#  h
Vacuum structure of the $\mathbb{Z}_{2}$ symmetric Georgi-Machacek model

Duarte Azevedo${}^{1,\,}$, Pedro Ferreira${}^{1,2\,}$111E-mail:
<EMAIL_ADDRESS>, Heather E. Logan${}^{3\,}$222E-mail:
<EMAIL_ADDRESS>, Rui Santos${}^{1,2\,}$333E-mail<EMAIL_ADDRESS>,
1Centro de Física Teórica e Computacional, Faculdade de Ciências,
Universidade de Lisboa, Campo Grande, Edifício C8 1749-016 Lisboa, Portugal
2ISEL - Instituto Superior de Engenharia de Lisboa,
Instituto Politécnico de Lisboa 1959-007 Lisboa, Portugal
3Ottawa-Carleton Institute for Physics, Carleton University,
1125 Colonel By Drive, Ottawa, Ontario K1S 5B6, Canada

###### Abstract

We discuss the vacuum structure of a version of the Georgi-Machecek model with
an exact $\mathbb{Z}_{2}$ symmetry acting on the triplet fields. Besides the
usual custodial-symmetric model, with $\rho=1$ at tree-level, a model with a
dark matter candidate is also viable. The other phases of the model lead to
electric charge breaking, a wrong pattern of electroweak symmetry breaking or
to $\rho\neq 1$ at tree-level. We derive conditions to have an absolute
minimum in each of the two viable phases, the custodial and the dark matter
phases.

##  1 Introduction

The Georgi-Machacek [1] (GM) model was proposed in 1985 in an attempt to
understand if patterns of electroweak symmetry breaking different than the one
proposed in the Standard Model (SM) were still viable. Besides the SM complex
doublet, the model has one real triplet and one complex triplet. Clearly the
model leads to different phenomenology at colliders but it is also able to
reproduce all the available experimental results. It is of particular
importance, because it is not trivial, that the SM ratio of charged to neutral
currents can be obtained at tree-level. This ratio is translated into the
relation $\rho=m_{W}^{2}/(m_{Z}^{2}\cos^{2}\theta_{W})=1$ at tree-level
($m_{W(Z)}$ is the $W(Z)$ boson mass) and is measured with great precision
allowing only for small deviations from higher order corrections. So far
experiments have neither confirmed nor excluded the spontaneous breaking
pattern predicted by the SM. Hence, all possibilities which agree with the
observed phenomenology should be considered.

In the same year a Higgs potential was written explicitly for the GM model by
Chanowitz and Golden [2]. The potential has only terms with mass dimension of
order 2 and order 4 and is invariant under a global $SU(2)_{L}\times
SU(2)_{R}$ symmetry. For this particular version of the model the vacuum
expectation values of the neutral components of the triplet can be arranged
such that a custodial $SU(2)$ symmetry survives after the spontaneous breaking
of the global $SU(2)_{L}\times SU(2)_{R}$ symmetry. This way, the equality of
the vacuum expectation values of the neutral fields which leads to $\rho=1$ at
tree-level is naturally preserved in the Higgs sector.

Several versions of GM model and its phenomenology have been discussed over
the years [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] and several
searches were proposed and performed at colliders and in particular at the
Large Hadron Collider (LHC). The latest constraints on the GM model can be
found in [18, 19]. There is also a public code available, GMCALC [20], which
allows to calculate the particle spectrum of the model together with the
couplings and decay widths. Checks of perturbative unitarity and boundedness
from below are included in the code. Starting from a parameter point that
constitutes a viable custodial-symmetric minimum of the scalar potential, the
code also checks for deeper alternative minima by performing a numerical scan
following the method proposed in Ref. [12]. A full study of the possible vacua
of the model has not been performed.

In this work we discuss the vacuum structure of the $\mathbb{Z}_{2}$ symmetric
model as proposed in ref. [2] which is a simple yet viable version of the GM
potential. The model with this potential and no further field additions has a
Dark Matter (DM) phase when spontaneous symmetry breaking leaves the
$\mathbb{Z}_{2}$ unbroken, that is, when the triplets have no vacuum
expectation values (VEVs). The model studied in the literature is the one
where custodial $SU(2)$ is preserved and $\rho=1$ at tree-level. The model has
however other types of minima that do not preserve the custodial symmetry that
range from electric charge breaking minima to models with just the wrong
pattern of spontaneous symmetry breaking (SSB). Besides, these non-custodial
vacua always give rise to extra Goldstone bosons. We will compare the depths
of the potential calculated at the different vacua at tree-level to understand
if either the DM or the Custodial vacuum are naturally stable and derive the
conditions for absolute stability in case they are not. The expressions found
can be used to ensure that the desired vacuum is the deepest one at tree-
level.

In this first attempt to understand the vacuum structure of the model we make
the analysis more tractable by considering only the real component of the (in
general complex) VEVs. In that sense the conditions we obtain for the
potential to be in an absolute minimum are necessary conditions.

The paper is organized as follows. In section 2 we briefly present the
potential and the vacuum structure of the model and in section 3 we present
the bilinear formalism. Models that retain the custodial symmetry after
spontaneous symmetry breaking (SSB) are discussed in section 4 while the ones
where said symmetry is broken are discussed in 5. In section 6 we discuss the
stability of the two viable models. We summarise our findings in section 7.

##  2 Scalar potential and vacuum structure

The Georgi-Machacek model [1] consists of the usual complex doublet
$(\phi^{+},\phi^{0})$ with hypercharge, $Y$, equal to 1, a real triplet
$(\xi^{+},\xi^{0},\xi^{-})$ with $Y=0$, and a complex triplet
$(\chi^{++},\chi^{+},\chi^{0})$ with $Y=2$. The fields can be written in the
following matrix form

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\phi^{0*}&\phi^{+}\\\
-\phi^{+*}&\phi^{0}\end{array}\right),$ (3) $\displaystyle\Xi$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\chi^{0*}&\xi^{+}&\chi^{++}\\\
-\chi^{+*}&\xi^{0}&\chi^{+}\\\
\chi^{++*}&-\xi^{+*}&\chi^{0}\end{array}\right).$ (7)

which transform under the global SU(2)${}_{L}\times$SU(2)R symmetry according
to

$\exp(iT^{a}\theta_{L}^{a})(\Phi\ {\rm or}\ \Xi)\exp(-iT^{b}\theta_{R}^{b}).$
(8)

The most general scalar potential invariant under the global
SU(2)${}_{L}\times$SU(2)R symmetry is

$\displaystyle V(\Phi,\Xi)$ $\displaystyle=$
$\displaystyle\frac{\mu_{2}^{2}}{2}{\rm
Tr}(\Phi^{\dagger}\Phi)+\frac{\mu_{3}^{2}}{2}{\rm
Tr}(\Xi^{\dagger}\Xi)+\lambda_{1}[{\rm
Tr}(\Phi^{\dagger}\Phi)]^{2}+\lambda_{2}{\rm
Tr}(\Phi^{\dagger}\Phi)\text{Tr}(\Xi^{\dagger}\Xi)$ (9)
$\displaystyle+\lambda_{3}{\rm
Tr}(\Xi^{\dagger}\Xi\Xi^{\dagger}\Xi)+\lambda_{4}[{\rm
Tr}(\Xi^{\dagger}\Xi)]^{2}-\lambda_{5}{\rm
Tr}(\Phi^{\dagger}\tau^{a}\Phi\tau^{b})\text{Tr}(\Xi^{\dagger}t^{a}\Xi t^{b})$

where we have further imposed a $\mathbb{Z}_{2}$ symmetry, $\Xi\to-\Xi$, which
eliminates the cubic terms ${\rm Tr}(\Phi^{\dagger}\tau^{a}\Phi\tau^{b})(U\Xi
U^{\dagger})_{ab}$ and ${\rm Tr}(\Xi^{\dagger}t^{a}\Xi t^{b})(U\Xi
U^{\dagger})_{ab}$ (for the definition of $U$ see Ref. [6]). This is the
potential originally proposed by Chanowitz and Golden [2] which not only
preserves $\rho=1$ at tree-level but also offers a new possibility that the
triplets have the dominant contribution to the W boson mass. The potential
must be bounded from below, and the necessary and sufficient conditions that
the quartic couplings $\lambda_{i}$ must obey for that to happen have been
established [12, 20]:

$\displaystyle\lambda_{1}$ $\displaystyle>$ $\displaystyle 0,$
$\displaystyle\lambda_{4}$ $\displaystyle>$
$\displaystyle\left\\{\begin{array}[]{l l}-\frac{1}{3}\lambda_{3}&{\rm for}\
\lambda_{3}\geq 0,\\\ -\lambda_{3}&{\rm for}\
\lambda_{3}<0,\end{array}\right.$ (12) $\displaystyle\lambda_{2}$
$\displaystyle>$ $\displaystyle\left\\{\begin{array}[]{l
l}\frac{1}{2}\lambda_{5}-2\sqrt{\lambda_{1}\left(\frac{1}{3}\lambda_{3}+\lambda_{4}\right)}&{\rm
for}\ \lambda_{5}\geq 0\ {\rm and}\ \lambda_{3}\geq 0,\\\
\omega_{+}(\zeta)\lambda_{5}-2\sqrt{\lambda_{1}(\zeta\lambda_{3}+\lambda_{4})}&{\rm
for}\ \lambda_{5}\geq 0\ {\rm and}\ \lambda_{3}<0,\\\
\omega_{-}(\zeta)\lambda_{5}-2\sqrt{\lambda_{1}(\zeta\lambda_{3}+\lambda_{4})}&{\rm
for}\ \lambda_{5}<0\,,\end{array}\right.$ (16)

with

$\omega_{\pm}(\zeta)=\frac{1}{6}(1-B)\pm\frac{\sqrt{2}}{3}\left[(1-B)\left(\frac{1}{2}+B\right)\right]^{1/2},$
(17)

and

$B\equiv\sqrt{\frac{3}{2}\left(\zeta-\frac{1}{3}\right)}\in[0,1];\qquad\zeta\in\left[\frac{1}{3},1\right].$
(18)

Due to the hermiticity of the potential, and its field structure, all of the
couplings in Eq. (9) are necessarily real, and therefore the CP symmetry
cannot be explicitly broken. As we will see later any attempt of spontaneous
CP breaking would lead to the breaking of the custodial symmetry.

