Emergent 2HDM in LSS Little-Higgs: Musings from Flavor and

Electroweak Physics

a) Nilanjana<EMAIL_ADDRESS>a) b)Soumya Sadhukhan
<EMAIL_ADDRESS>
a) Department of Physics and Astrophysics, University of Delhi, New Delhi,
110007, India
b) Ramakrishna Mission Residential College (Autonomous), Narendrapur, Kolkata
700103, India

###### Abstract

The low energy effective theory ($\sim$ TeV) of the little-Higgs model with
$SU(6)/Sp(6)$, as proposed by Low, Skiba and Smith (LSS), exhibits a two-Higgs
doublet model (2HDM) structure. The symmetry dictates interesting Yukawa
patterns, translating to non-trivial fermion couplings with both of the Higgs
doublets. The couplings of the scalars with the fermions can induce flavor
changing neutral currents (FCNC), which get constraints from flavor physics
observables such as BR$(B\rightarrow X_{s}\gamma)$, $B_{s}-\overline{B}_{s}$
mixing etc. The precision measurement of $Zb\overline{b}$ vertex, the top and
Higgs mass along with other Higgs coupling measurements at the Large Hadron
Collider (LHC) also enforce severe restrictions on the LSS model. Direct LHC
search results of beyond the Standard Model (BSM) particles also impose bounds
on the masses. We probe the LSS model in view of the above constraints through
a random scan in the multi-dimensional parameter space. We observe, on
contrary to the general 2HDM scenario, the emergent 2HDM from the LSS model is
less constrained from the flavor data and the $Zb\overline{b}$ measurement but
is severely constrained form the electroweak (EW) searches at the LHC. From
the flavor data and $Zb\overline{b}$, we find that the charged Higgs mass is
relaxed with $\tan\beta$ being restricted to $0.5-5$, whereas the charged
Higgs mass is pushed to larger than 1 TeV along with $\tan\beta$ being further
restricted to $<3$ when the LHC bounds are incorporated.

##  1 Introduction

Even if the Standard Model (SM) is the favorite candidate to explain all the
results obtained at the LHC, models with extended symmetries beyond the SM are
of great interest due to their elegant UV completion along with the power of
stabilizing the scalars against radiative corrections, i.e. solving the gauge
hierarchy problem [1]. There are different broad classes of models with
enhanced symmetries. Firstly, the models where the space-time symmetry is
enhanced to incorporate the spin, also to have symmetries that even relate the
bosons and the fermions; are called supersymmetric models [2]. Another set of
models also possess enhanced gauge symmetries which are systematically broken,
sometimes in multiple steps, to emerge the Higgs as a pseudo Nambu-Goldstone
boson (pNGB). In these models, such as little Higgs [3] and composite Higgs
[4] scenarios, the extended scalar sector manifests from the Goldstone bosons
and unbroken symmetry does contain the SM electro-weak group $SU(2)\times
U(1)$. In these sets of models, where the Higgs boson emerges as a “little”
part of a bigger representation and when some global symmetry is broken by the
interplay between two or more coupling constants, are termed as little Higgs
models (LH) [3] and many different variations of this model have been proposed
in literature [5, 6, 7, 8]. This collective symmetry breaking is the essential
ingredient in the little Higgs theories in order to free the Higgs mass-
squared from quadratic divergences at one loop. On the other hand, in some
models, when the enlarged global symmetry is broken by some strong dynamics
and the Higgs emerges as the pseudo Nambu-Goldstone boson of that enlarged
global symmetry, is termed composite Higgs models [9, 10, 11, 12]. Here, one
has to assume partial compositeness [13] of the SM fermions in order to
generate their masses.

Due to the presence of the Nambu Goldstone bosons (NGB) in the little Higgs
theories [14, 15] and the composite Higgs models [16, 17, 18, 19], it is
natural that they can usher in an extended scalar sector at TeV scale. Some
scenarios among these models represent an effective 2HDM [20] framework at the
lower energy scale ($\sim$ TeV). We take particular interest to study the case
where the global symmetry $SU(6)$ is broken to an $Sp(6)$ to provide an
extended LH group structure $SU(6)/Sp(6)$, which is one variation of the
little Higgs scenario developed by Low, Skiba and Smith (LSS Model) [21].
Similarly, there are composite Higgs models based on the same group structure
$SU(6)/Sp(6)$ [16, 17], where, similar to the little Higgs model, the low-
energy effective theory exhibits a 2HDM structure and in addition, beyond
standard model (BSM) particles exist. Even though the structure of these two
models are alike, there are some differences between these two sets of models.
The gauged subgroup in the $SU(6)/Sp(6)$ little Higgs model is more extended
than the electroweak (EW) subgroup and the additional gauge generators are
obtained via the symmetry breaking at the condensation scale ($f$). Also, the
technical difference in the little Higgs and composite Higgs scenario is the
generation of the scalar potential terms with different quartic terms due to
the non-identical Yukawa structures of these two models. Hence, the
dependencies of the emergent 2HDM scalar potential coefficients on different
set of model parameters are not similar in both cases. In this work, we choose
to explore the LSS model with an emergent 2HDM structure at low energy in the
view of recent flavor and LHC data.

In addition to the extended scalar sector, the BSM sector of the LSS model
includes new fermions, such as one vector-like quark doublet and two vector-
like quark singlets along with new gauge bosons. Natural realization of the
LSS model in the low energy effective theory can be executed through the
alignment limit of the emergent 2HDM [21, 22]. In the familiar 2HDM [20], the
Yukawa sector is organized arbitrarily using the $Z_{2}$ symmetry, according
to the phenomenological requirements, whereas, in the little Higgs models, a
larger symmetry dictates it. On the other hand, in the LSS scenario, the
Yukawa sector which is initially arranged in terms of the fermionic multiplets
aka sextets, directs the pattern of the Yukawa sector in emergent 2HDM after
the symmetry breaking. Moreover, ultraviolet completion of the 2HDM is
possible through the LSS model with an extended gauge sector $SU(6)/Sp(6)$.
Therefore, a study of this model can reveal the effects of larger symmetry on
the experimental observables, that are unlikely to manifest in ad hoc generic
2HDM. A comprehensive phenomenological analysis of the LSS model is shown in
Ref: [22], considering 8 TeV LHC results. But the extended scalar sector of
the LSS model demands a detailed study of collider phenomenology as well as
the effects of BSM scalars in flavor physics observables. In this paper, we
discuss the effective 2HDM emerging from the LSS model at low energy in the
context of flavor and electroweak (EW) observables, along with constraints
from 13 TeV LHC.

In light of the significant tension between the SM and experimental
measurements of lepton flavor universality (LFU) observables [23], it is
worthwhile to look for a NP model, capable of explaining the discrepancies. In
the LH models, the presence of new particles with possible non-trivial flavor
structure can potentially address the challenges in the flavor sector. The
flavor structure of different little Higgs models had been studied in the
literature through multiple variants [24, 25, 26], along with some
phenomenological studies [27]. The major focus of these models had been to
probe the effects of vectorlike fermions and the gauge bosons, significantly
modifying the flavor observables. The SM extensions with two Higgs doublets,
such as Refs: [28, 29] and various other scalar extensions of the SM [30, 31]
can also address the flavor anomalies with the presence of the extra scalars
in the theory. This motivates us to particularly investigate the effect of the
additional scalars of the LSS model in the flavor sector. It is worthwhile to
study how the neutral and the charged BSM scalars can modify the flavor and
other EW observables and therefore impose constraints on the emergent 2HDM
parameters derived from the LSS little Higgs model. At first, we assess the
impact of different flavor observables and the $Zb\overline{b}$ vertex
correction. Amongst all BSM scalars present, we take particular interest to
study the couplings of the charged Higgs to the third generation quarks, to
estimate the modifications in specific flavor observables. We observe that the
$B$-meson mixing, $B$-meson decays and correction to the $Zb\overline{b}$
vertex impose important constraints on the model parameter space.

In the LSS little-Higgs model, the lightest CP-even neutral state is
identified with 125 GeV Higgs, observed at the LHC. Precise measurements of
top and bottom mass along with other measurements in the Higgs sector place
non-trivial constraints on the model parameter space. Non-observation of new
states at the LHC [32, 33] and the EW precision tests (EWPT) [34] place strong
limits on the BSM sector and hence also constrain the LSS model. We determine
the allowed regions of $\tan\beta$ and charged Higgs mass in the emergent 2HDM
structure of the LSS model as compared to a generic 2HDM and study the crucial
phenomenological differences. In contrast to the generic 2HDM with specific
Yukawa structure (type-I,II,III 2HDM etc), where the flavor constraints play a
major role in ruling out a large parameter space, in the emergent 2HDM of the
LSS model, constraints from EW searches at the LHC are found to be more severe
than those coming from the flavor and $Zb\overline{b}$. This is due to the
predictive nature of the Yukawa sector in the LSS model that shrinks the
parameter space mainly to fix the Higgs mass, top quark mass and top quark
Yukawa coupling.

In section 2, we describe the Yukawa interaction in the LSS model to obtain
the couplings of the scalar to the SM and BSM fermions. Then we show how an
effective 2HDM framework can be worked out in the LSS model, connecting the
strong sector parameters of the LSS model to the entities of the emergent
2HDM. We also show how different Yukawa structures can be arranged in some
limits of the bottom quark Yukawa couplings. In section 3, we focus on the
different constraints from flavor and EW observables in the 2HDM framework of
LSS model. First, we consider all the relevant flavor and EW observables,
discussing their possible origin and effects. Then, we scan over the entire
model parameter space of the LSS model, curving out the region favored by the
flavor and EW observables. Next, we bring in the LHC measurements and discuss
their super-constraining effects on the parameter space, which is already
constrained from flavor and EW data. Finally, in section 4, we chart out
relative importance of our results in comparison to a generic 2HDM.

