# Unavoidable Higgs coupling deviations in the $Z_{2}$-symmetric Georgi-
Machacek model

Carlos Henrique de Lima<EMAIL_ADDRESS>Heather E.
Logan<EMAIL_ADDRESS>Ottawa-Carleton Institute for Physics,
Carleton University, Ottawa, ON K1S 5B6, Canada

(September 17, 2022)

###### Abstract

The $Z_{2}$-symmetric version of the Georgi-Machacek model does not possess a
decoupling limit in which all the new particles can be made arbitrarily heavy,
opening the possibility that the model can be _entirely_ excluded if
experiments reveal no deviations from the Standard Model. We explore this
model, focusing on the part of parameter space in which the vacuum expectation
value of the triplets, $\nu_{\chi}$, is small. In the small-$\nu_{\chi}$
limit, the second custodial-singlet scalar field $S$ necessarily becomes very
light and can contribute to the total width of the 125 GeV Higgs boson $h$ via
$h\to SS$. We show that this process, together with LHC measurements of the
$h\to\gamma\gamma$ rate, entirely excludes masses $m_{S}<m_{h}/2$ and thereby
severely constrains the parameter space, setting an experimental lower bound
$\nu_{\chi}\gtrsim 12.5$ GeV on the vacuum expectation value of the triplets.
This lower bound makes it impossible to avoid deviations from the Standard
Model in the couplings of $h$ to fermion and vector boson pairs. We study the
remaining parameter space after imposing constraints from direct searches for
the additional Higgs bosons, and show that it is on the edge of being fully
excluded at $95\%$ confidence level by LHC measurements of the 125 GeV Higgs
boson’s couplings. Measurements of these couplings at the future high-
luminosity run of the LHC will have sufficient precision to entirely exclude
the model at $5\sigma$ if no deviations from the Standard Model are observed.

##  I Introduction

The discovery of the 125 GeV Higgs boson at the CERN Large Hadron Collider
(LHC) [1, 2] opened a new avenue to test the Standard Model (SM) through
measurements of Higgs boson properties. So far the results indicate that the
SM remains in good agreement with experiments; while we know that the SM
cannot be the full story, we also know that any corrections from new physics
appear to be small. Many different models have been proposed as extensions to
the SM that take this into account. One of these extensions is the Georgi-
Machacek (GM) model [3, 4], which extends the scalar sector of the SM by
adding two SU(2)L triplet fields.

The GM model has several interesting features that motivate its proposal and
its use as a benchmark in LHC searches. One of them is the possibility to
enhance the Higgs coupling to vector bosons compared to its strength in the SM
[5, 6, 7, 8]. This enhancement can appear at tree level only in scalar sectors
containing representations of SU(2)L larger than the usual SM doublet.
Associated with this enhancement is the novel presence of doubly-charged,
singly-charged, and neutral Higgs bosons
$(H_{5}^{\pm\pm},H_{5}^{\pm},H_{5}^{0})$ transforming as a fiveplet under the
custodial symmetry, which couple to vector boson pairs with a strength
proportional to the vacuum expectation values (vevs) of the SU(2)L triplet
fields. The general GM model has been extensively studied in the literature
[9, 10, 11, 12, 13, 14, 15, 16, 17].

In this paper, we focus on a constrained version of the GM model in which the
scalar potential is invariant under a $Z_{2}$ symmetry under which the
SU(2)L-triplet scalar fields are odd. Imposing the $Z_{2}$ symmetry eliminates
two trilinear terms in the scalar potential. This version of the GM model,
which we will refer to as the Z2GM model, was in fact the original model
written down by Chanowitz and Golden in Ref. [4], in which they imposed the
$Z_{2}$ symmetry for simplicity. The Z2GM model was considered in all the
pioneering phenomenological studies until the scalar potential trilinear terms
were first written down by Aoki and Kanemura in Ref. [16]. One interesting
feature of the Z2GM model is that the $Z_{2}$ symmetry eliminates the
dimension-four operators by which the complex SU(2)L-triplet scalar would
generate a Majorana mass for the SM neutrinos, thereby eliminating the need to
require that the Yukawa couplings of these operators be extremely tiny. The
model also possesses a dark matter phase in which the SU(2)L triplets do not
acquire vevs so that the lightest $Z_{2}$-odd state is stable. In this paper,
we will however consider only the phase in which the triplet vevs are nonzero
and the $Z_{2}$ symmetry is spontaneously broken.111In our opinion the dark
matter phase of the Z2GM model is not particularly well-motivated as a model
for dark matter because the main motivation for the global symmetry structure
of the GM model in the first place is the prevention of large contributions to
the electroweak $\rho$ parameter. These contributions are absent when the vevs
of the SU(2)L triplets are zero, so there is no particular reason why a dark
matter model involving SU(2)L triplets would require the restrictive global
symmetry structure of the GM model.

The most important feature of the Z2GM model for our purposes is that it does
not possess a decoupling limit [9]. This stems from the fact that, in the
absence of the trilinear terms forbidden by the $Z_{2}$ symmetry, the scalar
potential involves only two dimensionful parameters, which can both be
eliminated in favour of the vevs of the doublet and triplet scalar fields.
These vevs are in turn bounded from above by the $W$ and $Z$ boson mass
relations. The masses of all the physical scalars in the Z2GM model can then
be expressed as proportional to these vevs multiplied by various combinations
of scalar quartic couplings. Imposing perturbative unitarity on the scalar
quartic couplings then bounds the masses of all the additional Higgs bosons in
the Z2GM model to lie below about 700 GeV [16]. This fact, together with the
increasing precision in the Higgs coupling measurements, can potentially allow
the model to be _entirely_ ruled out in future experiments. A similar analysis
[18, 19] of the non-decoupling $Z_{2}$-symmetric version of the two-Higgs-
doublet model has already entirely excluded that model.

A further curious feature of the Z2GM model, evident already in the mass
formulas of Ref. [4] but first studied explicitly in Ref. [8], is that in the
limit that the SU(2)L-triplet scalar vevs become very small, the mass of one
of the custodial-singlet Higgs bosons (which we will call $S$) also becomes
very small. This opens the possibility of a new decay mode for the 125 GeV
Higgs boson into a pair of these lighter states; i.e., $h\to SS$. We show that
this possibility is experimentally excluded by a combination of measurements
of the $h$ total width and the $h\to\gamma\gamma$ decay rate (the latter
indirectly constrains model parameters in a way that prevents the $hSS$
coupling from being dialed to zero). This imposes a lower bound of
$m_{S}>m_{h}/2$ and, due to the relationship between $m_{S}$ and the triplet
vevs in the Z2GM model, also puts an _experimental_ lower bound on the triplet
vevs, thereby guaranteeing nonzero deviations from the SM in the tree-level
couplings of $h$ to fermion and vector boson pairs. Based on projections of
Higgs coupling measurement precision at the high-luminosity LHC (HL-LHC) [20],
we show that the surviving parameter space of the Z2GM model will be entirely
excluded at $5\sigma$ if the measured central values of the Higgs couplings
remain SM-like.

The phenomenological implications of experimental constraints on the Z2GM
model, particularly from Higgs coupling measurements, were previously studied
in Refs. [8, 13, 21]. Our analysis updates the pioneering 2012–13 studies of
Refs. [8, 13] with the latest available LHC Higgs coupling data. It also
extends the recent analysis in Ref. [21] by including a detailed analysis of
the $m_{S}<m_{h}$ mass hierarchy, which was not considered in Ref. [21].
Furthermore, we refine the analysis in Ref. [21] by basing our Higgs coupling
constraints on the model-independent ATLAS measurements of ratios of Higgs
couplings [22], rather than using fits to the couplings themselves, which were
made by the LHC collaborations using model assumptions that do not hold in the
Z2GM model. As a cross-check we also analyze the constraints on the Z2GM model
from the code HiggsSignals [23], which implements the LHC Higgs signal
strength measurements.

In addition to the Higgs coupling measurements, we also impose all available
direct searches for additional neutral and charged Higgs bosons. We apply most
of these searches by using the public code HiggsBounds [24] to exclude model
points that violate the 95% confidence level (CL) experimental exclusion in
the most sensitive applicable search channel. There are additional important
search channels relevant to the Z2GM model which are not captured by
HiggsBounds, in particular those involving the doubly-charged Higgs and
involving Drell-Yan production of pairs of custodial-fiveplet states. For
these searches, we use the direct implementations within the public code
GMCALC [25]. These additional channels are vector boson fusion (VBF)
$H^{\pm\pm}_{5}\rightarrow W^{\pm}W^{\pm}\rightarrow\text{like-sign
dileptons}$ [26, 27], Drell-Yan $H^{\pm\pm}_{5}\rightarrow\text{like-sign
dileptons}$ [28, 29, 30], Drell-Yan $H_{5}^{0}H_{5}^{\pm}$ with
$H_{5}^{0}\rightarrow\gamma\gamma$ [31, 32] and Drell-Yan
$H_{5}^{++}H_{5}^{--}\rightarrow W^{+}W^{+}W^{-}W^{-}$ [33]. We also include
the indirect constraint from $b\rightarrow s\gamma$ [17], which excludes large
values of the triplet vev; most of the parameter region thereby excluded is
also excluded by the direct searches.

As we show, the combination of the direct search constraints with the Higgs
coupling modifier ratios from ATLAS excludes the entire parameter space at 95%
CL; using instead the HiggsSignals fit for the Higgs couplings results in a
small surviving region at this confidence level. Both methods lead to a
surviving parameter region at 99% CL. This surviving parameter region has
significant Higgs coupling modifications which can be decisively tested at the
HL-LHC and a light custodial singlet (below 200 GeV) which can be searched for
at future lepton colliders [34, 35].

This paper is organized as follows. In Sec. II we review the $Z_{2}$-symmetric
version of the GM model. In Sec. III we study the mass matrix for the two
custodial singlets and elucidate the physics that drives one of them light
when the triplet vev becomes very small. In Sec. IV we examine the
phenomenology of the 125 GeV Higgs boson decays to $SS$ and $\gamma\gamma$ and
their dependence on the underlying model parameters. In Sec. V we apply the
experimental constraints, first for the region of parameter space in which
$m_{S}<m_{h}/2$ in which we demonstrate that this region is entirely excluded,
and then for the remaining parameter space. We conclude in Sec. VI. Some
details about the behaviour of the custodial singlet mass matrix and a
comparison of the Z2GM model with the unconstrained GM model are given in
Appendix A. Finally the details of our implementation of the theoretical and
experimental constraints on the Z2GM model and of the scans over parameter
space are collected in Appendix B.

##  II $Z_{2}$-symmetric Georgi-Machacek model

The Georgi-Machacek model is an extension of the scalar sector of the SM. Its
scalar sector consists of the usual Higgs doublet ($\phi^{+}$, $\phi^{0}$)
with hypercharge $Y=1$, a complex triplet ($\chi^{++}$, $\chi^{+}$,
$\chi^{0}$) with hypercharge $Y=2$ and a real triplet ($\xi^{+}$, $\xi^{0}$,
$\xi^{-}$) with hypercharge $Y=0$ (here $\xi^{-}=-\xi^{+*}$ and we use the
convention $Q=T^{3}+Y/2$). This model is minimal in the sense of not extending
the gauge group of the SM and not using higher representations of SU(2)L
beyond triplets [30]. The $\rho$ parameter is preserved by imposing a global
SU(2)${}_{L}\times$SU(2)R symmetry on the scalar sector of the model, which
will break down to the custodial SU(2) symmetry upon electroweak symmetry
breaking.222This global symmetry is explicitly broken by hypercharge
interactions, and hence can be preserved only at tree level [36]. This means
that the model is better seen as an effective low-energy description of some
UV model such as a composite Higgs scenario [3]. Nevertheless, the loop-
induced custodial symmetry breaking is quantitatively small enough that the
tree-level custodial-symmetric GM model remains a useful effective theory for
experimental purposes at the LHC [37, 38, 39].

