Lepton-mass effects in the decays H →Z Z * →ℓ + ℓ - τ + τ - and H →W W * →ℓ ν τ ν τ

We consider $\ensuremath{\tau}$-lepton mass effects in the cascade decays $H\ensuremath{\rightarrow}Z(\ensuremath{\rightarrow}{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}})+{Z}^{*}(\ensuremath{\rightarrow}{\ensuremath{\tau}}^{+}{\ensuremath{\tau}}^{\ensuremath{-}})$ and $H\ensuremath{\rightarrow}{W}^{\ensuremath{-}}(\ensuremath{\rightarrow}{\ensuremath{\ell}}^{\ensuremath{-}}{\overline{\ensuremath{\nu}}}_{\ensuremath{\ell}})+{W}^{+*}(\ensuremath{\rightarrow}{\ensuremath{\tau}}^{+}{\ensuremath{\nu}}_{\ensuremath{\tau}})$. Since the scale of the problem is set by the off-shellness ${q}^{2}$ of the respective gauge bosons in the limits $({m}_{\ensuremath{\ell}}+{m}_{{\ensuremath{\ell}}^{\ensuremath{'}}}{)}^{2}\ensuremath{\le}{q}^{2}\ensuremath{\le}({m}_{H}\ensuremath{-}{m}_{W,Z}{)}^{2}$ and not by ${m}_{W,Z}^{2}$, lepton-mass effects are non-negligible for the $\ensuremath{\tau}$ modes in particular close to the threshold of the off-shell decays. Lepton-mass effects show up in the rate and in the three-fold joint angular decay distribution for the decays. Nonzero lepton masses lead to leptonic helicity-flip contributions which in turn can generate novel angular dependencies in the respective three-fold angular decay distributions. Lepton-mass effects are more pronounced in the $H\ensuremath{\rightarrow}Z(\ensuremath{\rightarrow}\ensuremath{\ell}\ensuremath{\ell}){Z}^{*}(\ensuremath{\rightarrow}\ensuremath{\tau}\ensuremath{\tau})$ mode which, in part, is due to the fact that the ratio of lepton helicity-flip/nonflip contributions in the decay ${Z}^{*}\ensuremath{\rightarrow}{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}$ is four times larger than in the decay ${W}^{+*}\ensuremath{\rightarrow}{\ensuremath{\ell}}^{+}\ensuremath{\nu}$. Overall the inclusion of $\ensuremath{\tau}$ mass effects leads to a 3.97% reduction in the leptonic $H\ensuremath{\rightarrow}Z{Z}^{*}$ rate. Lepton mass effects are quite pronounced for ${q}^{2}$ values from threshold up to $\ensuremath{\sim}200\text{ }\text{ }{\mathrm{GeV}}^{2}$. For example, at ${q}^{2}=50\text{ }\text{ }{\mathrm{GeV}}^{2}$ the transverse--longitudinal--scalar helicity composition of the off-shell $Z$--boson changes from $0.06:0.94:0$ to $0.04:0.65:0.31$ for the $\ensuremath{\tau}$ lepton. This has observational consequences for the angular decay distributions of the final-state leptons. We also briefly consider the corresponding off-shell--off-shell decays $H\ensuremath{\rightarrow}{Z}^{*}(\ensuremath{\rightarrow}{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}})+{Z}^{*}(\ensuremath{\rightarrow}{\ensuremath{\tau}}^{+}{\ensuremath{\tau}}^{\ensuremath{-}})$ and $H\ensuremath{\rightarrow}{W}^{\ensuremath{-}*}(\ensuremath{\rightarrow}{\ensuremath{\ell}}^{\ensuremath{-}}{\overline{\ensuremath{\nu}}}_{\ensuremath{\ell}})+{W}^{+*}(\ensuremath{\rightarrow}{\ensuremath{\tau}}^{+}{\ensuremath{\nu}}_{\ensuremath{\tau}})$.