##  2 Low-Skiba-Smith (LSS) Little Higgs Model

The little Higgs model proposed by Low, Skiba and Smith i.e. the LSS model
[21] exhibits a larger symmetry $SU(6)$ in the unbroken form, which is broken
to a residual symmetry $Sp(6)$ by the field condensates. The number of
massless Goldstone bosons that are expected to appear due to the symmetry
breaking, is equal to the difference in the number of generators, i.e.
$35-21=14$. The NGB’s ($\pi^{a}$) are contained in
$\Sigma=e^{[i\pi^{a}X^{a}/f]}\langle\Sigma\rangle$, where
$\langle\Sigma\rangle$ is the anti-symmetric condensate and $X^{a}$’s are the
broken generators, as given in [21]. These Goldstone bosons obtain relatively
small masses from the radiative corrections and therefore can be termed as
pseudo-Nambu Goldstone bosons (pNGB). Eight of these pNGB’s form two scalar
doublets ($\phi_{1}$ and $\phi_{2}$), which, at the TeV scale, is similar to
the scalar doublets of general 2HDM by construction. One major difference
between a general 2HDM and the emergent 2HDM in the LSS model is the
appearance of the vector like fermions and gauge bosons in the LSS model.
These BSM particles along with the 2HDM structure in the LSS model can
significantly alter the BSM scalar phenomenology, as shown in earlier works
[22, 35]. In the following, we discuss the Yukawa couplings of the scalars and
its effective 2HDM analysis. For a detailed description of the model we refer
to [21, 22].

###  2.1 Yukawa Sector: Diagonalization of Mass Matrices

Diverse nature of the Yukawa sector in the LSS model requires special
attention, as it has immense potential to churn out intricacies of flavor and
other EW physics. Here, construction of the Yukawa sector involves an
extension beyond the SM electroweak symmetry breaking. In the LSS model, the
$SU(6)$ symmetry is broken explicitly by the gauge and Yukawa couplings in
multiple steps: termed as collective symmetry breaking. In the fermion sector,
this is ensured by a special structure of the Yukawa couplings [21]. In this
model, we mainly focus on the couplings of the scalars with the third
generation of fermions. Also, in this study we have adopted a different
nomenclature for the fields compared to [22] and hence we show the following
steps for clarity. The Yukawa Lagrangian is,

$\displaystyle{\cal
L}_{Yuk}=y_{1}f\begin{pmatrix}Q^{\prime}&t^{\prime\prime}&(i\sigma^{2}Q)^{T}&0\end{pmatrix}(\Sigma)^{*}\begin{pmatrix}0\\\
t^{c}\end{pmatrix}+y_{2}f\begin{pmatrix}0&0&Q^{T}&0\end{pmatrix}(\Sigma)\begin{pmatrix}i\sigma^{2}Q^{\prime\,c}\\\
t^{\prime\prime\,c}\\\ 0\\\ b^{\prime\prime\,c}\end{pmatrix}$ $\displaystyle-
iy_{1b}f\begin{pmatrix}0&0&Q^{T}&0\end{pmatrix}(\Sigma)\begin{pmatrix}0\\\
0\\\ 0\\\
b^{c}\end{pmatrix}+iy_{2b}f\begin{pmatrix}0&0&(i\sigma_{2}Q)^{T}&0\end{pmatrix}(\Sigma)^{*}\begin{pmatrix}0\\\
b^{c}\\\ 0\\\ 0\end{pmatrix}+\ .{\rm h.c.}\ .$ (1)

The new fermions are one vector-like quark doublet Weyl-fermion pair
$Q^{\prime}=(t^{\prime},b^{\prime})^{T},{Q^{\prime}}^{c}=(-{b^{\prime}}^{c},{t^{\prime}}^{c})^{T}$
with $Y=1/6$ and EM charge $2/3$, one vector-like up-type quark singlet
$t^{\prime\prime},t^{\prime\prime\,c}$ with EM charge $\pm 2/3$, and one
vector-like down-type quark singlet $b^{\prime\prime},b^{\prime\prime\,c}$
with EM charge $\mp 1/3$. $Q=(t,b)^{T}$ is the SM SU(2) doublet. The $y_{i}$
and $y_{ib}$ (for $i=1,2$) are the dimensionless couplings in the top and
bottom sector respectively. Expanding the SU(2) structure of the Yukawa
couplings and including the vector-like fermion masses we get,

$\displaystyle{\cal L}_{\rm mass+Yuk}\supset$ $\displaystyle-
y_{1}\left(ft^{\prime\prime}t^{c}-i{Q^{\prime}}^{T}\phi_{2}^{*}t^{c}-iQ^{T}\cdot\phi_{1}t^{c}\right)+y_{2}\left(fQ^{T}\cdot{Q^{\prime}}^{c}+iQ^{T}\phi_{1}^{*}b^{\prime\prime\,c}+iQ^{T}\phi_{2}^{*}t^{\prime\prime\,c}\right)$
(2)
$\displaystyle+y_{3}f({Q^{\prime}}^{T}\cdot{Q^{\prime}}^{c})+y_{4}f(t^{\prime\prime\,c}t^{\prime\prime})+y_{5}f(b^{\prime\prime\,c}b^{\prime\prime})+y_{1b}(Q^{T}\phi_{1}^{*}b^{c})-y_{2b}(Q^{T}.\phi_{2}b^{c})+h.c..\
,$

Here $y_{3}$, $y_{4}$ and $y_{5}$ are also dimensionless constants. From Eq.
(2), we can infer the fermion mass matrices after electroweak symmetry
breaking (EWSB). The fermion mass matrices with EM charge $+2/3$ and $-1/3$
appears in the Lagrangian as,

${\cal
L}\supset\begin{pmatrix}t&t^{\prime}&t^{\prime\prime}\end{pmatrix}\begin{pmatrix}iy_{1}\frac{v_{1}}{\sqrt{2}}&y_{2}f&iy_{2}\frac{v_{2}}{\sqrt{2}}\\\
iy_{1}\frac{v_{2}}{\sqrt{2}}&y_{3}f&0\\\
-y_{1}f&0&y_{4}f\end{pmatrix}\begin{pmatrix}t^{c}\\\ {t^{\prime}}^{c}\\\
t^{\prime\prime\,c}\end{pmatrix}+\begin{pmatrix}b&b^{\prime}&b^{\prime\prime}\end{pmatrix}\begin{pmatrix}y_{ib}\frac{v_{i}}{\sqrt{2}}&y_{2}f&iy_{2}\frac{v_{1}}{\sqrt{2}}\\\
0&y_{3}f&0\\\ 0&0&y_{5}f\end{pmatrix}\begin{pmatrix}b^{c}\\\
{b^{\prime}}^{c}\\\ b^{\prime\prime\,c}\end{pmatrix}+{\rm h.c.}\ ,$ (3)

where $v_{i}=\\{v_{1},v_{2}\\}$, are $vev$’s of $\phi_{1}$ and $\phi_{2}$
respectively. After diagonalizing Eq. (3), we get couplings of the scalars to
the fermions for the top sector, in their mass basis. We implement a two-step
diagonalization, where first the $f$-dependent terms are diagonalized
analytically, and then the $v_{1,2}$ dependent terms (appear after EWSB) are
diagonalized. We define the rotations that diagonalize the $f$ dependent terms
as,

$\displaystyle\begin{pmatrix}t\\\ t^{\prime}\\\
t^{\prime\prime}\end{pmatrix}=\begin{pmatrix}c_{23}&-s_{23}&0\\\
s_{23}&c_{23}&0\\\ 0&0&1\end{pmatrix}\begin{pmatrix}t_{0}\\\ t_{1}\\\
t_{2}\end{pmatrix};\ \begin{pmatrix}t^{c}\\\ {t^{\prime}}^{c}\\\
t^{\prime\prime\,c}\end{pmatrix}=\begin{pmatrix}c_{14}&0&-s_{14}\\\ 0&1&0\\\
s_{14}&0&c_{14}\end{pmatrix}\begin{pmatrix}t_{0}^{c}\\\ t_{1}^{c}\\\
t_{2}^{c}\end{pmatrix};\ \begin{pmatrix}b\\\ b^{\prime}\\\
b^{\prime\prime}\end{pmatrix}=\begin{pmatrix}c_{23}&-s_{23}&0\\\
s_{23}&c_{23}&0\\\ 0&0&1\end{pmatrix}\begin{pmatrix}b_{0}\\\ b_{1}\\\
b_{2}\end{pmatrix},$ (4)

with $s_{23}\equiv\sin\theta_{23}=y_{2}/(\sqrt{y_{2}^{2}+y_{3}^{2}})$,
$c_{23}\equiv\cos\theta_{23}=-y_{3}/(\sqrt{y_{2}^{2}+y_{3}^{2}})$, and
$s_{14}\equiv\sin\theta_{14}=y_{1}/(\sqrt{y_{1}^{2}+y_{4}^{2}})$. After these
rotations, some field redefinitions are performed to make the mass matrix
entries real and positive.333 The field redefinitions are used, following the
notation in Ref: [22]: $t_{0}^{c}=i\tilde{t}_{0}^{c}$,
$t_{1}^{c}=-i\tilde{t}_{1}^{c}$, $t^{\prime\,c}=-\tilde{t}^{\prime\,c}$,
$\psi_{1}=i\tilde{\psi}_{1}$, and, $b_{1}=-\tilde{b}_{1}$,
$\psi_{2}^{c}=-i\tilde{\psi}_{2}^{c}$, and $\psi_{2}=i\tilde{\psi}_{2}$. For
brevity of notation, tildes on the fields are dropped and we denote the fields
$\tilde{\chi}_{i}$ simply as $\chi_{i}$ in the following. Hence the mass
matrices in the Lagrangian take the form,