As we are interested in the vacuum structure of the GM model we start by
writing the vacuum expectation values (VEVs) of the fields in the following
generic form

$\Phi=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}v_{1}-iv_{2}&v_{3}+iv_{4}\\\
-v_{3}+iv_{4}&v_{1}+iv_{2}\end{array}\right),\qquad\Xi=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}v_{8}-iv_{9}&v_{6}+iv_{7}&v_{12}+iv_{13}\\\
-v_{10}+iv_{11}&\sqrt{2}v_{5}&v_{10}+iv_{11}\\\
v_{12}-iv_{13}&-v_{6}+iv_{7}&v_{8}+iv_{9}\end{array}\right).$ (19)

There are 6 symmetry transformations in SU(2)${}_{L}\times$SU(2)R that can be
used to eliminate the redundant VEVs. We first use the three broken symmetry
transformations, $\theta_{R}^{a}=-\theta_{L}^{a}$ (these are the ones
orthogonal to the custodial SU(2)), to rotate away three of the four
components of the doublet, leaving only $v_{1}$. We can then use the three
custodial symmetry transformations, $\theta_{R}^{a}=\theta_{L}^{a}$, to rotate
away three of the VEVs in the triplet. There are several possibilities and we
choose to eliminate both the real and imaginary components of $\chi^{++}$ and
the imaginary component of $\xi^{+}$, which leads to the following form of the
vacuum configuration

$\Phi=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}v_{1}&0\\\
0&v_{1}\end{array}\right),\qquad\qquad\Xi=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}v_{8}-iv_{9}&v_{6}&0\\\
-v_{10}+iv_{11}&\sqrt{2}v_{5}&v_{10}+iv_{11}\\\
0&-v_{6}&v_{8}+iv_{9}\end{array}\right).$ (20)

We are therefore left with 7 non-redundant VEVs. The scalar potential in Eq.
(9) contains 4 distinct invariants under the SU(2)${}_{L}\times$SU(2)R global
symmetry. These were identified in Ref. [12] and we write them as

$\displaystyle{\rm Tr}(\Phi^{\dagger}\Phi)$ $\displaystyle=$ $\displaystyle
r^{2}\cos^{2}\gamma,$ (21) $\displaystyle{\rm Tr}(\Xi^{\dagger}\Xi)$
$\displaystyle=$ $\displaystyle r^{2}\sin^{2}\gamma,$ (22) $\displaystyle{\rm
Tr}(\Xi^{\dagger}\Xi\Xi^{\dagger}\Xi)$ $\displaystyle=$
$\displaystyle\zeta\cdot(r^{2}\sin^{2}\gamma)^{2},$ (23) $\displaystyle{\rm
Tr}(\Phi^{\dagger}\tau^{a}\Phi\tau^{b})\text{Tr}(\Xi^{\dagger}t^{a}\Xi t^{b})$
$\displaystyle=$
$\displaystyle\omega\cdot(r^{2}\cos^{2}\gamma)(r^{2}\sin^{2}\gamma)\,.$ (24)

The fact that the potential can be expressed in a four-dimensional subspace
spanned by these four invariants implies that there are three extra symmetries
besides SU(2)${}_{L}\times$SU(2)R that have not yet been identified.

##  3 Bilinear formalism and mass matrices for the GM model

In this section we will outline how a bilinear formalism may be applied to the
study of the vaccum structure of the GM model. In short, such a formalism
expresses the potential, and its minimisation conditions, in terms of
quadratic functions of its fields and/or VEVs, which has been proven to be an
effective strategy to compare the values of scalar potentials at different
stationary points, and ascertain their relative depths. In the context of the
two-Higgs doublet model (2HDM), a first version of the bilinear formalism was
first applied in [21, 22, 23], and then refined in [24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34], allowing for the study of the 2HDM’s vacuum structure and
possible global symmetries. The bilinear formalism was also applied to models
with different scalar content, to wit the 3HDM [35, 36], the complex singlet-
doublet model (CXSM) [37], the 2HDM with a real or complex singlet (N2HDM)
[38, 39], and the Higgs triplet model (HTM) [40]. For most models, the
bilinears are simply constructed as the four gauge-invariant quantities
allowed with the fields present; but in the case of the HTM, the bilinears
were simply the most convenient quadratic products of VEVs to express the
value of the potential at generic stationary points as a quadratic polynomial.
As we will shortly see, the same holds for the GM model — the presence of a
triplet, as in [40], yields such a structure of the potential that we have not
found it possible to express it as a product of quadratic gauge invariant
field products.

First however, in order to identify whether a given stationary point is a
minimum, we will need to study the second derivatives of the potential. In
order to obtain the mass matrices for all minima simultaneously we will
express the doublet $\Phi$ and the triplet $\Xi$ of Eq. (7) in terms of the
real components of the fields $\varphi_{i}$, according to

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}\varphi_{1}-i\varphi_{2}&\varphi_{3}+i\varphi_{4}\\\
-\varphi_{3}+i\varphi_{4}&\varphi_{1}+i\varphi_{2}\end{array}\right),$ (27)
$\displaystyle\Xi$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}\varphi_{8}-i\varphi_{9}&\varphi_{6}+i\varphi_{7}&\varphi_{12}+i\varphi_{13}\\\
-\varphi_{10}+i\varphi_{11}&\sqrt{2}\varphi_{5}&\varphi_{10}+i\varphi_{11}\\\
\varphi_{12}-i\varphi_{13}&-\varphi_{6}+i\varphi_{7}&\varphi_{8}+i\varphi_{9}\end{array}\right).$
(31)

With these conventions, the matrix elements of the $13\times 13$ mass matrix
are

$\left[M^{2}\right]_{ij}\,=\,\frac{\partial^{2}V}{\partial\varphi_{i}\partial\varphi_{j}}\,,$
(32)

with the potential given in Eq. (9). In order to make the analysis tractable
we will only consider the real part of the VEVs from Eq. (20), that is,

$\langle\varphi_{1}\rangle=v_{1}\;,\;\langle\varphi_{5}\rangle=v_{5}\;,\;\langle\varphi_{6}\rangle=v_{6}\;,\;\langle\varphi_{8}\rangle=v_{8}\;,\;\langle\varphi_{10}\rangle=v_{10}\;,\;$
(33)

with $v_{9}$ and $v_{11}$ set equal to zero. We leave a full analysis
including these imaginary parts of the vevs to future work. This leads to a
potential (evaluated at a stationary point for which these VEVs hold) equal to

$\displaystyle V$
$\displaystyle=v_{1}^{2}\left[\frac{\mu_{2}^{2}}{2}+\lambda_{1}v_{1}^{2}+\lambda_{2}v_{5}^{2}+\left(\lambda_{2}-\frac{\lambda_{5}}{4}\right)v_{8}^{2}+\lambda_{2}v_{6}^{2}+\lambda_{2}c_{10}^{2}-\frac{\lambda_{5}v_{5}v_{8}}{\sqrt{2}}-\frac{\lambda_{5}v_{10}v_{6}}{2}\right]$
(34)
$\displaystyle+v_{5}^{2}\left[\frac{\mu_{3}^{2}}{2}+(\lambda_{3}+\lambda_{4})v_{5}^{2}+2\lambda_{4}v_{8}^{2}+(2\lambda_{3}+2\lambda_{4})v_{6}^{2}+(2\lambda_{3}+2\lambda_{4})v_{10}^{2}\right]$
$\displaystyle+v_{8}^{2}\left[\frac{\mu_{3}^{2}}{2}+\left(\frac{\lambda_{3}}{2}+\lambda_{4}\right)v_{8}^{2}+(\lambda_{3}+2\lambda_{4})v_{6}^{2}+(\lambda_{3}+2\lambda_{4})v_{10}^{2}\right]$
$\displaystyle+v_{6}^{2}\left[\frac{\mu_{3}^{2}}{2}+(\lambda_{3}+\lambda_{4})v_{6}^{2}+2\lambda_{4}v_{10}^{2}\right]+v_{10}^{2}\left[\frac{\mu_{3}^{2}}{2}+(\lambda_{3}+\lambda_{4})v_{10}^{2}\right]-2\sqrt{2}\lambda_{3}v_{10}v_{5}v_{6}v_{8}.$

Let us now define the vector $A$ and the symmetric matrix $B$ as

$A=\left(\begin{array}[]{c}\mu_{2}^{2}\\\ \mu_{3}^{2}\\\ 0\\\ 0\\\ 0\\\
0\end{array}\right)\quad\quad,\quad\quad
B=\left(\begin{array}[]{cccccc}8\lambda_{1}&4\lambda_{2}&-\lambda_{5}&-\lambda_{5}&-\lambda_{5}&0\\\
4\lambda_{2}&8(\lambda_{3}+\lambda_{4})&0&0&0&0\\\
-\lambda_{5}&0&-2\lambda_{3}&0&-2\lambda_{3}&0\\\
-\lambda_{5}&0&0&-4\lambda_{3}&0&-2\lambda_{3}\\\
-\lambda_{5}&0&-2\lambda_{3}&0&-4\lambda_{3}&0\\\ 0&0&0&-2\lambda_{3}&0&0\\\
\end{array}\right),$ (35)

and the vector $X$, which depends on the VEVs as

$X=\left(\begin{array}[]{c}v_{1}^{2}/2\\\
\left(v_{10}^{2}+v_{5}^{2}+v_{6}^{2}+v_{8}^{2}\right)/2\\\
\sqrt{2}v_{5}v_{8}\\\ v_{8}^{2}/2\\\ v_{10}v_{6}\\\ v_{10}^{2}+v_{6}^{2}\\\
\end{array}\right).$ (36)

With these definitions the value of the potential at each stationary point
(SP) with VEVs $v_{i}$, is given by

$V_{SP}\;=\;A^{T}X_{SP}\,+\,\frac{1}{2}\,X^{T}_{SP}\,B\,X_{SP}\,.$ (37)

Thus we see that it is possible to express the value of the potential at a
given stationary point as a quadratic polynomial on $X$, that is, an order 2
polynomial expressed in terms of bilinears, quadratic combinations of vevs
444Though it is not possible, due to the $\lambda_{3}$ and $\lambda_{5}$ terms
in the potential, to express it as a quadratic polynomial of quadratic
products of fields.. Further, since the potential is the sum of two
homogeneous functions of order 2 and 4 ($V_{2}$ and $V_{4}$, the quadratic and
quartic terms in the vevs – linear and quadratic in $X$), we have

$\mbox{At any stationary point:}\,\frac{\partial
V}{\partial\varphi_{i}}\,=\,0\,\Longrightarrow\,\sum_{i}\varphi_{i}\,\frac{\partial
V}{\partial\varphi_{i}}\,=\,0\,\Longrightarrow 2\,V_{2}\,+\,4\,V_{4}\,=\,0\,.$
(38)

The value of the potential at a given stationary point, $V_{SP}$, is simply

$V_{SP}\;=\;\frac{1}{2}\,\left(V_{2}\right)_{SP}\;=\;-\,\left(V_{4}\right)_{SP}\;=\;\frac{1}{2}\,A^{T}X_{SP}\;=\;-\,X^{T}_{SP}\,B\,X_{SP}\,.$
(39)

Finally, we also need the vector $V^{\prime}$, defined as

$V^{\prime}\;=\;\frac{\partial V}{\partial X^{T}}\;=\;A\,+\,B\,X\,,$ (40)

and since $V^{\prime}$ is the gradient of the potential along the vector $X$,
the condition $X^{T}V^{\prime}=0$ has to hold. Let us now consider two
distinct vacua with vectors $X_{1}$ and $X_{2}$, respectively. Let
$V_{1}^{\prime}$ and $V_{2}^{\prime}$ be the gradient of the potential
considered at those stationary points. We can contract the vectors of the VEVs
from one minimum with the gradient of the other to obtain

$X_{1}^{T}V_{2}^{\prime}=X_{1}^{T}A+X_{1}^{T}BX_{2}.$ (41)

Since $B$ is symmetric, we have

$X_{1}^{T}BX_{2}=X_{2}^{T}BX_{1},$ (42)

which leads to

$X_{2}^{T}V_{1}^{\prime}-X_{2}^{T}A=X_{1}^{T}V_{2}^{\prime}-X_{1}^{T}A.$ (43)

The second term in each side of the equation is the quadratic part of the
potential computed at the corresponding stationary point. Therefore using Eq.
(39) we can rewrite Eq. (43) as

$V_{2}-V_{1}=\frac{1}{2}\left(X_{2}^{T}V_{1}^{\prime}-X_{1}^{T}V_{2}^{\prime}\right),$
(44)

which relates the relative height between the two minima with the VEV vectors
and gradients of the potential. We want to express the right hand side for
each case as a function of physical parameters from one vacuum as much as
possible. In this way, the stability of that vacuum is just a function of
quantities computed at that stationary point. In the next section we will
classify the different type of stationary points and write the corresponding
$X$ and $V^{\prime}$ vectors. The minimum conditions that led to the different
phases to be presented are shown in appendix A. The vacua classified as charge
breaking are the ones where the photon becomes massive and the mass matrix for
the gauge bosons is also given in appendix A.

