# The 2HDM Doppelganger

Baradhwaj Coleppa<EMAIL_ADDRESS>Agnivo Sarkar
<EMAIL_ADDRESS>IIT Gandhinagar, Palaj Campus, Gujarat 382355, India

(August 29, 2024)

###### Abstract

We discuss the structure of a model with an extended gauge symmetry group
$SU(2)\times SU(2)\times U(1)$ with a correspondingly rich Electroweak
Symmetry Breaking structure. In spite of the additional scalar degrees of
freedom in the model, the presence of the extra gauge group $SU(2)$ and its
associated heavy vector bosons ensures that the scalar spectrum of the model
after symmetry breaking is identical to that of the Two Higgs Doublet Models.
We construct the model and discuss its implications, specifically the
phenomenology associated with this class of models in contrast to the 2HDM.

###### pacs:

Valid PACS appear here

##  I Introduction

While the discovery of a Higgs boson whose properties are largely consistent
with that predicted in the Standard Model (SM) Aad  _et al._ (2012);
Chatrchyan  _et al._ (2012); Aad  _et al._ (2016a); ATL (2019); Sirunyan  _et
al._ (2019a, b) is undoubtedly a significant achievement of the ATLAS and CMS
experiments of the Large Hadron Collider (LHC), vexing questions surrounding
the stability of the Higgs boson mass, explanation of neutrino mass, and the
origin of dark matter still remain. Generic “Beyond the Standard Model” (BSM)
scenarios that have been constructed to answer one or more of such outstanding
theoretical questions usually necessitate the introduction of additional
degrees of freedom that should potentially be discovered at the LHC. BSM
scenarios typically involve enlarging the gauge structure of the SM, invoking
non-trivial patterns of Electroweak Symmetry Breaking (EWSB) involving
additional scalar particles, or enlarging the matter content of the SM - many
BSM scenarios admit more than one of these possibilities. The success or
failure of these models would ultimately be decided by the confirmation of
their predictions at the LHC. Given that there are a large number of such BSM
scenarios available, typically phenomenologists and experimentalists tend to
concentrate of explaining the results of the ATLAS and CMS data (null results
or otherwise) in the context of certain simplified scenarios that have
features that are rather generic. One such example in the context of extended
models with extended scalar sectors is the Two Higgs Doublet Model (2HDM) Lee
(1973); Akeroyd (1999); Aoki  _et al._ (2009); Baak  _et al._ (2012); Posch
(2011); Kominis (1994) (also see Branco  _et al._ (2012) for a comprehensive
review).

The 2HDM extends the SM by invoking a second Higgs doublet, and EWSB is
engineered by both the doublets via their vacuum expectation values (vev)
$v_{1}$ and $v_{2}$. After symmetry breaking, one spin-0 linear combination of
the fields is identified with the SM Higgs boson with mass 125 GeV, and the
second Higgs with different couplings to the gauge bosons and fermions, can be
either heavier or lighter than the SM one. There is a huge experimental
interest in discovering these extra Higgs bosons at the ATLAS Aad  _et al._
(2020) and CMS Sirunyan  _et al._ (2020a, b) experiments, partly because the
scalar sectors of the (Type II) 2HDM is identical to that of the Minimal
Supersymmetric Standard Model (MSSM). While the search for such extra Higgs
bosons is generic in the sense that experiments typically look for a
particular final state, the interpretation of the findings of these
experiments in a model-dependent scenario can have wildly different results
depending on the couplings of these extra Higgs bosons, and their production
cross-section and decay branching ratios. In this context, in this paper we
present the details of a model that has two interesting features: _i)_ an
extended electroweak gauge group $SU(2)\times SU(2)\times U(1)$, and _ii)_ a
scalar spectrum that is identical to that of the 2HDM. The model is a variant
of that presented in Ref. Sekhar Chivukula  _et al._ (2009); Chivukula  _et
al._ (2011) as the “Top Triangle Moose Model”, with one of the non-linear
sigma model fields being replaced by a linear Higgs field. As we demonstrate
below, this changes the scalar structure of the model in such a way as to
mimic the 2HDM (and MSSM) scenarios, and if the associated gauge bosons are
heavy, this model would look indistinguishable from the 2HDM at low energies.
The goal of the present work is to demonstrate the features of this model, and
sketch the rich and varied phenomenology this alternate scheme of EWSB allows
us to pursue.

This paper is organized as follows: in Sec. II, we present the details of the
model mostly restricting to those aspects of it that are distinct from that in
Refs. Sekhar Chivukula  _et al._ (2009); Chivukula  _et al._ (2011). In Sec.
III, we compute the couplings of the neutral and charged Higgs bosons in the
theory and identify possibilities for distinct LHC signatures. In Sec. IV, we
analyze the parameter space of the model that survives the LHC experimental
constraints, and present the cross-sections and decay branching ratios of the
Higgs bosons before concluding in Sec. V.

##  II The Model

The electroweak gauge structure of the model is a minimal extension of the SM
with the “221” structure of gauge groups Chivukula  _et al._ (2006); Abe  _et
al._ (2013); Coleppa  _et al._ (2018) $SU(2)_{0}\times SU(2)_{1}\times U(1)$.
Spontaneous symmetry breaking in the model is engineered via two Higgs
doublets $\Phi_{1}$ and $\Phi_{2}$ and a non-linear sigma model field
$\Sigma$. We illustrate this in Fig. 1 using the moose notation with the two
$SU(2)$ groups residing at “sites” 0 and 1 and the $U(1)$ group at site 2.

Figure 1: The model under consideration in the “moose” notation. “Sites” 0 and
2 contain the (mostly) SM gauge group $SU(2)\times U(1)$ with the other
$SU(2)$ residing at site 2. The gauge couplings of the $SU(2)\times
SU(2)\times U(1)$ groups are denoted by $g_{0},g_{1}$ and $g_{2}$
respectively. Symmetry breaking in this model is engineered by two Higgs
doublets $\Phi_{1,2}$ and a non-linear sigma model field $\Sigma$.

The non-linear sigma field $\Sigma$ breaks the $SU(2)_{0}\times SU(2)_{1}$
down to the diagonall $SU(2)_{V}$. The two additional scalar doublets in the
model transform as ($\textbf{2},\textbf{1},\frac{\textbf{1}}{\textbf{2}}$) and
($\textbf{1},\textbf{2},\frac{\textbf{1}}{\textbf{2}}$) respectively under the
full symmetry group of the model. We denote the vacuum expectation values
(vevs) of the different scalar fields as
$\langle\Phi_{1}\rangle=\frac{f_{1}}{\sqrt{2}}$,
$\langle\Phi_{2}\rangle=\frac{f_{2}}{\sqrt{2}}$ and $\langle\Sigma\rangle=F$
and follow the same parametrization as that in Refs. Sekhar Chivukula  _et
al._ (2009); Chivukula  _et al._ (2011):111While Refs. Sekhar Chivukula  _et
al._ (2009); Chivukula  _et al._ (2011) treated $\sin\beta$ as a small
parameter with the EWSB mostly engineered via a “higgsless” mechanism, in the
present paper we place no such constraints on $\sin\beta$ at the outset.

$f_{1}=v\sin\beta;\,\,F=f_{2}=\sqrt{2}v\cos\beta,$ (1)

where $v$ is the vev of the Higgs field in the SM (246 GeV). The explicit form
of the fields are given as

$\Sigma=\exp{\left(\frac{i\Pi^{a}_{\Sigma}\sigma^{a}}{F}\right)},\,\,\Phi_{i}=\begin{pmatrix}(f_{i}+H_{i}+i\Pi^{0}_{i})/\sqrt{2}\\\
i\Pi^{-}_{i}\end{pmatrix};\,i=1,2,$ (2)

where $\sigma^{a}$ are the three Pauli matrices. The gauge couplings for the
$SU(2)_{0}$, $SU(2)_{1}$, and $U(1)$ groups are parametrized as

$g=\frac{e}{\sin\theta\cos\phi};\,\,g_{1}=\frac{e}{\sin\theta\sin\phi},\,\,g_{2}=\frac{e}{\cos\theta}.$
(3)

###  II.1 Symmetry Breaking

To begin discussing the symmetry breaking structure of this particular 221
model, we write down the most general potential for the scalar fields
consistent with the gauge symmetries:

$\displaystyle\begin{split}V(\Phi_{1},\Phi_{2},\Sigma)&=m^{2}_{11}\Phi^{\dagger}_{1}\Phi_{1}+m^{2}_{22}\Phi^{\dagger}_{2}\Phi_{2}-(m^{2}_{12}\Phi^{\dagger}_{1}\Sigma\Phi_{2}+\textnormal{h.c.})+\frac{\beta_{1}}{2}(\Phi^{\dagger}_{1}\Phi_{1})^{2}+\frac{\beta_{2}}{2}(\Phi^{\dagger}_{2}\Phi_{2})^{2}\\\
&+\beta_{3}(\Phi^{\dagger}_{1}\Phi_{1})(\Phi^{\dagger}_{2}\Phi_{2})+\beta_{4}(\Phi^{\dagger}_{1}\Sigma\Phi_{2})(\Phi^{\dagger}_{2}\Sigma^{\dagger}\Phi_{1})+\bigg{[}\frac{\beta_{5}}{2}(\Phi^{\dagger}_{1}\Sigma\Phi_{2})^{2}+\textnormal{h.c.}\bigg{]}.\end{split}$
(4)

Through the following identification of parameters

$\displaystyle\begin{split}m^{2}_{11}&=-\bigg{[}\lambda_{1}f^{2}_{1}+\lambda_{3}(f^{2}_{1}+f^{2}_{2})\bigg{]};\,m^{2}_{22}=-\bigg{[}\lambda_{2}f^{2}_{2}+\lambda_{3}(f^{2}_{1}+f^{2}_{2})\bigg{]};\,m^{2}_{12}=\lambda_{5}\frac{f_{1}f_{2}}{2},\beta_{1}=2(\lambda_{1}+\lambda_{3});\\\
\beta_{2}&=2(\lambda_{2}+\lambda_{3});\,\beta_{3}=(\lambda_{4}+2\lambda_{3});\,\beta_{4}=\frac{1}{2}(\lambda_{5}+\lambda_{6})-\lambda_{4};\,\beta_{5}=\frac{1}{2}(\lambda_{5}-\lambda_{6}),\end{split}$
(5)

we could recast the Lagrangian to a form more commonly used in the 2HDM
literature222Note, however, that the $\Phi_{1,2}$ are charged under different
gauge groups unlike in the 2HDM; the combination $\Phi_{1}^{\dagger}\Phi_{2}$
is _not_ gauge invariant in the present model while the term
$\Phi_{1}^{\dagger}\Sigma\Phi_{2}$ is.:

$\displaystyle\begin{split}V(\Phi_{1},\Phi_{2},\Sigma)&=\lambda_{1}\bigg{[}\Phi^{\dagger}_{1}\Phi_{1}-\frac{f^{2}}{2}\bigg{]}^{2}+\lambda_{2}\bigg{[}\Phi^{\dagger}_{2}\Phi_{2}-\frac{F_{2}}{2}\bigg{]}^{2}+\lambda_{3}\bigg{[}\Phi^{\dagger}_{1}\Phi_{1}+\Phi^{\dagger}_{2}\Phi_{2}-\frac{f^{2}+F^{2}}{2}\bigg{]}^{2}\\\
&+\lambda_{4}\bigg{[}(\Phi^{\dagger}_{1}\Phi_{1})(\Phi^{\dagger}_{2}\Phi_{2})-(\Phi^{\dagger}_{1}\Sigma\Phi_{2})(\Phi^{\dagger}_{2}\Sigma^{\dagger}\Phi_{1})\bigg{]}+\lambda_{5}\bigg{[}\textnormal{Re}(\Phi^{\dagger}_{1}\Sigma\Phi_{2})-\frac{fF}{2}\bigg{]}^{2}\\\
&+\lambda_{6}\,\textnormal{Im}\bigg{[}\Phi^{\dagger}_{1}\Sigma\Phi_{2}\bigg{]}^{2},\end{split}$
(6)

where use has been made use of Eqn. 1, and we have relabeled $f_{1}\to f$ for
notational simplicity. We will fix all $\lambda_{i}$’s to be real parameters
to insure the hermiticity of the Lagrangian. Out of the eleven scalar degrees
of freedom, six are Goldstone modes that get eaten up by the massive gauge
bosons while five remain as physical scalar particles in the spectrum. In
keeping with the standard literature, these will be denoted as the charged
Higgs bosons $H^{\pm}$, two CP-even Higgses $(H,h)$ and a pseudo-scalar $A$.
The mass matrix of the two CP-even states can be written as

