Common origin of radiative neutrino mass, dark matter and leptogenesis in scotogenic Georgi-Machacek model

After spontaneous symmetry breaking, Higgs fields $\phi_{1}$ and $\phi_{2}$ acquire vacuum expectation values $v_{1}$ and $v_{2}$, such that $v=\sqrt{v_{1}^{2}+v_{2}^{2}}=246$ GeV. The visible sector is identical to 2HDM which contains two neutral physical scalars $(h,H)$, one pseudo-scalar particle $A$ and a pair of charged scalar $H^{\pm}$. Conditions for the minimization of the potential are

	$\displaystyle m_{11}^{2}=\frac{-\lambda_{1}v_{1}^{2}-(\lambda_{3}+\lambda_{4}+\lambda_{5})v_{2}^{2}}{2}\,\,,$
	$\displaystyle m_{22}^{2}=\frac{-\lambda_{2}v_{2}^{2}-(\lambda_{3}+\lambda_{4}+\lambda_{5})v_{1}^{2}}{2}\,.$		(8)

Different couplings $\lambda_{i}(i=1,...,5)$ are expressed in terms of physical masses $m_{h},~{}m_{H},~{}m_{A},~{}m_{H^{\pm}}$ and parameters $\alpha,~{}\beta$,

	$\displaystyle\lambda_{1}$	$\displaystyle=$	$\displaystyle\frac{1}{v^{2}c^{2}_{\beta}}~{}\Big{(}c^{2}_{\alpha}m^{2}_{H}+s^{2}_{\alpha}m^{2}_{h}\Big{)},$		(9)
	$\displaystyle\lambda_{2}$	$\displaystyle=$	$\displaystyle\frac{1}{v^{2}s^{2}_{\beta}}~{}\Big{(}s^{2}_{\alpha}m^{2}_{H}+c^{2}_{\alpha}m^{2}_{h}\Big{)},$		(10)
	$\displaystyle\lambda_{4}$	$\displaystyle=$	$\displaystyle\frac{1}{v^{2}}~{}(m^{2}_{A}-2m^{2}_{H^{+}}),$		(11)
	$\displaystyle\lambda_{5}$	$\displaystyle=$	$\displaystyle-\frac{m^{2}_{A}}{v^{2}},$		(12)
	$\displaystyle\lambda_{3}$	$\displaystyle=$	$\displaystyle\frac{1}{v^{2}s_{\beta}c_{\beta}}\left((m^{2}_{H}-m^{2}_{h})s_{\alpha}c_{\alpha}+m^{2}_{A}s_{\beta}c_{\beta}\right)-\lambda_{4}\,,$		(13)

where we denote $s_{\alpha,\beta}=\sin{\alpha},\sin{\beta},~{}c_{\alpha,\beta}=\cos{\alpha},\cos{\beta}$, with $\tan{\beta}=v_{2}/v_{1}$. Here $m_{h}=125$ GeV is the mass of SM Higgs and $m_{H}$ is denoted as the mass of heavy Higgs boson. After the SSB, the scalar fields $T$ and $\Delta$ acquire zero VEVs which resemble an unbroken residual $\mathcal{Z}_{2}$ symmetry. Therefore, the scalar fields $T$ and $\Delta$ are inert in nature. Mass terms for different inert scalar particles are given as follows