To make the global SU(2)${}_{L}\times$SU(2)R symmetry manifest, we express the
scalar fields as a bi-doublet $\Phi$ and a bi-triplet $X$,

$\displaystyle\Phi=\begin{pmatrix}\phi^{0*}&\phi^{+}\\\
-\phi^{+*}&\phi^{0}\end{pmatrix},\qquad
X=\begin{pmatrix}\chi^{0*}&\xi^{+}&\chi^{++}\\\ -\chi^{+*}&\xi^{0}&\chi^{+}\\\
\chi^{++*}&-\xi^{+*}&\chi^{0}\end{pmatrix},$ (1)

where $\xi^{0}$ is a real field and the rest are complex. We also impose an
additional $Z_{2}$ symmetry under which the bi-triplet is odd:

$\displaystyle\Phi\rightarrow\Phi\,,\qquad X\rightarrow-X\,.$ (2)

The $Z_{2}$ symmetry serves to eliminate the lepton-number-violating Yukawa
couplings of the complex triplet field to the lepton doublets, which would
give rise to a neutrino mass proportional to the vev of the complex triplet.
Since we will consider triplet vevs in excess of 10 GeV, this is desirable to
avoid small neutrino Yukawa couplings of order $10^{-10}$. The remaining
fermion masses are generated through Yukawa couplings of the Higgs doublet as
in the SM.

The most general renormalizable scalar potential obeying these global
symmetries can then be written as333We use the parameterization of Ref. [9]. A
translation table to the notations of other papers can be found in the
appendix of that reference.

$\displaystyle V(\Phi,X)$
$\displaystyle=\frac{\mu^{2}_{2}}{2}\Tr(\Phi^{\dagger}\Phi)+\frac{\mu^{2}_{3}}{2}\Tr(X^{\dagger}X)+\lambda_{1}[\Tr(\Phi^{\dagger}\Phi)]^{2}+\lambda_{2}\Tr(\Phi^{\dagger}\Phi)\Tr(X^{\dagger}X)$
$\displaystyle+\lambda_{3}\Tr(X^{\dagger}XX^{\dagger}X)+\lambda_{4}[\Tr(X^{\dagger}X)]^{2}-\lambda_{5}\Tr(\Phi^{\dagger}\tau^{i}\Phi\tau^{j})\Tr(X^{\dagger}t^{i}Xt^{j}).$
(3)

Notice the absence of trilinear couplings that would involve an odd number of
$X$ fields; these are forbidden in the Z2GM by imposing the $Z_{2}$ symmetry.
The generators for the doublet are given in terms of the standard Pauli
matrices, $\tau^{i}=\sigma^{i}/2$, and for the triplets they are the $3\times
3$ representation,

$\displaystyle t^{1}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&1&0\\\ 1&0&1\\\
0&1&0\end{pmatrix},\qquad t^{2}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&-i&0\\\
i&0&-i\\\ 0&i&0\end{pmatrix},\qquad t^{3}=\begin{pmatrix}1&0&0\\\ 0&0&0\\\
0&0&-1\end{pmatrix}.$ (4)

The spontaneous symmetry breaking of the electroweak gauge group is achieved
through the vevs of both the bi-doublet and the bi-triplet (the latter also
spontaneously breaks the $Z_{2}$ global symmetry):

$\displaystyle\expectationvalue{\Phi}=\frac{\nu_{\phi}}{\sqrt{2}}\mathds{1}_{2\times
2},\qquad\expectationvalue{X}=\nu_{\chi}\mathds{1}_{3\times 3}\,.$ (5)

The neutral components of the real and complex triplets must obtain the same
vev to preserve the custodial symmetry at tree-level. Applying these vevs in
the gauge sector yields the $W$ and $Z$ masses:

$M_{W}^{2}=g^{2}\nu^{2}/4,\qquad\qquad M_{Z}^{2}=(g^{2}+g^{\prime
2})\nu^{2}/4,$ (6)

where

$\displaystyle\nu^{2}=\frac{1}{\sqrt{2}G_{F}}$
$\displaystyle=\nu_{\phi}^{2}+8\nu_{\chi}^{2}\approx(246\,\text{GeV})^{2}\,.$
(7)

The next step is to perform the field redefinition in the rest of the theory
and decompose the neutral fields into real and imaginary components:

$\displaystyle\phi^{0}$
$\displaystyle=\frac{\nu_{\phi}}{\sqrt{2}}+\frac{1}{\sqrt{2}}(\phi^{0}_{R}+i\phi^{0}_{I}),\qquad\xi^{0}=\nu_{\chi}+\xi^{0}_{R},\qquad\chi^{0}=\nu_{\chi}+\frac{1}{\sqrt{2}}(\chi^{0}_{R}+i\chi^{0}_{I}).$

This change of variables is then applied to the potential. The minimization
condition for this model is:

$\displaystyle\partialderivative{V}{\nu_{\phi}}$
$\displaystyle=\nu_{\phi}\left[\mu^{2}_{2}+4\lambda_{1}\nu_{\phi}^{2}+3(2\lambda_{2}-\lambda_{5})\nu_{\chi}^{2}\right]=0\,,$
(8) $\displaystyle\partialderivative{V}{\nu_{\chi}}$
$\displaystyle=\nu_{\chi}\left[3\mu^{2}_{3}+3(2\lambda_{2}-\lambda_{5})\nu_{\phi}^{2}+12(\lambda_{3}+3\lambda_{4})\nu_{\chi}^{2}\right]=0\,.$
(9)

We can use Eqs. (8) and (9) to write $\mu_{2}^{2}$ and $\mu_{3}^{2}$ in terms
of the other couplings. This leaves us with 6 free parameters
($\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5},\nu_{\chi}$),444The
triplet vev $\nu_{\chi}$ can alternatively be traded for the dimensionless
mixing angle $s_{H}\equiv\sin\theta_{H}=2\sqrt{2}\nu_{\chi}/\nu$; we will
later exhibit our results in both parameterizations. with only $\nu_{\chi}$
being dimensionful, but having an upper bound $\nu_{\chi}\leq\nu/\sqrt{8}$
when all the contribution to $\nu$ comes from the triplets. We then trade
$\lambda_{1}$ to enforce the 125 GeV Higgs to have its measured mass, meaning
that we are left with 5 free parameters.

We next diagonalize the potential Eq. (II) to the mass eigenbasis. We can
expect, after the diagonalization, to have the following decomposition under
the custodial $SU(2)$:

$\displaystyle(\textbf{2},\textbf{2})=\textbf{1}\oplus\textbf{3},\qquad(\textbf{3},\textbf{3})=\textbf{1}\oplus\textbf{3}\oplus\textbf{5}.$
(10)

This means that, after the breaking, we expect to have 2 custodial singlets
($h,S$), 2 custodial triplets ($G,H_{3}$), and one custodial fiveplet
($H_{5}$). One of the custodial triplets ($G$) contains the Goldstone bosons
which will be eaten by the gauge fields. The custodial triplet sector
diagonalization is:

$\displaystyle G^{+}$
$\displaystyle=c_{H}\phi^{+}+s_{H}\frac{1}{\sqrt{2}}(\chi^{+}+\xi^{+})\,,$
(11) $\displaystyle G^{0}$
$\displaystyle=c_{H}\phi^{0}_{I}+s_{H}\chi^{0}_{I}\,,$ (12) $\displaystyle
H_{3}^{+}$
$\displaystyle=-s_{H}\phi^{+}+c_{H}\frac{1}{\sqrt{2}}(\chi^{+}+\xi^{+})\,,$
(13) $\displaystyle H_{3}^{0}$
$\displaystyle=-s_{H}\phi^{0}_{I}+c_{H}\chi^{0}_{I}\,,$ (14)

where $c_{H}$ and $s_{H}$ are:

$\displaystyle
c_{H}=\frac{\nu_{\phi}}{\nu}\,,\qquad\,s_{H}=2\sqrt{2}\frac{\nu_{\chi}}{\nu}\,.$
(15)

The custodial fiveplet diagonalization is:

$\displaystyle H^{++}_{5}$ $\displaystyle=\chi^{++}\,,$ (16) $\displaystyle
H^{+}_{5}$ $\displaystyle=\frac{1}{\sqrt{2}}(\chi^{+}-\xi^{+})\,,$ (17)
$\displaystyle H^{0}_{5}$
$\displaystyle=\sqrt{\frac{2}{3}}\xi^{0}_{R}-\sqrt{\frac{1}{3}}\chi^{0}_{R}\,.$
(18)

The custodial symmetry at tree level enforces that the masses inside any
custodial multiplet must be degenerate. In the Z2GM, the masses of the
custodial fiveplet and custodial triplet states are given respectively by:

$\displaystyle m_{5}^{2}$
$\displaystyle=\frac{3}{2}\lambda_{5}\nu_{\phi}^{2}+8\lambda_{3}\nu_{\chi}^{2}=\left(\frac{3}{2}\lambda_{5}c_{H}^{2}+\lambda_{3}s_{H}^{2}\right)\nu^{2}\,,$
(19) $\displaystyle m_{3}^{2}$
$\displaystyle=\frac{1}{2}\lambda_{5}\nu^{2}\,.$ (20)

Notice that the custodial triplet mass does not depend on $\nu_{\chi}$ (or
equivalently, $s_{H}$), and is only sensitive to one quartic coupling. The
experimental lower bound on $m_{3}$ will thus set a lower bound on
$\lambda_{5}$. The custodial fiveplet mass depends only on $\lambda_{3}$,
$\lambda_{5}$ and $\nu_{\chi}$ and has the following relation in the
$\nu_{\chi}\rightarrow 0$ (equivalently, $s_{H}\to 0$) limit:

$\displaystyle m_{5}=\sqrt{3}m_{3}\qquad\text{for}\,\,\nu_{\chi}\rightarrow
0\,.$ (21)

The remaining diagonalization is for the custodial singlet sector. Since this
is the focus of the paper and the source of our most interesting results, we
introduce the procedure here and develop it further in Section III. We first
define the gauge basis as:

$\displaystyle H_{1}^{0}$ $\displaystyle=\phi^{0}_{R}\,,$ (22) $\displaystyle
H_{1}^{0^{\prime}}$
$\displaystyle=\sqrt{\frac{1}{3}}\xi^{0}_{R}+\sqrt{\frac{2}{3}}\chi^{0}_{R}\,,$
(23)

where $H_{1}^{0^{\prime}}$ is the custodial singlet that appears in the
original bi-triplet. These states mix to form the physical Higgs $h$, which we
identify with the 125 GeV Higgs boson, and an additional CP-even custodial
singlet $S$. The mass diagonalization can be done using the following
orthogonal matrix:

$\displaystyle\begin{pmatrix}h\\\
S\end{pmatrix}=\begin{pmatrix}c_{\alpha}&-s_{\alpha}\\\
s_{\alpha}&c_{\alpha}\end{pmatrix}\begin{pmatrix}H_{1}^{0}\\\
H_{1}^{0^{\prime}}\end{pmatrix}\qquad({\rm for}\ m_{S}>m_{h}).$ (24)

Now we have to be careful with the definition of the angle $\alpha$. To be
consistent with Refs. [9, 16, 7, 8, 13, 14, 15] and with GMCALC [25], we
define the rotation in terms of the heavy and light mass eigenstates. This
means that for $m_{S}>m_{h}$ we have Eq. (24), while for $m_{S}<m_{h}$ we
have:

$\displaystyle\begin{pmatrix}S\\\
h\end{pmatrix}=\begin{pmatrix}c_{\alpha}&-s_{\alpha}\\\
s_{\alpha}&c_{\alpha}\end{pmatrix}\begin{pmatrix}H_{1}^{0}\\\
H_{1}^{0^{\prime}}\end{pmatrix}\qquad({\rm for}\ m_{S}<m_{h}).$ (25)

Since we will want $h$ to be mostly $H_{1}^{0}$ for consistency with the LHC
measurements of the Higgs couplings, this means that in the region where
$m_{S}>m_{h}$ the SM limit is $c_{\alpha}\to 1$, while in the region where
$m_{S}<m_{h}$ the SM limit is $s_{\alpha}\rightarrow 1$. In the next section,
we analyze the custodial singlet sector further, highlighting, in particular,
the region of parameter space in which one of the custodial singlets becomes
very light.

