# Neutral Higgs bosons phenomenology in the compact 341 model and
confrontations with the LHC results

M. Djouala111 Corresponding author. Laboratoire de Physique Mathématique et
Subatomique,
Frères Mentouri University, Constantine1, Algeria
<EMAIL_ADDRESS>N. Mebarki  Laboratoire de Physique Mathématique et Subatomique,
Frères Mentouri University, Constantine1, Algeria
<EMAIL_ADDRESS>
(Day Month Year; Day Month Year)

###### Abstract

The phenomenology of the neutral scalar bosons in the compact 341 model is
discussed. We show that the predictions of this model are fairly good and
compatible with the experimental data by calculating the signal strength of
$h_{1}$ for the channels $b\bar{b}$, $WW^{*}$, $ZZ^{*}$, $\tau\bar{\tau}$ and
$\gamma\gamma$. Moreover, we use the branching ratios to search for the heavy
neutral scalar bosons $h_{2}$ and $h_{3}$ where we take into account the
theoretical constraints to precise the allowed ranges for the fundamental
parameters of the scalar potential.

###### keywords:

Beyond the Standard Model; Higgs physics

PACS numbers:12.60.-i, 14.80.Cp

##  1 Introduction

The discovery of a new scalar resonance by the ATLAS and CMS at CERN with a
mass around 125 GeV [1, 2] in the Run 1 has paved the way for new directions
in high energy particle physics. Analyzing the properties of this particle is
compatible with the standard model (SM) predictions, that is a strong evidence
that it is the Higgs boson of the Standard Model. Currently the combined
analysis based on the Run 1 (7 and 8 TeV) LHC data shows that the couplings of
the SM Higgs boson with the vector bosons are found to be compatible with
those expected from the SM within a $\sim$ 10$\%$ uncertainty, whereas the
coupling of the SM higgs boson to the third generation fermions is compatible
within an uncertainty of $\sim$ 15-20$\%$. Thus, the current status of the
Higgs properties still allows to explore new interpretations of the
observation coming from new physics of different underlying structures.
The compact 341 model is among many BSM theories [3, 4, 7, 5, 6, 8] which is
constructed from the gauge group $SU(3)_{C}\otimes SU(4)_{L}\otimes U(1)_{X}$,
it predicts the existence of new particles namely the exotic quarks, scalar
and the gauge bosons, moreover, the physical scalar spectrum of this kind of
models is composed by three quadruplets $\chi$, $\eta$ and $\rho$ which lead
to three neutral CP-even scalars $h_{1,2,3}$ where the lightest neutral scalar
boson $h_{1}$ plays the role of the Standard Model Higgs boson, four simply
charged $h_{1,2}^{\mp}$ with two doubly charged scalar bosons $h^{\mp\mp}$.
The compact 341 model has a very specific arrangement of the fermions into
generations: for leptons, one has both right and left handed helecities
arranged in the same multiplet. In order to make the model anomaly free, the
second and third quarks families have to belong to the conjugate $4^{*}$
fundamental representation of the $SU(4)_{L}$ gauge group, while the first
family transforms as a quadruplet in the fundamental representation or vice
versa.
The motivations to study 341 models are various as, for example, explanation
of family replication [8, 9], electric charge quantization [10, 11], strong
CP-problem [12, 13, 14], incorporation of inflation[15] and others [16].
Moreover, the 341 models are able to deal with the current Higgs physics which
has an important role on the LHC to discover new physics. In this work we will
study the decay of the neutral scalar bosons in the compact 341 model and
confront our results to the experimental data.
The paper is organized as follow: section 2 reviews the fundamental features
of the compact 341 model by focusing on its particle content and its scalar
potential followed by the mass spectrum. In section 3, we review the
theoretical constraints on the scalar potential parameters which found from
the tree-level vacuum stability, perturbative unitarity and from the
positivity of the scalar boson masses. All couplings (Feynman rules) needed to
calculate the BRs and signal strengths of the neutral scalar bosons
$h_{1},h_{2},h_{3}$ are presented in section 4. In section 5, we calculate the
signal strength of $h_{1}$ using the various analytical expressions of the
partial decays width for different channels $\tau^{+}\tau^{-}$, $b\bar{b}$,
$WW^{*}$, $ZZ^{*}$, $\gamma\gamma$ and $Z\gamma$ and the branching ratios of
the other heavy scalar bosons $h_{i}$ (i=2,3). We present our numerical
results in section 6. We draw our conclusions in section 7. Finally, the
expressions of the theoretical constraints are given in the appendix.

##  2 Brief review of the model

In general, one of the most important parameters to distinguish different 341
models are denoted as $\beta$ and $\gamma$ which define the electric charges
of new particles through the following electric charge operator [17]:

$\begin{split}Q&=\frac{1}{2}\bigg{(}\lambda_{3}-\frac{1}{\sqrt{3}}\lambda_{8}-\frac{4}{\sqrt{6}}\lambda_{15}\bigg{)}+X\,.\end{split}$
(1)

Where

$\lambda_{3}=\text{diag}(1,-1,0,0)\,,~{}\lambda_{8}=\frac{1}{\sqrt{3}}\text{diag}(1,1,-2,0)\,,~{}\lambda_{15}=\frac{1}{\sqrt{6}}\text{diag}(1,1,1,-3)\,.$
(2)

The particle content of this model is represented in the following table [5]:

The complete anomaly free particle content of the compact 341 model where the
symbol $\sim$ refers to the quantum numbers of the $SU(3)_{C}$, $SU(4)_{L}$,
$U(1)_{X}$ respectively and F refers to flavors  Name 341 rep 331 rep
components F $\psi_{\alpha L}$00 (1,4,$0$)00 (1,3,$\frac{-1}{3}$)+(1,1,1)00
($\nu_{\alpha},l_{\alpha},\nu_{\alpha}^{c},l_{\alpha}^{c}$) 3 $Q_{iL}(i=2,3)$
(3,$\overline{4}$,$\frac{-1}{3})$ (3,$\bar{3}$,0)+(3,1,$\frac{-4}{3}$)
$(d_{i},u_{i},D_{i},J_{i})$ 2 $Q_{1L}$ (3,4,$\frac{2}{3})$
(3,3,$\frac{1}{3}$)+(3,1,$\frac{5}{3}$) $(u_{1},d_{1},U_{1},J_{1})$ 1
$u_{jR}$(j=1,i) ($\overline{3}$,1,$\frac{2}{3}$)
($\overline{3}$,1,$\frac{2}{3}$) $u_{jR}$ 4 $d_{jR}(j=1,i)$
($\overline{3}$,1,$\frac{-1}{3}$) ($\overline{3}$,1,$\frac{-1}{3}$) $d_{jR}$ 5
$J_{1R}$ ($\overline{3}$,1,$\frac{5}{3}$) ($\overline{3}$,1,$\frac{5}{3}$)
$J_{1R}$ 1 $J_{iR}(i=2,3)$ ($\overline{3}$,1,$\frac{-4}{3}$)
($\overline{3}$,1,$\frac{-4}{3}$) $J_{iR}$ 2 $\chi$ (1,4,-1) (1,3,
$\frac{-4}{3}$)+(1,1,0) ($\chi_{1}^{-}$,$\chi^{--}$,$\chi_{2}^{-}$,$\chi^{0})$
1 $\eta$ (1,4,0) (1,3,$\frac{-1}{3}$)+(1,1,1)
($\eta_{1}^{0}$,$\eta_{1}^{-}$,$\eta_{2}^{0}$,$\eta_{2}^{+})$ 1 $\rho$ (1,4,1)
(1,3,$\frac{2}{3}$)+(1,1,2)
($\rho_{1}^{+}$,$\rho^{0}$,$\rho_{2}^{+}$,$\rho^{++})$ 1

To generate masses for all particles, three quadruplets scalar fields
$SU(4)_{L}$ are introduced as we indicated in table 2 where:

$\displaystyle\chi^{0}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\upsilon_{\chi}+R_{\chi}+iI_{\chi}),$ (3)
$\displaystyle\eta^{0}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\upsilon_{\eta}+R_{\eta}+iI_{\eta}),$ (4)
$\displaystyle\rho^{0}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\upsilon_{\rho}+R_{\rho}+iI_{\rho}).$ (5)

with

$R_{\chi}=h_{1},~{}~{}R_{\eta}=\alpha h_{2}+\beta h_{3},~{}~{}R_{\rho}=\gamma
h_{2}+\sigma h_{3},~{}~{}$ (6)

where $\alpha$, $\beta$, $\gamma$ and $\sigma$ are real parameters:

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{-\sqrt{X^{2}+(Y-\sqrt{X^{2}+Y^{2}})^{2}}}{\sqrt{4(X^{2}+Y^{2})}},~{}~{}\beta=\frac{\sqrt{X^{2}+(Y+\sqrt{X^{2}+Y^{2}})^{2}}}{\sqrt{4(X^{2}+Y^{2})}}$
(7) $\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\frac{(Y+\sqrt{X^{2}+Y^{2}})(\sqrt{X^{2}+(Y-\sqrt{X^{2}+Y^{2}})^{2}})}{X\sqrt{4(X^{2}+Y^{2})}}$
(8) $\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{-(Y-\sqrt{X^{2}+Y^{2}})(\sqrt{X^{2}+(Y+\sqrt{X^{2}+Y^{2}})^{2}})}{X\sqrt{4(X^{2}+Y^{2})}}$
(9)

with

$X=\lambda_{5},~{}~{}~{}~{}Y=\lambda_{1}-\lambda_{3},$ (10)

and $I_{\chi}$, $I_{\eta}$ and $I_{\rho}$ represent the Goldostone bosons.
The most general scalar potential in the compact 341 model 222This scalar
potential is invariant under an additional discrete symmetry $Z_{3}$ [5] is
given by[5]:

$\displaystyle V(\eta,\rho,\chi)$ $\displaystyle=$
$\displaystyle\mu_{\eta}^{2}\eta^{{\dagger}}\eta+\mu_{\rho}^{2}\rho^{{\dagger}}\rho+\mu_{\chi}^{2}\chi^{{\dagger}}\chi+\lambda_{1}(\eta^{{\dagger}}\eta)^{2}+\lambda_{2}(\rho^{{\dagger}}\rho)^{2}+\lambda_{3}(\chi^{{\dagger}}\chi)^{2}$
(11) $\displaystyle+$
$\displaystyle\lambda_{4}(\eta^{{\dagger}}\eta)(\rho^{{\dagger}}\rho)+\lambda_{5}(\eta^{{\dagger}}\eta)(\chi^{{\dagger}}\chi)+\lambda_{6}(\rho^{{\dagger}}\rho)(\chi^{{\dagger}}\chi)+\lambda_{7}(\rho^{{\dagger}}\eta)(\eta^{{\dagger}}\rho)$
$\displaystyle+$
$\displaystyle\lambda_{8}(\chi^{{\dagger}}\eta)(\eta^{{\dagger}}\chi)+\lambda_{9}(\rho^{{\dagger}}\chi)(\chi^{{\dagger}}\rho),$

where $\mu^{2}_{\mu\rho\chi}$ are the mass dimension parameters and
$\lambda_{S}$ are dimensionless real coupling constants.
In the compact 341 model, the extension of the electroweak group leads to the
existence of nine scalar bosons, namely three neutral scalars $h_{i}$
(i=1,2,3) where the lightest one $h_{1}$ is assumed to be the observed scalar
of mass $\sim$ 125 GeV (Higgs boson) at the LHC, while, the other scalars
$h_{i}$ (i=2,3) should be heavier, four simply charged $h_{1}^{\mp}$,
$h^{\mp}_{2}$ scalars and two doubly charged $h^{\mp\mp}$ ones. After SSB the
scalar bosons acquire their masses [5]:

$\displaystyle M^{2}_{h_{1}}$ $\displaystyle\approx$
$\displaystyle\lambda_{2}\upsilon_{\rho}^{2}+\frac{\lambda_{3}\lambda^{2}_{4}+\lambda_{6}(\lambda_{1}\lambda_{6}-\lambda_{4}\lambda_{5})}{\lambda^{2}_{5}-4\lambda_{1}\lambda_{3}}\upsilon_{\rho}^{2},~{}~{}M_{h_{2}}^{2}\approx
c_{1}\upsilon_{\chi}^{2}+c_{2}\upsilon_{\rho}^{2}\approx
c_{1}\upsilon_{\chi}^{2},$ (12) $\displaystyle M_{h_{3}}^{2}$
$\displaystyle\approx$ $\displaystyle
c_{3}\upsilon_{\chi}^{2}+c_{4}\upsilon_{\rho}^{2}\approx
c_{3}\upsilon_{\chi}^{2},~{}~{}M^{2}_{h^{\mp}_{1}}=\frac{\lambda_{7}}{2}(\upsilon^{2}_{\eta}+\upsilon^{2}_{\rho}),~{}~{}M^{2}_{h^{\mp}_{2}}=\frac{\lambda_{8}}{2}(\upsilon^{2}_{\eta}+\upsilon^{2}_{\chi}),$
(13) $\displaystyle M^{2}_{h^{\mp\mp}}$ $\displaystyle=$
$\displaystyle\frac{\lambda_{9}}{2}(\upsilon^{2}_{\rho}+\upsilon^{2}_{\chi}),$
(14)

where

$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{(}\lambda_{1}+\lambda_{3}-\sqrt{(\lambda_{1}-\lambda_{3})^{2}+\lambda_{5}^{2}}\bigg{)}\upsilon_{\chi},$
(15) $\displaystyle c_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\bigg{(}\lambda_{1}+\lambda_{3}+\sqrt{(\lambda_{1}-\lambda_{3})^{2}+\lambda_{5}^{2}}\bigg{)}\upsilon_{\chi}.$
(16)

This model contains 15 electroweak gauge bosons, four of them are identified
as the SM gauge bosons and the remaining ones are heavy where some of them are
electrically charged and the others are neutral: $K^{0}$,$K^{\prime
0}$,$K^{\mp}_{1}$,$X^{\mp}$,$V^{\mp\mp}$ and $Y^{\mp}$ ,$Z^{\prime}$ and
$Z^{\prime\prime}$, their masses are found from the following Lagrangian:

$\mathcal{L}=(D_{\mu}\eta)^{\dagger}(D^{\mu}\eta)+(D_{\mu}\rho)^{\dagger}(D^{\mu}\rho)+(D_{\mu}\chi)^{\dagger}(D^{\mu}\chi),$
(17)

where

$D_{\mu}=\partial_{\mu}+igW_{\mu}^{a}\frac{\lambda_{a}}{2}+iXg_{X}W_{\mu}^{X},\qquad\text{with}\qquad
a=1....15.$ (18)

And by using the following combination:

$\displaystyle W^{\mp}=\frac{(W_{\mu}^{1}\pm
iW_{\mu}^{2})}{\sqrt{2}},~{}~{}K^{0},K^{\prime 0}=\frac{(W_{\mu}^{4}\mp
iW_{\mu}^{5})}{\sqrt{2}},~{}~{}K^{\mp}_{1}=\frac{(W_{\mu}^{6}\mp
iW_{\mu}^{7})}{\sqrt{2}},$ (19) $\displaystyle X^{\mp}=\frac{(W_{\mu}^{9}\mp
iW_{\mu}^{10})}{\sqrt{2}},~{}~{}V^{\mp\mp}=\frac{(W_{\mu}^{11}\mp
iW_{\mu}^{12})}{\sqrt{2}},~{}~{}Y^{\mp}=\frac{(W_{\mu}^{13}\mp
iW_{\mu}^{14})}{\sqrt{2}}.$ (20)

We get:

$\displaystyle M_{W{\mp}}^{2}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{4}\upsilon^{2}_{\rho},~{}~{}M_{K^{\prime
0},K^{0}}^{2}=\frac{g^{2}}{4}\upsilon^{2}_{\eta},~{}~{}M_{K_{1}^{\mp}}^{2}=\frac{g^{2}}{4}\upsilon^{2}_{\eta},~{}~{}M_{X^{\mp}}^{2}=\frac{g^{2}}{4}\upsilon^{2}_{\chi},$
(21) $\displaystyle M_{V^{\mp\mp}}^{2}$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{4}\upsilon^{2}_{\chi},~{}~{}M_{Z}^{2}=\frac{g^{2}\upsilon_{\rho}^{2}}{4c_{W}^{2}},~{}~{}M_{Y^{\mp}}^{2}=\frac{g^{2}}{4}(\upsilon^{2}_{\eta}+\upsilon^{2}_{\chi}),$
(22) $\displaystyle M_{Z^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{g^{2}c_{W}^{2}\upsilon_{\eta}^{2}}{h_{W}},~{}~{}M_{Z^{\prime\prime}}^{2}=\frac{g^{2}\upsilon_{\eta}^{2}\bigg{(}(1-4s_{W}^{2})^{2}+h_{W}^{2}\bigg{)}}{8h_{W}(1-4s_{W}^{2})}.$
(23)

##  3 Constraints

The theoretical constraints on the scalar parameters in the compact 341 model
are [18]:

  * •

to ensure the perturbative unitarity, in order to keep the scalar potential
bounded from below in the large fields limit for all possible directions in
the field space (the vacuum stability) constraints and to provide the vacuum
configuration $\langle\rho\rangle_{0}$, $\langle\chi\rangle_{0}$ and
$\langle\eta\rangle_{0}$ to be a minimum of the scalar potential one needs to
maintain the conditions which are given in the appendix A [18].

  * •

To maintain the perturbativity of the scalar potential, all the scalar quartic
couplings $\lambda_{1...9}$ have to satisfy:

$\displaystyle|\lambda_{i}|\leq 4\pi$ (24)

  * •

The positivity of the scalar bosons masses requires the following conditions:

$\displaystyle\lambda_{1}$ $\displaystyle+$
$\displaystyle\lambda_{3}-\sqrt{(\lambda_{1}-\lambda_{3})^{2}+\lambda_{5}^{2}}>0,~{}~{}\lambda_{1}+\lambda_{3}+\sqrt{(\lambda_{1}-\lambda_{3})^{2}+\lambda_{5}^{2}}>0,$
$\displaystyle\lambda_{2}$ $\displaystyle+$
$\displaystyle\frac{\lambda_{3}\lambda^{2}_{4}+\lambda_{6}(\lambda_{1}\lambda_{6}-\lambda_{4}\lambda_{5})}{\lambda^{2}_{5}-4\lambda_{1}\lambda_{3}}>0,~{}~{}\lambda_{7}>0,~{}~{}\lambda_{8}>0,~{}~{}\lambda_{9}>0.$
(25)

  * •

A new constraint over the parameters of the scalar potential is imposing by
identifying the lightest scalar $h_{1}$ as the Standard Model Higgs boson,
then we get:

$\lambda_{2}+\frac{\lambda_{3}\lambda^{2}_{4}+\lambda_{6}(\lambda_{1}\lambda_{6}-\lambda_{4}\lambda_{5})}{\lambda^{2}_{5}-4\lambda_{1}\lambda_{3}}=\frac{m^{2}_{h_{1}}}{\upsilon^{2}_{\rho}},$
(26)

where we take $\upsilon^{2}_{\rho}$=246 GeV.

  * •

Moreover, the existence of a Landau pole in our model $\Lambda$ at a scale
$\mu$ [5], all the VEVs and the particle masses $m_{i}$ are constrained by:

$m_{i}~{}~{}\text{ and VEVs}\leq\Lambda$ (27)

##  4 Feynman rules

From the scalar potential (11), we can derive all Higgs couplings $h_{j}ss$
where $h_{j}$ (j=1,2,3) and s=$h_{1}^{\mp},h_{2}^{\mp},h^{\mp\mp}$ and based
on the following Yukawa Lagrangian:

$\displaystyle\mathcal{L}_{Y}$ $\displaystyle=$
$\displaystyle\lambda_{11}^{J}\bar{Q}_{1L}\chi
J_{1R}+\lambda_{ij}^{J}\overline{Q}_{iL}\chi^{*}J_{jR}+\lambda_{1a}^{d}\overline{Q}_{1L}\rho
d_{aR}+\lambda_{ia}^{u}\overline{Q}_{iL}\rho^{*}u_{aR}$ (28) $\displaystyle+$
$\displaystyle\lambda_{11}^{U}\overline{Q}_{1L}\eta
U_{1R}+\lambda_{ij}^{D}\overline{Q}_{iL}\eta^{*}D_{jR}+\frac{\lambda_{1a}^{u}}{\Lambda^{2}}\epsilon_{mnop}\bigg{(}\overline{Q}_{1Lm}\rho^{*}_{n}\chi^{*}_{o}\eta^{*}_{p}\bigg{)}u_{aR}$
$\displaystyle+$
$\displaystyle\frac{\lambda_{ia}^{d}}{\Lambda^{2}}\epsilon_{mnop}\bigg{(}\overline{Q}_{iLm}\rho_{m}\chi_{o}\eta_{p}\bigg{)}d_{aR}+h.c,$

the couplings of the Higgs bosons $h_{j}$ with fermions $h_{j}\bar{f}f$ can be
determined. Furthermore, the couplings of Higgs bosons $h_{j}$ and gauge
bosons are contained in the covariant kinetic terms of the Higgs bosons (17).
Whereas, The triple couplings of three gauge bosons arise from the covariant
kinetic Lagrangian of the non-Abelian gauge bosons (Yang Mills Lagrangian):

$\mathcal{L}=-\frac{1}{4}W^{a}_{\mu\nu}W^{a\mu\nu},$ (29)

with

$W^{a\mu\nu}=\partial^{\mu}W^{a\nu}-\partial^{\nu}W^{a\mu}+gf^{abc}W^{b\mu}W^{c\nu},$
(30)

where a,b,c go from 1 to 15 and $f^{abc}$ are the structure constants of the
SU(4) group. The non vanishing Feynman rules are listed in the following
tables 333In those tables, we mention only couplings that contribute in our
discussion.:

Higgs $h_{1}$ interactions.  Interactions Couplings
$\overline{u}_{i}u_{i}h_{1}$ $\frac{M^{u_{i}}}{\upsilon_{\rho}}$ with
$u_{i}\equiv u,c,t$ $\overline{d}_{i}d_{i}h_{1}$
$\frac{M^{d_{i}}}{\upsilon_{\rho}}~{}\text{with}~{}d_{i}\equiv d,s,b$
$\overline{\ell}\ell h_{1}$ $\frac{m_{\ell}}{\upsilon_{\rho}}$
$W^{+}W^{-}h_{1}$ $\frac{g^{2}}{2}\upsilon_{\rho}$ $K^{-}_{1}K^{+}_{1}h_{1}$
$\frac{g^{2}}{2}\upsilon_{\rho}$ $V^{++}V^{--}h_{1}$
$\frac{g^{2}}{2}\upsilon_{\rho}$ $h^{++}h^{--}h_{1}$
$2\lambda_{2}\upsilon_{\rho}$ $h^{+}_{1}h^{-}_{1}h_{1}$
$2\lambda_{2}\upsilon_{\rho}$ $h^{+}_{2}h^{-}_{2}h_{1}$
$\frac{\upsilon_{\rho}}{2}(\lambda_{6}+\lambda_{4})$ $ZZh_{1}$
$\frac{g^{2}\upsilon_{\rho}}{4}\bigg{(}(C^{21})^{2}-\frac{2}{\sqrt{3}}C^{21}C^{22}-\frac{3}{\sqrt{6}}C^{21}C^{23}+\frac{(C^{22})^{2}}{3}+\frac{2}{\sqrt{18}}C^{22}C^{23}$+
$\frac{(C^{23})^{2}}{6}+\frac{4t^{2}}{4}(C^{24})^{2}-$$4tC^{24}C^{21}+4tC^{24}C^{22}+\frac{4t}{\sqrt{6}}C^{24}C^{23}\bigg{)}$