$M^{2}_{h,H}=\begin{bmatrix}2(\lambda_{1}+\lambda_{3})f^{2}+\frac{\lambda_{5}}{2}F^{2}&2(\lambda_{3}+\frac{\lambda_{5}}{4})fF\\\
&\\\
2(\lambda_{3}+\frac{\lambda_{5}}{4})fF&2(\lambda_{2}+\lambda_{3})F^{2}+\frac{\lambda_{5}}{2}f^{2}\end{bmatrix}$
(7)

with eigenvalues

$\displaystyle\begin{split}m^{2}_{H}&=\frac{1}{2}\bigg{[}(A_{s}+C_{s})+\sqrt{(A_{s}-C_{s})^{2}+4B^{2}_{s}}\bigg{]}\\\
m^{2}_{h}&=\frac{1}{2}\bigg{[}(A_{s}+C_{s})-\sqrt{(A_{s}-C_{s})^{2}+4B^{2}_{s}}\bigg{]},\end{split}$
(8)

where

$\qquad
A_{s}=2(\lambda_{1}+\lambda_{3})f^{2}+\frac{\lambda_{5}}{2}F^{2},\,B_{s}=2(\lambda_{3}+\frac{\lambda_{5}}{4})fF,\,\textrm{and}\,\,C_{s}=2(\lambda_{2}+\lambda_{3})F^{2}+\frac{\lambda_{5}}{2}f^{2}.$

The physical CP-even Higgs eigenstates are

$\displaystyle\begin{split}H&=\cos\alpha\,H_{1}+\sin\alpha\,H_{2}\\\
h&=-\sin\alpha\,H_{1}+\cos\alpha\,H_{2},\end{split}$ (9)

with the mixing angle $\alpha$ defined to be

$\sin 2\alpha=\frac{2B_{s}}{\sqrt{(A_{s}-C_{s})^{2}+4B^{2}_{s}}}.$ (10)

We will identify the $h$ with the SM-like Higgs of mass 126 GeV, while the $H$
would be a heavier Higgs. We will detail the production and decay of these
states in Sec. IV. Notice that while the mass term for the CP-even Higgses can
arise from terms like $\Phi_{i}^{\dagger}\Phi_{i}$, the $H^{\pm}$ mass comes
solely from the $\lambda_{4}$ term in Eqn. 6 \- the $\lambda_{5}$ and
$\lambda_{6}$ terms do not contain any terms quadratic in the pions and
describe interactions. The mass matrix for the charged scalars thus takes the
following form:

$M^{2}_{\pi^{\pm}}=\frac{\lambda_{4}}{2}\begin{bmatrix}f^{2}&-fF&f^{2}\\\
-fF&F^{2}&-fF\\\ f^{2}&-fF&f^{2}\end{bmatrix}.$ (11)

The corresponding eigenstates of the Goldstone modes $G^{\pm}_{1}$,
$G^{\pm}_{2}$ which get eaten by the $W^{\pm}_{\mu}$ and
$W^{{}^{\prime}\pm}_{\mu}$ are given by

$\displaystyle\begin{split}G_{1}^{\pm}&=-\frac{1}{\sqrt{2}}\Pi_{\Sigma}^{\pm}+\frac{1}{\sqrt{2}}\Pi_{2}^{\pm}\\\
G_{2}^{\pm}&=\frac{F}{2v}\Pi_{\Sigma}^{\pm}+\frac{f}{v}\Pi_{1}^{\pm}+\frac{F}{2v}\Pi_{2}^{\pm},\end{split}$
(12)

while the physical charged Higgs corresponds to the combination

$H^{\pm}=\frac{f}{\sqrt{2}v}\Pi_{\Sigma}^{\pm}-\frac{F}{\sqrt{2}v}\Pi_{1}^{\pm}+\frac{f}{\sqrt{2}v}\Pi_{2}^{\pm}$
(13)

with the corresponding mass eigen-value is
$M^{2}_{H^{\pm}}=\frac{\lambda_{4}}{2}(2f^{2}+F^{2})$. The neutral Goldstone
mode $G^{0}$ and the physical pseudoscalar $A$ can be obtained by the
following replacements in Eqns. 12, 13:

$\Pi^{\pm}_{\Sigma}\rightarrow\Pi^{0}_{\Sigma},\,\,\Pi^{\pm}_{1}\rightarrow\Pi^{0}_{1},\,\,\textrm{and}\,\,\,\Pi^{\pm}_{2}\rightarrow\Pi^{0}_{2}.$

We note that the scalar sector of our model as parametrized in Eqn. 6 requires
six free parameters (the $\lambda_{i}$) in addition to the two vevs, so we
need a total of eight parameters to write down the most general potential
invariant under $SU(2)_{0}\times SU(2)_{1}\times U(1)$. We can trade two of
the eight parameters to $\beta$ which is the ratio of two vev’s and the mixing
angle $\alpha$ between the two CP-even Higgs eigenstates. Five of the other
six $\lambda_{i}$s can be expressed in terms of the five physical Higgs masses
and the EWSB scale $v$. These expressions come in handy when imposing
theoretical and experimental constraints on the scalar couplings and we
present them below.

$\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2v^{2}\sin^{2}\beta}\bigg{[}(m^{2}_{H}\cos^{2}\alpha+m^{2}_{h}\sin^{2}\alpha)-\frac{1}{\sqrt{2}}\tan\beta\sin\alpha\cos\alpha(m^{2}_{H}-m^{2}_{h})\bigg{]}-\frac{\lambda_{5}}{2}(2\cot^{2}\beta-1).$
(14a) $\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4v^{2}\sin^{2}\beta}\bigg{[}(m^{2}_{H}\cos^{2}\alpha+m^{2}_{h}\sin^{2}\alpha)-\frac{\sqrt{2}}{\tan\beta}\sin\alpha\cos\alpha(m^{2}_{H}-m^{2}_{h})\bigg{]}-\frac{\lambda_{5}}{4}\bigg{(}\frac{\tan^{2}\beta}{2}-1\bigg{)}.$
(14b) $\displaystyle\lambda_{3}$ $\displaystyle=$
$\displaystyle\frac{\sin\alpha\cos\alpha}{2\sqrt{2}v^{2}\sin\beta\cos\beta}(m^{2}_{H}-m^{2}_{h})-\frac{\lambda_{5}}{2}.$
(14c) $\displaystyle\lambda_{4}$ $\displaystyle=$
$\displaystyle\frac{m^{2}_{H^{\pm}}}{v^{2}}.$ (14d) $\displaystyle\lambda_{6}$
$\displaystyle=$ $\displaystyle\frac{m^{2}_{A}}{v^{2}}.$ (14e)

###  II.2 Gauge Sector

The kinetic energy part of our $SU(2)\times SU(2)\times U(1)$ model can be
written down in the usual canonically normalized form

$\mathcal{L}_{K.E}=-\frac{1}{4}\sum_{i=0}^{1}F^{a}_{i\mu\nu}F^{a\mu\nu}_{i}-\frac{1}{4}B_{\mu\nu}B^{\mu\nu},$
(15)

where $F^{a\mu\nu}_{i}$ is the energy-momentum tensor for the non-abelian
gauge fields associated with the two $SU(2)$ gauge groups and
$B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}$ is the Abelian $U(1)$
counterpart. After spontaneous symmetry breaking six out of seven gauge bosons
become massive. To understand the gauge boson masses, we begin by writing down
the gauge invariant kinetic energy terms of the scalar fields:

$\mathcal{L}=\frac{F^{2}}{2}\textnormal{Tr}[D_{\mu}\Sigma^{\dagger}D^{\mu}\Sigma]+D_{\mu}\Phi^{\dagger}_{1}D^{\mu}\Phi_{1}+D_{\mu}\Phi^{\dagger}_{2}D^{\mu}\Phi_{2}.$
(16)

The covariant derivatives are given by

$\displaystyle\begin{split}D_{\mu}\Sigma&=\partial_{\mu}\Sigma+ig\tilde{W}_{0\mu}\Sigma-
ig_{1}\Sigma\tilde{W}_{1\mu}\\\
D_{\mu}\Phi_{1}&=\partial_{\mu}\Phi_{1}+ig\tilde{W}_{0\mu}\Phi_{1}-\frac{ig_{2}}{2}B_{2\mu}\Phi_{1}\\\
D_{\mu}\Phi_{2}&=\partial_{\mu}\Phi_{2}+ig_{1}\tilde{W}_{1\mu}\Phi_{2}-\frac{ig_{2}}{2}B_{2\mu}\Phi_{2},\end{split}$
(17)

where the matrix fields $\tilde{W}_{0\mu}=\frac{W^{a}_{0\mu}\sigma^{a}}{2}$
and $\tilde{W}_{1\mu}=\frac{W^{a}_{1\mu}\sigma^{a}}{2}$. Following Refs.
Sekhar Chivukula  _et al._ (2009); Chivukula  _et al._ (2011), we will treat
$\sin\phi$ in Eqn. 3 as a small parameter and diagonalize the gauge boson mass
matrices perturbatively in $\sin\phi$ (which we will relabel as $x$ henceforth
for notational simplicity). We will not reproduce the calculations already
laid out in Refs. Sekhar Chivukula  _et al._ (2009); Chivukula  _et al._
(2011) here - the salient features are the existence of heavy $W^{\prime}$ and
$Z^{\prime}$ bosons whose wavefunctions are peaked away from the $SU(2)_{0}$
and the $U(1)$ sites and the SM-like $W$ and $Z$ bosons that do not have a
significant overlap with the $SU(2)_{1}$ gauge group. For instance, the wave
functions of the charged gauge bosons are

$\displaystyle\begin{split}W^{\pm}_{\mu}&=(1-\frac{x^{2}}{8})W^{\pm}_{0\mu}+\frac{x}{2}W_{1\mu}\\\
W^{{}^{\prime}\pm}_{\mu}&=-\frac{x}{2}W_{0\mu}+(1-\frac{x^{2}}{8})W_{1\mu}\end{split}$
(18)

The photon remains massless and is given by

$A_{\mu}=\frac{e}{g}W^{3}_{0\mu}+\frac{e}{g_{1}}W^{3}_{1\mu}+\frac{e}{g_{2}}B_{2\mu}$
(19)

with the different coupling constants related by

$\frac{1}{e^{2}}=\frac{1}{g^{2}}+\frac{1}{g^{2}_{1}}+\frac{1}{g^{2}_{2}}.$
(20)

One could proceed to evaluate the various self-couplings between the gauge
fields - we relegate the results to Appendix A.

###  II.3 Fermion Sector

The model admits the SM-like fermions and also heavy vector-like fermions.
Charge assignments of the various species under the different gauge groups
can, in principle, be done in a number of ways. The assignment that we will
adopt is motivated by two factors:

  1. (i)

That the overall structure of the model stays as close to possible to the
original version presented in Refs. Sekhar Chivukula  _et al._ (2009);
Chivukula  _et al._ (2011).

  2. (ii)

That the spectrum and the pattern of couplings mimic one of the 2HDMs to the
extent possible so that a meaningful comparison can be made.

Admittedly one or both of these constraints can be relaxed that will lead to
different possibilities, but in this paper we will operate within the confines
of the aforementioned points. Notice that even $(ii)$ offers multiple choices
with regard to the couplings of the fermions to the scalar fields. We will
choose to follow the conventions of the so-called Type I 2HDM wherein both up-
type and down-type quarks and the charged leptons couple to one of the two
Higgs doublets, though both of them take part in EWSB. However, we mention at
the outset that any comparison to 2HDM cannot be one-to-one as the particle
content of this model in the gauge and fermionic sectors is richer than that
of the 2HDM.

The left-handed fermions are $SU(2)$ doublets which reside at site 0 and 1 -
we will denotes these as $\psi_{L0}$ and $\psi_{L1}$. In addition to the SM-
like $u_{R}$ and $d_{R}$ at site 2, there are also right-handed fermions
$\psi_{R2}$. that are doublets under $SU(2)_{1}$. The $\psi_{L0}$,
$\psi_{L1}$, and $\psi_{R1}$ have $U(1)$ charges similar to the SM doublets:
$\frac{1}{6}$ for quarks and $-\frac{1}{2}$ for leptons. The right-handed up
quark has a $U(1)$ charge $\frac{2}{3}$ while the down type quark carries
$-\frac{1}{3}$. For the case of leptons, the hypercharge assignments are
similar to the SM. We summarize all the electroweak charges for the fermions
in the Table 1.

| $\psi_{L0}$ |  $\psi_{L1}$ , $\psi_{R1}$ | $u_{R2},d_{R2}$
---|---|---|---
$SU(2)_{0}$ | 2 | 1 | 1
$SU(2)_{1}$ | 1 | 2 | 1
$U(1)$ | $\frac{1}{6}$ | $\frac{1}{6}$ |  $\frac{2}{3}$ or $-\frac{1}{3}$
Table 1: Fermionic charge assignments in the model - in addition to the SM-
like left-handed doublets and right-handed $SU(2)$ singlets, there is also a
right-handed fermion doublet under $SU(2)_{1}$.

