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)$}\to q_1\bar{q}2$describedbythepolarizeddecaystructurefunctions$H{mm}$. 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.