Higgs $h_{2}$ interactions.  Interactions Couplings $\overline{q}qh_{2}$
$\frac{M^{q}}{\upsilon_{\chi}}(\gamma+\frac{\upsilon_{\chi}}{\upsilon_{\eta}}\alpha)$
with $q\equiv u,s,b$ $\overline{J}Jh_{2}$
$\frac{M^{J}}{\upsilon_{\chi}}\gamma$ $\overline{U}Uh_{2}$
$\frac{M^{U}}{\upsilon_{\eta}}\alpha$ $\overline{\ell}\ell h_{2}$
$\frac{m_{\ell}\gamma}{\upsilon_{\chi}}$ $X^{+}X^{-}h_{2}$
$\frac{g^{2}}{2}\upsilon_{\chi}\gamma$ $V^{++}V^{--}h_{2}$
$\frac{g^{2}}{2}\upsilon_{\chi}\gamma$ $K^{\prime 0}K^{\prime 0}h_{2}$
$\frac{g^{2}}{2}\upsilon_{\eta}\alpha$ $K^{-}_{1}K^{+}_{1}h_{2}$
$\frac{g^{2}}{2}\upsilon_{\eta}\alpha$ $Y^{-}Y^{+}h_{2}$
$\frac{g^{2}}{2}(\upsilon_{\chi}\gamma+\upsilon_{\eta}\alpha)$
$h^{++}h^{--}h_{2}$
$\lambda_{4}\upsilon_{\eta}\alpha+(\lambda_{6}+\lambda_{9})\upsilon_{\chi}\gamma$
$h^{+}_{1}h^{-}_{1}h_{2}$
$(\lambda_{4}+\lambda_{7})\upsilon_{\eta}\alpha+\lambda_{6}\upsilon_{\chi}\gamma$
$h^{+}_{2}h^{-}_{2}h_{2}$
$\frac{\upsilon_{\chi}\gamma}{2}(2\lambda_{3}+\lambda_{5}+\lambda_{8})+\frac{\upsilon_{\chi}\alpha\lambda_{8}}{2}+\frac{\upsilon_{\eta}\alpha}{2}(\lambda_{5}+\lambda_{8}+2\lambda_{1})+\frac{\gamma\upsilon_{\eta}\lambda_{8}}{2}$
$ZZh_{2}$
$\frac{g^{2}\gamma}{4}\upsilon_{\chi}\bigg{(}\frac{3}{2}(C^{23})^{2}+4t^{2}(C^{24})^{2}+\frac{12t}{\sqrt{6}}C^{23}C^{24}\bigg{)}+\frac{g^{2}\alpha}{4}\upsilon_{\eta}\bigg{(}\frac{4}{3}(C^{22})^{2}+\frac{(C^{23})^{2}}{6}-\frac{4}{\sqrt{18}}C^{22}C^{23}\bigg{)}$
$Z^{\prime}Z^{\prime}h_{2}$
$\frac{g^{2}\gamma}{4}\upsilon_{\chi}\bigg{(}\frac{3}{2}(C^{33})^{2}+4t^{2}(C^{34})^{2}+\frac{12t}{\sqrt{6}}C^{33}C^{34}\bigg{)}+\frac{g^{2}\alpha}{4}\upsilon_{\eta}\bigg{(}\frac{4}{3}(C^{32})^{2}+\frac{(C^{33})^{2}}{6}-\frac{4}{\sqrt{18}}C^{32}C^{33}\bigg{)}$
$h_{1}h_{1}h_{2}$
$\frac{\lambda_{6}}{2}\upsilon_{\chi}\gamma+\frac{\lambda_{4}}{2}\upsilon_{\eta}\alpha$

Higgs $h_{3}$ interactions.  Interactions Couplings $\overline{q}qh_{3}$
$\frac{M^{q}}{\upsilon_{\chi}}(\sigma+\frac{\upsilon_{\chi}}{\upsilon_{\eta}}\beta)$
with $q\equiv u,s,b$ $\overline{J}Jh_{3}$
$\frac{M^{J}}{\upsilon_{\chi}}\sigma$ $\overline{U}Uh_{3}$
$\frac{M^{U}}{\upsilon_{\eta}}\beta$ $\overline{\ell}\ell h_{3}$
$\frac{m_{\ell}}{\upsilon_{\chi}}\sigma$ $X^{+}X^{-}h_{3}$
$\frac{g^{2}}{2}\upsilon_{\chi}\sigma$ $V^{++}V^{--}h_{3}$
$\frac{g^{2}}{2}\upsilon_{\chi}\sigma$ $K^{\prime 0}K^{0\prime}h_{3}$
$\frac{g^{2}}{2}\upsilon_{\eta}\beta$ $K^{-}K^{+}h_{3}$
$\frac{g^{2}}{2}\upsilon_{\eta}\beta$ $Y^{-}Y^{+}h_{3}$
$\frac{g^{2}}{2}(\upsilon_{\chi}\sigma+\upsilon_{\eta}\beta)$
$h^{++}h^{--}h_{3}$
$\lambda_{4}\upsilon_{\eta}\beta+\upsilon_{\chi}\sigma(\lambda_{6}+\lambda_{9})$
$h^{+}_{1}h^{-}_{1}h_{3}$
$\upsilon_{\eta}\beta(\lambda_{4}+\lambda_{7})+\lambda_{6}\upsilon_{\chi}\sigma$
$h^{+}_{2}h^{-}_{2}h_{3}$
$\frac{\upsilon_{\chi}\sigma}{2}(2\lambda_{3}+\lambda_{8}+\lambda_{5})+\frac{\upsilon_{\chi}\lambda_{8}}{2}\beta+\frac{\upsilon_{\eta}\beta}{2}(\lambda_{5}+\lambda_{8}+2\lambda_{1})+\frac{\upsilon_{\eta}\sigma\lambda_{8}}{2}$
$ZZh_{3}$
$\frac{g^{2}\upsilon_{\chi}\sigma}{4}\bigg{(}\frac{3}{2}(C^{23})^{2}+4t^{2}(C^{24})^{2}+\frac{12t}{\sqrt{6}}C^{23}C^{24}\bigg{)}+\frac{g^{2}\upsilon_{\eta}\beta}{4}\bigg{(}\frac{4}{3}(C^{22})^{2}+\frac{(C^{23})^{2}}{6}-\frac{4}{\sqrt{18}}C^{22}C^{23}\bigg{)}$
$Z^{\prime}Z^{\prime}h_{3}$
$\frac{g^{2}\upsilon_{\chi}\sigma}{4}\bigg{(}\frac{3}{2}(C^{33})^{2}+4t^{2}(C^{34})^{2}+\frac{12t}{\sqrt{6}}C^{33}C^{34}\bigg{)}+\frac{g^{2}\upsilon_{\eta}\beta}{4}\bigg{(}\frac{4}{3}(C^{32})^{2}+\frac{(C^{33})^{2}}{6}-\frac{4}{\sqrt{18}}C^{32}C^{33}\bigg{)}$
$h_{1}h_{1}h_{3}$
$\upsilon_{\chi}\frac{\lambda_{6}}{2}\sigma+\frac{\lambda_{4}}{2}\upsilon_{\eta}\beta$
$h_{2}h_{2}h_{3}$
$\frac{\lambda_{5}\upsilon_{\chi}}{2}(\alpha^{2}\sigma+2\alpha\beta\gamma)+\frac{\lambda_{5}\upsilon_{\eta}}{2}(\beta\gamma^{2}+2\alpha\gamma\sigma)$
$h_{2}h_{1}h_{3}$
$\lambda_{4}\upsilon_{\rho}\alpha\beta+\lambda_{6}\upsilon_{\rho}\gamma\sigma$

Couplings $\frac{g\gamma^{\mu}}{4}(g_{V}-\gamma_{5}g_{A})$ of the
Z$q\overline{q}$ vertex.  Interactions $g_{V}$ $g_{A}$ $\overline{u}uZ$
$\frac{-5}{3}S_{W}T_{W}+C_{W}$ $\frac{1}{C_{W}}$ $\overline{d}dZ$
$\frac{1}{3}S_{W}T_{W}-C_{W}$ $\frac{-1}{C_{W}}$ $\overline{s}sZ$
$C_{W}-\frac{5}{3}S_{W}T_{W}$ $C_{W}-3S_{W}T_{W}$ $\overline{b}bZ$
$C_{W}-\frac{5}{3}S_{W}T_{W}$ $C_{W}-3S_{W}T_{W}$ $\overline{c}cZ$
$-C_{W}+\frac{1}{3}S_{W}T_{W}$ $-\frac{1}{C_{W}}$ $\overline{t}tZ$
$-C_{W}+\frac{1}{3}S_{W}T_{W}$ $-\frac{1}{C_{W}}$ $\overline{U_{1}}U_{1}Z$
$\frac{-8}{3}S_{W}T_{W}$ 0 $\overline{J_{1}}J_{1}Z$ $\frac{-20}{3}S_{W}T_{W}$
0 $\overline{D_{2}}D_{2}Z$ $\frac{4}{3}S_{W}T_{W}$ 0 $\overline{D_{3}}D_{3}Z$
$\frac{4}{3}S_{W}T_{W}$ 0 $\overline{J_{2}}J_{2}Z$ $\frac{16}{3}S_{W}T_{W}$ 0
$\overline{J_{3}}J_{3}Z$ $\frac{16}{3}S_{W}T_{W}$ 0

Couplings ZVV, where V is the gauge bosons $X^{\mp}$, $Y^{\mp}$, $V^{\mp\mp}$,
$K^{\mp}_{1}$, $K^{\prime 0}$ and $W^{\mp}$, while,
$\sum_{\alpha\beta\mu}(p,k,q)=g_{\alpha\beta}(p-k)_{\mu}+g_{\beta\mu}(k-q)_{\alpha}+g_{\mu\alpha}(q-p)_{\beta}$.
Interactions Couplings $W^{+}_{\beta}W^{-}_{\alpha}Z_{\mu}$
$igC^{21}\sum_{\alpha\beta\mu}(p,k,q)$ $K^{\prime 0}_{\beta}K^{\prime
0}_{\alpha}Z_{\mu}$
$\frac{-ig}{2}(C^{21}+\sqrt{3}C^{22})\sum_{\alpha\beta\mu}(p,k,q)$
$X^{+}_{\beta}X^{-}_{\alpha}Z_{\mu}$
$\frac{-ig}{2\sqrt{3}}(\sqrt{3}C^{21}+C^{22}+C^{23})\sum_{\alpha\beta\mu}(p,k,q)$
$K^{+}_{1\beta}K^{-}_{1\alpha}Z_{\mu}$
$\frac{-ig}{2}(\sqrt{3}C^{22}-C^{21})\sum_{\alpha\beta\mu}(p,k,q)$
$V^{++}_{\beta}V^{--}_{\alpha}Z_{\mu}$
$\frac{-ig}{2\sqrt{3}}(-\sqrt{3}C^{21}+C^{22}+C^{23})\sum_{\alpha\beta\mu}(p,k,q)$
$Y^{+}_{\beta}Y^{-}_{\alpha}Z_{\mu}$
$\frac{-ig}{2\sqrt{3}}(-C^{22}+C^{23})\sum_{\alpha\beta\mu}(p,k,q)$

Where

$\displaystyle C^{21}$ $\displaystyle=$ $\displaystyle C_{W},\qquad
C^{22}=\frac{S_{W}T_{W}}{\sqrt{3}},\qquad
C^{23}=\frac{4S_{W}T_{W}}{\sqrt{6}},\qquad C^{24}=\frac{-S_{W}T_{W}}{t},$ (31)
$\displaystyle C^{32}$ $\displaystyle=$
$\displaystyle\sqrt{\dfrac{3-T^{2}_{W}}{3}},\qquad
C^{33}=\frac{-4}{\sqrt{6}}\frac{T^{2}_{W}}{\sqrt{3-T^{2}_{W}}},\qquad
C^{34}=\frac{T^{2}_{W}}{t\sqrt{3-T^{2}_{W}}},$ (32) $\displaystyle t$
$\displaystyle=$ $\displaystyle\frac{S_{W}}{\sqrt{1-4S^{2}_{W}}}.$ (33)

We mention that $S_{W}\equiv\sin\theta_{W}$, $C_{W}\equiv\cos\theta_{W}$,
$T_{W}\equiv\tan\theta_{W}$ and $h_{W}=3-4S_{W}^{2}$ where $\theta_{W}$ is the
Weinberg angle.

