Tree-level contributions toB¯→Xsγ

Abstract

Weak radiative decay $\overline{B}\ensuremath{\rightarrow}{X}_{s}\ensuremath{\gamma}$ is known to be a loop-generated process. However, it does receive tree-level contributions from Cabibbo-Kobayashi-Maskawa-suppressed $b\ensuremath{\rightarrow}u\overline{u}s\ensuremath{\gamma}$ transitions. In the present paper, we evaluate such contributions together with similar ones from the QCD penguin operators. For a low value of the photon energy cutoff ${E}_{0}\ensuremath{\simeq}{m}_{b}/20$ that has often been used in the literature, they can enhance the inclusive branching ratio by more than 10%. For ${E}_{0}=1.6\text{ }\text{ }\mathrm{GeV}$ or higher, the effect does not exceed 0.4%, which is due to phase-space suppression. Our perturbative results contain collinear logarithms that depend on the light quark masses ${m}_{q}$ ($q=u$, $d$, $s$). We have allowed ${m}_{b}/{m}_{q}$ to vary from 10 to 50, which corresponds to values of ${m}_{q}$ that are typical for the constituent quark masses. Such a rough method of estimation may be improved in the future with the help of fragmentation functions once the considered effects begin to matter in the overall error budget for $\mathcal{B}(\overline{B}\ensuremath{\rightarrow}{X}_{s}\ensuremath{\gamma})$.