The SM fermions (predominantly) derive their masses from coupling to the
$\Phi_{1}$ field. In view of this, we can write down the following term in the
fermion sector of the Lagrangian:

$\mathcal{L}=\lambda^{ij}_{u0}\bar{\psi}_{L0}\Phi_{1}u_{R2}+\lambda^{ij}_{d0}\bar{\psi}_{L0}\tilde{\Phi}_{1}d_{R2}+h.c.,$
(21)

where $\lambda^{ij}_{u0},\lambda^{ij}_{d0}$ are the Yukawa couplings that are
set by the masses of the up and the down-type quarks respectively and
$\tilde{\Phi}_{1}=-i\sigma_{2}\Phi_{1}$. The vector-like fermions at site 1
are unaffected by EWSB and thus admit a Dirac mass term
$\bar{\psi}_{L1}\psi_{R1}$. In addition, there are two gauge invariant
dimension-4 terms that we cannot ignore: the coupling of the $\psi_{L0}$ to
the $u_{R},d_{R}$ via the link field $\Sigma$ (which was the primary mass term
for the fermions in the Higgsless mechanism in Refs. Sekhar Chivukula  _et
al._ (2009); Chivukula  _et al._ (2011)) and $\bar{\psi}_{L1}\Phi_{2}f_{R2}$,
where $f_{R2}$ is either an up or a down type fermion residing at site 2. The
latter is a new term quite distinct from the Type I 2HDM that can arise here
due to the presence of the extra gauge group and the presence of vector-like
fermions. Putting everything together, the mass terms for the various quarks
and leptons in the model are given by the following terms in the Lagrangian:

$\mathcal{L}=-\lambda^{ij}_{u0}\bar{\psi}_{L0}\Phi_{1}u_{R2}-\lambda^{ij}_{d0}\bar{\psi}_{L0}\tilde{\Phi}_{1}d_{R2}-\lambda_{L}\bar{\psi}_{L0}\Sigma\psi_{R1}-M_{D}\bar{\psi}_{L1}\psi_{R1}-\lambda_{uR}\bar{\psi}_{L1}\Phi_{2}u_{R2}-\lambda_{dR}\bar{\psi}_{L1}\tilde{\Phi}_{2}d_{R2}+\textrm{h.c.},$
(22)

where we have written down the coupling of $\Phi_{2}$ with the up and down
type fermions with strengths $\lambda_{uR},\lambda_{dR}$. In order to mimic
the Type I 2HDM to the extent possible, the masses of all SM fermions arise
from the $\lambda^{ij}_{u0},\lambda^{ij}_{d0}$ terms and thus we will assume
this subdominant contribution does not affect the up-down mass splitting
appreciably. 333We use $u$ and $d$ generically to mean any up-type or down-
type quark.. Thus at the outset, we assume the hierarchy of coupling strengths
$\lambda_{uR,dR}\ll\lambda_{L}<\lambda_{u0,d0}$444While this is certainly not
necessary _a priori_ , this choice will enable us to write down the condition
for ideal fermion delocalization in this model identical to Ref. Sekhar
Chivukula  _et al._ (2009) \- see Eqn. 32.. This ensures that any deviations
in the Higgs-fermion interaction strengths compared to the SM arise
predominantly from the non-linear sigma model fields. To satisfy FCNC
constraints we have chosen $\lambda^{ij}_{L,R}=\lambda_{L,R}\delta^{ij}$ and
$M^{ij}_{D}=M_{D}\delta^{ij}$ and thus all the non-trivial flavor structure of
the model is encoded in $\lambda^{ij}_{u0},\lambda^{ij}_{d0}$. To quantify the
mixing between the SM fermions and their heavier partners, we define the
following parameters:

$\epsilon_{L}=\frac{\lambda_{L}}{M_{D}};\,\epsilon_{fR}=\frac{\lambda_{fR}F}{M_{D}},$
(23)

where $f$ is either $u$ or $d$. For notational simplicity, we define

$a_{f}=\frac{\lambda_{f0}v\sin\beta}{\sqrt{2}M_{D}}.$ (24)

The fermionic mass matrix takes the form

$M_{F}=M_{D}\begin{bmatrix}\epsilon_{L}&a_{f}\\\ 1&\epsilon_{R}\end{bmatrix}$
(25)

We require that the dominant contribution to the SM fermion mass arises out of
its coupling to the Higgs field $\Phi_{1}$. Accordingly, diagonalizing the
matrix pertubatively in $\epsilon_{L}$ and $\epsilon_{R}$, we get the light
fermion eigenvalue

$m_{f}=M_{D}a_{f}\bigg{[}1+\frac{\epsilon^{2}_{L}+\epsilon^{2}_{R}+\frac{2}{a_{f}}\epsilon_{L}\epsilon_{R}}{2(-1+a_{f}^{2})}\bigg{]}.$
(26)

In the limit $\epsilon_{L,R}\to 0$, choosing the Yukawa coupling $a_{f}$
appropriately yields the masses of the light fermions. For completeness, we
write down the mass the heavy partner below:

$m_{F}=M_{D}\bigg{[}1-\frac{\epsilon^{2}_{fL}+\epsilon^{2}_{fR}+2a_{f}\epsilon_{L}\epsilon_{R}}{2(-1+a_{f}^{2})}\bigg{]}$
(27)

The wave functions of the left and right handed SM fermion are:

$\displaystyle\begin{split}f_{L}&=f^{0}_{L}\psi^{f}_{L0}+f^{1}_{L}\psi^{f}_{L1}\\\
&=\bigg{(}1-\frac{(\epsilon_{L}+a_{f}\epsilon_{R})^{2}}{2(-1+a_{f}^{2})^{2}}\bigg{)}\psi_{L0}+\bigg{(}\frac{\epsilon_{L}+a_{f}\epsilon_{R}}{-1+a_{f}^{2}}\bigg{)}\psi_{L1}\\\
f_{R}&=f^{1}_{R}\psi^{f}_{R1}+f^{2}_{R}\psi^{f}_{R2}\\\
&=\bigg{(}\frac{a_{f}\epsilon_{L}+\epsilon_{R}}{-1+a_{f}^{2}}\bigg{)}\psi_{R1}+\bigg{(}1-\frac{(a_{f}\epsilon_{L}+\epsilon_{R})^{2}}{2(-1+a_{f}^{2})^{2}}\bigg{)}\psi_{R2}\end{split}$
(28)

and those of the heavy partners are orthogonal combinations of the above.

With all the mass eigenstates in place, we can compute the couplings of the SM
and heavy fermions with the gauge sector. For instance, the coupling of
$W^{\pm}$ with $tb$ is calculated as

$g^{L}_{Wtb}=gv^{0}_{W}t^{0}_{L}b^{0}_{L}+g_{1}v^{1}_{W}t^{1}_{L}b^{1}_{L},$
(29)

where $v_{W}^{i},t_{L}^{i},b_{L}^{i}$ are the amount of of the $W,t_{L},b_{L}$
wavefunctions at site $i$. Plugging in these values from Eqns. 18 and 28 and
taking $\sin\theta=\sin\theta_{W}\big{(}1+\frac{x^{2}}{8}\big{)}$ 555This is
evaluated using the definition of the Weinberg angle
$\cos^{2}\theta_{w}=\frac{M_{W}^{2}}{M_{Z}^{2}}$., we get

$g^{Wud}_{L}=\frac{e}{\sin\theta_{w}}\bigg{[}1+\frac{x^{2}}{4}-\frac{1}{2}\frac{(\epsilon_{L}+a_{f}\epsilon_{R})^{2}}{(-1+a_{f}^{2})^{2}}\bigg{]}.$
(30)

Following the argument below Eqn. 22, we neglect the $\epsilon_{R}$
contribution 666Note that the contribution to the $\rho$ parameter from the
heavy top-bottom is exactly as computed in Ref. Chivukula  _et al._ (2006) and
reads $\Delta\rho=\frac{M^{2}_{D}\epsilon^{4}_{fR}}{16\pi^{2}v^{2}}$. However,
since the $\epsilon_{fR}$ is not the dominant contribution to the light
fermion mass, this can be tuned as small as necessary to evade the
constraints.and rewrite the coupling as

$g^{Wud}_{L}=\frac{e}{\sin\theta_{w}}\bigg{[}1+\frac{x^{2}}{4}-\frac{\epsilon^{2}_{L}}{2}\bigg{]}.$
(31)

Thus the condition for the ideal fermion delocalization which renders the
$Wud$ coupling SM-like while concurrently making the $W^{\prime}ud$ coupling
zero can be written in exactly the same fashion as in Refs. Sekhar Chivukula
_et al._ (2009); Chivukula  _et al._ (2011):777We will also assume that the
top quark is also declocalized in exactly the same fashion because of
constraints arising from $Z\bar{b}_{L}b_{L}$ coupling - see Ref. Sekhar
Chivukula  _et al._ (2009) for details.

$\epsilon^{2}_{L}=\frac{x^{2}}{2}.$ (32)

We summarize the gauge-fermion couplings in Appendix A.

##  III Scalar Sector

In this section, we will present the couplings of the charged and neutral
Higgs bosons in the model to the gauge and fermion sectors. We will then
briefly discuss some of the constraints arising due to the demand of vacuum
stability on the Lagrangian parameters.

###  III.1 Couplings

We begin by computing the couplings of the Higgs bosons in the model to the
gauge bosons - there are two basic kinds of three point vertices: $VVX$ and
$VXX$, where $V$ generically means a (light or heavy) gauge boson and $X$ is a
(neutral or charged) higgs boson. These couplings arise from the kinetic
energy terms in the Lagrangian 16 as usual. We begin by tabulating all
couplings of the first type in Table 2.

Vertex | Strength
---|---
$hW^{+}_{\mu}W^{-}_{\nu}$ | $\frac{e^{2}v}{8\sin^{2}\theta_{w}}\bigg{(}\big{(}\sqrt{2}\cos\alpha\cos\beta-4\sin\alpha\sin\beta\big{)}-\frac{x^{2}}{4}\big{(}\sqrt{2}\cos\alpha\cos\beta+8\sin\alpha\sin\beta\big{)}\bigg{)}$
$HW^{+}_{\mu}W^{-}_{\nu}$ | $\frac{e^{2}v}{8\sin^{2}\theta_{w}}\bigg{(}\big{(}\sqrt{2}\cos\beta\sin\alpha+4\cos\alpha\sin\beta\big{)}-\frac{x^{2}}{4}\big{(}\sqrt{2}\cos\beta\sin\alpha-8\cos\alpha\sin\beta\big{)}\bigg{)}$
$hW^{{}^{\prime}+}_{\mu}W^{{}^{\prime}-}_{\nu}$ | $\frac{e^{2}v\cos\alpha\cos\beta}{\sqrt{2}x^{2}\sin^{2}\theta_{w}}\bigg{(}1-\frac{x^{2}}{2}\bigg{)}$
$HW^{{}^{\prime}+}_{\mu}W^{{}^{\prime}-}_{\nu}$ | $\frac{e^{2}v\cos\beta\sin\alpha}{\sqrt{2}x^{2}\sin^{2}\theta_{w}}\bigg{(}1-\frac{x^{2}}{2}\bigg{)}$
$hW^{+}_{\mu}W^{{}^{\prime}-}_{\nu}$ | $\frac{e^{2}v}{2\sqrt{2}x\sin^{2}\theta_{w}}\bigg{(}\cos\alpha\cos\beta+\frac{x^{2}}{2}\big{(}\sqrt{2}\sin\alpha\sin\beta-\frac{3}{4}\cos\alpha\cos\beta\big{)}\bigg{)}$
$HW^{+}_{\mu}W^{{}^{\prime}-}_{\nu}$ | $\frac{e^{2}v}{2\sqrt{2}x\sin^{2}\theta_{w}}\bigg{(}\sin\alpha\cos\beta-\frac{x^{2}}{2}\big{(}\sqrt{2}\cos\alpha\sin\beta+\frac{3}{4}\cos\beta\sin\alpha\big{)}\bigg{)}$
$AW^{+}_{\mu}W^{{}^{\prime}-}_{\nu}$ | $\frac{ie^{2}v\sin 2\beta}{4x\sin^{2}\theta_{w}}\bigg{(}1+\frac{x^{2}}{4}\bigg{)}$
$HZ_{\mu}Z_{\nu}$ | $\frac{e^{2}v}{16\sin^{2}\theta_{w}\cos^{2}\theta_{w}}\bigg{(}\sqrt{2}\cos\beta\sin\alpha+4\cos\alpha\sin\beta\bigg{)}$
$hZ_{\mu}Z_{\nu}$ | $\frac{e^{2}v}{16\sin^{2}\theta_{w}\cos^{2}\theta_{w}}\bigg{(}\sqrt{2}\cos\beta\cos\alpha-4\sin\alpha\sin\beta\bigg{)}$
$hZ^{{}^{\prime}}_{\mu}Z^{{}^{\prime}}_{\nu}$ | $\frac{e^{2}v\cos\alpha\cos\beta}{2\sqrt{2}x^{2}\sin^{2}\theta_{w}}\bigg{(}1+\frac{x^{2}}{8}\big{(}1-5\cos 2\theta_{w}\big{)}\sec^{2}\theta_{w}\bigg{)}$
$HZ^{{}^{\prime}}_{\mu}Z^{{}^{\prime}}_{\nu}$ | $\frac{e^{2}v\cos\beta\sin\alpha}{2\sqrt{2}x^{2}\sin^{2}\theta_{w}}\bigg{(}1+\frac{x^{2}}{8}\big{(}1-5\cos 2\theta_{w}\big{)}\sec^{2}\theta_{w}\bigg{)}$
$hZ_{\mu}Z^{{}^{\prime}}_{\nu}$ | $\frac{e^{2}v\sec^{3}\theta_{w}}{8\sqrt{2}x\sin^{2}\theta_{w}}\cos\alpha\cos\beta\bigg{(}5-2\cos 2\theta_{w}+\cos 4\theta_{w}\bigg{)}$
$HZ_{\mu}Z^{{}^{\prime}}_{\nu}$ | $\frac{e^{2}v\sec^{3}\theta_{w}}{8\sqrt{2}x\sin^{2}\theta_{w}}\sin\alpha\cos\beta\bigg{(}5-2\cos 2\theta_{w}+\cos 4\theta_{w}\bigg{)}$
$H^{-}W^{+}_{\mu}Z_{\nu}$ | $-\frac{ie^{2}x^{2}v}{32\cos^{3}\theta_{w}}\cos\beta\sin\beta$
$H^{-}W^{{}^{\prime}+}_{\mu}Z_{\nu}$ | $-\frac{ie^{2}v\sin 2\beta\sec\theta_{w}}{4x\sin^{2}\theta_{w}}\bigg{(}1+\frac{x^{2}}{4}\bigg{)}$
$H^{-}W^{+}_{\mu}Z^{{}^{\prime}}_{\nu}$ | $\frac{ie^{2}v\sin 2\beta}{4x\sin^{2}\theta_{w}}\bigg{(}1+\frac{x^{2}}{16}\big{(}3+\cos 2\theta_{w}\big{)}\sec^{2}\theta_{w}\bigg{)}$
$H^{-}W^{{}^{\prime}+}_{\mu}Z^{{}^{\prime}}_{\nu}$ | $-\frac{ie^{2}v\tan^{2}\theta_{w}\sin 2\beta}{8\sin^{2}\theta_{w}}\bigg{(}1+\frac{x^{2}}{4}\sec^{2}\theta_{w}\bigg{)}$