${\cal L}^{\rm
mass}\supset\begin{pmatrix}t_{0}&t_{1}&t_{2}\end{pmatrix}\begin{pmatrix}{\cal
M}^{t}_{11}&0&{\cal M}^{t}_{13}\\\ {\cal M}^{t}_{21}&{\cal M}^{t}_{22}&{\cal
M}^{t}_{23}\\\ 0&0&-{\cal M}^{t}_{33}\end{pmatrix}\begin{pmatrix}t_{0}^{c}\\\
t_{1}^{c}\\\
t_{2}^{c}\end{pmatrix}+\begin{pmatrix}b_{0}&b_{1}&b_{2}\end{pmatrix}\begin{pmatrix}{\cal
M}^{b}_{11}&0&{\cal M}^{b}_{13}\\\ {\cal M}^{b}_{21}&{\cal M}^{b}_{22}&{\cal
M}^{b}_{23}\\\ 0&0&{\cal M}^{b}_{33}\end{pmatrix}\begin{pmatrix}b_{0}^{c}\\\
b_{1}^{c}\\\ b_{2}^{c}\end{pmatrix}+{\rm h.c.}\ .$ (5)

After the field redefinitions, the new mass matrix entries ${\cal
M}^{t,b}_{ij}$ are,

$\displaystyle{\cal M}^{t}_{11}$ $\displaystyle=$
$\displaystyle\frac{y_{1}(y_{3}y_{4}v_{1}+y_{2}y_{3}v_{2}-y_{2}y_{4}v_{2})}{\sqrt{y}_{14}\sqrt{y}_{23}\sqrt{2}}\
,\qquad{\cal
M}^{t}_{13}=\frac{(y_{1}^{2}y_{3}v_{1}-y_{2}y_{3}y_{4}v_{2}-y_{1}^{2}y_{2}v_{2})}{\sqrt{y}_{14}\sqrt{y}_{23}\sqrt{2}}\
,$ $\displaystyle{\cal M}^{t}_{21}$ $\displaystyle=$
$\displaystyle\frac{y_{1}(y_{2}y_{4}v_{1}+y_{2}^{2}v_{2}+y_{3}y_{4}v_{2})}{\sqrt{y}_{14}\sqrt{y}_{23}\sqrt{2}}\
,\qquad\ {\cal
M}^{t}_{23}=\frac{(y_{1}^{2}y_{2}v_{1}-y_{2}^{2}y_{4}v_{2}+y_{1}^{2}y_{3}v_{2})}{\sqrt{y}_{14}\sqrt{y}_{23}\sqrt{2}}\
,$ $\displaystyle{\cal M}^{t}_{33}$ $\displaystyle=$
$\displaystyle-f\sqrt{y}_{14}\ ,\qquad{\cal M}^{t}_{22}=f\sqrt{y}_{23}\ ,$
$\displaystyle{\cal M}^{b}_{11}$ $\displaystyle=$ $\displaystyle
c_{23}y_{ib}\frac{v_{i}}{\sqrt{2}}\ ,\qquad{\cal
M}^{b}_{13}=y_{2}\frac{v_{1}}{\sqrt{2}}\ ,\qquad{\cal M}^{b}_{33}=y_{5}f\ ,$
$\displaystyle{\cal M}^{b}_{21}$ $\displaystyle=$ $\displaystyle
y_{ib}\frac{v_{i}}{\sqrt{2}}s_{23}\ ,\qquad{\cal
M}^{b}_{23}=-y_{2}\frac{v_{2}}{\sqrt{2}}s_{23},\ {\cal
M}^{b}_{22}=\sqrt{y}_{23}f,$ (6)

where $\sqrt{y}_{14}\equiv\sqrt{y_{1}^{2}+y_{4}^{2}}$,
$\sqrt{y}_{23}\equiv\sqrt{y_{2}^{2}+y_{3}^{2}}$.

In the next step, we diagonalize the $v_{1,2}$ dependent terms of the matrix
${\cal M}^{t}_{ij}$, where the entries of ${\cal M}^{t}_{ij}$ are functions of
intrinsic parameters of the $SU(6)/Sp(6)$ LSS model. This is achieved through
carrying out a bi-orthogonal transformation, constructing left and right
orthogonal matrices $R_{L}$ and $R_{R}$ respectively, such that,

$\displaystyle R_{L}{\cal M}^{t}_{ij}R_{R}^{T}\equiv({\cal M}^{t}_{ij})_{D},$
(7)

where $({\cal M}^{t}_{ij})_{D}$ is the diagonal matrix in the mass basis.
Instead of showing explicit dependences of $R_{L},R_{R}$, we perform the
diagonalization along with the parameter space scan; where for each set of
input parameters, the diagonalization process is being performed. This process
is iterated multiple times keeping the parameter dependence of these matrices.
The transformation of the fields are defined as,

$\displaystyle(t_{0}\ t_{1}\ t_{2})^{T}=R_{L}^{T}(T_{2}\ T_{1}\ t)^{T},$
$\displaystyle(t_{0}^{c}\ t_{1}^{c}\ t_{2}^{c})^{T}=R_{R}^{T}(T_{2}^{c}\
T_{1}^{c}\ t^{c})^{T},$ (8)

where $T_{i},\ t$ and $T_{i}^{c},\ t^{c}$ are the mass eigenstates in the top
sector. We identify $(t,t^{c})$ as the observed top-quark ($t$) mass
eigenstates and the others correspond to the mass eigenstates of the
vectorlike top partners ($T_{1}$, $T_{2}$). In the bottom sector, the off-
diagonal terms proportional to $v_{i}$’s are numerically insignificant due to
the smallness of $y_{ib}$, and hence we do not perform the mass-
diagonalization. We identify $(b_{0},b_{0}^{c})$ as the observed bottom quark
($b$) and the others correspond to vectorlike bottom partners
($B_{1}$,$B_{2}$). We also identify the mass eigenvalues as
$m_{t},M_{T_{1}},M_{T_{2}}$ in the top sector, and $m_{b},M_{B_{1}},M_{B_{2}}$
in the bottom sector. In order to identify the eigenvalues corresponding to
$m_{t},M_{T_{1}},M_{T_{2}}$ and $m_{b},M_{B_{1}},M_{B_{2}}$, it is ensured
that $m_{t}<M_{T_{1}}<M_{T_{2}}$ and $m_{b}<M_{B_{1}}<M_{B_{2}}$.

###  2.2 Yukawa Couplings of the Scalars

In this section, we are mainly interested in couplings of the SM like Higgs
$h$ and the charged Higgs $H^{\pm}$ to the fermions because those couplings
affect the flavor and EW observables the most as per the model parameter space
of LSS model is concerned. We will discuss this further at the end of this
section. In the following we present the structure of the relevant Yukawa
couplings, i.e the couplings of $h$ and $H^{\pm}$ in detail.

Before the mass diagonalization, the top-quark Yukawa coupling with Higgs-like
scalar ($h$) in the flavor basis can be written as,

${\cal
L}_{htt}=\frac{i\,h}{\sqrt{2}}\begin{pmatrix}t&\psi_{1}&t^{\prime}\end{pmatrix}\begin{pmatrix}y_{1}c_{\alpha}&-y_{2}s_{\alpha}&0\\\
0&0&0\\\ -y_{1}s_{\alpha}&0&0\end{pmatrix}\begin{pmatrix}t^{c}\\\
\psi_{1}^{c}\\\ t^{\prime\,c}\end{pmatrix}+{\rm h.c.}\ .$ (9)

Then we rewrite the Lagrangian in the basis where the $f$-terms are
diagonal444In our convention, we define the top-quark Yukawa coupling
$y_{htt}$ as ${\cal
L}_{htt}=(h/\sqrt{2})y_{htt}\hat{t}_{0}\hat{t}_{0}^{c}+{\rm h.c.}$. We define
this with a positive sign here since our field redefinitions made the fermion
mass terms positive. . Also, we implement the field redefinitions as stated
previously, to render the real fermion mass matrix. Hence, the Yukawa coupling
of $h$ in an intermediate basis now takes the form,

${\cal L}^{\rm
Yuk}_{h}\supset\frac{h}{\sqrt{2}}\left[y_{00}t_{0}t_{0}^{c}+y_{01}t_{0}t_{1}^{c}+y_{10}t_{1}t_{0}^{c}+y_{11}t_{1}t_{1}^{c}\right]+{\rm
h.c.}\ ,$ (10)

with
$y_{00}\equiv(-y_{1}c_{\alpha}c_{14}c_{23}+y_{1}s_{\alpha}c_{14}s_{23}+y_{2}s_{\alpha}s_{14}c_{23})$,
$y_{01}\equiv(-y_{1}c_{\alpha}s_{14}c_{23}+y_{1}s_{\alpha}s_{14}s_{23}-y_{2}s_{\alpha}c_{14}c_{23})$,
$y_{10}\equiv(y_{1}c_{\alpha}c_{14}s_{23}+y_{1}s_{\alpha}c_{14}c_{23}-y_{2}s_{\alpha}s_{14}s_{23})$,
$y_{11}\equiv(y_{1}c_{\alpha}s_{14}s_{23}+y_{1}s_{\alpha}s_{14}c_{23}+y_{2}s_{\alpha}c_{14}s_{23})$.
Yukawa couplings after the bi-orthogonal rotations to the mass-basis, as
defined in Eq.(8) take the form,

$y_{htt}=\left[y_{00}(R_{L})_{31}(R_{R})_{31}+y_{01}(R_{L})_{31}(R_{R})_{32}+y_{10}(R_{L})_{32}(R_{R})_{31}+y_{11}(R_{L})_{32}(R_{R})_{32}\right]\
,$ (11)

where $(R_{L})_{ij}$ and $(R_{R})_{ij}$ with $i,j=\\{1,2,3\\}$, are the
$(i,j)$-th entry of the rotation matrix $R_{L}$ and $R_{R}$ respectively. We
define the deviation of $htt$ from its SM value as,

$k_{tth}\equiv y_{htt}/y_{htt}^{SM},$ (12)

where, $y_{htt}^{SM}=m_{t}/v$. Similarly, Yukawa couplings of the CP even
neutral scalar ($H$) can be obtained from Eq.(10) by making the change
$c_{\alpha}\rightarrow s_{\alpha}$ and $s_{\alpha}\rightarrow-c_{\alpha}$. For
the couplings of the pseudo scalar ($A$), we refer to [22].

