# Constraining the Georgi-Machacek Model with a Light Higgs

Amine Ahriche ID<EMAIL_ADDRESS>Department of Applied Physics and
Astronomy, University of Sharjah, P.O. Box 27272 Sharjah, UAE.

###### Abstract

In this work, we investigate the viability of a light Higgs ($\eta$) scenario
in the Georgi-Machacek (GM) model, where we consider all theoretical and
experimental constraints such as the perturbativity, vacuum stability,
unitarity, electroweak precision tests, the Higgs di-photon and undetermined
decays and the Higgs total decay width. In addition, we consider more recent
experimental bounds from the searches for doubly-charged Higgs bosons in the
VBF channel $H_{5}^{++}\to W^{+}W^{+}$, Drell-Yan production of a neutral
Higgs boson $pp\to H_{5}^{0}(\gamma\gamma)H_{5}^{+}$, and for the light
scalars at LEP $e^{-}e^{+}\to Z\eta$, and at ATLAS and CMS in different final
states such as $pp\to\eta\to 2\gamma$ and $pp\to h\to\eta\eta\to 4\gamma,2\mu
2\tau,2\mu 2b,2\tau 2b$. By combining these bounds together, we found a
parameter space region that is significant as the case of the SM-like Higgs to
be the light CP-even eigenstate, and this part of the parameter space would be
tightened by the coming analyses.

##  I Introduction

Despite the discovery of a Standard Model (SM)-like 125 GeV Higgs boson at the
Large Hadron Collider (LHC) ATLAS:2012yve ; CMS:2012qbp , the SM is not
complete since it can not provide answers to many questions, such as the
hierarchy problem, fermions masses, strong CP problem, the dark matter nature,
baryon asymmetry at the Universe; and the neutrino oscillation data. The
discovered 125 GeV scalar seems to be the SM Higgs, however, it is not clear
how does the electroweak symmetry breaking (EWSB) occurs: via one single
scalar field or more?

In many SM extensions, the EWSB is achieved by more than one scalar, where
some of the new scalar fields acquire vacuum expectation values (VEV), and
then mix with the SM doublet, which make the 125 GeV Higgs is a composite
field. Among the popular models, the so-called Georgi-Machacek (GM) model
Georgi:1985nv , where the SM is extended by one complex and one real scalar
triplets, assigned by a global custodial $SU(2)_{V}$ symmetry, that is
preserved in the scalar potential after the EWSB. The model vacuum is defined
in a way that predicts a tree-level custodial symmetry, under which the scalar
spectrum manifests in multiplets: a quintet ($H_{5}$), a triplet ($H_{3}$) and
two CP-even singlets ($\eta$ and $h$).

The existence of extra scalar degrees of freedom and the non-trivial couplings
of the Higgs to the gauge bosons, make GM model phenomenologically very rich
Chanowitz:1985ug ; Gunion:1989ci ; Haber:1999zh ; Aoki:2007ah ; Godfrey:2010qb
; Low:2010jp ; Logan:2010en ; Chang:2012gn ; Kanemura:2013mc ; Englert:2013zpa
; Killick:2013mya ; Englert:2013wga ; Ghosh:2019qie ; Das:2018vkv ;
Hartling:2014zca ; Hartling:2014aga ; Chiang:2014bia ; Chiang:2015rva ;
Chang:2017niy ; Chiang:2015kka ; Chen:2022zsh . In addition, the GM model can
address some the SM open questions, such as the neutrino mass Chen:2020ark ,
dark matter Pilkington:2017qam , and the electroweak phase transition strength
Chiang:2014hia . The GM scalar sector has been confronted with the existing
data Ismail:2020zoz , where direct search constraints for extra Higgs bosons
and measurements of the SM-like Higgs properties are considered. In addition,
the authors derived bounds from the negative searches of the doubly-charged
Higgs bosons in the VBF channel $H_{5}^{++}\to W^{+}W^{+}$; and the Drell-Yan
production of a neutral Higgs boson $pp\to H_{5}^{0}(\gamma\gamma)H_{5}^{+}$.
In Bairi:2022adc , we have investigated the GM parameter space where the SM-
like Higgs is considered to be the light CP-even eigenstate $h=h_{125}$; and
the eigenstate $\eta$ is a heavier scalar $m_{\eta}>m_{h}$. We have considered
all the known theoretical and experimental constraints, including those from
the negative searches of the heavy scalar via $pp\to\eta\to hh,\tau\tau,ZZ$;
and we have found that a significant part of the parameter space is viable and
could be probed soon by future analyses with more data. One has to mention
that it turns out that two thirds of the parameter space allowed by the
constraints described in the literature, are excluded by some possibly
existing scalar potential minima (that either preserve or violate the CP and
electric charge symmetries) that are deeper than the electroweak (EW) vacuum
Bairi:2022adc . Here, we aim to investigate other part of the parameter space
where the SM-like Higgs is the heavy CP-even eigenstate, i.e., $h=h_{125}$ and
$m_{\eta}<m_{h}$. We will consider all the above mentioned constraints in
addition to constraints from the searches of light CP-even scalar whether are
direct at LEP $e^{-}e^{+}\to Z\eta$, and LHC $pp\to\eta\to\gamma\gamma$
CMS:2018cyk ; ATLAS:2018xad ; ATLAS:2022nkn ; or indirect $pp\to
h\to\eta\eta\to X\overline{X}Y\overline{Y}$ CMS:2018zvv ; CMS:2018qvj ;
CMS:2022xxa ; ATLAS:2021ypo ; ATLAS:2021hbr .

In section II, we review the GM model and present the constraints described in
the literature. We discuss the physics of a light CP-even scalar at colliders
in section III; and discuss our numerical results in section IV. In section V,
we give our conclusion.