Table 2: All the triple point $VVX$ couplings in the models where $V$ is a
(light or heavy) gauge boson and $X$ is a scalar. These have been computed to
$\mathcal{O}(x^{2})$.

A couple of comments are in order.

  1. 1.

The Alignment Limit: The $hWW$ coupling is changed from the corresponding SM
one by a rather unwieldy factor as opposed to the 2HDM where the scaling
factor is neatly given by $\sin(\beta-\alpha)$. This is due to the extended
nature of EWSB in the model, as the $\Sigma$ model field that breaks the
$SU(2)\times SU(2)$ down to a diagonal $SU(2)_{L}$ also feeds into the
mechanism. Nevertheless, the scaling is still controlled by the two parameters
$\alpha$ and $\beta$. The limit in which one extracts a SM-like higgs boson,
i.e., the decoupling limit, would thus not be the same as in the 2HDM. In Fig.
2 below, we show this alignment limit in the $\tan\beta-\sin(\beta-\alpha)$
plane commonly employed in the 2HDM literature. It can be seen that
identifying the lighter mass eigenstate as the SM-like Higgs and demanding
that its coupling to $WW$ take on the SM value forces the 2HDM to be confined
to the regions very close to $\sin(\beta-\alpha)=\pm 1$, while in this model
there is a fairly larger parameter space that opens up in the region
$-0.4\leq\sin(\beta-\alpha)\leq 0.3$ for all values of
$\tan\beta$888Interestingly the region around $\sin(\beta-\alpha)=0$ is the
alignment limit in the 2HDM corresponding to identifying the heavier $H$ as
the SM-like Higgs. See also Footnote [7]..

Figure 2: The alignment limit in the 2HDM (shown in red) and in this model
(shown as the shaded region within the blue contour). As can be seen, while
the 2HDM alignment limit is forced near the $\sin(\beta-\alpha)\approx\pm 1$
regions, the corresponding limit for this model occurs in the region
$-0.5\leq\sin(\beta-\alpha)<0.3$ owing to the nature of the $g_{hWW}$
coupling.

  2. 2.

Perturbativity: Couplings involving one (two) heavy gauge bosons are
proportional to $1/x$ ($1/x^{2}$) - treating $x$ as a small parameter as we
have done, we note that these couplings become large for sufficiently small
$x$ (or equivalently, a sufficiently large $m_{W^{\prime}}$). Thus, novel
decays involving these heavy gauge bosons need to be treated with care and are
not valid for arbitrarily massive $W^{\prime}$s as we would lose
perturbativity. As can be seen from Tables 7 and 8, a similar pattern also
emerges in the coupling of a $W^{\prime}$ or a $Z^{\prime}$ with two heavy
fermions.

Similarly we present the $VXX$ couplings in Table 3 \- again we see similar
features emerge where couplings involving a heavy gauge boson are enhanced by
a factor of $1/x$. The presence of these vertices makes the phenomenology of
this model rich - in addition to the usual decay channels for the heavy Higgs
bosons ($WW$, $ZZ$, $bb/\tau\tau$) etc., there exist possibilities of them
decaying into one of these heavy gauge bosons that would potentially lead to
stark signals at the LHC involving, for example, multi-lepton final states.
Since these heavy gauge bosons do not couple to the SM fermions, they evade
much of the direct bounds and the oblique ones and thus need not be terribly
massive. We discuss these possibilities further in Section IV.

Vertex | Strength
---|---
$A_{\mu}H^{+}H^{-}$ | $e\big{(}p_{H^{+}}-p_{H^{-}}\big{)}_{\mu}$
$Z_{\mu}H^{+}H^{-}$ | $\frac{e\cos 2\theta_{w}}{2\sin\theta_{w}\cos\theta_{w}}\bigg{(}1+\frac{x^{2}\sec^{2}\theta_{w}}{128}\big{(}12+8\cos 2\beta+2\cos 2(\beta-\theta_{w})+2\cos 2(\beta+\theta_{w})+3\cos 2\theta_{w}+4\cos 4\theta_{w}+\cos 6\theta_{w}\big{)}\bigg{)}\big{(}p_{H^{-}}-p_{H^{+}}\big{)}_{\mu}$
$Z^{{}^{\prime}}_{\mu}H^{+}H^{-}$ | $\frac{e\sec^{2}\theta_{w}}{2x\sin\theta_{w}}\bigg{(}\sin^{2}\beta-\frac{x^{2}}{64}\big{(}30+2\cos 2\beta-\cos 2(\beta-\theta_{w})+2\cos 2\theta_{w}-\cos 2(\beta+\theta_{w})\big{)}\bigg{)}\big{(}p_{H^{-}}-p_{H^{+}}\big{)}_{\mu}$
$W^{+}_{\mu}H^{-}A$ | $\frac{e}{2\sin\theta_{w}}\bigg{(}1+\frac{x^{2}}{32}\big{(}5+3\cos 2\beta\big{)}\bigg{)}\big{(}p_{H^{-}}-p_{A}\big{)}_{\mu}$
$W^{{}^{\prime}+}_{\mu}H^{-}A$ | $\frac{e}{2x\sin\theta_{w}}\bigg{(}\sin^{2}\beta-\frac{x^{2}}{2}\bigg{)}\big{(}p_{H^{-}}-p_{A}\big{)}_{\mu}$
$W^{+}_{\mu}H^{-}h$ | $\frac{ie\sin\alpha}{8\sin\theta_{w}}\bigg{(}\big{(}4\cos\beta+\sqrt{2}\sin\beta\big{)}+\frac{x^{2}}{8}\big{(}8\cos\beta-\sqrt{2}\sin\beta\big{)}\bigg{)}\big{(}p_{H^{-}}-p_{h}\big{)}_{\mu}$
$W^{+}_{\mu}H^{-}H$ | $-\frac{ie}{8\sin\theta_{w}}\bigg{(}\big{(}4\cos\alpha\cos\beta-\sqrt{2}\sin\alpha\sin\beta\big{)}+\frac{x^{2}}{8}\big{(}8\cos\alpha\cos\beta+\sqrt{2}\sin\alpha\sin\beta\big{)}\bigg{)}\big{(}p_{H^{-}}-p_{H}\big{)}_{\mu}$
$W^{{}^{\prime}+}_{\mu}H^{-}H$ | $\frac{ie}{2\sqrt{2}x\sin\theta_{w}}\bigg{(}\sin\alpha\sin\beta+\frac{x^{2}}{4\sqrt{2}}\big{(}4\cos\alpha\cos\beta-\sqrt{2}\sin\alpha\sin\beta\big{)}\bigg{)}\big{(}p_{H^{-}}-p_{H}\big{)}_{\mu}$
$W^{{}^{\prime}+}_{\mu}H^{-}h$ | $\frac{ie}{2\sqrt{2}x\sin\theta_{w}}\bigg{(}\sin\alpha\sin\beta-\frac{x^{2}}{4\sqrt{2}}\big{(}4\sin\alpha\cos\beta+\sqrt{2}\sin\alpha\sin\beta\big{)}\bigg{)}\big{(}p_{H^{-}}-p_{h}\big{)}_{\mu}$

Table 3: All the triple point $VXX$ couplings in the models where $V$ is a
(light or heavy) gauge boson and $X$ is a scalar. These have been computed to
$\mathcal{O}(x^{2})$.

We now turn to the Higgs-Fermion couplings - note that due to mixing, in
addition to terms like $h\bar{f}f$ and $H\bar{f}f$, there would also be scalar
couplings of the form $h\bar{f}F$ and $H\bar{f}F$. We list these couplings
below working in the limit $\epsilon_{R}\to 0$ and to $\mathcal{O}(x^{2})$.
Notice also that for all light fermions $a_{u,d}\propto m\approx 0$ \- however
we have retained the $a$ terms below as these are important for the top sector
(the generational index on the $\lambda$’s have been suppressed for brevity).

| $\bar{u}_{L}u_{R}$ | $\bar{u}_{L}U_{R}$ | $\bar{U}_{L}u_{R}$ | $\bar{U}_{L}U_{R}$
---|---|---|---|---
$h$ | $-\frac{\lambda_{u0}\sin\alpha}{\sqrt{2}}$ | $\frac{\lambda_{u0}\sin\alpha}{2}\frac{a_{u}x}{(a^{2}_{u}-1)}$ | $-\frac{\lambda_{u0}\sin\alpha}{2}\frac{x}{a^{2}_{u}-1}$ | $-\frac{\lambda_{u0}\sin\alpha}{2\sqrt{2}}\frac{a_{u}x^{2}}{a^{2}_{u}-1}$
$H$ | $\frac{\lambda_{u0}\cos\alpha}{\sqrt{2}}$ | $\frac{\lambda_{u0}\cos\alpha}{2}\frac{a_{u}x}{(a^{2}_{u}-1)}$ | $\frac{\lambda_{u0}\cos\alpha}{2}\frac{x}{a^{2}_{u}-1}$ | $\frac{\lambda_{u0}\cos\alpha}{2\sqrt{2}}\frac{a_{u}x^{2}}{a^{2}_{u}-1}$
$A$ | $-\frac{i\lambda_{u0}\cos\beta}{\sqrt{2}}$ | $\frac{i\lambda_{u0}\cos\beta}{2}\bigg{(}\frac{a_{u}x}{(a^{2}_{u}-1)}-\frac{x\tan^{2}\beta}{\sqrt{2}a_{u}}\bigg{)}$ | $-\frac{i\lambda_{u0}\cos\beta}{2}\frac{x}{a^{2}_{u}-1}$ | $-\frac{i\lambda_{u0}\cos\beta}{2}\frac{x}{a^{2}_{u}-1}$

Table 4: The couplings of the neutral scalars to the fermions in the model -
these have been evaluated in the limit $\epsilon_{R}\to 0$ and to
$\mathcal{O}(x^{2})$. The couplings with the down-type fermions can be
obtained via the substitutions $\lambda_{u0}\to\lambda_{d0}$ and $a_{u}\to
a_{d}$.

To leading order, $\lambda_{u0}=\frac{\sqrt{2}m_{u}}{v\sin\beta}$ (see Eqn.
26). Thus, it can be seen that the light fermion couplings to the neutral
scalars follows a Type I 2HDM-like pattern with the scaling factor (relative
to the SM) given by999In the Type I 2HDM, the $h(H)$ scaling has the
$\cos\alpha\,(\sin\alpha)$ \- this can, however, be accounted for by a
redefinition of the mixing angle $\alpha$. It is identical to Type I 2HDM in
the sense that the scaling does not differ between the up-type and down-type
fermions.