##  5 Higgs Decays

The main goal of our work is to study the signal strength of $h_{1}$ and the
branching ratio of the other heavy scalar bosons $h_{2}$ and $h_{3}$ for
different channels in the compact 341 model

###  5.1 $h_{1}$ decay

The first purpose of this paper is to study the deviation of our model from
the Standard Model by calculating the signal strength $\mu$ of the lightest
scalar boson $h_{1}$ and confront our results to the experimental data for
each individual channels $b\bar{b}$, $WW^{*}$, $ZZ^{*}$, $\tau\bar{\tau}$ and
$\gamma\gamma$.
The signal strength of any process with a giving initial state i producing an
Higgs h which decays to the final state f can be written as a product of the
Higgs boson production cross section and its branching ratio in units of the
corresponding value predicted by the SM [7]:

$\mu_{xy}=\frac{\sigma_{341}(pp\longrightarrow
h_{1})BR_{341}(h_{1}\longrightarrow xy)}{\sigma_{SM}(pp\longrightarrow
h)BR_{SM}(h\longrightarrow xy)},$ (34)

where the superscript SM and 341 refer to the Standard Model and the compact
341 model respectively, while, x and y are any finale state, $\sigma_{i}$ and
$\text{BR}_{i}$ (i=341, SM) are the corresponding cross section production
taking gluon-gluon fusion (ggF) is the dominant contribution of the Higgs
production and the branching ratio respectively. In the compact 341 model, the
coupling $tth_{1}$ is the same as tth of the Standard Model (SM), hence, the
cross section of the light Higgs $h_{1}$ production process is the same as the
SM at the LHC:

$\sigma_{341}(pp\longrightarrow h_{1})=\sigma_{SM}(pp\longrightarrow h),$ (35)

thus, the signal strength $\mu_{xy}$ becomes the ratio between the branching
ratio of the SM and of the compact 341 model:

$\mu_{xy}=\frac{BR_{341}}{BR_{SM}}=\frac{\Gamma_{SM}(h\longrightarrow
all)\Gamma_{341}(h_{i}\longrightarrow xy)}{\Gamma_{341}(h_{i}\longrightarrow
all)\Gamma_{SM}(h\longrightarrow xy)},$ (36)

where the total decay width $\Gamma_{341}(h_{1}\longrightarrow all)$ turns out
to be the same as the one of the SM Higgs boson $\Gamma_{SM}(h\longrightarrow
all)$:

$\displaystyle\Gamma_{341}(h_{1}\longrightarrow all)$ $\displaystyle=$
$\displaystyle\Gamma_{341}(h_{1}\longrightarrow
b\overline{b})+\Gamma_{341}(h_{1}\longrightarrow\tau^{+}\tau^{-})+\Gamma_{341}(h_{1}\longrightarrow
WW^{*})$ (37) $\displaystyle+$ $\displaystyle\Gamma_{341}(h_{1}\longrightarrow
ZZ^{*})+\Gamma_{341}(h_{1}\longrightarrow\gamma\gamma)+\Gamma_{341}(h_{1}\longrightarrow\gamma
Z)$ $\displaystyle+$ $\displaystyle\Gamma_{341}(h_{1}\longrightarrow gg).$

All the tree level couplings between the SM-like Higgs boson $h_{1}$ and all
the fermions and the W boson, are the same as the SM, that makes the partial
decay widths of $h_{1}$ into SM particles such as $b\bar{b}$,
$\tau^{+}\tau^{-}$, $WW^{*}$ and gg have the same expressions as the SM [20],
whereas, the partial decay width of $h_{1}$ into $ZZ^{*}$ is given by:

$\displaystyle\Gamma_{341}(h_{1}\longrightarrow Z^{*}Z)$ $\displaystyle=$
$\displaystyle\frac{g^{4}m_{h_{1}}}{2048\pi^{3}}G^{2}_{h_{1}ZZ}F\bigg{(}\frac{m_{Z}}{m_{h_{1}}}\bigg{)}\bigg{(}\sum_{j\equiv
u,d,c,s,b}(g_{jV}^{2}+g_{jA}^{2})$ (38) $\displaystyle+$
$\displaystyle\sum_{\ell\equiv\text{leptons}}(g_{\ell V}^{2}+g_{\ell
A}^{2})\bigg{)},$

with $G_{h_{1}ZZ}=4g_{h_{1}ZZ}/(g^{2}\upsilon_{\rho})$ where $g_{ZZh_{1}}$
represents the coupling between a pair of Z bosons and $h_{1}$ (see table 4)
and

$\displaystyle F(x)$ $\displaystyle=$
$\displaystyle-|1-x^{2}|(\frac{47}{2}x^{2}-\frac{13}{2}+\frac{1}{x^{2}})-\frac{3}{2}(1-6x^{2}+4x^{4})\ln(x)$
(39) $\displaystyle+$
$\displaystyle\frac{3(1-8x^{2}+20x^{4})}{\sqrt{4x^{2}-1}}\arccos(\frac{3x^{2}-1}{2x^{3}}),$

with the parameter $x=\frac{m_{Z}}{m_{h_{1}}}$.
The compact 341 model contains new charged particles including fermions,
scalars (the new singly and doubly charged scalar bosons) and gauge bosons
that will contribute to the decay amplitudes of the decays
$h_{1}\longrightarrow\gamma\gamma$ and $h_{1}\longrightarrow\gamma Z$ at one
loop level. In the case of $h_{1}$, besides the contribution of $W^{\mp}$
boson and the top quark, the processes
$\Gamma(h_{1}\longrightarrow\gamma\gamma(\gamma Z))$ receive new contributions
that come from new particles (gauge and scalar bosons) such as $V^{\mp\mp}$,
$K_{1}^{\mp}$, $h_{1}^{\mp}$, $h_{2}^{\mp}$ and $h^{\mp\mp}$, Furthermore, in
our model, the SM-like Higgs boson $h_{1}$ does not couple to the new heavy
charged fermions, hence, the exotic quarks do not contribute to the one-loop
decay amplitudes of the processes
$h_{2,3}\longrightarrow(\gamma\gamma,Z\gamma$, gg). Thus, the partial decay
widths of $h_{1}$ into $\gamma\gamma$ and Z$\gamma$ are given by [19, 20]:

$\displaystyle\Gamma(h_{1}\longrightarrow\gamma\gamma)$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}m_{h_{1}}^{3}}{1024\pi^{3}}\bigg{|}\sum_{V}\frac{g_{h_{1}VV}}{m^{2}_{V}}Q_{V}^{2}\mathcal{A}_{1}(\tau_{V})+\sum_{f}\frac{2g_{h_{1}ff}}{m_{f}}Q_{f}^{2}N_{c,f}\mathcal{A}_{\frac{1}{2}}(\tau_{f})$
(40) $\displaystyle+$
$\displaystyle\sum_{f}\frac{g_{h_{1}SS}}{m_{S}^{2}}Q_{S}^{2}N_{c,S}\mathcal{A}_{0}(\tau_{S})\bigg{|}^{2},$
$\displaystyle\Gamma(h_{1}\longrightarrow\gamma Z)$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}m_{h_{1}}^{3}}{512\pi^{3}}\bigg{(}1-\frac{M^{2}_{Z}}{M^{2}_{h_{1}}}\bigg{)}^{3}\bigg{|}\frac{2}{\upsilon}\frac{\mathcal{A}_{SM}}{\sin\theta_{W}}+\mathcal{A}\bigg{|}^{2},$
(41)

where the symbol V, f and S refer to Spin 1, Spin $\frac{1}{2}$ and Spin 0
particles respectively, $Q_{V},Q_{f}$, $Q_{S}$ are electric charges of the
vectors, fermions and scalars, $g_{h_{1}SS}$,$g_{h_{1}VV}$ and $g_{h_{1}ff}$
represent the couplings of the Higgs with scalars S, gauge bosons V and
fermions f, $N_{c,f},N_{c,S}$ are the number of fermion and scalar colors and
the loop functions for V,f and S particles $\mathcal{A}_{1}(\tau_{V})$,
$\mathcal{A}_{\frac{1}{2}}(\tau_{f})$ and $\mathcal{A}_{0}(\tau_{S})$ are
given by [19, 20]:

$\displaystyle A_{1}(x)$ $\displaystyle=$
$\displaystyle-x^{2}\bigg{(}2x^{-2}+3x^{-1}+3(2x^{-1}-1)f(x^{-1})\bigg{)},$
$\displaystyle A_{\frac{1}{2}}(x)$ $\displaystyle=$ $\displaystyle
2x^{2}\bigg{(}x^{-1}+(x^{-1}-1)f(x^{-1})\bigg{)},$ $\displaystyle A_{0}(x)$
$\displaystyle=$ $\displaystyle-x^{2}\bigg{(}x^{-1}-f(x^{-1})\bigg{)}.$ (42)

The parameter $\tau_{i}=\frac{4m_{i}^{2}}{m^{2}_{h_{i}}}$ where i represents
the corresponding particles in the loop and [20]:

$f(x)=\left\\{\begin{array}[]{ll}\vskip 12.0pt plus 4.0pt minus
4.0pt\arcsin^{2}\sqrt{x}\qquad\text{for}\qquad x\geq 1,\\\
\frac{-1}{4}\bigg{(}\ln(\frac{1+\sqrt{1-x^{-1}}}{1-\sqrt{1-x^{-1}}})-\imath\pi\bigg{)}^{2}\qquad\text{for}\qquad
x<1.\end{array}\right.$ (43)

The factors $\mathcal{A}$ and $\mathcal{A}_{SM}$ represent the contributions
of the new particles predicted by our model and the contributions coming from
the particles of the Standard Model respectively, their expressions are given
by:

$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\frac{g_{h_{1}VV}}{m^{2}_{V}}g_{ZVV}\mathcal{A}_{1}(\tau_{V},\lambda_{V})+\widetilde{N}_{c,f}\frac{2g_{h_{1}ff}}{m_{f}}2Q_{f}(g_{Z\ell\ell}+g_{Zrr})\mathcal{A}_{\frac{1}{2}}(\tau_{f},\lambda_{f})$
(44) $\displaystyle-$
$\displaystyle\widetilde{N}_{c,s}\frac{2g_{h_{1}SS}}{m^{2}_{S}}Q_{S}g_{ZSS}\mathcal{A}_{0}(\tau_{S},\lambda_{S}),$
$\displaystyle\mathcal{A}_{SM}$ $\displaystyle=$
$\displaystyle\cos\theta_{W}\mathcal{A}_{1}(\tau_{W},\lambda_{W})+N_{C}\frac{Q_{t}(2T_{3}^{t}-4Q_{t}\sin\theta_{W}^{2})}{\cos\theta_{W}}\mathcal{A}_{\frac{1}{2}}(\tau_{t},\lambda_{t}),$
(45)