	$\displaystyle m_{T^{0}}^{2}$	$\displaystyle=$	$\displaystyle M_{T}^{2}+\frac{1}{2}\kappa_{31}v_{1}^{2}+\frac{1}{2}\kappa_{32}v_{2}^{2}=m_{T^{+}}^{2}\,,$
	$\displaystyle m_{D^{0}}^{2}$	$\displaystyle=$	$\displaystyle M_{\Delta}^{2}+\frac{1}{2}(\kappa_{11}+\kappa_{21})v_{1}^{2}+\frac{1}{2}(\kappa_{12}+\kappa_{22})v_{2}^{2}=m_{A^{0}}^{2}\,,$
	$\displaystyle m_{D^{++}}^{2}$	$\displaystyle=$	$\displaystyle M_{\Delta}^{2}+\frac{1}{2}\kappa_{11}v_{1}^{2}+\frac{1}{2}\kappa_{12}v_{2}^{2}\,,$
	$\displaystyle m_{D^{+}}^{2}$	$\displaystyle=$	$\displaystyle M_{\Delta}^{2}+\frac{1}{2}(\kappa_{11}+\frac{1}{2}\kappa_{21})v_{1}^{2}+\frac{1}{2}(\kappa_{12}+\frac{1}{2}\kappa_{22})v_{2}^{2}\,,$
	$\displaystyle m_{TD}^{2}$	$\displaystyle=$	$\displaystyle-\left(\frac{\kappa_{41}v_{1}v_{2}}{2}+\frac{\kappa_{42}v_{1}v_{2}}{2}\right)\,,$
	$\displaystyle m_{T^{+}D^{+}}^{2}$	$\displaystyle=$	$\displaystyle\frac{\kappa_{41}v_{1}v_{2}}{2\sqrt{2}}+\frac{\kappa_{42}v_{1}v_{2}}{2\sqrt{2}}\,.$		(14)

The mass of inert pseudo-scalar is denoted as $m_{A^{0}}$ and $m_{D^{++}}$ is the mass of doubly charged scalar. Neutral parts of both the triplets mix with each other resulting two new physical neutral scalars $S_{1,2}$ and the mass matrix is given as

$\displaystyle M_{neutral}^{2}=\left(\begin{array}[]{cc}m_{T^{0}}^{2}&m_{TD}^{2}\\ m_{TD}^{2}&m_{D^{0}}^{2}\end{array}\right).{}$

(15)

We define a mixing angle $\gamma$ between these two inert scalar fields such that

	$\displaystyle S_{1}=T^{0}~{}\cos\gamma-D_{r}^{0}~{}\sin\gamma\,,$
	$\displaystyle S_{2}=T^{0}~{}\sin\gamma+D_{r}^{0}~{}\cos\gamma\,.$		(16)

Masses of new physical scalars are expressed as

$$m^{2}_{S_{1}/S_{2}}=\frac{m_{T^{0}}^{2}+m_{D^{0}}^{2}}{2}\mp\frac{m_{D^{0}}^{2}-m_{T^{0}}^{2}}{2}\sqrt{1+x^{2}},$$

(17)

where $x=\tan 2\gamma=\frac{2m_{TD}^{2}}{(m_{D^{0}}^{2}-m_{T^{0}}^{2})}$. Similarly, the charged parts also mix with each other and provides two physical charged scalars $S_{1,2}^{+}$. The mass matrix for the charged scalars is

$\displaystyle M_{charged}^{2}=\left(\begin{array}[]{cc}m_{T^{+}}^{2}&m_{T^{+}D^{+}}^{2}\\ m_{T^{+}D^{+}}^{2}&m_{D^{+}}^{2}\end{array}\right).{}$

(18)

Defining a new mixing angle $\delta$, we write physical charged scalars as

	$\displaystyle S_{1}^{+}=T^{+}~{}\cos\delta-D^{+}~{}\sin\delta\,,$
	$\displaystyle S_{2}^{+}=T^{+}~{}\sin\delta+D^{+}~{}\cos\delta\,.$		(19)

Masses of physical charged scalars are

$$m^{2}_{S_{1}^{+}/S_{2}^{+}}=\frac{m_{T^{+}}^{2}+m_{D^{+}}^{2}}{2}\mp\frac{m_{D^{+}}^{2}-m_{T^{+}}^{2}}{2}\sqrt{1+y^{2}},$$

(20)

with $y=\tan 2\delta=\frac{2m_{T^{+}D^{+}}^{2}}{(m_{D^{+}}^{2}-m_{T^{+}}^{2})}$.