The Yukawa coupling of the $b$-quark with $h$ and the mass of the $b$ quark
can be derived from Eq.(2) as,

${\cal
L}_{hbb}\supset\frac{c_{23}}{\sqrt{2}}\left[v\left(y_{1b}s_{\beta}+y_{2b}c_{\beta}\right)+h\left(y_{1b}c_{\alpha}-y_{2b}s_{\alpha}\right)\right]b_{0}b^{c}+{\rm
h.c.}\ .$ (13)

The coupling and the mass are thus identified to be;

$m_{b}\equiv v(y_{1b}s_{\beta}+y_{2b}c_{\beta})/\sqrt{2},~{}~{}~{}y_{hbb}\sim
m_{b}/v.$ (14)

Analogous mass and coupling expressions hold for the third generation lepton
($\tau$) with the replacement $y_{b}\rightarrow y_{\tau}$.

Similarly, couplings of the charged Higgs with the third generation quarks can
be obtained as,

${\cal L}^{Yuk}_{H^{\pm}}\supset
H^{+}\left(y^{+}_{00}b_{0}t_{0}^{c}+y^{+}_{01}b_{0}t_{1}^{c}+y^{+}_{10}b_{1}t_{0}^{c}+y^{+}_{11}b_{1}t_{1}^{c}\right)+H^{-}\left(y^{-}_{00}t_{0}b^{c}+y^{-}_{10}t_{1}b^{c}+y^{-}_{02}t_{0}\psi_{2}^{c}+y^{-}_{12}t_{1}\psi_{2}^{c}\right)+{\rm
h.c.}\ ,$ (15)

where,

$\displaystyle y^{+}_{00}$ $\displaystyle=$
$\displaystyle(y_{1}s_{\beta}s_{23}c_{14}-y_{1}c_{\beta}c_{23}c_{14}+y_{2}s_{\beta}c_{23}s_{14}),$
$\displaystyle y^{+}_{01}$ $\displaystyle=$
$\displaystyle(y_{1}s_{\beta}s_{23}s_{14}-y_{1}c_{\beta}c_{23}s_{14}-y_{2}s_{\beta}c_{23}c_{14}),$
$\displaystyle y^{+}_{10}$ $\displaystyle=$
$\displaystyle(-y_{1}s_{\beta}c_{23}c_{14}-y_{1}c_{\beta}s_{23}c_{14}+y_{2}s_{\beta}s_{23}s_{14}),$
$\displaystyle y^{+}_{11}$ $\displaystyle=$
$\displaystyle(-y_{1}s_{\beta}c_{23}s_{14}-y_{1}c_{\beta}s_{23}s_{14}-y_{2}s_{\beta}s_{23}c_{14}),$
$\displaystyle y^{-}_{00}$ $\displaystyle=$
$\displaystyle[(-y_{1b}c_{\beta}+y_{2b}s_{\beta})c_{23}],\hskip
5.69046pty^{-}_{10}=[(y_{1b}c_{\beta}-y_{2b}s_{\beta})s_{23}],$ $\displaystyle
y^{-}_{02}$ $\displaystyle=$ $\displaystyle(-y_{2}c_{\beta}c_{23}),\hskip
54.06006pty^{-}_{12}=(y_{2}c_{\beta}s_{23}).$

After the numerical computation of the rotation matrices ($R_{L},R_{R}$), that
diagonalize the off-diagonal terms proportional to $v_{1,2}$, we obtain the
Lagrangian involving the charged Higgs couplings with top and bottom quark as:

${\cal L}_{H^{\pm}tb}\supset
V_{tb}\frac{1}{v}H^{+}\overline{t}\left(y_{H^{\pm}t_{L}b_{R}}m_{t}P_{R}+y_{H^{\pm}t_{R}b_{L}}m_{b}P_{L}\right)b+{\rm
h.c.}\ ,$ (16)

where the charged Higgs Yukawa couplings are given by,

$\displaystyle
y_{H^{\pm}t_{L}b_{R}}=\left[y^{-}_{00}(R_{L})_{31}+y^{-}_{10}(R_{L})_{32}\right],$
$\displaystyle
y_{H^{\pm}t_{R}b_{L}}=\left[y^{+}_{00}(R_{R})_{31}+y^{+}_{01}(R_{R})_{32}\right].$
(17)

The relevant Lagrangian involving the coupling of top and strange quarks with
the charged Higgs can be obtained in a similar way,

${\cal L}_{H^{\pm}ts}\supset
V_{ts}\frac{1}{v}H^{+}\overline{t}\left(y_{H^{\pm}t_{L}s_{R}}m_{t}P_{R}+y_{H^{\pm}t_{R}s_{L}}m_{s}P_{L}\right)s+{\rm
h.c.}\ ,$ (18)

where the $y_{H^{\pm}t_{L}s_{R}},y_{H^{\pm}t_{R}s_{L}}$ can be derived from
Eq.(2.2) with the replacement of $y_{1b}\rightarrow y_{1s},y_{2b}\rightarrow
y_{2s}$ and then use them in analogous to $y^{-}_{00},y^{-}_{10}$ terms in
Eq.(17). Note that, the first parts of both the couplings are similar and
being proportional to $m_{t}$, they are the dominant terms. The second parts
of the Lagrangian, being proportional to $m_{b}$ or $m_{s}$, are not
significant due to smallness of the masses. Moreover, the couplings are of the
same order, i.e, $y_{1b}\sim y_{1s}$, $y_{2b}\sim y_{2s}$. The coupling
involving the down quark, $H^{\pm}td$, will also hold similar expression but
the coupling is relatively small due to the smallness of $V_{td}$. Overall,
the couplings $H^{\pm}td$ and $H^{\pm}ts$ will be smaller compared to
$H^{\pm}tb$ 555The couplings with lighter fermions that we neglect here,
appears in squares in the flavor observables, and their effect is much smaller
compared to the anomalies present in the flavor observables..

Among the other couplings of the charged Higgs, the $H^{\pm}cs$ and
$H^{\pm}\tau\nu$ couplings can be obtained as,

${\cal
L}^{Yuk}\supset\left(y_{1c}c_{\beta}-y_{2c}s_{\beta}\right)H^{+}sc^{c}+\left(-y_{1s}c_{\beta}+y_{2s}s_{\beta}\right)H^{-}cs^{c}+\left(-y_{1\tau}c_{\beta}+y_{2\tau}s_{\beta}\right)H^{-}\nu\tau^{c}+{\rm
h.c.}\ .$ (19)

The charged Higgs decays dominantly through $H^{\pm}\rightarrow tb$ mode for
most of the parameter space, unless there is an abrupt cancellation between
different contributions of Eq.(16) [22]. Otherwise, the second most prominent
decay mode $H^{\pm}\rightarrow cs$ takes over. The leptonic decay mode
$H^{\pm}\rightarrow\tau^{\pm}\nu$ is not found to be very significant. Hence
the tree level charged current contributions from the diagrams containing
$H^{\pm}$ is negligible in this model.

The couplings of the neutral pseudoscalar ($A$) and the heavy scalar ($H$) are
studied in detail in Ref: [22], which shows that $H$ and $A$ couple to third
generation quarks strongly. Their couplings to the leptons, as well as first
and second generation of quarks are negligible. Hence, tree level FCNC
processes in meson decays, neutral meson mixing and other lepton flavor
violating decays are suppressed in this model. We discuss different structures
of the Yukawa couplings in an effective 2HDM framework and their effects on
FCNC’s, in the next section.

###  2.3 Effective 2HDM Framework of LSS model

If the radiative corrections due to the presence of extra heavier gauge
bosons, vectorlike quarks and the singlet scalars are included in the LSS
model, an effective scalar potential at the TeV scale mimics that of the 2HDM,
albeit with the absence of a number of scalar field combinations. The scalar
potential generated at one loop in the LSS model [21] is,

$\displaystyle{\cal
V}_{LSS}=m_{1}^{2}|\phi_{1}|^{2}+m_{2}^{2}|\phi_{2}|^{2}+(b^{2}\phi_{1}^{T}\cdot\phi_{2}+{\rm
h.c.})+\lambda_{5}^{\prime}|\phi_{1}^{T}\cdot\phi_{2}|^{2}\ ,$ (20)

where $\phi_{1}$ and $\phi_{2}$ are the $SU(2)$ scalar doublets with
hypercharge +1/2 and -1/2 respectively, and
$\phi^{T}_{1}.\phi_{2}=\phi_{1}^{T}i\sigma^{2}\phi_{2}$ is the anti-symmetric
product of the fields. Similar to the 2HDM case, the ratio of the VEV’s of the
scalar doublets $\phi_{1,2}$ are presented in terms of

$\tan\beta\equiv v_{1}/v_{2}=\sqrt{m_{2}^{2}/m_{1}^{2}}\ ,$ (21)

which is a function of the LSS scalar potential parameters $m_{i}^{2}$. The
physical masses of the scalars that constitute the 2HDM structure can be
expressed as,

$\displaystyle m_{A}^{2}=2b^{2}/\sin{(2\beta)}\ ;\qquad
m_{H^{\pm}}^{2}=m_{A}^{2}-\lambda_{5}^{\prime}v^{2}/2\ ,$ $\displaystyle
m^{2}_{H,h}=\frac{1}{2}\left[m_{A}^{2}\pm\sqrt{m_{A}^{4}-4(m_{A}^{2}-m_{H^{\pm}}^{2})m_{H^{\pm}}^{2}\sin^{2}{(2\beta)}}\right]\
.$ (22)

The parameters $m_{1}^{2}$, $m_{2}^{2}$, $b^{2}$ and $\lambda_{5}^{\prime}$
are functions of the $SU(6)/Sp(6)$ model Lagrangian parameters and can be
expressed as,

