#  Phenomenology of an $E_{6}$ inspired extension of Standard Model: Higgs
sector

Sanchari<EMAIL_ADDRESS>_Department of Physics,
University of Calcutta_
_92, Acharya Prafulla Chandra Road, Kolkata 700009_ Anindya
<EMAIL_ADDRESS>_Department of Physics, University of Calcutta_
_92, Acharya Prafulla Chandra Road, Kolkata 700009_

###### Abstract

We investigate a variant of Left-Right symmetric model based $SU(3)_{C}\otimes
SU(2)_{L}\otimes U(1)_{L}\otimes SU(2)_{R}\otimes U(1)_{R}$ gauge group
$(32121)$. Spontaneous breaking of $32121$ down to the Standard Model (SM)
gauge group, requires a bi-doublet under $SU(2)_{L}\otimes SU(2)_{R}$, a
right-handed doublet scalar under $SU(2)_{R}$ along with a $SU(2)$ singlet
scalar boson. Symmetry breaking leads to several neutral and charged massive
gauge bosons apart from the SM $W$ and $Z$. The Large Hadron Collider (LHC)
results for the search of heavy gauge bosons can be used to constrain the
vacuum expectation values responsible for giving masses to these extra heavy
gauge bosons. Physical spectrum of the scalar bosons contains several neutral
CP-even and CP-odd states and a couple of charged scalars apart from the SM-
like Higgs boson. We have put constraints on the masses of some of these
scalars from the existing LHC data. The possible decay rates and production
cross-sections of these scalars have been investigated in some details.
Production cross-sections for some of the scalars look promising at 14 TeV and
27 TeV run of the LHC with high luminosity option. We keep, in our model, all
the fermions present in the $27$-dimensional fundamental representation of
$E_{6}$. Mass limit of one such exotic lepton has also been derived from
present LHC data. It is noted that some of these neutral exotic lepton or
neutral scalar bosons of this model can serve the purpose of a cold dark
matter.

##  1 Introduction

The Standard Model (SM) of Particle Physics has been extremely successful in
describing the interactions of elementary particles and fundamental forces
operative in microscopic world. Probably, the most subtle prediction of the SM
has been the existence of a scalar boson, the Higgs boson, responsible for
giving masses to all the elementary particles. With the discovery of Higgs
boson [1, 2] at the Large Hadron Collider (LHC), CERN; all the predictions of
the SM has been tested. In spite of its immense success, SM in its original
form, miserably fails to explain one important piece of experimental
observations, namely, the existence of Dark Matter (DM), a new kind of very
weakly interacting but massive matter pervading the whole universe. Moreover,
it is very crucial to know whether the discovered 125 GeV Higgs boson is the
sole agent for Electroweak Symmetry Breaking (EWSB) or a more rich scalar
sector is responsible for such an act. There are several theoretical studies
[3] which have been devoted to investigate the phenomenology of extended Higgs
sectors. It is important to note that any model with extended scalar sector
must contain a physical CP-even scalar boson with exactly the same properties
of the SM Higgs. The extended scalar sector may also be instrumental in
resolving some of the shortcomings of the SM. As for example, extended Higgs
sectors with singlet scalars may resolve the problem of dark matter [4]. Left-
Right (LR) symmetric triplet Higgs models are very popular in explaining
neutrino masses via seesaw mechanisms [5], [6]. Multi-Higgs doublet models
have been used in explaining flavour problems [7]. An added bonus for many
such models with extended Higgs sector is possibility of a stable neutral
scalar which may act as a suitable candidate for DM. On the experimental
front, signatures for extra neutral and charged scalar bosons have already
been in the top of the agenda for all the past and present experimental
programmes. Unfortunately, the evidence for the SM-like Higgs boson has been
observed so far. Present precision of experimental data on Higgs signal
strengths in different channels and our lack of experimental knowledge on
Higgs tri-linear coupling limit us from conclusively decide whether the 125
GeV boson is the only agent of EWSB [8]. In near future, with high luminosity
(HL) 14 (27) TeV run of the LHC we expect these pictures to be more clearer.
So it is of utmost priority for particle physics community to construct such
models with extended scalar and/or gauge sector and check whether these models
are consistent with the available and future experimental data from the LHC. A
number of such models have been proposed and their possible experimental
signatures at the LHC have been sought for. Unfortunately, many such efforts
have been futile so far. Non-observation of any signature of Physics beyond
the SM (whether supersymmetry [9] or extra-dimension [10]), only pushes the
scale of such new dynamics in upward direction.

In an endeavour to construct a model which has a rich scalar sector satisfying
the LHC data we turn our attention to an unifying gauge group $E_{6}$[11],
which can be broken down to $\left[SU(3)\right]^{3}$ followed by a further
breakdown to $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{L}\otimes
SU(2)_{R}\otimes U(1)_{R}$ ($32121$). We will not be interested in the
dynamics which may be responsible for the aforementioned symmetry breaking
chain, rather we will be investigating the appearance of SM gauge group from
$32121$ and the resulting phenomenology of additional particles and their
interactions among themselves or with the particles of SM. The main advantage
of working in a framework of unifying group like $E_{6}$ is the natural
appearance of right-handed neutrinos as well as 3 generations of heavy neutral
leptons, singlet under either of $SU(2)$ groups. The right-handed heavy
neutrinos may be responsible for neutrino mass generations via the (Type-I)
seesaw mechanism. This model contains a large number of neutral and charged
scalar bosons after the spontaneous breakdown of $SU(2)_{L}\otimes
U(1)_{L}\otimes SU(2)_{R}\otimes U(1)_{R}$ to EW gauge group of the SM. These
color singlet scalars originate from (1, 3, $\bar{\textbf{3}}$) representation
of $[SU(3)]^{3}$. The presence of such rich scalar sector is one of the
reasons behind the present investigation. Furthermore, several exotic fermions
will arise from the $27$-dimensional fundamental representation of $E_{6}$.
Some of the neutral Higgs bosons and charge neutral leptons may also serve the
purpose of dark matter. However, in this article we will mainly investigate
the phenomenology of the extended scalar sector. A separate study will be
devoted to the DM aspect of this model.

Although, the scalar sector of the model under consideration have many
similarities with that of the left-right model (LRM), the phenomenology of the
scalar sector has some distinct features which are different from left-right
symmetric model (LRSM). In the present work, we have banked on these features
of the model to distinguish it from the often studied models like LRSM. Let us
emphasise on the novelty of the present work in the following.

  * •

Although the $32121$ gauge group respects the LR symmetry, the model under
consideration is different from the conventional LRS model. The first hint of
this difference comes from the fact that both the bi-doublet vacuum
expectation values (vevs) $k_{1}$ and $k_{2}$ cannot be simultaneously set to
non-zero values unlike its more familiar variant based on $SU(2)_{L}\times
SU(2)_{R}\times U(1)_{B-L}$ unless we consider a tri-linear term in the scalar
potential. Even after adding such a term which comes out to be small along
with small value of $k_{2}$, phenomenology of this model remains practically
the same as $k_{2}=0$ case. This makes the phenomenology of the scalar sector
of $32121$ model different from LRSM.

  * •

We have considered the complete set of fermions arising from the $\bf
27$-dimensional representation of $E_{6}$. Some of these fermions are stable
but have electric charges. We have for the first time derived bounds on their
masses from the present LHC data. A detailed endeavour from the
experimentalist friends to study their signature including detector simulation
is presently unavailable and is urgently needed.

  * •

Although one of the charged Higgs bosons arising in the model has conventional
decay to $tb$, the other one is stable and produces a charged track in the
detector. Such stable charged bosons are not present in the LRSM and their
signatures have not been discussed in the present literature so far.

  * •

The pair-production of charged and neutral Higgs bosons arising from left-
handed doublet will produce either two charged tracks or a single charged
track in the detector, giving rise to a background free novel signature of
this model.

  * •

Presence of a heavy neutral gauge boson,$A^{\prime}$ apart from $Z^{\prime}$
(arising because of the presence of $SU(2)_{R}$, can be identified with the
$Z_{R}$ in LRSM) is also a hall mark of the model under the consideration.
This neutral boson arises due to the extra $U(1)$ factor in $32121$ and
couples to all the SM fermions, in contrast to several extensions of the SM by
an extra $U(1)$, where this heavy neutral state have selective coupling with
the SM fermions. Unlike the LRS model, where only one heavy neutral gauge
boson is present, $32121$ is characterised by two such states with similar
properties but having different masses.

  * •

Three possible Dark Matter candidates in forms of exotic fermion, a scalar and
a pseudoscalar originating from left-doublet, are present in the spectrum.
Although we will not discuss the direct detection bounds or relic density, an
lower bound on the mass of the scalar has been derived indirectly from the LHC
data in the present analysis.

Before we come to an end of this section, it is important to mention that we
have only guided by the framework of $E_{6}$ to study the phenomenology of the
a set of scalars whose masses are somehow interlaced due to the structure of
the scalar potential respecting the local $32121$ gauge symmetry. Similarly,
the gauge group and complete set of $27$-plet of of fermions are chosen from a
low energy point of view. Let us re-emphasize that we will not study the
symmetry breaking chain $E_{6}\rightarrow\left[SU(3)\right]^{3}\rightarrow
32121$, neither we start with an unifying Yukawa texture appropriate for
$E_{6}$ starting from GUT scale to generate the same at EW scale by
renormalization group equation (RGE) running. The Yukawa matrices in our
analysis, have been assumed to be non-diagonal. However, we will not discuss
the pattern of masses and mixings for the exotic fermions present in the
spectrum. The physical masses and mixings (in case of SM sector) have been
assumed to be in consistence with the observed values of SM fermions while,
for the exotic sector, they are free parameters defined at the EW scale. So,
our connection to $E_{6}$ is only confined to the choice of gauge group and
choice of fermions at low energy, without considering any effect from high
scale physics creeping in due to renormalization.

Plan of the article is the following. In Section 2, we discuss the symmetry
breaking mechanism and spectra of Higgs bosons after Spontaneous Symmetry
Breaking (SSB). This section will also contain a discussion of gauge boson
masses and fermion Yukawa interaction with the scalars. In the next section we
will discuss in detail the phenomenology of the physical Higgs bosons which
arise in the model after SSB. The decay branching ratios (BR) and production
cross-sections of such Higgs bosons are presented in Section 3 in the context
of 14 and 27 TeV run of the LHC. Finally, we conclude in Section 4 .

##  2 The description of $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{L}\otimes
SU(2)_{R}\otimes U(1)_{R}$ model

In this article, we will be interested in an extension of the SM whose root
can be traced back to the Grand Unified Group $E_{6}$. However at the energy
scale of LHC experiment in which we are interested, we keep all the fermions
and Higgs multiplets excepting few colored scalar bosons, which naturally
arise in the 27-dimensional fundamental representation of $E_{6}$. However, to
keep the number of matter fields in our model to a minimum, we will assume
that the colored scalars are too heavy (of the order of symmetry breaking
scale of $\left[SU(3)\right]^{3}$) to affect the phenomenology at the LHC
energy333If we intend to break the $\left[SU(3)\right]^{3}$ to $31221$ by
embedding the former into a 5-dimensional manifold and applying appropriate
orbifold boundary conditions, one may get rid of such colored scalars by
choosing appropriate boundary conditions on these fields at the orbifold
boundaries. [12]. Higgs multiplets will be instrumental in breaking down
$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{L}\otimes SU(2)_{R}\otimes U(1)_{R}$
to the SM the gauge group. While the complete set of fermions present in
27-plet are necessary for anomaly cancellation. However, this choice of
fermion representations used in our analysis is no way unique. One can have
vanishing anomaly contribution only by considering the left and right-handed
fermion doublets $L_{L}$, $L_{R}$, $Q_{L}$ and $Q_{R}$. Hyper-charge
assignments of the rest of the fermion multiplets listed in Table 1 can then
be done by the consideration of anomaly cancellation among the exotic fermions
and can be done in more than one ways. We have presented two such cases in the
Table 2. It is important to note that in such a situation, $U(1)_{L,R}$
charges for the Higgs multiplets will be different from the case presented in
this analysis. However, we will not consider such a situation and present our
phenomenological analysis with complete set of fermions arising from the
$27$-plet.

Gauge bosons present in this model automatically follows from the gauge group
of our interest. The matter and gauge fields which are present in our model
are listed in Table 1444Electric charge, $Q$ is defined through the relation,
$Q=T_{3L}+T_{3R}+Y_{L}/2+Y_{R}/2$. $L$ and $R$ carry their usual meaning.
These quantum numbers for each multiplet of $32121$ model have been noted in
the Table 1.

|  | $3_{C}$ | $2_{L}$ | $2_{R}$ | $1_{L}$ | $1_{R}$
---|---|---|---|---|---|---
| $L_{L}$ | $1$ | $2$ | $1$ | $-1/6$ | $-1/3$
| $\bar{L}_{R}$ | $1$ | $1$ | $2$ | $1/3$ | $1/6$
| $\bar{L}_{B}$ | $1$ | $2$ | $2$ | $-1/6$ | $1/6$
Fermions | $\bar{L}_{S}$ | $1$ | $1$ | $1$ | $1/3$ | $-1/3$
| $Q_{L}$ | $3$ | $2$ | $1$ | $1/6$ | $0$
| $\bar{Q}_{R}$ | $\bar{3}$ | $1$ | $2$ | $0$ | $-1/6$
| $\bar{Q}_{LS}$ | $\bar{3}$ | $1$ | $1$ | $-1/3$ | $0$
| $Q_{RS}$ | $3$ | $1$ | $1$ | $0$ | $1/3$
| $\Phi_{B}$ | $1$ | $2$ | $2$ | $1/6$ | $-1/6$
Bosons | $\Phi_{L}$ | $1$ | $2$ | $1$ | $1/6$ | $1/3$
| $\Phi_{R}$ | $1$ | $1$ | $2$ | $-1/3$ | $-1/6$
| $\Phi_{S}$ | $1$ | $1$ | $1$ | $-1/3$ | $1/3$
| $G^{i},\;i=1,...,8$ | $8$ | $1$ | $1$ | $0$ | $0$
| $W^{i}_{L},i=1,2,3$ | $1$ | $3$ | $1$ | $0$ | $0$
Gauge bosons | $W^{i}_{R},i=1,2,3$ | $1$ | $1$ | $3$ | $0$ | $0$
| $B_{L}$ | $1$ | $1$ | $1$ | $0$ | $0$
| $B_{R}$ | $1$ | $1$ | $1$ | $0$ | $0$
Table 1: Fermions and Bosons in $32121$ model with their respective quantum numbers | $3_{C}$ | $2_{L}$ | $2_{R}$ | $1_{L}$ | $1_{R}$
---|---|---|---|---|---
$L_{L}$ | $1$ | $2$ | $1$ | $1$ | $-3/2$
$\bar{L}_{R}$ | $1$ | $1$ | $2$ | $-1$ | $3/2$
$Q_{L}$ | $3$ | $2$ | $1$ | $-1/3$ | $1/2$
$\bar{Q}_{R}$ | $\bar{3}$ | $1$ | $2$ | $1/3$ | $-1/2$
$\Phi_{B}$ | $1$ | $2$ | $2$ | $0$ | $0$
$\Phi_{R}$ | $1$ | $1$ | $2$ | $1$ | $-3/2$
$\Phi_{S}$ | $1$ | $1$ | $1$ | $0$ | $0$
$\bar{L}_{B}$ | $1$ | $2$ | $2$ | $0$ | $0$
$\bar{Q}_{LS}$ | $\bar{3}$ | $1$ | $1$ | $1/3$ | $1/3$
$Q_{RS}$ | $3$ | $1$ | $1$ | $-1/3$ | $-1/3$
$\bar{L}_{S}$ | $1$ | $1$ | $1$ | $0$ | $0$
Table 2: Fermions and Scalars in $32121$ model with their respective quantum
numbers. The important thing to note here that, the $U(1)_{L}$ hypercharges of
$Q_{LS}$ and $Q_{RS}$ have to be same to cancel chiral anomaly. Similar for
the case of $U(1)_{R}$ too. But $U(1)_{L}$ and $U(1)_{R}$ hyperchages for each
of them may not be equal. That depends upon the charge assignment of
$Q_{LS},Q_{RS}$. However, we consider $Q_{LS},Q_{RS}$ have $2/3$ charge. So it
is an easy way to choose the $U(1)_{L}$ and $U(1)_{R}$ hypercharges of
$Q_{LS}$ as well as $Q_{RS}$ to be equal i.e., $1/3$.