##  II The Model: Parameters and Constraints

The GM model scalar sector includes a doublet $(\phi^{+},\,\phi^{0})^{T}$; and
two triplets $(\chi^{++},\,\chi^{+},\,\chi^{0})^{T}$ and
$(\xi^{+},\,\xi^{0},\,-\xi^{-})^{T}$ with the hypercharge $Y=1,2,0$,
respectively;

$\Phi=\left(\begin{array}[]{cc}\frac{\upsilon_{\phi}+h_{\phi}-ia_{\phi}}{\sqrt{2}}&\phi^{+}\\\
-\phi^{-}&\frac{\upsilon_{\phi}+h_{\phi}+ia_{\phi}}{\sqrt{2}}\end{array}\right),\quad\Delta=\left(\begin{array}[]{ccc}\frac{\upsilon_{\chi}+h_{\chi}-ia_{\chi}}{\sqrt{2}}&\xi^{+}&\chi^{++}\\\
-\chi^{-}&\upsilon_{\xi}+h_{\xi}&\chi^{+}\\\
\chi^{--}&-\xi^{-}&\frac{\upsilon_{\chi}+h_{\chi}+ia_{\chi}}{\sqrt{2}}\end{array}\right),$
(1)

where the vacuum expectation values (VEV’s) satisfy the relations
$\upsilon_{\chi}=\sqrt{2}\upsilon_{\xi}$ and
$\upsilon_{\phi}^{2}+8\upsilon_{\xi}^{2}\equiv\upsilon^{2}=(246.22\,\mathrm{GeV})^{2}$;
to ensure the tree-level custodial symmetry. The scalar potential that is
invariant under the global symmetry $SU(2)_{L}\times SU(2)_{R}\times U(1)_{Y}$
in the GM model is

$\displaystyle V(\varPhi,\Delta)$
$\displaystyle=\frac{m_{1}^{2}}{2}\mathrm{Tr}[\varPhi^{\dagger}\varPhi]+\frac{m_{2}^{2}}{2}\mathrm{Tr}[\Delta^{\dagger}\Delta]+\lambda_{1}(\mathrm{Tr}[\varPhi^{\dagger}\varPhi])^{2}+\lambda_{2}\mathrm{Tr}[\varPhi^{\dagger}\varPhi]\mathrm{Tr}[\Delta^{\dagger}\Delta]+\lambda_{3}\mathrm{Tr}[(\Delta^{\dagger}\Delta)^{2}]$
$\displaystyle+\lambda_{4}(\mathrm{Tr}[\Delta^{\dagger}\Delta])^{2}-\lambda_{5}\mathrm{Tr}[\varPhi^{\dagger}\frac{\sigma^{a}}{2}\varPhi\frac{\sigma^{b}}{2}]\mathrm{Tr}[\Delta^{\dagger}T^{a}\Delta
T^{b}]-\mu_{1}\mathrm{Tr}[\varPhi^{\dagger}\frac{\sigma^{a}}{2}\varPhi\frac{\sigma^{b}}{2}](U\Delta
U^{\dagger})_{ab}$
$\displaystyle-\mu_{2}\mathrm{Tr}[\Delta^{\dagger}T^{a}\Delta T^{b}](U\Delta
U^{\dagger})_{ab},$ (2)

with $\sigma^{1,2,3}$ are the Pauli matrices and $T^{1,2,3}$ correspond to the
generators of the $SU(2)$ triplet representation and the matrix $U$ is given
in Georgi:1985nv . After the EWSB, we are left with: three CP-even eiegenstaes
$\\{h,\eta,H_{5}^{0}\\}$, one CP-odd eigenstate $H_{3}^{0}$, two singly
charged scalars $\\{H_{3}^{\pm},H_{5}^{\pm}\\}$, and one doubly charged scalar
$H_{5}^{\pm\pm}$, that are defined as

$\displaystyle h$
$\displaystyle=c_{\alpha}h_{\phi}-\frac{s_{\alpha}}{\sqrt{3}}(\sqrt{2}h_{\chi}+h_{\xi}),\,\eta=s_{\alpha}h_{\phi}+\frac{c_{\alpha}}{\sqrt{3}}(\sqrt{2}h_{\chi}+h_{\xi}),\,H_{5}^{0}=\sqrt{\frac{2}{3}}h_{\xi}-\sqrt{\frac{1}{3}}h_{\chi},$
$\displaystyle H_{3}^{0}$
$\displaystyle=-s_{\beta}a_{\phi}+c_{\beta}a_{\chi},\,H_{3}^{\pm}=-s_{\beta}\phi^{\pm}+c_{\beta}\frac{1}{\sqrt{2}}(\chi^{\pm}+\xi^{\pm}),\,H_{5}^{\pm}=\frac{1}{\sqrt{2}}(\chi^{\pm}-\xi^{\pm}),\,H_{5}^{\pm\pm}=\chi^{\pm\pm},$
(3)

with $\tan\beta=2\sqrt{2}\upsilon_{\xi}/\upsilon_{\phi}$ and $\tan
2\alpha=2M_{12}^{2}/(M_{22}^{2}-M_{11}^{2})$, where $M^{2}$ is the mass
squared matrix in the basis
$\\{h_{\phi},\,\sqrt{\frac{2}{3}}h_{\chi}+\frac{1}{\sqrt{3}}h_{\xi}\\}$. The
quartic couplings $\lambda$’s can be expressed as

$\displaystyle\lambda_{1}$
$\displaystyle=\frac{\varrho_{1}c_{\alpha}^{2}+\varrho_{2}s_{\alpha}^{2}}{8\upsilon^{2}c_{\beta}^{2}},\,\lambda_{2}=\frac{m_{3}^{2}}{\upsilon^{2}}-\frac{c_{\alpha}s_{\alpha}}{\sqrt{6}\upsilon^{2}c_{\beta}s_{\beta}}(\varrho_{1}-\varrho_{2})-\frac{\mu_{1}}{2\sqrt{2}\upsilon
s_{\beta}},$ $\displaystyle\lambda_{3}$
$\displaystyle=\frac{\sqrt{2}(\mu_{1}c_{\beta}^{2}-3\mu_{2}s_{\beta}^{2})}{s_{\beta}^{3}\upsilon}-\frac{3c_{\beta}^{2}m_{3}^{2}}{s_{\beta}^{2}\upsilon^{2}}+\frac{m_{5}^{2}}{s_{\beta}^{2}\upsilon^{2}},\,\lambda_{5}=-\frac{\sqrt{2}\mu_{1}}{\upsilon
s_{\beta}}+\frac{2m_{3}^{2}}{\upsilon^{2}},$ $\displaystyle\lambda_{4}$
$\displaystyle=-\frac{\mu_{1}c_{\beta}^{2}-3\mu_{2}s_{\beta}^{2}}{\sqrt{2}s_{\beta}^{3}\upsilon}+\frac{c_{\beta}^{2}m_{3}^{2}}{s_{\beta}^{2}\upsilon^{2}}-\frac{m_{5}^{2}}{3s_{\beta}^{2}\upsilon^{2}}+\frac{\varrho_{1}s_{\alpha}^{2}+\varrho_{2}c_{\alpha}^{2}}{3s_{\beta}^{2}\upsilon^{2}},$
(4)

with $\varrho_{1}=\min(m_{h}^{2},m_{\eta}^{2})$ and
$\varrho_{2}=\max(m_{h}^{2},m_{\eta}^{2})$. The formulas of $\lambda_{1,2,4}$
here are valid for both cases of $m_{h}<m_{\eta}$ and $m_{h}>m_{\eta}$.