Where $\lambda_{i}=\frac{4m_{i}^{2}}{m^{2}_{Z^{2}}}$, $T_{3}=\frac{1}{2}$ is
the weak isospin of the top quark and $\mathcal{A}_{i}(x,y)$ are the loop
functions [19, 20]:

$\displaystyle A_{1}(x,y)$ $\displaystyle=$ $\displaystyle
4(3-\tan^{2}\theta_{W})I_{2}(x,y)+\bigg{(}(1+2x^{-1})\tan^{2}\theta_{W}-(5+2x^{-1})\bigg{)}I_{1}(x,y),$
$\displaystyle A_{\frac{1}{2}}(x,y)$ $\displaystyle=$ $\displaystyle
I_{1}(x,y)-I_{2}(x,y),$ $\displaystyle A_{0}(x,y)$ $\displaystyle=$
$\displaystyle I_{1}(x,y),$ (46)

where:

$\displaystyle I_{1}(x,y)$ $\displaystyle=$
$\displaystyle\frac{xy}{2(x-y)}+\frac{x^{2}y^{2}}{2(x-y)^{2}}\bigg{(}f(x^{-1})-f(y^{-1})\bigg{)}+\frac{x^{2}y}{(x-y)^{2}}\bigg{(}g(x^{-1})-g(y^{-1})\bigg{)},$
$\displaystyle I_{2}(x,y)$ $\displaystyle=$
$\displaystyle\frac{-xy}{2(x-y)}\bigg{(}f(x^{-1})-f(y^{-1})\bigg{)},$ (47)

with

$g(x)=\left\\{\begin{array}[]{ll}\vskip 12.0pt plus 4.0pt minus
4.0pt\sqrt{x^{-1}-1}\arcsin\sqrt{x}\qquad\text{for}\qquad x\geq 1\\\
\frac{\sqrt{1-x^{-1}}}{2}\bigg{(}\ln(\frac{1+\sqrt{1-x^{-1}}}{1-\sqrt{1-x^{-1}}})-\imath\pi\bigg{)}\qquad\text{for}\qquad
x<1\end{array}\right.$ (48)

where the couplings
$g_{h_{1}VV}$,$g_{ZVV}$,$g_{h_{1}ff}$,$g_{Zff}$,$g_{h_{1}SS}$ and $g_{ZSS}$
are given in tables 4,4,4.

###  5.2 The decay of the neutral heavy scalar bosons $h_{2}$ and $h_{3}$

For the neutral heavy scalar bosons $h_{2}$ and $h_{3}$, we calculate their
branching ratios (BRs) where:

$\text{BR}=\frac{\Gamma_{i}}{\Gamma(h\longrightarrow all)},$ (49)

with $\Gamma(h\longrightarrow all)=\sum\Gamma_{i}$ is the total decay rate
that represents the sum of the individual decay rates. The total decay width
of $h_{2}$ is determined by the following channels:

$\displaystyle\Gamma_{341}(h_{2}\longrightarrow all)$ $\displaystyle=$
$\displaystyle\Gamma_{341}(h_{2}\longrightarrow\tau^{+}\tau^{-},b\bar{b})+\Gamma_{341}(h_{2}\longrightarrow
ZZ)$ (50) $\displaystyle+$
$\displaystyle\Gamma_{341}(h_{2}\longrightarrow\gamma
Z)+\Gamma_{341}(h_{2}\longrightarrow\gamma\gamma)+\Gamma_{341}(h_{2}\longrightarrow
gg)$ $\displaystyle+$ $\displaystyle\Gamma_{341}(h_{2}\longrightarrow
h_{1}h_{1}).$

Furthermore, $h_{2}$ can also decay to new charged and neutral particles F
including fermions, scalar and gauge bosons, if kinematically allowed which
would require $m_{h_{2}}>2m_{F}$, therefore, in this case,
$\Gamma_{341}(h_{2}\longrightarrow all)$ has new contributions
$\Gamma_{341}(h_{2}\longrightarrow F)$ where F can be $K_{1}^{\mp}$,
$K^{\prime 0}$, $X^{\mp}$, $Y^{\mp}$, $V^{\mp\mp}$, $Z^{\prime}$ or exotic
quarks.
In our model, the heavy scalar bosons $h_{2,3}$ does not couple to the SM top
quark, therefore, the fermion contributions come only from exotic quarks in
the loop processes $\Gamma(h_{2}\longrightarrow gg,\gamma\gamma,\gamma Z)$,
this result in:

$\Gamma(h_{2}\longrightarrow
gg)=\frac{\alpha^{2}_{S}M^{3}_{h_{2}}}{128\pi^{3}\upsilon_{\eta}^{2}}\alpha^{2}\bigg{|}\sum_{i=1}^{3}A_{\frac{1}{2}}(\tau_{U})\bigg{|}^{2}+\frac{\alpha^{2}_{S}M^{3}_{h_{2}}}{128\pi^{3}\upsilon_{\chi}^{2}}\gamma^{2}\bigg{|}\sum_{i=1}^{3}A_{\frac{1}{2}}(\tau_{J})\bigg{|}^{2},$
(51)

where $\alpha_{S}$ is the strong coupling and $A_{\frac{1}{2}}(\tau_{f})$
represents the loop function of fermions. The one loop expressions for
$h_{2,3}$ decays into final states including massless bosons $\gamma\gamma$
and $Z\gamma$ can be mediated by new contributions that come from the new
charged particles namely $K_{1}^{\mp}$, $K^{\prime 0}$, $X^{\mp}$, $Y^{\mp}$,
$V^{\mp\mp}$, U, J, $h_{1}^{\mp}$, $h_{2}^{\mp}$ and $h^{\mp\mp}$, we note
that the contribution of the $W^{\mp}$ boson is not included in these
amplitudes, since there is no direct coupling between $h_{2,3}$ and the
$W^{\mp}$ boson.
Moreover, it is worth pointing out that the fermions and bosonic decays are
similar to the $h_{1}$ case just one needs to replace the couplings of table 4
by those of table 4 for $h_{2}$ and by table 4 for $h_{3}$.
Furthermore, the partial decay widths of $h_{2}$ into a pair of Higgs like-
boson $h_{1}$, into a pair of exotic quarks (U and J), into a pair of gauge
bosons $V_{i}$ ($V_{i}\equiv X^{\mp},V^{\mp\mp},K^{\prime
0},K_{1}^{\mp},Y^{\mp}$) and into a pair of Z and $Z^{\prime}$ are given
respectively by:

$\displaystyle\Gamma_{341}(h_{2}\longrightarrow h_{1}h_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi
m_{h_{2}}}(g_{h_{2}h_{1}h_{1}})^{2}\bigg{(}1-\frac{4m_{h_{1}}^{2}}{m^{2}_{h_{2}}}\bigg{)}^{\frac{1}{2}},$
(52) $\displaystyle\Gamma_{341}(h_{2}\longrightarrow UU)$ $\displaystyle=$
$\displaystyle\frac{3g^{2}}{32\pi}\frac{m_{U}^{2}m_{h_{2}}}{m_{W}^{2}}\bigg{(}1-\frac{4m^{2}_{U}}{m^{2}_{h_{3}}}\bigg{)}^{\frac{3}{2}}\bigg{(}\frac{\upsilon_{\rho}}{\upsilon_{\eta}}\alpha\bigg{)}^{2},$
(53) $\displaystyle\Gamma_{341}(h_{2}\longrightarrow JJ)$ $\displaystyle=$
$\displaystyle\frac{3g^{2}}{32\pi}\frac{m_{J}^{2}m_{h_{2}}}{m_{W}^{2}}\bigg{(}1-\frac{4m^{2}_{J}}{m^{2}_{h_{2}}}\bigg{)}^{\frac{3}{2}}\bigg{(}\frac{\upsilon_{\rho}}{\upsilon_{\chi}}\gamma\bigg{)}^{2},$
(54) $\displaystyle\Gamma_{341}(h_{2}\longrightarrow V_{i}V_{i})$
$\displaystyle=$
$\displaystyle\frac{k_{i}}{4\pi}\frac{M_{W}^{4}}{M_{h_{2}}\upsilon_{\rho}^{2}}\bigg{(}1-\frac{4M_{V_{i}}^{2}}{M_{h_{2}}^{2}}\bigg{)}^{\frac{1}{2}}\bigg{(}3+\frac{1}{4}\frac{M_{h_{2}}^{4}}{M^{4}_{V_{i}}}-\frac{M_{h_{2}}^{2}}{M^{2}_{V_{i}}}\bigg{)},$
(55) $\displaystyle\Gamma(h_{2}\longrightarrow ZZ)$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi}\frac{M_{Z}^{4}}{M_{h_{2}}\upsilon_{\rho}^{2}}\bigg{(}1-\frac{4M^{2}_{Z}}{M^{2}_{h_{2}}}\bigg{)}^{\frac{1}{2}}\bigg{(}3+\frac{M^{4}_{h_{2}}}{M_{Z}^{4}}-\frac{M^{2}_{h_{2}}}{M_{Z}^{2}}\bigg{)}G^{2}_{ZZh_{2}},$
(56) $\displaystyle\Gamma(h_{2}\longrightarrow Z^{\prime}Z^{\prime})$
$\displaystyle=$
$\displaystyle\frac{1}{8\pi}\frac{M_{Z}^{4}}{M_{h_{2}}\upsilon_{\rho}^{2}}\bigg{(}1-\frac{4M^{2}_{Z^{\prime}}}{M^{2}_{h_{2}}}\bigg{)}^{\frac{1}{2}}\bigg{(}3+\frac{M^{4}_{h_{2}}}{M_{Z^{\prime}}^{4}}-\frac{M^{2}_{h_{2}}}{M_{Z^{\prime}}^{2}}\bigg{)}G^{2}_{Z^{\prime}Z^{\prime}h_{2}}.$
(57)

where the expression of the trilinear coupling $h_{2}h_{1}h_{1}$ is given in
table 4, $m_{U}$, $m_{J}$ are the exotic quarks masses, $\upsilon_{\eta}$ is
the vacuum expectation value and the $k_{i}$ coefficients are given in the
following table:

$k_{i}$ coefficients.  Higgs $k_{X^{\mp}}$ $k_{V^{\mp\mp}}$ $k_{K^{\prime 0}}$
$k_{K_{1}^{\mp}}$ $k_{Y^{\mp}}$ $h_{2}$
$\frac{\upsilon_{\chi}}{\upsilon_{\rho}}\gamma$
$\frac{\upsilon_{\chi}}{\upsilon_{\rho}}\gamma$
$\frac{\upsilon_{\eta}}{\upsilon_{\rho}}\alpha$
$\frac{\upsilon_{\eta}}{\upsilon_{\rho}}\alpha$
$\frac{1}{\upsilon_{\rho}}(\upsilon_{\chi}\gamma+\upsilon_{\eta}\alpha)$

while the expressions of $G_{ZZh_{2}}$ and $G_{Z^{\prime}Z^{\prime}h_{2}}$ are
given by:

$\displaystyle G_{ZZh_{2}}$ $\displaystyle=$
$\displaystyle\frac{C_{W}^{2}}{2}\bigg{(}\frac{3}{2}\frac{\upsilon_{\chi}}{\upsilon_{\rho}}(C^{23})^{2}\gamma+4(C^{24})^{2}\frac{S_{W}^{2}}{1-4S_{W}^{2}}\frac{\upsilon_{\chi}}{\upsilon_{\rho}}\gamma+\frac{12}{\sqrt{6}}\frac{S_{W}}{\sqrt{1-4S_{W}^{2}}}\frac{\upsilon_{\chi}}{\upsilon_{\rho}}C^{23}C^{24}\gamma$
(58) $\displaystyle+$
$\displaystyle\frac{4}{3}(C^{22})^{2}\frac{\upsilon_{\eta}}{\upsilon_{\rho}}\alpha+\frac{(C^{23})^{2}}{6}\frac{\upsilon_{\eta}}{\upsilon_{\rho}}\alpha-\frac{4}{\sqrt{18}}C^{22}C^{23}\frac{\upsilon_{\eta}}{\upsilon_{\rho}}\alpha\bigg{)},$
$\displaystyle G_{Z^{\prime}Z^{\prime}h_{2}}$ $\displaystyle=$
$\displaystyle\frac{C_{W}^{2}}{2}\bigg{(}\frac{3}{2}\frac{\upsilon_{\chi}}{\upsilon_{\rho}}(C^{33})^{2}\gamma+4(C^{34})^{2}\frac{S_{W}^{2}}{1-4S_{W}^{2}}\frac{\upsilon_{\chi}}{\upsilon_{\rho}}\gamma+\frac{12}{\sqrt{6}}\frac{S_{W}}{\sqrt{1-4S_{W}^{2}}}\frac{\upsilon_{\chi}}{\upsilon_{\rho}}C^{33}C^{34}\gamma$
(59) $\displaystyle+$
$\displaystyle\frac{4}{3}(C^{32})^{2}\frac{\upsilon_{\eta}}{\upsilon_{\rho}}\alpha+\frac{(C^{33})^{2}}{6}\frac{\upsilon_{\eta}}{\upsilon_{\rho}}\alpha-\frac{4}{\sqrt{18}}C^{32}C^{33}\frac{\upsilon_{\eta}}{\upsilon_{\rho}}\alpha\bigg{)}.$

Regarding, the third neutral scalar boson $h_{3}$, its total decay width is
composed by the following decays:

$\displaystyle\Gamma_{341}(h_{3}\longrightarrow all)$ $\displaystyle=$
$\displaystyle\Gamma_{341}(h_{3}\longrightarrow\tau^{+}\tau^{-},b\bar{b})+\Gamma_{341}(h_{3}\longrightarrow\text{exotic~{}quarks})$
(60) $\displaystyle+$
$\displaystyle\Gamma_{341}(h_{3}\longrightarrow\gamma\gamma)+\Gamma_{341}(h_{3}\longrightarrow
Z\gamma)+\Gamma_{341}(h_{3}\longrightarrow gg)$ $\displaystyle+$
$\displaystyle\Gamma_{341}(h_{3}\longrightarrow
h_{2}h_{2})+\Gamma_{341}(h_{3}\longrightarrow h_{1}h_{1})$ $\displaystyle+$
$\displaystyle\Gamma_{341}(h_{3}\longrightarrow
h_{2}h_{1})+\Gamma_{341}(h_{3}\longrightarrow VV),$

where V represents the gauge bosons $X^{\pm}$, $V^{\pm\pm}$, $K^{\prime 0}$,
$K_{1}^{\pm}$, $Y^{\mp}$, Z and $Z^{\prime}$. It is worth to point out that
$h_{3}$ may decay into a pair of $h_{1}h_{1}$, $h_{2}h_{2}$ and $h_{1}h_{2}$
where their decay widths are given by:

$\displaystyle\Gamma_{341}(h_{3}\longrightarrow h_{1}h_{1})$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi
m_{h_{3}}}(g_{h_{3}h_{1}h_{1}})^{2}\bigg{(}1-\frac{4m_{h_{1}}^{2}}{m^{2}_{h_{3}}}\bigg{)}^{\frac{1}{2}},$
(61) $\displaystyle\Gamma_{341}(h_{3}\longrightarrow h_{2}h_{2})$
$\displaystyle=$ $\displaystyle\frac{1}{16\pi
m_{h_{3}}}(g_{h_{3}h_{2}h_{2}})^{2}\bigg{(}1-\frac{4m_{h_{2}}^{2}}{m^{2}_{h_{3}}}\bigg{)}^{\frac{1}{2}},$
(62) $\displaystyle\Gamma_{341}(h_{3}\longrightarrow h_{2}h_{1})$
$\displaystyle=$ $\displaystyle\frac{1}{16\pi
m_{h_{3}}^{2}}(g_{h_{3}h_{2}h_{1}})^{2}\bigg{(}\frac{m_{h_{2}}^{4}}{m_{h_{3}}^{2}}-\frac{2m_{h_{1}}^{2}m_{h_{2}}^{2}}{m^{2}_{h_{3}}}-2m_{h_{2}}^{2}+\frac{m_{h_{1}}^{4}}{m_{h_{3}}^{2}}-2m_{h_{1}}^{2}$
(63) $\displaystyle+$ $\displaystyle m_{h_{3}}^{2}\bigg{)}^{\frac{1}{2}},$

where $g_{h_{3}h_{2}h_{1}}$, $g_{h_{3}h_{1}h_{1}}$ and $g_{h_{3}h_{2}h_{2}}$
represents trilinear terms $h_{3}h_{2}h_{1}$, $h_{3}h_{1}h_{1}$ and
$h_{3}h_{2}h_{2}$ respectively, their expressions are given in table 4.
The partial decay width of $h_{3}$ into a pair of gauge bosons $V_{i}$ is
given by Eq (55) where in the case of $h_{3}$, the coefficients $k_{i}$ are
given by:

$k_{i}$ coefficients.  Higgs $k_{X^{\mp}}$ $k_{V^{\mp\mp}}$ $k_{K^{\prime 0}}$
$k_{K_{1}^{\mp}}$ $k_{Y^{\mp}}$ $h_{3}$
$\frac{\upsilon_{\chi}}{\upsilon_{\rho}}\sigma$
$\frac{\upsilon_{\chi}}{\upsilon_{\rho}}\sigma$
$\frac{\upsilon_{\eta}}{\upsilon_{\rho}}\beta$
$\frac{\upsilon_{\eta}}{\upsilon_{\rho}}\beta$
$\frac{1}{\upsilon_{\rho}}(\upsilon_{\chi}\sigma+\upsilon_{\eta}\beta)$

The partial decay width of $h_{3}$ into a pair of SM Z and $Z^{\prime}$ gauge
bosons are given by the Eqs (56) and (57) respectively only one needs to
replace $m_{h_{2}}$ with $m_{h_{3}}$ and both $G_{ZZh_{2}}$ and
$G_{Z^{\prime}Z^{\prime}h_{2}}$ with $G_{ZZh_{3}}$ and
$G_{Z^{\prime}Z^{\prime}h_{3}}$ respectively, where:

$\displaystyle G_{ZZh_{3}}$ $\displaystyle=$
$\displaystyle\frac{C^{2}_{W}}{2}\bigg{(}\frac{3}{2}\frac{\upsilon_{\chi}}{\upsilon_{\rho}}(C^{23})^{2}\sigma+4(C^{24})^{2}\frac{S_{W}^{2}}{1-4S_{W}^{2}}\frac{\upsilon_{\chi}}{\upsilon_{\rho}}\sigma+\frac{12}{\sqrt{6}}\frac{S_{W}}{\sqrt{1-4S_{W}^{2}}}\frac{\upsilon_{\chi}}{\upsilon_{\rho}}C^{23}C^{24}\sigma$
(64) $\displaystyle+$
$\displaystyle\frac{4}{3}(C^{22})^{2}\frac{\upsilon_{\eta}}{\upsilon_{\rho}}\beta+\frac{(C^{23})^{2}}{6}\frac{\upsilon_{\eta}}{\upsilon_{\rho}}\beta-\frac{4}{\sqrt{18}}C^{22}C^{23}\frac{\upsilon_{\eta}}{\upsilon_{\rho}}\beta\bigg{)},$
$\displaystyle G_{Z^{\prime}Z^{\prime}h_{3}}$ $\displaystyle=$
$\displaystyle\frac{C^{2}_{W}}{2}\bigg{(}\frac{3}{2}\frac{\upsilon_{\chi}}{\upsilon_{\rho}}(C^{33})^{2}\sigma+4(C^{34})^{2}\frac{S_{W}^{2}}{1-4S_{W}^{2}}\frac{\upsilon_{\chi}}{\upsilon_{\rho}}\sigma+\frac{12}{\sqrt{6}}\frac{S_{W}}{\sqrt{1-4S_{W}^{2}}}\frac{\upsilon_{\chi}}{\upsilon_{\rho}}C^{33}C^{34}\sigma$
(65) $\displaystyle+$
$\displaystyle\frac{4}{3}(C^{32})^{2}\frac{\upsilon_{\eta}}{\upsilon_{\rho}}\beta+\frac{(C^{33})^{2}}{6}\frac{\upsilon_{\eta}}{\upsilon_{\rho}}\beta-\frac{4}{\sqrt{18}}C^{32}C^{33}\frac{\upsilon_{\eta}}{\upsilon_{\rho}}\beta\bigg{)}.$

The partial decay widths of the $h_{3}$ into gluons and into a pair of exotic
quarks are given respectively by:

$\displaystyle\Gamma(h_{3}\longrightarrow gg)$ $\displaystyle=$
$\displaystyle\frac{\alpha^{2}_{S}M^{3}_{h_{3}}}{128\pi^{3}\upsilon_{\eta}^{2}}\beta^{2}\bigg{|}\sum_{i=1}^{3}AU_{\frac{1}{2}}(\tau_{U})\bigg{|}^{2}+\frac{\alpha^{2}_{S}M^{3}_{h_{3}}}{128\pi^{3}\upsilon_{\chi}^{2}}\sigma^{2}\bigg{|}\sum_{i=1}^{3}AJ_{\frac{1}{2}}(\tau_{J})\bigg{|}^{2},$
(66) $\displaystyle\Gamma_{341}(h_{3}\longrightarrow UU)$ $\displaystyle=$
$\displaystyle\frac{3g^{2}}{32\pi}\frac{m_{U}^{2}m_{h_{3}}}{m_{W}^{2}}\bigg{(}1-\frac{4m^{2}_{U}}{m^{2}_{h_{3}}}\bigg{)}^{\frac{3}{2}}\bigg{(}\frac{\upsilon_{\rho}}{\upsilon_{\eta}}\beta\bigg{)}^{2},$
(67) $\displaystyle\Gamma_{341}(h_{3}\longrightarrow JJ)$ $\displaystyle=$
$\displaystyle\frac{3g^{2}}{32\pi}\frac{m_{J}^{2}m_{h_{3}}}{m_{W}^{2}}\bigg{(}1-\frac{4m^{2}_{J}}{m^{2}_{h_{3}}}\bigg{)}^{\frac{3}{2}}\bigg{(}\frac{\upsilon_{\rho}}{\upsilon_{\chi}}\sigma\bigg{)}^{2}.$
(68)

##  6 Numerical Analysis

In this section, we discuss our numerical results of the signal strength of
the Higgs like-boson $h_{1}$ and the branching ration (BR) of the other
scalars $h_{2}$ and $h_{3}$. All the expressions of the compact 341 model are
related to $\upsilon_{\chi}$, $\upsilon_{\eta}$, $\upsilon_{\rho}$, and to the
scalar parameters $\lambda_{1..9}$.
In our work, we consider the following scenario:
$\upsilon_{\rho}$=246 GeV, $\upsilon_{\chi}=\upsilon_{\eta}$=2 TeV and
$m_{\text{exotic~{}quarks}}$=750 GeV [21], while, for the scalar parameters
$\lambda_{1..9}$, we make random choices where we constrained them using the
theoretical constraints that we have discussed in the text.