The first block represents the minimal matter contents in the fermion sector,
i.e., SM-fermions with right-handed neutrino. The scalars in the next block
will provide us a gauge invariant Yukawa Lagrangian generating the masses of
the SM-fermions and Majorana mass for $\nu_{R}$. We can generate the masses of
all the exotic fermions with $\Phi_{S}$.

The third block represents the other beyond Standard Model (BSM) fermions
present in 27-plet and separately cancels chiral anomaly. $L_{B}$ is SU(2)
doublet but U(1) singlet, $L_{S}$ is pure singlet. Here are two cases. For
$Q_{LS}$ as well as $Q_{RS}$ separately,

$(i)$ If $U(1)_{L}$ hypercharge is equal to $U(1)_{R}$ hypercharge (i.e., same
value/sign), $Q_{LS},Q_{RS}$ will be vector-like and they will not contribute
in anomaly cancellation.

$(ii)$ If $U(1)_{L}$ hypercharge is not equal to $U(1)_{R}$ hypercharge (i.e.,
different value/sign), $Q_{LS},Q_{RS}$ will be chiral fermions and will
contribute in anomaly cancellation.

###  2.1 Scalar sector of $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{L}\otimes
SU(2)_{R}\otimes U(1)_{R}$ model

We now briefly discuss the Higgs multiplets responsible for symmetry breaking
and their interactions in this model. The scalar sector of the $32121$ model
contains one Higgs bi-doublet ($\Phi_{B}$), one left-handed ($\Phi_{L}$), one
right-handed ($\Phi_{R}$) weak doublets and a singlet Higgs boson ($\Phi_{S}$)
with non-zero $U(1)$ charges. These scalars arise from the (1, 3,
$\bar{\textbf{3}}$) representation of $\left[SU(3)\right]^{3}$. For a complete
symmetry breaking mechanism from $32121\longrightarrow SU(3)_{C}\otimes
SU(2)_{L}\otimes U(1)_{Y}\longrightarrow SU(3)_{C}\otimes U(1)_{EM}$, the
alignments of Higgs bi-doublet, right (left)-handed doublet and the singlet
will be the following.

$\displaystyle\Phi_{B}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\frac{1}{\sqrt{2}}(k_{1}+h_{1}^{0}+i\xi_{1}^{0})&h_{1}^{+}\\\
h_{2}^{-}&\frac{1}{\sqrt{2}}(k_{2}+h_{2}^{0}+i\xi_{2}^{0})\end{pmatrix},$
$\displaystyle\vspace{1cm}\Phi_{L}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}h_{L}^{+}\\\
\frac{1}{\sqrt{2}}(v_{L}+h_{L}^{0}+i\xi_{L}^{0})\end{pmatrix},\Phi_{R}=\begin{pmatrix}\frac{1}{\sqrt{2}}(v_{R}+h_{R}^{0}+i\xi_{R}^{0})\\\
h_{R}^{-}\end{pmatrix},\Phi_{S}=\frac{1}{\sqrt{2}}(v_{S}+h_{S}^{0}+i\xi_{S}^{0})$
(1)

The Higgs potential, $\cal V$ which obeys symmetry of the gauge group can be
written as sum of two parts ${\cal V}_{1}$ and ${\cal V}_{2}$, where,

$\displaystyle\mathcal{V}_{1}=$ $\displaystyle-$
$\displaystyle\mu_{1}^{2}Tr\left({\Phi_{B}}^{\dagger}\Phi_{B}\right)-\mu_{3}^{2}\left({\Phi_{L}}^{\dagger}\Phi_{L}+{\Phi_{R}}^{\dagger}\Phi_{R}\right)-\mu_{4}^{2}{\Phi_{S}}^{\dagger}\Phi_{S}$
(2) $\displaystyle+$
$\displaystyle\lambda_{1}Tr\left[({\Phi_{B}}^{\dagger}\Phi_{B})^{2}\right]+\lambda_{3}\left(Tr\left[{\Phi_{B}}^{\dagger}\tilde{\Phi}_{B}\right]Tr\left[\tilde{\Phi}_{B}^{\dagger}\Phi_{B}\right]\right)$
$\displaystyle+$
$\displaystyle\alpha_{1}(\Phi_{S}^{\dagger}\Phi_{S})^{2}+\beta_{1}Tr\left[{\Phi_{B}}^{\dagger}\Phi_{B}\right](\Phi_{S}^{\dagger}\Phi_{S})+\gamma_{1}\left[(\Phi_{L}^{\dagger}\Phi_{L})+(\Phi_{R}^{\dagger}\Phi_{R})\right](\Phi_{S}^{\dagger}\Phi_{S})$
$\displaystyle+$
$\displaystyle\rho_{1}\left[(\Phi_{L}^{\dagger}\Phi_{L})^{2}+(\Phi_{R}^{\dagger}\Phi_{R})^{2}\right]+\rho_{3}\left[(\Phi_{L}^{\dagger}\Phi_{L})(\Phi_{R}^{\dagger}\Phi_{R})\right]+c_{1}Tr\left[{\Phi_{B}}^{\dagger}\Phi_{B}\right]\left[(\Phi_{L}^{\dagger}\Phi_{L})+(\Phi_{R}^{\dagger}\Phi_{R})\right]$
$\displaystyle+$ $\displaystyle
c_{3}\left[(\Phi_{L}^{\dagger}\Phi_{B}\Phi_{B}^{\dagger}\Phi_{L})+(\Phi_{R}^{\dagger}\Phi_{B}^{\dagger}\Phi_{B}\Phi_{R})\right]+c_{4}\left[(\Phi_{L}^{\dagger}\tilde{\Phi}_{B}\tilde{\Phi}_{B}^{\dagger}\Phi_{L})+(\Phi_{R}^{\dagger}\tilde{\Phi}_{B}^{\dagger}\tilde{\Phi}_{B}\Phi_{R})\right]$

and,

$\mathcal{V}_{2}=\mu_{BS}Tr\left[{\Phi^{\dagger}_{B}}\tilde{\Phi}_{B}\right]\Phi_{S}^{\ast}+h.c.$
(3)

All the parameters in $\cal V$ are considered to be real excluding any
possibility of CP-violation via the Higgs sector. $\mathcal{V}$ has a symmetry
under $L\longleftrightarrow R$ exchange. In the above,
$\tilde{\Phi}_{B}\equiv\sigma_{2}\Phi_{B}^{*}\sigma_{2}$.

We note that ${\cal V}_{1}$ has a symmetry corresponding to global phase
transformations on the fields

$\Phi_{B}\rightarrow e^{i\theta_{B}}\;\Phi_{B};~{}~{}\Phi_{L}\rightarrow
e^{i\theta_{L}}\;\Phi_{L};~{}~{}\Phi_{R}\rightarrow
e^{i\theta_{R}}\;\Phi_{R}{~{}\rm and}~{}\Phi_{S}\rightarrow
e^{i\theta_{S}}\;\Phi_{S}.$ (4)

However, ${\cal V}_{2}$ which is proportional to $\mu_{BS}$, breaks this
global symmetry explicitly (for example see ref. [13]) otherwise respecting
the symmetries of $32121$ gauge group. This results into appearance of
bilinear terms like $h^{0}_{1}h^{0}_{2}$, $h^{+}_{1}h^{-}_{2}$. However, with
both $k_{1},k_{2}\neq 0$ such bilinear terms also could be generated from the
term proportional to $\lambda_{3}$. Setting one of these vevs to zero
automatically prohibits the appearance of such bilinear terms in the scalar
potential. In other words, setting both $k_{1}$ and $k_{2}$ to their non-zero
values excluding ${\cal V}_{2}$, would break the global symmetry in Eq. 4
spontaneously which results into undesirable extra massless scalar modes. One
can of course have non-zero $k_{1}$ and $k_{2}$ simultaneously, however in
such a case, a tri-linear term proportional to $\mu_{BS}$ is necessary to
break the global symmetry explicitly and thus avoiding the appearance of extra
Goldstone modes.

The kinetic Lagrangian for scalars is,

$\mathcal{L}_{\Phi}=Tr[{(\mathcal{D}_{\mu}\Phi_{B})}^{\dagger}(\mathcal{D}^{\mu}\Phi_{B})]+{(\mathcal{D}_{\mu}\Phi_{L})}^{\dagger}(\mathcal{D}^{\mu}\Phi_{L})+{(\mathcal{D}_{\mu}\Phi_{R})}^{\dagger}(\mathcal{D}^{\mu}\Phi_{R})+{(\mathcal{D}_{\mu}\Phi_{S})}^{\dagger}(\mathcal{D}^{\mu}\Phi_{S})$
(5)

It is needless to mention that covariant derivatives acting on different Higgs
multiplets, are not same and contain appropriate gauge bosons in them.

Minimization conditions we obtain are the following,

$\displaystyle
2\sqrt{2}\mu_{BS}k_{2}v_{S}+k_{1}(2\lambda_{1}k_{1}^{2}+2(\lambda_{1}+2\lambda_{3})k_{2}^{2}-2\mu_{1}^{2}+(c_{1}+c_{4})v_{L}^{2}+(c_{1}+c_{3})v_{R}^{2}+\beta_{1}v_{S}^{2})=0$
(6) $\displaystyle
2\sqrt{2}\mu_{BS}k_{1}v_{S}+k_{2}(2\lambda_{1}k_{2}^{2}+2(\lambda_{1}+2\lambda_{3})k_{1}^{2}-2\mu_{1}^{2}+(c_{1}+c_{3})v_{L}^{2}+(c_{1}+c_{4})v_{R}^{2}+\beta_{1}v_{S}^{2})=0$
(7) $\displaystyle
v_{L}\;[(c_{1}+c_{4})k_{1}^{2}+(c_{1}+c_{3})k_{2}^{2}-2\mu_{3}^{2}+2\rho_{1}v_{L}^{2}+\rho_{3}v_{R}^{2}+\gamma_{1}v_{S}^{2}]=0$
(8) $\displaystyle
v_{R}\;[(c_{1}+c_{3})k_{1}^{2}+(c_{1}+c_{4})k_{2}^{2}-2\mu_{3}^{2}+2\rho_{1}v_{R}^{2}+\rho_{3}v_{L}^{2}+\gamma_{1}v_{S}^{2}]=0$
(9) $\displaystyle
2\sqrt{2}k_{1}k_{2}\mu_{BS}+v_{S}(\beta_{1}(k_{1}^{2}+k_{2}^{2})-2\mu_{4}^{2}+\gamma_{1}(v_{L}^{2}+v_{R}^{2})+2\alpha_{1}v_{S}^{2})=0$
(10)

From Eqs. 6 and 7,

$\displaystyle\mu_{1}^{2}=\dfrac{1}{2k_{1}}\left(2\sqrt{2}\mu_{BS}k_{2}v_{S}+k_{1}(2k_{1}^{2}\lambda_{1}+2(\lambda_{1}+2\lambda_{3})k_{2}^{2}+(c_{1}+c_{4})v_{L}^{2}+(c_{1}+c_{3})v_{R}^{2}+\beta_{1}v_{S}^{2})\right)$
(11)
$\displaystyle\mu_{1}^{2}=\dfrac{1}{2k_{2}}\left(2\sqrt{2}\mu_{BS}k_{1}v_{S}+k_{2}(2k_{2}^{2}\lambda_{1}+2(\lambda_{1}+2\lambda_{3})k_{1}^{2}+(c_{1}+c_{3})v_{L}^{2}+(c_{1}+c_{4})v_{R}^{2}+\beta_{1}v_{S}^{2})\right)$
(12)

Using Eqs. 11 and 12 for $k_{1},k_{2}\neq 0$, we have,

$\mu_{BS}=\dfrac{1}{\sqrt{2}v_{S}}\dfrac{k_{1}k_{2}}{k_{2}^{2}-k_{1}^{2}}\left((c_{3}-c_{4})\dfrac{v_{L}^{2}-v_{R}^{2}}{2}-2\lambda_{3}(k_{2}^{2}-k_{1}^{2})\right)$
(13)

Spontaneous breaking of Left-Right symmetry demands, $v_{R}\neq 0$. Thus Eq.
(9) results into,

$\mu_{3}^{2}=\frac{1}{2}[(c_{1}+c_{3})k_{1}^{2}+\rho_{3}v_{L}^{2}+2\rho_{1}v_{R}^{2}+\gamma_{1}v_{S}^{2}].$
(14)

The choice $v_{L}\neq 0$ leads to appearance of an extra massless scalar mode
which is undesirable555This is related to the spontaneous breakdown of a
global symmetry of the defined in Eq. 4. scalar potential. So we stick to the
case with $v_{L}=0$.