It has been shown in Bairi:2022adc  that the scalar potential (2) could
acquire some minima that could violate the CP-symmetry and/or electric charge,
where they could be deeper than the electroweak vacuum
$\\{\upsilon_{\phi},\sqrt{2}\upsilon_{\xi},\upsilon_{\xi}\\}$. Then, this part
of the parameter space would be ignored. Here, we impose the constraints from
(1) vacuum stability, (2) unitarity, (3) the electroweak precision tests, (4)
the di-photon and undetermined Higgs branching ratios and total decay width;
in addition to (5) the constraints from negative searches for light scalar
resonances at LEP OPAL:2002ifx . For the constraints (1-3), we used the
results described in Bairi:2022adc .

In this setup, the SM-like Higgs $h$ (the CP-even scalar
$m_{h}=125.18\,\mathrm{GeV}$) decays mainly into pairs of fermions
$(cc,\,\mu\mu,\,\tau\tau,\,b\overline{b})$ and gauge bosons $WW^{*}$ and
$ZZ^{*}$, in addition to a pair of light scalars $\eta\eta$ when kinematically
allowed. Since the Higgs couplings to SM fields are scaled by the coefficients

$\displaystyle\kappa_{\mathfrak{\mathrm{F}}}$
$\displaystyle=\frac{g_{hff}^{GM}}{g_{hff}^{SM}}=\frac{c_{\alpha}}{c_{\beta}},\,\kappa_{V}=\frac{g_{hVV}^{GM}}{g_{hVV}^{SM}}=c_{\alpha}c_{\beta}-\sqrt{\frac{8}{3}}s_{\alpha}s_{\beta},$
(5)

then, its total decay width can be written as

$\displaystyle\varGamma_{h}^{tot}$
$\displaystyle=\Gamma_{h}^{SM}\sum_{X=SM}\kappa_{X}^{2}\mathcal{B}^{SM}(h\to
XX)+\Theta(m_{h}-2m_{\eta})\frac{g_{h\eta\eta}^{GM}}{32\pi
m_{h}}\big{(}1-4m_{\eta}^{2}/m_{h}^{2}\big{)}^{1/2},$ (6)

where the last term represents the partial decay width $\Gamma(h\to\eta\eta)$,
$\varGamma_{h}^{SM}=4.08\,\mathrm{MeV}$ ParticleDataGroup:2020ssz  and
$\mathcal{B}^{SM}(h\to XX)$ are the SM values for total decay width and the
branching ratios for the Higgs, respectively. Here, $g_{h\eta\eta}^{GM}$ is
the scalar triple coupling $h\eta\eta$. Since the light scalar $\eta$ can be
seen at detectors via its decay to light fermions $\eta\to f\bar{f}$, then,
the Higgs decay $h\to\eta\eta$ does not match any of the known SM final
states, and hence called undetermined channel, which is constrained by ATLAS
as $\mathcal{B}_{und}<0.22$ ATLAS:2019cid ; ATLAS:2019nkf . The total Higgs
decay width recent measurements based on the off-shell Higgs production in the
final state $h\to ZZ^{*}\to\ell\ell\nu\nu$ give the upper bound
$\Gamma_{h}=4.6_{-2.5}^{+2.6}~{}\textrm{MeV}$ ATLAS-CONF-2022-068 .

In previous analysis of the GM parameter space Bairi:2022adc , it has been
shown that the measurements of the Higgs signal strength modifiers imply
constraints on the coefficients $\kappa_{F,V}$. Here, another factor is
constrained in addition, which is the undetermined Higgs decay; and
consequently the scalar $\eta$ mass and the triple coupling
$g_{h\eta\eta}^{GM}$. Then, the partial Higgs signal strength modifier for the
channel $h\to XX$ ($X=f,V$) can be simplified in this setup as

$\mu_{XX}=\frac{\sigma(pp\to h)\times\mathcal{B}(h\to XX)}{\sigma^{SM}(pp\to
h)\times\mathcal{B}^{SM}(h\to
XX)}=\kappa_{F}^{2}\kappa_{X}^{2}(1-\mathcal{B}_{und}),$ (7)

where $\sigma(gg\to h)\,[\sigma^{SM}(gg\to h)]$ is the $ggF$ production cross
section in the GM [SM] model.

For the di-photon Higgs strength modifier, it can be written as

$\displaystyle\mu_{\gamma\gamma}=\kappa_{F}^{2}(1-\mathcal{B}_{und})\left|\frac{\frac{\upsilon}{2}\sum_{i}\frac{g_{hii}^{GM}}{m_{i}^{2}}Q_{i}^{2}A_{0}^{\gamma\gamma}(4m_{i}^{2}/m_{h}^{2})+\kappa_{V}A_{1}^{\gamma\gamma}(4m_{W}^{2}/m_{h}^{2})+\kappa_{F}\frac{4}{3}A_{1/2}^{\gamma\gamma}(4m_{t}^{2}/m_{h}^{2})}{A_{1}^{\gamma\gamma}(4m_{W}^{2}/m_{h}^{2})+\frac{4}{3}A_{1/2}^{\gamma\gamma}(4m_{t}^{2}/m_{h}^{2})}\right|^{2},$
(8)

where $i=H_{3}^{+},\,H_{5}^{+},\,H_{5}^{++}$ stands for all charged scalars
inside the loop diagrams, $Q_{i}$ is the electric charge of the field $i$ in
units of $|e|$, $g_{hii}^{GM}$ are the Higgs triple couplings to the charged
scalars; and the functions $A_{i}^{\gamma\gamma}$ are given in the literature
Djouadi:2005gi . According to the partial Higgs strength modifier formulas in
(7) and (8), one expects that the experimental constraints would enforce the
coefficients $\kappa_{F,V}$ to lie around unity. In our numerical scan, we
consider the recent experimental values ParticleDataGroup:2022pth  for the
partial Higgs strength modifiers (7) and (8).

