lambda_{2}}{4}\left(Tr\Sigma_{2}^{2}\right)^{2}-\frac{\lambda}{2}Tr\Sigma_{1}^{2}Tr\Sigma_{2}^{2}$
(2.83)

We redefine the couplings as

$\displaystyle\lambda^{\prime}_{1,2}=(4\pi)^{2}\frac{\tilde{\lambda}^{\prime}_{1,2}}{N_{c_{1,2}}}\times\frac{1}{t}$
,
$\displaystyle\lambda_{1,2}=(4\pi)^{2}\frac{\tilde{\lambda}_{1,2}}{N_{c_{1,2}}^{2}}\times\frac{1}{t}$
(2.84)
$\displaystyle\lambda=(4\pi)^{2}\frac{\tilde{\lambda}}{N_{c_{1}}N_{c_{2}}}\times\frac{1}{t}$
, $\displaystyle
g_{1,2}^{2}=(4\pi)^{2}\frac{\tilde{\alpha}_{1,2}}{N_{c_{1,2}}}\times\frac{1}{t}$
(2.85)

with all tilded quantities constants, and eventually we will take the large
$N_{c_{1,2}}$ limit.

As shown in [75], the potential (2.83) is bounded from below if the parameters
satisfy the following inequalities:

$\lambda_{i}+\frac{\lambda_{i}^{\prime}}{k_{i}}>0\quad\left(1\leq k_{i}\leq
N_{c_{i}}\right)\quad,\quad\left(\lambda_{1}+\frac{\lambda_{1}^{\prime}}{k_{1}}\right)\left(\lambda_{2}+\frac{\lambda_{2}^{\prime}}{k_{2}}\right)>\lambda^{2}$
(2.86)

If $\lambda_{i}^{\prime}>0$, then it is enough to check the above for
$k_{i}=N_{c_{i}}$, while if $\lambda_{i}^{\prime}<0$, $k_{i}=1$ suffices.
However, for large $N_{c_{i}}$, the second case is impossible, since

$\lambda_{i}+\lambda_{i}^{\prime}>0\to\tilde{\lambda}_{i}^{\prime}>0$ (2.87)

which is in contradiction with the original assumption of
$\lambda_{i}^{\prime}<0$.

So the only possibility is just $\lambda_{1}^{\prime}>0$,
$\lambda_{2}^{\prime}>0$:

$\tilde{\lambda}_{1}+\tilde{\lambda}_{1}^{\prime}>0\quad,\quad\tilde{\lambda}_{2}+\tilde{\lambda}_{2}^{\prime}>0\quad,\quad\left(\tilde{\lambda}_{1}+\tilde{\lambda}_{1}^{\prime}\right)\left(\tilde{\lambda}_{2}+\tilde{\lambda}_{2}^{\prime}\right)>\tilde{\lambda}^{2}$
(2.88)

The thermal mass is

$V_{T}=(4\pi)^{2}\frac{T^{2}}{48\log{T}}\left(\left(\tilde{\lambda}_{1}+2\tilde{\lambda}_{1}^{\prime}-\frac{N_{c_{2}}}{N_{c_{1}}}\tilde{\lambda}+12\tilde{\alpha}_{1}\right)Tr\Sigma_{1}^{2}+\left(\tilde{\lambda}_{2}+2\tilde{\lambda}_{2}^{\prime}-\frac{N_{c_{1}}}{N_{c_{2}}}\tilde{\lambda}+12\tilde{\alpha}_{2}\right)Tr\Sigma_{2}^{2}\right)$
(2.89)

We provide here an existence proof for a negative thermal mass with parameters
satisfying the boundedness of the potential constraint.

First one can show that only one sector would not work, as expected. This
means that if
$\tilde{\lambda}=\tilde{\alpha}_{2}=\tilde{\lambda}_{2}=\tilde{\lambda}_{2}^{\prime}=0$,
there is no solution of the above fixed flow RG equations for real
$\tilde{\alpha}_{1}$, $\tilde{\lambda}_{1}$, $\tilde{\lambda}_{1}^{\prime}$
assuming $\tilde{\lambda}_{1}+\tilde{\lambda}_{1}^{\prime}>0$ (boundedness)
and $\tilde{\lambda}_{1}+2\tilde{\lambda}_{1}^{\prime}<0$ (negative thermal
mass).

However, a solution for bounded potential with negative thermal mass square
exists for

$\displaystyle\tilde{\alpha}_{1}=\tilde{\alpha}_{2}$ $\displaystyle=$
$\displaystyle\frac{2+\sqrt{2}}{2}$ (2.90)
$\displaystyle\tilde{\lambda}_{1}^{\prime}=\tilde{\lambda}_{2}^{\prime}$
$\displaystyle=$ $\displaystyle 2$ (2.91) $\displaystyle\tilde{\lambda}_{1}$
$\displaystyle=$ $\displaystyle 12\left(2+\sqrt{2}\right)-26$ (2.92)
$\displaystyle\tilde{\lambda}_{2}$ $\displaystyle=$ $\displaystyle 16$ (2.93)
$\displaystyle\tilde{\lambda}$ $\displaystyle=$
$\displaystyle\sqrt{120\left(2+\sqrt{2}\right)-392}$ (2.94)
$\displaystyle\frac{N_{c_{2}}}{N_{c_{1}}}$ $\displaystyle=$ $\displaystyle 16\
.$ (2.95)

This is therefore another relevant example of symmetry non-restoration at
arbitrary high temperature.

##  3 Asymptotic safety at high temperature

Another way to achieve a UV complete theory, up to gravity, is via the
presence of an interacting ultraviolet fixed point in all couplings. In fact,
one can have a combination of safe and free couplings for the model to be well
defined at all scales.

Due to the fact that the discovery of asymptotically safe quantum field theory
is relatively recent [55] the issue of symmetry non-restoration for this
relevant class of models has never been investigated before.

We will consider here examples classified according to whether we can re-use
part of the results and reasoning employed above for the complete
asymptotically free theories or we need a separate in depth analysis of the
safe model.

For the first class we consider theories structurally similar to the one
considered above albeit with sufficient matter fields such that asymptotic
freedom is lost while assuming that perturbative asymptotic safety occurs.

To transform the previous proof valid for asymptotically free theories to the
equivalent potential asymptotically safe case we need to

  * •

replace all tilded quantities with untilded ones;

  * •

eliminate the $\log{T}$ in the denominator of the thermal mass;

  * •

replace the $16\pi^{2}d\alpha_{i}/dt$ in the left-hand-sides of the RGEs with
a zero.

This means that in the theories investigated in the previous section, once
asymptotic freedom is lost and potential asymptotic safety appears, symmetry
restoration is a must.

###  3.1 Explicit examples of asymptotic safety

We now consider explicit constructions of asymptotically safe quantum field
theories that cannot be reduced to the example above because they either have
multiple gauge singlet scalar quartic terms or/and have gauged scalars.
Interestingly we anticipate that in both examples the symmetry is restored at
high temperature.

####  3.1.1 The Litim-Sannino (LS) model

The first model we consider here is the one put forward in [55] in which
asymptotically safe quantum field theories and their structure was first
discovered and understood. The Lagrangian reads:

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}Tr\left(F^{\mu\nu}F_{\mu\nu}\right)+Tr\left(\bar{Q}i\not{D}Q\right)+Tr\left(\partial_{\mu}H^{\dagger}\partial^{\mu}H\right)$
(3.1) $\displaystyle+$ $\displaystyle
y\,Tr\left(\bar{Q}_{L}HQ_{R}+\bar{Q}_{R}H^{\dagger}Q_{L}\right)-u\,Tr\left(H^{\dagger}H\right)^{2}-v\,\left(TrH^{\dagger}H\right)^{2}\
,$

with symmetry

$G=SU(N_{C})\times SU(N_{F})\times SU(N_{F})\times U_{V}(1)\ ,$ (3.2)

under which the fields transform as

$\displaystyle Q_{L}$ $\displaystyle\sim$
$\displaystyle\left(N_{C},N_{F},1,1\right)\ ,$ (3.3) $\displaystyle Q_{R}$
$\displaystyle\sim$ $\displaystyle\left(N_{C},1,N_{F},1\right)\ ,$ (3.4)
$\displaystyle H$ $\displaystyle\sim$
$\displaystyle\left(1,N_{F},\overline{N_{F}},0\right)\ .$ (3.5)

We assume the Veneziano limit, needed to ensure the rigorousness of the result

$N_{F},N_{C}\to\infty\;\;\;,\;\;\;\frac{N_{F}}{N_{C}}=\frac{11}{2}+\epsilon\
,$ (3.6)

with $\epsilon\ll 1$ to control the size of the UV fixed point couplings that
at the relevant order in perturbation theory read

$\displaystyle\alpha_{g}$ $\displaystyle\equiv$
$\displaystyle\frac{g^{2}N_{C}}{(4\pi)^{2}}=\frac{26}{57}\epsilon+{\cal
O}(\epsilon^{2})\ ,$ (3.7) $\displaystyle\alpha_{y}$ $\displaystyle\equiv$
$\displaystyle\frac{y^{2}N_{C}}{(4\pi)^{2}}=\frac{4}{19}\epsilon+{\cal
O}(\epsilon^{2})\ ,$ (3.8) $\displaystyle\alpha_{h}$ $\displaystyle\equiv$
$\displaystyle\frac{uN_{F}}{(4\pi)^{2}}=\frac{\sqrt{23}-1}{19}\epsilon+{\cal
O}(\epsilon^{2})\ ,$ (3.9) $\displaystyle\alpha_{v}$ $\displaystyle\equiv$
$\displaystyle\frac{vN_{F}^{2}}{(4\pi)^{2}}=-\frac{1}{19}\left(2\sqrt{23}-\sqrt{20+6\sqrt{23}}\right)\epsilon+{\cal
O}(\epsilon^{2}).$ (3.10)

The $T^{2}$ term of the $H$ mass square is

$\displaystyle m_{T}^{2}$ $\displaystyle=$
$\displaystyle(4\pi)^{2}\,\frac{T^{2}}{24}\,\left(8\alpha_{h}+4\alpha_{v}+2\alpha_{y}\right)$
(3.11) $\displaystyle\approx$ $\displaystyle 9.7\,\epsilon\,T^{2}>0\ ,$

so that the symmetry is restored at high temperature. Therefore we arrive at
the conclusion that the original model of an asymptotically safe quantum field
theory is also safe with respect to global symmetries.