To break the extra $U(1)$ we have to opt for $v_{S}\neq 0$ resulting into
(from Eq. 10),

$\mu_{4}^{2}=\dfrac{1}{2v_{S}}\left(2\sqrt{2}k_{1}k_{2}\mu_{BS}+v_{S}(\beta_{1}(k_{1}^{2}+k_{2}^{2})+\gamma_{1}(v_{L}^{2}+v_{R}^{2})+2\alpha_{1}v_{S}^{2})\right)$
(15)

Once we fix the minimisation conditions of the Higgs potential, we are ready
to note the Higgs mass matrices under such alignment of vacuum. However before
delving into the details of scalar mass matrices let us make some brief
comments on an important issue related to the minimum of the scalar potential.
It is important to note that a scalar potential such as Eq. 2 depending on so
many fields may have more than one minima having varying depths. Thus merely
satisfying the minimisation condition (by scalar potential parameters) does
not ascertain that one is at the deepest minimum of the potential. In
principle, different choices of the scalar potential parameters correspond to
minima of different depths. Moreover, radiative corrections can significantly
change to the structure of scalar potential and consequently change the depths
of different minima of the potential. Hence, it is expected that one must at
least incorporate one-loop corrections to the scalar potential to before
looking for the deepest minima. However the exercise of calculating an
effective potential at one loop for our model is beyond the scope of the
present analysis. So we stick to the tree level potential and have not tried
to look for its deepest minima. As long as the tunnelling time from the false
vacuum to the true (deepest) vacuum is larger than the lifetime of the
Universe, sitting at a minimum other than the deepest one is not always
hazardous. However a realistic estimation of this tunnelling time also
requires a one loop corrected effective potential of our model. A recent study
[14] has been devoted to the analysis of the tree level scalar potential with
particular emphasis on vacuum alignment and structure of minima of the
potential. We would like to note that alignment of the vacuum used in our
analysis satisfies the criterion of a good vacuum a la [14].

In the following, we note the CP-even, CP-odd and charged scalar mass matrices
after replacing $\mu_{1}$, $\mu_{BS}$, $\mu_{3}$ and $\mu_{4}$ using Eqs. 11,
13, 14 and 15 respectively.

In a basis, defined by the fields
$\left\\{h_{1}^{0},h_{2}^{0},h_{L}^{0},h_{R}^{0},h_{S}^{0}\right\\}$ the
square of CP-even mass matrix (${M_{r}^{0}}^{2}$) is,

$\displaystyle\begin{pmatrix}2\lambda_{1}k_{1}^{2}+k_{2}^{2}\Delta^{\prime}&2\lambda_{1}k_{1}k_{2}+k_{1}k_{2}\Delta^{\prime}&0&(c_{1}+c_{3})k_{1}v_{R}&\beta_{1}k_{1}v_{S}-\dfrac{k_{1}k_{2}^{2}\Delta^{\prime}}{v_{S}}\\\
2\lambda_{1}k_{1}k_{2}+k_{1}k_{2}\Delta^{\prime}&2\lambda_{1}k_{2}^{2}+k_{1}^{2}\Delta^{\prime}&0&(c_{1}+c_{4})k_{2}v_{R}&\beta_{1}k_{2}v_{S}-\dfrac{k_{1}^{2}k_{2}\Delta^{\prime}}{v_{S}}\\\
0&0&\dfrac{1}{2}\left((c_{4}-c_{3})k_{-}^{2}+(\rho_{3}-2\rho_{1})v_{R}^{2}\right)&0&0\\\
(c_{1}+c_{3})k_{1}v_{R}&(c_{1}+c_{4})k_{2}v_{R}&0&2\rho_{1}v_{R}^{2}&\gamma_{1}v_{R}v_{S}\\\
\beta_{1}k_{1}v_{S}-\dfrac{k_{1}k_{2}^{2}\Delta^{\prime}}{v_{S}}&\beta_{1}k_{2}v_{S}-\dfrac{k_{1}^{2}k_{2}\Delta^{\prime}}{v_{S}}&0&\gamma_{1}v_{R}v_{S}&2\alpha_{1}v_{S}^{2}+\dfrac{k_{1}^{2}k_{2}^{2}\Delta^{\prime}}{v_{S}^{2}}\end{pmatrix}$
(16)

where, $k_{\pm}^{2}=k_{1}^{2}\pm k_{2}^{2}$ and
$\Delta^{\prime}=\dfrac{(4\lambda_{3}k_{-}^{2}+(c_{4}-c_{3})v_{R}^{2})}{2k_{-}^{2}}$.

While, the square of CP-odd mass matrix (${M_{i}^{0}}^{2}$) in
$\left\\{\xi_{1}^{0},\xi_{2}^{0},\xi_{L}^{0},\xi_{R}^{0},\xi_{S}^{0}\right\\}$
basis is,

$\displaystyle\begin{pmatrix}k_{2}^{2}\Delta^{\prime}&k_{1}k_{2}\Delta^{\prime}&0&0&\dfrac{k_{1}k_{2}^{2}\Delta^{\prime}}{v_{S}}\\\
k_{1}k_{2}\Delta^{\prime}&k_{1}^{2}\Delta^{\prime}&0&0&\dfrac{k_{1}^{2}k_{2}\Delta^{\prime}}{v_{S}}\\\
0&0&\dfrac{1}{2}[(c_{4}-c_{3})(k_{1}^{2}-k_{2}^{2})+(\rho_{3}-2\rho_{1})v_{R}^{2}]&0&0\\\
0&0&0&0&0\\\
\dfrac{k_{1}k_{2}^{2}\Delta^{\prime}}{v_{S}}&\dfrac{k_{1}^{2}k_{2}\Delta^{\prime}}{v_{S}}&0&0&\dfrac{k_{1}^{2}k_{2}^{2}\Delta^{\prime}}{v_{S}^{2}}\end{pmatrix}$
(17)

Once we diagonalise the above matrix, the three zero eigenvalues of CP-odd
mass matrix corresponds to three Goldstone bosons responsible for giving
masses to heavy neutral gauge bosons.

Square of the charged scalar mass matrix (${M^{\pm}}^{2}$), in the basis
$\left\\{h_{1}^{+},h_{2}^{+},h_{L}^{+},h_{R}^{+}\right\\}$ is the following,

$\displaystyle\begin{pmatrix}\dfrac{(c_{4}-c_{3})k_{1}^{2}v_{R}^{2}}{2k_{-}^{2}}&\dfrac{(c_{4}-c_{3})k_{1}k_{2}v_{R}^{2}}{2k_{-}^{2}}&0&\dfrac{1}{2}(c_{3}-c_{4})k_{1}v_{R}\\\
\dfrac{(c_{4}-c_{3})k_{1}k_{2}v_{R}^{2}}{2k_{-}^{2}}&\dfrac{(c_{4}-c_{3})k_{2}^{2}v_{R}^{2}}{2k_{-}^{2}}&0&\dfrac{1}{2}(c_{3}-c_{4})k_{2}v_{R}&\\\
0&0&\dfrac{1}{2}(\rho_{3}-2\rho_{1})v_{R}^{2}&0\\\
\dfrac{1}{2}(c_{3}-c_{4})k_{1}v_{R}&\dfrac{1}{2}(c_{3}-c_{4})k_{2}v_{R}&0&\dfrac{1}{2}(c_{4}-c_{3})k_{-}^{2}\end{pmatrix}$
(18)

Diagonalisation of the above matrix gives us two massive charged scalars and
two massless Goldstones corresponding to a couple of heavy charged gauge
bosons.

We note that, with non-zero $k_{1}$ and $k_{2}$, $W$ (and $Z$)-masses get
contribution proportional to $\left(k_{1}^{2}+k_{2}^{2}\right)^{1\over 2}$
while $W_{L}-W_{R}$ mixing is proportional to $\frac{k_{1}k_{2}}{v_{R}^{2}}$
(see Eq. 27). Experimental limit on the $W_{L}-W_{R}$ mixing [15] forces one
to choose any one of these vevs to be very small666e.g. If we set the mixing
angle of $W_{L}$ \- $W_{R}$ at its maximum allowed value, $k_{2}$ will be of
the order of $0.27~{}\rm GeV$, assuming $k_{1}>k_{2}$. compared to other
keeping $\left(k_{1}^{2}+k_{2}^{2}\right)^{1\over 2}$ fixed at 246 GeV. One
can then safely assume $k_{1}^{2}+k_{2}^{2}\approx k_{1}^{2}-k_{2}^{2}\approx
k_{1}^{2}$.

Thus in small $k_{2}$ limit, we can rewrite the scalar mass matrices as,

$\displaystyle{M_{r}^{0}}^{2}=\begin{pmatrix}2\lambda_{1}k_{1}^{2}&0&0&(c_{1}+c_{3})k_{1}v_{R}&\beta_{1}k_{1}v_{S}\\\
0&\frac{1}{2}[4\lambda_{3}k_{1}^{2}+(c_{4}-c_{3})v_{R}^{2}]&0&0&0\\\
0&0&\frac{1}{2}[(c_{4}-c_{3})k_{1}^{2}+(\rho_{3}-2\rho_{1})v_{R}^{2}]&0&0\\\
(c_{1}+c_{3})k_{1}v_{R}&0&0&2\rho_{1}v_{R}^{2}&\gamma_{1}v_{R}v_{S}\\\
\beta_{1}k_{1}v_{S}&0&0&\gamma_{1}v_{R}v_{S}&2\alpha_{1}v_{S}^{2}\end{pmatrix}$
(19)

$\displaystyle{M_{i}^{0}}^{2}=\begin{pmatrix}0&0&0&0&0\\\
0&\frac{1}{2}[4\lambda_{3}k_{1}^{2}+(c_{4}-c_{3})v_{R}^{2}]&0&0&0\\\
0&0&\frac{1}{2}[(c_{4}-c_{3})k_{1}^{2}+(\rho_{3}-2\rho_{1})v_{R}^{2}]&0&0\\\
0&0&0&0&0\\\ 0&0&0&0&0\end{pmatrix}$ (20)

$\displaystyle{M^{\pm}}^{2}=\begin{pmatrix}\frac{1}{2}(c_{4}-c_{3})v_{R}^{2}&0&0&\frac{1}{2}(c_{3}-c_{4})k_{1}v_{R}\\\
0&0&0&0\\\ 0&0&\frac{1}{2}(\rho_{3}-2\rho_{1})v_{R}^{2}&0\\\
\frac{1}{2}(c_{3}-c_{4})k_{1}v_{R}&0&0&\frac{1}{2}(c_{4}-c_{3})k_{1}^{2}\end{pmatrix}$
(21)

It is also evident that inspite of setting $k_{2}=0$ and as a consequence
$\mu_{BS}=0$, the elements of scalar mass matrices, mass eigenvalues and
mixing matrices practically remains the same as before. It is easy to verify
that in the limit $v_{R},v_{S}>>k_{1}>>k_{2}$ (the first inequality arises
from the experimental lower limits on heavy gauge boson masses, discussed in a
following section), mass matrix defined in Eq. 16 will practically produce the
same eigenvalues and mixing among the scalars as has been resulted from Eq.
19. Similarly Eqs. 17 and 18 will generate same masses and mixings as Eqs. 20
and 21 will do respectively.

As mentioned above, the masses and mixings among the scalars in $k_{2}\neq
0,~{}\mu_{BS}\neq 0$ case are nearly same as the $k_{2}=0$ case, Although, a
non-zero $k_{2}$ would result into some new couplings among the scalars which
are not present in the later case. Some new decay channels will open up for
the scalars like $h^{0}_{2}$ and $H^{0}_{S}$. However, these new decay modes
will not affect the decay patterns of the physical scalars in a significant
way as the decay rates will be proportional to $k_{2}^{2}$. We will not
discuss them any further. All the following analysis will be done in $k_{2}=0$
limit which could be viewed as some special but not phenomenologically
different from the more general situation with both $k_{1}$ and $k_{2}$ set to
non-zero values.

The scalar potential has $10$ real parameters, $\lambda_{1}$, $\lambda_{3}$,
$\rho_{1}$, $\rho_{3}$, $c_{1}$, $c_{3}$, $c_{4}$, $\alpha_{1}$, $\beta_{1}$
and $\gamma_{1}$. In order to find a set of acceptable values of the physical
Higgs boson masses and the potential to be stable at least at classical level,
the parameters of scalar potential must obey the following conditions.

$\lambda_{1},~{}(\lambda_{1}+2\lambda_{3}),~{}\rho_{1},~{}\rho_{3},~{}(c_{1}+c_{3}),~{}(c_{1}+c_{4}),~{}\alpha_{1},~{}\beta_{1},~{}\gamma_{1}~{}>~{}0$
(22)

The condition that the physical charged Higgs mass squares be positive,
demands

$c_{4}-c_{3}>0{~{}\rm and}~{}\rho_{3}-2\rho_{1}>0$

Values of $(c_{4}-c_{3})$ and $(\rho_{3}-2\rho_{1})$ can be constrained from a
model independent experimental limit of charged Higgs boson mass.

From the CP-even scalar mass matrix we notice that it is effectively a
$3\times 3$ mass matrix in $\\{h_{1}^{0},h_{R}^{0},h_{S}^{0}\\}$ basis and
thus difficult to diagonalise analytically. However, one linear combination of
$h_{1}^{0},h_{R}^{0}$ and $h_{S}^{0}$ will be definitely like the SM Higgs
boson with mass 125 GeV and having similar properties with this.

$\displaystyle{M_{r}^{0}}^{2}_{3\times
3}=\begin{pmatrix}2\lambda_{1}k_{1}^{2}&(c_{1}+c_{3})k_{1}v_{R}&\beta_{1}k_{1}v_{S}\\\
(c_{1}+c_{3})k_{1}v_{R}&2\rho_{1}v_{R}^{2}&\gamma_{1}v_{R}v_{S}\\\
\beta_{1}k_{1}v_{S}&\gamma_{1}v_{R}v_{S}&2\alpha_{1}v_{S}^{2}\end{pmatrix}$
(23)

We will denote the eigenstates of mass matrix (Eq. 23) by
$h^{0},H_{R}^{0},H_{S}^{0}$. The rest of the two massive CP-even and two
massive CP-odd scalars do not mix with others, and we shall use the same
notation to identify the mass eigenstates as we have used to define gauge
eigenstates. For the charged Higgs sector, the two massive eigenstates will be
denoted by $H_{1}^{\pm}$ (which is a linear combinations of $h_{1}^{\pm}$ and
$h_{R}^{\pm}$) and $H_{L}^{\pm}$.