Besides the above mentioned constraints, the negative searches for doubly-
charged Higgs bosons in the VBF channel $H_{5}^{++}\to W^{+}W^{+}$; and from
Drell-Yan production of a neutral Higgs boson $pp\to
H_{5}^{0}(\gamma\gamma)H_{5}^{+}$; give strong bounds on the parameter space
Ismail:2020zoz . It has been shown in Ismail:2020zoz , that the doubly-charged
Higgs bosons in the VBF channel leads to a constraint from CMS on
$s_{\beta}^{2}\times\mathcal{B}(H_{5}^{++}\to W^{+}W^{+})$ CMS:2017fhs .
Clearly, non vanishing values for the branching ratios
$\mathcal{B}(H_{5}^{++}\to W^{+}H_{3}^{+})$ (for $m_{5}>m_{3}+m_{W}$) and
$\mathcal{B}(H_{5}^{++}\to H_{3}^{+}H_{3}^{+})$ (for $m_{5}>2m_{3}$), will
significantly help to relax this bounds. While the relevant quantity for the
constraints on $H_{5}^{0}\to\gamma\gamma$ is the fiducial cross section times
branching ratio
$\sigma_{fid}=\big{(}\sigma_{H_{5}^{0}H_{5}^{+}}\times\epsilon_{+}+\sigma_{H_{5}^{0}H_{5}^{-}}\times\epsilon_{-}\big{)}\mathcal{B}(H_{5}^{0}\to\gamma\gamma)$,
that is constrained by ATLAS at 8 TeV ATLAS:2014jdv  and at 13 TeV
ATLAS:2017ayi . Here, we used the decay rate formulas, the cross section and
efficiency values used in Ismail:2020zoz  to include these constraints in our
numerical analysis.

Since part of the charged triplet $H_{3}^{\pm}$ is coming from the SM doublet
as shown in (3), then it should couple the up and down quarks as the W gauge
bosons does. This interaction leads to flavor violating processes such as the
$b\to s$ transition ones, which depend only on the charged triplet mass
$m_{3}$ and the mixing angle $\beta$ (and consequently the triplet VEV
$\upsilon_{\chi}$). In our numerical scan, we consider the bounds on the
$m_{3}$-$\upsilon_{\chi}$ plan shown in Hartling:2014aga .

##  III the light scalar $\eta$ in the Collider

After the discovery of the Higgs boson with $m_{h}=125.18\,\mathrm{GeV}$,
efforts have been devoted to search for light neutral scalar boson through
different channels over a wide range of mass. Such results can also be used to
impose constraints on models with many neutral scalars such as the GM model.

The two CP-even eigenstates $h$ and $\eta$ are defined through a mixing angle
$\alpha$ and $(m_{\eta}<m_{h})$, where the heavy eigenstate $h$ is identified
to be the SM-like Higgs with the measured mass $m_{h}=125.18\,\mathrm{GeV}$.
Here, the light scalar $\eta$ has similar couplings as the SM Higgs, but
modified with the factors

$\displaystyle\zeta_{V}$ $\displaystyle=\frac{g_{\eta
VV}^{GM}}{g_{hVV}^{SM}}=s_{\alpha}c_{\beta}+\sqrt{\frac{8}{3}}c_{\alpha}s_{\beta},\,\zeta_{F}=\frac{g_{\eta
FF}^{GM}}{g_{hFF}^{SM}}=\frac{s_{\alpha}}{c_{\beta}},$ (9)

then, the partial decay width of the light scalar $\eta$ into SM final states
can be written as $\Gamma(\eta\to
X\overline{X})=\zeta_{X}^{2}\Gamma^{SM}(\eta\to X\overline{X})$, where
$\Gamma^{SM}(\eta\to X\overline{X})$ is the Higgs partial decay width
estimated at $m_{h}\to m_{\eta}$ Higgs . Thus, its total decay width can be
written as

$\varGamma_{\eta}^{tot}=\Gamma_{\eta}^{SM}\sum_{X=SM}\zeta_{X}^{2}\mathcal{B}^{SM}(\eta\to
X\overline{X}),$ (10)

where $\Gamma_{\eta}^{SM}$ and $\mathcal{B}^{SM}(\eta\to X\overline{X})$ are
the Higgs total decay width and the branching ratios estimated at $m_{h}\to
m_{\eta}$ Higgs . At colliders, there have been many searches for light
scalars which are translated into constraints on the light scalar mass and its
couplings to SM particles. In what follows, we will focus on two types of
searches of the light scalar $\eta$: (1) direct production like
$e^{-}e^{+},pp\to\eta+X$, where the scalar could be identified via one of its
SM-like decays $\eta\to\gamma\gamma,\mu\mu,\tau\tau,cc,bb$, and (2) indirect
production via the Higgs decay $pp\to h\to\eta\eta\to
X\overline{X}Y\overline{Y}$, where the light scalar is identified via its SM
decays $X,Y=\gamma,\mu,\tau,c,b$. Here, we will consider the constraints from
the negative searches for $pp\to\eta\to\gamma\gamma$ at CMS at 8+13 TeV
CMS:2018cyk , and at ALTAS at 13 TeV with integrated luminosity
$80~{}\mathrm{fb}^{-1}$ ATLAS:2018xad ; and at $138~{}\mathrm{fb}^{-1}$
ATLAS:2022nkn .

At LEP, many searches for Higgs at low mass range $m_{h}<100~{}\textrm{GeV}$
have been performed, and bounds on the form factor OPAL:2002ifx ; that can be
simplified in our setup as

$\kappa_{Z\eta}=\frac{\sigma(e^{-}e^{+}\to~{}\eta~{}Z)}{\sigma^{SM}(e^{-}e^{+}\to~{}\eta~{}Z)}=\zeta_{V}^{2}.$
(11)

A similar search for light SM-like Higgs in the di-photon channel with masses
in the range $70-110~{}\mathrm{GeV}$ has been done by CMS at 8 $\mathrm{TeV}$
and 13 $\mathrm{TeV}$ CMS:2018cyk , where some upper bounds are established on
the production cross section scaled by its SM value
$\zeta_{F}^{2}.\kappa_{\gamma\gamma}^{\eta}=\frac{\sigma(pp\to\eta)\times\mathcal{B}(\eta\to\gamma\gamma)}{\sigma^{SM}(pp\to\eta)\times\mathcal{B}^{SM}(\eta\to\gamma\gamma)}$.
In our setup, the parameter $\kappa_{\gamma\gamma}^{\eta}$ can be simplified
as

$\kappa_{\gamma\gamma}^{\eta}=\zeta_{F}^{2}\frac{\Gamma_{SM}^{tot}(\eta)}{\Gamma^{tot}(\eta)}\left|\frac{\frac{\upsilon}{2}\sum_{i}\frac{g_{\eta
ii}^{GM}}{m_{i}^{2}}Q_{i}^{2}A_{0}^{\gamma\gamma}\big{(}4m_{i}^{2}/m_{\eta}^{2}\big{)}+\zeta_{V}A_{1}^{\gamma\gamma}\big{(}4m_{W}^{2}/m_{\eta}^{2}\big{)}+\zeta_{F}\frac{4}{3}A_{1/2}^{\gamma\gamma}\big{(}4m_{t}^{2}/m_{\eta}^{2}\big{)}}{A_{1}^{\gamma\gamma}\big{(}4m_{W}^{2}/m_{\eta}^{2}\big{)}+\frac{4}{3}A_{1/2}^{\gamma\gamma}\big{(}4m_{t}^{2}/m_{\eta}^{2}\big{)}}\right|^{2},$
(12)

with $i=H_{3}^{\pm},~{}H_{5}^{\pm},~{}H_{5}^{\pm\pm}$, $g_{\eta ii}^{GM}$ are
the scalar triple couplings of the scalar $\eta$ to the charged scalars; and
the functions $A_{0,1,1/2}^{\gamma\gamma}$ are given in the literature
Djouadi:2005gi .