##  4 Models with custodial symmetry after SSB

###  4.1 The custodial vacuum

The GM model was first proposed due to the very nice property that there is a
solution of the minimization conditions with no charged VEVs where the triplet
neutral VEVs (those of the real fields $\varphi_{5}$ and $\varphi_{8}$) are
proportional. This condition preserves the custodial symmetry and maintains
the SM structure of the gauge bosons mass matrix, that is, the Weinberg angle
is preserved at tree-level. As discussed in the introduction, the equality of
vacuum expectation values is naturally preserved in the Higgs sector because
the custodial $SU(2)$ survives the spontaneous breaking of the global
$SU(2)_{L}\times SU(2)_{R}$ symmetry. In this scenario the VEVs are chosen as

$\langle\varphi_{1}\rangle=v_{1}\;,\;\langle\varphi_{5}\rangle=v_{5}\;,\;\langle\varphi_{6}\rangle=0\;,\;\langle\varphi_{8}\rangle=\sqrt{2}v_{5}\;,\;\langle\varphi_{10}\rangle=0\;,\;$
(45)

leading to the minimization conditions

$\displaystyle\mu_{2}^{2}$
$\displaystyle=-4v_{1}^{2}\lambda_{1}+3v_{5}^{2}(\lambda_{5}-2\lambda_{2})\,,$
(46) $\displaystyle\mu_{3}^{2}$
$\displaystyle=v_{1}^{2}(\lambda_{5}-2\lambda_{2})-4v_{5}^{2}(\lambda_{3}+3\lambda_{4})\,.$

The mass spectrum is the following: there are two custodial singlet states
that mix, $h$ and $H$, with masses given by

$\displaystyle m_{h,\,H}^{2}$ $\displaystyle=$ $\displaystyle
4\lambda_{1}v_{1}^{2}+4v_{5}^{2}(\lambda_{3}+3\lambda_{4})\mp$ (47)
$\displaystyle
2\sqrt{4\lambda_{1}^{2}v_{1}^{4}+v_{1}^{2}v_{5}^{2}\left[3(\lambda_{5}-2\lambda_{2})^{2}-8\lambda_{1}(\lambda_{3}+3\lambda_{4})\right]+4v_{5}^{4}(\lambda_{3}+3\lambda_{4})^{2}}\,,$

a custodial triplet, $H_{3}^{0},H_{3}^{\pm}$, with masses given by

$m_{3}^{2}=\frac{1}{2}\lambda_{5}\left(v_{1}^{2}+8v_{5}^{2}\right)\,,$ (48)

and a custodial 5-plet, $H_{5}^{0},H_{5}^{\pm},H_{5}^{\pm\pm}$, with masses

$m_{5}^{2}=\frac{3\lambda_{5}v_{1}^{2}}{2}+8\lambda_{3}v_{5}^{2}\,.$ (49)

In order that the gauge bosons have the observed mass the VEVs have to obey

$v_{1}^{2}+8v_{5}^{2}=v^{2}=\frac{1}{\sqrt{2}G_{F}}\approx(246\,{\rm
GeV})^{2}\,.$ (50)

There are also two mixing angles usually taken as input parameters and that
parametrize the mixing between the doublet and the triplet components [3].

Since the cubic terms are absent due to the $\mathbb{Z}_{2}$ symmetry,
$\Xi\to-\Xi$, the mass scale of the scalar states is set by the electroweak
scale and the model does not possess a decoupling limit. The upper bounds on
the scalar masses from perturbative unitarity were first studied in Ref. [6]
which found the following constraints on the masses: $m_{3}\lesssim 400$ GeV,
$m_{5}\lesssim 700$ GeV. Despite the absence of a decoupling limit, the model
in this phase is still not excluded by data [41].

Finally, the vectors $X$ and $V^{\prime}$ can be written as

$X_{C}=\left(\begin{array}[]{c}v_{1}^{2}/2\\\ 3v_{5}^{2}/2\\\ 2v_{5}^{2}\\\
v_{5}^{2}\\\ 0\\\ 0\\\ \end{array}\right),\qquad
V^{\prime}_{C}=A+BX_{C}=\left(\begin{array}[]{c}0\\\ 2\bar{m}^{2}\\\
-\bar{m}^{2}\\\ -\bar{m}^{2}\\\ -\bar{m}^{2}\\\ -2\lambda_{3}v_{5}^{2}\\\
\end{array}\right),$ (51)

with $\bar{m}^{2}=\frac{1}{2}\lambda_{5}v^{2}_{1}+4\lambda_{3}\,v_{5}^{2}$.

###  4.2 The dark matter vacuum

If only the doublet $\Phi$ develops a VEV, the Lagrangian will still be
exactly $\mathbb{Z}_{2}$ symmetric after SSB. This in turn means that all the
particles from the triplet do not interact with fermions nor do they have
triple vertices with two gauge bosons. The lightest of these particles is thus
stable and is therefore a good dark matter candidate. Such a situation is
indeed a possible solution of the minimization equations of the potential,
corresponding to VEVs $v_{1}\equiv v$ and all others zero, in Eq. (33). The
only minimization condition is

$v=\sqrt{-\frac{\mu_{2}^{2}}{4\lambda_{1}}}\,.$ (52)

In this scenario the custodial symmetry is still preserved and therefore the
physical particles fall into the same custodial representations as in the
custodial vacuum. The two custodial singlet states no longer mix; the 125 GeV
Higgs boson is identified with $h=\varphi_{1}$ with mass given by

$m_{h}^{2}=8\lambda_{1}v^{2}.$ (53)

The $\mathbb{Z}_{2}$-odd states comprise the familiar custodial singlet,
triplet, and fiveplet with masses given by

$\displaystyle m_{1}^{2}$ $\displaystyle=$
$\displaystyle(\mu_{3}^{2}+2\lambda_{2}v^{2})-\lambda_{5}v^{2},$
$\displaystyle m_{3}^{2}$ $\displaystyle=$
$\displaystyle(\mu_{3}^{2}+2\lambda_{2}v^{2})-\frac{1}{2}\lambda_{5}v^{2},$
$\displaystyle m_{5}^{2}$ $\displaystyle=$
$\displaystyle(\mu_{3}^{2}+2\lambda_{2}v^{2})+\frac{1}{2}\lambda_{5}v^{2},$
(54)

respectively.

When $\lambda_{5}$ is positive the real custodial-singlet neutral scalar with
mass $m_{1}$ is the lightest of the $\mathbb{Z}_{2}$-odd states and is the
dark matter candidate. When $\lambda_{5}$ is negative the custodial-fiveplet
states, comprising one neutral, one charged, and one doubly-charged scalar,
are the lightest. The question now is if we can have just one neutral state as
the dark matter candidate. As shown in [42] the mass degeneracy between
charged and neutral states can be lifted via quantum corrections, which tend
to make the charged components slightly heavier than the neutral ones.
Therefore one would expect that in this phase of the GM model quantum
corrections also lift the degeneracy and that the neutral state would be the
dark matter candidate. The model has a fully dark sector that only
communicates with the visible sector via couplings to the SM Higgs and to the
gauge bosons in quartic couplings. The visible sector is indistinguishable
from the SM from the point of view of the (tree-level) Higgs couplings to
fermions and gauge bosons, though the loop-induced coupling to photon pairs
will be modified due to contributions from loops of the $\mathbb{Z}_{2}$-odd
states.

Because $\mu_{3}^{2}$ remains a free parameter that can be taken arbitrarily
large, this phase of the model possesses a decoupling limit, in which all of
the $\mathbb{Z}_{2}$-odd states can be taken heavy with masses of order
$\sqrt{\mu_{3}^{2}}$.

We end this section writing the X and $V^{\prime}$ vectors for this stationary
point, as defined in Eqs. (36) and (40),

$X_{DM}=\left(\begin{array}[]{c}m_{h}^{2}/(16\lambda_{1})\\\ 0\\\ 0\\\ 0\\\
0\\\ 0\\\ \end{array}\right),\qquad
V^{\prime}_{DM}=A+BX_{DM}=\left(\begin{array}[]{c}0\\\
m_{1}^{2}+\bar{k}^{2}\\\ \bar{k}^{2}\\\ \bar{k}^{2}\\\ \bar{k}^{2}\\\ 0\\\
\end{array}\right),$ (55)

with $\bar{k}^{2}=-\frac{\lambda_{5}m_{h}^{2}}{16\lambda_{1}}.$

##  5 Models where the custodial symmetry is broken after SSB

###  5.1 The real non-custodial vacuum

Let us now consider the scenario where the solution of the minimization
equations has $\langle\varphi_{8}\rangle\neq\sqrt{2}\langle\varphi_{5}\rangle$
(but still no charged VEVs) which means that the custodial symmetry is broken.
The absence of the dimension 3 soft breaking terms will then lead to two
massless scalars (the result of spontaneous breaking of continuous global
symmetries of the model). Let the VEV choice be

$\langle\varphi_{1}\rangle=v_{1}\;,\;\langle\varphi_{5}\rangle=v_{5}\;,\;\langle\varphi_{6}\rangle=0\;,\;\langle\varphi_{8}\rangle=v_{8}\;,\;\langle\varphi_{10}\rangle=0\;,\;$
(56)

which leads to the minimization conditions

$\displaystyle\mu_{2}^{2}$
$\displaystyle=-4\lambda_{1}v_{1}^{2}-2\lambda_{2}\left(v_{5}^{2}+v_{8}^{2}\right)+\frac{2\lambda_{3}v_{8}^{2}\left(2\sqrt{2}v_{5}+v_{8}\right)\left(v_{5}v_{8}^{2}-2v_{5}^{3}\right)}{v_{1}^{2}\left(\sqrt{2}v_{5}^{2}+v_{5}v_{8}-\sqrt{2}v_{8}^{2}\right)}\,,$
(57) $\displaystyle\mu_{3}^{2}$
$\displaystyle=-2\lambda_{2}v_{1}^{2}-4v_{5}^{2}(\lambda_{3}+\lambda_{4})-4v_{8}^{2}(\lambda_{3}+\lambda_{4})+\frac{4\lambda_{3}v_{8}^{3}}{\sqrt{2}v_{5}+2v_{8}}\,,$
$\displaystyle\lambda_{5}$
$\displaystyle=\frac{4\lambda_{3}v_{5}v_{8}\left(v_{8}^{2}-2v_{5}^{2}\right)}{v_{1}^{2}\left(\sqrt{2}v_{5}^{2}+v_{5}v_{8}-\sqrt{2}v_{8}^{2}\right)}\,.$