The $3\times 3$ block of neutral CP-even mass matrix (see mass matrix (23)),
can be diagonalised numerically. We must keep in mind that one of the
eigenstates must correspond to the SM-like Higgs boson $h^{0}$. This implies
that the mass eigenvalue and the corresponding eigenvector must be consistent
with the measured value of SM Higgs boson mass and its signal strengths to
different decay channels at the LHC. We have done a scan over the parameters
of the mass matrix (Eq. 23) over a range keeping the values of $k_{1}$,
$v_{R}$ and $v_{S}$ fixed. We will see in the next section that the value of
$k_{1}$ is fixed from the $W$-boson mass, while a lower limit on the values of
$v_{R}$ and $v_{S}$ can be obtained from the consideration of the masses of
heavy gauge bosons $W_{R}$ and $A^{\prime}$ arising in this model. While
scanning over the parameters we have set the values of $v_{R}$ and $v_{S}$ at
their lower limits of 14.7 TeV and 13 TeV respectively.

The result of the scan is presented in Fig. 1. For the points in the plot,
value of one of the eigenstates satisfies the SM Higgs mass condition and
Higgs signal strength to $b\bar{b}$ decay mode [15]. It can be seen from the
plot, that relatively larger values of the parameters controlling the off-
diagonal terms of the mass matrix are possible. This in turn implies the
eigenstates (particularly the one which can be identified with $h^{0}$) are
linear combinations all three gauge states
$\\{h_{1}^{0},h_{R}^{0},h_{S}^{0}\\}$. while performing this scan over a large
range of parameter space, we have observed that in most cases it keeps
$\lambda_{1}$ more or less fixed close to the value
$\frac{m_{h^{0}}^{2}}{2k_{1}^{2}}$. But the values of $\rho_{1}$ and
$\alpha_{1}$ are completely unconstrained. Instead of diagonalising the mass
matrix numerically we have restricted ourselves to the values of $c_{1}+c_{3}$
and $\gamma_{1}$ such that the corresponding off-diagonal terms in the mass
matrix can be neglected with respect to the diagonal terms. In this limit,
large values of $\beta_{1}$ forces one to accept large values of $\alpha_{1}$
so that SM Higgs signal strengths as calculated from the model is in agreement
with experimental observation. Furthermore, we keep a tiny value for
$\beta_{1}$ consistent with the above scan result. Under such assumptions
about the values of these parameters, mass (squared) eigenvalues can be
approximated by the following expressions,

$\displaystyle m_{h^{0}}^{2}$ $\displaystyle=$
$\displaystyle\lambda_{1}k_{1}^{2}+\alpha_{1}v_{S}^{2}-\sqrt{\Delta}$
$\displaystyle m_{H_{R}^{0}}^{2}$ $\displaystyle\simeq$ $\displaystyle
2\rho_{1}v_{R}^{2}$ $\displaystyle m_{H_{S}^{0}}^{2}$ $\displaystyle=$
$\displaystyle\lambda_{1}k_{1}^{2}+\alpha_{1}v_{S}^{2}+\sqrt{\Delta}$ (24)

with the eigenstate corresponding to the eigenvalue $m_{h^{0}}^{2}$ will be
identified with the SM-like Higgs boson with mass 125 GeV. Here,
$\Delta={(\alpha_{1}v_{S}^{2}-\lambda_{1}k_{1}^{2})}^{2}+\beta_{1}^{2}k_{1}^{2}v_{S}^{2}$.
Mixing angle, $\theta$, (operative between $h^{0}$ and $H_{S}^{0}$) in small
$\beta_{1}$ limit, can be written as,

$\tan(2\theta)=\dfrac{\beta_{1}k_{1}v_{S}}{\alpha_{1}v_{S}^{2}-\lambda_{1}k_{1}^{2}}$
(25)

Figure 1: Allowed parameter space for ($c_{1}+c_{3}$) and $\gamma_{1}$ for
some fixed values of $\beta_{1}$. The side bar represents the values of
$\beta_{1}$ in $log_{10}$ scale.

We will be mainly interested in a study considering in the above mentioned
simplified version of the parameter space where we have only considered that
$H^{0}_{S}$ has a tiny mixing (proportional to $\beta_{1}$) with SM-like Higgs
boson $h^{0}$. At the end, we will make comment about the possible outcome of
a study with non-negligible values of $c_{1}+c_{3}$ and $\gamma_{1}$.

###  2.2 Gauge sector of $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{L}\otimes
SU(2)_{R}\otimes U(1)_{R}$ model

The gauge sector of $32121$ model consists of $16$ gauge bosons namely, the
eight gluons ($G^{a^{\prime}}$, $a^{\prime}=1,...,8)$, $SU(2)_{L,R}$ gauge
bosons, $W_{L}^{a},W_{R}^{a},(a=1,2,3)$ and two $U(1)$ gauge bosons $B_{L}$
and $B_{R}$. Their interactions are governed by 5 gauge coupling constants
$g_{3}$, $g_{2L}$, $g_{2R}$, $g_{1L}$ and $g_{1R}$.

The gauge kinetic Lagrangian can be expressed in terms of field strength
tensor in usual way,

$\mathcal{L}_{GK}=-\frac{1}{4}G^{a^{\prime}\mu\nu}G^{a^{\prime}}_{\mu\nu}-\frac{1}{4}W_{L}^{a\mu\nu}W^{a}_{L\mu\nu}-\frac{1}{4}W_{R}^{a\mu\nu}W^{a}_{R\mu\nu}-\frac{1}{4}B_{L}^{\mu\nu}B_{L\mu\nu}-\frac{1}{4}B_{R}^{\mu\nu}B_{R\mu\nu}-\frac{\epsilon}{2}B_{L}^{\mu\nu}B_{R\mu\nu}$
(26)

The last term in Eq. (26) represents the $U(1)_{L,R}$ kinetic mixing
proportional to a dimensionless parameter $\epsilon$. A non-zero value of the
kinetic mixing coefficient $\epsilon$ would modify the extra heavy neutral
gauge boson coupling to a pair of fermions [16]. However, focus of our present
study is not in that direction and we will use $\epsilon=0$ in our following
analysis.

The charged gauge bosons mass-matrix (in $W_{L}-W_{R}$ basis) follows from the
Higgs kinetic Lagrangian (Eq. (5)) in $k_{1},k_{2}\neq 0$ limit:

$\displaystyle
M_{W^{\pm}}^{2}=\frac{1}{4}\begin{pmatrix}g_{2L}^{2}(k_{1}^{2}+k_{2}^{2})&-2g_{2L}g_{2R}k_{1}k_{2}\\\
-2g_{2L}g_{2R}k_{1}k_{2}&g_{2R}^{2}(k_{1}^{2}+k_{2}^{2}+v_{R}^{2})\end{pmatrix}$
(27)

which in small $k_{2}$ limit appears as,

$\displaystyle
M_{W^{\pm}}^{2}=\frac{1}{4}\begin{pmatrix}g_{2L}^{2}k_{1}^{2}&0\\\
0&g_{2R}^{2}(k_{1}^{2}+v_{R}^{2})\end{pmatrix}$ (28)

Eigenvalues of the already diagonalised mass matrix provide $W_{L}$ and
$W_{R}$ masses. The experimentally measured value of $W_{L}$ mass fixes
$k_{1}$ at 246 GeV, if we set at $g_{2L}=e/\sin\theta_{W}$ where, $\theta_{W}$
is the Weinberg angle. Throughout our article, we shall denote $W_{L}$ as the
SM $W$ boson with a mass $80.379$ GeV [15]. Experimentally measured value of
$W$ mass would fix the value of $k_{1}$ and an experimental lower limit on
$W_{R}$ mass [17] provides a lower bound on $v_{R}(>14.7$ TeV).

One can similarly obtain the mass matrix for neutral gauge bosons in
$W_{3L},W_{3R},B_{L},B_{R}$ basis, with $k_{2}\neq 0$, $M_{NG}^{2}$:

$\displaystyle\frac{1}{2}\begin{pmatrix}\dfrac{1}{2}g_{2L}^{2}k_{+}^{2}&-\frac{1}{2}g_{2L}g_{2R}k_{+}^{2}&\frac{1}{6}g_{1L}g_{2L}k_{-}^{2}&-\frac{1}{6}g_{1R}g_{2L}k_{-}^{2}\\\
-\frac{1}{2}g_{2L}g_{2R}k_{+}^{2}&\frac{1}{2}g_{2R}^{2}(k_{+}^{2}+v_{R}^{2})&-\frac{1}{3}g_{1L}g_{2R}(\frac{1}{2}k_{-}^{2}+v_{R}^{2})&\frac{1}{6}g_{1R}g_{2R}(k_{-}^{2}-v_{R}^{2})\\\
\frac{1}{6}g_{1L}g_{2L}k_{-}^{2}&-\frac{1}{3}g_{1L}g_{2R}(\frac{1}{2}k_{-}^{2}+v_{R}^{2})&g_{1L}^{2}(\frac{1}{18}k_{+}^{2}+\frac{2}{9}v_{R}^{2}+\frac{2}{9}v_{S}^{2})&g_{1L}g_{1R}(-\frac{1}{18}k_{+}^{2}+\frac{1}{9}v_{R}^{2}-\frac{2}{9}v_{S}^{2})\\\
-\frac{1}{6}g_{1R}g_{2L}k_{-}^{2}&\frac{1}{6}g_{1R}g_{2R}(k_{-}^{2}-v_{R}^{2})&g_{1L}g_{1R}(-\frac{1}{18}k_{+}^{2}+\frac{1}{9}v_{R}^{2}-\frac{2}{9}v_{S}^{2})&g_{1R}^{2}(\frac{1}{18}k_{+}^{2}+\frac{1}{18}v_{R}^{2}+\frac{2}{9}v_{S}^{2})\end{pmatrix}$
(29)

which in small $k_{2}$ scenario practically becomes,

$\displaystyle
M_{NG}^{2}=\frac{1}{2}\begin{pmatrix}\frac{1}{2}g_{2L}^{2}k_{1}^{2}&-\frac{1}{2}g_{2L}g_{2R}k_{1}^{2}&\frac{1}{6}g_{1L}g_{2L}k_{1}^{2}&-\frac{1}{6}g_{1R}g_{2L}k_{1}^{2}\\\
-\frac{1}{2}g_{2L}g_{2R}k_{1}^{2}&\frac{1}{2}g_{2R}^{2}(k_{1}^{2}+v_{R}^{2})&-\frac{1}{3}g_{1L}g_{2R}(\frac{1}{2}k_{1}^{2}+v_{R}^{2})&\frac{1}{6}g_{1R}g_{2R}(k_{1}^{2}-v_{R}^{2})\\\
\frac{1}{6}g_{1L}g_{2L}k_{1}^{2}&-\frac{1}{3}g_{1L}g_{2R}(\frac{1}{2}k_{1}^{2}+v_{R}^{2})&g_{1L}^{2}(\frac{1}{18}k_{1}^{2}+\frac{2}{9}v_{R}^{2}+\frac{2}{9}v_{S}^{2})&g_{1L}g_{1R}(-\frac{1}{18}k_{1}^{2}+\frac{1}{9}v_{R}^{2}-\frac{2}{9}v_{S}^{2})\\\
-\frac{1}{6}g_{1R}g_{2L}k_{1}^{2}&\frac{1}{6}g_{1R}g_{2R}(k_{1}^{2}-v_{R}^{2})&g_{1L}g_{1R}(-\frac{1}{18}k_{1}^{2}+\frac{1}{9}v_{R}^{2}-\frac{2}{9}v_{S}^{2})&g_{1R}^{2}(\frac{1}{18}k_{1}^{2}+\frac{1}{18}v_{R}^{2}+\frac{2}{9}v_{S}^{2})\end{pmatrix}$
(30)

In practical, the presence of this small $k_{2}$ will not sensitively affect
the masses and mixings in the neutral gauge sector .

Before we make predictions about the masses of the neutral gauge bosons, let
us make further assumption about the four gauge coupling constants. We will
identify the $SU(2)_{L}$ of $32121$ with the weak isospin group of the
Standard Model. It follows automatically that $U(1)_{Y}$ of the SM will arise
due to breaking of $SU(2)_{R}\otimes U(1)_{L}\otimes U(1)_{R}$. Consequently,
one can identify $g_{Y}$ (the $U(1)_{Y}$ gauge coupling constant with
$g_{Y}=e/\cos\theta_{W}$) of the SM via the following relation: .

$\dfrac{1}{g_{Y}^{2}}=\dfrac{1}{g_{2R}^{2}}+\dfrac{1}{g_{1L}^{2}}+\dfrac{1}{g_{1R}^{2}}$
(31)

Above relation among the gauge couplings allows us to choose any two of
$g_{2R},g_{1L}$ and $g_{1R}$ independently. In order to keep our Lagrangian
manifestly LR symmetric, we assume $g_{2L}=g_{2R}$ and $g_{1L}=g_{1R}$. All
our analysis presented in the following will be based on this assumption.

To completely determine the gauge boson masses we need to know the values of
the gauge coupling constants and three non-zero vacuum expectation values
(vevs). The gauge coupling constants have been already fixed from the symmetry
breaking condition and demand of manifest Left-Right symmetry. The value or
allowed range of values of $v_{S}$ remains to be known for evaluation of the
gauge boson masses from Eqs. (28), (30). It is to be noted that $v_{S}$ plays
a crucial role in breaking $U(1)_{L}\otimes U(1)_{R}$. The tree level relation
among $m_{Z}$, $m_{W}$ and $\cos\theta_{W}$ has been ensured by identifying
the massless eigenstate (of $M^{2}_{NG}$) with the photon, which has equal
coupling to left- and right-chiral fermions.

We have implemented the model Lagrangian in SARAH [19] as well as in FeynRules
[20]. In the following analysis all the cross-sections will be calculated with
help of Madgraph5(v2.6.6) [21] using NNPDF23NLO parton distribution functions
[22] with the factorisation scale set equal to the average mass of the final
state particles.

Upon diagonalisation, one of the eigenvalues of Eq. (30) will give a zero
eigenvalue corresponding to the photon. Another eigenvalue comes out to be
nearly equal to 91.2 GeV, which we identify with the $Z-$boson. Other two
eigenvalues correspond to two heavy neutral gauge bosons which we identify as
$Z^{\prime}$ and $A^{\prime}$, the last one being a hall mark of an extra
$U(1)$ gauge symmetry.