Concerning the indirect searches via the Higgs decay into a pairs of light
scalars in the channels $h\to HH\to X\overline{X}Y\overline{Y}$ ($H$ could be
a CP-odd scalar $a$ or a CP-even one like $\eta$ in our setup), many searches
by ATLAS and CMS at $8\,\mathrm{TeV}$ and/or $13\,\mathrm{TeV}$ and at
different values for the integrated luminosity values; have been performed in
many final states, among them that decay into photons or light fermions. For
instance, negative searches for $pp\to h\to\eta\eta\to
X\overline{X}Y\overline{Y}$ established bounds either on the branching ratio
$\mathcal{B}(h\to\eta\eta\to X\overline{X}Y\overline{Y})$, the ratio
$\frac{\sigma(pp\to h)}{\sigma^{SM}(pp\to h)}\mathcal{B}(h\to\eta\eta\to
X\overline{X}Y\overline{Y})$ and/or the cross section section $\sigma(pp\to
h)\times\mathcal{B}(h\to\eta\eta\to X\overline{X}Y\overline{Y})$. In our work,
we will consider the constraints coming from the CMS and ATLAS searches for
the Higgs decay into a pair of light pseudoscalars in the final state $h\to
2a\to 2b2\tau$ CMS:2018zvv , $h\to 2a\to 2\tau 2\mu$ at
$\sqrt{s}=13~{}\mathrm{TeV}$ and $\mathcal{L}=35.9~{}\mathrm{fb}^{-1}$
CMS:2018qvj , $h\to 2a\to 4\gamma$ at CMS $\sqrt{s}=13~{}TeV$ and
$\mathcal{L}=132~{}\mathrm{fb}^{-1}$ CMS:2022xxa , $h\to 2a\to 2b2\mu$ at
$\sqrt{s}=13~{}\mathrm{TeV}$ and $\mathcal{L}=36.1~{}\mathrm{fb}^{-1}$
ATLAS:2021ypo , $h\to 2a\to 2b2\mu$ at ATLAS $\sqrt{s}=13~{}\mathrm{TeV}$ and
$\mathcal{L}=139~{}\mathrm{fb}^{-1}$ ATLAS:2021hbr . All these analyses were
performed using the ggF Higgs production mode at the LHC.

##  IV Numerical Analysis and Discussion

Here, we have considered the heavy CP-even scalar to be the
$125\,\textrm{GeV}$ SM-like Higgs; and have taken into account the different
theoretical and experimental constraints described in Sections II and III,
such as the constraints from perturbativity, vacuum stability Hartling:2014zca
; Arhrib:2011uy , electroweak precision tests Hartling:2014aga  , the di-
photon and undetermined Higgs decays, the total Higgs decay width; and the B
physics flavor constraints. In addition, we have considered also the
constraints from the fact that the EW vacuum
$(\upsilon_{\phi},\,\sqrt{2}\upsilon_{\xi},\upsilon_{\xi})$ must be the
deepest among possible minima that may preserve or violate the CP and electric
charge symmetries as described in Bairi:2022adc . As a first step of this
numerical study, we perform a full numerical scan over the GM model parameter
space, then, in a next step we will imposes the constraints from the negative
searches for doubly-charged Higgs bosons in the VBF channel $H_{5}^{++}\to
W^{+}W^{+}$; and from Drell-Yan production of a neutral Higgs boson $pp\to
H_{5}^{0}(\gamma\gamma)H_{5}^{+}$; negative searches for a light Higgs at LEP
$e^{-}e^{+}\to~{}\eta~{}Z$ OPAL:2002ifx , the direct search for a light
resonance at the LHC CMS:2018cyk  and the indirect searches for light
resonance via the final states $h\to\eta\eta\to X\overline{X}Y\overline{Y}$
CMS:2018zvv ; CMS:2018qvj ; CMS:2022xxa ; ATLAS:2021ypo ; ATLAS:2021hbr .

The GM parameter space is described by the free parameters
$\lambda_{2},\;\lambda_{4},\;m_{\eta},m_{3},\;m_{5},\,s_{\alpha}$ and
$t_{\beta}=\tan\beta\equiv 2\sqrt{2}\upsilon_{\xi}/\upsilon_{\phi}$, within
the ranges

$78~{}\textrm{GeV}<m_{3,5}<3~{}\textrm{TeV},\,10~{}\textrm{GeV}<m_{\eta}<m_{h},\,\left|\lambda_{2,4}\right|\leq
20,\,|t_{\beta}|\leq 3,$ (13)

where for the lower bound on $m_{3,5}$, we considered lower mass bound on this
singly charged scalar obtained from the direct search at the LEP II
LEPHiggsWorkingGroupforHiggsbosonsearches:2001ogs .

Here, the negative values of $t_{\beta}$ are considered for the following
reason. In the GM model, there exists an invariance under the transformation
$(\upsilon_{\xi},\mu_{1,2})\to(-\upsilon_{\xi},-\mu_{1,2})$, which means
$V(\Phi,\Delta,\mu_{1,2})=V(\Phi,-\Delta,-\mu_{1,2})$. Consequently, the
scalar mass matrix elements also remain invariant under this transformation.
However, because the physical scalar eigenstates are mixtures of the
components of $\Phi$ and $\Delta$, most of the physical vertices that involves
scalars are not invariant under
$(\upsilon_{\xi},\mu_{1,2})\to(-\upsilon_{\xi},-\mu_{1,2})$. This means that
these vertices change; and therefore any two benchmark points (BPs) with the
same input parameters but with different signs of $(\pm
t_{\beta},\pm\mu_{1,2})$ are physically different. This can be seen in the
scaling factors (5) and (9). This makes the BPs with negative $t_{\beta}$
values in (13) independent part of the parameter space that should not be
ignored.

After combining all the first step constraints, we show in Fig. 1 the viable
parameters space and the different physical observables using 34.7k BPs.

Figure 1: Different physical observables estimated in the GM model by taking
into account the constraints from perturbativity, vacuum stability,
electroweak precision tests, the di-photon and undetermined Higgs decays; and
the total Higgs decay width.

