  In our previous work, we studied the polar distribution of the scaled energy
of bottom-flavored hadrons from polarized top quark decays
$t(\uparrow)\rightarrow W^++b(\rightarrow X_b)$, using two different helicity
coordinate systems. Basically, the energy distributions are governed by the
unpolarized, polar and azimuthal rate functions which are related to the
density matrix elements of the decay $t(\uparrow)\rightarrow W^++b$. Here we
present, for the first time, the analytical expressions for the ${\cal
O}(\alpha_s)$ radiative corrections to the differential azimuthal decay rates
of the partonic process $t(\uparrow)\rightarrow b+W^+(+g)$ in two helicity
systems, which are needed to study the azimuthal distribution of the energy
spectrum of the B-hadron produced in polarized top quark decays. Our
predictions of the hadron energy distributions enable us to deepen our
knowledge of the hadronization process and to determine the polarization states
of top quarks.
