Z^\ast X \to H^\pm X$ [@Asakawa:2005gv] and the decay process $H^\pm\to W^\mp Z$ [@Kanemura:1997ej] at the CERN LHC, and further via the processes $e^+e^-\to W^\mp
H^\pm$ [@Godbole:1994np; @Cheung:1994rp; @Kanemura:1999tg] and $e^+e^-\to \nu \bar \nu W^{\pm\ast} Z^\ast \to \nu \bar \nu
H^\pm$ [@Kanemura:2000cw] at the ILC. Another striking feature of models with complex isospin-triplets, such as the GM model, the left-right symmetric model, the littlest Higgs model, and some models motivated by neutrino masses, is the appearance of doubly-charged states $H^{\pm\pm}$. At hadron colliders, such doubly-charged Higgs bosons are studied via the pair production mechanism [@Akeroyd:1998zr; @Gunion:1989in; @Dion:1998pw] as well as the single production mechanism [@Dion:1998pw; @Akeroyd:2005gt] and the $W$-boson fusion mechanism [@Vega:1989tt; @Huitu:2000ut]. They can also be investigated at the ILC and its $e^-e^-$, $e^-\gamma$ and $\gamma\gamma$ option in various scenarios [@Gunion:1998ii].

The S-matrix for two-body elastic scatterings
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In this section, we calculate the transition matrix of elastic scatterings of two scalar-boson states in the GM model. The transition matrix $T(\varphi_1 \varphi_2 \to \varphi_3 \varphi_4)$ is equivalent to the S-wave amplitude $\langle \varphi_3 \varphi_4 | a^0 | \varphi_1 \varphi_2
\rangle$ at high energies ($\sqrt{s}\gg m_W^{2}$), where $\varphi_i$ represent longitudinally-polarized weak bosons or physical Higgs bosons of the model. The 