We adopt the criteria of copositivity of symmetric matrices to get the conditions of vacuum stability Chakrabortty:2013mha . Only the quartic terms in the potential should be considered, since these terms dominate at large field value. It is to be noted that vacuum stability conditions don’t give any constraints to couplings $\kappa_{41}$ and $\kappa_{42}$. This is because we can make these couplings positive by applying a phase rotation of a field or field redefinition. We parameterize the fields as,

$$\begin{split}&\phi_{1}^{\dagger}\phi_{1}=|h_{1}|^{2}\qquad\phi_{2}^{\dagger}\phi_{2}=|h_{2}|^{2}\qquad\Delta^{\dagger}\Delta=|\delta|^{2}\qquad T^{\dagger}T=|t|^{2}\qquad\phi_{1}^{\dagger}\phi_{2}=f|h_{1}||h_{2}|e^{i\theta_{1}}\\ &\Delta^{\dagger}T=g|\delta||t|e^{i\theta_{2}}\qquad\Delta^{\dagger}\phi_{1}=m|\delta||\phi_{1}|e^{i\theta_{3}}\qquad\Delta^{\dagger}\phi_{2}=n|\delta||\phi_{2}|e^{i\theta_{4}}\end{split}$$

(21)

where $f,g,m,n\in[0,1]$. According to the definition above, considering the quartic terms out and ignoring $\kappa_{41}$ and $\kappa_{42}$ terms, the potential of Eq. (5) can be expressed as

$$\begin{split}V(h_{1},h_{2},\delta,t)=&\frac{\lambda_{1}}{2}h_{1}^{4}+\frac{\lambda_{2}}{2}h_{2}^{4}+\lambda_{3}\,h_{1}^{2}h_{2}^{2}+\lambda_{4}\,\rho^{2}h_{1}^{2}h_{2}^{2}+\frac{\lambda_{5}}{2}f^{2}\,\cos 2\theta_{1}\,h_{1}^{2}h_{2}^{2}+(\rho_{1}+\rho_{2})\,\delta^{4}\\ &+\rho_{3}\,t^{4}+(\rho_{4}+\rho_{5}\,g^{2})\delta^{2}t^{2}+(\kappa_{11}+\kappa_{21}m^{2})\delta^{2}h_{1}^{2}+\frac{\kappa_{31}}{2}t^{2}h_{1}^{2}\\ &+(\kappa_{11}+\kappa_{22}n^{2})\delta^{2}h_{2}^{2}+\frac{\kappa_{32}}{2}t^{2}h_{2}^{2}\end{split}$$

(22)

The symmetric matrix of quartic couplings can be represented in basis $(h_{1}^{2},h_{2}^{2},\delta^{2},t^{2})$ as

$$M=\begin{pmatrix}\frac{1}{2}\lambda_{1}&\frac{1}{2}\lambda_{345}&\frac{1}{2}\left(\kappa_{11}+m^{2}\kappa_{21}\right)&\frac{1}{4}\kappa_{31}\\ &\frac{1}{2}\lambda_{2}&\frac{1}{2}\left(\kappa_{12}+n^{2}\kappa_{22}\right)&\frac{1}{4}\kappa_{32}\\ &&\rho_{1}+\rho_{2}&\frac{1}{2}\left(\rho_{4}+g^{2}\rho_{5}\right)\\ &&&\rho_{3}\\ \end{pmatrix}$$

(23)

where $\lambda_{345}=\lambda_{3}+f^{2}(\lambda_{4}-|\lambda_{5}|)$. If $(\lambda_{4}-|\lambda_{5}|)\geq 0$, the minimum of the potential is obtained for $f=0$, whereas for $(\lambda_{4}-|\lambda_{5}|)<0$ the minimum obtained assuming $f=1$. Similar convention can be employed to parameters $g,m,n$ to apply the copositivity criteria upon the matrix $M$. Copositive conditions for which vacuum of the scalar potential in Eq. (22) becomes stable are:

$$\begin{split}&\lambda_{1}\geq 0,\quad\lambda_{2}\geq 0,\quad\rho_{1}+\rho_{2}\geq 0,\quad\rho_{3}\geq 0,\\ &\kappa_{11}+\kappa_{21}+\sqrt{2\lambda_{1}(\rho_{1}+\rho_{2})}\geq 0,\\ &\kappa_{11}+\sqrt{2\lambda_{1}(\rho_{1}+\rho_{2})}\geq 0,\\ &\kappa_{12}+\kappa_{22}+\sqrt{2\lambda_{1}(\rho_{1}+\rho_{2})}\geq 0,\\ &\kappa_{12}+\sqrt{2\lambda_{1}(\rho_{1}+\rho_{2})}\geq 0,\\ &\lambda_{3}+\lambda_{4}-|\lambda_{5}|+\sqrt{\lambda_{1}\lambda_{2}}\geq 0,\\ &\lambda_{3}+\sqrt{\lambda_{1}\lambda_{2}}\geq 0,\\ &8\lambda_{1}\rho_{3}-\kappa_{31}^{2}\geq 0,\\ &8\lambda_{2}\rho_{3}-\kappa_{32}^{2}\geq 0,\\ &8(\lambda_{3}+\lambda_{4}-|\lambda_{5}|)\rho_{3}-\kappa_{31}\kappa_{32}+2\sqrt{(8\lambda_{1}\rho_{3}-\kappa_{31}^{2})(8\lambda_{2}\rho_{3}-\kappa_{32}^{2})}\geq 0,\\ &8\lambda_{3}\rho_{3}-\kappa_{31}\kappa_{32}+2\sqrt{(8\lambda_{1}\rho_{3}-\kappa_{31}^{2})(8\lambda_{2}\rho_{3}-\kappa_{32}^{2})}\geq 0,\end{split}$$

(24)

where we used a theorem of copositivity criteria to get copositive matrix conditions PING1993109 . The conditions mentioned above are derived assuming $\rho_{4}\geq 0$ and $\rho_{5}\geq 0$.

In addition, scalar and fermion couplings must also remain within the perturbative limit for which following conditions must be satisfied

$\displaystyle\lambda_{i},\kappa_{1i,2i,3i,4i},\rho_{i}<4\pi,\hskip 14.22636pt\lambda,y<\sqrt{4\pi}\,.$

(25)

The neutrino masses are generated via one-loop diagram as shown in Fig. 1. The Yukawa interaction terms in Eq. (7) and scalar terms with $\kappa_{41}$ and $\kappa_{42}$ appearing in Eq. (5) are responsible for generation of light neutrino mass. The $\mathcal{Z}_{4}$ symmetry in the model assures that two different scalar doublets are necessary in order to generate tiny neutrino mass in one-loop.

Neutrino mass in the present model can further be realized as self-energy corrections. However, there will be contributions from two different diagrams involving neutral scalars and charged scalars as shown in Fig. 2. The expression of neutrino mass is given as

	$\displaystyle({\cal M}_{\nu})_{ij}$	$\displaystyle=$	$\displaystyle\cos\gamma\sin\gamma\sum_{k=1}^{3}{[y_{ik}\lambda_{jk}+\lambda_{ik}y_{jk}]\over 32\pi^{2}}M_{\Sigma_{k}}\left[{m_{S_{1}}^{2}\over m_{S_{1}}^{2}-M_{\Sigma_{k}}^{2}}\ln{m_{S_{1}}^{2}\over M_{\Sigma_{k}}^{2}}-{m_{S_{2}}^{2}\over m_{S_{2}}^{2}-M_{\Sigma_{k}}^{2}}\ln{m_{S_{2}}^{2}\over M_{\Sigma_{k}}^{2}}\right]$		(26)
			$\displaystyle-\cos\delta\sin\delta\sum_{k=1}^{3}{[y_{ik}\lambda_{jk}+\lambda_{ik}y_{jk}]\over 32\pi^{2}}M_{\Sigma_{k}}\left[{m_{S_{1}^{+}}^{2}\over m_{S_{1}^{+}}^{2}-M_{\Sigma_{k}}^{2}}\ln{m_{S_{1}^{+}}^{2}\over M_{\Sigma_{k}}^{2}}-{m_{S_{2}^{+}}^{2}\over m_{S_{2}^{+}}^{2}-M_{\Sigma_{k}}^{2}}\ln{m_{S_{2}^{+}}^{2}\over M_{\Sigma_{k}}^{2}}\right]\ .$

It is to be noted that, in the present scenario if the mixing angle between neutral physical scalar $\sin\gamma\rightarrow 0$, then mixing angle between charged scalars ($\sin\delta$) also becomes zero. The conditions for which those mixing angles become zero are

$$\kappa_{41}=\kappa_{42}=0\,,\hskip 14.22636pt\kappa_{41}=-\kappa_{42}\,.$$

(27)

Therefore, in order to generate tiny neutrino mass one must have $\kappa_{41,42}\neq 0$ and $\kappa_{41}\neq-\kappa_{42}$.