$\displaystyle\begin{split}\xi_{h}^{u,d}&=\sin\alpha/\cos\beta,\\\
\xi_{H}^{u,d}&=\cos\alpha/\cos\beta,\\\ \xi_{A}^{u,d}&=\cot\beta.\end{split}$
(33)

In addition, the charged Higgs couplings to the light fermions are given (to
leading order) by the following expressions:

$\displaystyle\begin{split}H^{+}\bar{u}_{R}d_{L}+h.c.&=i\lambda^{ij}_{u0}\cos\beta,\\\
H^{+}\bar{u}_{L}d_{R}+h.c.&=-i\lambda^{ij}_{d0}\cos\beta.\end{split}$ (34)

###  III.2 Vacuum Stability

Let us begin by writing the potential of the scalar fields again:

$\displaystyle V(\Phi_{1},\Phi_{2},\Sigma)$
$\displaystyle=m^{2}_{11}\Phi^{\dagger}_{1}\Phi_{1}+m^{2}_{22}\Phi^{\dagger}_{2}\Phi_{2}-(m^{2}_{12}\Phi^{\dagger}_{1}\Sigma\Phi_{2}+\textnormal{h.c.})+\frac{\beta_{1}}{2}(\Phi^{\dagger}_{1}\Phi_{1})^{2}+\frac{\beta_{2}}{2}(\Phi^{\dagger}_{2}\Phi_{2})^{2}$
$\displaystyle+\beta_{3}(\Phi^{\dagger}_{1}\Phi_{1})(\Phi^{\dagger}_{2}\Phi_{2})+\beta_{4}(\Phi^{\dagger}_{1}\Sigma\Phi_{2})(\Phi^{\dagger}_{2}\Sigma^{\dagger}\Phi_{1})+\bigg{[}\frac{\beta_{5}}{2}(\Phi^{\dagger}_{1}\Sigma\Phi_{2})^{2}+\textnormal{h.c.}\bigg{]}.$

Defining $\Phi^{\dagger}_{1}\Phi_{1}=a$, $\Phi^{\dagger}_{2}\Phi_{2}=b$,
$\textnormal{Re}[\Phi^{\dagger}_{1}\Sigma\Phi_{2}]=c$ and
$\textnormal{Im}[\Phi^{\dagger}_{1}\Sigma\Phi_{2}]=d$, the quartic part of the
Lagrangian (which needs to be a manifestly positive quantity so the potential
can be bounded from below) can now be recast in the following form:

$\displaystyle\begin{split}V_{4}&=\frac{1}{2}\big{(}\sqrt{\beta_{1}}a-\sqrt{\beta_{2}}b\big{)}^{2}+\big{(}\beta_{3}+\sqrt{\beta_{1}\beta_{2}}\big{)}(ab-c^{2}-d^{2})+2(\beta_{3}+\beta_{4}+\sqrt{\beta_{1}\beta_{2}})c^{2}\\\
&+\big{(}\textnormal{Re}[\beta_{5}]-\beta_{3}-\beta_{4}-\sqrt{\beta_{1}\beta_{2}}\big{)}(c^{2}-d^{2})-2cd\,\textnormal{Im}[\beta_{5}].\end{split}$
(35)

With the field identifications $\Phi_{1}\to\Phi_{1}$ and
$\Phi_{2}\to\Sigma\Phi_{2}$, this is identical to the corresponding potential
in the 2HDM literature - see, for instance Bhattacharyya and Das (2016). We
can now write down the (Cauchy-Schwarz) inequality for the combination of
fields in the following form:

$ab\geq c^{2}+d^{2}.$ (36)

By considering various choices of field directions and ensuring that $V_{4}$
does not run negative for large values of the parameters $a,b$ leads us to the
following constraints on the various $\beta$’s:

$\displaystyle\begin{split}&\beta_{1}>0\\\ &\beta_{2}>0\\\
&(\beta_{3}+\sqrt{\beta_{1}\beta_{2}})>0\\\
&\beta_{3}+\beta_{4}+\sqrt{\beta_{1}\beta_{2}}>|\beta_{5}|.\end{split}$ (37)

Translated into the space of the $\lambda$’s as given in Eqn. 5, these
constraints read

$\displaystyle\begin{split}&\lambda_{1}+\lambda_{3}>0\\\
&\lambda_{2}+\lambda_{3}>0\\\
&2\lambda_{3}+\lambda_{4}+2\sqrt{(\lambda_{1}+\lambda_{3})(\lambda_{2}+\lambda_{3})}>0\\\
&2\lambda_{3}+\frac{1}{2}(\lambda_{5}+\lambda_{6})-\frac{1}{2}|\lambda_{5}-\lambda_{6}|+2\sqrt{(\lambda_{1}+\lambda_{3})(\lambda_{2}+\lambda_{3})}>0.\end{split}$
(38)

These conditions guarantee that the electroweak vacuum is stable in this model
- the results are formally identical to that found in the 2HDM literature. An
extensive discussion of the stability bounds on the 2HDM potential can be
found in Klimenko (1985); Maniatis  _et al._ (2006); Nie and Sher (1999).

###  III.3 Unitarity

Insuring perturbative unitarity in the $2\rightarrow 2$ longitudinal vector
boson scattering is an important aspect of every BSM scenario. This amplitude
can be written in the schematic form

$\mathcal{M}(W_{L}W_{L}\to
W_{L}W_{L})=a_{0}+a_{1}\frac{E^{4}}{M_{W}^{4}}+a_{2}\frac{E^{2}}{M_{W}^{2}},$
(39)

with a constant piece $a_{0}$ and terms with grow with energy as $E^{2}$ and
$E^{4}$. In the SM, the relation between the four point and the three-point
vertices arising out of gauge invariance cancels the $E^{4}$ growth, while the
cancelation of the $E^{2}$ piece requires the higgs exchange diagrams - a fact
that was used to first constrain the mass of the higgs boson Lee  _et al._
(1977). Before the discovery of the higgs, “higgsless” models of electroweak
symmetry breaking emerged that guaranteed perturbative unitarity using
exchange of a tower of KK bosons in lieu of a higgs Chivukula  _et al._
(2003); Csaki  _et al._ (2004). The present model has both higgs and heavy
gauge bosons and so unitarization of the longitudinal vector boson scattering
can proceed via both channels. We will analyse this for the process
$W_{L}Z_{L}\rightarrow W_{L}Z_{L}$. Writing the amplitude in the form

$\mathcal{M}(W^{\pm}_{L}Z_{L}\rightarrow
W^{\pm}_{L}Z_{L})=\mathcal{M}_{4}+\mathcal{M}_{s}+\mathcal{M}_{t}+\mathcal{M}_{u},$

we see that the amplitudes arise from all tree-level processes listed in Fig[
3. The $\mathcal{O}(E^{4})$ growth is canceled by imposing the sum rule

$g_{WWZZ}=g^{2}_{WWZ}+g^{2}_{WW^{{}^{\prime}}Z}.$ (40)

The first term is reminiscent of the SM wherein the relation between the four-
and three-point couplings is guaranteed by gauge invariance while the second
arises from the $W^{\prime}$ exchange diagram. Similarly, to cancel all the
quadratic order energy growth, all the three-point vertices involving gauge
bosons and Higgs bosons must be constrained by

$\bigg{(}\frac{M^{4}_{Z}}{M^{2}_{W}}g^{2}_{WWZ}+\frac{M^{4}_{Z}}{M^{2}_{W^{{}^{\prime}}}}g^{2}_{WW^{{}^{\prime}}Z}\bigg{)}=g_{HWW}g_{HZZ}+g_{hWW}g_{hZZ}-g^{2}_{H^{\pm}W^{\mp}Z}.$
(41)

We note here that this involves both gauge and higgs mediated diagrams. In
Fig. 4 we show the allowed region in the $M_{W^{{}^{\prime}}}-\sin\beta$ plane
after imposing these sum rules for various choices of $\sin\alpha$.101010The
choice of the $\sin\alpha$ values anticipates Fig. 6, wherein we have put in
the LHC constraints on the $h$-125 higgs; $\sin\alpha=0.95$ and $-0.5$ are
allowed, while $\sin\alpha=0.3$ is not.

Figure 3: Gauge Boson and Scalars mediated Feynman diagrams contributing to
the scattering process $W^{+}Z\rightarrow W^{+}Z$.

Figure 4: The yellow shaded region corresponds to the allowed benchmark points
after recasting the unitarity condition for our model considering different
value of $\sin\alpha$.

##  IV LHC Phenomenology

Finally with all the couplings and other relevant model details in place, we
turn to the phenomenological issue of production and discovery of the various
particles in the model. In this section, we concentrate on the scalar sector
of the model and detail the cross-sections and branching ratios of the various
higgses.111111See Ref. Chen  _et al._ (2020) for a recent analysis in the
Type-I 2HDM case.We postpone a detailed collider analysis of this model
including the gauge and the fermion sectors to a future work Coleppa and
Sarkar (2020).

###  IV.1 The 125 GeV Higgs

As is true in any model with multiple CP-even scalars, the phenomenological
analysis of the model will be different depending on which one is chosen to be
SM-like. There are two choices in this model: choosing the lighter of the two
mass eigenstates to be the SM-like Higgs boson will mean that we have another
heavy Higgs while choosing the heavier mass eigenstate to be the 125 GeV Higgs
would mean that there is another higgs _lighter_ than the SM one. We begin
with an understanding of how the parameter spaces of the model will look like
in the two distinct cases.

We first recall the scaling to be employed to facilitate the comparison to the
SM case. Fixing, for instance, the lighter higgs to be SM-like, we have

$\frac{\sigma(gg\rightarrow h)}{\sigma(gg\rightarrow
h_{\textrm{SM}})}=\frac{\Gamma(h\rightarrow
gg)}{\Gamma(h_{\textrm{SM}}\rightarrow gg)}.$ (42)

Since in the SM, the decay width of $h\rightarrow gg$ takes the form

$\Gamma(h_{\textrm{SM}}\rightarrow
gg)=\frac{\alpha^{2}m^{3}_{h}N^{2}_{C}}{256\pi^{3}v^{3}}|\sum_{i}e^{2}_{i}F_{i}|^{2},$
(43)

we find

$\qquad\frac{\Gamma(h\rightarrow gg)}{\Gamma(h_{\textrm{SM}}\rightarrow
gg)}=\frac{|\sum_{q_{i},Q_{i}}\xi^{i}_{hff}e^{2}_{i}F_{i}|^{2}}{|\sum_{q_{i}}e^{2}_{i}F_{i}|^{2}},$
(44)

where $F_{1/2}=-2\tau\big{[}1+(1-\tau)f(\tau)\big{]}$,
$\tau=\frac{4m^{2}_{f}}{m^{2}_{h}}$ and

$f(\tau)=\left\\{\begin{array}[]{ll}[\sin^{-1}\big{(}\sqrt{\tau^{-1}}\big{)}]^{2}&\mbox{
$\tau\geq 1$}\\\
-\frac{1}{4}\big{|}\ln\frac{1+\sqrt{1-\tau}}{1-\sqrt{1-\tau}}-i\pi\big{|}^{2}&\mbox{$\tau<1$}.\end{array}\right.$

We have also defined the scaling factor
$\xi_{hff}=\frac{g_{hff}}{g^{SM}_{hff}}$, and summed over both the SM and the
heavy vector fermions in the loop. The latter are a non-negligible
contribution because even as $M_{D}\to\infty$ and the heavy quarks formally
decouple, the loop factor $F_{\frac{1}{2}}(\tau)$ contributes the asymptotic
value $-\frac{4}{3}$. We use Eqns. 42 and 44 to compute the rates in our model
as simple multiples of the corresponding ones in the SM. For example, Fig. 5
shows the range of parameter space available after imposing the reported SM
value of the $gg\to h$ cross-section for the observed 125 GeV higgs in the
$\sin\alpha-\sin\beta$ parameter space.121212We note here that since the
scaling of the higgs couplings in our model do not neatly factor as
$\sin(\beta-\alpha)$, we choose a different parameter space from the one
commonly employed in the 2HDM literature. As can be seen readily, there is a
wide range of the mixing angle $\alpha$ and the ratio $f/F$ that are
admissible for both cases.

Figure 5: The yellow shaded region corresponds to the 125 GeV lighter higgs
$h$ (left) and the 125 GeV heavier higgs $H$ (right) yielding the reported
value of gluon-fusion production cross-section.

Of course, to make a realistic comparison, we need to compute
$\sigma\times\textrm{BR}$ for the higgs in various channels and check that we
consistently can reproduce the reported SM-higgs values for various choices of
parameter values. A 125 GeV higgs primarily decays to $b\bar{b}$
($\tau\bar{\tau}$), with small contributions from the off-shell channels like
$WW^{*}$, $ZZ^{*}$ and finally the loop induced decay channels like
$\gamma\gamma$, $Z\gamma$ and $gg$. In Fig. 6, we show the regions of
parameter space available after demanding that the $\sigma\times\textrm{BR}$
values corresponding to the $\gamma\gamma$, and the $WW^{*}/ZZ^{*}$ channel
rates match the reported experimental value ATL (2020); CMS (2020) \-
superimposed on each plot is the corresponding region in Type I 2HDM.

Figure 6: The extent of parameter space admissible (regions in yellow) after
demanding that the rates for the $gg\rightarrow\gamma\gamma$ (left), and the
$gg\rightarrow h\rightarrow WW^{*}/ZZ^{*}$ (right) match the experimentally
reported values for the 125 GeV higgs for the case of the light higgs $h$
being SM-like. Also overlaid in each plot are the corresponding regions in the
Type I 2HDM (regions enclosed by the blue contours). The dashed red contours
are the ATLAS $h\to b\bar{b}$ limits.