Figure 2: Weak dependence of $Z$ mass on $v_{S}$

Fig. 2 shows the weak dependence of $Z$ mass on $v_{S}$ whereas, Fig. 3
reveals a strong correlation between the mass of $A^{\prime}$ with $v_{S}$.
Taking a hint from this fact, we would like to find an allowed range of
$v_{S}$ from the LHC data itself. In such an effort, an experimental search of
a heavy neutral gauge boson at the LHC and its subsequent decay to a pair of
leptons would be helpful. ATLAS collaboration at the LHC [18] has looked for a
pair of high $p_{T}$ leptons ($e$ and $\mu$) to put an upper limit on the
production cross-section times the branching ratio of a heavy neutral gauge
boson at 13 TeV. We have translated this upper limit on the $\sigma\times BR$
to the mass of $A^{\prime}$. In our model $A^{\prime}$ couples to both quarks
and leptons with couplings proportional to their $U(1)_{L,R}$ charges. We
present the $\sigma\times BR$ of $A^{\prime}$ in Fig. 4. The black solid and
dashed lines represent the observed and expected 95% C.L. upper limit on
cross-section times branching ratio by ATLAS respectively. While the blue
solid line gives the $\sigma\times BR$ of $A^{\prime}$ in $32121$ model as a
function of $A^{\prime}$ mass. One can find a lower limit on $A^{\prime}$ mass
equals to 3.5 TeV.

Figure 3: Dependence of $A^{\prime}$ mass on $v_{S}$ (left panel) and on
$g_{1L}$ and $g_{1R}$, with $g_{1L}=g_{1R}$ for a fixed $v_{S}$ (right panel).

Knowledge of a lower limit on $A^{\prime}$ mass enables one to get a lower
limit on $v_{S}$. $m_{A^{\prime}}$ is a function of the gauge coupling
constants and three non-zero vevs necessary for symmetry breaking. Mass of
$A^{\prime}$ has a very weak dependence on $k_{1}$ and $v_{R}$ in comparison
to $v_{S}$. Values of the gauge couplings and $k_{1}$ are fixed. And we set
value of $v_{R}$ at its lower limit while obtaining a lower limit on $v_{S}$.
We thus arrive a lower limit on $v_{S}$ which equals to 12.61 TeV.
$m_{A^{\prime}}$ is a slowly increasing function of $v_{R}$. So one cannot
arrive at an absolute lower limit on $v_{S}$. The allowed region of
$v_{R}-v_{S}$ space has been presented in Fig. 5.

Figure 4: Production cross-section, $\sigma\times BR$ plot for a heavy neutral
gauge boson production at the LHC. The black solid (dashed) line represents
the observed (expected) 95% C.L. upper limit on $\sigma\times BR$ from ATLAS
(with $\sqrt{s}=13$ TeV, $36.1$ fb-1) considering dilepton decay channel of
the produced gauge boson and the blue line corresponds to the prediction of
$32121$ model.

$Z^{\prime}$ mass on the other hand, is mainly controlled by $v_{R}$ and it
has a weak dependence on $v_{S}$. With $v_{R}$ at its lower limit,
$Z^{\prime}$ mass comes out to be 5.9 TeV. For such a massive $Z^{\prime}$,
cross-section times its branching ratio to a pair of leptons is of the order
of $10^{-3}$ fb. This rate is well below the upper limit of cross-section
times BR for an heavy neutral gauge boson by ATLAS collaboration [18] and
presented in Fig. 4.

Figure 5: Allowed region in $v_{R}$ \- $v_{S}$ space, obtained from the limits
on $A^{\prime}$ mass. $m_{A^{\prime}}$ has a strong dependence on $v_{S}$ and
have a milder dependence on $v_{R}$.

###  2.3 Fermion sector of $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{L}\otimes
SU(2)_{R}\otimes U(1)_{R}$ model

The gauge quantum numbers of the fermions have been already listed in Table 1.
In the following we note the fermions with their chiral components.

$\displaystyle L_{L}$ $\displaystyle=$ $\displaystyle\begin{pmatrix}\nu_{L}\\\
e_{L}\end{pmatrix},\hskip 34.14322pt\hskip
36.98866ptL_{R}=\begin{pmatrix}\nu_{R}\\\ e_{R}\end{pmatrix}$ $\displaystyle
Q_{L}$ $\displaystyle=$ $\displaystyle\begin{pmatrix}u_{L}\\\
d_{L}\end{pmatrix},\hskip 68.28644ptQ_{R}=\begin{pmatrix}u_{R}\\\
d_{R}\end{pmatrix}$ $\displaystyle Q_{LS}$ $\displaystyle=$ $\displaystyle
q_{SL},\hskip 14.22636ptQ_{RS}=q_{SR},\hskip 14.22636ptL_{S}=l_{S}\hskip
14.22636pt\mbox{and,}$ $\displaystyle L_{B}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}N_{1}&E_{1}\\\ E_{2}&N_{2}\end{pmatrix}\hskip
8.5359pt\mbox{with}\hskip
19.91684pt\bar{L}_{B}=\begin{pmatrix}\bar{N}_{1}&\bar{E}_{2}\\\
\bar{E}_{1}&\bar{N}_{2}\end{pmatrix}$ (32)

Here, $L_{L},L_{R},Q_{L},Q_{R}$ represent SM fermions along with a right-
handed neutrino ($\nu_{R}$). $Q_{LS}$ and $Q_{RS}$ are color triplets and
$SU(2)$ gauge singlet exotic quarks having $U(1)_{L}$ and $U(1)_{R}$ hyper-
charges respectively. They together form a $4$-component Dirac spinor $q_{S}$.
$N_{1}$ and $N_{2}$ are neutral heavy leptons while $E_{1}$ and $E_{2}$ are
singly charged heavy leptons. They pair-wise form 4-component Dirac spinors,
$N$ and $E$ respectively. $l_{S}$ is a neutral exotic singlet fermion but
carrying $U(1)_{L}$ and $U(1)_{R}$ gauge quantum numbers.

The fermionic sector of this model consists of several heavy leptons and
quarks apart from theirs SM counterparts. We would like to spend few words on
them. The presence of $\nu_{R}$ facilitates us to write a Dirac or Majorana
mass for the neutrinos [6], [23]777We have only noted down a possible Dirac
mass term for the neutrinos in Eq. (33). Heavy charged lepton $E^{\pm}$ and
heavy neutrino $N$ arise from the $SU(2)$ bi-doublet $L_{B}$. These will
couple to SM gauge bosons and thus can be produced at the LHC. Similarly,
$SU(2)$ singlet quarks $Q_{LS}$ and $Q_{RS}$ form a heavy quark of electric
charge $+{1\over 3}$ of Dirac type. Finally, there remains a $SU(2)_{L,R}$
singlet lepton of zero electric charge. This could well be a candidate for
Dark Matter. The assignment of $U(1)$ charges for the fermions, from the
requirement of anomaly cancellation, is such that the exotic fermions can only
couple to the gauge bosons but do not have any mixing with the SM fermions.
This feature will play a crucial role in determining the possible signatures
of these fermions at colliders.

Fermions get their masses via their interactions with Higgs fields. The
relevant Yukawa Lagrangian is noted below.

$\displaystyle\mathcal{L}_{Yukawa}$ $\displaystyle=$ $\displaystyle
y_{qij}\bar{Q}_{iL}\Phi_{B}Q_{jR}+\tilde{y}_{qij}\bar{Q}_{iR}\tilde{\Phi}_{B}Q_{jL}+y_{lij}\bar{L}_{iL}\Phi_{B}L_{jR}+\tilde{y}_{lij}\bar{L}_{iR}\tilde{\Phi}_{B}L_{jL}$
(33) $\displaystyle+$ $\displaystyle
y_{sij}\bar{Q}_{iLS}\Phi_{S}Q_{jRS}+y_{LBij}\;Tr\left[\bar{L}_{iB}\tilde{L}_{jB}\right]\Phi_{S}^{c}+\frac{y_{LSij}}{\Lambda}\bar{L}_{iS}L_{jS}^{c}\Phi_{S}\Phi_{S}$
$\displaystyle+$ $\displaystyle
y_{BBij}\;Tr\left[\bar{L}_{iB}\tilde{\Phi}_{B}\right]\L_{jS}^{c}+H.C.$

where, $i,j=1,2,3$ are generation numbers and $y$(s) are Yukawa coupling
constants. $\Phi_{S}^{c}$ is complex conjugate of $\Phi_{S}$ and
$\tilde{L}_{B}=\sigma_{2}L_{B}^{*}\sigma_{2}$.

In general the Yukawa coupling matrices, $y_{q}$, $y_{l}$, $y_{LB},y_{s}$ are
non-diagonal 888In general, in an unifying model like $E_{6}$, all the Yukawa
couplings at the low energy will be generated from a single (and possibly a
non-diagonal) Yukawa texture at the GUT scale [24]. . The diagonalisation of
the Yukawa matrices in the first line of Eq. 33 will give rise to the SM-
fermion masses and mixing in the form of $V_{CKM}$ and $V_{PMNS}$. There are
no term present in the Yukawa Lagrangian leading to exotic fermion SM-fermion
mixing. Thus while considering the phenomenology of the some of the exotic
fermions, we have used their physical masses as the free parameters of the
analysis and derive possible bounds on them from LHC itself. The last term in
Eq. 33 introduces a mixing between the singlet lepton and the neutral lepton
from the bi-doublet. In this work, we shall not be investigating the
phenomenological implications of this term.

It is important to note a dimension-4 mass term for the singlet lepton $L_{S}$
(a Weyl spinor) cannot be written as it transform non-trivially under
$U(1)_{L,R}$. Using the singlet Higgs field $\Phi_{S}$, we are able write a
dimension-5 operator, which in turn generates mass for $L_{S}$. It is well
known that any one of the Higgs bosons from ${\bf 27}$-plet of $E_{6}$ cannot
give mass to $L_{S}$. To generate a mass using Higgs mechanism, one must
employ a Higgs from a multiplet of $E_{6}$ other than 27 [25]. So $\Lambda$
may be identified with the vev of such a Higgs boson. Or we can simply assume
that $L_{S}$ has acquired mass from a Higgs belonging to other rep. of $E_{6}$
and we treat its mass as a free parameter in our analysis.

Figure 6: Observed (line with dots) and expected (dashed) 95% C.L.
experimental upper limit on the cross-section ($\sigma$) of heavy charged
lepton pair-production at the 13 TeV run of the LHC. Also shown in the plot
the theoretical prediction from the $32121$ model (blue line).

One can constrain the masses/Yukawa couplings of exotic quarks and leptons
from the direct search limits on their masses at the LHC [26, 27]. As for
example, ATLAS collaboration has searched for long-lived heavy charged lepton
at 13 TeV run with a collected luminosity of 36.1 fb-1. We have estimated the
pair-production cross-section of $E^{+}E^{-}$ at the 13 TeV and compared it
with the experimental 95% C.L. upper limit on the same cross-section obtained
by ATLAS collaboration. The plots have been presented in Fig. 6. One can see
that $E$ mass in 32121 model cannot be less than 1.091 TeV at 95% C.L.

##  3 Phenomenology of new Higgs bosons of $32121$ model at the LHC

Apart from the SM-like Higgs boson, $32121$ model contains a number of neutral
and charged scalar states. We will now discuss the possible interactions and
signatures of such states at the LHC in this section.

###  3.1 Phenomenology of the scalars arising from the bi-doublet in $32121$
model

$h_{2}^{0}$ ($\xi^{0}_{2}$) is the neutral CP-even (odd) scalar originating
from the Higgs bi-doublet, $\Phi_{B}$. From Eqs. (19) and (20), we can easily
see their masses are equal. They do not decay to a pair of gauge bosons as the
vev $k_{2}$ has been set to zero. For the very same reason, $h^{0}_{2}$ or
$\xi^{0}_{2}$ does not couple to a pair of any other scalars.

Figure 7: Feynman diagrams for $h_{2}^{0}$ and $\xi_{2}^{0}$ production via
gluon gluon fusion through b quark loop

$h^{0}_{2}$ ($\xi^{0}_{2}$) couples to a pair SM fermions via Yukawa coupling
(see Eq. (33)). It is interesting to note that the coupling of $h^{0}_{2}$
($\xi^{0}_{2}$) to a pair of top quark is proportional to bottom-Yukawa
coupling and vice-versa. ATLAS and CMS have searched for a heavy neutral Higgs
boson produced in association with $b$ quarks followed by its decay to a pair
of $b$ quarks at $\sqrt{s}=13$ TeV [29, 30]. We consider the production of an
$h_{2}^{0}$ in association with a pair of $b$ quarks and its decay to a pair
of $b$ quarks. The resulting rate of events can be compared with the measured
rate by ATLAS Collaboration to set a lower limit on the mass of
$h_{2}^{0}~{}(\xi_{2}^{0})$. The calculated (in 32121 model) and (95% C.L.
upper limit on the) measured cross-sections are presented in the Fig. 8. The
95% C.L. lower limit on $m_{h_{2}^{0}/\xi_{2}^{0}}$ comes out to be greater
than $800$ GeV. While estimating the $h^{0}_{2}$ ($\xi^{0}_{2}$) production
cross-section in association with a pair of b-quarks, we have incorporated the
QCD K-factor ($\sim 1.1$) following the ref. [31, 32]. However, the lower
limit derived in the above, depends on the charged Higgs boson ($H^{\pm}_{1}$)
mass in the following way. A careful look into the branching ratios of
$h^{0}_{2}$ reveals that it dominantly decays to a pair of $b$-quarks, unless
a decay to $H^{\pm}_{1}W^{\pm}$ is kinematically allowed,. Consequently, mass
of $H^{\pm}_{1}$ plays a crucial role in determining the rate of $4b$ final
state from considered above. A heavier charged Higgs (when
$h^{0}_{2}\rightarrow H^{\pm}_{1}W^{\mp}$ is disallowed) will push the lower
limit on $h^{0}_{2}$ mass in upward direction and vice versa.

In Fig. 8, we have presented the $\sigma(pp\rightarrow
b\bar{b}h^{0}_{2})\rightarrow b\bar{b}(b\bar{b})$ in two cases. The red line
represents the rate when $m_{H^{\pm}_{1}}>m_{h^{0}_{2}}$ and the later decays
to a pair of $b\bar{b}$ with 100% BR. While the blue curve represents the case
when $h^{0}_{2}$ can also decay to $H^{\pm}_{1}$ thus having a reduced decay
rate to $b\bar{b}$. A charged Higgs mass of 750 GeV has been assumed while
making this plot. The sudden change in the slope of the blue curve due to
onset of $h_{2}^{0}\rightarrow H^{\pm}_{1}W^{\mp}$ decay mode around
$m_{h^{0}_{2}}\simeq 850$ GeV (see Fig. 9) is evident.