We consider vector fermions to be much heavier than scalar particles $M_{\Sigma}\gg m_{S_{1,2}},m_{S_{1,2}^{+}}$, and further assume ${m_{S_{1}}^{2}}\simeq{m_{S_{2}}^{2}}$, ${m_{S_{1}^{+}}^{2}}\simeq{m_{S_{2}^{+}}^{2}}$. With the above simplified choice, following Eqs. (14), (17) and Eq. (20) we can rewrite the neutrino mass matrix elements as

$\displaystyle({\cal M}_{\nu})_{ij}$

$\displaystyle=$

$\displaystyle\left(\frac{\kappa_{41}+\kappa_{42}}{2}\right)v_{1}v_{2}\left[\sum_{k=1}^{3}{[y_{ik}\lambda_{jk}+\lambda_{ik}y_{jk}]\over 32\pi^{2}}\frac{1}{M_{\Sigma_{k}}}\ln{m_{S_{1}}^{2}\over M_{\Sigma_{k}}^{2}}+\sum_{k=1}^{3}{[y_{ik}\lambda_{jk}+\lambda_{ik}y_{jk}]\over 32\pi^{2}}\frac{1}{\sqrt{2}M_{\Sigma_{k}}}\ln{m_{S_{1}^{+}}^{2}\over M_{\Sigma_{k}}^{2}}\right]\ .$

(28)

Therefore, from Eq. (28), one can realize seesaw like radiative neutrino mass expressed as

$\displaystyle{\cal M}_{\nu}$

$\displaystyle=$

$\displaystyle\frac{(2+\sqrt{2})(\kappa_{41}+\kappa_{42})v_{1}v_{2}}{128\pi^{2}}(y\zeta^{-1}\lambda^{T}+\lambda\zeta^{-1}y^{T})\,,$

(29)

where we assumed that the mass matrix of vector fermion $\zeta={\rm diag}\{\zeta_{1},\zeta_{2},\zeta_{3}\}$ and the diagonal matrix elements are given as

$$\zeta_{i}\equiv M_{\Sigma_{i}}\left[\ln{m_{S_{1}}^{2}\over M_{\Sigma_{i}}^{2}}\right]^{-1}=M_{\Sigma_{i}}\left[\ln{m_{S_{1}^{+}}^{2}\over M_{\Sigma_{i}}^{2}}\right]^{-1}\,.$$

(30)

Therefore, if one considers $(\kappa_{41}+\kappa_{42})\sim\mathcal{O}(1)$, $y\sim\lambda\simeq\mathcal{O}(10^{-2})$, $\tan\beta\sim 1$ and $M_{\Sigma}\sim\mathcal{O}(10^{10})$ GeV, we get neutrino mass scale at $m_{\nu}\simeq\mathcal{O}(0.05)$ eV for TeV scale triplets in sGM framework. The order of Yukawa coupling also controls parameters significant for the process of leptogenesis which will be discussed later. The mass matrix ${\mathcal{M}}_{\nu}$ can be diagonalized by Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix $U$

$${{\mathcal{M}}_{\nu}}=U^{*}\cdot\hat{\mathcal{M}}_{\nu}\cdot U^{\dagger}$$

(31)

where $\hat{\mathcal{M}}_{\nu}={\rm{diag}}(m_{1},m_{2},m_{3})$.