From Fig. 1, one notices a significant parameter space comparable to the case
where the SM-like Higgs is the light CP-even eigenstate. The parameter space
in the plan $\\{m_{3},m_{5}\\}$ is different than the case of GM model with
heavy scalar $\eta$, while in the plans $\\{t_{\beta},s_{\alpha}\\}$,
$\\{\kappa_{F},\kappa_{V}\\}$ and $\\{\zeta_{F},\zeta_{V}\\}$ they are
similar. One has to notice that imposing different theoretical and
experimental constraints, especially the Higgs total width, the Higgs signal
strength modifiers and the B physics flavor constraints, makes the parameter
space well constrained. It is separated into three distinct islands in the
plan $\\{t_{\beta},s_{\alpha}\\}$, where the first one corresponds to positive
$t_{\beta}$ values, the second corresponds to negative $t_{\beta}$ and
positive $s_{\alpha}$ values, while the region of negative $t_{\beta}$ and
negative $s_{\alpha}$ values represents the third island. The Higgs fermions
couplings ($hf\bar{f}$) are always positive; and can be modified by a ratio up
to 15%, while the Higgs gauge couplings ($hVV$) can be modified by few
percent; and could be negative for negative $s_{\alpha}$. The second and third
islands that were not considered previously in the literature (negative
$t_{\beta}$) correspond to negative ($hf\bar{f}$) couplings. The magnitude of
these couplings can be enhanced up to 120%, which makes the searches for SM-
like light scalars in the light fermions channels efficient to probe this part
of the viable parameter space.

Here, one has to mention that most of the allowed light scalar mass values are
for $m_{\eta}>m_{h}/2$ due to the conflict between the constraints from the
undetermined ($h\to\eta\eta$) and di-photon ($h\to\gamma\gamma$) Higgs decays.

Some of these 34.7k BPs are in agreement all the above mentioned constraints,
including those are considered in the second step of our analysis. For
instance, we show in Fig. 2 some of the observables like the form factor (11)
that is constrained by OPAL OPAL:2002ifx , the ratio
$s_{\beta}^{2}\times\mathcal{B}(H_{5}^{++}\to W^{+}W^{+})$ constrained by CMS
CMS:2017fhs ; and cross section at 8+13 TeV $\sigma(pp\to\eta\to\gamma\gamma)$
constrained by CMS CMS:2018cyk .

Figure 2: Left: the form factor (11) versus the light scalar mass
$m_{\eta},$where the palette shows the light scalar total decay width. The
green curve represents the OPAL bounds OPAL:2002ifx . Middle: the ratio
$s_{\beta}^{2}\times\mathcal{B}(H_{5}^{++}\to W^{+}W^{+})$ compared to the CMS
bounds CMS:2017fhs , where the yellow (blue) region corresponds to 68% (95%)
CL, and the palette shows the branching ratio $\mathcal{B}(H_{5}^{++}\to
W^{+}W^{+})$. Right: the combined cross section at 8+13 TeV
$\sigma(pp\to\eta\to\gamma\gamma)$ scaled by the SM values compared to the CMS
bounds for the mass range $80\,\mathrm{GeV}<m_{\eta}<110\,\mathrm{GeV}$
CMS:2018cyk , where the palette shows the factor (12) that represents the
enhancement effect on the decay $\eta\to\gamma\gamma$ due to the coupling with
charged scalars. The yellow (blue) region corresponds to 68% (95%) CL.

From Fig. 2-left and -right, one learns that the first and second islands in
Fig. 1-top-right is in agreement with the OPAL bounds (i.e., the lower green
island in Fig. 2-left that corresponds to $|\zeta_{V}|\lesssim 0.7$). The
third island is also in agreement with OPAL since it corresponds to
$80~{}\mathrm{GeV}\leq m_{\eta}\leq 120~{}\mathrm{GeV}$, i.e., the pink island
in Fig. 2-left.

From Fig. 2-right, one notices that the BPs with the $\eta$ mass in the range
$80~{}\mathrm{GeV}-110~{}\mathrm{GeV}$ that are in agreement with the bound
form $pp\to\eta\to\gamma\gamma$ are those that belong to the first and second
islands. From Fig. 2-middle, one remarks that the majority of the BPs with
open decay channels $H_{5}^{++}\to W^{+}H_{3}^{+},H_{3}^{+}H_{3}^{+}$, which
makes the branching ratio $\mathcal{B}(H_{5}^{++}\to W^{+}W^{+})$
significantly smaller than unity. Indeed, some of the BPs with
$\mathcal{B}(H_{5}^{++}\to W^{+}W^{+})=1$ are also in agreement with the
bounds on $s_{\beta}^{2}\times\mathcal{B}(H_{5}^{++}\to W^{+}W^{+})$
CMS:2017fhs .

Clearly, the constraints we have considered in our second step analysis seems
to interesting and efficient. For instance, the constraints from the doubly-
charged Higgs bosons in the VBF channel $H_{5}^{++}\to W^{+}W^{+}$ excludes
41.75% of the BPs; and those from the Drell-Yan production of a neutral Higgs
boson $pp\to H_{5}^{0}(\gamma\gamma)H_{5}^{+}$ excludes only 1.7%. The
negative direct searches of the scalar $\eta$ exclude 0.9% of the BPs. By
combining all these constraints, we got 44.1% of the BPs excluded. In Fig. 3,
we reproduce Fig. 1 by considering only the viable 19.4k BPs.

Figure 3: Different physical observables estimated in the GM model by taking
into account all the constraints.

From Fig. 3-left, one remarks that the majority of the BPs with
$m_{5}>200~{}\mathrm{GeV}$ are excluded, and those that survive have the decay
channels $H_{5}^{++}\to H_{3}^{+}H_{3}^{+},H_{3}^{+}W^{+}$ open, i.e., the
branching ratio for $\mathcal{B}(H_{5}^{++}\to W^{+}W^{+})$ is significantly
smaller than unity, as shown in Fig. 2-middle. Most of the BPs with
$m_{5}<200~{}\mathrm{GeV}$ are not constrained by the searches of the same
sign leptons from the doubly-charged Higgs bosons production in the VBF
channel $H_{5}^{++}\to W^{+}W^{+}$. Therefore, future analysis should consider
the range $78~{}\mathrm{GeV}<m_{5}<200~{}\mathrm{GeV}$. It is clear that this
viable parameter space would tightened by taking into account future analyses
with more data.