$\displaystyle\lambda_{5}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{cg_{1}^{2}\left[g_{2}^{2}+(c^{\prime}/c)y_{2}^{2}\right])}{g_{1}^{2}+g_{2}^{2}+(c^{\prime}/c)y_{2}^{2}}\
,\quad
b^{2}=\frac{3f^{2}}{8\pi^{2}}y_{1}^{2}y_{2}(y_{3}-y_{4})\log{\frac{\Lambda^{2}}{M_{f}^{2}}}\
,$ $\displaystyle m^{2}_{1\,f}$ $\displaystyle=$
$\displaystyle\frac{3f^{2}}{8\pi^{2}}(y_{1}^{2}-y_{2}^{2})(y_{3}^{2}-y_{4}^{2})\log{\frac{\Lambda^{2}}{M_{f}^{2}}}\
,\quad
m^{2}_{2\,f}=\frac{3f^{2}}{8\pi^{2}}(y_{1}^{2}y_{2}^{2}+y_{2}^{2}y_{5}^{2}-y_{2}^{2}y_{3}^{2}-y_{1}^{2}y_{4}^{2})\log{\frac{\Lambda^{2}}{M_{f}^{2}}}\
,$ $\displaystyle m^{2}_{1g}$ $\displaystyle=$ $\displaystyle
m^{2}_{2g}=\frac{3}{64\pi^{2}}\left[3g^{2}M_{g}^{2}\log{\frac{\Lambda^{2}}{M_{g}^{2}}}+g^{\prime\,2}M_{g^{\prime}}^{2}\log{\frac{\Lambda^{2}}{M_{g^{\prime}}^{2}}}\right]\
,\quad
m^{2}_{1s}=m^{2}_{2s}=\frac{\lambda_{5}^{\prime}}{16\pi^{2}}M_{s}^{2}\log{\frac{\Lambda^{2}}{M_{S}^{2}}}\
.$ (23)

$\lambda_{5}^{\prime}$ gets gauge (proportional to $g_{1}$ and $g_{2}$) as
well as fermionic contribution (proportional to $y_{2}$), whereas $b^{2}$
receives only fermionic contribution (proportional to $y_{i}$). In Eq. 23,
$m_{1}^{2}$ and $m_{2}^{2}$ get contributions from the scalars (s), gauge
bosons (g) and fermions (f) in the loop. The degrees of constructive or
destructive interference depend on the sign of the coefficients. $\Lambda$ is
the cut-off which is taken to be $4\pi f$, where $f$ is the energy scale
associated with $SU(6)/Sp(6)$ breaking. $M_{f}$ is the heavy vector-like
fermion mass-scale. The heavy gauge boson masses are
$M_{g}=f\sqrt{(g^{2}_{1}+g^{2}_{2})/2}$ and
$M_{g^{\prime}}=f\sqrt{(g^{\prime\,2}_{1}+g^{\prime\,2}_{2})/2}$. The singlet
scalar mass is $M_{s}=f\sqrt{c(g_{1}^{2}+g_{2}^{2})+c^{\prime}y_{2}^{2}}$,
where $c$ and $c^{\prime}$ are $O(1)$ parameters that depend on the details of
UV completion, as explained in Ref:[22].

How the two scalar doublets $\phi_{1}$ and $\phi_{2}$ couple to other lighter
fermions (lighter than the top) in the 2HDM, determines presence or absence of
FCNC’s in the theory. As tree level FCNC’s are tightly bound by experimental
absence, they can place stringent constraints on the model. The tree level
FCNC’s are absent in this emergent 2HDM from the LSS model, if the $Z_{2}$
symmetry is not broken. Symmetries in the LSS model compel top-quark to couple
to both $\phi_{1}$ and $\phi_{2}$ (see Eq.(1)), which breaks the $Z_{2}$
symmetry of the 2HDM in the top sector. This type of models with Type III 2HDM
flavor structure results in non-trivial FCNC’s in the third generation.

In the light fermionic Yukawa sector, $Z_{2}$ symmetry can be enforced, i.e.
either $\phi_{1}$ or $\phi_{2}$ couples to the fermions. With the Yukawa
interactions $y_{1(b,\tau,c)}\neq 0$ and $y_{2(b,\tau,c)}=0$, i.e. light down
type fermions being coupled only to $\phi_{1}$, stringent constraints from the
LHC such as $h\rightarrow b\overline{b},\tau\overline{\tau}$ measurements
become important. The alternate possibility is $y_{1(b,\tau,c)}=0$ and
$y_{2(b,\tau,c)}\neq 0$, which relaxes the earlier constraints. If this Yukawa
structure is adopted for the up type light fermions as well, this will
resemble a Type I 2HDM framework in the light fermion sector, while the top
sector will break it. The earlier framework $y_{1(b,\tau,c)}\neq 0$ and
$y_{2(b,\tau,c)}=0$, along with the up type coupling with $\phi_{1}$ is also a
Type I framework for the light fermions, but that is not so tenable from the
LHC constraints.

On the other hand, an alternative Yukawa pattern can be explored where
$y_{1}\neq 0$ for the up-type light fermions, them being coupled only to
$\phi_{1}$ and the down-type fermions only couple to $\phi_{2}$ i.e.
$y_{2}\neq 0$ for the down type. The light fermion sector in this scenario
will look like a Type II 2HDM set up. If the top couples to both $\phi_{1}$
and $\phi_{2}$, it breaks the Type II structure, as seen earlier. The
constraints from the $h\rightarrow b\overline{b},\tau\overline{\tau}$ at the
LHC are relaxed in this scenario. Hence these type of peculiarity in the
flavor structure are potentially ripe for non trivial implications in the
flavor physics sector. A detailed analysis of the impact of few hitherto
important flavor observables in the 2HDM sector is the core of this work.

##  3 Constraints from Flavor and EW physics

The LSS model can exhibit different Yukawa patterns, leading to different
2HDM-like scenarios. Depending on the flavor structure of the Yukawa
couplings, flavor changing-neutral currents can place important constraints on
the model. We study the FCNC’s involving different quarks, among which the
third generation ones are in particular non-trivial. In this model, the top-
quark couples to both $\phi_{1}$ and $\phi_{2}$. The Yukawa coupling involving
the bottom-quark depends on the model parameters $y_{1b}$ and $y_{2b}$, which
play a major role in the calculation of the flavor observables. The choice of
$y_{1b}\neq 0$, $y_{2b}\neq 0$ reflects the Type-III 2HDM-like scenario. We
have also considered alternative scenarios when $y_{1b}=0$, $y_{2b}\neq 0$ or
$y_{1b}=0$, $y_{2b}\neq 0$, which can lead to different 2HDM scenarios, as
mentioned in section 2.3 666For a detailed study of the flavor structure of
general 2HDM models, one might look at Ref [36].. Hence, in the LSS model we
study three cases:

Case I: $y_{1b}\neq 0$, $y_{2b}\neq 0$,

Case II: $y_{1b}=0$, $y_{2b}\neq 0$,

Case III: $y_{1b}\neq 0$, $y_{2b}=0$.

###  3.1 Flavor Observables and $Zb\overline{b}$

As reflected in the previous section, the effect of the Yukawa couplings
involving the third generation quarks have dominant contribution in flavor and
EW observables. Also, as mentioned earlier, the contribution of the tree level
FCNC can be avoided in a large model parameter space of the LSS model. Hence,
while discussing the FCNC’s, we will be focusing only on the loop level FCNC’s
in the following text. Note that, the vectorlike fermions also have
significant impact on the flavor and EW observables [37], but, in this note we
focus on the scalar sector of the effective 2HDM in the LSS model. We also
include constraints from $Zb\overline{b}$ measurement, while the S and T
parameters are kept well within limit by keeping the new gauge degrees of
freedom in the heavier side [22]. We discuss different flavor observables
dividing them into few broad categories.

#### Radiative B-meson decays:

In context of the 2HDM, the most stringent constraint comes from the radiative
$B$-meson decays, $B\rightarrow X_{s}\gamma$ ($B\rightarrow X_{d}\gamma$). The
latest experimental and the theoretical value of BR($B\rightarrow
X_{s}\gamma$) show discrepancy as indicated in Table 1. In the SM, the quark-
level transition is mediated by $W$ boson and $t$ quark exchange via
electromagnetic penguin diagram. The matrix element for this process at the
electroweak scale is governed by the dipole operator. The effective
Hamiltonian for this process is given by,

$\mathcal{H}_{eff}=-\frac{4G_{F}}{\sqrt{2}}V^{*}_{ts}V_{tb}\sum_{i=1}^{8}C_{i}(\mu)O_{i}(\mu).$
(24)

Here, $V_{ij}$ represents the relevant Cabibbo-Kobayashi-Maskawa (CKM)
factors. The $O_{i}$ are a complete set of renormalized dimension six
operators. They consist of six four-quark operators, $O_{1-6}$, the
electromagnetic dipole operator, $O_{7}$, and the chromo-magnetic dipole
operator, $O_{8}$. These operators are evolved from the electroweak scale down
to bottom mass scale using Renormalization Group Equations (RGE). The partial
decay width of the quark level transition is given by,

$\Gamma(b\rightarrow s\gamma)=\frac{\alpha
G_{F}^{2}m_{b}^{5}}{128\pi^{4}}|V^{*}_{ts}V_{tb}C_{7}(\mu_{b})|^{2}.$ (25)

The deviation in the observable is proportional to $\delta C_{7}(\mu_{b})$,
which can be expressed in terms of the effective Wilson coefficients at the
matching scale $(\mu_{W})$, given as $\delta C_{7}(\mu_{W})$ and $\delta
C_{8}(\mu_{W})$. The explicit expressions for $C_{7,8}$ can be found in the
literature [38]. In 2HDM like scenarios, such as in the LSS model, additional
contribution comes from the charged Higgs coupling to the quarks, given by

$\mathcal{L}=\frac{\sqrt{2}}{v}H^{+}\overline{t}[V_{ij}m_{ui}\lambda_{u}^{i}P_{L}d_{j}+V_{ij}m_{dj}\lambda_{d}^{j}P_{R}d_{j}]+h.c.\
.$ (26)

Now, the contribution of the charged Higgs in $\delta C_{7,8}$ can be
expressed as $\delta C_{7}(x)$ and $\delta C_{8}(x)$, computed as [39] [40],

$\displaystyle\delta C_{7,8}=\left[{\lambda_{t}^{2}\over
3}F^{(1)}_{7,8}(x)-\lambda_{t}\lambda_{b}F^{(2)}_{7,8}(x)\right],$ (27)

where, $x=\frac{\overline{m}_{t}^{2}(\mu_{W})}{m_{H^{\pm}}^{2}}$,
$\lambda_{t}\equiv y_{H^{\pm}t_{L}b_{R}}$ and $\lambda_{b}\equiv
y_{H^{\pm}t_{R}b_{L}}$, as given in Eq.(17). In the following text, we choose
a simpler notation and denote these effective Yukawa couplings as
$y_{Ht_{L}b_{R}}$ and $y_{Ht_{R}b_{L}}$. The detailed expression of the
function “$F$” is given in Ref [41][28].