In what follows we will perform numerical scans of the full parameter space of
the Z2GM model, imposing the usual theoretical constraints (perturbative
unitarity, boundedness from below, and absence of deeper custodial-violating
minima) on the parameters of the scalar potential as implemented in GMCALC.
Details of these constraints, as well as of our scan procedure, are given in
Appendix B. We will also apply from the start the experimental lower bound on
the custodial-triplet and -fiveplet masses [30, 40] (see Appendix B for
details),

$\displaystyle m_{3},m_{5}\geq 76~{}\text{GeV}\,.$ (26)

Perturbative unitarity of the quartic couplings in the scalar potential,
together with the measured value of the Fermi constant $G_{F}$, also lead to
upper bounds on these masses of $m_{3}\lesssim 400$ GeV and $m_{5}\lesssim
700$ GeV [16].

##  III Diagonalization of the custodial singlet sector

###  III.1 Features of the mass matrix

In the custodial singlet sector, the mass matrix before diagonalizing has the
form:

$\displaystyle\mathcal{M}^{2}=\begin{pmatrix}\mathcal{M}_{11}^{2}&\mathcal{M}_{12}^{2}\\\
\mathcal{M}_{12}^{2}&\mathcal{M}_{22}^{2}\end{pmatrix}\,,$ (27)

where:

$\displaystyle\mathcal{M}_{11}^{2}$
$\displaystyle=8\lambda_{1}\nu_{\phi}^{2}\,,$ (28)
$\displaystyle\mathcal{M}_{12}^{2}$
$\displaystyle=2\sqrt{3}\nu_{\chi}\nu_{\phi}(2\lambda_{2}-\lambda_{5})\,,$
(29) $\displaystyle\mathcal{M}_{22}^{2}$
$\displaystyle=8\nu_{\chi}^{2}(\lambda_{3}+3\lambda_{4})\,.$ (30)

It is clear from Eqs. (27)–(30) that in the $\nu_{\chi}\rightarrow 0$
limit555We explicitly take this limit coming from positive nonzero values of
$\nu_{\chi}$. At $\nu_{\chi}=0$ a phase transition occurs to the dark matter
phase of the theory in which the $Z_{2}$ symmetry is unbroken; in this dark
matter phase the SU(2)L triplets are $Z_{2}$-odd, their masses are controlled
by $\mu_{3}^{2}$, and they can be decoupled by taking $\mu_{3}^{2}\gg\nu^{2}$.
We do not consider the dark matter phase in this paper. the Higgs boson $h$
acquires a mass-squared of the SM form $8\lambda_{1}\nu^{2}$, the mixing goes
to zero, and the second custodial singlet becomes massless. The small
$\nu_{\chi}$ region is thus populated by a low-mass custodial singlet $S$. We
defined the diagonalization matrix depending on the mass hierarchy in Eq. (24)
and Eq. (25). For both cases the mixing angle $\alpha$ can be computed using:

$\displaystyle\sin(2\alpha)$
$\displaystyle=\frac{2\mathcal{M}^{2}_{12}}{|m_{h}^{2}-m_{S}^{2}|}\,,$ (31)
$\displaystyle\cos(2\alpha)$
$\displaystyle=\frac{\mathcal{M}^{2}_{22}-\mathcal{M}^{2}_{11}}{|m_{h}^{2}-m_{S}^{2}|}\,.$
(32)

Note that, after fixing $\nu$ using the Fermi constant, the custodial-singlet
sector depends on only four combinations of parameters: $\nu_{\chi}$,
$\lambda_{1}$, $(2\lambda_{2}-\lambda_{5})$ and $(\lambda_{3}+3\lambda_{4})$.
We can simplify the analysis by defining the combinations:

$\displaystyle\lambda_{25}=2\lambda_{2}-\lambda_{5},\qquad\lambda_{34}=\lambda_{3}+3\lambda_{4}\,,$
(33)

so that

$\displaystyle\mathcal{M}^{2}_{12}=2\sqrt{3}\nu_{\chi}\nu_{\phi}\lambda_{25},\qquad\mathcal{M}^{2}_{22}=8\nu_{\chi}^{2}\lambda_{34}.$
(34)

Because the mass of $h$ is already measured to be 125 GeV, we use this to fix
the value of $\lambda_{1}$. This inversion is independent of the mass
hierarchy between $h$ and $S$:

$\displaystyle\lambda_{1}=\frac{1}{8\nu_{\phi}^{2}}\left(m_{h}^{2}+\frac{(\mathcal{M}^{2}_{12})^{2}}{\mathcal{M}_{22}^{2}-m_{h}^{2}}\right)\,.$
(35)

It is important to remember that $\lambda_{1}$ is constrained by perturbative
unitarity666The upper bound on $\lambda_{1}$ from perturbative unitarity
depends on the values of $\lambda_{2}$ and $\lambda_{5}$, but it cannot exceed
$\pi/3$ [9]. Boundedness-from-below of the scalar potential also constrains
$\lambda_{1}>0$. and this inversion may lead to a disallowed value for
$\lambda_{1}$, in which case the point is discarded.

The variables $\lambda_{25}$ and $\lambda_{34}$ are constrained by
perturbative unitarity and vacuum stability to lie in the ranges [9]

$\displaystyle\lambda_{25}\in\left(-\frac{8\pi}{3},\frac{4\pi}{3}\right),\qquad\lambda_{34}\in\left(0,\pi\right).$
(36)

The lower bound on $\lambda_{34}$ can also be quickly obtained from the
requirement that the physical masses-squared are positive: using the trace and
determinant of the mass matrix,

$\displaystyle{\rm Tr}(\mathcal{M}^{2})$
$\displaystyle=m_{h}^{2}+m_{S}^{2}\,,$ (37)
$\displaystyle\det(\mathcal{M}^{2})$ $\displaystyle=m_{h}^{2}m_{S}^{2}\,,$
(38)

requiring that $m_{S}^{2}>0$ imposes that the determinant is positive, and as
a consequence, $\lambda_{34}$ needs to be positive.

The next step is to find an expression for $m_{S}$ in terms of these
combinations of couplings that is independent of whether $m_{S}$ is greater or
less than $m_{h}$. Using Eq. (35) in Eq. (37) we can find an expression for
$m_{S}$:

$\displaystyle
m_{S}^{2}=8\lambda_{34}\nu_{\chi}^{2}+\frac{16\lambda_{25}^{2}\nu_{\phi}^{2}\nu_{\chi}^{2}}{8\lambda_{34}\nu_{\chi}^{2}-m_{h}^{2}}\,.$
(39)

We can subtract the Higgs mass-squared from Eq. (39) to find the hierarchy
relation:

$\displaystyle
m_{S}^{2}-m_{h}^{2}=\mathcal{K}+\frac{16\lambda_{25}^{2}\nu_{\chi}^{2}\nu_{\phi}^{2}}{\mathcal{K}}\,,$
(40)

where we define the combination of parameters:

$\displaystyle\mathcal{K}=8\lambda_{34}\nu_{\chi}^{2}-m_{h}^{2}\,.$ (41)

Figure 1:  The separation of the two possible mass hierarchies for the
custodial singlets in the space of $\nu_{\chi}$ and $\lambda_{34}$, as
controlled by the sign of $\mathcal{K}$. In dark blue we have points with
$m_{S}>m_{h}$ ($\mathcal{K}>0$) and in light blue we have points with
$m_{S}<m_{h}$ ($\mathcal{K}<0$). The boundary between these two regions is
defined by $\mathcal{K}=0$.

We can see that the sign of $\mathcal{K}$ determines the hierarchy between the
Higgs mass and the other custodial singlet mass. This happens because the
numerator of the second term in Eq. (40) is always positive. We generalize
this result in Appendix A, where we derive this relation for a general
Hermitian matrix and also apply it to the general GM model with explicitly
broken $Z_{2}$ to highlight the differences compared to the $Z_{2}$-symmetric
version. When $\mathcal{K}$ is negative the Higgs $h$ is the heavier state,
i.e., $m_{S}<m_{h}$. When $\mathcal{K}$ is positive the Higgs $h$ is the
lighter state. We can see these two regions in Fig. 1.

An additional relation can be found between $\lambda_{25}$ and $\lambda_{34}$
using the condition that the determinant is positive semi-definite (i.e., that
$m_{S}^{2}\geq 0$). Assuming that $\nu_{\chi}\neq 0$, we obtain a bound on how
large $\absolutevalue{\lambda_{25}}$ can be:

$\displaystyle 16\lambda_{1}\lambda_{34}-3\lambda_{25}^{2}\geq 0\,.$ (42)

This imposes an additional upper bound on $\absolutevalue{\lambda_{25}}$. We
can also express this condition using Eq. (39) when $m_{S}<m_{h}$ (i.e.,
$\mathcal{K}<0$) by requiring that $m_{S}^{2}\geq 0$:

$\displaystyle\lambda_{25}^{2}\leq\frac{\lambda_{34}}{2\nu_{\phi}^{2}}|\mathcal{K}|=\frac{\lambda_{34}}{2\nu_{\phi}^{2}}\left|8\lambda_{34}\nu_{\chi}^{2}-m_{h}^{2}\right|.$
(43)

Choosing values for $\lambda_{34}$ and $\nu_{\chi}$, this puts an additional
bound on $\lambda_{25}$. This relation is just the requirement that the
lighter custodial-singlet mass-squared is positive semi-definite.

###  III.2 The light custodial singlet region

We now focus in on the region with $m_{S}<m_{h}$. In this region of the
parameter space we have $\mathcal{K}$ negative, which from the definition of
$\mathcal{K}$ means that:

$\displaystyle m_{h}^{2}>8\lambda_{34}\nu_{\chi}^{2}\,.$ (44)

If we make $\lambda_{34}$ in Eq. (44) as large as possible subject to
perturbative unitarity and bounded-from-below constraints (i.e.,
$\lambda_{34}^{\rm max}=\pi$), we obtain that for
$\nu_{\chi}<m_{h}/\sqrt{8\pi}\simeq 24.9~{}\text{GeV}$ we _always_ have
$m_{S}<m_{h}$. This value of $\nu_{\chi}$ is controlled by $\lambda_{34}$
alone: if the upper bound on $\lambda_{34}$ is reduced, then the value of
$\nu_{\chi}$ below which only the hierarchy $m_{S}<m_{h}$ is possible becomes
larger. We can see this interesting region in Fig. 2 in which we show the
entire parameter space as a function of $\nu_{\chi}$ and $m_{S}$. Again, the
light blue points have $m_{S}<m_{h}$ and the dark blue points have
$m_{S}>m_{h}$. There is no overlap of the two colours in Fig. 2. From this
figure we see clearly that a lower bound on $m_{S}$ will translate directly
into a lower bound on $\nu_{\chi}$ according to $\nu_{\chi}^{\rm
min}=m_{S}^{\rm min}/\sqrt{8\pi}$ as long as $m_{S}^{\rm min}\leq m_{h}$.

Figure 2:  Allowed regions of $m_{S}$ versus $\nu_{\chi}$ after imposing the
theoretical constraints and the mass bound $m_{3},m_{5}\geq 76$ GeV. We use
the same color scheme as in Fig. 1, i.e., light blue points have $m_{S}<m_{h}$
and dark blue points have $m_{S}>m_{h}$. The region with $\nu_{\chi}$ below
about 24.9 GeV is populated only by $m_{S}<m_{h}$. For values of $\nu_{\chi}$
smaller than about 12.5 GeV we have only points with $m_{S}<m_{h}/2$.