In this setup, the light scalar $\eta$ can be probed due to production modes
both at the HL-LHC HL-LHC  and at future lepton colliders such as the ILC
Fujii:2017vwa , CLIC Abramowicz:2016zbo , the CEPC Chen:2016zpw  and the FCCee
Dawson:2013bba  running at the Z-pole and above the Zh production threshold.
In future searches, more severe constraints on the parameters
$m_{\eta},~{}\kappa_{V,F},~{}\zeta_{V,F}$ would be established. Distinguish
the GM model among other similar models that involves a light scalar such by
just relying on the Higgs measurements and the $\eta$-discovery (or $\eta$
negative searches) is not trivial. However, in the GM model, a correlation
between the scalar-gauge and scalar-fermions couplings is absent unlike most
of the SM multi-scalar extensions, i.e., no correlation between
$\kappa_{V}-\kappa_{F}$ and $\zeta_{V}-\zeta_{F}$ exists. This feature could
help to probe the GM model in some cases where both ratios
$\kappa_{V}/\kappa_{F}$ and $\zeta_{V}/\zeta_{F}$ are significantly different
than unity. In addition, the custodial triplet and fiveplet members have many
production modes, which makes the GM model more interesting for any search.

##  V Conclusion

In this work, we studied the GM scalar sector in the case where the SM-like
Higgs corresponds to the heavy CP-even eigenstate. We have shown the viability
of an important region of the parameter space, that is significant as the case
of the light CP-even scalar to be the SM-like Higgs. In our analysis, we
considered the constraints from perturbativity, unitarity, boundness from
below, the electroweak precision tests, the di-photon and undetermined Higgs
decays; and the total Higgs decay width. For this we generated around 34.7k
BPs that fulfill all the previously mentioned constraints. In addition, we
have imposed more bounds from the searches for (1) doubly-charged Higgs bosons
in the VBF channel $H_{5}^{++}\to W^{+}W^{+}$, (2) Drell-Yan production of a
neutral Higgs boson $pp\to H_{5}^{0}(\gamma\gamma)H_{5}^{+}$, and for the
light scalars by ATLAS and CMS in different final states such as (3) $pp\to
h\to\eta\eta\to 4\gamma,2\mu 2\tau,2\mu 2b,2\tau 2b$. We found that only 55.9%
of the BPs survives against these three constraints, where they exclude
41.75%, 1.7%, and 0.9% of the BPs, respectively.

After imposing all constraints, we found that the negative searches of the
doubly-charged Higgs bosons in the VBF channel $H_{5}^{++}\to W^{+}W^{+}$
exclude all BPs with $\mathcal{B}(H_{5}^{++}\to W^{+}W^{+})=1$ and
$m_{5}>200\,\mathrm{GeV}$, while all surviving BPs have the decay channels
$H_{5}^{++}\to H_{3}^{+}H_{3}^{+},H_{3}^{+}W^{+}$ open, $m_{5}>2m_{3}$. It is
important if the future searches for doubly charged scalars would consider
masses below $200\,\mathrm{GeV}$ to probe this scenario. In the near future,
the coming analyses with more data will make the parameter space more
constrained.

The light scalar $\eta$ can be produced and detected at future hadron/lepton
colliders, which is the case of many light scalars in multi-scalar models.
Then, more severe constraints on the parameters
$m_{\eta},~{}\kappa_{V,F},~{}\zeta_{V,F}$ would be established. However,
distinguishing the GM light scalar among other scalars involved in other
multi-scalar models could be possible; and relies on the measurement of the
$\eta$ couplings to gauge bosons ($\zeta_{V}$) and fermions ($\zeta_{F}$). The
reason for this is that, unlike in many multi-scalar models, a correlation
between the scalar-gauge and scalar-fermions couplings is absent in the GM
model. Although, considering some future neutral/charged scalar searches may
put more constraints on the custodial triplet and fiveplet members, which
makes the GM model more interesting.

Acknowledgments: This work is funded by the University of Sharjah under the
research projects No 21021430100 “Extended Higgs Sectors at Colliders:
Constraints & Predictions” and No 21021430107 “Hunting for New Physics at
Colliders”.

