Analysis of Br(B-->X_s gamma) at NNLO with a Cut on Photon Energy

TL;DR

This work delivers the first NNLO prediction for the inclusive radiative decay $Br(\bar{B}\to X_s\gamma)$ with a photon-energy cut $E_\gamma\ge 1.6$ GeV by implementing a renormalization-group improved, multi-scale operator product expansion that cleanly factorizes hard, jet, and soft physics across scales $\mu_h\sim m_b$, $\mu_i\sim \sqrt{m_b\Delta}$, and $\mu_0\sim \Delta$. The authors derive and evaluate a master formula for the cut rate $F(E_0)$ that resums logarithms via the jet and soft functions $\tilde{j}$ and $\tilde{s}$ and a fixed-order hard function $h$, including two-loop corrections at the intermediate and soft scales and carefully estimate residual scale dependence. They obtain a partial branching fraction $Br(\bar{B}\to X_s\gamma)$ at $E_0=1.6$ GeV of $$(2.98_{-0.17}^{+0.13}{}_{pert}\pm 0.16_{hadr}\pm 0.11_{pars}\pm 0.09_{m_c})\times 10^{-4}$, with the total result consistent with the current world average but about 1.4 standard deviations lower, highlighting significant low-scale perturbative uncertainties and the potential for New Physics contributions in rare radiative $B$ decays.

Abstract

By combining a recent estimate of the total B-->X_s gamma branching fraction at O(alpha_s^2) with a detailed analysis of the effects of a cut E_gamma>1.6GeV on photon energy, a prediction for the partial B-->X_s gamma branching fraction at next-to-next-to-leading order in renormalization-group improved perturbation theory is obtained, in which contributions from all relevant scales are properly factorized. The result Br(B-->X_s gamma)=(2.98+-0.26)x10^{-4} is about 1.4 sigma lower than the experimental world average. This opens a window for significant New Physics contributions in rare radiative B decays.

Figures (1)

• Figure 1: Scale dependence of the ratio $T$. The curves show the effect of varying the hard scale $\mu_h$ (dotted), intermediate scale $\mu_i$ (dashed), soft scale $\mu_0$ (dash-dotted), and the reference scale $\mu$ (solid) by a factor of $\sqrt{2}$ about their default values. In the left plot the resummation is performed for the leading-power terms only, while in the right plot the $p_3$ term is also included (see text for further explanation).