###  3.2 A gauged scalar variant of the LS model

Here we consider an interesting example featuring a two-scalar sector with one
of the scalars being gauged while the full theory remains asymptotically safe
[71]. This model allows for a relevant test of symmetry (non)restoration and
the Lagrangian of the model reads:

$\displaystyle{\cal L}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}Tr\left(F^{\mu\nu}F_{\mu\nu}\right)+Tr\left(\bar{Q}i\not{D}Q\right)+Tr\left(\partial_{\mu}H^{\dagger}\partial^{\mu}H\right)+Tr\left(D_{\mu}\tilde{S}^{\dagger}D^{\mu}\tilde{S}\right)$
(3.12) $\displaystyle+$
$\displaystyle\left(\frac{y}{\sqrt{2}}\,Tr\left(\tilde{Q}HQ\right)+h.c.\right)-u_{2}\,Tr\left(H^{\dagger}H\right)^{2}-u_{1}\,\left(TrH^{\dagger}H\right)^{2}$
$\displaystyle-$ $\displaystyle
w_{2}\,Tr\left(\tilde{S}^{\dagger}\tilde{S}\right)^{2}-w_{1}\,\left(Tr\tilde{S}^{\dagger}\tilde{S}\right)^{2}\
,$

where the fields transform under the gauge and 3 global symmetries
($N_{S}=N_{C}-2$)

$G=SU(N_{C})\times SU(N_{F})_{L}\times SU(N_{F})_{R}\times SU(N_{S})\ ,$
(3.13)

as

$\displaystyle Q$ $\displaystyle\sim$ $\displaystyle(N_{C},N_{F},1,1)\ ,$
(3.14) $\displaystyle\tilde{Q}$ $\displaystyle\sim$
$\displaystyle(\overline{N_{C}},1,\overline{N_{F}},1)\ ,$ (3.15)
$\displaystyle H$ $\displaystyle\sim$
$\displaystyle(1,\overline{N_{F}},N_{F},1)\ ,$ (3.16) $\displaystyle\tilde{S}$
$\displaystyle\sim$ $\displaystyle(\overline{N_{C}},1,1,\overline{N_{S}})\ .$
(3.17)

For small and positive

$\epsilon=\frac{N_{F}}{N_{C}}-\frac{11}{2}+\frac{N_{S}}{4N_{C}}\to\frac{N_{F}}{N_{C}}-\frac{21}{4}\
,$ (3.18)

the following relations are satisfied [71] at the UV fixed point:

$\displaystyle\alpha_{g}\equiv\frac{N_{C}g^{2}}{(4\pi)^{2}}$ $\displaystyle=$
$\displaystyle\frac{25}{18}\epsilon\ ,$ (3.19)
$\displaystyle\alpha_{y}\equiv\frac{N_{C}y^{2}}{(4\pi)^{2}}$ $\displaystyle=$
$\displaystyle\frac{24}{25}\alpha_{g}\ ,$ (3.20)
$\displaystyle\alpha_{u_{1}}\equiv\frac{N_{F}^{2}u_{1}}{(4\pi)^{2}}$
$\displaystyle=$
$\displaystyle\frac{-6\sqrt{22}+3\sqrt{19+6\sqrt{22}}}{100}\alpha_{g}\ ,$
(3.21) $\displaystyle\alpha_{u_{2}}\equiv\frac{N_{F}u_{2}}{(4\pi)^{2}}$
$\displaystyle=$ $\displaystyle\frac{3}{25}\left(\sqrt{22}-1\right)\alpha_{g}\
,$ (3.22) $\displaystyle\alpha_{w_{1}}\equiv\frac{N_{C}^{2}w_{1}}{(4\pi)^{2}}$
$\displaystyle=$
$\displaystyle\frac{3\pm\sqrt{3\left(4\sqrt{2}-5\right)}}{16\sqrt{2}}\alpha_{g}\
,$ (3.23) $\displaystyle\alpha_{w_{2}}\equiv\frac{N_{C}w_{2}}{(4\pi)^{2}}$
$\displaystyle=$ $\displaystyle\frac{1}{16}\left(2-\sqrt{2}\right)\alpha_{g}\
.$ (3.24)

Following the analysis of the LS case but now generalised to both scalars we
arrive at

$\displaystyle m_{T}^{2}(H)$ $\displaystyle=$
$\displaystyle(4\pi)^{2}\frac{T^{2}}{48}\left(2\alpha_{y}+16\alpha_{u_{2}}+8\alpha_{u_{1}}\right)\approx
38.4\,\epsilon T^{2}>0\ ,$ (3.25) $\displaystyle m_{T}^{2}(S)$
$\displaystyle=$
$\displaystyle(4\pi)^{2}\frac{T^{2}}{24}\left(8\alpha_{w_{2}}+4\alpha_{w_{1}}+3\alpha_{g}\right)\approx
37.2\,\epsilon T^{2}>0\ .$ (3.26)

This implies that no symmetries can be broken at high temperature.

##  4 Conclusions

In this paper we analysed Weinberg’s symmetry non-restoration idea within UV
complete theories of either asymptotically free or safe nature.

The reason why these are natural models to investigate is that only for UV
complete theories it is consistent to consider the arbitrary large temperature
limit.

Safe and free theories share short scale conformality that insures a well
defined behaviour at arbitrary high energies. Because of this, they belong to
a special subset of all possible quantum field theories. The remaining field
theories should be considered as effective low energy descriptions that cannot
be complete without quantum gravity possibly modifying their high energy
behaviour. In any event, given the fact that we do not yet have a complete
theory of quantum gravity, for these theories the symmetry non-restoration
test cannot be performed at arbitrary high temperatures.

As complete asymptotically free templates we commenced our investigation with
$SU(N_{c})$ gauge-Yukawa theories featuring $N_{f}$ fundamental Dirac fermions
and two singlet scalars coupled via Yukawa interactions to the fermions. We
demonstrated that symmetry is restored for this class of asymptotically free
theories. We then generalised the result to arbitrary (Weyl) fermion
representations and to certain multiple singlet scalar theories. It was
sufficient to demonstrate the incompatibility between the request of negative
thermal mass squared for one of the scalars and the simultaneous need for its
coupling to be asymptotically free.

We then moved to investigate the case of gauge scalars and have shown that
high temperature symmetry non-restoration appeared for the case of two gauged
scalars transforming according to the fundamental representation of two
independent gauge sectors. Fermions in the fundamental representation were
included as well but without Yukawa couplings.

We then moved to investigate the case of asymptotically safe theories starting
by noticing that the symmetry restoration results discovered for the singlet
scalars discussed above could be extended to potentially safe theories.

Two more relevant examples were investigated in the asymptotically safe
scenario in which either multiple quartic scalar field terms were present in
the Lagrangian [55] and/or some of the scalar were gauged [71]. In these
models symmetries restore at high temperature.

As an interesting class of UV complete theories featuring symmetry non-
restoration at arbitrary high temperatures we discovered the one featuring two
gauged scalars, each in a fundamental representation of its own $SU(N_{ci})$
gauge group: for large enough ratios of colours, one scalar thermal mass can
be negative.

So far we discussed UV complete theories before adding quantum gravity. We can
imagine that a possible safe and free completion of the standard model occurs
few orders of magnitude below the scale above which quantum gravity cannot be
ignored. In this case our analysis still applies. It can even happen that
quantum gravity is, per se, asymptotically free [76], and in this case we can
ignore it.

The simplicity of the UV complete models discovered here featuring arbitrary
high temperature symmetry non-restoration phenomenon invites for further
theoretical and phenomenological investigations. For example, it would be
interesting to investigate whether UV complete grand-unified theories of the
Pati-Salam type exist and that can feature the phenomenon of symmetry non-
restoration. Additionally there could be dark sectors that are gravitationally
coupled to us that can be UV complete and feature early universe phase
transitions from a symmetric to a broken one as the temperature increases.

#### Acknowledgments

BB is very grateful to Goran Senjanović for having been introduced to the
fascinating subject of symmetry non-restoration at high temperature, as well
as for illuminating interminable discussions on related subjects. BB
acknowledges the financial support from the Slovenian Research Agency
(research core funding No. P1-0035).

#### Note added

While we were completing the present work, a related paper appeared [74] in
which explicit examples of Banks-Zaks type CFTs were considered in which
symmetry nonrestoration occurred at nonzero temperature. Differently and in a
complementary manner of [74] our work investigates, rather than theories
around IR fixed points, models featuring either Gaussians (complete
asymptotically free) or interacting (completely asymptotically safe) UV fixed
points such that we can investigate the infinite temperature limit within a
given UV complete quantum field theory.

