Fully analytical O(\alpha_s) results for on-shell and off-shell polarized W-boson decays into massive quark pairs

We provide analytical $O(\alpha_s)$ results for the three polarized decay structure functions $H_{++},\,H_{00}$ and $H_{--}$ that describe the decay of a polarized $W$ boson into massive quark--antiquark pairs. As an application we consider the decay $t\to b+W^+$ involving the helicity fractions $\rho_{mm}$ of the $W^+$ boson followed by the polarized decay \hbox{W^+(\uparrow)}\to q_1\bar{q}_2 described by the polarized decay structure functions $H_{mm}$. We thereby determine the $O(\alpha_s)$ polar angle decay distribution of the cascade decay process $t\to b+W^+(\to q_1\bar{q}_2)$. As a second example we analyze quark mass and off-shell effects in the cascade decays $H\to W^{-}+W^{\ast +}(\to q_1\bar{q}_2)$ and $H\to Z+Z^{\ast}(\to q\bar{q})$. For the decays $H\to W^{-}+W^{\ast +}(\to c\bar b)$ and $H\to Z+Z^{\ast}(\to b\bar{b})$ we find substantial deviations from the mass-zero approximation in particular in the vicinity of the threshold region.