$B_c$ meson production at the Tevatron Revisited

CDF recently measured the quantity $\frac{\sigma(B_c^+)}{\sigma(B^+)} \frac{{\rm BR}(B_c^+ \to J/\psi \ell^+ \nu)}{{\rm BR}(B^+ \to J/\psi K^+)}$, {}from which we determine the ratio $\frac{\sigma(B_c^+)}{\sigma(\bar b)}$ to be $(2.08\err{1.06}{0.95})\times 10^{-3}$. In this note, we show that the ratio $\frac{\sigma(B_c^+)}{\sigma(\bar b)}$ obtained by dividing the $\sigma(B_c^+)$ by the leading order $\sigma(\bar b)$ is consistent with this derived CDF measurement. We calculate the cross section $\sigma(B_c^+)$ using the perturbative QCD fragmentation functions of Braaten, Cheung, and Yuan and the corresponding induced gluon fragmentation functions, with the charm-quark mass $m_c$ as a parameter. We also estimate the parameter $m_c$ from the CDF data and then predict the production rate at RunII.