##  Appendix A  The 1-loop RG equations

In this section we summarise the relevant one loop RG equations used in the
main text starting with the normalisation of the fields given by

${\cal
L}_{kin}=-\frac{1}{4}F_{\mu\nu}^{A}F^{A\mu\nu}+i\overline{\Psi}\not{D}\Psi+\frac{1}{2}D^{\mu}\Phi^{a}D_{\mu}\Phi^{a}$
(A.1)

The gauge RG equation is

$(4\pi)^{2}\beta_{g}\equiv(4\pi)^{2}\mu\frac{dg}{d\mu}=-b_{0}g^{3}$ (A.2)

with

$b_{0}=\frac{11}{3}T(G)-\frac{2}{3}T(F)-\frac{1}{6}T(S)$ (A.3)

where $G,F,S$ stand for gauge bosons, Weyl fermions and real scalars,
respectively, and $T(R)$ is the Dynkin index of the representation $R$,
defined as

$Tr\left(T^{A}(R)T^{B}(R)\right)=T(R)\delta^{AB}$ (A.4)

In SU($N_{c}$) we will need the following:

$T(fundamental)=\frac{1}{2}\;\;\;,\;\;\;T(adjoints)=N_{c}$ (A.5)

The Yukawa RG equations for Dirac fermions $\Psi_{i}$

${\cal L}_{Yukawa}=\sum_{i,j}Y^{a}_{ij}\overline{\Psi}_{i}\phi^{a}\Psi_{j}$
(A.6)

are [77] ($\kappa=1$ for Dirac fermions and $\kappa=1/2$ for Weyl fermions)

$\displaystyle(4\pi)^{2}\beta_{Y}^{a}\equiv(4\pi)^{2}\mu\frac{dY^{a}}{d\mu}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left(Y^{b}Y^{b\dagger}Y^{a}+Y^{a}Y^{b\dagger}Y^{b}\right)+2Y^{b}Y^{a\dagger}Y^{b}$
$\displaystyle+$ $\displaystyle\kappa
Y^{b}Tr\left(Y^{b\dagger}Y^{a}+Y^{a\dagger}Y^{b}\right)-3g^{2}\left(C_{2}(F)Y^{a}+Y^{a}C_{2}(F)\right)$

where $\phi^{a}$ are real scalars and

$\left(C_{2}(F)\right)_{ij}=\sum_{kA}T^{A}_{ik}T^{A}_{kj}$ (A.8)

where the generators $T^{A}$ are in the (in general reducible) representation
of the fermions.

Here and in the following a repeated index gets summed ($a$, $b$ over real
scalars, $\alpha$ over SU($N_{c}$) generators, $i$, $j$, $k$ over
(bi-)spinors) even when the explicit sum is not written.

Notice that the Yukawa matrices in (A.6) are Hermitean by definition.

The scalar sector is defined by

$V=\frac{1}{4!}\lambda_{abcd}\phi_{a}\phi_{b}\phi_{c}\phi_{d}$ (A.9)

Following [78] we introduce the completely symmetric tensors

$\displaystyle\Lambda^{2}_{abcd}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\sum_{perm}\lambda_{abef}\lambda_{efcd}$ (A.10)
$\displaystyle\Lambda^{Y}_{abcd}$ $\displaystyle=$
$\displaystyle\frac{1}{12}\sum_{perm}Tr\left(Y^{a\dagger}Y^{e}+Y^{e\dagger}Y^{a}\right)\lambda_{ebcd}$
(A.11) $\displaystyle H_{abcd}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{perm}Tr\left(Y^{a\dagger}Y^{b}Y^{c\dagger}Y^{d}\right)$
(A.12) $\displaystyle\Lambda^{S}_{abcd}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\sum_{perm}\sum_{A=1}^{N_{c}^{2}-1}\left(T^{A}(S)T^{A}(S)\right)_{ae}\lambda_{ebcd}$
(A.13) $\displaystyle A_{abcd}$ $\displaystyle=$
$\displaystyle\frac{1}{8}\sum_{perm}\sum_{A,B=1}^{N_{c}^{2}-1}\left\\{T^{A}(S),T^{B}(S)\right\\}_{ab}\left\\{T^{A}(S),T^{B}(S)\right\\}_{cd}$
(A.14)

where the sum over ”perm” means that we sum over all $4!$ permutations of the
indices $a$, $b$, $c$ and $d$ so to make the left-hand sides completely
symmetric in all indices. The matrices $T^{A}(S)$ are the Hermitean
SU($N_{c}$) generators in the representation of the scalars. Since $\phi^{a}$
are taken real, these generators are imaginary and anti-symmetric. For real
representations of SU($N_{c}$) this is automatic, while for complex
representations one has to work out the form of these matrices. More
precisely, they are found in the covariant derivative:

$D_{\mu}\phi^{a}=\partial_{\mu}\phi^{a}-igW_{\mu}^{A}{\left(T^{A}(S)\right)^{a}}_{b}\phi^{b}$
(A.15)

For the case of more gauge couplings $g_{\alpha}$ of gauge groups with
generators $T^{A}_{\alpha}$, one should remember that

$A_{abcd}\frac{\phi^{a}\phi^{b}\phi^{c}\phi^{d}}{4!}=\left(M^{2}_{W}\right)^{AB}\left(M^{2}_{W}\right)^{AB}$
(A.16)

with the $W$ mass

$\left(M^{2}_{W}\right)^{AB}=\frac{1}{2}\phi^{a}g_{\alpha}g_{\beta}\left\\{T_{\alpha}^{A}(S),T_{\beta}^{B}(S)\right\\}_{ab}\phi^{b}$
(A.17)

The 1-loop RG equations then read [78]

$16\pi^{2}\frac{d\lambda_{abcd}}{dt}=\Lambda^{2}_{abcd}+2\kappa\Lambda^{Y}_{abcd}-8\kappa
H_{abcd}-3g^{2}\Lambda^{S}_{abcd}+3g^{4}A_{abcd}$ (A.18)

Finally, at high temperature the thermal mass matrix is given by [3] (see also
[79])

$m^{2}_{ab}(T)=\frac{T^{2}}{24}\left(\lambda_{abcc}+2\kappa
Tr\left(Y^{a\dagger}Y^{b}+Y^{b\dagger}Y^{a}\right)+6g^{2}\left(T^{A}(S)T^{A}(S)\right)_{ab}\right)$
(A.19)

It is useful to rewrite the above formulae by multiplying the various
quantities by constant $\phi^{a}\phi^{b}\phi^{c}\phi^{d}/4!$ and summing over
the indices $a,b,c,d$. We thus define

$\displaystyle V_{\Lambda^{2}}$
$\displaystyle\equiv\Lambda^{2}_{abcd}\frac{\phi^{a}\phi^{b}\phi^{c}\phi^{d}}{4!}=\frac{1}{2}\frac{\partial^{2}V}{\partial\phi^{a}\partial\phi^{b}}\frac{\partial^{2}V}{\partial\phi^{a}\partial\phi^{b}}$
(A.20) $\displaystyle V_{\Lambda^{Y}}$ $\displaystyle\equiv
2\kappa\Lambda^{Y}_{abcd}\frac{\phi^{a}\phi^{b}\phi^{c}\phi^{d}}{4!}=\kappa\phi^{a}Tr\left(Y^{a\dagger}Y^{e}+Y^{e\dagger}Y^{a}\right)\frac{\partial
V}{\partial\phi^{e}}$ (A.21) $\displaystyle V_{H}$
$\displaystyle\equiv-8\kappa
H_{abcd}\frac{\phi^{a}\phi^{b}\phi^{c}\phi^{d}}{4!}=-2\kappa
Tr\left(Y^{a\dagger}Y^{b}Y^{c\dagger}Y^{d}\right)\phi^{a}\phi^{b}\phi^{c}\phi^{d}$
(A.22) $\displaystyle V_{\Lambda^{S}}$
$\displaystyle\equiv-3g^{2}\Lambda^{S}_{abcd}\frac{\phi^{a}\phi^{b}\phi^{c}\phi^{d}}{4!}=-3g^{2}\phi^{a}\left(T^{A}(S)T^{A}(S)\right)_{ae}\frac{\partial
V}{\partial\phi^{e}}$ (A.23) $\displaystyle V_{A}$ $\displaystyle\equiv
3g^{4}A_{abcd}\frac{\phi^{a}\phi^{b}\phi^{c}\phi^{d}}{4!}=\frac{3}{8}g^{4}\left(\phi^{a}\left\\{T^{A}(S),T^{B}(S)\right\\}_{ab}\phi^{b}\right)\left(\phi^{c}\left\\{T^{A}(S),T^{B}(S)\right\\}_{cd}\phi^{d}\right)$
(A.24)

Eq. (A.18) can thus be written as

$16\pi^{2}\frac{\phi^{a}\phi^{b}\phi^{c}\phi^{d}}{4!}\frac{d\lambda_{abcd}}{dt}=V_{\Lambda^{2}}+V_{\Lambda^{Y}}+V_{H}+V_{\Lambda^{S}}+V_{A}$
(A.25)

while the equivalent of (A.19) is (for vanishing Yukawa)

$\Delta
V(T)\equiv\frac{1}{2}m_{ab}^{2}(T)\phi^{a}\phi^{b}=\frac{T^{2}}{48}\left(2\frac{\partial^{2}V}{\partial\phi^{a}\partial\phi^{a}}+6g^{2}\phi^{a}\left(T^{A}(S)T^{A}(S)\right)_{ab}\phi^{b}\right)$
(A.26)

##  Appendix B SU($N_{c}$) with two singlet scalars and fundamental fermions

In this model the two singlet scalar couple through Yukawa couplings to
$N_{f_{1}}$ ($N_{f_{2}}$) Dirac fermions in the fundamental representation of
$SU(N_{c})$. We further allow for $N_{f_{0}}$ Dirac fermions in the
fundamental representation of the gauge group that are inert with respect to
the scalars, i.e. do not possess Yukawa couplings.