It is interesting to observe that while there is a significant overlap between
the two models, there are striking differences as well. Since there are
additional particles that run in the loop - both heavy quarks (whose mass is
set at $1.2$ TeV) and heavy gauge bosons (whose mass is set at 500 GeV), both
the production rates and decay branching ratios are different from the 2HDM
case. We see from Figs. 6 and 7 that the $\gamma\gamma$ rate can be
accommodated in almost the entire region of parameter space, while the $WW/ZZ$
case is more restricted regardless of which higgs is chosen to be SM-like.
Type I 2HDM shows precisely the opposite behavior wherein the $WW/ZZ$
constraint is a little more relaxed in terms of the allowed parameter space.
Putting in the LHC constraints, for the case of a light higgs being the 125
GeV particle, we find that the range $-0.3<\sin\alpha<-1$ covers almost the
entire range of $\sin\beta$ values whereas for positive values of
$\sin\alpha$, a smaller portion of admissible range opens up at
$\sin\alpha\approx 0.7$ for values of $\sin\beta\approx 0.6$ and above. Both
these regions shrink if we also impose the latest $h\to b\bar{b}$ (with the
higgs produced via associated production) constraints ATL (2018) \- these are
shown by the dashed red contours in Fig. 6.

It is clear from the foregoing analysis that a simple measurement of rates in
different channels alone would not serve to tell the models apart in spite of
the present model having a richer EWSB structure. A precise determination of
the decay widths of the higgs would point to interesting physics beyond simple
two-doublet extensions of the SM as it would reveal the $\mathcal{O}(x^{2})$
corrections inherent in the couplings. Since $x\approx m_{W}/m_{W^{\prime}}$,
this would be more efficient if the $W^{\prime}$ is not too massive - we
remind the reader that in this model since there are no tree level couplings
of the light fermions to the extra heavy gauge bosons,131313The $Z^{\prime}$
does have a small hypercharge coupling $\propto x$ to the SM fermions. the
heavy gauge bosons can comfortably evade all direct bounds coming from
experiments.

In Fig. 7 below, we perform a similar analysis fixing the heavier of the two
eigenstates to be the SM-like higgs boson and compute the parameter space
available after demanding SM-like rates. Comparing this to Fig. 6, we see that
the admissible parameter spaces in the two cases are largely exclusive of each
other in spite of the production cross-sections having a significant overlap
(see Fig. 5).

Figure 7: The extent of parameter space admissible (regions in yellow) after
demanding that the rates for the $gg\rightarrow\gamma\gamma$ (left), and the
$gg\rightarrow h\rightarrow WW^{*}/ZZ^{*}$ (right) match the experimentally
reported values for the 125 GeV higgs for the case of the heavy higgs $H$
being SM-like. Also overlaid in each plot are the corresponding regions in the
Type I 2HDM (regions enclosed by the blue contours). The dashed red contours
are the ATLAS $h\to b\bar{b}$ limits.

###  IV.2 The Heavy Higgs $H$

We now turn our attention to the task of identifying promising channels at the
LHC that would uncover the particular new physics signals associated with this
model. In all our calculations below, we fix $M_{D}=1.2$ TeV, and
$M_{W^{{}^{\prime}\pm}}=400$ GeV and work in the $\sin\beta-\sin\alpha$ plane.
While relaxing these would undoubtedly introduce more decay channels
particularly for lighter $H$, in this section we restrict our attention to
these benchmark points.

The ATLAS and CMS experiments have looked for a heavy higgs in the diboson
channel Aaboud  _et al._ (2018), and have placed upper limits on the
corresponding $\sigma\times$BR for the process for a wide range of heavy higgs
masses - in Fig. 8, we translate these limits to the case of a $H$ that is
light ($M_{H}=200$ GeV), moderately heavy ($M_{H}=600$ GeV), and heavy
($M_{H}=1000$ GeV). It is seen that the limits are stronger on lighter
higgses, and that the allowed region saturates in a band around
$-1\leq\sin\alpha\leq 0.3$ and a thin strip around $\sin\alpha=1$. It is seen
from Fig. 6 that these regions also correspond to the ones that are allowed by
the h-125 data - thus there is potential in this model for a heavier higgs to
be discovered for a wide range of masses. We then turn to the question of
understanding the best channels to do so.

Figure 8: The extent of parameter space admissible (regions in yellow) after
imposing the cross-section bound from ATLAS in the $H\to ZZ$ channel for
$M_{H}=$200 GeV (left), 600 GeV (middle), and 1 TeV (right). It is seen that
the admissible region grows with increasing mass as the limit on the cross-
section gets understandably weaker.

We start by listing out the decay widths in all the available channels: in
addition to the usual $VV$ and $ff$, there also exist decays into the lighter
higgses, and $H^{+}H^{-}$, and $WW^{\prime}/ZZ^{\prime}$ channels (if
kinematically allowed). In what follows, we have defined
$\xi_{HVV}=\frac{g_{HVV}}{g^{SM}_{hVV}}$ ($V=W^{\pm},Z$),
$\xi_{HVV^{{}^{\prime}}}=\frac{g_{HVV^{{}^{\prime}}}}{g^{SM}_{hVV}}$
($V=W^{\pm},Z$ and $V^{{}^{\prime}}=W^{{}^{\prime}\pm},Z^{{}^{\prime}}$,
$\lambda_{HSS}$ is the Higgs self coupling ($S=A,h$), and finally
$\xi_{Hf\bar{f}}=\frac{g_{Hf\bar{f}}}{g^{SM}_{hf\bar{f}}}$ and $N_{c}=3\,(1)$
for the quarks (leptons).

$\displaystyle\Gamma(H\rightarrow VV)=\frac{m^{3}_{H}}{16\pi
v^{2}}\sqrt{1-\frac{4m^{2}_{V}}{m^{2}_{H}}}\bigg{(}1-\frac{4m^{2}_{V}}{m^{2}_{H}}+\frac{3}{4}\bigg{(}\frac{4m^{2}_{V}}{m^{2}_{H}}\bigg{)}^{2}\bigg{)}\xi^{2}_{HVV}$
(45a) $\displaystyle\Gamma(H\rightarrow
VV^{{}^{\prime}})=\frac{m^{2}_{V}m^{2}_{V^{{}^{\prime}}}}{4\pi
v^{2}m_{H}}\xi^{2}_{HVV^{{}^{\prime}}}\bigg{(}2+\frac{(m^{2}_{H}-m^{2}_{V^{{}^{\prime}}}-m^{2}_{V})}{4m^{2}_{V}m^{2}_{V^{{}^{\prime}}}}\bigg{)}\sqrt{1-2\bigg{(}\frac{m^{2}_{V}+m^{2}_{V^{{}^{\prime}}}}{m^{2}_{H}}\bigg{)}+\bigg{(}\frac{m^{2}_{V^{{}^{\prime}}}-m^{2}_{V}}{m^{2}_{H}}\bigg{)}^{2}}$
(45b) $\displaystyle\Gamma(H\rightarrow SS)=\frac{\lambda^{2}_{HSS}}{32\pi
m_{H}}\sqrt{1-\frac{4m^{2}_{S}}{m^{2}_{H}}}$ (45c)
$\displaystyle\Gamma(H\rightarrow
H^{+}H^{-})=\frac{\lambda^{2}_{HH^{+}H^{-}}}{32\pi
m_{H}}\sqrt{1-\frac{4m^{2}_{H^{\pm}}}{m^{2}_{H}}}$ (45d)
$\displaystyle\Gamma(H\rightarrow f\bar{f})=\frac{N_{c}m^{2}_{f}m_{H}}{8\pi
v^{2}}\bigg{(}1-\frac{4m^{2}_{f}}{m^{2}_{H}}\bigg{)}^{\frac{3}{2}}.$ (45e)

In Fig. 9, we plot the branching ratio of the $H$ into various channels for
two different choices of $\sin\alpha$ and $\sin\beta$. While the qualitative
picture seems largely consistent between the two plots, it is seen that the
relative weights of the $Htt$ and the $WW/ZZ$ is quite different for the
entire range of heavy higgs mass. This is due to the coupling $g_{HWW}$ being
numerically smaller for the second choice, and
$g_{Ht\bar{t}}\propto\cos\alpha$ being larger. The combined effect renders
BR$(H\to t\bar{t})$ maximal thus also making it difficult experimentally to
discover the heavy higgs for smaller values of the mixing angle $\sin\alpha$.
However, there are other novel channels which in spite of lower decay
branching ratios would nevertheless be of potentially great interest at the
LHC. In terms of the collider signatures, we can deduce the following:

  1. 1.

For a relatively light higgs ($m_{H}<400$ GeV), in addition to the $tt$
channel (after it opens up kinematically), the most promising channels are the
$WW$ and $ZZ$ corresponding to either multi-leptonic or multi-jet final
states. The channel $H\to ZZ\to 4\ell$ with four hard leptons seems
promising141414The ATLAS search limit imposed in Fig. 8 corresponds to the
$\ell\ell jj$ channel. as it aids in full reconstruction of the parent
particle without any associated hadronic background, but comes with the price
of tagging efficiency of the four leptons and the associated low
$Z\to\ell\ell$ branching ratios. Also, in spite of a smaller branching ratio,
the channel $H\to hh$ might also be viable - the presence of heavily boosted
$b$ collinear quarks in the final state could invite employing fat jet
techniques for reconstruction.

  2. 2.

For a heavier higgs ($m_{H}>400$ GeV), we see that in addition to the channels
above (except the $hh$), there is the interesting possibility of it decaying
into the charged higgs and a $W$, opening up interesting cascade decays. A
particularly novel collider signature in this model is the decay chain $H\to
H^{+}W^{-}$ with $H^{+}\to W^{\prime}Z$ assuming the mass hierarchy
$m_{H}>m_{H^{+}}>m_{W^{\prime}}$. We discuss this and the $H^{+}\to WZ$
channel in more detail following Fig. 11.

Figure 9: The decay branching ratio of the heavy higgs $H$ into different
final states for two different choices of $\sin\alpha$ and $\sin\beta$.

###  IV.3 The Charged Higgs $H^{\pm}$

The charged higgs has been extensively looked for in the $tb$ and other
channels for $m_{H^{\pm}}>m_{t}$ and in the $\tau\nu$ and $m_{H^{\pm}}<m_{t}$
whence it can be produced as a decay product of the top quark. The Tevatron
D$\cancel{0}$ collaboration, has searched for the charged Higgs as a decay
product of the top-quark, with the $H^{\pm}$ further decaying to either
$c\bar{s}$ or $\tau^{\pm}\nu_{\tau}$ Abazov  _et al._ (2009). The non-
observation of the $H^{\pm}$ puts an upper bound on the branching ratio,
BR$(t\rightarrow H^{\pm}t)\leq$ 0.12 - 0.26, depending upon the charged Higgs
mass. In our model, the decay width of the top quark to charged Higgs and the
bottom quark takes the form

$\Gamma(t\rightarrow H^{\pm}b)=\bigg{(}\frac{m_{t}}{16\pi
v^{2}}\bigg{)}\big{[}m^{2}_{t}R^{2}+\mathcal{O}(m^{2}_{b})\big{]}\bigg{(}1-\frac{m^{2}_{H^{\pm}}}{m^{2}_{t}}\bigg{)}^{2},$

where

$R=\bigg{(}\frac{\lambda_{t}}{\lambda^{\textrm{SM}}_{t}}\bigg{)}\cos\beta=\cot\beta\bigg{(}\frac{2(1-\frac{\epsilon^{2}_{L}}{2})M^{2}_{D}-2m^{2}_{t}}{M^{2}_{D}-m^{2}_{t}}\bigg{)}.$
(46)

Note that for a sufficiently large Dirac mass, the quantity $R$ is insensitive
to all the model parameters except $\sin\beta$ \- in Fig. 10 on the left, we
display the branching ratio BR$(t\rightarrow H^{\pm}t)$ for various values of
$\sin\beta$. Overlaid on the plot is a line corresponding to the BR value of
0.2151515The precise bound of the BR depends on the $H^{\pm}$ mass - however,
here we simply show the upper limit to extract the main qualitative features.
\- it is seen that a charged Higgs with mass $m_{H^{\pm}}\leq 140$ GeV is
mostly disallowed for the entire range of $\sin\beta$. Typically one would
need very low values of $R$, or equivalently large $\sin\beta$, for a light
charged Higgs to escape the Tevatron bounds.

The LHC collaborations have also looked for a heavy charged higgs for a wide
range of masses in the $tb$ channel Aad  _et al._ (2016b), and have imposed
upper limits on the $\sigma\times$BR value - on the right in Fig. 10, we
display this limit (black curve) and also show the corresponding limit in our
model for two values of $\sin\beta$. Given that a heavy charged higgs in our
model has multiple decay channels, folding in the correct branching ratio to
$tb$ gives a number that is well within the experimental bounds for the entire
range of masses.

Figure 10: (Left): The black dashed line shows the maximally allowed branching
ratio for $t\rightarrow H^{\pm}b$ from the Tevatron experiments. It is seen
that increasing the value of $\sin\beta$ can relax the lower mass bound for
the charged Higgs, albeit only slightly. (Right): The LHC experimental
exclusion limit in the channel $H^{\pm}\to t\bar{b}$ \- we have translated the
limits to the present model for two different values of $\sin\beta$.