The $B\rightarrow X_{s}\gamma$ branching ratio receives large contributions
from the charged Higgs couplings with the top and bottom quark. We impose the
limits on the Wilson coefficient to be in the range $-0.063\leq\delta
C_{7}+0.242~{}\delta C_{8}\leq 0.073~{}$. The theoretical and experimental
uncertainties are combined in a quadrature while deriving the limits. It is
interesting to note that the relative minus sign between the two contributions
(from $W$ and $H^{\pm}$ in the loop) in general 2HDM model gives a destructive
interference for some values of the model parameters [28]. Whereas, in the LSS
model, the coefficients ($\lambda_{t,b}$) depend on the parameters of the
strong sector and both destructive and constructive interference are possible
depending on the values of the LSS model parameters. This is also true for the
other observables as well. Charged Higgs also contributes to $b\rightarrow
d\gamma$ decay in the same manner. The coupling of the charged Higgs to the
top and down quark is also similar to Eq.(16), but this coupling is suppressed
in this model as $V_{ts}\gg V_{td}$.

#### Neutral meson mixing:

In the SM, neutral meson mixing occurs due to the box diagrams with two $W$
exchanges in the loop level. In the case of $B_{d}$ and $B_{s}$ mesons, the
hierarchical structure of the CKM matrix and the large mass of the top quark
imply that the mixing is dominated by the diagrams involving the top quarks.
In 2HDM like scenarios, the observables related to the neutral-meson mixing
receive charged Higgs contributions [42]. Additional diagrams are included by
replacing the $W$ lines with charged Higgs ones, yielding the contribution to
the mixing as,

$\Delta
m_{q}=\frac{G_{F}^{2}}{24\pi^{2}}(V_{tq}V^{*}_{tb})^{2}~{}\eta_{B}~{}m_{B}~{}m_{t}~{}f^{2}_{Bq}~{}I_{tot}(x_{W},x_{H},x_{H}),$
(28)
$I_{\textrm{tot}}=I_{WW}(x_{W})+A_{t}^{4}I_{HH}(x_{H},x_{H})+2A_{t}^{2}I_{WH}(x_{W},x_{H}),$
(29)

where, $x_{W}=\overline{m}_{t}^{2}/m_{W}^{2}$,
$x_{H}=\overline{m}_{t}^{2}/m_{H}^{2}$. $I_{WW}$, $I_{HH}$ and $I_{WH}$
indicate respectively the internal bosonic lines of the corresponding diagrams
with an external light quark $q=d,s$. For details of these Inami-Lim functions
and other parameters we refer to Ref [28, 43]. We consider the charged Higgs
contributions to the $B_{s}^{0}-\overline{B}_{s}^{0}$ mixing only, as the
constraints are stronger than $B_{d}^{0}-\overline{B}_{d}^{0}$ mixing [23].
Also, as mentioned earlier, the charged Higgs coupling to the top ($t$) and
down ($d$) quark is suppressed in our model. Both experimental measurements
and the SM predictions for $B_{s}^{0}-\overline{B}_{s}^{0}$ mixing are given
in Table:1. We obtain the total contributions to the mass splitting from the
$W^{\pm}$ bosons, Goldstones and the charged Higgs boson ($H^{\pm}$) through
the box graphs. Normalizing $\Delta m_{B_{s}}$ with respect to its SM
prediction, we obtain the allowed range at $2\sigma$ to be $0.675\leq{\Delta
m_{B_{s}}\over\Delta m^{\textrm{SM}}_{B_{s}}}\leq 1.265~{}$.

Table 1:  Standard model vs. experimental values of the flavor observables. Observables | SM value | Experimental value
---|---|---
$\textrm{Br}(B\rightarrow X_{s}\gamma)$ |  $(3.36\pm 0.23)\times 10^{-4}$ [23] |  $(3.32\pm 0.16)\times 10^{-4}$ [39]
$\Delta m_{B_{s}}$ |  $(17.757\pm 0.021)~{}\textrm{ps}^{-1}$ [23] |  $(18.3\pm 2.7)~{}\textrm{ps}^{-1}$ [23]
$R_{b}$ |  $0.21581\pm 0.00011$[44] |  $0.21629\pm 0.00066$ [34]
$B_{s}\rightarrow\mu\mu$ |  $(3.65\pm 0.23)\times 10^{-9}$ [45] |  $(3.0\pm 0.6^{+0.3}_{-0.2})\times 10^{-9}$ [46]

#### $\mathbf{Z\rightarrow b\overline{b}}$ vertex ($\mathbf{R_{b}}$):

The $Z\rightarrow b\overline{b}$ vertex has provided opportunities to search
for new physics contributions, due to the heavy masses involved in the loop.
The radiative corrections at the vertex might imply charged Higgs exchanges in
addition to the standard $W$ boson coupling with top and bottom. Precision
measurement of electroweak precision observable $Z\rightarrow b\overline{b}$
branching ratio is measured as a ratio,

$R_{b}={\Gamma(Z\rightarrow b\overline{b})\over\Gamma(Z\rightarrow\rm
hadrons)}~{}.$ (30)

The modifications in $R_{b}$ due to the charged Higgs contributions at one
loop is given by [30, 47],

$\displaystyle\delta R_{b}\simeq-0.7785~{}\delta g^{L}_{\textrm{new}}~{}.$
(31)

Here $\delta g^{L}_{\textrm{new}}$ is the modification in the
$Zb_{L}\overline{b}_{L}$ coupling, calculated from a combination of triangle
graphs where $H^{\pm}$ and the charged Goldstones float inside the loop. We
neglect the modification in the $Zb_{R}\overline{b}_{R}$ coupling because it
is proportional to $m_{b}^{2}$, compared to $Zb_{L}\overline{b}_{L}$ coupling
which is proportional to $m_{t}^{2}$ [30]. We constrain $\delta R_{b}$ in the
$2\sigma$ range, $-0.00086\leq\delta R_{b}\leq 0.00182~{}$.

#### Other flavor observables:

The flavor observables such as $B_{s}\rightarrow\mu\mu$ and
$B_{d}\rightarrow\mu\mu$ can also show sizable effect in generic 2HDM. But
these observables will have low impact in the LSS model, as $H^{\pm}l\nu$
coupling is small, resulting in very tiny BR$(H^{\pm}\rightarrow l\nu)$ as
discussed in the previous section. In general, standard 2HDM scenarios require
very large values of the $H^{+}l\nu$ couplings with small charged Higgs mass
[36] in order to explain the current experimental values (Table1) of
$B_{s,d}\rightarrow\mu\mu$. This is very unlikely to happen in the LSS model.
The other robust predictions from flavor observables such as $R_{D}$,
$R_{D^{*}}$, $R_{K}$, $R_{K^{*}}$ and meson decays depend on BSM scalar
couplings, $H^{\pm}l\nu$, $H^{+}H^{-}(Z/\gamma)$, $(H^{0}/h/A^{0})l^{+}l^{-}$
($l=\mu,\tau$), in both tree level and loop level contributions. These
couplings, being small in the LSS scenario, give no effective contribution to
the aforementioned flavor observables. Hence, the constraining power of these
observables are weaker, and we do not include them in our study.

###  3.2 Constraints from flavor physics and $R_{b}$ (Scan A)

The flavor and electroweak observables depend on the following LSS model
parameters,

$y_{1}$, $y_{2}$, $y_{3}$, $y_{4}$, $y_{5}$, $c$, $c^{\prime}$, $g_{1}$,
$g_{1}^{\prime}$, $y_{1b}$, $y_{2b}$, $f$ and $M$.

The values of the top sector Yukawa couplings $y_{i},i=1,2..5$ are taken to be
order of unity, which is a natural choice as they, as a combination, provide
the top quark mass with VEV’s $v_{1},v_{2}\sim 100~{}$GeV. Three cases of the
LSS model (case I, II and III) have different bottom Yukawa structure,
depending on values Yukawa couplings $y_{1b},y_{2b}$. The Yukawa couplings in
the bottom sector, $y_{1b}$ and $y_{2b}$ are responsible for the bottom quark
mass. To generate bottom mass, which is two orders of magnitude smaller than
the top mass, $y_{1b}$ and $y_{2b}$ are varied in an optimum range, much
smaller than the other Yukawa couplings. $y_{1b}$ and $y_{2b}$ are varied over
a range $|y_{ib}|\leq 0.10$. The intermediate Yukawa couplings
$y_{00}^{-},y_{10}^{-}$ have combinations of the form $y_{1b}\cos\beta-
y_{2b}\sin\beta$, which chart out relative contributions of the
$y_{1b},y_{2b}$. The rest of the parameters, $c$, $c_{1}$, $g_{1}$ and
$g_{1}^{\prime}$ are considered to be $\mathcal{O}$(1).

