Renormalization-group improved calculation of the $B\to X_s\gamma$ branching ratio

Abstract

Using results on soft-collinear factorization for inclusive B-meson decay distributions, a systematic study of the partial $B\to X_s\gamma$ decay rate with a cut $E_\gamma > E_0$ on photon energy is performed. For values of $E_0$ below about 1.9 GeV, the rate can be calculated without reference to shape functions using a multi-scale operator product expansion (MSOPE). The transition from the shape-function region to the MSOPE region is studied analytically. The resulting prediction for the $B\to X_s\gamma$ branching ratio depends on three large scales: $m_b$, $\sqrt{m_b\Delta}$, and $\Delta=m_b-2E_0$. Logarithms associated with these scales are resummed at next-to-next-to-leading logarithmic order. While power corrections in $\Lambda_{QCD}/\Delta$ turn out to be small, the sensitivity to the scale $\Delta\approx 1.1$ GeV (for $E_0\approx 1.8$ GeV) introduces significant perturbative uncertainties, which so far have been ignored. The new theoretical prediction for the $B\to X_s\gamma$ branching ratio with $E_\gamma\ge 1.8$ GeV is $Br(B\to X_s\gamma)=(3.38_{-0.42-0.30}^{+0.31+0.32})\times 10^{-4}$, where the first error is an estimate of perturbative uncertainties and the second one reflects uncertainties in input parameters. With this cut $(89_{-7}^{+6}\pm 1)%$ of all events are contained. The implications of larger theory uncertainties for New Physics searches are briefly explored with the example of the type-II two-Higgs-doublet model, for which the lower bound on the charged-Higgs mass is reduced compared with previous estimates to approximately 200 GeV at 95% confidence level.