Generalized screening theorem for Higgs decay processes in the two-doublet model

The radiative corrections to the decay processes of the neutral ($CP$-even) Higgs boson ($H$) into a longitudinal gauge boson pair, {it i.e.}, $H rightarrow Z_{L}Z_{L}$ and $H rightarrow W_{L}^{+}W_{L}^{-}$ are analyzed in the two-Higgs doublet model by assuming that all of the Higgs boson masses are much greater than the $W$ and $Z$ bosons. These calculations are motivated to see if one could see potentially large virtual effects to these decay rates due to the charged and $CP$-odd neutral Higgs boson masses ($m_{G}$ and $m_{A}$, respectively) which are supposed to be larger than $m_{H}$. It is pointed out that, although the radiative corrections to the decay width $Gamma (Hrightarrow W_{L}^{+}W_{L}^{-})$ depend sensitively in general on $m_{G}$ and $m_{A}$, there occurs a screening effect, {it i.e.,} cancellation in leading terms once we set $m_{G}=m_{A}$, so that the radiative corrections tend to be minimized. It is also pointed out that the decay rate $Gamma (Hrightarrow Z_{L}Z_{L})$ is fairly insensitive to the other heavier Higgs masses and is possibly a good measuring tool of the Higgs mixing angle. The mechanism of these screening phenomena in the Higgs decays is explained on the basis of a new screening theorem, which we postulate with reference to the custodial symmetry in the Higgs potential.