  With respect to the two theoretical approaches for estimating the hadronic
productions of the mesons $B_c$ and $B_c^*$: the full $\alpha_s^4$ calculation
and the fragmentation approximation, with a thorough comparative study a quite
remarkable conclusion may be drawn. In hadronic collisions the condition for
the applicability of the fragmentation approximation is the transverse momentum
$P_T$ of the produced $B_c$ and $B_c^*$ being much greater than the meson
masses, and the higher the energy of the relevant subprocess the higher the
$P_T$ is requested. The mechanisms for the productions are also clarified. In
order to show the resultant differences of the two approaches, newly introduced
distribution of the fraction of the $B_c$ (or $B_c^*$) meson energy (a
measurable observable) in the center of mass of the subprocess is emphasized.

High Energy Physics - Phenomenology