To understand the best discovery modes for a charged higgs in this model, we
begin with the light case ($m_{H^{\pm}}<$ 200 GeV) - in Fig. 11, we present
the branching fractions into the various available channels. Before the $tb$
opens up, the decay is dominated by the $\tau\nu$ and the $cs$ channels. One
interesting feature of this model is the presence of the $H^{\pm}WZ$ vertex,
and thus the decay $H^{\pm}\to WZ$, though highly suppressed after the $tb$
becomes kinematically available, is still non-zero. There is a small
interesting region around $m_{H^{\pm}}\approx$ 170 - 180 GeV where this decay
is quite appreciable (with a 10-20% BR). While the kinematic range in which
this decay is appreciable is indeed very narrow, it can still serve as a
useful discriminant from models in which the $SU(2)_{L}\times U(1)_{Y}$
breaking is carried out only by Higgs fields in the doublet representation (as
in the 2HDM). In all such models, regardless of the number of higgs doublets,
the $ZW^{\pm}H^{\mp}$ vertex is absent at tree level - see Ref. Johansen  _et
al._ (1982) for details.

Figure 11: In the left digram we have presented the branching ratio for the
different channels which are kinematically allowed for the light charged
Higgs. In the figure at right, The thick black curve showed the region
excluded for the from the recent experimental searches conducted by both ATLAS
and CMS.

The situation becomes quite interesting for more massive charged Higgses: in
addition to the usual channels one probes, there are interesting new final
states allowed in this model owing to the extended electroweak gauge sector.
We begin by first listing out the partial decay widths in all the channels
like in the previous heavy Higgs case.

$\displaystyle\Gamma(H^{\pm}\rightarrow
VV^{{}^{\prime}})=\frac{m^{2}_{V}m^{2}_{V^{{}^{\prime}}}}{4\pi
v^{2}m_{H^{\pm}}}\xi^{2}_{H^{\pm}VV^{{}^{\prime}}}\bigg{(}2+\frac{(m^{2}_{H^{\pm}}-m^{2}_{V}-m^{2}_{V^{{}^{\prime}}})}{4m^{2}_{V}m^{2}_{V^{{}^{\prime}}}}\bigg{)}\sqrt{1-2\bigg{(}\frac{m^{2}_{V}+m^{2}_{V^{{}^{\prime}}}}{m^{2}_{H^{\pm}}}\bigg{)}+\bigg{(}\frac{m^{2}_{V^{{}^{\prime}}}-m^{2}_{V}}{m^{2}_{H^{\pm}}}\bigg{)}^{2}}$
(47a) $\displaystyle\Gamma(H^{\pm}\rightarrow VS)=\frac{m^{3}_{H^{\pm}}}{8\pi
v^{2}}\xi^{2}_{H^{\pm}VS}\bigg{[}1-\frac{(m_{S}+m_{V})^{2}}{m^{2}_{H^{\pm}}}\bigg{]}^{\frac{3}{2}}\bigg{[}1-\frac{(m_{V}-m_{S})^{2}}{m^{2}_{H^{\pm}}}\bigg{]}^{\frac{3}{2}}$
(47b) $\displaystyle\Gamma(H^{\pm}\rightarrow
ff^{{}^{\prime}})=\frac{N_{C}\lambda^{\frac{1}{2}}}{8\pi
v^{2}m^{3}_{H^{\pm}}}\big{[}(m^{2}_{H^{\pm}}-m^{2}_{f}-m^{2}_{f^{{}^{\prime}}})(m^{2}_{f}+m^{2}_{f^{{}^{\prime}}})\bigg{(}\frac{\lambda_{f}}{\lambda^{SM}_{f}}\bigg{)}^{2}-4m^{2}_{f}m^{2}_{f^{{}^{\prime}}}\big{]},$
(47c)

where the various $\xi_{H^{\pm}VV^{\prime}}$ are given by

$\xi_{H^{\pm}W^{{}^{\prime}\mp}Z}=\frac{\sin\beta}{2}\bigg{(}1+\frac{x^{2}}{4}\bigg{)};\,\,\xi_{H^{\pm}W^{\mp}Z}=\frac{x^{2}\cos\beta\sin\beta}{16\sin^{2}\theta_{w}\cos^{2}\theta_{w}};\,\,\xi_{H^{\pm}W^{\mp}Z^{{}^{\prime}}}=\frac{\sin\beta\cos\theta_{w}}{2}\bigg{(}1+\frac{x^{2}}{8}\bigg{)},$

the $\xi_{H^{\pm}VS}$ by

$\displaystyle\xi_{H^{\pm}W^{{}^{\prime}\mp}A}=\frac{1}{x}\bigg{(}\sin^{2}\beta-\frac{x^{2}}{2}\bigg{)};\,\,\xi_{H^{\pm}W^{{}^{\prime}\mp}h}=\frac{1}{\sqrt{2}x}\big{[}\sin\alpha\sin\beta-\frac{x^{2}}{32}(4\sin\alpha\cos\beta+\sqrt{2}\sin\alpha\sin\beta)\big{]};$
$\displaystyle\xi_{H^{\pm}W^{\mp}A}=1+\frac{x^{2}}{32}(5+3\cos
2\beta);\,\,\xi_{H^{\pm}W^{\mp}h}=\frac{1}{4}\big{[}(4\sin\alpha\cos\beta+\sqrt{2}\sin\alpha\sin\beta)+\frac{x^{2}}{32}(8\sin\alpha\cos\beta-\sqrt{2}\sin\alpha\sin\beta)\big{]},$

and the Kallen function
$\lambda(m_{H^{\pm}},m_{f},m_{f^{{}^{\prime}}})=\big{[}m^{2}_{H^{\pm}}-(m_{f}+m_{f^{{}^{\prime}}})^{2}\big{]}\big{[}m^{2}_{H^{\pm}}-(m_{f}-m_{f^{{}^{\prime}}})^{2}\big{]}$.

In Fig. 12, we display the branching ratio of a heavy charged Higgs
($m_{H^{\pm}}>300$ GeV) into various channels for three different values of
$\sin\beta$ with $\sin\alpha$ fixed at $-0.7$.161616The reason for this choice
is that $\sin\alpha=-0.7$ allows for a wide range of $\sin\beta$ values - see
Fig. 6. and for $M_{W^{\prime}}=400$ GeV.

Figure 12: In the above figure, we have presented the branching ratios for
different dominating channels. From left to right we have shown the changing
value of the branching ratio for the increasing value of $\sin\beta$ while
keeping the $\sin\alpha$ value fix at $-0.7$.

Certain qualitative features of these plots are worth noting.

  * •

For small values of $\sin\beta$, the $H^{+}tb$ coupling becomes large (see
Eqn. 46), and hence this decay mode dominates over the entire range of masses.

  * •

For moderate values of $\sin\beta$, there are other competing modes that open
up beyond $M_{H^{\pm}}\approx 750$ GeV. It is interesting to note that the
presence of a light $W^{\prime}$ (with mass 400 GeV as chosen) facilitates the
discovery of the charged Higgs via the channel $H^{+}\to W^{\prime+}Z$. Since
the $W^{\prime}$ in our model has no couplings to SM fermions at tree level,
we are left with the decay chain $H^{\pm}\to W^{\prime\pm}Z\to W^{\pm}ZZ$ with
a potentially interesting final state topology with multiple hard leptons
and/or jets.

  * •

The $H^{\pm}W^{\mp}Z^{\prime}$ coupling has an additional $\cos\theta_{w}$
factor compared to $\xi_{H^{\pm}W^{{}^{\prime}\mp}Z}$ and thus is subdominant
in the entire parameter space. However, the final decay chain for the process
$H^{\pm}\to W^{\pm}Z^{\prime}\to W^{\pm}ZZ$ could also be potentially probed
at the LHC. For both these processes, there could also be interesting cascade
decays possible with the heavy vector boson decaying to a higgs and a lighter
vector boson.

##  V Conclusions

In this paper, we presented a gauge extension of the SM inspired by the “Top
Triangle Moose” construction in Ref. Sekhar Chivukula  _et al._ (2009) and
modified it minimally by replacing one of the two non-linear sigma models with
a Higgs field. The pattern of symmetry breaking pattern aided by both Higgs
and non-linear sigma model fields results in a scalar spectrum that is
identical to the well-known 2HDM class in spite of markedly different gauge
and scalar sectors. The particle content includes heavy vector, scalar, and
fermionic resonances in addition to the SM particles. We have chosen to
construct the model so that it mimics the Type I 2HDM in terms of the pattern
of the neutral higgs couplings to the SM fermions.

The heavy and light neutral higgses in the model, while coupling differently
to the SM $W,Z$ exhibit a different alignment limit compared to the 2HDM. We
have imposed the constraints coming from the 125 GeV higgs discovery and find
that there is a fair amount of parameter space that can potentially be
directly probed at the LHC. The model retains the ideal fermion delocalization
aspect of “higgsless” models that came to fore prior to the higgs discovery
effectively rendering the heavy gauge bosons fermiophobic. Taken together with
the fact that the heavy neutral and charged higgses can decay into the
$W^{\prime}$ and $Z^{\prime}$, we find that this model admits many striking
collider signatures that involve multiple SM gauge bosons in the final state.
The presence of the additional $SU(2)$ group affects the higgs phenomenology
in interesting ways, a particular example being the $WZ$ and $W^{\prime}Z$
decay modes of the charged Higgs that are absent in 2HDM and other higgs
doublet models.

This work (re)emphasizes that it is always possible to construct models which
have distinct UV-completion but apparently look indistinguishable at low
energy regime. Favoring or rejecting a particular class of models from others
merely considering the discovery of a new particle in the future collider
experiments might not be a viable tactic going forward as many of these do
have parameter spaces wherein the cross-section numbers look very similar. For
a successful “inverse collider program”, one should simultaneously test a
theory from multiple experimental fronts. This work reinforces that the future
paradigm of the collider experiments is one where both the energy and
precision frontiers coexist on an equal standpoint. In that spirit, we will
undertake the full collider study of this model in an upcoming work.

###### Acknowledgements.

We thank Sekhar Chivukula and Elizabeth Simmons for a careful reading of the
manuscript and providing useful suggestions.

##  Appendix A Gauge and Fermion Couplings

We will start by calculating all the self-couplings to $\mathcal{O}(x^{2})$.
We start with an example each of triple gauge vertex and a four point
coupling:

$\displaystyle G_{WWZ}$
$\displaystyle=g\big{(}v^{W}_{0}\big{)}^{2}v^{Z}_{0}+g_{1}\big{(}v^{W}_{1}\big{)}^{2}v^{Z}_{1}$
$\displaystyle=-\frac{e\cos\theta_{w}}{\sin\theta_{w}}\bigg{(}1+\frac{x^{2}}{8\cos^{2}\theta_{w}}\bigg{)},$
$\displaystyle G_{WWZ\gamma}$
$\displaystyle=g^{2}\big{(}v^{W}_{0}\big{)}^{2}v^{Z}_{0}\big{(}\frac{e}{g}\big{)}+g_{1}^{2}\big{(}v^{W}_{1}\big{)}^{2}v^{Z}_{1}\big{(}\frac{e}{g_{1}}\big{)}$
$\displaystyle=e\big{[}g\big{(}v^{W}_{0}\big{)}^{2}v^{Z}_{0}+g_{1}\big{(}v^{W}_{1}\big{)}^{2}v^{Z}_{1}\big{]}$
$\displaystyle=eG_{WWZ}.$

Here the $v^{Z}_{i}$ etc. are the amount of overlap of the particular gauge
boson in site $i$ \- see Ref. Sekhar Chivukula  _et al._ (2009); Chivukula
_et al._ (2011) . In a similar fashion, we can calculate all three point and
four point couplings - we list the results in Tables 5 and 6.