We consider the generalised form of GM model (Eq. (5)) since it is well known that custodial symmetry in GM model is broken at the one-loop level by hypercharge interactions Gunion:1990dt ; Blasi:2017xmc ; Keeshan:2018ypw . After symmetries of the scalar potential breaks of spontaneously, $\mathcal{Z}_{2}$ symmetry is partially conserved by triplet scalars and both $Y=0$ and $Y=1$ triplet fields remain inert. At this stage (after SSB), Higgs doublets and new inert triplets doesn’t have any impact on $\rho$ parameter and $\rho=1$ is satisfied at tree level. However, one-loop corrections parameter to $\rho$ parameter must be taken into account which can also be obtained in terms of ${\bar{T}}=\alpha T=\Delta\rho$ due to new scalars and fermions. Firstly, let us consider the contribution from new fermions within the model. For a single generation of vector doublet fermions, there will be new contribution to ${\bar{T}}$ parameter which is given as Cynolter:2008ea

$\displaystyle\begin{split}{\bar{T}}^{\rm VLF}=-\frac{g_{2}^{2}}{8\pi^{2}m_{W}^{2}}~{}\Pi(m_{1},m_{2})\,,\end{split}$

(32)

where

$\displaystyle\begin{split}\Pi(m_{1},m_{2})&=-\frac{1}{2}\left(m_{1}^{2}+m_{2}^{2}\right)\left(\text{div}+\log\left(\frac{\mu_{EW}^{2}}{m_{1}m_{2}}\right)\right)\\ &+m_{1}m_{2}\left(\text{div}+\frac{\left(m_{1}^{2}+m_{2}^{2}\right)\log\left(\frac{m_{2}^{2}}{m_{1}^{2}}\right)}{2\left(m_{1}^{2}-m_{2}^{2}\right)}+\log\left(\frac{\mu_{EW}^{2}}{m_{1}m_{2}}\right)+1\right)\\ &-\frac{1}{4}\left(m_{1}^{2}+m_{2}^{2}\right)-\frac{\left(m_{1}^{4}+m_{2}^{4}\right)\log\left(\frac{m_{2}^{2}}{m_{1}^{2}}\right)}{4\left(m_{1}^{2}-m_{2}^{2}\right)}\,.\end{split}$

(33)

In the present work we include three vector fermion doublets. However, since vector fermions do not mix with each other and mass of charged and neutral fermions are degenerate (i.e.; $M_{\Sigma^{+}}=M_{\Sigma^{0}}=M_{\Sigma}$ for a single generation), the contribution ${\bar{T}}^{\rm VLF}=0$.

Let us now discuss how electroweak precision observables modify in presence of new scalars. In case of 2HDM within alignment limit $(\beta-\alpha)=\pi/2$, contribution to ${\bar{T}}$ parameter reads as  He:2001tp ; Grimus:2007if ; Grimus:2008nb

$\displaystyle{\bar{T}}^{\rm 2HDM}=\frac{g_{2}^{2}}{64\pi^{2}m_{W}^{2}}\left(\xi\left(m_{H^{\pm}}^{2},m_{A}^{2}\right)+\xi\left(m_{H^{\pm}}^{2},m_{H}^{2}\right)-\xi\left(m_{A}^{2},m_{H}^{2}\right)\right),$

(34)

where

$\displaystyle\xi\left(x,y\right)=\begin{cases}\frac{x+y}{2}-\frac{xy}{x-y}\ln\left(\frac{x}{y}\right),&\text{if $x\neq y$}.\\ 0,&\text{if $x=y$}.\\ \end{cases}$

(35)

From Eq. (34), it can be concluded that ${\bar{T}}^{\rm 2HDM}$ vanishes for $m_{A}=m_{H^{\pm}}$ or $m_{H}=m_{H^{\pm}}$. Therefore, if one considers above conditions, new contribution to ${\bar{T}}$ parameter will arise from triplet scalar fields $T$ and $\Delta$ only. It is to noted that, in the present formalism there exists mixing between neutral and charged components of the triplet fields, which must be taken into account to calculate ${\bar{T}}$. Considering the effects of mixing, additional contribution to ${\bar{T}}$ parameter is

