Energy spectrum of bottom- and charmed-flavored mesons from polarized top quark decay $t(\uparrow)\rightarrow W^+ + B/D+X$ at ${\cal O}(\alpha_s)$

We consider the decay of a polarized top quark into a stable $W^+$ boson and charmed-flavored (D) or bottom-flavored (B) hadrons, via $t(\uparrow)\rightarrow W^++D/B+X$. We study the angular distribution of the scaled-energy of B/D-hadrons at next-to-leading order (NLO) considering the contribution of bottom and gluon fragmentations into the heavy mesons B and D. To obtain the energy spectrum of B/D-hadrons we present our analytical expressions for the parton-level differential decay widths of $t(\uparrow)\rightarrow b+W^+(+g)$ at NLO. Comparison of our predictions with data at the LHC enable us to test the universality and scaling violations of the B- and D-hadron fragmentation functions (FFs). These can also be used to determine the polarization states of top quarks and since the energy distributions depend on the ratio $m_W /m_t$ we advocate the use of such angular decay measurements for the determination of the top quark's mass.