Vertex | Computed as | Strength
---|---|---
$G_{WW\gamma}$ | $g(v^{W}_{0})^{2}v^{\gamma}_{0}+g_{1}(v^{W}_{1})^{2}v^{\gamma}_{1}$ | $e$
$G_{W^{{}^{\prime}}W^{{}^{\prime}}\gamma}$ | $g(v^{W^{{}^{\prime}}}_{0})^{2}v^{\gamma}_{0}+g_{1}(v^{W^{{}^{\prime}}}_{1})^{2}v^{\gamma}_{1}$ | $e$
$G_{WW^{{}^{\prime}}\gamma}$ | $gv^{W}_{0}v^{W^{{}^{\prime}}}_{0}v^{\gamma}_{0}+g_{1}v^{W}_{1}v^{W^{{}^{\prime}}}_{1}v^{\gamma}_{1}$ | 0
$G_{WWZ}$ | $g(v^{W}_{0})^{2}v^{Z}_{0}+g_{1}(v^{W}_{1})^{2}v^{Z}_{1}$ | $-\frac{e\cos\theta_{w}}{\sin\theta_{w}}\bigg{[}1+\frac{x^{2}}{8\cos^{2}\theta_{w}}\bigg{]}$
$G_{WWZ^{{}^{\prime}}}$ | $g(v^{W}_{0})^{2}v^{Z^{{}^{\prime}}}_{0}+g_{1}(v^{W}_{1})^{2}v^{Z^{{}^{\prime}}}_{1}$ | $\frac{ex}{4\sin\theta_{w}}$
$G_{W^{{}^{\prime}}W^{{}^{\prime}}Z^{{}^{\prime}}}$ | $g(v^{W^{{}^{\prime}}}_{0})^{2}v^{Z^{{}^{\prime}}}_{0}+g_{1}(v^{W^{{}^{\prime}}}_{1})^{2}v^{Z^{{}^{\prime}}}_{1}$ | $-\frac{e}{x\sin\theta_{w}}\bigg{[}1-\frac{x^{2}}{8}\big{(}3+\sec^{2}\theta_{w}\big{)}\bigg{]}$
$G_{W^{{}^{\prime}}W^{{}^{\prime}}Z}$ | $g(v^{W^{{}^{\prime}}}_{0})^{2}v^{Z}_{0}+g_{1}(v^{W^{{}^{\prime}}}_{1})^{2}v^{Z}_{1}$ | $-\frac{e}{2\sin\theta_{w}\cos\theta_{w}}\bigg{[}1+\frac{x^{2}}{8}\sec^{2}\theta_{w}-2\cos^{2}\theta_{w}\bigg{]}$
$G_{WW^{{}^{\prime}}Z}$ | $gv^{W}_{0}v^{W^{{}^{\prime}}}_{0}v^{Z}_{0}+g_{1}v^{W}_{1}v^{W^{{}^{\prime}}}_{1}v^{Z}_{1}$ | $\frac{ex}{4\sin\theta_{w}\cos\theta_{w}}$
$G_{WW^{{}^{\prime}}Z^{{}^{\prime}}}$ | $gv^{W}_{0}v^{W^{{}^{\prime}}}_{0}v^{Z^{{}^{\prime}}}_{0}+g_{1}v^{W}_{1}v^{W^{{}^{\prime}}}_{1}v^{Z^{{}^{\prime}}}_{1}$ | $-\frac{e}{2\sin\theta_{w}}\bigg{[}1+\frac{x^{2}}{2}\big{(}1+\frac{\sec^{2}\theta_{w}}{4}\big{)}\bigg{]}$
Table 5: Triple gauge boson couplings in the model - these have been evaluated perturbatively in $x$ to $\mathcal{O}(x^{2})$. Vertex | Strength
---|---
$G_{WW\gamma\gamma}$ | $e^{2}$
$G_{W^{{}^{\prime}}W^{{}^{\prime}}\gamma\gamma}$ | $e^{2}$
$G_{WW^{{}^{\prime}}\gamma\gamma}$ | $0$
$G_{WWZ\gamma}$ | $e\,G_{WWZ}$
$G_{WWZ^{{}^{\prime}}\gamma}$ | $e\,G_{WWZ^{{}^{\prime}}}$
$G_{WW^{{}^{\prime}}Z\gamma}$ | $e\,G_{WW^{{}^{\prime}}Z}$
$G_{WW^{{}^{\prime}}Z^{{}^{\prime}}\gamma}$ | $e\,G_{WW^{{}^{\prime}}Z^{{}^{\prime}}}$
$G_{W^{{}^{\prime}}W^{{}^{\prime}}Z\gamma}$ | $e\,G_{W^{{}^{\prime}}W^{{}^{\prime}}Z}$
$G_{W^{{}^{\prime}}W^{{}^{\prime}}Z^{{}^{\prime}}\gamma}$ | $e\,G_{W^{{}^{\prime}}W^{{}^{\prime}}Z^{{}^{\prime}}}$
Table 6: Four point gauge couplings in the model - these have been evaluated
perturbatively in $x$ to $\mathcal{O}(x^{2})$.

The gauge-fermion couplings can be calculated similarly and we list the
charged current and neutral current interactions in Tables 7 and 8.

Coupling | Computed As | Strength
---|---|---
$g^{Wtb}_{L}$ | $gv^{0}_{W}t^{0}_{L}b^{0}_{L}+g_{1}v^{1}_{W}t^{1}_{L}b^{1}_{L}$ | $\frac{e}{\sin\theta_{w}}$
$g^{WTb}_{L}=g^{WtB}$ | $gv^{0}_{W}T^{0}_{L}b^{0}_{L}+g_{1}v^{1}_{W}T^{1}_{L}b^{1}_{L}$ | $-\frac{ex}{2\sqrt{2}\sin\theta_{w}}$
$g^{WTB}_{L}$ | $gv^{0}_{W}T^{0}_{L}B^{0}_{L}+g_{1}v^{1}_{W}T^{1}_{L}B^{1}_{L}$ | $\frac{e}{2\sin\theta_{w}}\big{(}1+\frac{3}{8}x^{2}\big{)}$
$g^{Wtb}_{R}$ | $g_{1}v^{1}_{W}t^{1}_{R}b^{1}_{R}$ | 0
$g^{WTb}_{R}=g^{WtB}_{R}$ | $g_{1}v^{1}_{W}T^{1}_{R}b^{1}_{R}$ | 0
$g^{WTB}_{R}$ | $g_{1}v^{1}_{W}T^{1}_{R}B^{1}_{R}$ | $\frac{e}{2\sin\theta_{w}}\big{(}1-\frac{x^{2}}{8}\big{)}$
$g^{W^{{}^{\prime}}tb}_{L}$ | $gv^{0}_{W^{{}^{\prime}}}t^{0}_{L}b^{0}_{L}+g_{1}v^{1}_{W^{{}^{\prime}}}t^{1}_{L}b^{1}_{L}$ | 0
$g^{W^{{}^{\prime}}Tb}_{L}=g^{W^{{}^{\prime}}tB}_{L}$ | $gv^{0}_{W^{{}^{\prime}}}T^{0}_{L}b^{0}_{L}+g_{1}v^{1}_{W^{{}^{\prime}}}T^{1}_{L}b^{1}_{L}$ | $\frac{e}{\sqrt{2}\sin\theta_{w}}$
$g^{W^{{}^{\prime}}TB}_{L}$ | $gv^{0}_{W^{{}^{\prime}}}T^{0}_{L}B^{0}_{L}+g_{1}v^{1}_{W^{{}^{\prime}}}T^{1}_{L}B^{1}_{L}$ | $\frac{e}{x\sin\theta_{w}}\big{(}1-\frac{3}{4}x^{2}\big{)}$
$g^{W^{{}^{\prime}}tb}_{R}$ | $g_{1}v^{0}_{W}t^{1}_{R}b^{1}_{R}$ | 0
$g^{W^{{}^{\prime}}Tb}=g^{W^{{}^{\prime}}tB}$ | $g_{1}v^{0}_{W}T^{1}_{R}b^{1}_{R}$ | 0
$g^{W^{{}^{\prime}}TB}_{R}$ | $g_{1}v^{1}_{W^{{}^{\prime}}}T^{1}_{R}B^{1}_{R}$ | $\frac{e}{x\sin\theta_{w}}\big{(}1-\frac{x^{2}}{4}\big{)}$

Table 7: Charged current interactions in the model calculated to
$\mathcal{O}(x^{2})$.

Coupling | Computed As | Strength
---|---|---
$g^{Ztt}_{L}$ | $(gv^{0}_{Z}(t^{0}_{L})^{2}+g_{1}v^{1}_{Z}(t^{1}_{L})^{2})T_{3}+g_{2}v^{2}_{Z}((t^{0}_{L})^{2}+(t^{1}_{L})^{2}))(Q-T_{3})$ | $-\frac{e}{\sin\theta_{w}\cos\theta_{w}}(T_{3}-\sin^{2}\theta_{w}Q)$
$g^{ZtT}_{L}$ | $(gv^{0}_{Z}t^{0}_{L}T^{0}_{L}+g_{1}v^{1}_{Z}t^{1}_{L}T^{1}_{L})+g_{2}v^{2}_{Z}(t^{0}_{L}T^{0}_{L}+t^{1}_{L}T^{1}_{L})(Q-T_{3})$ | $\frac{exT_{3}}{2\sqrt{2}\sin\theta_{w}\cos\theta_{w}}$
$g^{ZTT}_{L}$ | $(gv^{0}_{Z}(T^{0}_{L})^{2}+g_{1}v^{1}_{Z}(T^{1}_{L})^{2})+g_{2}v^{2}_{Z}((T^{0}_{L})^{2}+(T^{1}_{L})^{2})(Q-T_{3})$ | $-\frac{e}{\sin\theta_{w}\cos\theta_{w}}\bigg{(}\frac{1}{2}\big{[}1+\frac{x^{2}}{8}(4-\sec^{2}\theta_{w})\big{]}T_{3}-Q\sin^{2}\theta_{w}\bigg{)}$
$g^{Ztt}_{R}$ | $g_{1}v^{1}_{Z}(t^{1}_{R})^{2}T_{3}+g_{2}v^{2}_{Z}((t^{1}_{R})^{2}+(t^{2}_{R})^{2})(Q-T_{3})$ | $e\tan\theta_{w}(Q-T_{3})$
$g^{ZtT}_{R}$ | $g_{1}v^{1}_{Z}(t^{1}_{R}T^{1}_{R})T_{3}+g_{2}v^{2}_{Z}((t^{1}_{R}T^{1}_{R})+(t^{2}_{R}T^{2}_{R}))(Q-T_{3})$ | $0$
$g^{ZTT}_{R}$ | $g_{1}v^{1}_{Z}(T^{1}_{R})^{2}T_{3}+g_{2}v^{2}_{Z}((T^{1}_{R})^{2}+(T^{2}_{R})^{2})(Q-T_{3})$ | $-\frac{e}{\sin\theta_{w}\cos\theta_{w}}\bigg{(}\frac{1}{2}\big{[}(1-\frac{x^{2}}{8}\sec^{2}\theta_{w})T_{3}\big{]}-Q\sin^{2}\theta_{w}\bigg{)}$
$g^{Z^{{}^{\prime}}tt}_{L}$ | $(gv^{0}_{Z^{{}^{\prime}}}(t^{0}_{L})^{2}+g_{1}v^{1}_{Z^{{}^{\prime}}}(t^{1}_{L})^{2})T_{3}+g_{2}v^{2}_{Z^{{}^{\prime}}}((t^{0}_{L})^{2}+(t^{1}_{L})^{2}))(Q-T_{3})$ | $\frac{ex\tan\theta_{w}}{2\cos\theta_{w}}(Q-T_{3})$
$g^{Z^{{}^{\prime}}tT}_{L}$ | $(gv^{0}_{Z^{{}^{\prime}}}t^{0}_{L}T^{0}_{L}+g_{1}v^{1}_{Z^{{}^{\prime}}}t^{1}_{L}T^{1}_{L})+g_{2}v^{2}_{Z^{{}^{\prime}}}(t^{0}_{L}T^{0}_{L}+t^{1}_{L}T^{1}_{L})(Q-T_{3})$ | $-\frac{e}{\sqrt{2}\sin\theta_{w}}\bigg{(}1-\frac{x^{2}}{8}\tan^{2}\theta_{w}\bigg{)}T_{3}$
$g^{Z^{{}^{\prime}}TT}_{L}$ | $(gv^{0}_{Z^{{}^{\prime}}}(T^{0}_{L})^{2}+g_{1}v^{1}_{Z^{{}^{\prime}}}(T^{1}_{L})^{2})+g_{2}v^{2}_{Z^{{}^{\prime}}}((T^{0}_{L})^{2}+(T^{1}_{L})^{2})(Q-T_{3})$ | $-\frac{e}{x\sin\theta_{w}}\bigg{[}1-\frac{x^{2}}{8}\big{(}4-3\tan^{2}\theta_{w}\big{)}\bigg{]}T_{3}+\frac{ex\tan\theta_{w}}{2\cos\theta_{w}}Q$
$g^{Z^{{}^{\prime}}tt}_{R}$ | $g_{1}v^{1}_{Z^{{}^{\prime}}}(t^{1}_{R})^{2}T_{3}+g_{2}v^{2}_{Z^{{}^{\prime}}}((t^{1}_{R})^{2}+(t^{2}_{R})^{2})(Q-T_{3})$ | $\frac{ex\tan\theta_{w}}{2\cos\theta_{w}}(Q-T_{3})$
$g^{Z^{{}^{\prime}}tT}_{R}$ | $g_{1}v^{1}_{Z^{{}^{\prime}}}(t^{1}_{R}T^{1}_{R})T_{3}+g_{2}v^{2}_{Z^{{}^{\prime}}}((t^{1}_{R}T^{1}_{R})+(t^{2}_{R}T^{2}_{R}))(Q-T_{3})$ | $0$
$g^{Z^{{}^{\prime}}TT}_{R}$ | $g_{1}v^{1}_{Z}(T^{1}_{R})^{2}T_{3}+g_{2}v^{2}_{Z}((T^{1}_{R})^{2}+(T^{2}_{R})^{2})(Q-T_{3})$ | $-\frac{e}{x\sin\theta_{w}}\bigg{[}1-\frac{3x^{2}}{8}\tan^{2}\theta_{w}\bigg{]}T_{3}+\frac{ex\tan\theta_{w}}{2\cos\theta_{w}}Q$

Table 8: Neutral current interactions in the model calculated to
$\mathcal{O}(x^{2})$.

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