	$\displaystyle{\bar{T}^{new}}=\frac{g_{2}^{2}}{64\pi^{2}m_{W}^{2}}\Big{(}s_{\delta}^{2}\xi(m_{D^{\pm\pm}}^{2},m_{S_{1}^{\pm}}^{2})+c_{\delta}^{2}\xi(m_{D^{\pm\pm}}^{2},m_{S_{2}^{\pm}}^{2})+(c_{\delta}s_{\gamma}/\sqrt{2}+c_{\gamma}s_{\delta})^{2}\xi(m_{S_{2}^{\pm}}^{2},m_{S_{1}}^{2})$
	$\displaystyle+(c_{\delta}c_{\gamma}/\sqrt{2}-s_{\gamma}s_{\delta})^{2}\xi(m_{S_{2}^{\pm}}^{2},m_{S_{2}}^{2})+(s_{\delta}s_{\gamma}/\sqrt{2}-c_{\gamma}c_{\delta})^{2}\xi(m_{S_{1}^{\pm}}^{2},m_{S_{1}}^{2})$
	$\displaystyle+(s_{\delta}c_{\gamma}/\sqrt{2}+s_{\gamma}c_{\delta})^{2}\xi(m_{S_{2}}^{2},m_{S_{1}^{\pm}}^{2})+s_{\delta}^{2}\xi(m_{S_{1}^{\pm}}^{2},m_{A^{0}}^{2})/2+c_{\delta}^{2}\xi(m_{S_{2}^{\pm}}^{2},m_{A^{0}}^{2})/2$
	$\displaystyle-2s_{\delta}^{2}c_{\delta}^{2}\xi(m_{S_{2}^{\pm}}^{2},m_{S_{1}^{\pm}}^{2})-c_{\gamma}^{2}\xi(m_{S_{2}}^{2},m_{A^{0}}^{2})-s_{\gamma}^{2}\xi(m_{S_{1}}^{2},m_{A^{0}}^{2})\Big{)}\,.$		(36)

In the above Eq. (36), $s_{\theta}=\sin\theta$ and $c_{\theta}=\cos\theta$ where $\theta=\gamma,~{}\delta$. Using the above expression, bounds on the mixing angles or mass splittings between scalar particles can be obtained in the present model. We use the value $T=\frac{\bar{T}^{new}}{\alpha}=0.07\pm 0.12$ PhysRevD.98.030001 to constrain the model parameter space.

In this section, we briefly discuss collider constraints on different visible and dark sector particles of the model. LEP excludes a new charged scalar of mass less than 80 GeV from charged scalar decay into $cs$ and $\nu\tau$ final state Abbiendi:2013hk . Similarly bound on charged fermion mass is 101.2 GeV Abdallah:2003xe ; Achard:2001qw from the decay of charged fermion into $\nu W^{\pm}$ final state. Let us now consider the bounds obtained from LHC. In the present model we have singly charged scalars $S_{1,2}^{\pm}$ and one doubly charged scalar $D^{\pm\pm}$ in dark sector. These particles can contribute to the Higgs to diphoton decay process. The decay rate for the process $h\rightarrow\gamma\gamma$ is given as

	$\displaystyle\Gamma(h\rightarrow\gamma\gamma)$	$\displaystyle=$	$\displaystyle\frac{\alpha^{2}G_{F}m_{h}^{3}}{128\sqrt{2}\pi^{3}}\bigg{|}\sum_{f}N_{c}Q_{f}^{2}g_{hf\bar{f}}A^{h}_{1/2}(\tau_{f})+g_{hW^{+}W^{-}}A^{h}_{1}(\tau_{W})+\frac{\lambda_{hH^{\pm}\,H^{\mp}}v}{2m^{2}_{H^{\pm}}}A^{h}_{0}(\tau_{H^{\pm}})$		(37)
			$\displaystyle\hskip 42.67912pt+\frac{\lambda_{hS_{1}^{\pm}\,S_{1}^{\mp}}v}{2m^{2}_{S_{1}^{\pm}}}A^{h}_{0}(\tau_{S_{1}^{\pm}})+\frac{\lambda_{hS_{2}^{\pm}\,S_{2}^{\mp}}v}{2m^{2}_{S_{2}^{\pm}}}A^{h}_{0}(\tau_{S_{2}^{\pm}})+4{\frac{\lambda_{hD^{\pm\pm}D^{\mp\mp}}v}{2m^{2}_{D^{\pm\pm}}}A^{h}_{0}(\tau_{D^{\pm\pm}})}\bigg{|}^{2}\,.$