Figure 8: The black solid (dashed) line represents the observed (expected) 95%
C.L. upper limit on the production cross-section ($\sigma$) times branching
ratio to $b\bar{b}$, of heavy neutral Higgs boson in association with $b$
quarks as a function of Higgs boson mass at $\sqrt{s}=13$ TeV with 27.8 fb-1
integrated luminosity. The blue line corresponds to $\sigma(pp\rightarrow
h_{2}^{0}b\bar{b})\times BR(h_{2}^{0}\rightarrow b\bar{b})$ whereas the red
line represents the same but considering $m_{H_{1}^{\pm}}>m{h_{2}^{0}}$.

A dominant production mechanism for such a Higgs boson at the LHC will be via
gluon gluon fusion (Fig. 7). Unlike the SM Higgs boson, in this case, gluon
gluon fusion cross-section is dominated by the bottom quark loop. We present
the production cross-section (considering NLO QCD correction for this
production process, see [33]) and decay branching ratios of $h^{0}_{2}$ in
Fig. 9.

Figure 9: $h^{0}_{2}$ production cross section ($\sigma$) via gluon fusion at
LHC (left panel) for 14 and 27 TeV proton proton center of mass energy. The
right panel shows the branching ratios of $h^{0}_{2}$ to different final
states. $h^{0}_{2}$ and $\xi^{0}_{2}$ have the same masses and coupling
strengths

At the 14 TeV run of the LHC, $h^{0}_{2}$ production cross-section varies from
14 fb for $m_{h_{2}^{0}}=800$ GeV to 0.2 fb for 1.5 TeV mass of this scalar.
Production cross-section at 27 TeV is even higher and it varies from 77 fb at
$m_{h_{2}^{0}}=800$ GeV to 1.5 fb for $m_{h_{2}^{0}}=1500$ GeV. Once produced,
$h^{0}_{2}$ dominantly decays to a pair of $b$-quarks, unless it decay to
$H_{1}^{\pm}W^{\mp}$. The later decay mode will only be allowed when
$m_{h_{2}^{0}}$ is sufficiently higher than $m_{H_{1}^{\pm}}+m_{W}$. This
choice of mass ordering depends on the choice of parameters namely
$\lambda_{3}$ and $c_{4}-c_{3}$. In the plot presented in Fig. 9, a certain
choice of these parameters have been assumed, so that the
$h^{0}_{2}\rightarrow H_{1}^{\pm}W^{\mp}$ has been kinematically allowed, with
an additional assumption about the mass of $H_{1}^{\pm}$ (750 GeV).

In the pseudo-scalar sector, $\xi^{0}_{2}$ arises from the Higgs bi-doublet
and has a mass equal to the mass of $h_{2}^{0}$. It has exactly the similar
coupling strengths to the SM fermions as the $h^{0}_{2}$. The choice of
vanishing $k_{2}$ forbids its coupling to a pair of gauge bosons or the
scalars. Consequently the production and decay mechanism and their rate of
$\xi^{0}_{2}$ is exactly the same as $h^{0}_{2}$. We will not present these
numbers separately.

Charged Higgs boson, $H_{1}^{\pm}$ arises from the Higgs bi-doublet,
$\Phi_{B}$. From the expression (21), one can see,
$m_{H_{1}^{\pm}}^{2}=\frac{1}{2}(c_{4}-c_{3})(k_{1}^{2}+v_{R}^{2})$. This
massive charged Higgs couples to SM fermions and decays to a top and a bottom
quark with nearly 100% branching ratio. It also couples to the heavy gauge
bosons of $32121$ model but the coupling of $H_{1}^{\pm}$ to the SM gauge
bosons ($W^{\pm},Z$) is proportional to $k_{2}$, hence identically vanishes.
It can be singly produced at LHC with a top and bottom quark or pair-produced
via Drell-Yan process or via vector boson fusion process.

Figure 10: The black solid and dashed line represent observed and expected 95%
C.L. experimental upper limit on the cross-section ($\sigma$) times BR of
heavy charged scalar production via $pp\rightarrow\bar{t}bH_{1}^{+}\times
BR(H_{1}^{+}\rightarrow t\bar{b})$ at the 13 TeV run of the LHC with 139 fb-1
integrated luminosity[34]. Also shown in the plot the theoretical prediction
of $\sigma\times BR$ for $H^{+}_{1}$ production in the $32121$ model (blue
line).

In Fig. 11, we have presented the cross-section of associated production of
$H_{1}^{\pm}$ with a top and a bottom at the LHC and branching ratio of
$H_{1}^{\pm}$. In case of $H_{1}^{\pm}$ production, the main contribution will
be from $gg\rightarrow\bar{t}bH_{1}^{\pm}$. ATLAS and CMS collaborations have
searched for a heavy charged Higgs boson decaying to a top and bottom at 13
TeV run [34, 35, 36]. Using the most recent upper limit on the $\sigma\times
BR$ provided by ATLAS, we put a lower limit on the mass of the charged Higgs
$m_{H_{1}^{\pm}}>$ 720 GeV (see Fig. 10). The sudden change in the slope of
the blue curve representing $\sigma_{(}pp\rightarrow\bar{t}bH_{1}^{+})\times
BR(H_{1}^{+}\rightarrow t\bar{b})$ is due to the sudden decrease of
$BR(H_{1}^{+}\rightarrow t\bar{b})$ around $m_{H_{1}^{\pm}}=900$ GeV (see Fig.
11). We have set the mass of $h_{2}^{0}~{}(\xi_{2}^{0})$ at its lower limit of
800 GeV following the Fig. 9 corresponding to the scenario when
$m_{H_{1}^{\pm}}>m_{h_{2}^{0}}$ (the red line).

We have presented cross-section for $H^{\pm}tb$ production (in this process
NLO QCD correction and running mass for b quark can be important, see [37,
32]) at centre of mass energies of 14 and 27 TeV. In Fig. 11, the right panel
shows the branching ratios of $H_{1}^{\pm}$ to different final states. Until
kinematically allowed for the decay to $h_{2}^{0}~{}W^{+}$ and
$\xi_{2}^{0}W^{+}$, $H_{1}^{+}$ dominantly decays to $t\bar{b}$
($BR(H_{1}^{+}\rightarrow t\bar{b})\sim 0.999$). For large $m_{H_{1}^{\pm}}$
the branching ratios to $h_{2}^{0}~{}(\xi_{2}^{0})W^{+}$ channel become more
dominant.

Figure 11: $H_{1}^{\pm}$ production cross-section ($\sigma$) via
$pp\rightarrow\bar{t}bH_{1}^{+}$ process at LHC (left panel) for 14 and 27 TeV
proton proton center of mass energy. The right panel shows the branching
ratios of $H_{1}^{\pm}$ to different final states setting the mass of
$h_{2}^{0}~{}(\xi_{2}^{0})$ at its lowest limit ($800$ GeV).

The cross-section for $H^{\pm}tb$ production at centre of mass energy
approximately 14 (27) TeV varies from 0.15 (1) pb for $m_{H_{1}^{\pm}}=720$
GeV to 0.005 (0.06) pb for $m_{H_{1}^{\pm}}=1500$ GeV. After being produced,
$H_{1}^{\pm}$ will decay further and considering respective decay channels
(e.g., $tb$ or $h_{2}^{0}~{}(\xi_{2}^{0})~{}W^{\pm}$) one can expect a good
amount of events at HL-LHC. However, one needs to consider further decays of
top or $h_{2}^{0}~{}(\xi_{2}^{0})$.

###  3.2 Phenomenology of the scalars arising from Left-handed Higgs Doublet

In this section, our primary concern will be the neutral and charged states
originating from the left-handed doublet scalar. Among the neutral CP-even
scalar $h_{L}^{0}$, neutral CP-odd scalar $\xi_{L}^{0}$ and charged scalars
$H_{L}^{\pm}$, The first two have equal masses (see Eqs. (19), (20), (21)) and
do not mix with other neutral states. These three states can be pair-produced
at the LHC via quark anti-quark fusion mediated by one of the electroweak
gauge bosons. However as we set $v_{L}$ to be zero, neither of these states
decays to a pair of SM particles.

As already pointed out, we will not vary all the parameters of the mass matrix
independently to study the masses of the scalars. We will treat the physical
masses as free parameters of our analysis. However some caveats are to be
imposed on some combinations of parameters of the mass matrices. As for
example, $\rho_{3}-2\rho_{1}$ will always assumed to be a positive quantity
which is ascertained from the positivity of charged Higgs boson ($H_{L}^{\pm}$
) mass (squared). Now the other charged Higgs boson ($H_{1}^{\pm}$ ) mass
squared is proportional to $c_{4}-c_{3}$. This in turn forces us to take this
combination also to be positive. As a consequence, masses of $h_{L}^{0}$ and
$\xi_{L}^{0}$ are always greater than mass of $H_{L}^{+}$. However,
$(m_{h^{0}_{L}}-m_{H^{+}_{L}})$ can be controlled by choosing a proper
magnitude of the combination $(c_{4}-c_{3})\frac{k_{1}^{2}}{2}$. From the
expressions (19) and (21),
$m_{h_{L}^{0}}^{2}=(c_{4}-c_{3})\frac{k_{1}^{2}}{2}+m_{H_{L}^{\pm}}^{2}$. We
will see in the following that $h^{0}_{L}$ will decay to $H_{L}^{+}W^{-}$ if
kinematically allowed. So in order to make $h_{L}^{0}$ stable, one needs to
set $(c_{4}-c_{3})\frac{k_{1}^{2}}{2}<m_{W}^{2}+2m_{W}m_{H_{L}^{\pm}}$.
However, an unstable $h^{0}_{L}$ implies that the mass of $H_{1}^{\pm}$
becomes too heavy in the ballpark of 17 TeV.

In the following analysis, $h^{0}_{L}$ and $\xi^{0}_{L}$ are assumed to be
stable. Thus they could be potential candidates for DM. $H_{L}^{\pm}$ also do
not have any decay mode. Once produced at colliders, it passes through the
detector without decaying. However being a charged particle, it leaves its
signature in the tracker and e. m. calorimeter before leaving the detector.
ATLAS collaboration has searched for long-lived stau $(\tilde{\tau}$, the
super-symmetric partner of $\tau$-lepton) which are very similar to the
$H_{L}^{\pm}$ [27]. So the upper limit of the cross-section of pair-production
of such long-lived $\tilde{\tau}$s at LHC centre of mass energy of 13 TeV, as
quoted by ATLAS collaboration can be used in our case to constrain the
$m_{H^{\pm}_{L}}$ which is the only free parameter that controls the
$H_{L}^{\pm}$ pair-production. In Fig. 12, we present the variation of
$H_{L}^{\pm}$ pair-production cross-section (blue solid line) as a function of
its mass. Over-layed are the observed and expected upper limits on the pair-
production of long-lived stau (black solid and dashed lines). The intersection
of these two curves gives us a 95% C.L. lower limit of 494 GeV, on the left-
handed charged Higgs boson ($H_{L}^{\pm}$) mass.

Figure 12: Observed (line with dots) and expected (dashed) 95% C.L.
experimental upper limit on the cross-section ($\sigma$) of heavy stable
charged scalar pair-production at the 13 TeV run of the LHC. Also shown in the
plot the theoretical prediction for $H^{+}_{L}H^{-}_{L}$ pair-production
cross-section in the $32121$ model (blue line).

Let us now concentrate on the possible production and decay signatures of
charged and neutral Higgs bosons arising from the left-handed doublet. As
mentioned above, these can be pair-produced at the LHC, via a mechanism
similar to Drell-Yan. In Fig. 13, the pair-production cross-sections have been
presented with Higgs masses at 14 (27) TeV run of LHC. One can see from Fig.
13, production cross-section for $H_{L}^{\pm}$ varies from 0.4 (1.8) fb at 500
GeV to 0.005 (0.06) fb at 1.5 TeV at the center of mass energy 14 (27) TeV.
$H_{L}^{\pm}$ being stable, does not decay any further and we are left with
two ionising tracks of heavy particle in the detector [27], [28]. At HL-LHC,
such a cross-section results into 15 background free events even for a
$H_{L}^{\pm}$ mass of 1.5 TeV. This particular signature is unique and cannot
arise from the SM. Thus we hope to explore stable charged Higgs masses upto
1.5 TeV at the 14 TeV HL run of LHC.

Figure 13: The red solid line corresponds to $H_{L}^{\pm}$ pair-production
cross-section ($\sigma$) at LHC at $\sqrt{s}=14$ TeV (left panel) and at
$\sqrt{s}=27$ TeV (right panel) whereas the blue solid line represents the
combined production cross-section of one charged ($H_{L}^{\pm}$) and one
neutral (CP-even or CP-odd) scalar at LHC at $\sqrt{s}=14$ TeV (left panel)
and at $\sqrt{s}=27$ TeV (right panel)

.

We now turn to the production of a $H_{L}^{\pm}$ in association with either a
$h_{L}^{0}$ or $\xi_{L}^{0}$. The production mechanism at the LHC is same as
above but with a small difference. The scalar current in the later case is
connected to initial state left-handed quark current by a $W^{\pm}$
propagator. Consequently the cross-section for $h_{L}^{0}H_{L}^{\pm}$ is lower
than the $H_{L}^{\pm}$ pair-production. However, when we combine the
$\xi_{L}^{0}H_{L}^{\pm}$ with it, total cross-section of associated production
becomes comparable with pair-production rate of charged Higgs bosons. In Fig.
13, associated production cross-section has been presented. One can see that
at 14 (27) TeV run of LHC, the cross-section varies from 1.4 (5.9) fb to 0.002
(0.038) fb when the charged Higgs mass varies from 0.5 TeV to 1.5 TeV.

While discussing the possible signatures of the associated production, we have
to be careful about the mass ordering between $H_{L}^{+}$ and
$h_{L}^{0}~{}(\xi_{L}^{0})$. When kinematically allowed, $h_{L}^{0}$ will
decay (with 100% branching ratio) to $W^{-}H_{L}^{+}$. Depending the further
decay of the $W$-boson, associated production will result into two charged
tracks + 2 jets or 2 charged tracks with a lepton and
$E_{T}\\!\\!\\!\\!\\!\\!/$. On the other hand when $h_{L}^{0}$ is stable,
associated production would result into a signal, comprising of a single
charged track (from $H^{\pm}_{L}$) in association with
$E_{T}\\!\\!\\!\\!\\!\\!/$ (arising from $h^{0}_{L}$ and $\xi^{0}_{L}$).