The gauge coupling 1-loop RGE is

$16\pi^{2}\frac{dg}{dt}=-b_{0}g^{3}\,,$ (B.1)

with

$b_{0}=\frac{11}{3}N_{c}-\frac{2}{3}\left(N_{f_{0}}+N_{f_{1}}+N_{f_{2}}\right)\,.$
(B.2)

The solution is

$\alpha_{g}=\frac{g^{2}}{(4\pi)^{2}}=\frac{\tilde{\alpha}_{g}}{t}\,,$ (B.3)

with

$\tilde{\alpha}_{g}=\frac{1}{2b_{0}}\,.$ (B.4)

The Yukawa RGE are

$16\pi^{2}\frac{dy_{i}}{dt}=\left(3+2N_{c}N_{f_{i}}\right)y_{i}^{3}-3g^{2}\frac{N_{c}^{2}-1}{N_{c}}y_{i}\;\;\;,\;\;\;i=1,2\,.$
(B.5)

Assuming the ansatz

$\alpha_{y_{i}}=\frac{y_{i}^{2}}{(4\pi)^{2}}=\frac{\tilde{\alpha}_{y_{i}}}{t}\,,$
(B.6)

the fixed flow solution is given by

$\tilde{\alpha}_{y_{i}}=\frac{6\frac{N_{c}^{2}-1}{N_{c}}\tilde{\alpha}_{g}-1}{2\left(3+2N_{c}N_{f_{i}}\right)}\;\;\;,\;\;\;i=1,2\,,$
(B.7)

and have positive solutions only if the gauge coupling is big enough

$6\tilde{\alpha}_{g}\frac{N_{c}^{2}-1}{N_{c}}-1>0\,,$ (B.8)

which reduces to a constraint on the number of Dirac fermion fundamentals :

$\frac{22}{4}N_{c}-\frac{9}{2}\left(N_{c}-\frac{1}{N_{c}}\right)<N_{f_{0}}+N_{f_{1}}+N_{f_{2}}<\frac{22}{4}N_{c}\,.$
(B.9)

The RG equations for the scalar couplings are

$\displaystyle 16\pi^{2}\frac{d\lambda_{1}}{dt}$ $\displaystyle=$
$\displaystyle
18\lambda_{1}^{2}+2\lambda^{2}-8N_{c}N_{f_{1}}y_{1}^{4}+8N_{c}N_{f_{1}}y_{1}^{2}\lambda_{1}$
(B.10) $\displaystyle 16\pi^{2}\frac{d\lambda_{2}}{dt}$ $\displaystyle=$
$\displaystyle
18\lambda_{2}^{2}+2\lambda^{2}-8N_{c}N_{f_{2}}y_{2}^{4}+8N_{c}N_{f_{2}}y_{2}^{2}\lambda_{2}$
(B.11) $\displaystyle 16\pi^{2}\frac{d\lambda}{dt}$ $\displaystyle=$
$\displaystyle-8\lambda^{2}+6\lambda(\lambda_{1}+\lambda_{2})+4N_{c}\left(N_{f_{1}}y_{1}^{2}+N_{f_{2}}y_{2}^{2}\right)\lambda$
(B.12)

The ansatz

$\alpha_{\lambda_{i}}=\frac{\lambda_{i}}{(4\pi)^{2}}=\frac{\tilde{\alpha}_{\lambda_{i}}}{t}\;\;\;,\;\;\;\alpha_{\lambda}=\frac{\lambda}{(4\pi)^{2}}=\frac{\tilde{\alpha}_{\lambda}}{t}$
(B.13)

reduces the system of ODEs (B.10)-(B.12) to a system of algebraic equations

$\displaystyle-\tilde{\alpha}_{\lambda_{1}}$ $\displaystyle=$ $\displaystyle
18\tilde{\alpha}_{\lambda_{1}}^{2}+2\tilde{\alpha}_{\lambda}^{2}-8N_{c}N_{f_{1}}\tilde{\alpha}_{y_{1}}^{2}+8N_{c}N_{f_{1}}\tilde{\alpha}_{y_{1}}\tilde{\alpha}_{\lambda_{1}}$
(B.14) $\displaystyle-\tilde{\alpha}_{\lambda_{2}}$ $\displaystyle=$
$\displaystyle
18\tilde{\alpha}_{\lambda_{2}}^{2}+2\tilde{\alpha}_{\lambda}^{2}-8N_{c}N_{f_{2}}\tilde{\alpha}_{y_{2}}^{2}+8N_{c}N_{f_{2}}\tilde{\alpha}_{y_{2}}\tilde{\alpha}_{\lambda_{2}}$
(B.15) $\displaystyle-\tilde{\alpha}_{\lambda}$ $\displaystyle=$
$\displaystyle-8\tilde{\alpha}_{\lambda}^{2}+6\tilde{\alpha}_{\lambda}\left(\tilde{\alpha}_{\lambda_{1}}+\tilde{\alpha}_{\lambda_{2}}\right)+4N_{c}\left(N_{f_{1}}\tilde{\alpha}_{y_{1}}+N_{f_{2}}\tilde{\alpha}_{y_{2}}\right)\tilde{\alpha}_{\lambda}$
(B.16)

To this we add (B.4) and (B.7). We look for strictly positive solutions for
all 6 couplings
$\tilde{\alpha}_{g,y_{1},y_{2},\lambda_{1},\lambda_{2},\lambda}$, with

$N_{c}>1\;\;\;,\;\;\;N_{f_{0,1,2}}\geq 0\;\;\;,\;\;N_{f_{1}}>0\;\;{\rm
or}\;\;N_{f_{2}}>0$ (B.17)

and $N_{f_{0}}+N_{f_{1}}+N_{f_{2}}$ in the interval (B.9).

Once this is obtained one can compute the thermal mass for the scalar scalars:

$m_{i}^{2}(T)=(4\pi)^{2}\frac{T^{2}}{12\log{T}}\left(3\tilde{\alpha}_{\lambda_{i}}-\tilde{\alpha}_{\lambda}+2N_{c}N_{f_{i}}\tilde{\alpha}_{y_{i}}\right)$
(B.18)

It turns out that there are 1784 inequivalent (we do not count those obtained
by $N_{f_{1}}\leftrightarrow N_{f_{2}}$) choices of colours and flavours which
satisfy (B.17) and (B.9). However we are not only looking for fixed flow
solutions, what we also need is that they lead to a negative thermal mass.

We will now prove in general that there are no solutions with symmetry non-
restoration.

Let it be $m_{1}^{2}(T)<0$. To be so one needs

$\tilde{\alpha}_{\lambda}-2N_{c}N_{f_{1}}\tilde{\alpha}_{y_{1}}>3\tilde{\alpha}_{\lambda_{1}}>0$
(B.19)

We can now rewrite (B.14) as

$2\left(\tilde{\alpha}_{\lambda}^{2}-4N_{c}N_{f_{1}}\tilde{\alpha}_{y_{1}}^{2}\right)+\tilde{\alpha}_{\lambda_{1}}+18\tilde{\alpha}_{\lambda_{1}}^{2}+8N_{c}N_{f_{1}}\tilde{\alpha}_{y_{1}}\tilde{\alpha}_{\lambda_{1}}=0$
(B.20)

All the terms except the first one are manifestly positive, so to satisfy the
equation, the first term should be negative. However, the first term can be
rewritten as

$\tilde{\alpha}_{\lambda}^{2}-4N_{c}N_{f_{1}}\tilde{\alpha}_{y_{1}}^{2}=\left(\tilde{\alpha}_{\lambda}-2N_{c}N_{f_{1}}\tilde{\alpha}_{y_{1}}\right)\left(\tilde{\alpha}_{\lambda}+2N_{c}N_{f_{1}}\tilde{\alpha}_{y_{1}}\right)+4N_{c}N_{f_{1}}\left(N_{c}N_{f_{1}}-1\right)\tilde{\alpha}_{y_{1}}^{2}$
(B.21)

This is positive, since the last term is non-negative, while the first product
is positive due to (B.19). Equation (B.20) thus cannot have a solution.

We conclude the section summarising the result for the model presented: there
is no fixed flow solution once a negative thermal mass is assumed.

##  Appendix C Gauged scalars

We consider in this appendix various examples of scalars in non-trivial
representations of the gauge group.

###  C.1  $SU(2)$ with two scalar triplets

First we take the two scalar fields as gauge SU(2) triplets, coupled each to
one fermion SU(2) doublet ($N_{f_{1}}=N_{f_{2}}=1$). To use almost all of the
old results we still keep the $Z_{2}\times Z_{2}$ discrete symmetry. There is
now an extra quartic term:

$V=\frac{\lambda_{1}}{4}\left(\vec{\varphi}_{1}\cdot\vec{\varphi}_{1}\right)^{2}+\frac{\lambda_{2}}{4}\left(\vec{\varphi}_{2}\cdot\vec{\varphi}_{2}\right)^{2}-\frac{\lambda_{11}}{2}\left(\vec{\varphi}_{1}\cdot\vec{\varphi}_{1}\right)\left(\vec{\varphi}_{2}\cdot\vec{\varphi}_{2}\right)-\frac{\lambda_{12}}{2}\left(\vec{\varphi}_{1}\cdot\vec{\varphi}_{2}\right)^{2}$
(C.1)

Denoting

$\phi=(\vec{\varphi_{1}},\vec{\varphi_{2}})$ (C.2)

we compute the quartic couplings directly from the definition

$V=\frac{\lambda_{abcd}}{4!}\phi^{a}\phi^{b}\phi^{c}\phi^{d}$ (C.3)

i.e.