###  6.1 The signal strength

Any extension of the SM must possess a scalar with 125 GeV of mass, in our
model, we identify $h_{1}$ as the SM Higgs boson, therefore, the signal
strength is discussed to clarify if the lightest scalar boson $h_{1}$ recovers
the SM Higgs boson and to check the validity of our model.
We calculated the signal strength of $h_{1}$ where we take into account that
the most dominant production channel of $h_{1}$ comes from the gluon-gluon
fusion, while, its decay is into several channels namely $b\bar{b}$, $WW^{*}$,
$ZZ^{*}$, $\gamma\gamma$ and $\tau^{+}\tau^{-}$.
The fermionic decays and the bosonic decays are possible at the tree-level,
whereas we apply one-loop expressions for the decays into final states
including massless bosons (that is gg, $\gamma\gamma$ and Z$\gamma$).
Table 6.1, shows the results of our model and the available experimental
signal strength values from LHC Run 1 for the combination of ATLAS and CMS,
and separately for each experiment, for the combined $\sqrt{s}$ = 7 and 8 TeV
data for different Higgs boson decay channels. These results are obtained
assuming that the Higgs boson production process cross sections at s = 7 and 8
TeV are the same as in the SM [22].
Figure 1 shows the signal strengths for the various decay modes $b\bar{b}$,
$WW^{*}$, $ZZ^{*}$, $\gamma\gamma$ and $\tau^{+}\tau^{-}$ in the compact 341
model with the data reported at ATLAS, CMS and the combined ATLAS+ CMS Run1.
Our results can fit the current data withing the experimental errors which
makes the compact 341 model in perfect agreement with the values measured by
ATLAS and CMS. That ensures the viability of the compact 341 model to be an
available model in the future work at the LHC.

The SM Higgs boson signal strengths of the official ATLAS, CMS and ATLAS and
CMS combination for Run 1 [22], based on 25 $\text{fb}^{-1}$ of integrated
luminosity and of the compact 341 model.  Decay channel ATLAS CMS ATLAS+CMS
The compact 341 model $\mu^{\gamma\gamma}$ $1.14^{+0.27}_{-0.25}$
$1.11^{+0.25}_{-0.23}$ $1.14^{+0.19}_{-0.18}$ 1.03 $\mu^{ZZ}$
$1.52^{+0.40}_{-0.34}$ $1.04^{+0.32}_{-0.26}$ $1.29^{+0.26}_{-0.23}$ 1.06
$\mu^{WW}$ $1.22^{+0.23}_{-0.21}$ $0.90^{+0.23}_{-0.21}$
$1.09^{+0.18}_{-0.16}$ 0.99 $\mu^{\tau\tau}$ $1.41^{+0.40}_{-0.36}$
$0.88^{+0.30}_{-0.28}$ $1.11^{+0.24}_{-0.22}$ 0.99 $\mu^{bb}$
$0.62^{+0.37}_{-0.37}$ $0.81^{+0.45}_{-0.43}$ $0.70^{+0.29}_{-0.27}$ 0.99

Figure 1:  The signal strength results of $h_{1}$ for various decay modes of
the compact 341 model (red points) and the experimental data (blue points)

###  6.2 Searches for $h_{i}$ i=2,3

Figure 2 shows BR$(h_{2}\longrightarrow h_{1}h_{1})$ as a function of the
heavy Higgs $h_{2}$ mass for $v_{\chi}=v_{\eta}$=2 TeV where in the allowed
parameter space, the mass of $m_{h_{2}}$ ranges from 500 GeV to 1.3 TeV. Its
clear that in the region 500-1300 GeV , the decay channel
$h_{2}\longrightarrow h_{1}h_{1}$ is the most important one with a BR $\geq
90\%$, this happens because the trilinear coupling $h_{1}h_{2}h_{2}$ increases
about one order of magnitude $v_{\eta}$. Interestingly, one can has a new
production mechanism for $h_{2}$ namely $pp\longrightarrow
h_{2}\longrightarrow h_{1}h_{1}$. This production channel might be useful to
increase the signal of the double Higgs production at the LHC.
As the mass becomes larger than 1300 GeV, we can notice that the decay modes
of $h_{2}$ into the heavy gauge bosons $K^{\prime 0}$, $K_{1}^{\mp}$,
$Y^{\mp}$, $V^{\mp\mp}$, $X^{\mp}$ and $Z^{\prime}$ and into exotic quarks are
kinetically allowed, in this case, the total decay width contains additional
channels $h_{2}\longrightarrow VV$ and $h_{2}\longrightarrow Q\overline{Q}$
where V$\equiv K^{\prime 0}$, $K_{1}^{\mp}$, $Y^{\mp}$, $V^{\mp\mp}$,
$X^{\mp}$ and $Z^{\prime}$ and Q$\equiv$ exotic quarks. The maximum value of
the branching ratio of the process $h_{2}$ into gauge bosons $K^{\prime 0}$,
$K_{1}^{\mp}$, $V^{\mp\mp}$ and $X^{\mp}$ plateaus close to $\sim$0.35,
$\sim$0.26 as we shown in tables 6.2 and 6.2 and in figures 3 and 4,
therefore, one may notice that those decays are the most significant decay
channel and the decay channel $h_{2}\longrightarrow h_{1}h_{1}$ becomes weak
and contributing roughly $20\%$.
The branching ratios of the remaining decay modes are tabulated in the
following tables:

The branching ratios (BRs) of the heavy scalar $h_{2}$ for differnet channels.
Decay channel BR where $m_{h_{2}}\in$ [500-1300](GeV) BR where $m_{h_{2}}\in$
[1300-3500](GeV) $b\bar{b}$ $\sim\mathcal{O}(10^{-5})$ [$10^{-5}-10^{-6}$]
$\tau^{+}\tau^{-}$ $\sim\mathcal{O}(10^{-7})$ $\sim 10^{-7}$ $\gamma\gamma$
$\sim\mathcal{O}(10^{-7})$ $\sim 10^{-7}$ $\gamma Z$
$\sim\mathcal{O}(10^{-3})$ [$10^{-3}-2.6.10^{-3}$] $gg$
$\sim\mathcal{O}(10^{-4})$ [$10^{-4}-10^{-5}$] ZZ $\sim\mathcal{O}(10^{-32})$
[$10^{-32}-10^{-31}$] $Z^{\prime}Z^{\prime}$ $\sim\mathcal{O}(10^{-4})$
[$10^{-3}-10^{-2}$] $K^{\prime 0}K^{\prime 0}$ 0 [$10^{-3}$-0.26] $Y^{+}Y^{-}$
0 [$10^{-4}$-$10^{-2}$] $X^{+}X^{-}$ 0 [$10^{-3}$-0.35] $V^{++}V^{--}$ 0
[$10^{-3}$-0.34]

The branching ratios (BRs) of the heavy scalar $h_{2}$ for differnet channels.
Decay channel BR where $m_{h_{2}}\in$ [500-1300](GeV) BR where $m_{h_{2}}\in$
[1300-3500](GeV) $K_{1}^{+}K_{1}^{-}$ 0 [$10^{-3}$-0.26] $U\bar{U}$ 0
[$10^{-4}$-$10^{-2}$] $J\bar{J}$ 0 [$10^{-4}$-$10^{-1}$]

Figure 2:  The branching ratio of $h_{2}\longrightarrow h_{1}h_{1}$ versus the
heavy Higgs $h_{2}$ mass. Figure 3:  The branching ratio of
$h_{2}\longrightarrow K^{0\prime}K^{0\prime}$ versus the heavy Higgs $h_{2}$
mass.

Figure 4:  The branching ratio of $h_{2}\longrightarrow X^{+}X^{+}$,
$h_{2}\longrightarrow K_{1}^{+}K_{1}^{-}$ and $h_{2}\longrightarrow
V^{++}V^{--}$ versus the heavy Higgs $h_{2}$ mass.

Now we turn our discussion to $h_{3}$ decay, Figure 5 shows
BR$(h_{3}\longrightarrow ZZ)$ as a function of the heavy Higgs $h_{3}$ mass
for $v_{\chi}=v_{\eta}$=2 TeV where in the allowed parameter space, the mass
of $m_{h_{3}}$ ranges from 3 TeV to 4 TeV. The decay channel
$h_{3}\longrightarrow ZZ$ is the most important one with a BR $\geq 80\%$
which is about $\sim$0.8, while, the BR in $h_{1}h_{1}$ is the second most
significant decay channel contributing roughly $20\%$ as we shown in figure 6,
whereas the lowest value of the branching ratio in that region is just below
$\sim 10^{-9}$.
The branching ratios of the other processes are shown in the following table:

The branching ratios (BRs) of the heavy scalar $h_{3}$ for differnet channels.
Decay channel BR where $m_{h_{3}}\in$ [3000-4000](GeV) $b\bar{b}$
$\sim\mathcal{O}(10^{-5})$ $\tau^{+}\tau^{-}$ [$10^{-7}-10^{-9}$]
$\gamma\gamma$ $[10^{-7}-10^{-5}]$ $\gamma Z$ $[10^{-3}-10^{-2}]$ $gg$
$[10^{-5}-10^{-4}$] ZZ $\sim 0.80$ $Z^{\prime}Z^{\prime}$ $[10^{-7}-10^{-5}]$
$K^{\prime 0}K^{\prime 0}$ [$1.5.10^{-6}-1.9.10^{-4}$] $Y^{+}Y^{-}$
[$6.10^{-6}-5.10^{-5}$] $X^{+}X^{-}$ [$10^{-6}-3.10^{-5}$] $V^{++}V^{--}$
[$10^{-6}-10^{-5}$] $K_{1}^{+}K_{1}^{-}$ [$1.5.10^{-6}-1.9.10^{-4}$]
$U\bar{U}$ [$5.10^{-2}-0.25$] $J\bar{J}$ [$5.10^{-2}-10^{-3}$] $h_{1}h_{1}$
$\sim 0.20$ $h_{1}h_{2}$ [$10^{-4}-10^{-2}$] $h_{2}h_{2}$
[$5.10^{-4}-10^{-3}$]

Figure 5:  The branching ratio of $h_{3}\longrightarrow ZZ$ as a function of
the heavy Higgs $h_{3}$ mass. Figure 6:  The branching ratio of
$h_{3}\longrightarrow h_{1}h_{1}$ as a function of the heavy Higgs $h_{3}$
mass.

At the end, we give the total decay width for the two CP-even scalars $h_{2}$
and $h_{3}$ in figure 7. It is well known that the SM Higgs with a mass of 125
GeV has a very narrow width almost equal to $\Gamma_{h}\sim$ 4 MeV.
In our case, the total width of $h_{2}$ is in the range 4.1.$10^{-5}$-1.014
(TeV), we notice a very narrow width of $h_{2}$ when the decay of $h_{2}$ into
the heavy gauge bosons and exotic quarks are closed and it can decay only into
a pair of leptons, quarks, scalars and $\gamma\gamma(Z\gamma)$, while, when
they opened the total width is large and it achieves the value $\sim$ 1.014
TeV. Whereas, the total width of $h_{3}$ can be located between 0.0649 and
1.488 TeV.

Figure 7:  The total decay width of the $h_{2}$ versus the one of $h_{3}$.

##  7 Conclusion

The compact 341 model is a gauge extension of the Standard Model with only
three quadruplets scalar fields composed by three neutral scalars, four singly
charged and two doubly charged scalar bosons. In this work we discussed some
phenomenology of the scalar sector.
We have used a set of the theoretical constraints such as: perturbative
unitarity, boundedness from below constraints and the positivity of the scalar
bosons masses condition to constrain the scalar parameters.
We have calculated the signal strength of $h_{1}$ for different channels where
we obtained a good fit to the data as can be seen in figure 1. In particular,
with $\upsilon_{\chi}$= 2 TeV, we have shown that the compact 341 model is
achieved to a good result that are compatible to the measurement of LHC.
The second focus of our paper is the calculation of the branching ratio of the
other heavy scalar bosons $h_{2}$ and $h_{3}$, we found that $h_{2}$ decay
preferentially into a pair of Higgs-like particles with a branching ratio
$\sim$ 1 when $m_{h_{2}}\in$ [500,1300] GeV, a feature not easily obtained in
other extensions of the Standard Model and in the second region where
$m_{h_{2}}\in$ [1300,3500] GeV, $h_{2}$ prefers to decay into the heavy gauge
bosons $X^{\mp},K_{1}^{\mp},K^{\prime 0},V^{\mp\mp}$. Moreover, we found that
$h_{3}$ decay preferentially into a pair of Z bosons where $m_{h_{3}}\in$
[3000,4000] GeV .
Finally, we have studied the total decay width of $h_{2}$ and $h_{3}$, we
found that $\Gamma_{h_{2}}$ and $\Gamma_{h_{3}}$ can achieve 1.014 TeV and
1.488 TeV respectively.
In conclusion, we have studied the scalar sector of the compact 341 model and
showed that at a scale of a few TeV this model is a compelling alternative to
the SM once, it is able to reproduce the experimental data regarding the
signal strength.
The production process $gg\longrightarrow h_{2,3}$ followed by the decays
$h_{2,3}\longrightarrow h_{1}h_{1}$ could be sizeable and could be an
important source of double $h_{1}$ production where it is rather difficult to
produce it using the conventional channel.