The vectors $X$ and $V^{\prime}$ are

$X_{NC}=\left(\begin{array}[]{c}v_{1}^{2}/2\\\
1/2\left(v_{5}^{2}+v_{8}^{2}\right)\\\ \sqrt{2}v_{5}v_{8}\\\ v_{8}^{2}/2\\\
0\\\ 0\\\ \end{array}\right)\,,\quad
V^{\prime}_{NC}=\left(\begin{array}[]{c}0\\\
\frac{4\lambda_{3}v_{8}^{3}}{\sqrt{2}v_{5}+2v_{8}}\\\
-\frac{2\lambda_{3}v_{5}v_{8}^{2}}{v_{5}+\sqrt{2}v_{8}}\\\
2\lambda_{3}v_{8}\left(\frac{2v_{5}^{3}-v_{5}v_{8}^{2}}{\sqrt{2}v_{5}^{2}+v_{5}v_{8}-\sqrt{2}v_{8}^{2}}-v_{8}\right)\\\
-\frac{2\lambda_{3}v_{5}v_{8}^{2}}{v_{5}+\sqrt{2}v_{8}}\,,\\\
\lambda_{3}\left(-v_{8}^{2}\right)\\\ \end{array}\right)\,.$ (58)

###  5.2 Charge breaking vacua

We now consider the scenarios where charge breaking (CB) occurs spontaneously,
and where therefore the photon becomes massive after SSB. A priori, it would
seem that there would be a large number of possible CB vacua, but in fact – as
will be explained in appendix A – the minimization equations of the GM model
curtail most of those possibilities. Indeed, there are but five distinct vacua
that lead to charge breaking,

  1. 1.

$v_{5}=v_{8}=0$ and
$v_{10}=-\frac{\lambda_{5}}{\lambda_{3}}\frac{v_{1}^{2}}{8v_{6}}$.

  2. 2.

$v_{5}=v_{8}=0$ and $v_{10}=-v_{6}$.

  3. 3.

$v_{5}=v_{8}=0$ and $v_{10}=v_{6}$.

  4. 4.

$v_{1}=v_{5}=v_{8}=0$ and $v_{6}=-v_{10}$.

  5. 5.

$v_{1}=v_{5}=v_{8}=0$ and $v_{6}=v_{10}$.

To insist on this point, these are the only combinations of real VEVs
originating CB which are possible solutions of the extremum equations of the
GM model. The minimum conditions then lead to the following relations for
vacuum number 1,

$\displaystyle\mu_{2}^{2}$
$\displaystyle=-\frac{\lambda_{2}\lambda_{5}^{2}v_{1}^{4}}{32\lambda_{3}^{2}v_{6}^{2}}-\frac{v_{1}^{2}\left(32\lambda_{1}\lambda_{3}+\lambda_{5}^{2}\right)}{8\lambda_{3}}-2\lambda_{2}\,,$
(59) $\displaystyle\mu_{3}^{2}$
$\displaystyle=-\frac{\lambda_{5}^{2}v_{1}^{4}(\lambda_{3}+\lambda_{4})}{16\lambda_{3}^{2}v_{6}^{2}}-2\lambda_{2}v_{1}^{2}-4v_{6}^{2}(\lambda_{3}+\lambda_{4})\,.$

For vacuum number 2 we have

$\displaystyle\mu_{2}^{2}$
$\displaystyle=-4\lambda_{1}v_{1}^{2}-v_{6}^{2}(4\lambda_{2}+\lambda_{5})\,,$
(60) $\displaystyle\mu_{3}^{2}$
$\displaystyle=-\frac{1}{2}v_{1}^{2}(4\lambda_{2}+\lambda_{5})-4v_{6}^{2}(\lambda_{3}+2\lambda_{4})\,.$

For vacuum number 3 we obtain

$\displaystyle\mu_{2}^{2}$
$\displaystyle=-4\lambda_{1}v_{1}^{2}+v_{6}^{2}(-4\lambda_{2}+\lambda_{5})\,,$
(61) $\displaystyle\mu_{3}^{2}$
$\displaystyle=\frac{1}{2}v_{1}^{2}(-4\lambda_{2}+\lambda_{5})-4v_{6}^{2}(\lambda_{3}+2\lambda_{4})\,.$

Finally for vacua 4 and 5 the condition is

$\mu_{3}^{2}=-4v_{6}^{2}(\lambda_{3}+2\lambda_{4})\,.$ (62)

It should be clear to the reader that the values of the VEVs shown here is
almost certainly different from vacuum to vacuum, though we are not labeling
them differently. Following the bilinear formalism, we can then write $X$ and
$V^{\prime}$ vectors for each CB vacuum:

  * •

Charge-breaking 1:

$X_{CB1}=\left(\begin{array}[]{c}v_{1}^{2}/2\\\ \bar{l}^{2}/2\\\ 0\\\ 0\\\
-v_{1}^{2}\lambda_{5}/(8\lambda_{3})\\\ \bar{l}^{2}\\\
\end{array}\right),\qquad
V^{\prime}_{CB1}=A+BX_{CB1}=\left(\begin{array}[]{c}0\\\ 0\\\
-v_{1}^{2}\lambda_{5}/4\\\ -v_{1}^{2}\lambda_{5}/2-2\lambda_{3}\bar{l}^{2}\\\
0\\\ 0\\\ \end{array}\right),$ (63)

with
$\bar{l}^{2}=v_{6}^{2}+\frac{v_{1}^{4}\lambda_{5}^{2}}{64v_{6}^{2}\lambda_{3}^{2}}$.

  * •

Charge-breaking 2:

$X_{CB2}=\left(\begin{array}[]{c}v_{1}^{2}/2\\\ v_{6}^{2}\\\ 0\\\ 0\\\
-v_{6}^{2}\\\ 2v_{6}^{2}\\\ \end{array}\right),\qquad
V^{\prime}_{CB2}=A+BX_{CB2}=\left(\begin{array}[]{c}0\\\
4v_{6}^{2}\lambda_{3}-v_{1}^{2}\lambda_{5}/2\\\
2v_{6}^{2}\lambda_{3}-v_{1}^{2}\lambda_{5}/2\\\
-4v_{6}^{2}\lambda_{3}-v_{1}^{2}\lambda_{5}/2\\\
4v_{6}^{2}\lambda_{3}-v_{1}^{2}\lambda_{5}/2\\\ 0\\\ \end{array}\right).$ (64)

  * •

Charge-breaking 3:

$X_{CB3}=\left(\begin{array}[]{c}v_{1}^{2}/2\\\ v_{6}^{2}\\\ 0\\\ 0\\\
v_{6}^{2}\\\ 2v_{6}^{2}\\\ \end{array}\right),\qquad
V^{\prime}_{CB3}=A+BX_{CB3}=\left(\begin{array}[]{c}0\\\
4v_{6}^{2}\lambda_{3}+v_{1}^{2}\lambda_{5}/2\\\
-2v_{6}^{2}\lambda_{3}-v_{1}^{2}\lambda_{5}/2\\\
-4v_{6}^{2}\lambda_{3}-v_{1}^{2}\lambda_{5}/2\\\
-4v_{6}^{2}\lambda_{3}-v_{1}^{2}\lambda_{5}/2\\\ 0\\\ \end{array}\right).$
(65)

  * •

Charge-breaking 4:

$X_{CB4}=\left(\begin{array}[]{c}0\\\ v_{6}^{2}\\\ 0\\\ 0\\\ -v_{6}^{2}\\\
2v_{6}^{2}\\\ \end{array}\right),\qquad
V^{\prime}_{CB4}=A+BX_{CB4}=\left(\begin{array}[]{c}\mu_{2}^{2}+v_{6}^{2}(4\lambda_{2}+\lambda_{5})\\\
4v_{6}^{2}\lambda_{3}\\\ 2v_{6}^{2}\lambda_{3}\\\ -4v_{6}^{2}\lambda_{3}\\\
4v_{6}^{2}\lambda_{3}\\\ 0\\\ \end{array}\right).$ (66)

  * •

Charge-breaking 5:

$X_{CB5}=\left(\begin{array}[]{c}0\\\ v_{6}^{2}\\\ 0\\\ 0\\\ v_{6}^{2}\\\
2v_{6}^{2}\\\ \end{array}\right),\qquad
V^{\prime}_{CB5}=A+BX_{CB5}=\left(\begin{array}[]{c}\mu_{2}^{2}+v_{6}^{2}(4\lambda_{2}-\lambda_{5})\\\
4v_{6}^{2}\lambda_{3}\\\ -2v_{6}^{2}\lambda_{3}\\\ -4v_{6}^{2}\lambda_{3}\\\
-4v_{6}^{2}\lambda_{3}\\\ 0\\\ \end{array}\right).$ (67)

###  5.3 Wrong-Electroweak vacuum

There is a set of unphysical solutions of the minimization equations that do
not break the electroweak symmetry correctly, leading to a massless $Z$ boson
– a curious situation, only made possible by the possibility of having VEVs
for the triplet fields with zero hypercharge. The two possible minima is this
category are

  * •

W1:
$\langle\varphi_{1}\rangle=0\;,\;\langle\varphi_{5}\rangle=0\;,\;\langle\varphi_{6}\rangle=\frac{1}{2}\sqrt{-\frac{\mu_{3}^{2}}{\lambda_{3}+\lambda_{4}}}\;,\;\langle\varphi_{8}\rangle=0\;,\;\langle\varphi_{10}\rangle=0\;,\;$

  * •

W2:
$\langle\varphi_{1}\rangle=0\;,\;\langle\varphi_{5}\rangle=v_{5}\;,\;\langle\varphi_{6}\rangle=v_{6}\;,\;\langle\varphi_{8}\rangle=0\;,\;\langle\varphi_{10}\rangle=0\;,\;$

with the further condition

$\mu_{3}^{2}=-4(\lambda_{3}+\lambda_{4})\left(v_{5}^{2}+v_{6}^{2}\right)\,.$
(68)

Notice, in fact, how the above stationary points correspond to VEVs for the
fields $\xi^{0}$ and $\xi^{+}$ from Eq. (7), which carry no hypercharge,
allowing the $Z$ boson to remain massless 555It may seem peculiar to have a
VEV for $\xi^{+}$ and still have a massless photon in these vacua, but in fact
the definition of electric charge is different in this situation.. Indeed, the
symmetry group left invariant after SSB for these vacua is $U(1)\times U(1)$ –
one of these $U(1)$ is the unbroken hypercharge group, the other is
“extracted” from $SU(2)$. The $X$ and $V^{\prime}$ vectors in this scenario
are