Here $G_{F}$ is the Fermi coupling constant, $\alpha$ is the fine-structure constant, $N_{c}=3(1)$ for quarks (leptons), $Q_{f}$ is the electric charge of the fermion in the loop, and $\tau_{i}=m_{h}^{2}/4m_{i}^{2}~{}(i=f,W,S_{1}^{\pm},H^{\pm},S_{2}^{\pm},D^{\pm\pm})$. Couplings of SM Higgs $h$ with different charged scalars in Eq. (37) are listed in Appendix. The relevant loop functions are given by

	$\displaystyle A^{h}_{1/2}(\tau)$	$\displaystyle=$	$\displaystyle 2\left[\tau+(\tau-1)f(\tau)\right]\tau^{-2},$		(38)
	$\displaystyle A^{h}_{1}(\tau)$	$\displaystyle=$	$\displaystyle-\left[2\tau^{2}+3\tau+3(2\tau-1)f(\tau)\right]\tau^{-2},$		(39)
	$\displaystyle A^{h}_{0}(\tau)$	$\displaystyle=$	$\displaystyle-[\tau-f(\tau)]\tau^{-2}\,,$		(40)

and the function $f(\tau)$ is given by

$\displaystyle f(\tau)=\left\{\begin{array}[]{ll}\displaystyle\left[\sin^{-1}\left(\sqrt{\tau}\right)\right]^{2},&(\tau\leq 1),\\ \displaystyle-\frac{1}{4}\left[\log\left(\frac{1+\sqrt{1-\tau^{-1}}}{1-\sqrt{1-\tau^{-1}}}\right)-i\pi\right]^{2},&(\tau>1)\,.\end{array}\right.$

(43)

One can calculate the quantity called signal strength $R_{\gamma\gamma}$ given as

$\displaystyle R_{\gamma\gamma}=\frac{\sigma(pp\rightarrow h)}{\sigma(pp\rightarrow h)^{\rm SM}}\frac{Br(h\rightarrow\gamma\gamma)}{Br(h\rightarrow\gamma\gamma)^{\rm SM}}$

(44)

Using the latest and future constraints from LHC on the $R_{\gamma\gamma}$ signal strength Aaboud:2018xdt ; Sirunyan:2018koj , masses of charged particles in our model can be constrained. It is to be noted that for heavier masses of charged inert scalar particles with mass around TeV, one can safely work with 2HDM constraints. In the present framework, we consider only one Higgs doublet $\phi_{2}$ couples to the SM sector which resembles the standard type-I 2HDM. For type-I 2HDM, Higgs signal strength does not provide any limit on $\tan\beta$ with alignment limit $(\beta-\alpha)=\pi/2$. However, deviation from the alignment limit puts stringent bound on $\tan\beta$ for type-I 2HDM Arcadi:2018pfo ; Khachatryan:2014jba ; Aad:2015gba ; Bauer:2017fsw . Apart from Higgs signal strength measurement, the most stringent bound on charged scalar mass comes from flavor physics when the $B\rightarrow X_{s}\gamma$ decay is taken into account Amhis:2016xyh ; Arbey:2017gmh . It is found that for all types of 2HDM including type-I 2HDM, inclusive decay $b\rightarrow s\gamma$ excludes charged Higgs mass below 650 GeV for $\tan\beta<1$  Arbey:2017gmh . Therefore, in the present framework, with the alignment limit $(\beta-\alpha)={\pi}/{2}$, we conservatively work with following conditions

$$m_{H}=500~{}{\rm GeV},~{}m_{A}=m_{H^{\pm}}=650~{}{\rm GeV},\,1\leq\tan\beta\leq 10\,.$$

(45)