At this point, we can understand why the authors of Ref. [21] obtained a lower
bound on $\nu_{\chi}$ by requiring that $m_{S}>m_{h}$. The low $\nu_{\chi}$
region is _only_ populated by the mass hierarchy $m_{S}<m_{h}$. This can be
seen intuitively if we look at the custodial-singlet mass matrix in Eq. (27).
Low values of $\nu_{\chi}$ mean that the off-diagonal terms and the $(2,2)$
element are small. These elements of the matrix determine the mass of $S$,
which thus becomes small as well. If we take $\lambda_{25}=0$, so that the
off-diagonal terms vanish, this is even more evident since then
$\mathcal{M}_{22}^{2}=8\nu_{\chi}^{2}\lambda_{34}$ is identified with
$m_{S}^{2}$ and it is proportional to $\nu_{\chi}^{2}$.

An _experimental lower bound_ on $\nu_{\chi}$ can therefore be achieved if we
can set a lower bound on $m_{S}$. For the phenomenology of this model, a lower
bound on $\nu_{\chi}$ has important implications for all the couplings of the
theory, in particular those of the 125 GeV Higgs boson $h$.

In what follows we will demonstrate that the entirety of the parameter space
with $m_{S}<m_{h}/2$ is excluded by a combination of the experimental
constraints on $h\to SS$ and $h\to\gamma\gamma$, thereby setting a lower bound
$\nu_{\chi}\gtrsim 12.5$ GeV, or equivalently $s_{H}\gtrsim 0.143$. We first
examine how the underlying parameters control these two Higgs decay
observables and how the limited parameter freedom makes it impossible to
accommodate them both simultaneously.

##  IV Higgs decay phenomenology in the light singlet region

In this section, we examine the underlying model parameters that control the
125 GeV Higgs boson $h$ decays to $SS$ in the region $m_{S}<m_{h}/2$ and the
contributions of the singly- and doubly-charged Higgs bosons to the loop-
induced $h\to\gamma\gamma$ decay. Focusing on the small-$\nu_{\chi}$ region,
we show that there is not enough parameter freedom in the Z2GM model to
simultaneously tune the $h\to SS$ width to be sufficiently small and the
$h\to\gamma\gamma$ width to be sufficiently SM-like to be able to satisfy the
experimental constraints.

###  IV.1 Higgs decay to $SS$

Assuming that the second custodial singlet is lighter than $m_{h}/2$, it is
possible to have an additional decay channel for the 125 GeV Higgs boson,
$h\rightarrow SS$. The Lagrangian that controls this process can be written
as:

$\displaystyle\mathcal{L}_{hSS}=-\frac{g_{hSS}}{2}hS^{2}\,,$ (45)

where the relevant coupling (for $m_{S}<m_{h}$) is:

$\displaystyle
g_{hSS}=2s_{\alpha}\nu_{\phi}\left[12c_{\alpha}^{2}\lambda_{1}+\lambda_{25}(s_{\alpha}^{2}-2c_{\alpha}^{2})\right]+2\sqrt{3}c_{\alpha}\nu_{\chi}\left[\lambda_{25}(c_{\alpha}^{2}-2s_{\alpha}^{2})+4s_{\alpha}^{2}\lambda_{34}\right]\,.$
(46)

Here $\lambda_{25}$ and $\lambda_{34}$ are the same combinations of couplings
that appear above in the custodial singlet sector mass matrix.

Since we will be primarily interested in the low $\nu_{\chi}$ region, we can
understand the behaviour of the model in this region analytically by examining
the residual coupling in the $\nu_{\chi}\rightarrow 0$ limit:777Remember that
from the diagonalization matrix the SM limit in the $m_{S}<m_{h}$ region is
$s_{\alpha}\rightarrow 1$.

$\displaystyle g_{hSS}\to 2\lambda_{25}\nu\qquad(\rm{for}\ \nu_{\chi}\to 0).$
(47)

Because $\nu\simeq 246$ GeV, the coupling $\lambda_{25}$ would need to be
extremely small to satisfy the experimental bounds on the $h$ total width when
$m_{S}<m_{h}/2$. We can write the decay rate for this channel as

$\displaystyle\Gamma_{hSS}=|g_{hSS}|^{2}\frac{1}{32\pi
m_{h}}\left(1-\frac{4m_{S}^{2}}{m_{h}^{2}}\right)^{1/2}\,,$ (48)

and add it to the total width of the Higgs into SM channels, where we take the
SM value to be $\Gamma_{SM}=4.09$ MeV [41]:

$\displaystyle\Gamma_{T}$
$\displaystyle=\kappa_{H}^{2}\Gamma_{SM}+\Gamma_{hSS}\,.$ (49)

In Eq. (49), $\kappa_{H}^{2}$ encodes the modifications to the decay widths to
SM final states. We can use $\kappa_{H}\to 1$888The tree-level couplings of
$h$ to fermion and vector boson pairs, and its loop-induced coupling to gluon
pairs, indeed go to their SM values in the $\nu_{\chi}\to 0$ limit. The loop-
induced coupling of $h$ to photon pairs and to $Z\gamma$ do not approach their
SM values in this limit due to the presence of additional light singly- and
doubly-charged scalars; however, the contributions of these decays to the
total Higgs width are so small (less than a percent) that we ignore them in
this qualitative analysis. We do of course include them in our full numerical
analysis. for $\nu_{\chi}\rightarrow 0$ to understand the expected behaviour
in this limit.

A generic constraint can be obtained from the indirect bound on the Higgs
total width from an analysis of on- and off-shell $ZZ$ production [42] (see
Appendix B for details). Taking a conservative 99.7% confidence level
($3\sigma$) constraint, we have $\Gamma_{T}<19.1~{}\text{MeV}$. Setting
$\kappa_{H}\rightarrow 1$ in the $\nu_{\chi}\to 0$ limit we thus get an upper
bound on the allowed decay width for the $h\to SS$ channel of
$\Gamma_{hSS}<15.0~{}\text{MeV}$.999This is an extremely conservative
constraint since such a large new contribution to the Higgs total width would
suppress the Higgs branching ratios to all SM final states by more than a
factor of 4, resulting in strong disagreement with experimental data. This
translates into an upper bound at $99.7\%$ CL for
$\absolutevalue{\lambda_{25}}$ in the $\nu_{\chi}\to 0$ limit of:

$\displaystyle\absolutevalue{\lambda_{25}}\leq 0.028\,.$ (50)

Moving away from $\nu_{\chi}\rightarrow 0$ we will see that $\lambda_{25}$
still needs to be relatively small.

In Fig. 3 we show the total width of $h$, scanning over the entire Z2GM
parameter space for which $m_{S}<m_{h}/2$. We can see that generic parameter
values typically lead to enormous $h\to SS$ decay widths, but that a well-
populated region of parameter space nevertheless exists in which the $hSS$
coupling is sufficiently suppressed to satisfy the indirect constraint
$\Gamma_{T}<19.1$ MeV.

Figure 3:  Total width of the 125 GeV Higgs as a function of the triplet vev
$\nu_{\chi}$ from a scan over the Z2GM model parameter space in which
$m_{S}<m_{h}/2$. The green points are excluded by experimental constraints on
$b\to s\gamma$. The light blue points are excluded by direct searches for the
non-SM Higgs bosons as encoded in HiggsBounds and GMCALC (see Appendix B for
details). The maroon points are allowed after considering these constraints.
The pink band at the bottom of the plot indicates the allowed range
$\Gamma_{T}<19.1$ MeV at 99.7% CL.

###  IV.2 Higgs to diphoton decay

In the Z2GM model, the 125 GeV Higgs to diphoton decay rate is modified by the
presence of the singly- and doubly-charged scalars $H_{3}^{\pm}$,
$H_{5}^{\pm}$ and $H_{5}^{\pm\pm}$ running in the loop, as well as by
modifications of the couplings of the Higgs to $W$ boson and fermion pairs. We
adopt the usual normalization, [43]

$\displaystyle\mathcal{L}_{int}=-\frac{gm_{f_{i}}}{2M_{W}}\kappa_{f_{i}}\bar{f}_{i}f_{i}h+gM_{W}\kappa_{W}W^{+}W^{-}h-\frac{gm_{H_{i}}^{2}}{M_{W}}\kappa_{H_{i}}H_{i}H_{i}^{*}h\,,$
(51)

where the $\kappa_{i}$ parameterize the couplings of each particle to the 125
GeV Higgs boson (in the SM, $\kappa_{W}=\kappa_{f}=1$). This parameterization
yields the expression for the partial width of $h\to\gamma\gamma$, [43]

$\displaystyle\Gamma(h\rightarrow\gamma\gamma)=\frac{\alpha^{2}g^{2}m_{h}^{3}}{1024\pi^{3}M_{W}^{2}}\absolutevalue{\sum_{i}N_{i}Q_{i}^{2}\kappa_{i}F_{i}(\tau_{i})}^{2}\,,$
(52)

where $N_{i}$ is the number of colours of particle $i$, $Q_{i}$ is its
electric charge in units of $e$, and $F_{i}(\tau_{i})$ is a loop function that
depends on the spin of particle $i$ according to: [43]

$\displaystyle F_{1}(\tau_{i})$
$\displaystyle=2+3\tau_{i}+3\tau_{i}\left(2-\tau_{i}\right)f(\tau_{i})\,,$
(53) $\displaystyle F_{1/2}(\tau_{i})$
$\displaystyle=-2\tau_{i}\left[1+(1-\tau_{i})f(\tau_{i})\right]\,,$ (54)
$\displaystyle F_{0}(\tau_{i})$
$\displaystyle=\tau_{i}\left[1-\tau_{i}f(\tau_{i})\right]\,,$ (55)

with $\tau_{i}=4m_{i}^{2}/m_{h}^{2}$ and

$\displaystyle f(\tau_{i})$
$\displaystyle=\begin{cases}\left[\sin^{-1}\left(\sqrt{\frac{1}{\tau_{i}}}\right)\right]^{2}&\
{\rm for}\ \tau_{i}\geq 1,\\\
-\frac{1}{4}\left[\ln(\frac{1+\sqrt{1-\tau_{i}}}{1-\sqrt{1-\tau_{i}}})-i\pi\right]^{2}&\
{\rm for}\ \tau_{i}<1.\end{cases}$ (57)

We can then define an effective coupling modification factor $\kappa_{\gamma}$
for this decay amplitude relative to the SM:

$\displaystyle\kappa_{\gamma}=\frac{\sum_{GM}N_{i}Q_{i}^{2}\kappa_{i}F_{i}(\tau_{i})}{\sum_{SM}N_{i}Q_{i}^{2}F_{i}(\tau_{i})}\,,$
(58)

such that
$\kappa_{\gamma}^{2}=\Gamma_{GM}(h\to\gamma\gamma)/\Gamma_{SM}(h\to\gamma\gamma)$.
Writing out the couplings explicitly for the Z2GM model yields the expression:

$\displaystyle\kappa_{\gamma}=\frac{\kappa_{V}F_{1}(\tau_{W})+\frac{4}{3}\kappa_{f}F_{1/2}(\tau_{t})+\kappa_{3}F_{0}(\tau_{3})+5\kappa_{5}F_{0}(\tau_{5})}{F_{1}(\tau_{W})+\frac{4}{3}F_{1/2}(\tau_{t})}\,,$
(59)

where in both the Z2GM and SM amplitudes we have kept only the dominant
fermionic contribution coming from the top quark. We have also used the fact
that $\kappa_{W}=\kappa_{Z}\equiv\kappa_{V}$ in the Z2GM model and that the
$H_{5}^{+}H_{5}^{-}h$ and $H_{5}^{++}H_{5}^{--}h$ couplings are controlled by
the same coupling factor $\kappa_{5}$.