We have used the relation $M=1.5\times f$ where $f$ takes value upto 2 TeV amd
we have fixed the VEV at 246 GeV. Hence, we perform a random multi-parameter
scan in 12 parameter space. As noted in Ref: [21], to prevent the VEV’s
running away to $\infty$, sufficient conditions are imposed, given as
$m_{1,2}^{2}>0$, $(m_{1}^{2}m_{2}^{2}-b^{4})<0$ and $b^{2}$ is real. We have
implemented these criteria in the multi-parameter scan as well. As the flavor
constraints are expected to be overwhelmed by the LHC constraints, we work in
a more relaxed framework in terms of LHC parameters in order to emphasize the
flavor intricacies of this type of set-up. The top and Higgs masses are fixed
in the scan within a liberal window of 20 GeV and 10 GeV respectively 777Note
that we impose a much stronger constraint on these parameters in the next
scan, Scan B.. In order to see the effects emanating from the flavor sector,
the flavor observables are kept within their current allowed limits as listed
in Table. 1. Along with that, the bottom mass is also varied within 3-5 GeV.
We list the values of the parameters and constraints in the first scan, (Scan
A) in Table. 2.

Table 2:  Values of the parameters and constraints from flavor observables used in the multiparameter Scan A. Parameter | Value
---|---
$y_{i}$(i=1….5), $c$, $c_{1}$, $g_{1}$, $g_{1}^{\prime}$ |  $\mathcal{O}$(1)
$y_{1b}$, $y_{2b}$ | $\leq 0.10$
$f$ |  $\leq 2$ TeV
$M$ |  $\leq 3$ TeV
VEV | 246 GeV
Quantity | Constraint
$\delta C_{7}+0.242~{}\delta C_{8}$ | ($-0.063,0.073$)
${\Delta m_{B_{s}}\over\Delta m^{\textrm{SM}}_{B_{s}}}$ | ($0.675,1.265$)
$\delta R_{b}~{}$ | ($-0.00086,0.00182$)
Table 3:  The constraints used in the multiparameter Scan B. Quantity | Constraint
---|---
$|\sin(\beta-\alpha)|$ | $\sim 1$
$k_{tth}$ | $0.7-1.4$
$m_{t}$ |  $156-170$ GeV
$m_{h}$ |  $123-127$ GeV
$m_{b}$ |  $3-5$ GeV
$m_{H}^{\pm}$ | $>m_{t}$
$m_{B}$, $m_{T}$ |  $>1.4$ TeV

Figure 1: Scan A: Variation of $\tan\beta$ as a function of $y_{1b}$ and
$y_{2b}$ in three cases, case-I (upper panel), case-II (lower panel left) and
case-III (lower panel right), after imposing constraints from $R_{b}$ and
$B_{s}-\overline{B_{s}}$ combined (green) and $B\rightarrow X_{s}\gamma$
(red).

Figure 2: Scan A: Variation of $\tan\beta$ with the effective Yukawa couplings
and $k_{tth}$ in three cases, case-I (left column), case-II (middle column)
and case-III (right column), after imposing constraints from $R_{b}$ and
$B_{s}-\overline{B_{s}}$ combined (green) and $B\rightarrow X_{s}\gamma$
(red).

In the following, we present Fig 1 and Fig 2 for three different cases, as
specified earlier. In Fig 1, we plot case I at the top panel and case II
(left) and III (right) at the bottom panel. In Fig 2 we plot case I, II and
III in the first, second and third column respectively. In these figures, the
yellow points are generated by the model parameter space that is allowed after
imposition of constraints on them, as given in the top part of Table. 2. With
further imposition of constraints from the flavor and EW observables, green
and red points depict the projection of allowed parameter region that satisfy
the constraints (Table. 2) from $R_{b}$ and $B_{s}-\overline{B_{s}}$ mixing
together and $B\rightarrow X_{s}\gamma$ respectively. It is found that $R_{b}$
and $B_{s}-\overline{B_{s}}$ satisfy almost the same parameter space. Hence we
choose to impose combined constraints in the plots. The flavor bound from
$B\rightarrow X_{s}\gamma$ decay is found to be the most constraining for the
LSS model. Hence, in the following discussion about the model parameter space
constrained from flavor observables, we shall refer mainly to the strongest
constraint that comes from $B\rightarrow X_{s}\gamma$.

We show the variation of the Yukawa parameters with $\tan\beta$ in Fig 1 and
2. In this model, both $m_{1}^{2},m_{2}^{2}$ get contributions of same order
of magnitude, through some combinations of the Yukawa couplings,
$y_{i},i=1,2...5$, which are of the same order. Hence, $\tan\beta$ in the LSS
model, being written in terms of the ratio of $m_{1}^{2},m_{2}^{2}$, is
expected to lie in the vicinity of unity. It is reflected from the plots that
overall small $\tan\beta$ values are preferred. As $\tan\beta$ values
gradually increase from $<1$ to $>1$, upto $\tan\beta\sim 5$, relative
dominance of the $\sin\beta$ and $\cos\beta$ dependent terms in Eq.(2.2) and
(17) increase and decrease respectively. In Fig 1, in the case-I scenario,
where both $y_{ib}\neq 0$, flavor constraints seem to rule out the higher
$\tan\beta$ region for higher $y_{1b}$ values along with lower $\tan\beta$
region for higher $y_{2b}$ values. This happens as the contributions to
$y_{1b}\cos\beta-y_{2b}\sin\beta$ do not exactly cancel each other, thus
paving way for increased contribution to the observables ($B\rightarrow
X_{s}\gamma$ etc), constrained by experimental bounds.

From Fig 1, it is clear that the maximum value of $\tan\beta$ can be 5.0 for
case-II, whereas for case-III, $\tan\beta$ upto 6 is mostly preferred by the
flavor constraints 888We do not show the region for $\tan\beta\geq 6$, as this
region is less favored by the LSS model as well as the flavor observables,
resulting in a very few points in the plot. In case-II, for $y_{1b}=0$, the
$\lambda_{t}$ values in Eq.(27) remain significant only for larger $y_{2b}$
and $\sin\beta$, which is related to $\tan\beta$. Only with large $y_{2b}$ and
$\sin\beta$, $\lambda_{t}$ becomes larger which is paramount to fit the flavor
data. Similarly, for case-III with $y_{2b}=0$, after a certain $y_{1b}(\geq
0.02)$, smaller values of $\tan\beta$ are preferred with increasing $y_{1b}$
as $\cos\beta$ becomes significant in that range and fails to provide the
required values of $\lambda_{t}$ to satisfy the flavor constraints. Moreover,
in case-III, $\tan\beta$ cannot obtain a very small value, $\tan\beta<0.5$
region is disfavored from the flavor observables. Therefore, case-II with
$y_{1b}=0$ differs significantly from the case-I with all Yukawas non-zero,
whereas case-III with $y_{2b}=0$ remains relatively unaffected.

The effective Yukawa pattern subsequently reveals the dominance of either
$y_{1b}$ or $y_{2b}$ through their contributions in Eq.(17). From Fig:2, we
can see different ranges of $y_{Ht_{R}b_{L}}$ and $y_{Ht_{L}b_{R}}$. The
allowed $y_{Ht_{R}b_{L}}$ values are one order of magnitude higher than
$y_{Ht_{L}b_{R}}$ values, up to $\mathcal{O}$(1.0) as opposed to
$\mathcal{O}$(10-1). This is due to the fact that $y_{Ht_{R}b_{L}}$ and
$y_{Ht_{L}b_{R}}$ are primarily dominated by the top sector and bottom sector
Yukawa couplings ($y_{i},i=1,2$ vs. $y_{ib},i=1,2$) respectively. They are
also function of left and right top sector mixing elements in $R_{L}$ and
$R_{R}$, which are of similar orders of magnitude, being function of $y_{i}$’s
($i=1,2..5$). The effective Yukawa coupling $y_{Ht_{R}b_{L}}$ is a combination
of top sector Yukawas and there is no direct $y_{1b},y_{2b}$ contribution.
Hence $y_{Ht_{R}b_{L}}$ almost produces similar allowed regions after flavor
constraints in three different cases. This is reflected in Fig. 2, top row
where we plot $y_{Ht_{R}b_{L}}$ vs. $\tan\beta$. The flavor-allowed regions in
case-III is somewhat similar to that of case-I, but case-II stands apart,
where relatively larger values of $\tan\beta$ ($>4.5$) are not allowed.

The middle row of Fig. 2 shows the allowed regions of the effective Yukawa
coupling $y_{Ht_{L}b_{R}}$ with $\tan\beta$ for three cases. Large values of
$y_{Ht_{L}b_{R}}$ ($\sim 0.1$) are allowed for $\tan\beta<1.5$ in case-I.
Larger values of $\tan\beta$ ($\sim 4-5$) are favored in the region where
$y_{Ht_{L}b_{R}}$ is $>0.02$ in case-II whereas in case-III, $\tan\beta\leq 1$
is favored. In case-III, smaller values of $y_{1b}$ ($<0.02$) prefers large
$\tan\beta$. Another interesting feature of case-III is that a very small
value of $\tan\beta$ ($<0.2$) is disfavored by $B\rightarrow X_{s}\gamma$, as
shown in all the plots Fig. 2(right column).

We also plot the $\tan\beta$ dependence of $k_{tth}$ in Fig: 2. It is found
that, in the experimentally allowed window of $0.7<k_{tth}<1.4$, strongly
prefers $\tan\beta$ within the range $0.2-4.0$ in case-I and III, but values
of $\tan\beta$ larger than 4 are also allowed by $B\rightarrow X_{s}\gamma$.
In case-II, the upper limit on $\tan\beta$ is 4.0. The allowed values of the
charged Higgs mass is almost same in these three cases. In the next section,
we discuss the LHC constraints, starting from combined flavor constraints on
the charged Higgs mass.

###  3.3 Constraints from LHC (Scan B)

Figure 3: Scan B:LSS model parameter space after imposing constraints from
flavor observables and $R_{b}$ (Magenta) and further imposition of LHC
observables (Blue) in three cases, case-I (left column), case-II (middle
column) and case-III (right column).