$\lambda_{abcd}=\frac{\partial^{4}V}{\partial\phi^{a}\partial\phi^{b}\partial\phi^{c}\partial\phi^{d}}\;\;\;,\;\;\;a,b,c,d=1,\ldots,6$
(C.4)

For the Yukawa term we take

${\cal
L}_{Yukawa}=\sum_{i=1}^{2}y_{i}\bar{\psi}_{i}\left(\frac{\vec{\tau}}{2}\cdot\vec{\varphi}_{i}\right)\psi_{i}$
(C.5)

with $\tau^{A}$, $A=1,2,3$ the Pauli matrices.

The (reducible) generators for the fermions (two fundamental representations
of SU(2))

$\Psi=(\psi_{1},\psi_{2})$ (C.6)

are

$T^{A}=\frac{1}{2}\left(\begin{array}[]{cc}\tau^{A}&0\\\
0&\tau^{A}\end{array}\right)\;\;\;,\;\;\;A=1,2,3$ (C.7)

The fixed flow RGE are

$\displaystyle\tilde{\alpha}_{g}$ $\displaystyle=$ $\displaystyle
2b_{0}\tilde{\alpha}_{g}^{2}$ (C.8) $\displaystyle-\tilde{\alpha}_{y_{i}}$
$\displaystyle=$
$\displaystyle\frac{5}{2}\tilde{\alpha}_{y_{i}}^{2}-9\tilde{\alpha}_{g}\tilde{\alpha}_{y_{i}}$
(C.9) $\displaystyle-\tilde{\alpha}_{\lambda_{i}}$ $\displaystyle=$
$\displaystyle+22\tilde{\alpha}_{\lambda_{i}}^{2}+6\tilde{\alpha}_{\lambda_{11}}^{2}+4\tilde{\alpha}_{\lambda_{11}}\tilde{\alpha}_{\lambda_{12}}+2\tilde{\alpha}_{\lambda_{12}}^{2}-\tilde{\alpha}_{y_{i}}^{2}+4\tilde{\alpha}_{\lambda_{i}}\tilde{\alpha}_{y_{i}}$
(C.10)
$\displaystyle+12\tilde{\alpha}_{g}^{2}-24\tilde{\alpha}_{g}\tilde{\alpha}_{\lambda_{i}}$
$\displaystyle\tilde{\alpha}_{\lambda_{11}}$ $\displaystyle=$ $\displaystyle
6\tilde{\alpha}_{g}^{2}+24\tilde{\alpha}_{g}\tilde{\alpha}_{\lambda_{11}}+8\tilde{\alpha}_{\lambda_{11}}^{2}-10\tilde{\alpha}_{\lambda_{11}}(\tilde{\alpha}_{\lambda_{1}}+\tilde{\alpha}_{\lambda_{2}})$
(C.11)
$\displaystyle+2\tilde{\alpha}_{\lambda_{12}}^{2}-2\tilde{\alpha}_{\lambda_{12}}(\tilde{\alpha}_{\lambda_{1}}+\tilde{\alpha}_{\lambda_{2}})-2\tilde{\alpha}_{\lambda_{11}}(\tilde{\alpha}_{y_{1}}+\tilde{\alpha}_{y_{2}})$
$\displaystyle\tilde{\alpha}_{\lambda_{12}}$ $\displaystyle=$ $\displaystyle
6\tilde{\alpha}_{g}^{2}+24\tilde{\alpha}_{g}\tilde{\alpha}_{\lambda_{12}}+16\tilde{\alpha}_{\lambda_{11}}\tilde{\alpha}_{\lambda_{12}}$
(C.12)
$\displaystyle+10\tilde{\alpha}_{\lambda_{12}}^{2}-4\tilde{\alpha}_{\lambda_{12}}(\tilde{\alpha}_{\lambda_{1}}+\tilde{\alpha}_{\lambda_{2}})-2\tilde{\alpha}_{\lambda_{12}}(\tilde{\alpha}_{y_{1}}+\tilde{\alpha}_{y_{2}})$

The thermal mass square results

$m_{i}^{2}(T)=(4\pi)^{2}\frac{T^{2}}{12\log{T}}\left(\tilde{\alpha}_{y_{i}}+5\tilde{\alpha}_{\lambda_{i}}+6\tilde{\alpha}_{g}-3\tilde{\alpha}_{\lambda_{11}}-\tilde{\alpha}_{\lambda_{12}}\right)$
(C.13)

The gauge beta function is known,

$b_{0}=\frac{22}{3}-\frac{2}{3}\left(N_{f_{0}}+2\right)-\frac{2}{3}=\frac{16-2N_{f_{0}}}{3}\to\tilde{\alpha}_{g}=\frac{3}{4(8-N_{f_{0}})}$
(C.14)

from where, to get $\tilde{\alpha}_{g}>1/9$, see (C.9), we need

$2\leq N_{f_{0}}<8$ (C.15)

By explicit search on can find that there are no solutions of the fixed flow
RGE for positive $\tilde{\alpha}_{g}$, $\tilde{\alpha}_{y_{1,2}}$,
$\tilde{\alpha}_{\lambda_{1,2}}$ and real $\tilde{\alpha}_{\lambda_{11,12}}$.

###  C.2  $SU(2)$ with one scalar singlet and one scalar triplet

We take now one adjoint scalar and one singlet scalar that couple to the
fermions (again in the fundamental representation, $N_{f_{1}}=N_{f_{2}}=1$)
with the following Yukawa term:

${\cal
L}_{Yuk}=y_{1}\bar{\psi}_{1}{\phi}_{1}\psi_{1}+y_{2}\bar{\psi}_{2}\left(\frac{\vec{\tau}}{2}\cdot\vec{\varphi}_{2}\right)\psi_{2}\
.$ (C.16)

Now the first scalar is singlet, the second is triplet. Obviously
$\lambda_{12}$ cannot appear now. We will again call the remaining mixed
constant $\lambda_{11}=\lambda$ in this section.

The fixed flow RGE are now

$\displaystyle\tilde{\alpha}_{g}$ $\displaystyle=$ $\displaystyle
2b_{0}\tilde{\alpha}_{g}^{2}$ (C.17) $\displaystyle-\tilde{\alpha}_{y_{1}}$
$\displaystyle=$ $\displaystyle
14\tilde{\alpha}_{y_{1}}^{2}-9\tilde{\alpha}_{g}\tilde{\alpha}_{y_{1}}$ (C.18)
$\displaystyle-\tilde{\alpha}_{y_{2}}$ $\displaystyle=$
$\displaystyle\frac{5}{2}\tilde{\alpha}_{y_{2}}^{2}-9\tilde{\alpha}_{g}\tilde{\alpha}_{y_{2}}$
(C.19) $\displaystyle-\tilde{\alpha}_{\lambda_{1}}$ $\displaystyle=$
$\displaystyle
18\tilde{\alpha}_{\lambda_{1}}^{2}+6\tilde{\alpha}_{\lambda}^{2}-16\tilde{\alpha}_{y_{1}}^{2}+16\tilde{\alpha}_{\lambda_{1}}\tilde{\alpha}_{y_{1}}$
(C.20) $\displaystyle-\tilde{\alpha}_{\lambda_{2}}$ $\displaystyle=$
$\displaystyle
12\tilde{\alpha}_{g}^{2}-24\tilde{\alpha}_{g}\tilde{\alpha}_{\lambda_{2}}+2\tilde{\alpha}_{\lambda}^{2}+22\tilde{\alpha}_{\lambda_{2}}^{2}-\tilde{\alpha}_{y_{2}}^{2}+4\tilde{\alpha}_{\lambda_{2}}\tilde{\alpha}_{y_{2}}$
(C.21) $\displaystyle\tilde{\alpha}_{\lambda}$ $\displaystyle=$ $\displaystyle
12\tilde{\alpha}_{g}\tilde{\alpha}_{\lambda}-6\tilde{\alpha}_{\lambda_{1}}\tilde{\alpha}_{\lambda}+8\tilde{\alpha}_{\lambda}^{2}-10\tilde{\alpha}_{\lambda}\tilde{\alpha}_{\lambda_{2}}-8\tilde{\alpha}_{\lambda}\left(\tilde{\alpha}_{y_{1}}+\frac{1}{4}\tilde{\alpha}_{y_{2}}\right)$
(C.22)

while the thermal masses are

$\displaystyle m_{1}^{2}(T)$ $\displaystyle=$
$\displaystyle(4\pi)^{2}\frac{T^{2}}{12\log{T}}\left(4\tilde{\alpha}_{y_{1}}+3\tilde{\alpha}_{\lambda_{1}}-3\tilde{\alpha}_{\lambda}\right)$
(C.23) $\displaystyle m_{2}^{2}(T)$ $\displaystyle=$
$\displaystyle(4\pi)^{2}\frac{T^{2}}{12\log{T}}\left(\tilde{\alpha}_{y_{2}}+5\tilde{\alpha}_{\lambda_{2}}+6\tilde{\alpha}_{g}-\tilde{\alpha}_{\lambda}\right)$
(C.24)

The gauge beta function is

$b_{0}=\frac{22}{3}-\frac{2}{3}\left(N_{f_{0}}+2\right)-\frac{1}{3}=\frac{17-2N_{f_{0}}}{3}\to\tilde{\alpha}_{g}=\frac{3}{2(17-2N_{f_{0}})}$
(C.25)

from where, to get $\tilde{\alpha}_{g}>1/9$, see (C.18) or (C.19), we need

$2\leq N_{f_{0}}\leq 8$ (C.26)

We find only two solutions:

$\displaystyle N_{f_{0}}=8$
$\displaystyle:\left(\tilde{\alpha}_{g},\tilde{\alpha}_{y_{1}},\tilde{\alpha}_{y_{2}},\tilde{\alpha}_{\lambda_{1}},\tilde{\alpha}_{\lambda_{2}},\tilde{\alpha}_{\lambda}\right)=\left(1.5,0.893,5.0,0.518,0.182,0\right)$
(C.27) $\displaystyle N_{f_{0}}=8$
$\displaystyle:\left(\tilde{\alpha}_{g},\tilde{\alpha}_{y_{1}},\tilde{\alpha}_{y_{2}},\tilde{\alpha}_{\lambda_{1}},\tilde{\alpha}_{\lambda_{2}},\tilde{\alpha}_{\lambda}\right)=\left(1.5,0.893,5.0,0.518,0.5,0\right)$
(C.28)

Since both have $\tilde{\alpha}_{\lambda}=0$, symmetry is always restored at
high enough $T$.