  * •

W1:

$X_{W1}=\left(\begin{array}[]{c}0\\\
-\frac{\mu_{3}^{2}}{8(\lambda_{3}+\lambda_{4})}\\\ 0\\\ 0\\\ 0\\\
-\frac{\mu_{3}^{2}}{4(\lambda_{3}+\lambda_{4})}\\\ \end{array}\right),\qquad
V^{\prime}_{W1}=A+BX_{W1}=\left(\begin{array}[]{c}\mu_{2}^{2}-\frac{\lambda_{2}\mu_{3}^{2}}{2(\lambda_{3}+\lambda_{4})}\\\
0\\\ 0\\\ \frac{\lambda_{3}\mu_{3}^{2}}{2(\lambda_{3}+\lambda_{4})}\\\ 0\\\
0\\\ \end{array}\right)\,.$ (69)

  * •

W2:

$X_{W2}=\left(\begin{array}[]{c}0\\\
\frac{1}{2}\left(v_{5}^{2}+v_{6}^{2}\right)\\\ 0\\\ 0\\\ 0\\\ v_{6}^{2}\\\
\end{array}\right),\qquad
V^{\prime}_{W2}=A+BX_{W2}=\left(\begin{array}[]{c}\mu_{2}^{2}+2\lambda_{2}\left(v_{5}^{2}+v_{6}^{2}\right)\\\
0\\\ 0\\\ -2\lambda_{3}\left(v_{5}^{2}+v_{6}^{2}\right)\\\ 0\\\ 0\\\
\end{array}\right)\,.$ (70)

###  5.4 Tantamount vacua

There is a set of vacua that produces three massive gauge bosons and thus is
tantamount to the SM electroweak breaking vacuum. The difference lies in the
different gauge bosons’ mass structure. In the SM breaking we obtain
$m_{W}=gv/2$ and $m_{Z}=\sqrt{g^{2}+g^{\prime 2}}v/2$ while in this case the
neutral component of the $SU(2)_{L}$ gauge bosons remains massless (“the
photon”) and the “$Z$ boson” is just the (massive) hypercharge gauge boson.
These vacua would therefore yield an unrealistic value for the $\rho$
parameter, as well as unrealistic couplings of the $Z$ boson to fermions, and
are thus in disagreement with current experimental data.

The vacua that produce these results are

  * •

T1:
$\langle\varphi_{1}\rangle=0\;,\;\langle\varphi_{5}\rangle=v_{5}\;,\;\langle\varphi_{6}\rangle=0\;,\;\langle\varphi_{8}\rangle=0\;,\;\langle\varphi_{10}\rangle=v_{10}\;,\;$
with the condition

$\mu_{3}^{2}=-4(\lambda_{3}+\lambda_{4})\left(v_{10}^{2}+v_{5}^{2}\right)\,.$
(71)

In this vacuum the gauge boson masses are
$m_{W}=g\sqrt{v_{10}^{2}+v_{5}^{2}}$, $m_{Z}=g^{\prime}v_{10}$.

  * •

T2:
$\langle\varphi_{1}\rangle=0\;,\;\langle\varphi_{5}\rangle=0\;,\;\langle\varphi_{6}\rangle=0\;,\;\langle\varphi_{8}\rangle=0\;,\;\langle\varphi_{10}\rangle=v_{10}\;,\;$
with the condition

$\mu_{3}^{2}=-4(\lambda_{3}+\lambda_{4})v_{10}^{2}\,.$ (72)

In this vacuum the gauge boson masses are $m_{W}=gv_{10}$,
$m_{Z}=g^{\prime}v_{10}$.

The corresponding $X$ and $V^{\prime}$ vectors are

  * •

T1:

$X_{T1}=\left(\begin{array}[]{c}0\\\
-\frac{\mu_{3}^{2}}{8(\lambda_{3}+\lambda_{4})}\\\ 0\\\ 0\\\ 0\\\
-\frac{\mu_{3}^{2}}{4(\lambda_{3}+\lambda_{4})}-v_{5}^{2}\\\
\end{array}\right),\qquad
V^{\prime}_{T1}=A+BX_{T1}=\left(\begin{array}[]{c}\mu_{2}^{2}-\frac{\lambda_{2}\mu_{3}^{2}}{2(\lambda_{3}+\lambda_{4})}\\\
0\\\ 0\\\
\frac{\lambda_{3}\mu_{3}^{2}}{2(\lambda_{3}+\lambda_{4})}+2\lambda_{3}v_{5}^{2}\\\
0\\\ 0\\\ \end{array}\right)\,.$ (73)

  * •

T2:

$X_{T2}=\left(\begin{array}[]{c}0\\\
-\frac{\mu_{3}^{2}}{8(\lambda_{3}+\lambda_{4})}\\\ 0\\\ 0\\\ 0\\\
-\frac{\mu_{3}^{2}}{4(\lambda_{3}+\lambda_{4})}\\\ \end{array}\right),\qquad
V^{\prime}_{T2}=A+BX_{T2}=\left(\begin{array}[]{c}\mu_{2}^{2}-\frac{\lambda_{2}\mu_{3}^{2}}{2(\lambda_{3}+\lambda_{4})}\\\
0\\\ 0\\\ \frac{\lambda_{3}\mu_{3}^{2}}{2(\lambda_{3}+\lambda_{4})}\\\ 0\\\
0\\\ \end{array}\right)\,.$ (74)

##  6 Stability of the viable models

###  6.1 Stability of dark matter minima

Let us now consider the coexistence of a dark matter minimum with any of the
other stationary points (minima or not) listed above. The starting point of
this analysis is that the minimization equations of the potential admit
solutions of two different types simultaneously. We force the dark matter
solution to be a minimum while the stationary point to be compared has no
constraints on the second derivatives. The bilinear formalism provides an easy
method to compute and compare the values of the potential at each of those two
stationary points.

Let us start by comparing the DM vacuum with all the CB ones. We will analyse
in detail two of the cases, the arguments will be the same for all the others.

  * •

CB1 vs. DM:

    * –

for $\lambda_{5}>0$

$V_{CB1}-V_{DM}=\frac{1}{2}(v_{6}^{2}+v_{10}^{2})m_{H^{+}_{1}}^{2}+\frac{1}{4}v_{1}^{2}\lambda_{5}(v_{6}-v_{10})^{2}>0\,,$
(75)

    * –

for $\lambda_{5}<0$

$V_{CB1}-V_{DM}=\frac{1}{2}(v_{6}^{2}+v_{10}^{2})m_{H^{+}_{2}}^{2}-\frac{1}{4}v_{1}^{2}\lambda_{5}(v_{6}+v_{10})^{2}>0\,,$
(76)

where all explicit VEVs are from the CB1 stationary point, and all masses are
from the DM minimum. Given our initial hypothesis of the DM phase being a
minimum, the squared scalar mass $m_{H^{+}_{1}}^{2}$ is necessarily positive;
likewise, we see that all VEV combinations figuring in the above expressions
are perforce positive – and the signs affecting the $\lambda_{5}$ coupling
render all terms where it appears also positive. Thus, the difference between
the potential at the different SPs, $V_{CB1}-V_{DM}$ is always positive – this
means that, if there is a DM minimum, any CB1 extremum will necessarily lie
above it.

  * •

CB2 vs. DM:

$V_{\text{CB2}}-V_{\text{DM}}=\frac{m_{H^{+}_{1}}^{2}v_{6}^{2}}{2}\,,$ (77)

where all explicit VEVs are from CB2 and all masses are from the DM phase.
This difference is clearly always positive when the DM phase is a minimum.

  * •

CB3 vs. DM:

$V_{\text{CB3}}-V_{\text{DM}}=\frac{m_{H^{+}_{2}}^{2}v_{6}^{2}}{2}\,,$ (78)

where again all explicit VEVs are from CB3 and all masses are from the DM
phase. And this difference is again always positive.

  * •

CB4 and CB5 vs. DM:

$V_{\text{CB4/5}}-V_{\text{DM}}=\frac{m_{h}^{4}+32\lambda_{1}\mu_{3}^{2}v_{6}^{2}}{64\lambda_{1}}\,,$
(79)

where all explicit VEVs are from CB4 or CB5 and all masses are from the DM
phase. This difference can be either positive or negative – notice how the
quadratic coupling $\mu_{3}^{2}$ can be negative, and therefore the sign of
the potential difference between the two stationary points is not determined.
In fact, a numerical study of this case found regions of parameter space for
which $V_{\text{CB4/5}}-V_{\text{DM}}>0$, but also other regions where
$V_{\text{CB4/5}}-V_{\text{DM}}<0$.

  * •

Custodial vacuum vs. DM:

$V_{\text{C}}-V_{\text{DM}}=\frac{3m_{2}^{2}v_{5}^{2}}{4}\,,$ (80)

where all explicit VEVs are from the custodial vacuum and all masses are from
the DM phase. This difference is thus always positive.

  * •

Non-custodial vacuum vs. DM:

$V_{NC}-V_{DM}=\frac{1}{12}\left[m_{1}^{2}\left(\sqrt{2}v_{5}-v_{8}\right)^{2}+m_{2}^{2}\left(v_{5}+\sqrt{2}v_{8}\right)^{2}\right]\,,$
(81)

where all explicit VEVs are from the non-custodial vacuum and the masses are
from the DM minimum. This difference is thus always positive.

  * •

W1, W2, T1, T2 vs. DM:

$V_{W1,W2,T1,T2}-V_{DM}=\frac{1}{16}\left(\frac{\mu_{2}^{4}}{\lambda_{1}}-\frac{\mu_{3}^{4}}{\lambda_{3}+\lambda_{4}}\right)\,,$
(82)

The difference in values of the potential can be positive or negative,
depending on the magnitudes of both quadratic coefficients666Notice that, due
to the bounded from below conditions of Eq. (16), the combinations of
$\lambda$ coefficients in Eq. (82) are necessarily positive..

Therefore, taking into account all the conditions above, in order to force the
DM vacuum to be an absolute minimum at tree-level we just need to impose the
two conditions:

$\displaystyle\frac{8\lambda_{1}\mu_{3}^{4}}{\lambda_{3}+2\lambda_{4}}$
$\displaystyle\leq$ $\displaystyle m_{h}^{4}\,,$ (83)
$\displaystyle\frac{\mu_{2}^{4}}{\lambda_{1}}-\frac{\mu_{3}^{4}}{\lambda_{3}+\lambda_{4}}$
$\displaystyle\geq$ $\displaystyle 0\,,$ (84)

where, remember, in the DM minimum $h$ is supposed to represent the SM-like
Higgs boson and as such $m_{h}=125$ GeV. Using Eq. (52) and Eq. (53), we can
cast these two equations in a different form, as they become lower bounds on
combinations of $\lambda_{3}$ and $\lambda_{4}$:

$\mbox{min}\left(\frac{1}{2}\lambda_{3}+\lambda_{4}\,,\,\lambda_{3}+\lambda_{4}\right)\;\geq\;\frac{\mu_{3}^{4}}{2v^{2}m^{2}_{h}}\,.$
(85)

Notice that the bounded from below conditions of Eq. (16) force these two
combinations of $\lambda_{3}$ and $\lambda_{4}$ to be positive, this
constraint is therefore more restrictive777 Though this constraint presupposes
the existence of some Wrong-electroweak or Tantamount stationary point
coexisting with the DM minimum..

