Polarization and kinematic properties of the splitting functions $q\to W^{\pm }+q^{\mathrm{\prime }}$ and $q\to Z^0+q$

We consider the processes $g\to W+q^{\mathrm{\prime }}$ and $q\to Z+q$ and derive the respective splitting functions as functions of two kinematic variables: the longitudinal momentum fraction $z$ and transverse momentum $p_T$ of the produced bosons with respect to the parent quark. We take into account kinematic (phase space) restrictions connected with nonzero masses of the gauge bosons and with limited initial energy. We separately consider three different polarization states of the bosons.