Testing the Higgs triplet model with the mass difference at the LHC

In general, there can be mass differences among scalar bosons of the Higgs triplet field with the hypercharge of Y=1. In the Higgs triplet model, when the vacuum expectation value $v_\Delta$ of the triplet field is much smaller than that $v$ ($\simeq 246$ GeV) of the Higgs doublet field as required by the electroweak precision data, a characteristic mass spectrum $m_{H^{++}}^2-m_{H^+}^2\simeq m_{H^+}^2-m_{\phi^0}^2(\equiv \xi)$ appears, where $m_{H^{++}}$, $m_{H^+}$, $m_{\phi^0}$ are the masses of the doubly-charged ($H^{++}$), the singly-charged ($H^+$) and the neutral ($\phi^0=H^0$ or $A^0$) scalar bosons, respectively. It should be emphasized that phenomenology with $\xi\neq 0$ is drastically different from that in the case with $\xi=0$ where the doubly-charged scalar boson decays into the same sign dilepton $\ell^+\ell^+$ or the diboson $W^+W^+$ depending on the size of $v_\Delta$. We find that, in the case of $\xi > 0$, where $H^{++}$ is the heaviest, $H^{++}$ can be identified via the cascade decays such as $H^{++}\to H^+W^{+(*)}\to \phi^0 W^{+(*)}W^{+(*)}\to b\bar{b}\ell^+\nu\ell^+\nu$. We outline how the Higgs triplet model can be explored in such a case at the LHC. By the determination of the mass spectrum, the model can be tested and further can be distinguished from the other models with doubly-charged scalar bosons.