###  C.3 $SU(N_{c})$ with two scalar fundamentals

The potential is

$\displaystyle V$ $\displaystyle=$
$\displaystyle\frac{\lambda_{1}}{2}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)^{2}+\frac{\lambda_{2}}{2}\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)^{2}+\lambda_{3}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)$
(C.29) $\displaystyle+$
$\displaystyle\lambda_{4}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{2}\right)\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{1}\right)+\frac{\lambda_{5}}{2}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{2}\right)^{2}+\frac{\lambda_{5}^{*}}{2}\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{1}\right)^{2}$

with $\lambda_{1,2,3,4}$ real and in general $\lambda_{5}$ complex.

The relation between the complex and the real basis is as usual

$\varphi_{\alpha k}=\frac{1}{\sqrt{2}}\left(R_{\alpha k}+iI_{\alpha
k}\right)\quad,\quad\alpha=1,2\quad,\quad k=1,\ldots,N_{c}$ (C.30)

so that

$\phi^{a}=\left(R_{1}^{k},I_{1}^{k},R_{2}^{k},I_{2}^{k}\right)^{T}$ (C.31)

We get

$\displaystyle V_{\Lambda^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\frac{\partial^{2}V}{\partial\phi^{a}\partial\phi^{b}}\frac{\partial^{2}V}{\partial\phi^{b}\partial\phi^{a}}$
$\displaystyle=$
$\displaystyle\sum_{\alpha,\beta=1}^{2}\sum_{k,l=1}^{N_{c}}\left(\frac{\partial^{2}V}{\partial{\varphi_{\alpha}}^{k}\partial\varphi_{\beta
l}^{*}}\frac{\partial^{2}V}{{\partial\varphi_{\beta}}^{l}\partial\varphi_{\alpha
k}^{*}}+\frac{\partial^{2}V}{\partial{\varphi_{\alpha}}^{k}\partial{\varphi_{\beta}}^{l}}\frac{\partial^{2}V}{{\partial\varphi_{\beta
l}}^{*}\partial\varphi_{\alpha k}^{*}}\right)$ $\displaystyle=$ $\displaystyle
Tr\left(M_{1}M_{1}^{\dagger}+2M_{2}M_{2}^{\dagger}+M_{3}M_{3}^{\dagger}+N_{1}N_{1}^{\dagger}+2N_{2}N_{2}^{\dagger}+N_{3}N_{3}^{\dagger}\right)$

with

$\displaystyle M_{1}$ $\displaystyle=$
$\displaystyle\left(\lambda_{1}\left({\vec{\varphi}_{1}}^{*}\cdot\vec{\varphi}_{1}\right)+\lambda_{3}\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)\right)\mathbbm{1}+\lambda_{1}\vec{\varphi}_{1}^{*}\otimes\vec{\varphi}_{1}+\lambda_{4}\vec{\varphi}_{2}^{*}\otimes\vec{\varphi}_{2}$
(C.33) $\displaystyle M_{2}$ $\displaystyle=$
$\displaystyle\left(\lambda_{4}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{2}\right)+\lambda_{5}^{*}\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{1}\right)\right)\mathbbm{1}+\lambda_{3}\vec{\varphi}_{1}^{*}\otimes\vec{\varphi}_{2}+\lambda_{5}^{*}\vec{\varphi}_{2}^{*}\otimes\vec{\varphi}_{1}$
(C.34) $\displaystyle M_{3}$ $\displaystyle=$
$\displaystyle\left(\lambda_{2}\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)+\lambda_{3}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)\right)\mathbbm{1}+\lambda_{4}\vec{\varphi}_{1}^{*}\otimes\vec{\varphi}_{1}+\lambda_{2}\vec{\varphi}_{2}^{*}\otimes\vec{\varphi}_{2}$
(C.35) $\displaystyle N_{1}$ $\displaystyle=$
$\displaystyle\lambda_{1}\vec{\varphi}_{1}\otimes\vec{\varphi}_{1}+\lambda_{5}\vec{\varphi}_{2}\otimes\vec{\varphi}_{2}$
(C.36) $\displaystyle N_{2}$ $\displaystyle=$
$\displaystyle\lambda_{3}\vec{\varphi}_{1}\otimes\vec{\varphi}_{2}+\lambda_{4}\vec{\varphi}_{2}\otimes\vec{\varphi}_{1}$
(C.37) $\displaystyle N_{3}$ $\displaystyle=$
$\displaystyle\lambda_{2}\vec{\varphi}_{2}\otimes\vec{\varphi}_{2}+\lambda_{5}^{*}\vec{\varphi}_{1}\otimes\vec{\varphi}_{1}$
(C.38)

This gives

$\displaystyle V_{\Lambda^{2}}$
$\displaystyle=\left(\left(2N_{c}+8\right)\lambda_{1}^{2}+2N_{c}\lambda_{3}^{2}+4\lambda_{3}\lambda_{4}+2\lambda_{4}^{2}+2\left|\lambda_{5}\right|^{2}\right)\frac{1}{2}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)^{2}$
$\displaystyle+\left(\left(2N_{c}+8\right)\lambda_{2}^{2}+2N_{c}\lambda_{3}^{2}+4\lambda_{3}\lambda_{4}+2\lambda_{4}^{2}+2\left|\lambda_{5}\right|^{2}\right)\frac{1}{2}\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)^{2}$
$\displaystyle+\left(2\left(N_{c}+1\right)\left(\lambda_{1}+\lambda_{2}\right)\lambda_{3}+4\lambda_{3}^{2}+2\left(\lambda_{1}+\lambda_{2}\right)\lambda_{4}+2\lambda_{4}^{2}+2\left|\lambda_{5}\right|^{2}\right)\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)$
$\displaystyle+\left(2\left(\lambda_{1}+\lambda_{2}\right)\lambda_{4}+8\lambda_{3}\lambda_{4}+2N_{c}\lambda_{4}^{2}+\left(4+2N_{c}\right)\left|\lambda_{5}\right|^{2}\right)\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{2}\right)\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{1}\right)$
$\displaystyle+2\left(\lambda_{1}+\lambda_{2}+4\lambda_{3}+2\left(N_{c}+1\right)\lambda_{4}\right)\lambda_{5}\frac{1}{2}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{2}\right)^{2}$
$\displaystyle+2\left(\lambda_{1}+\lambda_{2}+4\lambda_{3}+2\left(N_{c}+1\right)\lambda_{4}\right)\lambda_{5}^{*}\frac{1}{2}\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{1}\right)^{2}$
(C.39)

We easily find

$\displaystyle V_{\Lambda^{S}}$ $\displaystyle=$
$\displaystyle-3g^{2}\phi^{a}\left(T^{A}(S)T^{A}(S)\right)_{ab}\frac{\partial
V}{\partial\phi^{a}}$ (C.40) $\displaystyle=$
$\displaystyle-3\frac{N_{c}^{2}-1}{2N_{c}}g^{2}\left(\varphi_{\alpha}^{k}\frac{\partial
V}{\partial\varphi_{\alpha}^{k}}+\varphi_{\alpha k}^{*}\frac{\partial
V}{\partial\varphi_{\alpha k}^{*}}\right)=-6\frac{N_{c}^{2}-1}{N_{c}}g^{2}V$

Using

$\phi^{a}\left\\{T^{A}(S),T^{B}(S)\right\\}_{ab}\phi^{b}=2g^{2}\varphi^{*}_{\alpha
a}{\left\\{T^{A},T^{B}\right\\}^{a}}_{b}\varphi_{\alpha}^{b}$ (C.41)

and the usual

${\left(T^{A}\right)^{a}}_{b}{\left(T^{A}\right)^{c}}_{d}=\frac{1}{2}\left({\delta^{a}}_{d}{\delta^{c}}_{b}-\frac{1}{N_{c}}{\delta^{a}}_{b}{\delta^{c}}_{d}\right)$
(C.42)

we get

$\displaystyle V_{A}$ $\displaystyle=$
$\displaystyle\frac{3}{4}g^{4}\frac{N_{c}^{2}+2}{N_{c}^{2}}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}+\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)^{2}$
$\displaystyle+$
$\displaystyle\frac{3}{4}g^{4}\frac{N_{c}^{2}-4}{N_{c}}\left(\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)^{2}+2\left(\vec{\varphi}_{1}^{*}\vec{\varphi}_{2}\right)\left(\vec{\varphi}_{2}^{*}\vec{\varphi}_{1}\right)+\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{3}{4}g^{4}\frac{N_{c}^{3}+N_{c}^{2}-4N_{c}+2}{N_{c}^{2}}\left(\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)^{2}+\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)^{2}\right)$
$\displaystyle+$
$\displaystyle\frac{3}{2}g^{4}\frac{N_{c}^{2}+2}{N_{c}^{2}}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)+\frac{3}{2}g^{4}\frac{N_{c}^{2}-4}{N_{c}}\left(\vec{\varphi}_{1}^{*}\vec{\varphi}_{2}\right)\left(\vec{\varphi}_{2}^{*}\vec{\varphi}_{1}\right)$

