$B_c$ Meson Productions Via Induced Gluon Fragmentation

Gluons cannot directly fragment into $B_c$ mesons until at order $\alpha_s^3$. However, the Altarelli-Parisi evolution of the $\bar b$-quark fragmentation functions $D_{\bar b \to B_c}(z)$ and $D_{\bar b \to B_c^*}(z)$ from the heavy quark mass scales up to the collider energy scale $Q$ can induce the gluon fragmentation functions $D_{g \to B_c}(z)$ and $D_{g \to B_c^*}(z)$, respectively, through the gluon splitting $g\to \bar b$ at the order $\alpha_s^3 \, {\rm log} (Q/m_{b,c})$. We will determine these induced gluon fragmentation functions. The $B_c$ meson productions due to the induced gluon fragmentation will be evaluated and compared to the productions by the direct $\bar b$-quark fragmentation. The contribution from the induced gluon fragmentation is found to be a significant portion of the total $B_c$ and $B_c^*$ meson productions, and therefore cannot be ignored.