
phenomenology {#sec:pheno}
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Dark matter {#subsec:DM}
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If $({\mathcal M}_0)_{\eta\eta} < ({\mathcal M}_0)_{ss}$, the dark matter candidate ${\mathcal H}_1^0$ is given by $\eta_r^0$ approximately because we assume small mixing. See, e.g., Ref. [@i-doublet] for studies about the inert doublet scalar. Let us assume $m_{{\mathcal H}_1^0}^{}\simeq 75\,{{\text{GeV}}}$ and $m_{{\mathcal A}^0}^{} \gtrsim 125\,{{\text{GeV}}}$. As shown in Ref. [@Lundstrom:2008ai], these values satisfy constraints from the LEP experiments [@:2005ema; @EspiritoSanto:2003by] and the WMAP experiment [@Komatsu:2008hk]. The mass splitting ($m_{{\mathcal A}^0}^{} - m_{{\mathcal H}_1^0}^{} \gtrsim 50\,{{\text{GeV}}}$) suppresses quasi-elastic scattering on nuclei (${\mathcal H}_1^0 N \to {\mathcal A}^0 N$ mediated by the $Z$ boson) enough to satisfy constraints from direct search experiments of the DM [@Aprile:2011hi]. By using eqs.  and , we obtain $$\begin{aligned}
\frac{ \lambda_{s\Phi\eta}^2 v_s^2 }{ ({\mathcal M}_0)_{ss}^2 }
\simeq
 \frac{ 2}{ v^2 }
 \left( m_{{\mathcal A}^0}^2 - m_{{\mathcal H}_1^0}^2 \right)
\gtrsim 0.3 .
\label{eq:lambda}\end{aligned}$$ In order to be consistent with our assumption of small $\theta_0^\prime$ (e.g., $\simeq 0.1$), $({\mathcal M}_0)_{ss} \gtrsim 3\,{{\text{TeV}}}$ is required. The value in eq.  results in $$\begin{aligned}
\frac{\mu}{\mu_\eta} \gtrsim 10^{-4} .
\label{eq:mu-50GeV}\end{aligned}$$ For the greater value of $m_{{\mathcal A}^0}^{}$, the larger $\mu/\mu_\eta$ is predicted. In particular, by taking $m_{{\mathcal A}^0}^{}$ to be the TeV scale, we obtain $\mu/\mu_\eta \sim 10^{-2}$, which yields $v_\Delta^{} \sim 1\,{{\text{GeV}}}$ for $\mu_\eta^{}$ and $m_\Delta^{}$ to be at the electroweak scale. Such a value for $v_\Delta^{}$ is suggested in the recent study of radiative corrections to the electroweak parameters [@Kanemura:2012rs].

On the contrary, if we take $m_{{\mathcal A}^0}^{} \simeq 83\,{{\text{GeV}}}$ which is allowed in a tiny region [@Lundstrom:2008ai]