On the hadronic production of $B_c$ and $B_c^*$ mesons

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.