###  6.2 Stability of the custodial vacuum

As in the previous section for the DM phase, our starting point now will be to
assume the existence of a custodial minimum, coexisting with other types of
stationary points. Let us go through the several possibilities:

  * •

DM vs. Custodial:

$V_{DM}-V_{C}=\frac{3v_{1}^{2}v_{5}^{2}(4\lambda_{1}\mu_{3}^{2}-2\lambda_{2}\mu_{2}^{2}+\lambda_{5}\mu_{2}^{2})^{2}}{4\lambda_{1}m_{h}^{2}m_{H}^{2}}\,,$
(86)

where please notice that the masses figuring in this expression are now those
obtained in the custodial vacuum (see Eq. (47), Eq. (48) and Eq. (49)). This
difference in values of the potential at the two stationary points is clearly
positive, which indicates that, whenever there is a custodial minimum, any
existing DM stationary point will necessarily lie above it. Notice, though,
that we had already concluded, in Eq. (80), that a DM minimum implied that any
custodial stationary point needed to lie above it. The two conclusions are not
contradictory, rather they imply that if a DM (custodial) minimum exists, any
coexisting custodial (DM) stationary point lies above it, and is necessarily a
saddle point. In fact, comparing Eq. (80) and Eq. (86), we see that when the
minimum corresponds to the DM phase, then $V_{DM}-V_{C}<0$ which, from the
equation above, implies that one of the squared scalar masses of the custodial
stationary point, $m_{h}^{2}$ or $m_{H}^{2}$, must perforce be negative, while
the other is positive – thus we conclude that not only the custodial
stationary point lies above the DM minimum, it is also necessarily a saddle
point. Analogously, if we have a custodial minimum then $V_{DM}-V_{C}>0$ and
from Eq. (80) we see that one must necessarily have one of the squared masses
computed at the DM stationary point, $m_{2}^{2}$, be negative – it is easy to
then show that in that case the DM stationary point is also a saddle point,
lying above the custodial minimum.

  * •

Non-custodial vs. custodial:

$V_{NC}-V_{C}=\frac{A^{NC/C}}{B^{NC/C}}$ (87)

with

$\displaystyle A^{NC/C}$ $\displaystyle=$
$\displaystyle-\lambda_{3}\left(2v_{5}^{2}-v_{8}^{2}\right)\left\\{-v_{1}^{4}\left(-32v_{5}^{6}v_{8}^{2}-4\sqrt{2}v_{5}^{5}v_{8}^{3}+50v_{5}^{4}v_{8}^{4}-11\sqrt{2}v_{5}^{3}v_{8}^{5}-26v_{5}^{2}v_{8}^{6}\right.\right.$
(88)
$\displaystyle\left.+4\sqrt{2}v_{5}^{7}v_{8}+8v_{5}^{8}+14\sqrt{2}v_{5}v_{8}^{7}-4v_{8}^{8}\right)$
$\displaystyle\times\left\\{v_{5}^{2}\left[\lambda_{2}^{2}-4\lambda_{1}(\lambda_{3}+\lambda_{4})\right]-2v_{8}^{2}\left[\lambda_{2}^{2}-2\lambda_{1}(\lambda_{3}+2\lambda_{4})\right]\right\\}$
$\displaystyle\left.+4\lambda_{2}\lambda_{3}v_{1}^{2}v_{5}v_{8}^{2}\left(2v_{5}^{2}-v_{8}^{2}\right)\right.$
$\displaystyle\times\left(-12v_{5}^{5}v_{8}^{2}-8\sqrt{2}v_{5}^{4}v_{8}^{3}+17v_{5}^{3}v_{8}^{4}+3\sqrt{2}v_{5}^{2}v_{8}^{5}+4\sqrt{2}v_{5}^{6}v_{8}+4v_{5}^{7}-10v_{5}v_{8}^{6}+2\sqrt{2}v_{8}^{7}\right)$
$\displaystyle+2\lambda_{3}v_{5}^{2}v_{8}^{2}\left(2v_{5}^{2}-v_{8}^{2}\right)\left[24v_{5}^{8}(\lambda_{3}+\lambda_{4})-16v_{5}^{6}v_{8}^{2}(5\lambda_{3}+6\lambda_{4})-12\sqrt{2}v_{5}^{5}v_{8}^{3}(\lambda_{3}+\lambda_{4})\right.$
$\displaystyle+6v_{5}^{4}v_{8}^{4}(17\lambda_{3}+25\lambda_{4})-\sqrt{2}v_{5}^{3}v_{8}^{5}(17\lambda_{3}+33\lambda_{4})$
$\displaystyle-6v_{5}^{2}v_{8}^{6}(7\lambda_{3}+13\lambda_{4})+12\sqrt{2}v_{5}^{7}v_{8}(\lambda_{3}+\lambda_{4})$
$\displaystyle\left.\left.+6\sqrt{2}v_{5}v_{8}^{7}(3\lambda_{3}+7\lambda_{4})-4v_{8}^{8}(\lambda_{3}+3\lambda_{4})\right]\right\\}\,,$
$\displaystyle B^{NC/C}$ $\displaystyle=$ $\displaystyle
2\sqrt{2}\left(v_{5}+\sqrt{2}v_{8}\right)^{2}\left(\sqrt{2}v_{5}^{2}+v_{5}v_{8}-\sqrt{2}v_{8}^{2}\right)$
(89)
$\displaystyle\times\left\\{v_{1}^{4}\left(-3v_{5}^{2}v_{8}^{2}+2\sqrt{2}v_{5}^{3}v_{8}+2v_{5}^{4}-2\sqrt{2}v_{5}v_{8}^{3}+2v_{8}^{4}\right)\left[3\lambda_{2}^{2}-4\lambda_{1}(\lambda_{3}+3\lambda_{4})\right]\right.$
$\displaystyle\left.+12\lambda_{2}\lambda_{3}v_{1}^{2}v_{5}v_{8}\left(2v_{5}^{2}-v_{8}^{2}\right)\left(\sqrt{2}v_{5}^{2}+v_{5}v8-\sqrt{2}v_{8}^{2}\right)+12\lambda_{3}^{2}v_{5}^{2}v_{8}^{2}\left(v_{8}^{2}-2v_{5}^{2}\right)^{2}\right\\}\,.\qquad$

The VEVs are from the NC vacuum. The difference can be positive or negative.

  * •

CB1 vs. Custodial:

$V_{CB1}-V_{C}=\frac{A^{CB1/C}}{B^{CB1/C}},$ (90)

with

$\displaystyle
A^{CB1/C}=\mu_{3}^{4}\left[1024\lambda_{1}^{2}\lambda_{3}^{2}+128\lambda_{1}\lambda_{3}\lambda_{5}(\lambda_{5}-3\lambda_{2})+3\lambda_{5}^{2}(\lambda_{5}-2\lambda_{2})^{2}\right]$
$\displaystyle+8\mu_{2}^{2}\mu_{3}^{2}\left\\{32\lambda_{1}\lambda_{3}\left[3\lambda_{5}(\lambda_{3}+\lambda_{4})-4\lambda_{2}\lambda_{3}\right]+3\lambda_{5}(2\lambda_{2}-\lambda_{5})\left[4\lambda_{2}\lambda_{3}-\lambda_{5}(\lambda_{3}+\lambda_{4})\right]\right\\}$
$\displaystyle+16\mu_{2}^{4}\left[16\lambda_{2}^{2}\lambda_{3}^{2}-24\lambda_{2}\lambda_{3}\lambda_{5}(\lambda_{3}+\lambda_{4})+\lambda_{5}^{2}(\lambda_{3}+\lambda_{4})(7\lambda_{3}+3\lambda_{4})\right]\,,\qquad$
(91) $\displaystyle
B^{CB1/C}=16\left[3(\lambda_{5}-2\lambda_{2})^{2}-16\lambda_{1}(\lambda_{3}+3\lambda_{4})\right]\left[8\lambda_{2}^{2}\lambda_{3}-(\lambda_{3}+\lambda_{4})\left(32\lambda_{1}\lambda_{3}+\lambda_{5}^{2}\right)\right]\,.\qquad$
(92)

Numerically, we found that a coexisting CB1 vacuum can have an energy above or
below the Custodial vacuum.

  * •

CB2 vs. Custodial:

$V_{CB2}-V_{C}=\frac{1}{8}\left[\frac{9m_{3}^{2}\left(v_{C,1}^{2}v_{CB,6}^{2}+2v_{CB,1}^{2}v_{C,5}^{2}\right)}{v_{C,1}^{2}+8v_{C,5}^{2}}+m_{5}^{2}v_{CB,6}^{2}\right]\,,$
(93)

where VEVs are marked as either being charge-breaking or custodial and all
masses are from the custodial vacuum. The difference is thus always positive.

  * •

CB3 vs. Custodial:

$V_{CB3}-V_{C}=\frac{1}{8}\left[\frac{m_{3}^{2}\left(v_{C,1}^{2}v_{CB,6}^{2}+6v_{CB,1}^{2}v_{C,5}^{2}\right)}{v_{C,1}^{2}+8v_{C,5}^{2}}+m_{5}^{2}v_{CB,6}^{2}\right]\,,$
(94)

where VEVs are marked as either being charge-breaking or custodial and all
masses are from the custodial vacuum. The difference is always positive.

  * •

CB4/5 vs. Custodial:

$V_{CB4/5}-V_{C}=-\frac{v_{1}^{2}(-2\lambda_{2}\mu_{3}^{2}+2\lambda_{3}\mu_{2}^{2}+4\lambda_{4}\mu_{2}^{2}+\lambda_{5}\mu_{3}^{2})+2\lambda_{3}\mu_{3}^{2}v_{5}^{2}}{8(\lambda_{3}+2\lambda_{4})}\,,$
(95)

where VEVs are from the custodial vacuum. The difference can be positive or
negative.

  * •

W1, W2, T1 and T2 vs. Custodial. For all these scenarios the difference in
height in the potential is given by the expression

$V_{W1,W2,T1,T2}-V_{C}=\frac{v^{2}\left(2\mu_{3}^{2}v_{5}^{2}\left(m_{5}^{2}-2\lambda_{2}v_{1}^{2}\right)+\mu_{2}^{2}v_{1}^{2}\left(m_{5}^{2}+8\lambda_{4}v_{5}^{2}\right)\right)-m_{3}^{2}\left(3\mu_{2}^{2}v_{1}^{4}+2\mu_{3}^{2}v_{1}^{2}v_{5}^{2}\right)}{12m_{3}^{2}v_{1}^{2}-4v^{2}\left(m_{5}^{2}+8\lambda_{4}v_{5}^{2}\right)}\,,$
(96)

where VEVs and masses are from the Custodial vacuum. The difference can be
positive or negative.