## References

  * (1) G. Aad et al. [ATLAS], Phys. Lett. B 716, 1-29 (2012) [arXiv:1207.7214 [hep-ex]].
  * (2) S. Chatrchyan et al. [CMS], Phys. Lett. B 716 (2012), 30-61 [arXiv:1207.7235 [hep-ex]].
  * (3) H. Georgi and M. Machacek, Nucl. Phys. B 262, 463-477 (1985)
  * (4) M. S. Chanowitz and M. Golden, Phys. Lett. B 165, 105-108 (1985)
  * (5) J. F. Gunion, R. Vega and J. Wudka, 1673-1691 (1990)
  * (6) H. E. Haber and H. E. Logan, Phys. Rev. D 62, 015011 (2000) [arXiv:hep-ph/9909335 [hep-ph]].
  * (7) M. Aoki and S. Kanemura, Phys. Rev. D 77, no.9, 095009 (2008) [erratum: Phys. Rev. D 89, no.5, 059902 (2014)] [arXiv:0712.4053 [hep-ph]].
  * (8) S. Godfrey and K. Moats, Phys. Rev. D 81, 075026 (2010) [arXiv:1003.3033 [hep-ph]].
  * (9) I. Low and J. Lykken, JHEP 10, 053 (2010) [arXiv:1005.0872 [hep-ph]].
  * (10) H. E. Logan and M. A. Roy, Rev. D 82, 115011 (2010) [arXiv:1008.4869 [hep-ph]].
  * (11) S. Chang, C. A. Newby, N. Raj and C. Wanotayaroj, Phys. Rev. D 86, 095015 (2012) [arXiv:1207.0493 [hep-ph]].
  * (12) S. Kanemura, M. Kikuchi and K. Yagyu, Phys. Rev. D 88, 015020 (2013) [arXiv:1301.7303 [hep-ph]].
  * (13) C. Englert, E. Re and M. Spannowsky, Rev. D 87, no.9, 095014 (2013) [arXiv:1302.6505 [hep-ph]].
  * (14) R. Killick, K. Kumar and H. E. Logan, 033015 (2013) [arXiv:1305.7236 [hep-ph]].
  * (15) C. Englert, E. Re and M. Spannowsky, 035024 (2013) [arXiv:1306.6228 [hep-ph]].
  * (16) N. Ghosh, S. Ghosh and I. Saha, Phys. Rev. D 101 (2020) no.1, 015029 [arXiv:1908.00396 [hep-ph]].
  * (17) D. Das and I. Saha, Phys. Rev. D 98 (2018) no.9, 095010 [arXiv:1811.00979 [hep-ph]].
  * (18) K. Hartling, K. Kumar and H. E. Logan,Phys. Rev. D 90 (2014) no.1, 015007 [arXiv:1404.2640 [hep-ph]].
  * (19) K. Hartling, K. Kumar and H. E. Logan, Phys. Rev. D 91 (2015) no.1, 015013 [arXiv:1410.5538 [hep-ph]].
  * (20) C. W. Chiang, S. Kanemura and K. Yagyu, Phys. Rev. D 90 (2014) no.11, 115025 [arXiv:1407.5053 [hep-ph]].
  * (21) C. W. Chiang, S. Kanemura and K. Yagyu, Phys. Rev. D 93 (2016) no.5, 055002 [arXiv:1510.06297 [hep-ph]].
  * (22) J. Chang, C. R. Chen and C. W. Chiang, JHEP 03 (2017), 137 [arXiv:1701.06291 [hep-ph]].
  * (23) C. W. Chiang and K. Tsumura, JHEP 04 (2015), 113 [arXiv:1501.04257 [hep-ph]].
  * (24) T. K. Chen, C. W. Chiang, C. T. Huang and B. Q. Lu, [arXiv:2205.02064 [hep-ph]].
  * (25) S. L. Chen, A. Dutta Banik and Z. K. Liu, Nucl. Phys. B 966 (2021), 115394 [arXiv:2011.13551 [hep-ph]].
  * (26) T. Pilkington, [arXiv:1711.04378 [hep-ph]].
  * (27) C. W. Chiang and T. Yamada, Phys. Lett. B 735 (2014), 295-300 [arXiv:1404.5182 [hep-ph]].
  * (28) A. Ismail, H. E. Logan and Y. Wu, [arXiv:2003.02272 [hep-ph]].
  * (29) Z. Bairi and A. Ahriche, [arXiv:2207.00142 [hep-ph]].
  * (30) A. M. Sirunyan et al. [CMS], Phys. Lett. B 793 (2019), 320-347 [arXiv:1811.08459 [hep-ex]].
  * (31) [ATLAS], “Search for resonances in the 65 to 110 GeV diphoton invariant mass range using 80 fb-1 of $pp$ collisions collected at $\sqrt{s}=13$ TeV with the ATLAS detector,” ATLAS-CONF-2018-025.
  * (32) [ATLAS], “Search for boosted diphoton resonances in the 10 to 70 GeV mass range using 138 $fb^{-1}$ of 13 TeV $pp$ collisions with the ATLAS detector,” ATLAS-CONF-2022-018.
  * (33) A. M. Sirunyan et al. [CMS], Phys. Lett. B 785 (2018), 462 [arXiv:1805.10191 [hep-ex]].
  * (34) A. M. Sirunyan et al. [CMS], JHEP 11 (2018), 018 [arXiv:1805.04865 [hep-ex]].
  * (35) [CMS], “Search for the exotic decay of the Higgs boson into two light pseudoscalars with four photons in the final state in proton-proton collisions at $\sqrt{s}$ = 13 TeV,” [arXiv:2208.01469 [hep-ex]].
  * (36) [ATLAS], “Search for Higgs boson decays into two spin-0 particles in the $bb\mu\mu$ final state with the ATLAS detector in $pp$ collisions at $\sqrt{s}=13$ TeV,” ATLAS-CONF-2021-009.
  * (37) G. Aad et al. [ATLAS], Phys. Rev. D 105 (2022) no.1, 012006 [arXiv:2110.00313 [hep-ex]].
  * (38) G. Abbiendi et al. [OPAL], Eur. Phys. J. C 27 (2003), 311-329 [arXiv:hep-ex/0206022 [hep-ex]].
  * (39) P. A. Zyla et al. [Particle Data Group], PTEP 2020 (2020) no.8, 083C01
  * (40) M. Aaboud et al. [ATLAS], Phys. Rev. Lett. 122 (2019) no.23, 231801 [arXiv:1904.05105 [hep-ex]].
  * (41) M. Aaboud et al. [ATLAS], Phys. Rev. D 101 (2020) no.1, 012002 [arXiv:1909.02845 [hep-ex]].
  * (42) “Evidence of off-shell Higgs boson production and constraints on the total width of the Higgs boson in the $ZZ\to 4\ell$ and $ZZ\to 2\ell 2\nu$ decay channels with the ATLAS detector“, ATLAS-CONF-2022-068.
  * (43) A. Djouadi, Phys. Rept. 457 (2008), 1-216 [arXiv:hep-ph/0503172 [hep-ph]].
  * (44) R. L. Workman et al. [Particle Data Group], PTEP 2022 (2022), 083C01.
  * (45) A. M. Sirunyan et al. [CMS], Phys. Rev. Lett. 120 (2018) no.8, 081801 [arXiv:1709.05822 [hep-ex]].
  * (46) G. Aad et al. [ATLAS], Phys. Rev. Lett. 113 (2014) no.17, 171801 [arXiv:1407.6583 [hep-ex]].
  * (47) M. Aaboud et al. [ATLAS], Phys. Lett. B 775 (2017), 105-125 [arXiv:1707.04147 [hep-ex]].
  * (48) https://twiki.cern.ch/twiki/bin/view/LHCPhysics/LHCHWG.
  * (49) A. Arhrib, R. Benbrik, M. Chabab, G. Moultaka, M. C. Peyranere, L. Rahili and J. Ramadan, Phys. Rev. D 84 (2011), 095005 [arXiv:1105.1925 [hep-ph]].
  * (50) [LEP Higgs Working Group for Higgs boson searches, ALEPH, DELPHI, L3 and OPAL], [arXiv:hep-ex/0107031 [hep-ex]].
  * (51) O. Bruning et al. Rep. Prog. Phys. 85 (2022) 046201.
  * (52) K. Fujii, C. Grojean, M. E. Peskin, T. Barklow, Y. Gao, S. Kanemura, H. Kim, J. List, M. Nojiri and M. Perelstein, et al. [arXiv:1710.07621 [hep-ex]].
  * (53) H. Abramowicz, A. Abusleme, K. Afanaciev, N. A. Tehrani, C. Balazs, Y. Benhammou, M. Benoit, B. Bilki, J. J. Blaising and M. J. Boland, et al. Eur. Phys. J. C 77 (2017) no.7, 475 doi:10.1140/epjc/s10052-017-4968-5 [arXiv:1608.07538 [hep-ex]].
  * (54) Z. Chen, Y. Yang, M. Ruan, D. Wang, G. Li, S. Jin and Y. Ban, Chin. Phys. C 41 (2017) no.2, 023003 [arXiv:1601.05352 [hep-ex]].
  * (55) S. Dawson, A. Gritsan, H. Logan, J. Qian, C. Tully, R. Van Kooten, A. Ajaib, A. Anastassov, I. Anderson and D. Asner, et al. [arXiv:1310.8361 [hep-ex]].