###  3.3 Phenomenology of the scalar arising from Right-handed Higgs Doublet

Next, in our agenda, is the heavy neutral Higgs boson, $H_{R}^{0}$. Due to
non-zero $v_{R}$, it couples to a pair of neutral heavy gauge bosons. But it
cannot have any coupling to SM fermions999It may couple to the SM fermions if
we allow a possible mixing between $H^{0}_{R}$ with SM Higgs boson. The plot
(Fig. 14) showing the branching ratios of $H_{R}^{0}$ reveals that it
dominantly decays to a pair of SM Higgs bosons or to a pair of $H_{L}^{\pm}$
or $h_{L}^{0}~{}(\xi_{L}^{0})$ once these decays are kinematically allowed.
Decay to a pair of heavy neutral gauge bosons are kinematically disallowed.
Furthermore, coupling of $H_{R}^{0}$ to a pair of $Z$ bosons conspires to be
small hence its decay rate to a pair $Z$ bosons is negligible. $H_{R}^{0}$ can
have an effective coupling at one-loop ($H^{\pm}_{L}$, $W^{\pm}_{R}$ and
$H^{\pm}_{1}$ running in the triangle loop) to a pair of photons. The decay
branching ratio can be as high as $10^{-5}$ over a wide mass range of
$H^{0}_{R}$, and is thus not phenomenologically very interesting.

Figure 14: Associated production cross section ($\sigma$) of $H^{0}_{R}$ along
with a vector boson at LHC (left panel) for 13, 14 and 27 TeV proton proton
center of mass energy. The right panel shows the branching ratios of
$H^{0}_{R}$ to different final states with an assumption of
$m_{h_{L}^{0}}=m_{\xi_{L}^{0}}=500$ GeV and $m_{H_{S}^{0}}=700$ GeV.

The main production mechanism for $H^{0}_{R}$ is in association with a gauge
boson ($Z$, $Z^{\prime}$, $A^{\prime}$ and $W_{R}$) via annihilation of quark
anti-quark pair. It can also be produced in vector boson fusion mechanism. In
this article, we will only consider the production of $H_{R}^{0}$ in
association with a vector boson (Fig. 15).

In Fig. 14 (right panel) we have presented the combined cross-section of
production of a $H^{0}_{R}$ in association with $Z^{\prime}$, $A^{\prime}$ and
$W_{R}$, with the heavy gauge boson masses set at their experimental lower
limits. Among these three production channels, contribution of
$\sigma(H^{0}_{R}A^{\prime})$ is nearly 70% of combined cross-section
presented in Fig. 14. Combined cross-section of $H_{R}^{0}$ associated
production at the LHC varies from $0.4$ fb to $0.22$ fb for a range of
$m_{H_{R}}:400$ to $1500$ GeV at a center of mass energy of $27$ TeV. At $14$
TeV run of the LHC, the cross-section is quite small. It is in the ballpark of
$0.005$ fb (Fig. 14) for a $H^{0}_{R}$ of 1 TeV mass. The kinematic
suppression due to the presence of a heavy gauge boson in the final state can
be one of the reasons behind the smallness of the total cross-section.

Figure 15: Feynman diagram for $H_{R}^{0}$ production in association with a
vector boson where $V,V^{\prime}$ represent any vector boson of $32121$ model
($Z$, $Z^{\prime}$, $A^{\prime}$ and $W_{R}$).

Before we close this sub-section, let us make some brief comments about the
possible signature of $H^{0}_{R}$ at the LHC. The most promising, signature,
in our opinion will arise when $H^{0}_{R}$ can decay to a pair of
$H^{\pm}_{L}$. As mentioned before, if will produce two charged tracks in the
detector with their invariant mass peaking at the mass of $H^{0}_{R}$. Along
with a pair of charged tracks heavy gauge boson decay will probably give rise
to a pair of high mass jets or leptons. As for example, at 27 TeV run of the
HL-LHC, one expects around 30 two charged tracks two lepton events for a
$m_{H^{0}_{R}}=1$ TeV101010The branching ratio for di-lepton decay
($e,\mu,\tau$) of $A^{\prime}$ is $\sim 10\%$ and branching ratio of
$H_{R}^{0}\rightarrow H_{L}^{+}H_{L}^{-}$ is around 50%. So considering almost
$70\%$ contribution of $\sigma(H_{R}^{0}A^{\prime})$, at $27$ TeV run with
$3000$ fb-1 integrated luminosity, for the process $\sigma(pp\rightarrow
H_{R}^{0}A^{\prime})\times BR(A^{\prime}\rightarrow ll)\times
BR(H_{R}^{0}\rightarrow H_{L}^{+}H_{L}^{-})$ one can expect around 30 events
for 1 TeV $H_{R}^{0}$ mass.. While at the 14 TeV run with high luminosity
option, detection of such events seems to be very challenging even for a 500
GeV $H^{0}_{R}$.

###  3.4 Phenomenology of the $SU(2)_{L}\otimes SU(2)_{R}$ Singlet scalar in
$32121$ model

Next, in our agenda, is the scalar arising from $SU(2)_{L}\times SU(2)_{R}$
singlet $\Phi_{S}$. $\beta_{1}$ being small in order to satisfy the SM Higgs
bosons signal strength (see Fig. 1), it does not have any significant role in
the phenomenology of the singlet scalar and we can treat mass of the singlet
scalar itself as the free parameter of our analysis. In the following we
intend to study the decays and dominant production channel of the singlet
scalar boson.

In Fig. 16, we present branching ratios of $H_{S}^{0}$ to its available decay
channels. To remind, $WW$, $ZZ$, $t\bar{t}$, $b\bar{b}$ decays of $H_{S}^{0}$
take place via the mixing with the SM Higgs boson. While rest of the decays
are driven via the direct couplings of $H_{S}^{0}$ to the decaying particles.

The dominant contribution to $H_{S}^{0}\rightarrow gg,\gamma\gamma$ decay,
arise from triangle loops of heavy exotic quarks and leptons. Charged Higgs
states arising from $\Phi_{B}$, $\Phi_{L}$ also contribute to singlet Higgs
decay to $\gamma\gamma$. $H_{S}^{0}\rightarrow gg$ is important as production
cross-section of $H_{S}^{0}$ via gluon fusion is directly proportional to this
decay width. However, $v_{S}$ being large, singlet Higgs Yukawa to exotic
leptons/quarks are tiny (see Eq. (33) in section 2.3). Consequently, decay
width to $gg$ is small. Similar arguments can be given to understand the
smallness of $H_{S}^{0}\rightarrow\gamma\gamma$ decay rate.

The branching ratios to several decay channels are moderately sensitive to
$\beta_{1}$. With a higher value of $\beta_{1}(\sim 10^{-3})$ one can satisfy
all the constraints from SM Higgs boson signal strengths. However,
$\beta_{1}>10^{-3}$ 111111$\beta_{1}>0.01$ is excluded as the singlet
component in SM-like Higgs will be too high to satisfy the experimentally
measured signal strengths. will lead to a singlet Higgs boson mass of 700 GeV
and above. Furthermore, a higher value of $\beta_{1}$ leads to a larger mixing
between the singlet and the SM-like Higgs boson. Thus the singlet Higgs boson
decay rates to $tt$, $bb$, $WW$ and $ZZ$ channels will increase slightly. The
variation of branching ratios over a wide mass range of $H_{S}^{0}$ for a
fixed $\beta_{1}$ have been shown in the Fig. 16.

Figure 16: Branching ratio of $H_{S}^{0}$ to different channels including
exotic quarks and exotic leptons for $\beta_{1}=1.5\times 10^{-4}$ and
$v_{S}=13$ TeV and exotic quark mass 1.3 TeV.

In this section we present singlet Higgs production cross-section via gluon
gluon fusion over a range of singlet Higgs mass The production mechanism is
the same as the SM Higgs production via gluon fusion. However, the triangle
loop (see Fig. 17) is driven by exotic quarks which are heavy in mass. There
will be a very tiny contribution from the standard model top-quark through the
mixing of singlet Higgs with the SM Higgs boson. While estimating this cross-
section we have incorporated a K-factor following ref. [33], assuming higher
order QCD correction to the production of a singlet Higgs boson will be of
similar order that of a SM Higgs boson production via gluon fusion. For our
illustration we have assumed $\beta_{1}$ (mixing parameter) to be equal to 1.5
$\times 10^{-4}$. This value of $\beta_{1}$ is consistent with the measured
values of SM Higgs boson signal strengths to different channels.

Figure 17: Feynman diagram for $H_{S}^{0}$ production via gluon gluon fusion
quark loops, $q_{S}$ is the exotic singlet quark and $q$ represents any SM
quark Figure 18: $H_{S}^{0}$ production cross-section ($\sigma$) via gluon
fusion for $\beta_{1}=1.5\times 10^{-4}$ and $v_{S}=13$ TeV and exotic quark
mass 1.3 TeV.

In Fig. 18, we have presented the singlet Higgs boson production cross-section
at proton proton centre of mass energies 13, 14 and 27 Tev respectively. At 14
(27) TeV run of LHC, production cross-section varies from 15.9 (57.6) fb to
10-3 (0.056) fb when $H_{S}^{0}$ mass changes from 0.3 to 4 TeV. Although the
production mechanism is similar to the SM Higgs boson production via gluon
fusion, the cross-section for $H_{S}^{0}$ production is order of magnitude
smaller than a SM-like Higgs boson of same mass, even after considering the
contribution from 3 species of $SU(2)$ singlet exotic quarks. This can be
explained by the small Yukawa coupling of these exotic quarks to the Singlet
Higgs boson (see Eq. (33)). We have assumed the exotic quark mass to be equal
to 1.3 TeV [26]121212The lower limit on the mass of a heavy stable quark
following the ref. [26] is 200 GeV, obtained from 8 TeV run of LHC. Due to non
availability of any further updated analysis at 13 TeV, we have assumed that
mass limit, on such an object, is in the ballpark of a TeV. The mass limit on
heavy stable lepton [27] is 1.09 TeV. Assuming the quarks will have a higher
production cross-section at the LHC, we have assumed they must be heavier than
the exotic leptons.. Before we conclude, let us make some qualitative comments
about the possible signatures of $H_{S}^{0}$ at the LHC. For a low mass
($<700$ GeV), $WW$ decay mode can be exploited to look for possible signatures
of this Higgs boson. However, once, $m_{H_{S}^{0}}$ becomes greater than a
TeV, gluon gluon decay of $H_{S}^{0}$ becomes dominant and detection of such a
scalar will be difficult due to a possible large QCD background. However, for
higher singlet Higgs masses ($>2.2$ TeV), it can decay to a pair of exotic
leptons, thus will produce a unique signature of two charged tracks with their
invariant mass peaking at the singlet Higgs mass. This signal will be
background free and probably the best bet for detection of such a scalar
boson. As for example, at the 14 TeV LHC, decay of a 2.2 TeV $H_{S}^{0}$ will
approximately produce 48 events with a pair of charged tracks with 3 ab-1
integrated luminosity. While we expect to have 15 such events for a 3 TeV
$H_{S}^{0}$ at 14 TeV run with same luminosity. At 27 TeV run, the situation
will improve drastically, we can expect to see 80 such events even for a 4 TeV
singlet Higgs boson.

Finally we would like to make a brief comment about the situation when
$c_{1}+c_{3}\neq 0$ and $\gamma_{1}\neq 0$. Making $c_{1}+c_{3}$ non-zero,
would introduce mixing between right-handed neutral scalar with SM-like Higgs
boson. However, we have to satisfy the experimentally observed signal
strengths of $h^{0}$. This in turn limits the above mixing and the $H^{0}_{R}$
would possibly have small decay channels to the SM fermions and SM gauge
bosons. On the other hand a non-zero $\gamma_{1}$ would have more prominent
effect on $H^{0}_{R}$ \- $H^{0}_{S}$ phenomenology. A $\gamma_{1}$ induced
mixing between $H^{0}_{R}$ and $H^{0}_{S}$ would lead to $H^{0}_{R}$ decays to
SM fermions along with exotic fermions when kinematically allowed. At the same
time, both these states could be produced via gluon fusion.

The production of the BSM scalars in 32121 model and the possible backgrounds
have been discussed in Table 3 very briefly.

Scalars | Production at LHC | Possible final state | Possible backgrounds
---|---|---|---
$h_{2}^{0}(\xi_{2}^{0})$ | $h_{2}^{0}(\xi_{2}^{0})b\bar{b}$ | $b\bar{b}b\bar{b}l^{+}l^{-}\nu_{l}\bar{\nu}_{l}$ |  $t\bar{t}b\bar{b},t\bar{t}h,$ DY + jets
$H_{1}^{\pm}$ | $H_{1}^{+}\bar{t}b$ | $b\bar{b}b\bar{b}l^{+}l^{-}\nu_{l}\bar{\nu}_{l}$ |  $t\bar{t}b\bar{b},t\bar{t}h,$ DY + jets
$H_{L}^{\pm}$ |  $H_{L}^{+}H_{L}^{-}$ pair-production | $H_{L}^{+}H_{L}^{-}$ | Stable heavy charged particle
|  |  | creating two charged tracks and
|  |  | possibly background free
$-$ | $H_{L}^{+}h_{L}^{0}(\xi_{L}^{0})$ | $H_{L}^{+}h_{L}^{0}(\xi_{L}^{0})$ | Being stable and neutral $h_{L}^{0}(\xi_{L}^{0})$
|  |  | will remain undectected and $H_{L}^{+}$
|  |  | will create one charged track
|  |  | and possibly background free
$H_{R}^{0}$ | $H_{R}^{0}A^{\prime}$ | $H_{L}^{+}H_{L}^{-}l^{+}l^{-}$ | Two oppositely chagred tracks of
|  |  | heavy stable particles, invariant
|  |  | mass distribution of $H_{L}^{+}H_{L}^{-}$ should
|  |  | peak at $M_{H_{R}^{0}}$, background free
$H_{S}^{0}$ |  $H_{S}^{0}$ via gluon fusion | $E^{+}E^{-}$ | Stable heavy charged particle
|  |  | creating two charged tracks and
|  |  | possibly background free
|  | $q_{S}\bar{q}_{S}$ | Stable heavy charged colored
|  |  | particles will hadronize
Table 3: Significant production processes of BSM scalars of 32121 at the LHC
and their possible backgrounds

##  4 Conclusions

To summarise, we have investigated phenomenological implications of a LR
symmetric model based on $E_{6}$ inspired gauge group $SU(3)_{C}\otimes
SU(2)_{L}\otimes U(1)_{L}\otimes SU(2)_{R}\otimes U(1)_{R}$. The later
symmetry group can be a result of two step breaking of $E_{6}$. We have
studied the phenomenology of the Higgs bosons, responsible for the symmetry
breaking of 32121 gauge group down to the SM gauge group. The model is
hallmarked by the presence of a complete family of 27-plet of fermions
belonged to the fundamental representation of $E_{6}$. Apart from these TeV
scale fermions, a weak bi-doublet (under $SU(2)_{L}\times SU(2)_{R}$), a
right-handed Higgs doublet and a singlet is necessary for complete symmetry
breaking. The measured value of $W$-boson mass fixes one of the bi-doublet
vacuum expectation values, which we identify with the SM Higgs vev. The
experimental lower limits on the mass of right-handed charged gauge boson
$W_{R}$ in turn constrain the vacuum expectation value, $v_{R}$, of the
neutral member of the right-handed Higgs doublet. $v_{R}$ comes out to be
greater than 14 TeV. The second bi-doublet vev $k_{2}$ is set to zero to avoid
a possible admixture of $W_{R}$ in the SM $W$ boson. Appearance of an
additional massive neutral gauge boson is a result of the breaking of an extra
$U(1)$ symmetry. The experimental lower limit from the LHC, on the mass of
such an extra $U(1)$ gauge boson puts a lower limit of 12.61 TeV on the vacuum
expectation value $v_{S}$ of the singlet scalar boson.