Therefore, we can finally state the conditions to have the custodial extremum
as the absolute minimum. Besides having all masses positive to guarantee that
we are indeed in a minimum, we need to force the parameters of the potential
to comply with the following conditions

$\displaystyle V_{NC}-V_{C}\geq 0,$ (97) $\displaystyle V_{CB1}-V_{C}\geq 0,$
(98)
$\displaystyle-\frac{v_{1}^{2}(-2\lambda_{2}\mu_{3}^{2}+2\lambda_{3}\mu_{2}^{2}+4\lambda_{4}\mu_{2}^{2}+\lambda_{5}\mu_{3}^{2})+2\lambda_{3}\mu_{3}^{2}v_{5}^{2}}{8(\lambda_{3}+2\lambda_{4})}\geq
0,$ (99)
$\displaystyle\frac{v^{2}\left(2\mu_{3}^{2}v_{5}^{2}\left(m_{5}^{2}-2\lambda_{2}v_{1}^{2}\right)+\mu_{2}^{2}v_{1}^{2}\left(m_{5}^{2}+8\lambda_{4}v_{5}^{2}\right)\right)-m_{3}^{2}\left(3\mu_{2}^{2}v_{1}^{4}+2\mu_{3}^{2}v_{1}^{2}v_{5}^{2}\right)}{12m_{3}^{2}v_{1}^{2}-4v^{2}\left(m_{5}^{2}+8\lambda_{4}v_{5}^{2}\right)}\geq
0.$ (100)

The first equation, Eq. (97), is written using Eq. (87), Eq. (88) and Eq.
(89). They are written as a function of the non-custodial VEVs that obey Eqs.
(57). The remaining three equations Eq. (98), Eq. (99), Eq. (100) are written
as a function of the parameters of the potential, the masses in the custodial
phase and the custodial VEVs. In particular, Eq. (98) needs Eq. (90), Eq. (91)
and Eq. (92).

##  7 Conclusions

We have analysed the vacuum structure of a simple version of the GM model
where the scalar potential is not only invariant under $SU(2)_{L}\times
SU(2)_{R}$ but is also invariant under the $\mathbb{Z}_{2}$ symmetry of the
triplet fields, $\Xi\to-\Xi$. We have then used the invariance freedom to
reduce the number of possible VEVs to 7. In this first attempt to understand
the vacuum structure of the model we have further reduced the number to 5 by
considering only the real component of the VEVs. In that sense the conditions
obtained for the potential to be in an absolute minimum are necessary
conditions.

We have classified the phases of the model in viable phases, the Custodial and
DM phases, and the non-viable. The non-viable are the non-custodial vacuum
where 5 Goldstone bosons are generated, the charge-breaking vacua (CB1,2 with
6 Goldstone bosons and CB3,4,5 with 5 Goldstone bosons), and the wrong-
electroweak and tantamount vacua with 4 Goldstone bosons. We have then derived
a set of analytical formulae that allows to compare the difference in the
depths of two stationary points from two distinct phases. If we want to be
certain that we have an absolute minimum in the DM phase, we just need that
all masses be positive in that phase and to impose the conditions given in
equation (85). These are the two conditions for which we have checked
numerically that the difference in depths could be either positive or
negative. Hence, we need to force the DM minimum to be below the other phases’
stationary point. If in turn we want the custodial vacuum to be a minimum we
impose positivity of the masses in that phase together with the conditions
expressed by equations (97), (98), (99) and (100). Note that there are many
cases where the minimum is always below the stationary point of the other
phase at tree-level. This is for instance the case where the DM minimum is
compared with the Custodial stationary point or when the Custodial minimum is
compared with the DM stationary point.

The Georgi-Machacek model has therefore an elaborate vacuum structure – vacua
with Dark Matter candidates; vacua which preserve or not custodial symmetry;
and patterns of electroweak breaking with remaining symmetry group $U(1)\times
U(1)$, with massless photon and $Z$, or a wrong value predicted for the
Weinberg angle. Crucially, though, this foray into the many possible vacua of
the model revealed that neither of the vacua of interest – the Dark Matter
minimum or the minimum which preserves custodial symmetry – are automatically
stable, as occurs for instance in the 2HDM [22, 23, 26, 30], a model for which
it was proven that minima which break different symmetries cannot coexist. For
the GM model, though, we have shown that, though the Dark Matter minimum
cannot coexist with deeper Charge Breaking, Custodial or Non-Custodial vacua,
it can nonetheless have deeper stationary points with incorrect electroweak
symmetry breaking (yielding massless $Z$’s or the wrong value of
$\theta_{W}$). Further, the Custodial minimum is not safe from eventual
tunnelling to a deeper charge breaking vacuum of the type CB1, or indeed to a
non-custodial vacuum (this last conclusion is known, see [12, 20]). The
situation will of course become even more complex when one starts analysing
the GM model with soft breaking terms of the $\mathbb{Z}_{2}$ considered in
this paper – the more common form of the GM model, studied for instance in
[12]. Such terms will obviously change the possible vacua and, being cubic in
the fields, will necessarily complicate the algebra. Many of the stranger
vacua discussed here – the Wrong-Electroweak and Tantamount ones – may well no
longer be possible when soft breaking terms appear in the potential. On the
other hand, taking the 2HDM as an example, new types minima may become
possible with those soft breaking terms (such as for instance spontaneous CP
breaking vacua), and coexistence of minima impossible with exact symmetries
may be possible when they are softly broken [27, 30].

##  Appendix A Vacua

We have presented the simplest and most general vacuum structure in Eq. (20).
We will now show the possible types of vacua in the $\mathbb{Z}_{2}$ symmetric
model. The most general vacuum configuration is written as

$\Phi=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}v_{1}&0\\\
0&v_{1}\end{array}\right),\qquad\qquad
X=\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}v_{8}&v_{6}&0\\\
-v_{10}&\sqrt{2}v_{5}&v_{10}\\\ 0&-v_{6}&v_{8}\end{array}\right)\,,$ (101)

and leads to the following minimum conditions:

$\displaystyle\frac{\partial V}{\partial\varphi_{1}}\Bigg{|}_{0}$
$\displaystyle=$ $\displaystyle
v_{1}\left[\mu_{2}^{2}+4\lambda_{1}v_{1}^{2}+2\lambda_{2}(v_{5}^{2}+v_{6}^{2}+v_{8}^{2}+v_{10}^{2})-\lambda_{5}(v_{10}v_{6}+\sqrt{2}v_{5})v_{8}+\frac{v_{8}^{2}}{2}\right],$
(102) $\displaystyle\frac{\partial V}{\partial\varphi_{3}}\Bigg{|}_{0}$
$\displaystyle=$
$\displaystyle-\frac{\lambda_{5}}{\sqrt{2}}v_{1}v_{8}(v_{6}+v_{10}),$ (103)
$\displaystyle\frac{\partial V}{\partial\varphi_{5}}\Bigg{|}_{0}$
$\displaystyle=$ $\displaystyle
v_{5}\left[\mu_{3}^{2}+4\lambda_{4}v_{8}^{2}+4(v_{5}^{2}+v_{6}^{2}+v_{10}^{2})(\lambda_{3}+\lambda_{4})-2\sqrt{2}\lambda_{3}\frac{v_{6}v_{8}v_{10}}{v_{5}}\right.$
(104)
$\displaystyle\left.+v_{1}^{2}\left(2\lambda_{2}-\frac{\lambda_{5}v_{8}}{\sqrt{2}v_{5}}\right)\right],$
$\displaystyle\frac{\partial V}{\partial\varphi_{6}}\Bigg{|}_{0}$
$\displaystyle=$ $\displaystyle
v_{6}\left[\mu_{3}^{2}+4\lambda_{4}v_{10}^{2}+4(v_{5}^{2}+v_{6}^{2})(\lambda_{3}+\lambda_{4})+2v_{8}^{2}(\lambda_{3}+2\lambda_{4})-2\sqrt{2}\lambda_{3}\frac{v_{5}v_{8}v_{10}}{v_{6}}\right.$
(105)
$\displaystyle\left.+v_{1}^{2}\left(2\lambda_{2}-\frac{\lambda_{5}v_{10}}{2v_{6}}\right)\right],$
$\displaystyle\frac{\partial V}{\partial\varphi_{8}}\Bigg{|}_{0}$
$\displaystyle=$ $\displaystyle
v_{8}\left\\{\mu_{3}^{2}+4\lambda_{4}v_{5}^{2}+2(v_{6}^{2}+v_{8}^{2}+v_{10}^{2})(\lambda_{3}+2\lambda_{4})-2\sqrt{2}\lambda_{3}\frac{v_{5}v_{6}v_{10}}{v_{8}}\right.$
(106)
$\displaystyle\left.+v_{1}^{2}\left[2\lambda_{2}-\left(1+\sqrt{2}\frac{v_{5}}{v_{8}}\right)\frac{\lambda_{5}}{2}\right]\right\\},$
$\displaystyle\frac{\partial V}{\partial\varphi_{10}}\Bigg{|}_{0}$
$\displaystyle=$ $\displaystyle
v_{10}\left[\mu_{3}^{2}+4\lambda_{4}v_{6}^{2}+4(v_{5}^{2}+v_{10}^{2})(\lambda_{3}+\lambda_{4})+2v_{8}^{2}(\lambda_{3}+2\lambda_{4})-2\sqrt{2}\lambda_{3}\frac{v_{5}v_{6}v_{8}}{v_{10}}+\right.$
(107)
$\displaystyle\left.v_{1}^{2}\left(2\lambda_{2}-\frac{\lambda_{5}v_{6}}{2v_{10}}\right)\right],$
$\displaystyle\frac{\partial V}{\partial\varphi_{11}}\Bigg{|}_{0}$
$\displaystyle=$ $\displaystyle
2\lambda_{3}\left(\sqrt{2}v_{5}v_{6}v_{10}-v_{8}v_{10}^{2}-v_{6}^{2}v_{8}\right),$
(108)

and the conditions for the remaining fields are identically zero.