By taking

$g^{2}=\frac{16\pi^{2}\tilde{\alpha}}{N_{c}t}\quad,\quad\lambda_{i}=\frac{16\pi^{2}\tilde{\lambda}_{i}}{N_{c}t}$
(C.44)

we get for constant $\alpha,\lambda_{i}$ in the large $N_{c}$ limit the
following fixed flow RGEs:

$\displaystyle-\tilde{\lambda}_{1}$ $\displaystyle=$ $\displaystyle
2\tilde{\lambda}_{1}^{2}+2\tilde{\lambda}_{3}^{2}-6\tilde{\alpha}\tilde{\lambda}_{1}+\frac{3}{2}\tilde{\alpha}^{2}$
(C.45) $\displaystyle-\tilde{\lambda}_{2}$ $\displaystyle=$ $\displaystyle
2\tilde{\lambda}_{2}^{2}+2\tilde{\lambda}_{3}^{2}-6\tilde{\alpha}\tilde{\lambda}_{2}+\frac{3}{2}\tilde{\alpha}^{2}$
(C.46) $\displaystyle-\tilde{\lambda}_{3}$ $\displaystyle=$ $\displaystyle
2\left(\tilde{\lambda}_{1}+\tilde{\lambda}_{2}\right)\tilde{\lambda}_{3}-6\tilde{\alpha}\tilde{\lambda}_{3}$
(C.47) $\displaystyle-\tilde{\lambda}_{4}$ $\displaystyle=$ $\displaystyle
2\tilde{\lambda}_{4}^{2}+2\left|\tilde{\lambda}_{5}\right|^{2}-6\tilde{\alpha}\tilde{\lambda}_{4}+\frac{3}{2}\tilde{\alpha}^{2}$
(C.48) $\displaystyle-\tilde{\lambda}_{5}$ $\displaystyle=$ $\displaystyle
4\tilde{\lambda}_{4}\tilde{\lambda}_{5}-6\tilde{\alpha}\tilde{\lambda}_{5}$
(C.49)

The thermal potential is

$\displaystyle\Delta V_{T}$ $\displaystyle=$
$\displaystyle\frac{T^{2}}{48}\left(2\frac{\partial^{2}V}{\partial\phi^{a}\partial\phi^{a}}+6\phi^{a}\left(T^{A}(S)T^{A}(S)\right)_{ab}\phi^{b}\right)$
$\displaystyle=$
$\displaystyle\frac{T^{2}}{48}\sum_{i=1}^{2}\sum_{a=1}^{N_{c}}\left(4\frac{\partial^{2}V}{\partial\varphi_{i}^{a}\partial\varphi_{ia}^{*}}+6g^{2}\frac{N_{c}^{2}-1}{N_{c}}\varphi_{i}^{a}\varphi_{ia}^{*}\right)$

Using (C.33) and (C.35)

$\displaystyle\sum_{i=1}^{2}\sum_{a=1}^{N_{c}}\frac{\partial^{2}V}{\partial\varphi_{i}^{a}\partial\varphi_{ia}^{*}}$
$\displaystyle=$ $\displaystyle Tr\left(M_{1}+M_{3}\right)$ (C.51)
$\displaystyle=$
$\displaystyle\left(\left(N_{c}+1\right)\lambda_{1}+N_{c}\lambda_{3}+\lambda_{4}\right)\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)$
$\displaystyle+$
$\displaystyle\left(\left(N_{c}+1\right)\lambda_{2}+N_{c}\lambda_{3}+\lambda_{4}\right)\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)$

At large $N_{c}$

$\Delta
V_{T}=(4\pi)^{2}\frac{T^{2}}{24\log{T}}\left(\left(2\left(\tilde{\lambda}_{1}+\tilde{\lambda}_{3}\right)+3\tilde{\alpha}\right)\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)+\left(2\left(\tilde{\lambda}_{2}+\tilde{\lambda}_{3}\right)+3\tilde{\alpha}\right)\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)\right)$
(C.52)

###  C.4  $SU(N_{c_{1}})\times SU(N_{c_{2}})$ with two scalar fundamentals

The model we will study now is similar to the previous one, but now we have
two simple groups, $SU(N_{c_{1}})\times SU(N_{c_{2}})$, so that each
$\varphi_{i}$ is in a fundamentals representation of its $SU(N_{ci})$ and a
singlet under the other one. The most general potential is

$V=\frac{\lambda_{1}}{2}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)^{2}+\frac{\lambda_{2}}{2}\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)^{2}-\lambda\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)$
(C.53)

As before we derive the various pieces of the RGE using (A.20), (A.23),
(A.24):

$\displaystyle V_{\Lambda^{2}}$ $\displaystyle=$
$\displaystyle\left(\left(2N_{c1}+8\right)\lambda_{1}^{2}+2N_{c2}\lambda^{2}\right)\frac{1}{2}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)^{2}$
$\displaystyle+\left(\left(2N_{c2}+8\right)\lambda_{2}^{2}+2N_{c1}\lambda^{2}\right)\frac{1}{2}\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)^{2}$
$\displaystyle-\left(2\left(N_{c1}\lambda_{1}+N_{c2}\lambda_{2}\right)\lambda+2\left(\lambda_{1}+\lambda_{2}\right)\lambda-4\lambda^{2}\right)\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)$
$\displaystyle V_{\Lambda^{S}}$ $\displaystyle=$
$\displaystyle-6\frac{N_{c1}^{2}-1}{N_{c1}}g_{1}^{2}\left(\frac{\lambda_{1}}{2}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)^{2}-\frac{\lambda}{2}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)\right)$
(C.55)
$\displaystyle-6\frac{N_{c2}^{2}-1}{N_{c2}}g_{2}^{2}\left(\frac{\lambda_{2}}{2}\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)^{2}-\frac{\lambda}{2}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)\right)$
$\displaystyle V_{A}$ $\displaystyle=$
$\displaystyle\frac{3}{4}g_{1}^{4}\frac{N_{c1}^{3}+N_{c1}^{2}-4N_{c1}+2}{N_{c1}^{2}}\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)^{2}$
(C.56)
$\displaystyle+\frac{3}{4}g_{2}^{4}\frac{N_{c2}^{3}+N_{c2}^{2}-4N_{c2}+2}{N_{c2}^{2}}\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)^{2}$

Defining

$\displaystyle i=1,2$ $\displaystyle:$ $\displaystyle
g_{i}^{2}=\frac{16\pi^{2}\tilde{\alpha}_{i}}{N_{ci}t}\quad,\quad\lambda_{i}=\frac{16\pi^{2}\tilde{\lambda}_{i}}{N_{ci}t}$
(C.57)
$\displaystyle\lambda=\frac{16\pi^{2}\tilde{\lambda}}{\sqrt{N_{c1}N_{c2}}t}$
(C.58)

with constant we get for the RG equations at large $N_{ci}$

$\displaystyle-\tilde{\lambda}_{1}$ $\displaystyle=$ $\displaystyle
2\tilde{\lambda}_{1}^{2}+2\tilde{\lambda}^{2}-6\tilde{\alpha}_{1}\tilde{\lambda}_{1}+\frac{3}{2}\tilde{\alpha}_{1}^{2}$
(C.59) $\displaystyle-\tilde{\lambda}_{2}$ $\displaystyle=$ $\displaystyle
2\tilde{\lambda}_{2}^{2}+2\tilde{\lambda}^{2}-6\tilde{\alpha}_{2}\tilde{\lambda}_{2}+\frac{3}{2}\tilde{\alpha}_{2}^{2}$
(C.60) $\displaystyle-\tilde{\lambda}$ $\displaystyle=$ $\displaystyle
2\left(\tilde{\lambda}_{1}+\tilde{\lambda}_{2}\right)\tilde{\lambda}-3\left(\tilde{\alpha}_{1}+\tilde{\alpha}_{2}\right)\tilde{\lambda}$
(C.61)

The thermal effective potential

$\Delta
V_{T}=\frac{T^{2}}{48}\sum_{i=1}^{2}\sum_{a=1}^{N_{c}}\left(4\frac{\partial^{2}V}{\partial\varphi_{i}^{a}\partial\varphi_{ia}^{*}}+6g_{i}^{2}\frac{N_{ci}^{2}-1}{N_{ci}}\varphi_{i}^{a}\varphi_{ia}^{*}\right)\\\
$

becomes at large $N_{ci}$

$\displaystyle\Delta V_{T}=(4\pi)^{2}\frac{T^{2}}{24\log{T}}$
$\displaystyle\left(\left(2\left(\tilde{\lambda}_{1}-\sqrt{\frac{N_{c2}}{N_{c1}}}\tilde{\lambda}\right)+3\tilde{\alpha}_{1}\right)\left(\vec{\varphi}_{1}^{*}\cdot\vec{\varphi}_{1}\right)\right.$
$\displaystyle\left.+\left(2\left(\tilde{\lambda}_{2}-\sqrt{\frac{N_{c1}}{N_{c2}}}\tilde{\lambda}\right)+3\tilde{\alpha}_{2}\right)\left(\vec{\varphi}_{2}^{*}\cdot\vec{\varphi}_{2}\right)\right)$
(C.62)