The expressions for the $\kappa$’s can be read directly from the Lagrangian by
matching the definitions:

$\displaystyle g_{hW^{+}W^{-}}$ $\displaystyle=gM_{W}\kappa_{W}\,,\qquad
g_{hff}=g\frac{m_{f}}{2M_{W}}\kappa_{f}\,,$ (60) $\displaystyle
g_{hH^{+}_{3}H^{-}_{3}}$
$\displaystyle=g\frac{m_{3}^{2}}{M_{W}}\kappa_{3}\,,\qquad
g_{hH^{+}_{5}H^{-}_{5}}=g\frac{m_{5}^{2}}{M_{W}}\kappa_{5}\,,\qquad
g_{hH^{++}_{5}H^{--}_{5}}=g\frac{m_{5}^{2}}{M_{W}}\kappa_{5}\,.$ (61)

Because of the definitions of the mixing angle $\alpha$, the explicit
expressions for the $\kappa$ factors depend on the mass hierarchy between $S$
and $h$. Since in this section we are interested in the light singlet region,
we give here the formulas valid for $m_{S}<m_{h}$. The case of $m_{S}>m_{h}$
can be obtained from these by the replacements $s_{\alpha}\to c_{\alpha}$,
$c_{\alpha}\to-s_{\alpha}$. The gauge and fermionic couplings are:

$\displaystyle\kappa_{V}=s_{\alpha}\frac{\nu_{\phi}}{\nu}+\frac{8}{\sqrt{3}}c_{\alpha}\frac{\nu_{\chi}}{\nu}\,,\qquad\kappa_{f}=s_{\alpha}\frac{\nu}{\nu_{\phi}}.\qquad\qquad({\rm
for}\ m_{S}<m_{h})$ (62)

The (dimensionless) coupling factors of $h$ to charged custodial-triplet and
-fiveplet scalar pairs are:

$\displaystyle\kappa_{3}$
$\displaystyle=\frac{1}{6m_{3}^{2}\nu}\Bigg{[}3s_{\alpha}\nu_{\phi}^{3}(4\lambda_{2}-\lambda_{5})+8\sqrt{3}c_{\alpha}\nu_{\phi}^{2}\nu_{\chi}(\lambda_{3}+3\lambda_{4}+\lambda_{5})$
$\displaystyle\qquad\qquad\quad+24s_{\alpha}\nu_{\phi}\nu_{\chi}^{2}(8\lambda_{1}+\lambda_{5})+16\sqrt{3}c_{\alpha}\nu_{\chi}^{3}(6\lambda_{2}+\lambda_{5})\Bigg{]}\,,$
(63) $\displaystyle\kappa_{5}$
$\displaystyle=\frac{\nu}{2m_{5}^{2}}\left[s_{\alpha}\nu_{\phi}(4\lambda_{2}+\lambda_{5})+8\sqrt{3}c_{\alpha}\nu_{\chi}(\lambda_{3}+\lambda_{4})\right].\qquad\qquad({\rm
for}\ m_{S}<m_{h}).$ (64)

The expressions for $\kappa_{3}$ and $\kappa_{5}$ depend on the masses $m_{3}$
and $m_{5}$, which are themselves dependent upon the underlying quartic
couplings and vevs as given in Eqs. (20) and (19), respectively. To connect
the analysis of this sector to the preceding calculations, we define two
additional linear combinations of quartic scalar couplings,

$\displaystyle\bar{\lambda}_{25}=4\lambda_{2}+\lambda_{5},\qquad\bar{\lambda}_{34}=\lambda_{3}+\lambda_{4}.$
(65)

These are linearly independent from the combinations
$\lambda_{25}=2\lambda_{2}-\lambda_{5}$ and
$\lambda_{34}=\lambda_{3}+3\lambda_{4}$ that we defined before. Using these
new couplings we can write the masses $m_{3}$ and $m_{5}$ as:

$\displaystyle m_{3}^{2}$
$\displaystyle=(\bar{\lambda}_{25}-2\lambda_{25})\frac{\nu^{2}}{6}\,,\qquad
m_{5}^{2}=(\bar{\lambda}_{25}-2\lambda_{25})\frac{\nu_{\phi}^{2}}{2}+4(3\bar{\lambda}_{34}-\lambda_{34})\nu_{\chi}^{2}\,.$
(66)

We can then study the limit $\nu_{\chi}\rightarrow 0$ (which also implies
$s_{\alpha}\to 1$) to gain some intuition about the behaviour of
$\kappa_{\gamma}$. In this limit, $\kappa_{V}$ and $\kappa_{f}$ both go to 1,
i.e., the tree-level gauge and Yukawa couplings of $h$ become SM-like. The
deviation of $\kappa_{\gamma}$ from unity is then dependent solely on the
contributions of the singly- and doubly-charged scalars in the loop:

$\displaystyle\kappa_{\gamma}\to
1+\frac{\kappa_{3}F_{0}(\tau_{3})+5\kappa_{5}F_{0}(3\tau_{3})}{F_{1}(\tau_{W})+\frac{4}{3}F_{1/2}(\tau_{t})}\,,\qquad({\rm
for}\ \nu_{\chi}\to 0)$ (67)

where we also used the fact that in this limit $m_{5}\to\sqrt{3}m_{3}$ as in
Eq. (21). Applying this limit to Eqs. (63) and (64) we get:

$\displaystyle\kappa_{3}\to\frac{\bar{\lambda}_{25}+4\lambda_{25}}{\bar{\lambda}_{25}-2\lambda_{25}}\,,\qquad\kappa_{5}\to\frac{\bar{\lambda}_{25}}{\bar{\lambda}_{25}-2\lambda_{25}}\,.\qquad({\rm
for}\ \nu_{\chi}\to 0)$ (68)

To obtain $\kappa_{\gamma}\to 1$, in this limit we would require either that
both $\kappa_{3},\kappa_{5}\to 0$, or for a cancellation to occur between the
amplitude contributions from $H_{3}^{+}$ and $H_{5}^{+,++}$. Neither of these
is possible in the $\nu_{\chi}\to 0$ limit. Instead, the requirement of very
small $\lambda_{25}$ needed to suppress the $h\to SS$ width in this limit
drives both $\kappa_{3}$ and $\kappa_{5}$ to 1. One way around this would be
to take $\bar{\lambda}_{25}$ to be even smaller than $\lambda_{25}$, in which
case $\kappa_{5}\to 0$ while $\kappa_{3}\to-2$, which still leads to a non-SM
rate for $h\to\gamma\gamma$. This latter possibility however drives $m_{3}$
and $m_{5}$ to zero and is hence precluded by the experimental lower bounds on
$m_{3}$ and $m_{5}$ of 76 GeV from a combination of LEP-2 and ATLAS data (see
Appendix B for details).

In the following section, we apply the Higgs total width and
$h\to\gamma\gamma$ signal rate constraints quantitatively to show that the
region with $m_{S}<m_{h}/2$ is entirely excluded at more than the $3\sigma$
level, thereby setting an experimental lower bound on $m_{S}$, and hence on
$\nu_{\chi}$. We will then proceed to study the constraints on the remaining
parameter space of the Z2GM model.

##  V Experimental status of the Z2GM model

We now proceed to perform full numerical scans of the parameter space of the
Z2GM model subject to the theoretical constraints from perturbative unitarity
and vacuum stability. We then sequentially apply the experimental constraints
from direct searches for the non-SM Higgs bosons in the Z2GM model and LHC
measurements of the signal strengths of the 125 GeV Higgs boson $h$. Details
of the scan procedure and the sources and implementation of the theoretical
and experimental constraints are collected in Appendix B.

###  V.1 Experimental status of the region with $m_{S}<m_{h}/2$

We begin with a quantitative analysis of the parameter region with
$m_{S}<m_{h}/2$. To characterize the phenomenology of this region, we first
show in Fig. 4 the full parameter region with $m_{S}<m_{h}/2$ subject to the
theoretical constraints and the experimental lower bound $m_{3},m_{5}\geq 76$
GeV, which comes from a combination of LEP-2 searches for charged Higgs boson
pair production (constraining $m_{3}$) and an ATLAS search at 8 TeV for like-
sign dimuon production (constraining $m_{5}$).

Figure 4:  Full parameter region with $m_{S}<m_{h}/2$ subject to theoretical
constraints and the lower bounds $m_{3},m_{5}\geq 76$ GeV. Green points are
excluded by the constraint on $b\to s\gamma$, which eliminates the
high-$\nu_{\chi}$ region in the $m_{3}$–$\nu_{\chi}$ plane (left plot). Blue
points are excluded by direct searches for the non-SM Higgs bosons in the Z2GM
model, as implemented in GMCALC and HiggsBounds, together with the requirement
that the 125 GeV Higgs total width is below the indirect bound of 19.1 MeV as
discussed in the previous section. The maroon points survive the above
constraints (but will be eliminated by the 125 GeV Higgs coupling measurements
in Fig. 5).

Applying the direct searches for non-SM Higgs bosons implemented in
HiggsBounds and GMCALC, together with the indirect 99.7% CL upper bound on the
125 GeV Higgs total width of 19.1 MeV discussed in the previous section,
eliminates all but the maroon points in Fig. 4. Additionally, we can see that
the $b\rightarrow s\gamma$ bound is redundant in this case, signalling the
power of the direct searches in this region. The remaining parameter space is
clustered at small $\nu_{\chi}$, mainly due to the suppression of the single-
production cross sections of the additional Higgs bosons through vector boson
fusion, gluon fusion, and fermion-antifermion fusion at small $\nu_{\chi}$,
allowing them to evade the direct searches.

While the masses $m_{3}$, $m_{5}$, and $m_{S}$ still individually populate
most of their theoretically-allowed ranges, the bound on the 125 GeV Higgs
total width strongly restricts the internal parameters of the model. Because
the contributions of the singly- and doubly-charged scalars to the
$h\to\gamma\gamma$ rate involve different combinations of the same parameters
that control $h\to SS$, imposing the experimental constraint on the
$h\to\gamma\gamma$ rate will serve to entirely exclude this remaining region
of the parameter space.

Figure 5:  Surviving parameter space (maroon) from Fig. 4 as a function of the
coupling ratio $|\lambda_{\gamma V}|$ and $\nu_{\chi}$. The brown points show
the surviving parameter space after imposing the $3\sigma$ constraints on
$\lambda_{fV}$ and $\kappa_{fV}$. The beige shaded stripe shows the $3\sigma$
allowed ATLAS range for this ratio; since no points remain in this range, the
entire parameter region with $m_{S}<m_{h}/2$ is excluded. The pink shaded
stripe shows the $5\sigma$ projected sensitivity to this coupling ratio at the
HL-LHC for comparison.

To do this quantitatively, we use the ATLAS fit of ratios of Higgs coupling
modifiers from Ref. [22], as described in Appendix B. Combining what amount to
independent measurements of the same underlying parameters in the Z2GM model,
we obtain bounds on the combinations $\lambda_{\gamma
V}\equiv\kappa_{\gamma}/\kappa_{V}$,
$\lambda_{fV}\equiv\kappa_{f}/\kappa_{V}$, and
$\kappa_{fV}\equiv\kappa_{f}\kappa_{V}/\kappa_{H}$, where $\kappa_{H}$ is
defined in terms of the total Higgs width according to
$\Gamma_{T}=\kappa_{H}^{2}\Gamma_{T}^{SM}$.

In order to visualize the exclusion, we plot the surviving points from Fig. 4
(in maroon) as a function of $|\lambda_{\gamma V}|$ and $\nu_{\chi}$ in Fig.
5. We further require that $\lambda_{fV}$ and $\kappa_{fV}$ lie within their
$3\sigma$ (99.7% CL) allowed ranges, shown in light brown; this restriction
preserves most of the previously allowed parameter region in this projection.
However, we see that the allowed parameter space exhibits a very sizeable
deviation from the SM in $\lambda_{\gamma V}$, with the allowed points lying
in the range $|\lambda_{\gamma V}|\sim 0.5$–0.7. This is mainly due to the
suppression of $\kappa_{\gamma}$ caused by the additional singly- and doubly-
charged Higgs bosons in the loop. _All_ remaining points lie outside the
current $3\sigma$ allowed range of $|\lambda_{\gamma V}|$, as shown by the
beige shaded stripe in Fig. 5. For completeness, we also show as a pink shaded
stripe the projected HL-LHC $5\sigma$ sensitivity to this coupling ratio [44],
which will exclude the entire parameter region with $m_{S}<m_{h}/2$ even more
strongly if no deviation from the SM is found.