##  Appendix A Perturbative unitarity and Vacuum stability conditions

The perturbative unitarity conditions are [18]:

$\displaystyle\alpha^{2}\lambda_{4}$ $\displaystyle+$
$\displaystyle\gamma^{2}(\lambda_{6}+\lambda_{9})<16\pi,$
$\displaystyle\beta^{2}\lambda_{4}$ $\displaystyle+$
$\displaystyle\sigma^{2}(\lambda_{6}+\lambda_{9})<16\pi,$
$\displaystyle\alpha\beta\lambda_{4}$ $\displaystyle+$
$\displaystyle\gamma\sigma(\lambda_{6}+\lambda_{9})<8\pi,$
$\displaystyle(\alpha^{2}$ $\displaystyle+$ $\displaystyle
2\alpha\beta)\lambda_{4}+\gamma^{2}\lambda_{6}<32\pi,$
$\displaystyle\lambda_{4}$ $\displaystyle+$ $\displaystyle\lambda_{6}<32\pi,$
$\displaystyle\beta^{2}\lambda_{4}$ $\displaystyle+$
$\displaystyle\sigma^{2}\lambda_{6}<32\pi,$ $\displaystyle\sigma\lambda_{9}$
$\displaystyle+$ $\displaystyle\beta\lambda_{7}<16\pi,$
$\displaystyle\gamma\lambda_{9}$ $\displaystyle+$
$\displaystyle\alpha\lambda_{7}<16\pi,$ $\displaystyle\gamma\sigma\lambda_{6}$
$\displaystyle<$ $\displaystyle 16\pi,$ $\displaystyle\alpha^{2}\lambda_{4}$
$\displaystyle+$
$\displaystyle\gamma^{2}\lambda_{6}+\alpha^{2}\lambda_{7}<16\pi,$
$\displaystyle\beta^{2}\lambda_{4}$ $\displaystyle+$
$\displaystyle\sigma^{2}\lambda_{6}+\beta^{2}\lambda_{7}<16\pi,$
$\displaystyle\alpha\beta\lambda_{4}$ $\displaystyle+$
$\displaystyle\sigma\gamma\lambda_{6}+\alpha\beta\lambda_{7}<8\pi,$
$\displaystyle 2\alpha^{2}\lambda_{1}$ $\displaystyle+$ $\displaystyle
2\gamma^{2}\lambda_{3}+\lambda_{5}(\alpha^{2}+\gamma^{2})+\lambda_{8}(\alpha^{2}+\gamma^{2}+(\sqrt{2}+1)\gamma\alpha)<32\pi,$
$\displaystyle 2\beta^{2}\lambda_{1}$ $\displaystyle+$ $\displaystyle
2\sigma^{2}\lambda_{3}+\lambda_{5}(\beta^{2}+\sigma^{2})+\lambda_{8}(\beta^{2}+\sigma^{2}+\sigma\beta)<32\pi,$
$\displaystyle 2\beta\alpha\lambda_{1}$ $\displaystyle+$ $\displaystyle
4\sigma\gamma\lambda_{3}+2\lambda_{5}(\beta\alpha+\gamma^{2})+\lambda_{8}(2\beta\alpha+2\sigma\alpha+\beta+\sigma\gamma+\beta\gamma+\sigma\beta)<32\pi,$
$\displaystyle\lambda_{1}$ $\displaystyle+$
$\displaystyle\lambda_{3}+\lambda_{5}<32\pi,$
$\displaystyle\alpha^{4}\lambda_{1}$ $\displaystyle+$
$\displaystyle\gamma^{2}\lambda_{3}<32\pi,$
$\displaystyle\beta^{4}\lambda_{1}$ $\displaystyle+$
$\displaystyle\sigma^{4}\lambda_{3}+\sigma^{2}\beta^{2}\lambda_{5}<32\pi,$
$\displaystyle 6\beta^{2}\alpha^{2}\lambda_{1}$ $\displaystyle+$
$\displaystyle
6\sigma^{2}\gamma^{2}\lambda_{3}+\lambda_{5}(4\alpha\beta\gamma\sigma+\gamma^{2}\alpha^{2}+\gamma^{2}\beta^{2}+\alpha^{2}\sigma^{2})<32\pi,$
$\displaystyle 2\alpha^{3}\beta\lambda_{1}$ $\displaystyle+$ $\displaystyle
2\lambda_{3}\gamma^{3}\sigma+\lambda_{5}(\beta\alpha\gamma^{2}+\alpha^{2}\gamma\sigma)<16\pi,$
$\displaystyle 2\beta^{3}\alpha\lambda_{1}$ $\displaystyle+$ $\displaystyle
2\lambda_{3}\sigma^{3}\gamma+\lambda_{5}(\beta\alpha\sigma^{2}+\beta^{2}\gamma\sigma)<16\pi,$
$\displaystyle\lambda_{4}$ $\displaystyle+$
$\displaystyle\lambda_{6}+\lambda_{9}<16\pi,$ $\displaystyle\lambda_{4}$
$\displaystyle+$ $\displaystyle\lambda_{6}+\lambda_{7}<16\pi.$ (69)

The vacuum stability and the minimization conditions are satisfied only if
[18]:

$\displaystyle\lambda_{1}$ $\displaystyle>$ $\displaystyle
0,\qquad\lambda_{2}>0,\qquad\lambda_{3}>0,$ $\displaystyle\lambda_{4}$
$\displaystyle+$ $\displaystyle 2\sqrt{\lambda_{1}\lambda_{2}}>0.$
$\displaystyle\lambda_{4}$ $\displaystyle+$
$\displaystyle\lambda_{7}+2\sqrt{\lambda_{1}\lambda_{2}}>0$
$\displaystyle-2\sqrt{\lambda_{1}\lambda_{3}}$ $\displaystyle<$
$\displaystyle\lambda_{5}<2\sqrt{\lambda_{1}\lambda_{3}},$
$\displaystyle-2\sqrt{\lambda_{2}\lambda_{3}}$ $\displaystyle<$
$\displaystyle\lambda_{6}<2\sqrt{\lambda_{2}\lambda_{3}},$
$\displaystyle-2\sqrt{\lambda_{1}\lambda_{3}}$ $\displaystyle<$
$\displaystyle\lambda_{5}+\lambda_{8}<2\sqrt{\lambda_{1}\lambda_{3}},$
$\displaystyle-2\sqrt{\lambda_{2}\lambda_{3}}$ $\displaystyle<$
$\displaystyle\lambda_{6}+\lambda_{9}<2\sqrt{\lambda_{2}\lambda_{3}},$
$\displaystyle 4(\lambda_{4}$ $\displaystyle+$
$\displaystyle\lambda_{7})\lambda_{3}-2(\lambda_{5}+\lambda_{8})(\lambda_{6}+\lambda_{9})+2\sqrt{\Lambda_{1}}>0,$
$\displaystyle 4\lambda_{4}\lambda_{3}$ $\displaystyle-$ $\displaystyle
2\lambda_{5}\lambda_{6}+2\sqrt{\Lambda_{2}}>0,$ $\displaystyle
4\lambda_{4}\lambda_{3}$ $\displaystyle-$ $\displaystyle
2\lambda_{5}(\lambda_{6}+\lambda_{9})+2\sqrt{\Lambda_{3}}>0,$ $\displaystyle
4\lambda_{4}\lambda_{3}$ $\displaystyle-$ $\displaystyle
2(\lambda_{5}+\lambda_{8})(\lambda_{6}+\lambda_{9})+2\sqrt{\Lambda_{4}}>0,$
$\displaystyle 4\lambda_{4}\lambda_{3}$ $\displaystyle-$ $\displaystyle
2(\lambda_{5}+\lambda_{8})\lambda_{6}+2\sqrt{\Lambda_{5}}>0,$ $\displaystyle
4(\lambda_{4}$ $\displaystyle+$
$\displaystyle\lambda_{7})\lambda_{3}-2\lambda_{5}\lambda_{6}+2\sqrt{\Lambda_{6}}>0,$
$\displaystyle 4(\lambda_{4}$ $\displaystyle+$
$\displaystyle\lambda_{7})\lambda_{3}-2\lambda_{5}(\lambda_{6}+\lambda_{9})+2\sqrt{\Lambda_{7}}>0,$
$\displaystyle 4(\lambda_{4}$ $\displaystyle+$
$\displaystyle\lambda_{7})\lambda_{3}-2(\lambda_{5}+\lambda_{8})\lambda_{6}+2\sqrt{\Lambda_{8}}>0,$
(70)

where:

$\displaystyle\Lambda_{1}$ $\displaystyle=$
$\displaystyle(4\lambda_{1}\lambda_{3}-(\lambda_{5}+\lambda_{8})^{2})(4\lambda_{2}\lambda_{3}-(\lambda_{6}+\lambda_{9})^{2}),$
$\displaystyle\Lambda_{2}$ $\displaystyle=$
$\displaystyle(4\lambda_{1}\lambda_{3}-\lambda_{5}^{2})(4\lambda_{2}\lambda_{3}-\lambda_{6}^{2}),$
$\displaystyle\Lambda_{3}$ $\displaystyle=$
$\displaystyle(4\lambda_{1}\lambda_{3}-\lambda_{5}^{2})(4\lambda_{2}\lambda_{3}-(\lambda_{6}+\lambda_{9})^{2}),$
$\displaystyle\Lambda_{4}$ $\displaystyle=$
$\displaystyle(4\lambda_{1}\lambda_{3}-(\lambda_{5}+\lambda_{8})^{2})(4\lambda_{2}\lambda_{3}-(\lambda_{6}+\lambda_{9})^{2}),$
$\displaystyle\Lambda_{5}$ $\displaystyle=$
$\displaystyle(4\lambda_{1}\lambda_{3}-(\lambda_{5}+\lambda_{8})^{2})(4\lambda_{2}\lambda_{3}-\lambda_{6}^{2}),$
$\displaystyle\Lambda_{6}$ $\displaystyle=$
$\displaystyle(4\lambda_{1}\lambda_{3}-\lambda_{5}^{2})(4\lambda_{2}\lambda_{3}-\lambda_{6}^{2}),$
$\displaystyle\Lambda_{7}$ $\displaystyle=$
$\displaystyle(4\lambda_{1}\lambda_{3}-\lambda_{5}^{2})(4\lambda_{2}\lambda_{3}-(\lambda_{6}+\lambda_{9})^{2}),$
$\displaystyle\Lambda_{8}$ $\displaystyle=$
$\displaystyle(4\lambda_{1}\lambda_{3}-(\lambda_{5}+\lambda_{8})^{2})(4\lambda_{2}\lambda_{3}-\lambda_{6}^{2}).$
(71)

## Acknowledgments

We are very grateful to the Algerian ministry of higher education and
scientific research and DGRSDT for the financial support.

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