All fermions present in the $27$ dimensional fundamental representation of
$E_{6}$ are considered to be present in our model. We have written down the
relevant dimension-4 Yukawa interactions of these fermions either with the bi-
doublet scalar or the singlet scalar field, excepting the singlet lepton field
$L_{S}$, for which we write a dimension-5 Yukawa term involving singlet scalar
$\Phi_{S}$. Furthermore, using the LHC data on the search of heavy charged
long-lived particles, we have put a lower limit of 1.09 TeV on the mass of
heavy exotic charged lepton.

After discussing the symmetry breaking pattern in some details, we have mainly
devoted ourselves on the phenomenology of the scalars (CP-even, CP-odd and
charged Higgs) present in this model. We investigated their decay modes, and
possible production processes at the LHC. Without going into the details of
signal background analysis we have discussed possible signatures of the Higgs
bosons arising from this model.

Apart from the SM-like Higgs boson $h^{0}$, two more neutral scalars
($h^{0}_{2}$ and $\xi^{0}_{2}$) of same mass and having similar couplings to
the SM fermions will originate from the bi-doublet after SSB. Lower limit on
the masses of these scalars have been obtained from the LHC data and they must
be heavier than 800 GeV. Their dominant production mechanism at the LHC will
be in association with a pair of $b$-quarks. Once produced they will mainly
decay to a pair of $b$-quarks. The production cross-section of such scalars
via gluon fusion vary from $14~{}(77)$ fb to $0.2~{}(1.5)$ fb for
$m_{h_{2}^{0}}=800$ and $1500$ GeV respectively at 14 (27) TeV run of LHC.

Three more neutral Higgs bosons arise after SSB from the left and right-handed
doublets. Two of them have their origin in the left-handed doublet and one in
the right-handed doublet. The neutral Higgses originating from the left-handed
doublet are stable and once produced in the collider they can only contribute
to missing energy signature. These scalars can be good candidate for relic of
the Universe. While the scalar which arise from right-handed doublet can be
produced at the LHC in association with any of the neutral gauge boson ($Z$,
$Z^{\prime}$ or $A^{\prime}$) or via vector boson fusion process. The
production cross-section varies from $0.4$ fb to $0.22$ fb for a range of
Higgs mass $400$ to $1500$ GeV.

Two charged Higgs bosons will be the hallmark of the model. One of them,
$H_{1}^{\pm}$ comes from the bi-doublet and this particular charged state
mainly couples to a $t$ and a $b$ quark via Yukawa interactions. A lower limit
of 720 GeV has been derived on its mass from the LHC data. The estimated
cross-section for $H_{1}^{\pm}$ production in association with $tb$ varies
from $0.15~{}(1)$ pb and $0.005~{}(0.06)$ pb for $m_{H_{1}^{\pm}}=720$ and
$1500$ respectively GeV at 14 (27) TeV center of mass energy at the LHC. The
rest of the charged states have origin in the left-handed doublet. They can be
produced at the LHC in a mechanism similar to Drell-Yan. Once produced they
will not decay. But being charged, they will leave their signature in the
detector via an ionising track. In SUSY models, similar signal are produced by
stable/long-lived stau. Such a signal has been looked for at the LHC by ATLAS
collaboration. A lower limit on the mass of the charged Higgs
$H_{L}^{\pm}~{}(>494)$ GeV has been derived using the ATLAS data. We further
investigate the pair-production of $H_{L}^{\pm}$ and associated production of
$H_{L}^{\pm}~{}H_{L}^{0}~{}(\xi_{L}^{0})$ at LHC.

The last menu in our list is the singlet Higgs. It decays dominantly to a pair
gluons and exotic fermions. We consider its production via gluon fusion at the
LHC. However, production cross-section of a singlet Higgs via gluon fusion is
inversely proportional to singlet vev, $v_{S}^{2}$. However, $v_{S}$ being in
the ball park of 13 TeV, singlet Higgs production via gluon fusion fall below
the level of a fb for a 1.5 TeV singlet Higgs even at 27 TeV run of the LHC.

Finally we would like to point out that, the Higgs sector of this model
promises interesting phenomenology. A detail signal-background analysis has
been already in our agenda [38]. Finally, the neutral $SU(2)$ singlet lepton
$N$, Higgs bosons $h^{0}_{L}$ and $\xi^{0}_{L}$ can serve the purpose of
relic. It is important to see whether they can satisfactorily fulfil the
constraints from the experimental data on relic density and direct detection
of dark matter [39].

Acknowledgement : SB acknowledges financial support from DST-SERB, Govt. of
India in form of an INSPIRE-Senior Research Fellowship. Both the authors are
grateful to Prof. Joydeep Chakrabortty for introducing the subject to them and
taking part in the initial stage of the work. They also thank Dr. Triparno
Bandyopadhyay for several insightful discussions and comments. SB also
acknowledges Dr. Tapoja Jha and Debabrata Bhowmik for useful discussions. Both
of us are grateful to Prof. Amitava Raychaudhuri for several discussions on
the symmetries of scalar potential.

##  5 Appendix:

##  Appendix A Masses and mixings in the particle sector of 32121 model:

Neutral CP-even scalars:

$\displaystyle\begin{pmatrix}h_{1}^{0}\\\ h_{2}^{0}\\\ h_{L}^{0}\\\
h_{R}^{0}\\\
h_{S}^{0}\end{pmatrix}=\begin{pmatrix}\cos\theta&0&0&0&\sin\theta\\\
0&1&0&0&0\\\ 0&0&1&0&0\\\ 0&0&0&1&0\\\
-\sin\theta&0&0&0&\cos\theta\end{pmatrix}\begin{pmatrix}h^{0}\\\ h_{2}^{0}\\\
h_{L}^{0}\\\ H_{R}^{0}\\\ H_{S}^{0}\end{pmatrix}$ (34)

where,

$\theta=\dfrac{1}{2}\tan^{-1}\left(\dfrac{\beta_{1}k_{1}v_{S}}{\alpha_{1}v_{S}^{2}-\lambda_{1}k_{1}^{2}}\right)$

The basis on the LHS is the gauge eigenstates of the neutral CP-even scalars
and the basis on the RHS shows the mass eigenstates, where $\theta$ is the
mixing angle.

Charged scalars:

$\displaystyle\begin{pmatrix}h_{1}^{+}\\\ h_{2}^{+}\\\ h_{L}^{+}\\\
h_{R}^{+}\end{pmatrix}=\begin{pmatrix}c_{11}&0&0&c_{14}\\\ 0&1&0&0\\\
0&0&1&0\\\ c_{41}&0&0&c_{44}\end{pmatrix}\begin{pmatrix}H_{1}^{+}\\\
H_{2}^{+}\\\ H_{L}^{+}\\\ H_{R}^{+}\end{pmatrix}$ (35)

where,

$c_{11}=\sqrt{1-\dfrac{k_{1}^{2}}{v_{R}^{2}}}=c_{44},c_{14}=\dfrac{k_{1}}{v_{R}}=-c_{41}$

Neutral gauge sector:

$\displaystyle\begin{pmatrix}W_{3L}\\\ W_{3R}\\\ B_{L}\\\
B_{R}\end{pmatrix}=\begin{pmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\\
a_{21}&a_{22}&a_{23}&a_{24}\\\ a_{31}&a_{32}&a_{33}&a_{34}\\\
a_{41}&a_{42}&a_{43}&a_{44}\end{pmatrix}\begin{pmatrix}Z\\\ Z^{\prime}\\\
A^{\prime}\\\ A\end{pmatrix}$ (36)

where $A$ is the photon and $a_{ij}$ are the elements of the mixing matrix or
the rotational matrix that rotates the gauge basis to mass basis.

$\displaystyle
a_{11}=\cos\theta_{W},a_{21}=\dfrac{-g^{\prime}\sin\theta_{W}}{g_{2R}},a_{31}=\dfrac{-g^{\prime}\sin\theta_{W}}{g_{1L}},a_{41}=\dfrac{-g^{\prime}\sin\theta_{W}}{g_{1R}}$
$\displaystyle
a_{14}=\sin\theta_{W},a_{24}=\sin\theta_{W},a_{34}=\dfrac{\sqrt{\cos
2\theta_{W}}}{\sqrt{2}},a_{44}=\dfrac{\sqrt{\cos 2\theta_{W}}}{\sqrt{2}}$
$\displaystyle a_{12}=-1.643\times
10^{-4},a_{22}=0.704,a_{32}=-0.707,a_{42}=5.457\times 10^{-2}$ $\displaystyle
a_{13}=2.255\times 10^{-5},a_{23}=-0.450,a_{33}=-0.386,a_{43}=0.804$

##  Appendix B Couplings of the BSM scalars in 32121 model:

For $h_{2}^{0}/\xi_{2}^{0}$:

$M_{h_{2}^{0}}$ (GeV) | $\dfrac{1}{\sqrt{2}}\sqrt{4\lambda_{3}k_{1}^{2}+(c_{4}-c_{3})v_{R}^{2}}$ |  $M_{\xi_{2}^{0}}$ (GeV) | $\dfrac{1}{\sqrt{2}}\sqrt{4\lambda_{3}k_{1}^{2}+(c_{4}-c_{3})v_{R}^{2}}$
---|---|---|---
$h_{2}^{0}b\bar{b}$ coupling | $\dfrac{y_{t}}{\sqrt{2}}$ | $\xi_{2}^{0}b\bar{b}$ | $\dfrac{y_{t}}{\sqrt{2}}\gamma^{5}$
$h_{2}^{0}H^{\pm}W^{\mp}$ | $\dfrac{g_{2L}}{2}\sqrt{1-\dfrac{k_{1}^{2}}{v_{R}^{2}}}$ | $\xi_{2}^{0}H^{\pm}W^{\mp}$ | $\dfrac{g_{2L}}{2}\sqrt{1-\dfrac{k_{1}^{2}}{v_{R}^{2}}}$
$h_{2}^{0}t\bar{t}$ | $\dfrac{y_{b}}{\sqrt{2}}$ | $\xi_{2}^{0}t\bar{t}$ | $\dfrac{y_{b}}{\sqrt{2}}\gamma^{5}$

For $H_{R}^{0}$:

$M_{H_{R}^{0}}$ (GeV) | $\sqrt{2\rho_{1}v_{R}^{2}}$
---|---
$H_{R}^{0}h^{0}h^{0}$ | $-(c_{1}+c_{4})\cos^{2}\theta v_{R}-\gamma_{1}\sin^{2}\theta v_{R}$
$H_{R}^{0}H_{L}^{\pm}H_{L}^{\mp}$ | $-\rho_{3}v_{R}$
$H_{R}^{0}h_{L}^{0}h_{L}^{0}/H_{R}^{0}\xi_{L}^{0}\xi_{L}^{0}$ | $-\rho_{3}v_{R}$
$H_{R}^{0}h^{0}H_{S}^{0}$ | $(\gamma_{1}-c_{1}-c_{4})\sin\theta\cos\theta v_{R}$

For $H_{S}^{0}$:

$M_{H_{S}^{0}}$ (GeV) | $\sqrt{2\alpha_{1}v_{S}^{2}}$
---|---
$H_{S}^{0}ZZ$ | $[\dfrac{g_{2L}^{2}\cos^{2}\theta_{W}k_{1}}{2}+g_{2L}g^{\prime}\sin\theta_{W}\cos\theta_{W}k_{1}+\dfrac{g^{\prime 2}\sin^{2}\theta_{W}k_{1}}{2}]\sin\theta$
$H_{S}^{0}h^{0}h^{0}$ | $-\beta_{1}\sin^{3}\theta k_{1}+2(\beta_{1}-3\lambda_{1})\sin\theta\cos^{2}\theta k_{1}$
| $+2(\beta_{1}-3\alpha_{1})\sin^{2}\theta\cos\theta
v_{S}-\cos^{3}\theta\beta_{1}v_{S}$
$H_{S}^{0}W^{\pm}W^{\mp}$ | $\dfrac{g_{2L}^{2}\sin\theta k_{1}}{2}$
$H_{S}^{0}t\bar{t}$ | $\dfrac{y_{t}\sin\theta}{\sqrt{2}}$
$H_{S}^{0}E^{\pm}E^{\mp}(H_{S}^{0}NN)$ | $\sqrt{2}y_{LB}\cos\theta$
$H_{S}^{0}q_{S}\bar{q}_{S}$ | $\dfrac{y_{s}\cos\theta}{\sqrt{2}}$

For $H_{1}^{\pm}$:

$M_{H_{1}^{\pm}}$ (GeV) | $\dfrac{1}{\sqrt{2}}\sqrt{(c_{4}-c_{3})(k_{1}^{2}+v_{R}^{2})}$
---|---
$H_{1}^{+}\bar{t}b$ | $-(y_{t}+y_{b})\sqrt{1-\dfrac{k_{1}^{2}}{v_{R}^{2}}}$
$H_{1}^{+}h_{2}^{0}W^{-}/H_{1}^{+}\xi_{2}^{0}W^{-}$ | $\dfrac{g_{2L}}{2}\sqrt{1-\dfrac{k_{1}^{2}}{v_{R}^{2}}}$

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