The exclusion of the entire parameter region with $m_{S}<m_{h}/2$ entirely
excludes the possibility of very small $\nu_{\chi}$, setting a lower bound of
$\nu_{\chi}>m_{h}/2\sqrt{8\pi}\simeq 12.5$ GeV (or equivalently, $s_{H}\gtrsim
0.143$). This experimental lower bound on $\nu_{\chi}$ in turn forces the
couplings of $h$ to deviate non-negligibly from the SM. This can be easily
understood by fixing $\kappa_{f}=1$ and trying to see how close to 1 one can
make $\kappa_{V}$. For $m_{S}<m_{h}$, $\kappa_{f}=1$ requires
$s_{\alpha}=\nu_{\phi}/\nu$. Plugging this into the expression for
$\kappa_{V}$ and simplifying yields

$\kappa_{V}=1-8\left(1\pm\sqrt{\frac{8}{3}}\right)\frac{\nu_{\chi}^{2}}{\nu^{2}}\qquad({\rm
for}\ \kappa_{f}=1),$ (69)

where the $\pm$ accounts for the two possible quadrants of the mixing angle
$\alpha$. This last expression is valid for both $m_{S}<m_{h}$ and
$m_{S}>m_{h}$. This means that it is impossible to achieve both $\kappa_{f}=1$
and $\kappa_{V}=1$ unless $\nu_{\chi}\to 0$, a possibility which we have just
excluded.

We now turn to the analysis of the rest of the parameter space to further
investigate the consequences of the lower bound on $\nu_{\chi}$.

###  V.2 Experimental status of the rest of the parameter space

Figure 6:  Full parameter region subject to theoretical constraints and the
lower bounds $m_{3},m_{5}>76$ GeV and $m_{S}>m_{h}/2$. Green points are
excluded by the constraint on $b\to s\gamma$, which eliminates the
high-$\nu_{\chi}$ region in the $m_{3}$–$\nu_{\chi}$ plane (left plot). Blue
points are excluded by direct searches for the non-SM Higgs bosons in the Z2GM
model, as implemented in GMCALC and HiggsBounds. The maroon points survive
these constraints.

We begin by plotting in Fig. 6 the masses $m_{3}$, $m_{5}$, and $m_{S}$ as a
function of $\nu_{\chi}$ over the entire parameter space of the Z2GM model as
allowed by theoretical constraints, subject to the experimental lower bounds
$m_{3},m_{5}\geq 76$ GeV and the new bound found in the previous section
$m_{S}>m_{h}/2$. The lower bound on $\nu_{\chi}$ of about 12.5 GeV, imposed by
the experimental constraint $m_{S}>m_{h}/2$, is clearly visible. Points with
large values of $\nu_{\chi}$ are excluded by $b\to s\gamma$; because the new-
physics contribution to $b\to s\gamma$ depends only on $m_{3}$ and
$\nu_{\chi}$, we show the excluded region shaded in green in the left panel of
Fig. 6, while in the remaining panels the points excluded by $b\to s\gamma$
are shown in green. The interplay between the $b\to s\gamma$ constraint and
the upper bound on $m_{3}$ from perturbative unitarity and vacuum stability
constraints in the Z2GM model entirely excludes $\nu_{\chi}$ values above
about 58 GeV.

Figure 7:  Surviving parameter space after applying direct searches for new
Higgs bosons in the Z2GM model (the maroon points from Fig. 6), showing the
ranges of the coupling modifier ratios $|\lambda_{fV}|$, $|\kappa_{fV}|$, and
$|\lambda_{\gamma V}|$ (upper plots), as well as the masses $m_{3}$, $m_{5}$,
and $m_{S}$ (lower plots), as a function of $\nu_{\chi}$. The $2\sigma$ and
$3\sigma$ allowed ranges of the coupling modifier ratios from ATLAS are shown
as horizontal bands in the upper plots. Points allowed at $3\sigma$ by all
three coupling modifier ratio measurements are shown in grey. No points are
allowed at $2\sigma$ by all three coupling modifier ratios.

We then apply the constraints from direct searches for non-SM Higgs bosons
using the searches implemented in GMCALC and HiggsBounds. The points excluded
by these searches are shown in blue. The most important of these searches in
GMCALC are a search for vector boson fusion production of $H_{5}^{\pm\pm}$ for
masses above 200 GeV with decays into like-sign $W$ boson pairs [45], which
together with the upper bound on $m_{5}$ from perturbative unitarity and
vacuum stability constraints in the Z2GM model entirely excludes $\nu_{\chi}$
values above about 40 GeV; a search for Drell-Yan production of
$H_{5}^{++}H_{5}^{--}$ with decays to like-sign $W$ pairs [33], which leads to
an even stronger constraint for $m_{5}$ between 200 and 300 GeV; and theorist-
recast constraints on $H_{5}^{\pm\pm}\to W^{\pm}W^{\pm}$ [27] and
$H_{5}^{0}\to\gamma\gamma$ [46, 47], which constrain the parameter space for
$m_{5}$ below 200 GeV. The sculpting of the surviving parameter space (shown
in maroon) by these constraints on $H_{5}$ can be clearly seen in the middle
panel of Fig. 6. The most important of the searches implemented through
HiggsBounds are LHC searches for $S$ decaying into $WW$, $ZZ$, or $hh$, which
combine with the searches implemented in GMCALC to exclude $m_{S}$ values
above about 175 GeV, and LEP searches for $e^{+}e^{-}\to SZ$ which are
important when $m_{S}<m_{h}$. Additional details of the direct searches are
given in Appendix B.

We now consider the constraints from the 125 GeV Higgs boson coupling
measurements. Within the parameter space that survives the direct searches for
additional Higgs bosons, very large deviations of the $h$ couplings from their
SM values are possible. These are shown in the upper three plots of Fig. 7,
where we plot the absolute values of
$\lambda_{fV}\equiv\kappa_{f}/\kappa_{V}$,
$\kappa_{fV}\equiv\kappa_{f}\kappa_{V}/\kappa_{H}$, and $\lambda_{\gamma
V}\equiv\kappa_{\gamma}/\kappa_{V}$ as a function of $\nu_{\chi}$. The
$2\sigma$ and $3\sigma$ allowed ranges of these observables based on the ATLAS
Higgs coupling fit of Ref. [22] are shown respectively by the orange and grey
horizontal stripes (see Appendix B for details of our choice and handling of
this coupling fit). Points that fall within the $3\sigma$ range of all three
coupling combinations are shown in grey. No points fall within the $2\sigma$
range of all three coupling combinations, which means that the Z2GM model is
_entirely_ excluded at the $2\sigma$ level based on the Higgs coupling
measurements of Ref. [22]. The effect of the Higgs coupling constraints on the
allowed ranges of $m_{3}$, $m_{5}$, and $m_{S}$ is shown in the lower three
plots of Fig. 7.

Figure 8:  As in Fig. 7 but showing the constraints from a HiggsSignals fit of
$h$ signal strengths. Magenta points have $p>0.05$ (allowed at 95% CL) and
yellow points have $p>0.01$ (allowed at 99% CL).

As a cross-check, instead of directly applying the constraints from the
coupling ratios we instead perform the fit of $h$ signal strengths to ATLAS
and CMS data as implemented in HiggsSignals. The advantage of using
HiggsSignals is that it includes CMS data, as well as a more up-to-date
collection of experimental inputs than the ATLAS coupling fit of Ref. [22].
The disadvantages of using HiggsSignals are that the fit is less transparent,
returning a $p$-value rather than providing insight into which observables are
driving any discrepancies with experimental data, and that the statistical
treatment implemented in HiggsSignals necessarily handles potentially
correlated systematic uncertainties in a less sophisticated way than the
dedicated coupling fits performed by the experiments themselves. We show the
results of the HiggsSignals fit in Fig. 8 in the same form as Fig. 7, except
that we show points with $p>0.05$ in magenta and points with $p>0.01$ in
yellow. HiggsSignals finds a small but not-insignificant parameter region with
$p>0.05$ (allowed at 95% CL). For this reason, we conclude that the Z2GM model
is _on the edge_ of being excluded, but cannot yet be said to be _fully_
excluded. Notice that the remaining allowed region from the HiggsSignals fit
has $\lambda_{\gamma V}\sim 0.6$–0.85, which is considerably smaller than the
SM value and will be further tested as the LHC collects additional data.

We can get a better understanding of the effect of the Higgs coupling fit by
examining the correlations among the coupling modifier ratios. To that end, in
Fig. 9 we plot pairs of the coupling modifier ratios $|\lambda_{fV}|$,
$|\kappa_{fV}|$, and $|\lambda_{\gamma V}|$ against each other. The colour
scheme is the same as in Fig. 8. We see that none of the pairs of coupling
modifier ratios can be simultaneously SM-like (the black star at (1,1) in each
panel of Fig. 9), though $|\lambda_{fV}|$ and $|\kappa_{fV}|$ come close.
$|\lambda_{\gamma V}|$, in particular, is rather far from being simultaneously
SM-like with either of the other two coupling ratios; indeed, as shown in the
rightmost panel of Fig. 9, none of the surviving parameter points are
simultaneously within the $2\sigma$ allowed ranges of $|\lambda_{\gamma V}|$
and $|\lambda_{fV}|$. This explains the absence of orange points in Fig. 7.
This behaviour ultimately derives from the non-decoupling nature of the Z2GM
model.

Figure 9:  Correlations among the coupling modifier ratios. Colours are the
same as in Fig. 8. The central values for the ATLAS coupling fit are indicated
with a small black square and the SM prediction of (1,1) with a small black
star.

We finally consider the prospects for improved measurements of the Higgs
coupling modifier ratios at the HL-LHC [20]. In the left panel of Fig. 10 we
reproduce the magenta points from the upper right panel of Fig. 7, showing
also in orange the points for which $\lambda_{fV}$ and $\kappa_{fV}$ lie
within their $2\sigma$ ranges from the ATLAS coupling fit [22]. The horizontal
orange stripe shows the $2\sigma$ allowed range for $|\lambda_{\gamma V}|$,
which is plotted on the $y$-axis. As shown before, the model is just barely
excluded at $2\sigma$ by this analysis. For comparison, in the right panel of
Fig. 10 we plot the points that survive the current LHC constraints from
direct searches and are within the projected $5\sigma$ range of $\lambda_{fV}$
and $\kappa_{fV}$ at the HL-LHC [20], assuming that their experimental central
values will be SM-like. The horizontal shaded brown band in this plot shows
the projected $5\sigma$ range for $|\lambda_{\gamma V}|$. The fact that the
points are well outside this band indicates that the HL-LHC will decisively
exclude the entire Z2GM model assuming that no deviations of these couplings
from their SM values are found. In this case, if the GM model occurs in
nature, an explicit breaking of the $Z_{2}$ symmetry would be required.

Figure 10:  (Left) Surviving parameter space after applying direct searches
for new Higgs bosons in the Z2GM model (the maroon points from Fig. 6), and in
orange those points for which $\lambda_{fV}$ and $\kappa_{fV}$ are both within
their $2\sigma$ experimental ranges. The horizontal shaded orange band shows
the $2\sigma$ allowed range of $|\lambda_{\gamma V}|$, showing that (just
barely) no points satisfy all three coupling ratio constraints. (Right) The
projection for the HL-LHC showing those points for which $\lambda_{fV}$ and
$\kappa_{fV}$ are both within their expected $5\sigma$ ranges, along with the
projected $5\sigma$ allowed range of $|\lambda_{\gamma V}|$, assuming that no
deviations from the SM are found, showing that the HL-LHC will decisively
exclude the entire Z2GM model in this scenario.