###  C.5  $SU(N_{c_{1}})\times SU(N_{c_{2}})$ with two scalar adjoints

We now present a model again with two simple gauge groups,
$SU(N_{c_{1}})\times SU(N_{c_{2}})$, and one adjoint for each gauge group. The
potential is parametrised by

$V=\frac{\lambda_{1}^{\prime}}{4}Tr\Sigma_{1}^{4}+\frac{\lambda_{2}^{\prime}}{4}Tr\Sigma_{2}^{4}+\frac{\lambda_{1}}{4}\left(Tr\Sigma_{1}^{2}\right)^{2}+\frac{\lambda_{2}}{4}\left(Tr\Sigma_{2}^{2}\right)^{2}-\frac{\lambda}{2}Tr\Sigma_{1}^{2}Tr\Sigma_{2}^{2}$
(C.63)

The 1-loop corrections are (A.20), (A.23), (A.24):

$\displaystyle V_{\Lambda^{2}}$
$\displaystyle=\frac{1}{8}\left(Tr\Sigma_{1}^{4}\left(12\lambda_{1}\lambda_{1}^{\prime}+\lambda_{1}^{\prime
2}\frac{2N_{c_{1}}^{2}-18}{N_{c_{1}}}\right)-Tr\Sigma_{1}^{2}Tr\Sigma_{2}^{2}\left(\lambda_{1}\lambda\left(2N_{c_{1}}^{2}+2\right)+\lambda\lambda_{1}^{\prime}\frac{4N_{c_{1}}^{2}-6}{N_{c_{1}}}\right)\right.$
$\displaystyle\left.+\left(Tr\Sigma_{1}^{2}\right)^{2}\left(\lambda_{1}^{2}\left(N_{c_{1}}^{2}+7\right)+\lambda_{1}\lambda_{1}^{\prime}\frac{4N_{c_{1}}^{2}-6}{N_{c_{1}}}+\lambda_{1}^{\prime
2}\frac{3N_{c_{1}}^{2}+9}{N_{c_{1}}^{2}}\right)+\left(Tr\Sigma_{2}^{2}\right)^{2}\lambda^{2}\left(N_{c_{1}}^{2}-1\right)\right)$
$\displaystyle+Tr\Sigma_{1}^{2}Tr\Sigma_{2}^{2}\lambda^{2}$ (C.64)
$\displaystyle+\frac{1}{8}\left(Tr\Sigma_{2}^{4}\left(12\lambda_{2}\lambda_{2}^{\prime}+\lambda_{2}^{\prime
2}\frac{2N_{c_{2}}^{2}-18}{N_{c_{2}}}\right)-Tr\Sigma_{1}^{2}Tr\Sigma_{2}^{2}\left(\lambda_{2}\lambda\left(2N_{c_{2}}^{2}+2\right)+\lambda\lambda_{2}^{\prime}\frac{4N_{c_{2}}^{2}-6}{N_{c_{2}}}\right)\right.$
$\displaystyle\left.+\left(Tr\Sigma_{2}^{2}\right)^{2}\left(\lambda_{2}^{2}\left(N_{c_{2}}^{2}+7\right)+\lambda_{2}\lambda_{2}^{\prime}\frac{4N_{c_{2}}^{2}-6}{N_{c_{2}}}+\lambda_{2}^{\prime
2}\frac{3N_{c_{2}}^{2}+9}{N_{c_{2}}^{2}}\right)+\left(Tr\Sigma_{1}^{2}\right)^{2}\lambda^{2}\left(N_{c_{2}}^{2}-1\right)\right)$

$\displaystyle V_{\Lambda^{S}}=-$ $\displaystyle
3g_{1}^{2}N_{c_{1}}\left(\lambda_{1}^{\prime}Tr\Sigma_{1}^{4}+\lambda_{1}\left(Tr\Sigma_{1}^{2}\right)^{2}-\lambda
Tr\Sigma_{1}^{2}Tr\Sigma_{2}^{2}\right)$ $\displaystyle-$ $\displaystyle
3g_{2}^{2}N_{c_{2}}\left(\lambda_{2}^{\prime}Tr\Sigma_{2}^{4}+\lambda_{2}\left(Tr\Sigma_{2}^{2}\right)^{2}-\lambda
Tr\Sigma_{1}^{2}Tr\Sigma_{2}^{2}\right)$ (C.65)

$V_{\Lambda_{A}}=3g_{1}^{4}\left(N_{c_{1}}Tr\Sigma_{1}^{4}+3\left(Tr\Sigma_{1}^{2}\right)^{2}\right)+3g_{2}^{4}\left(N_{c_{2}}Tr\Sigma_{2}^{4}+3\left(Tr\Sigma_{2}^{2}\right)^{2}\right)$
(C.66)

We redefine the constants as

$\displaystyle\lambda^{\prime}_{1,2}=(4\pi)^{2}\frac{\tilde{\lambda}^{\prime}_{1,2}}{N_{c_{1,2}}}\times\frac{1}{t}$
,
$\displaystyle\lambda_{1,2}=(4\pi)^{2}\frac{\tilde{\lambda}_{1,2}}{N_{c_{1,2}}^{2}}\times\frac{1}{t}$
(C.67)
$\displaystyle\lambda=(4\pi)^{2}\frac{\tilde{\lambda}}{N_{c_{1}}N_{c_{2}}}\times\frac{1}{t}$
, $\displaystyle
g_{1,2}^{2}=(4\pi)^{2}\frac{\tilde{\alpha}_{1,2}}{N_{c_{1,2}}}\times\frac{1}{t}$
(C.68)

with all tilded quantities constants, and eventually took the large
$N_{c_{1,2}}$ limit.

In the Veneziano limit the RGE are

$\displaystyle-\tilde{\lambda}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\tilde{\lambda}_{1}^{2}+2\tilde{\lambda}_{1}\tilde{\lambda}_{1}^{\prime}+\frac{3}{2}\tilde{\lambda}_{1}^{\prime
2}+\frac{1}{2}\tilde{\lambda}^{2}-12\tilde{\alpha}_{1}\tilde{\lambda}_{1}+36\tilde{\alpha}_{1}^{2}$
(C.69) $\displaystyle-\tilde{\lambda}_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\tilde{\lambda}_{2}^{2}+2\tilde{\lambda}_{2}\tilde{\lambda}_{2}^{\prime}+\frac{3}{2}\tilde{\lambda}_{2}^{\prime
2}+\frac{1}{2}\tilde{\lambda}^{2}-12\tilde{\alpha}_{2}\tilde{\lambda}_{2}+36\tilde{\alpha}_{2}^{2}$
(C.70) $\displaystyle-\tilde{\lambda}$ $\displaystyle=$
$\displaystyle\tilde{\lambda}\left(\frac{1}{2}\left(\tilde{\lambda}_{1}+\tilde{\lambda}_{2}\right)+\tilde{\lambda}_{1}^{\prime}+\tilde{\lambda}_{2}^{\prime}-6\left(\tilde{\alpha}_{1}+\tilde{\alpha}_{2}\right)\right)$
(C.71) $\displaystyle-\tilde{\lambda}_{1}^{\prime}$ $\displaystyle=$
$\displaystyle\tilde{\lambda}_{1}^{\prime
2}-12\tilde{\alpha}_{1}\tilde{\lambda}_{1}^{\prime}+12\tilde{\alpha}_{1}^{2}$
(C.72) $\displaystyle-\tilde{\lambda}_{2}^{\prime}$ $\displaystyle=$
$\displaystyle\tilde{\lambda}_{2}^{\prime
2}-12\tilde{\alpha}_{2}\tilde{\lambda}_{2}^{\prime}+12\tilde{\alpha}_{2}^{2}$
(C.73)

The thermal mass is

$\displaystyle V_{T}=\frac{T^{2}}{48}$
$\displaystyle\left(\left(\lambda_{1}\left(N_{c_{1}}^{2}+1\right)+\lambda_{1}^{\prime}\frac{2N_{c_{1}}^{2}-3}{N_{c_{1}}}-\lambda\left(N_{c_{2}}^{2}-1\right)+12N_{c_{1}}g_{1}^{2}\right)Tr\Sigma_{1}^{2}\right.$
$\displaystyle\left.\left(\lambda_{2}\left(N_{c_{2}}^{2}+1\right)+\lambda_{2}^{\prime}\frac{2N_{c_{2}}^{2}-3}{N_{c_{2}}}-\lambda\left(N_{c_{1}}^{2}-1\right)+12N_{c_{2}}g_{2}^{2}\right)Tr\Sigma_{2}^{2}\right)$
(C.74)

and becomes in the Veneziano limit

$V_{T}=(4\pi)^{2}\frac{T^{2}}{48\log{T}}\left(\left(\tilde{\lambda}_{1}+2\tilde{\lambda}_{1}^{\prime}-\frac{N_{c_{2}}}{N_{c_{1}}}\tilde{\lambda}+12\tilde{\alpha}_{1}\right)Tr\Sigma_{1}^{2}+\left(\tilde{\lambda}_{2}+2\tilde{\lambda}_{2}^{\prime}-\frac{N_{c_{1}}}{N_{c_{2}}}\tilde{\lambda}+12\tilde{\alpha}_{2}\right)Tr\Sigma_{2}^{2}\right)$
(C.75)

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