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We provide analytical $O(\alpha_s)$ results for the three polarized decaystructure functions $H_{++},\,H_{00}$ and $H_{--}$ that describe the decay of apolarized $W$ boson into massive quark--antiquark pairs. As an application weconsider the decay $t\to b+W^+$ involving the helicity fractions $\rho_{mm}$ ofthe $W^+$ boson followed by the polarized decay $\hbox{$W^+(\uparrow)$}\toq_1\bar{q}_2$ described by the polarized decay structure functions $H_{mm}$. Wethereby determine the $O(\alpha_s)$ polar angle decay distribution of thecascade decay process $t\to b+W^+(\to q_1\bar{q}_2)$. As a second example weanalyze quark mass and off-shell effects in the cascade decays $H\toW^{-}+W^{\ast +}(\to q_1\bar{q}_2)$ and $H\to Z+Z^{\ast}(\to q\bar{q})$. Forthe decays $H\to W^{-}+W^{\ast +}(\to c\bar b)$ and $H\to Z+Z^{\ast}(\tob\bar{b})$ we find substantial deviations from the mass-zero approximation inparticular in the vicinity of the threshold region.

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