##  VI Conclusions

In this paper, we analyzed the phenomenology of a constrained version of the
Georgi-Machacek (GM) model with a $Z_{2}$ symmetry enforced in the scalar
potential (Z2GM). Unlike the full GM model, when the triplet vacuum
expectation value $\nu_{\chi}$ is nonzero the Z2GM model does _not_ possess a
decoupling limit in which the additional scalars can all be taken heavy while
the couplings of the remaining 125 GeV Higgs boson approach their SM values.
This implies that the Z2GM model will exhibit some irreducible deviations from
the SM, and could potentially be entirely excluded by current or near-future
experiments.

A key component of this analysis stemmed from the fact that in the limit of
small $\nu_{\chi}$ in the Z2GM model, the second custodial singlet $S$ becomes
very light, opening the possibility of a significant decay width of $h\to SS$.
The limited parameter freedom of the Z2GM model also prevents a simultaneous
suppression of the $hSS$ coupling and the $h$ couplings to the singly- and
doubly-charged scalars which modify the loop-induced $h\gamma\gamma$ coupling.
These two features allow current constraints on the Higgs total width and the
$h\to\gamma\gamma$ rate to entirely exclude the parameter region with
$m_{S}<m_{h}/2$ at $99.7\%$ confidence level. This in turn puts an absolute
lower bound on $\nu_{\chi}$ of about 12.5 GeV.

This lower bound on $\nu_{\chi}$ _enforces_ a nonzero minimal deviation of the
tree-level couplings of $h$ from their SM values. The loop-induced coupling of
$h$ to $\gamma\gamma$ also receives significant modifications from the
presence of the non-decoupling singly- and doubly-charged scalars in the loop.
Combining constraints on the model from direct searches for non-SM Higgs
bosons and measurements of the couplings of the 125 GeV Higgs boson, we showed
that the Z2GM model is on the verge of being fully excluded by current
experimental data. The remaining parameter space exhibits sizeable deviations
in the 125 GeV Higgs couplings, particularly in $\lambda_{\gamma
V}\equiv\kappa_{\gamma}/\kappa_{V}$, which is suppressed by 15–40% compared to
its SM value. It also restricts $S$ to be lighter than about 175 GeV, which
could be probed by a future $e^{+}e^{-}$ collider. We show that the addition
of the projected HL-LHC measurements of the 125 GeV Higgs couplings will
completely exclude the model at more than the $5\sigma$ level, assuming that
no deviation from the SM is found. In this case, if the GM model occurs in
nature, an explicit breaking of the $Z_{2}$ symmetry would be required.

###### Acknowledgements.

We thank João G. Alencar Caribé for helpful comments and discussions. This
work was supported by the Natural Sciences and Engineering Research Council of
Canada (NSERC).

##  Appendix A  Generalization of the mass hierarchy relation

The relation derived in the Z2GM model for the mass hierarchy between $h$ and
$S$, Eq. (40), can be generalized for an arbitrary $2\times 2$ hermitian
matrix of which one eigenvalue is fixed. This means that we can also do this
analysis for the GM model without the extra $Z_{2}$ symmetry and compare the
results to the more constrained Z2GM case. In this appendix, we derive this
result generically and then apply it to both the Z2GM and GM models to
illustrate the unique feature created by the $Z_{2}$ symmetry. The core of the
argument is the level repulsion phenomenon for matrix eigenvalues.

Let $\Lambda$ be a $2\times 2$ hermitian matrix that has the general form:

$\displaystyle\Lambda=\begin{pmatrix}\lambda_{11}&\lambda_{12}\\\
\lambda_{12}^{*}&\lambda_{22}\end{pmatrix}\,.$ (70)

The eigenvalues of the matrix will be called $\lambda_{a}$ and $\lambda_{b}$.
In principle, they can be any real numbers. However, for our setup, we want to
fix $\lambda_{a}$ to be a specific value, $\lambda_{fix}$. We can implement
this by adjusting $\lambda_{11}$ such that $\lambda_{a}=\lambda_{fix}$. This
procedure is uniquely determined and does not depend on which eigenvalue is
larger or smaller:

$\displaystyle\lambda_{11}=\lambda_{fix}+\frac{\absolutevalue{\lambda_{12}}^{2}}{\lambda_{22}-\lambda_{fix}}\,.$
(71)

We then insert this relation into the trace of $\Lambda$:

$\displaystyle\Tr(\Lambda)$
$\displaystyle=\lambda_{11}+\lambda_{22}=\lambda_{fix}+\frac{\absolutevalue{\lambda_{12}}^{2}}{\lambda_{22}-\lambda_{fix}}+\lambda_{22}=\lambda_{fix}+\lambda_{b}.$
(72)

This gives us a unique result for the other eigenvalue $\lambda_{b}$:

$\displaystyle\lambda_{b}=\lambda_{22}+\frac{\absolutevalue{\lambda_{12}}^{2}}{\lambda_{22}-\lambda_{fix}}\,.$
(73)

Given this relation we can subtract $\lambda_{fix}$ from both sides to obtain
the hierarchy between the eigenvalues:

$\displaystyle\lambda_{b}-\lambda_{fix}=\mathcal{K}+\frac{\absolutevalue{\lambda_{12}}^{2}}{\mathcal{K}}\,,\qquad{\rm
where}\ \mathcal{K}=\lambda_{22}-\lambda_{fix}\,.$ (74)

This result is the generalization of what we obtained in Eq. (40). Since the
matrix $\Lambda$ is hermitian, the numerator of the second term in Eq. (74) is
always positive. This means that the sign of the right-hand side is controlled
by the sign of $\mathcal{K}$. The eigenvalue $\lambda_{b}$ is larger than the
fixed one $\lambda_{fix}$ when $\mathcal{K}>0$ and smaller when
$\mathcal{K}<0$.

We can understand why this result happens if we look at the matrix when
$\lambda_{12}=0$. In this case the eigenvalues are directly determined
$\lambda_{11}=\lambda_{a}$, $\lambda_{22}=\lambda_{b}$. Now we adjust
$\lambda_{11}$ such that $\lambda_{a}=\lambda_{fix}$. This give us the system
with the eigenvalues $\lambda_{fix}$ and $\lambda_{22}=\lambda_{b}$. Now,
turning on the off-diagonal term these eigenvalues will repel by an amount
$\Delta$. This means that the new eigenvalues are $\lambda_{fix}+\Delta$ and
$\lambda_{22}-\Delta$. In this example we chose $\lambda_{fix}$ to be the
larger eigenvalue; in the opposite situation, the sign of $\Delta$ will be
flipped.

Because we want one of the eigenvalues to be equal to $\lambda_{fix}$, we
again need to adjust $\lambda_{11}$ to enforce this. This changes the value of
$\Delta$, but the second eigenvalue remains repelled from its initial value.
This means that the hierarchy between the two eigenvalues is preserved when
$\lambda_{12}$ is nonzero, simply because $|\lambda_{12}|^{2}$ is always
positive.

We now use this result to compare the $m_{S}<m_{h}$ region in the Z2GM model
to that in the GM model without the $Z_{2}$ symmetry. Using the explicit
expressions for the custodial-singlet mass matrix in the two models, we obtain

$\displaystyle\mathcal{K}_{Z2GM}$
$\displaystyle=8\lambda_{34}\nu_{\chi}^{2}-m_{h}^{2}\,,$ (75)
$\displaystyle\mathcal{K}_{GM}$
$\displaystyle=\frac{M_{1}\nu_{\phi}^{2}}{4\nu_{\chi}}-6M_{2}\nu_{\chi}+8\lambda_{34}\nu_{\chi}^{2}-m_{h}^{2}\,,$
(76)

where the scalar potential for the GM model is identical to that in Eq. (II)
with the addition of the two $Z_{2}$-breaking terms, [9]

$V_{GM}=V_{Z2GM}-M_{1}{\rm
Tr}(\Phi^{\dagger}\tau^{a}\Phi\tau^{b})(UXU^{\dagger})_{ab}-M_{2}{\rm
Tr}(X^{\dagger}t^{a}Xt^{b})(UXU^{\dagger})_{ab}.$ (77)

Here $U$ is a unitarity matrix given in Ref. [9] that rotates $X$ into the
Cartesian basis.

We plot $\mathcal{K}$ (normalized by $m_{h}^{2}$ to make it dimensionless)
against $\nu_{\chi}$ in Fig. 11 for the two models from scans over the model
parameters. The red line shows $\mathcal{K}=0$ in order to highlight the
region of negative $\mathcal{K}$ values, which give rise to the hierarchy
$m_{S}<m_{h}$. In the Z2GM model (left panel of Fig. 11), the region with
small $\nu_{\chi}$ is populated _only_ with values of $\mathcal{K}$ less than
zero. This is enforced by the upper bound $\lambda_{34}<\pi$ from perturbative
unitarity, which gives rise to the parabolic shape of the upper left edge of
the populated parameter region. In contrast, the GM model (right panel of Fig.
11) is well populated with positive values of $\mathcal{K}$ all the way down
to $\nu_{\chi}=0$ (negative values of $\mathcal{K}$ also appear, as shown in
the inset). This happens because the first term in Eq. (76) can easily be
larger than $m_{h}^{2}$ even when $\nu_{\chi}$ is very small.

Figure 11:  Parameter space scans showing $\mathcal{K}$ as a function of
$\nu_{\chi}$ for the Z2GM (left) and GM (right) models. The red line indicates
$\mathcal{K}=0$, which divides the two mass hierarchies $m_{S}>m_{h}$
(positive $\mathcal{K}$) from $m_{S}<m_{h}$ (negative $\mathcal{K}$).

##  Appendix B Theoretical and experimental constraints applied

In this appendix, we collect the details of the theoretical and experimental
constraints applied to the Z2GM model in our analysis. We also describe the
strategy that we used to scan the parameter space. In our analysis we used the
public codes GMCALC version 1.5.3 [25], HiggsBounds version 5.3.2 beta [24],
and HiggsSignals version 2.2.3 [23]. HiggsBounds and HiggsSignals were called
from within GMCALC.

###  B.1 Theoretical and indirect constraints

We require that the parameters of the scalar potential satisfy the constraints
from perturbative unitarity of two-to-two scalar scattering amplitudes [16]
and that the scalar potential is bounded from below [9]. While these
conditions were originally found for the unconstrained GM model (without the
additional $Z_{2}$ symmetry), they do not depend on the $Z_{2}$-breaking terms
and hence also apply to the Z2GM model. Together with the requirement that all
the squared masses of physical scalars are positive, these conditions restrict
the allowed ranges of the quartic couplings so that all allowed values are
captured by the scan ranges for $\lambda_{2}$, $\lambda_{3}$, $\lambda_{4}$,
and $\lambda_{5}$ described in Ref. [25], with the additional constraint
$\lambda_{5}>0$ in the Z2GM model from the requirement that $m_{3}^{2}>0$. The
full set of perturbative unitarity and boundedness-from-below constraints are
then applied to each point by GMCALC.

We also apply a check in GMCALC, implemented using a numerical scan of vevs,
that rejects points for which the potential possesses a deeper minimum in
which the custodial symmetry is spontaneously broken [9, 48] (i.e., we require
that the custodial-symmetric vacuum is the global minimum of the scalar
potential).

The Z2GM model is also subject to constraints from $B$ physics, arising from
one-loop processes involving the top quark and the charged scalar $H_{3}^{+}$.
These constraints depend only on $m_{3}$ and $\nu_{\chi}$ (or alternatively
$s_{H}$), and the most important of them is from $b\to s\gamma$ [49], which is
also applied via its implementation in GMCALC (we apply the “loose” bound as
described in Ref. [49]). This excludes parameter points with large values of
$\nu_{\chi}$. The same points will also be excluded by the direct searches for
additional Higgs bosons that we discuss below.

