The 2HDM Doppelganger

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)

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.

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.

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.

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.

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]..

   

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.