The minimum conditions for $\varphi_{3}$ can be used as a starting point of
three main subgroups with either $v_{1}=0$, $v_{8}=0$ or $v_{6}=-v_{10}$. We
then found step by step all configurations that led to stationary points of
the potential. All charge breaking minima have in common the following
conditions

$v_{5}=v_{8}=0,\qquad v_{6}\neq 0,\text{ and }v_{10}\neq 0,$ (109)

that is, the neutral VEVs from the triplet have to vanish while the charged
ones have to be non-zero. There are also Charge-Breaking minima that obey all
the above conditions plus $v_{1}=0$. In principle there could be other vacuum
configurations that would break electric charge but we found that they are
never stationary points. By taking $v_{5}=v_{8}=0$ only three stationarity
equations survive, the ones related to $\varphi_{1}$, $\varphi_{6}$ and
$\varphi_{10}$. This imposes conditions on three (arbitrary) Lagrangian
parameters, and we choose

$\displaystyle\mu_{2}^{2}$
$\displaystyle=-4\lambda_{1}v_{1}^{2}-\frac{8\lambda_{3}v_{10}^{2}v_{6}^{2}}{v_{1}^{2}}-2\lambda_{2}\left(v_{10}^{2}+v_{6}^{2}\right)\,,$
(110) $\displaystyle\mu_{3}^{2}$
$\displaystyle=-2\left(\lambda_{2}v_{1}^{2}+2(\lambda_{3}+\lambda_{4})\left(v_{10}^{2}+v_{6}^{2}\right)\right)\,,$
$\displaystyle\lambda_{5}$
$\displaystyle=-\frac{8\lambda_{3}v_{10}v_{6}}{v_{1}^{2}}\,.$

The condition $v_{5}=v_{8}=0$ together with $v_{6}\neq 0\text{ and }v_{10}\neq
0$ leads to the following mass matrix for the gauge bosons in the
$W^{1},W^{2},W^{3},B$ basis.

$\left(\begin{array}[]{cccc}\frac{1}{4}g^{2}\left(v_{1}^{2}+4v_{10}^{2}\right)&0&0&0\\\
0&\frac{1}{4}g^{2}\left(v_{1}^{2}+4\left(v_{6}^{2}+v_{10}^{2}\right)\right)&0&0\\\
0&0&\frac{1}{4}g^{2}\left(v_{1}^{2}+4v_{6}^{2}\right)&-\frac{1}{4}v_{1}^{2}gg^{\prime}\\\
0&0&-\frac{1}{4}v_{1}^{2}gg^{\prime}&\frac{1}{4}g^{\prime
2}\left(v_{1}^{2}+4v_{10}^{2}\right)\\\ \end{array}\right)$ (111)

Clearly there is no zero eigenstate and thus the photon becomes massive. There
is another Charge-Breaking stationary point with the conditions
$v_{1}=v_{5}=v_{8}=0$ together with $v_{6}\neq 0\text{ and }v_{10}\neq 0$.
Again for this case we see that no zero mass eigenstate emerges.

#### Acknowledgments

DA, PF and RS are supported by FCT, Contracts UIDB/00618/2020,
UIDP/00618/2020, PTDC/FIS-PAR/31000/2017, CERN/FISPAR/0002/2017, CERN/FIS-
PAR/0014/2019, and by the HARMONIA project, contract UMO-2015/18/M/ST2/0518.
HEL is supported by the Natural Sciences and Engineering Research Council of
Canada (NSERC). This work was also supported by the grant H2020-MSCA-RISE-2014
No. 645722 (NonMinimalHiggs).

## References

  * [1] H. Georgi and M. Machacek, _DOUBLY CHARGED HIGGS BOSONS_ , _Nucl. Phys. B_ 262 (1985) 463.
  * [2] M.S. Chanowitz and M. Golden, _Higgs Boson Triplets With M ( $W$) = M ($Z$) $\cos\theta\omega$_, _Phys. Lett. B_ 165 (1985) 105.
  * [3] J. Gunion, R. Vega and J. Wudka, _Higgs triplets in the standard model_ , _Phys. Rev. D_ 42 (1990) 1673.
  * [4] J. Gunion, R. Vega and J. Wudka, _Naturalness problems for rho = 1 and other large one loop effects for a standard model Higgs sector containing triplet fields_ , _Phys. Rev. D_ 43 (1991) 2322.
  * [5] K.-m. Cheung, R.J. Phillips and A. Pilaftsis, _Signatures of Higgs triplet representations at TeV e+ e- colliders_ , _Phys. Rev. D_ 51 (1995) 4731 [hep-ph/9411333].
  * [6] M. Aoki and S. Kanemura, _Unitarity bounds in the Higgs model including triplet fields with custodial symmetry_ , _Phys. Rev. D_ 77 (2008) 095009 [0712.4053].
  * [7] H.E. Logan and M.-A. Roy, _Higgs couplings in a model with triplets_ , _Phys. Rev. D_ 82 (2010) 115011 [1008.4869].
  * [8] C.-W. Chiang and K. Yagyu, _Testing the custodial symmetry in the Higgs sector of the Georgi-Machacek model_ , _JHEP_ 01 (2013) 026 [1211.2658].
  * [9] C. Englert, E. Re and M. Spannowsky, _Pinning down Higgs triplets at the LHC_ , _Phys. Rev. D_ 88 (2013) 035024 [1306.6228].
  * [10] C.-W. Chiang, A.-L. Kuo and K. Yagyu, _Enhancements of weak gauge boson scattering processes at the CERN LHC_ , _JHEP_ 10 (2013) 072 [1307.7526].
  * [11] S. Kanemura, K. Tsumura, K. Yagyu and H. Yokoya, _Fingerprinting nonminimal Higgs sectors_ , _Phys. Rev. D_ 90 (2014) 075001 [1406.3294].
  * [12] K. Hartling, K. Kumar and H.E. Logan, _The decoupling limit in the Georgi-Machacek model_ , _Phys. Rev. D_ 90 (2014) 015007 [1404.2640].
  * [13] C.-W. Chiang, S. Kanemura and K. Yagyu, _Novel constraint on the parameter space of the Georgi-Machacek model with current LHC data_ , _Phys. Rev. D_ 90 (2014) 115025 [1407.5053].
  * [14] C. Degrande, K. Hartling, H.E. Logan, A.D. Peterson and M. Zaro, _Automatic predictions in the Georgi-Machacek model at next-to-leading order accuracy_ , _Phys. Rev. D_ 93 (2016) 035004 [1512.01243].
  * [15] J. Chang, C.-R. Chen and C.-W. Chiang, _Higgs boson pair productions in the Georgi-Machacek model at the LHC_ , _JHEP_ 03 (2017) 137 [1701.06291].
  * [16] B. Keeshan, H.E. Logan and T. Pilkington, _Custodial symmetry violation in the Georgi-Machacek model_ , _Phys. Rev. D_ 102 (2020) 015001 [1807.11511].
  * [17] N. Ghosh, S. Ghosh and I. Saha, _Charged Higgs boson searches in the Georgi-Machacek model at the LHC_ , _Phys. Rev. D_ 101 (2020) 015029 [1908.00396].
  * [18] C.-W. Chiang, G. Cottin and O. Eberhardt, _Global fits in the Georgi-Machacek model_ , _Phys. Rev. D_ 99 (2019) 015001 [1807.10660].
  * [19] A. Ismail, H.E. Logan and Y. Wu, _Updated constraints on the Georgi-Machacek model from LHC Run 2_ , 2003.02272.
  * [20] K. Hartling, K. Kumar and H.E. Logan, _GMCALC: a calculator for the Georgi-Machacek model_ , 1412.7387.
  * [21] J. Velhinho, R. Santos and A. Barroso, _Tree level vacuum stability in two Higgs doublet models_ , _Phys. Lett._ B322 (1994) 213.
  * [22] P.M. Ferreira, R. Santos and A. Barroso, _Stability of the tree-level vacuum in two Higgs doublet models against charge or CP spontaneous violation_ , _Phys. Lett._ B603 (2004) 219 [hep-ph/0406231].
  * [23] A. Barroso, P.M. Ferreira and R. Santos, _Charge and CP symmetry breaking in two Higgs doublet models_ , _Phys. Lett._ B632 (2006) 684 [hep-ph/0507224].
  * [24] C.C. Nishi, _CP violation conditions in N-Higgs-doublet potentials_ , _Phys. Rev._ D74 (2006) 036003 [hep-ph/0605153].
  * [25] M. Maniatis, A. von Manteuffel, O. Nachtmann and F. Nagel, _Stability and symmetry breaking in the general two-Higgs-doublet model_ , _Eur. Phys. J._ C48 (2006) 805 [hep-ph/0605184].
  * [26] I.P. Ivanov, _Minkowski space structure of the Higgs potential in 2HDM_ , _Phys. Rev._ D75 (2007) 035001 [hep-ph/0609018].
  * [27] A. Barroso, P.M. Ferreira and R. Santos, _Neutral minima in two-Higgs doublet models_ , _Phys. Lett._ B652 (2007) 181 [hep-ph/0702098].
  * [28] C.C. Nishi, _The Structure of potentials with N Higgs doublets_ , _Phys. Rev._ D76 (2007) 055013 [0706.2685].
  * [29] M. Maniatis, A. von Manteuffel and O. Nachtmann, _CP violation in the general two-Higgs-doublet model: A Geometric view_ , _Eur. Phys. J._ C57 (2008) 719 [0707.3344].
  * [30] I.P. Ivanov, _Minkowski space structure of the Higgs potential in 2HDM. II. Minima, symmetries, and topology_ , _Phys. Rev._ D77 (2008) 015017 [0710.3490].
  * [31] M. Maniatis, A. von Manteuffel and O. Nachtmann, _A New type of CP symmetry, family replication and fermion mass hierarchies_ , _Eur. Phys. J._ C57 (2008) 739 [0711.3760].
  * [32] C.C. Nishi, _Physical parameters and basis transformations in the Two-Higgs-Doublet model_ , _Phys. Rev._ D77 (2008) 055009 [0712.4260].
  * [33] M. Maniatis and O. Nachtmann, _On the phenomenology of a two-Higgs-doublet model with maximal CP symmetry at the LHC_ , _JHEP_ 05 (2009) 028 [0901.4341].
  * [34] P. Ferreira, M. Maniatis, O. Nachtmann and J.P. Silva, _CP properties of symmetry-constrained two-Higgs-doublet models_ , _JHEP_ 08 (2010) 125 [1004.3207].
  * [35] I. Ivanov and C. Nishi, _Properties of the general NHDM. I. The Orbit space_ , _Phys. Rev. D_ 82 (2010) 015014 [1004.1799].
  * [36] I. Ivanov and C. Nishi, _Symmetry breaking patterns in 3HDM_ , _JHEP_ 01 (2015) 021 [1410.6139].
  * [37] P. Ferreira, _The vacuum structure of the Higgs complex singlet-doublet model_ , _Phys. Rev. D_ 94 (2016) 096011 [1607.06101].
  * [38] P. Ferreira, M. Mühlleitner, R. Santos, G. Weiglein and J. Wittbrodt, _Vacuum Instabilities in the N2HDM_ , _JHEP_ 09 (2019) 006 [1905.10234].
  * [39] I. Engeln, P. Ferreira, M.M. Mühlleitner, R. Santos and J. Wittbrodt, _The Dark Phases of the N2HDM_ , _JHEP_ 08 (2020) 085 [2004.05382].
  * [40] P. Ferreira and B. Gonçalves, _Stability of neutral minima against charge breaking in the Higgs triplet model_ , _JHEP_ 02 (2020) 182 [1911.09746].
  * [41] D. Das and I. Saha, _Cornering variants of the Georgi-Machacek model using Higgs precision data_ , _Phys. Rev. D_ 98 (2018) 095010 [1811.00979].
  * [42] M. Cirelli, N. Fornengo and A. Strumia, _Minimal dark matter_ , _Nucl. Phys. B_ 753 (2006) 178 [hep-ph/0512090].