Finally, in the analysis of the parameter region with $m_{S}<m_{h}/2$, we use
the indirect bound on the 125 GeV Higgs boson total width from analyses of on-
and off-shell production in the four-lepton final state [42],

$\displaystyle\Gamma_{T}<19.1~{}{\rm MeV}\qquad{\rm at\ 99.7\%\ CL}.$ (78)

This can be compared to the SM prediction for the 125 GeV Higgs boson total
width of 4.09 MeV [41]. Such a large enhancement of the Higgs total width due
to the non-SM decay $h\to SS$ would dramatically modify the Higgs branching
ratios to SM final states and hence would also be excluded by Higgs signal
strength measurements.

###  B.2 Constraints from direct searches for additional Higgs bosons

####  B.2.1 Direct searches implemented in GMCALC

GMCALC implements several direct searches for the additional Higgs bosons of
the Z2GM model, including dedicated LHC searches for the doubly-charged Higgs
$H_{5}^{\pm\pm}$ and searches recast by theorists to constrain the model.
These are as follows:

  * •

Production of $H_{5}^{\pm\pm}$ in vector boson fusion (VBF) with decays to
$W^{\pm}W^{\pm}\to$ like-sign dileptons. The cross-section is proportional to
$\nu_{\chi}^{2}$, so this process directly constrains large triplet vevs. We
include the latest dedicated CMS search for this process [45] valid for
$m_{5}\geq 200$ GeV, as well as a theorist recast [27] of an ATLAS search for
like-sign $W$ boson production in VBF at 8 TeV [50] which constrains this
process for $m_{5}$ as low as 100 GeV. The implementation of these searches in
GMCALC accounts for the fact that BR($H_{5}^{++}\to W^{+}W^{+}$) can be less
than one (due to $H_{5}\to H_{3}W$ and $H_{5}\to H_{3}H_{3}$ decays).

  * •

Drell-Yan production of $H_{5}^{++}H_{5}^{--}$ or $H_{5}^{\pm\pm}H_{5}^{\mp}$
with $H_{5}^{\pm\pm}$ decaying into like-sign $W$ boson pairs. We include a
recent dedicated ATLAS search [33] valid for $m_{5}\geq 200$ GeV that
_entirely_ excludes $m_{5}$ values between 200 and 350 GeV under the
assumption that BR($H_{5}^{++}\to W^{+}W^{+})=1$; the implementation of this
search in GMCALC accounts for the possibility that this branching ratio is
less than one, which indeed occurs in the Z2GM model, so that some parameter
space in this mass range survives this direct constraint. We also include a
theory recast [29, 30] of ATLAS like-sign dimuon data at 8 TeV [28], which
puts a lower bound on $m_{5}$ of 76 GeV under the assumption that
BR($H_{5}^{++}\to W^{+}W^{+})=1$; i.e., that $H_{5}^{\pm\pm}\to
W^{\pm}H_{3}^{\pm}$ does not compete with decays of $H_{5}^{++}$ into like-
sign $W$ pairs. Together with LEP-2 searches for pair production of singly-
charged Higgs bosons [40] (interpreted here as $e^{+}e^{-}\to
H_{3}^{+}H_{3}^{-}$), which exclude the possibility of $m_{3}<78$ GeV assuming
that $H_{3}^{\pm}$ decays entirely into a combination of $\tau\nu$ and $cs$
final states and hence remove the possibility of $H_{5}^{++}$ decays to
$W^{+}H_{3}^{+}$, this allows us to impose an _absolute_ lower bound
$m_{3},m_{5}\geq 76$ GeV.

  * •

Drell-Yan production of $H_{5}^{0}H_{5}^{\pm}$ with
$H_{5}^{0}\rightarrow\gamma\gamma$. We include a theory recast [46, 47] of an
ATLAS diphoton resonance search at 8 TeV [31], which significantly constrains
the model for $m_{5}$ below about 120 GeV.

####  B.2.2 Direct searches from HiggsBounds

HiggsBounds implements a very large number of direct search limits for neutral
and singly-charged Higgs bosons from LEP, Tevatron, and LHC experiments. We
apply the HiggsBounds constraints to the additional Higgs bosons $S$,
$H_{3}^{0}$, $H_{5}^{0}$, and $H_{3}^{\pm}$. We do not apply the HiggsBounds
constraints to the 125 GeV Higgs boson $h$ decays into SM final states because
HiggsBounds bases its exclusions on applying the single most sensitive
experimental analysis to any given model point; therefore, a downward
fluctuation in the 125 GeV Higgs boson event rate in a single measurement
could exclude model points that would more properly be allowed based on a
global combination of Higgs signal strengths. We will later use HiggsSignals
to constrain the 125 GeV Higgs boson’s production and decay rates in SM
channels.

After applying the direct searches implemented in GMCALC as discussed above,
the additional parameter regions excluded by HiggsBounds are almost entirely
due to searches for the second custodial singlet $S$. These include
constraints from LEP searches for $e^{+}e^{-}\rightarrow SZ$ (with $S\to
b\bar{b}$, $\gamma\gamma$, and inclusive final states) when $S$ is
sufficiently light, as well as LHC searches involving $S\to WW$, $ZZ$, and
$hh$ at heavier $S$ masses.

###  B.3 Constraints from production and decay rates of the 125 GeV Higgs
boson

In the Z2GM model the couplings of $h$ to fermion pairs, $W$ and $Z$ boson
pairs, and photon pairs are modified compared to their values in the SM. We
can therefore use LHC measurements of Higgs production and decay rates, which
are sensitive to these couplings, to constrain the parameter space. We use two
different strategies to apply these constraints and compare their results in
the text.

####  B.3.1 Ratios of coupling modifiers

In an ideal world, we would test each model point by applying the experimental
constraints on the Higgs coupling modification factors $\kappa_{f}$,
$\kappa_{V}$, and $\kappa_{\gamma}$. This $\kappa$-framework is useful when we
have new states around the electroweak scale, which is the case in the Z2GM
model. Unfortunately, direct fits of the Higgs boson couplings based on LHC
data necessarily require assumptions to be made in order to eliminate flat
directions. In particular, the most common fits are made assuming
modifications to $\kappa_{f}$ and $\kappa_{V}$ (but no new particles in the
loops contributing to $\kappa_{g}$ or $\kappa_{\gamma}$), or modifications to
$\kappa_{g}$ and $\kappa_{\gamma}$ due to new particles in the loops (but no
modification of $\kappa_{f}$ and $\kappa_{V}$). Because the Z2GM model
predicts modifications to $\kappa_{f}$ and $\kappa_{V}$ along with new
particles in the loops for $\kappa_{\gamma}$, we are restricted to using
coupling fits that accommodate this possibility. The only such coupling fit
that exists is for the generic parameterization of six ratios of coupling
modifiers $\lambda_{ij}\equiv\kappa_{i}/\kappa_{j}$ together with one overall
measure of the signal rate $\kappa_{gZ}\equiv\kappa_{g}\kappa_{Z}/\kappa_{H}$,
where $\kappa_{H}$ parameterizes modifications to the total width of the Higgs
according to $\Gamma_{T}=\kappa_{H}^{2}\Gamma_{T}^{SM}$. We therefore take as
input the most recent such fit by ATLAS in Table 12 of Ref. [51].

For added statistical power we can take advantage of the fact that the Z2GM
model obeys $\kappa_{t}=\kappa_{b}=\kappa_{\tau}=\kappa_{g}\equiv\kappa_{f}$
and $\kappa_{Z}=\kappa_{W}\equiv\kappa_{V}$, so that several of these coupling
modifier ratios represent independent measurements of the same underlying
combination of model parameters. We statistically combine these “redundant”
measurements assuming that the uncertainties are Gaussian distributed (we
symmetrize asymmetric uncertainty ranges by taking their average) and ignoring
the fact that some of the systematic uncertainties are correlated; in
particular, we combine $\lambda_{\tau Z}$, $\lambda_{bZ}$ and
$\lambda_{Zg}^{-1}$ to obtain $\lambda_{fV}$. By this method we obtain,

$\displaystyle\kappa_{fV}$
$\displaystyle=\frac{\kappa_{f}\kappa_{V}}{\kappa_{H}}=1.06\pm 0.07\,,$ (79)
$\displaystyle\lambda_{\gamma V}$
$\displaystyle=\frac{\kappa_{\gamma}}{\kappa_{V}}=0.94\pm 0.07\,,$ (80)
$\displaystyle\lambda_{fV}$
$\displaystyle=\frac{\kappa_{f}}{\kappa_{V}}=0.92\pm 0.08\,.$ (81)

In our analysis of the Z2GM parameter space we require that each of these
observables _separately_ lies within $2$ or $3\sigma$ of its central value;
i.e., we do not combine their likelihoods.

We also consider the anticipated precision of these measurements at the HL-LHC
[20]. Combining channels in the same way, we obtain the anticipated $1\sigma$
experimental uncertainties,

$\displaystyle\delta\kappa_{fV}=0.034\,,\qquad\delta\lambda_{\gamma
V}=0.024\,,\qquad\delta\lambda_{fV}=0.034\,.\qquad{\rm(HL-LHC)}$ (82)

####  B.3.2 HiggsSignals

As an independent cross-check using experimental inputs from both ATLAS and
CMS and different statistical methods, we also apply the global Higgs signal
strength fit in HiggsSignals [23]. We treat each parameter point as its own
model (with zero free parameters) to extract the $p$-value. We apply the peak-
centred $\chi^{2}$ method implemented in HiggsSignals incorporating all
neutral scalars in order to capture total signal rates when one of the other
scalars is close in mass to the 125 GeV Higgs. We show regions with $p>0.05$
(allowed at 95% CL) and $p>0.01$ (allowed at 99% CL). Overall, the region of
parameter space allowed by the HiggsSignals global fit is similar to that
obtained using the coupling modifier fit based on ATLAS data alone, which
gives us higher confidence in the robustness of our conclusions.

###  B.4 Numerical scan procedure

The Z2GM model contains 7 parameters; after fixing $G_{F}$ and $m_{h}$, we are
left with 5 free parameters that must be scanned. We choose the parameters
$\mu_{3}^{2}$, $\lambda_{2}$, $\lambda_{3}$, $\lambda_{4}$, and $\lambda_{5}$
as the free parameters and fix $\mu_{2}^{2}$ and $\lambda_{1}$ in terms of
these and the measured values of $G_{F}$ and $m_{h}$. This corresponds to
INPUTSET = 2 in GMCALC (we also set $M_{1}=M_{2}=0$ in GMCALC in order to
implement the $Z_{2}$ symmetry). We also require that the global minimum has
nonzero $\nu_{\chi}$; i.e., that the $Z_{2}$ symmetry is spontaneously
broken.101010In this paper we do not analyze the dark matter phase of the
theory in which $\nu_{\chi}=0$ and the $Z_{2}$ symmetry is preserved. All of
our numerical scans begin by imposing the theoretical constraints
(perturbative unitarity, boundedness from below, and the absence of deeper
minima) together with the lower bound $m_{3},m_{5}\geq 76$ GeV from direct
searches.

The scans are performed in two steps. First, we use GMCALC to generate random
samples uniformly distributed in the variables $\sqrt{|\mu_{3}^{2}|}$,
$\lambda_{2}$, $\lambda_{3}$, $\lambda_{4}$, and $\lambda_{5}$ and apply the
relevant constraints to discard excluded points. Then, to better populate the
allowed parameter regions and improve the efficiency of the scans, we use the
surviving points as input to the machine learning implementation
LearnDistribution in MATHEMATICA [52]. This allows us to efficiently generate
a very large number of points in MATHEMATICA concentrated in the vicinity of
the relatively small surviving regions of parameter space, which we finally
feed back through GMCALC to calculate physical observables and impose the
relevant constraints.

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