At the LHC, properties of the Higgs boson are measured in several channels
[48], which indicates that the couplings of the 125 GeV Higgs to the gauge
bosons ($hVV$) and to the fermions ($hf\overline{f}$) are consistent with the
SM prediction [20, 49, 50, 51], even though some mismatches are still there in
the fermionic coupling measurements. These measurements are found to be in
favor of the alignment limit [52, 53] in a 2HDM-like setup i.e.
$|\sin(\beta-\alpha)|\sim 1$. The non observation of anomalous nature of the
$hVV$ coupling strongly indicates that any BSM contribution is suppressed in
the Higgs-gauge boson interaction, implying $|\cos(\beta-\alpha)|\sim 0$ [54,
55]. Hence, the $hVV$ coupling in the LSS model, which exhibits a 2HDM-like
structure, are suppressed. Among the fermionic couplings, we only constrain
the Higgs coupling to the third generation of quarks. We allow both positive
and negative $hVV$ and $ht\overline{t}$ couplings leading to both constructive
or destructive interference in the Higgs observables. ATLAS and CMS
experiments have imposed bounds on the couplings of the SM Higgs [48], by
looking at different production and decay modes of the Higgs boson. The
current limit on $htt$ coupling is dependent on different assumptions held
during the statistical fit, as shown in Ref: [48]. After inspecting the limits
carefully, we perform our search in the window of $0.70<k_{tth}<1.4$, where
$k_{tth}$ is defined as the ratio of the measured $htt$ coupling compared to
its SM value (see Eq(12)). In the alignment limit, the $hb\overline{b}$
coupling is $\sim m_{b}/v$, which is maintained throughout the multi-parameter
scan.

We derive the masses of the SM and BSM particles in terms of the model
parameters. We calculate the top mass in the $\overline{MS}$ scheme [56, 57]
and constrain it in the range $156-170$ GeV around
$m_{t}(\overline{MS})=163~{}$GeV. The Higgs mass [58, 59] is considered in the
range $123-127$ GeV. LEP experiment excludes the charged Higgs masses below 80
GeV [60]. At the LHC, searches for the charged Higgs have been performed
through various decay channels, $H^{\pm}\rightarrow cs$ [61, 62], $tb$ [63,
64] and $\nu\tau^{\pm}$ [65], and most of these searches exclude
$m_{H}^{\pm}<m_{t}$. However, the experimental searches [66] in
$H^{\pm}\rightarrow tb$ channel is favored at higher masses of $H^{\pm}$, as
predicted in other BSM models, for example, in supersymmetric scenarios [67].
Incidentally, also in the LSS model we get the $BR(H^{\pm}\rightarrow tb\sim
1)$ satisfying a large parameter space of the model. Electroweak precision
measurements [34, 44] require the charged Higgs mass to be close to the mass
of one the neutral Higgses, which is also satisfied in our model (also shown
in [22]). Different searches at the LHC have imposed limits on the masses of
the vector like quarks as well, and typically they are in the range of 1 to
1.4 TeV [32]. We list the values of all the constraining observables of the
second Scan B in Table. 4. The model parameters follow the same values as
given in Table. 2.

Table 4:  The experimental constraints, used in the multiparameter Scan B. Quantity | Constraints
---|---
$|\sin(\beta-\alpha)|$ | $\sim 1$
$k_{tth}$ | $0.7-1.4$
$m_{t}$ | ($156,170$) GeV
$m_{h}$ | ($123,127$) GeV
$m_{b}$ | ($3,5$) GeV
$m_{H}^{\pm}$ | $>m_{t}$
$m_{B}$, $m_{T}$ |  $>1.4$ TeV

In Fig 3, the magenta region shows the model parameter space allowed by
combination of the flavor constraints and the $R_{b}$ (allowed points of Scan
A). We constrain the magenta region further with observables of the Scan B as
given in Table. 4. The result of Scan B is shown by the blue points. The first
row of Fig 3 shows the allowed parameter space of the Yukawa couplings, which
contribute to the couplings of charged Higgs to the third generation quarks.
Interestingly, after both the scans, we get very distinct patterns for the
three cases. Parameter space that satisfies all the constraints (blue points)
are uniformly distributed in case I. In case II, negative values of the
$y_{Ht_{R}b_{L}}$ are more preferred and in case III, very small values of
$y_{Ht_{L}b_{R}}$ are not allowed after the imposition of LHC constraints.

In the second row of Fig 3 we show the plot of $\tan\beta$ vs. the mass of the
charged Higgs boson. Mass of the charged Higgs is a function of, $\tan\beta$,
$\lambda^{\prime}_{5}$ and $b^{2}$, which depend intrinsically on other model
parameters. The most important constraint form the LHC is on the $htt$
coupling, which again depends on the same set of model parameters. The
parameter space that survives after Scan B, predicts such ranges of
$m_{H}^{\pm}$ as shown by blue points in Fig 3. The preferred values of
charged Higgs mass stay well above 1 TeV, the smallest allowed value being
$\sim 1.3$ TeV, from Scan B. Case-I shows more preference towards smaller
values of $m_{H}^{\pm}$ compared to case-II. Overall, the plots in the first
and second row reflect mainly the correlation among the flavor physics
observables, $R_{b}$ and Higgs Yukawa measurements at the LHC, in the allowed
LSS model parameter space.

The bottom row confirms that the top-like VLQ partners are well above 1.3 TeV
after Scan B and the allowed region is almost same in the three cases. Our
results also predict that the similar inference is true for the bottom-like
VLQ’s. In general, we find that the limits on VLQ masses and the charged Higgs
mass are more stringent after the imposition of the LHC bounds, i.e, Scan B,
specifically due to the alignment condition. Similar conclusion can be made
for the extra neutral and charged gauge bosons, such as $B^{\prime}$ and
$W^{\prime}$, where the allowed masses are pushed to even higher values, $\geq
2.2~{}$TeV. This also takes care of the constraints coming from electroweak
precision tests (EWPT) [22].

##  4 Conclusion

We re-introduce the LSS little Higgs model that acts as an UV complete model
to produce an effective 2HDM at the EW scale, along with other BSM species,
such as vector like fermions, extra gauge bosons and scalars. Compared to the
generic 2HDM, emergent 2HDM from the LSS model will be predictive in nature,
as the Yukawa structure here will be dictated by a bigger symmetry. In this
work, emergent 2HDM from the LSS model is discussed from the viewpoint of
standard 2HDM to determine the crucial phenomenological differences. We bring
out both flavor and LHC constraints together from the arsenal, ably aided by
EW precision observable $R_{b}$ to pin down the allowed parameter space of the
LSS model. In the context of an emergent 2HDM, qualitative features of an
$SU(6)/Sp(6)$ composite Higgs model are expected to be similar to those for
the LSS model, albeit with obvious difference in parametric dependences.

In the flavor sector, variants of models are constructed from the bottom
Yukawa perspective, and we have found that the case with both $y_{ib}\neq 0$
is relatively less constrained compared to the cases with either of them put
to zero. We deploy different flavor observables, potentially important to the
charged Higgs sector. Among them the $B\rightarrow X_{s}\gamma$ branching
ratio is found to be the most constraining, while $B_{s}-\overline{B}_{s}$
mixing and EWPT observable $R_{b}$ constraints are relatively liberate, though
are not widely off. While the flavor constraints keep the charged Higgs fairly
relaxed, the $\tan\beta$ gets more restricted ($\leq 5$) compared to the usual
2HDM. The effective charged Higgs Yukawas, $y_{Ht_{L}b_{R}}$ and
$y_{Ht_{R}b_{L}}$ show very distinct $\tan\beta$ dependences in different
cases (case-I,II,III), manifesting different flavor patterns.

With respect to the combined flavor and LHC constraints, charged Higgs mass
($m_{H}^{\pm}$) and $\tan\beta$ are bound tighter in the emergent 2HDM from
the LSS model than in the usual 2HDM. Even when some of the constrained 2HDM
scenarios rule out the charged Higgs only at sub-TeV region, here in the LSS
model, combined flavor and LHC bounds push the charged Higgs mass lower bound
to 1.3 TeV. Similarly, $\tan\beta$ is spread over a wider range in general
2HDM as opposed to a narrow range of (0.5-3) in the different LSS scenarios.
This is a reflection of the predictive nature of the Yukawa sector in the LSS
model, where the Higgs mass, top mass and top Yukawa couplings are fixed in
terms of strong sector parameters of the LSS model. Whereas, these quantities
were easily arranged in the construction of the general 2HDM through the
enforcement of a $Z_{2}$ symmetry.

These type of LH models with a large number of parameters are often very fine
tuned, as multiple parameters contribute to a single observable, such as the
Higgs mass, which is very precisely known. It is worthwhile to know in future
how the three different flavor scenarios, as discussed above, can have an
impact on the charged Higgs phenomenology. In general, we find that the
charged Higgs along with other neutral BSM particles are placed at masses well
above one TeV. Hence, the decay products of them are expected to be highly
energetic and that one can use to improvise different LHC search techniques.
In one of the earlier LHC searches [68], heavy charged Higgs similar to this
scenario which dominantly decays through $tb$ mode, was probed in all jet
final states, using boosted properties of top and $W$. These kind of searches
will have better prosepct at the advanced run of the LHC with enhanced
luminosity and energy.

## Acknowledgments

SS thanks UGC for the DS Kothari postdoctoral fellowship grant with award
letter No.F.4-2/2006 (BSR)/PH/17-18/0126. N.K. acknowledges the support from
the Dr. D. S. Kothari Postdoctoral scheme (201819-PH/18-19/0013). SS thanks
Dr. Shrihari Gopalakrishna for his guidance as PhD supervisor to study this
model. SS also thanks Dr. Shrihari Gopalakrishna and Dr. Tuhinsubhra Mukherjee
for their inputs on the LSS model construction and phenomenology during the
course of an earlier work [22]. We also sincerely thank the referee for a
number of important, thought-provoking suggestions that have immensely helped
us to improve our manuscript.

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