Advanced Predictions for Moments of the B->X_s+gamma Photon Spectrum
Matthias Neubert
TL;DR
This work develops a QCD factorization framework to predict moments of the inclusive $\bar{B}\to X_s\gamma$ photon spectrum with a photon-energy cut, achieving NNLO accuracy through RG-improved perturbation theory and separating the hard, hard-collinear, and soft scales. By deriving exact leading-power expressions and incorporating power corrections within HQET, the authors show that the first two moments are governed by low-scale dynamics and can yield precise extractions of the $b$-quark mass and kinetic-energy parameter in the shape-function scheme. They perform a thorough numerical analysis with realistic inputs, quantify perturbative and scheme-related uncertainties, and assess frame-boost effects to match $\Upsilon(4S)$-frame measurements. The results provide high-precision predictions for the mean photon energy and its variance, enabling robust determinations of heavy-quark parameters from experimental data.
Abstract
Based on a new, exact QCD factorization formula for the partial B->X_s+gamma decay rate with a restriction on large photon energy, improved predictions are presented for the partial moments <E_gamma> and <E_gamma^2>-<E_gamma>^2 of the photon spectrum defined with a cut E_gamma>E_0. In the region where Delta=m_b-2E_0 is large compared with Lambda_{QCD}, a theoretical description without recourse to shape functions can be obtained. However, for Delta<<m_b it is important to separate short-distance contributions arising from different scales. The leading terms in the heavy-quark expansion of the moments receive contributions from the scales Delta and \sqrt{m_b Delta} only, but not from the hard scale m_b. For these terms, a complete scale separation is achieved at next-to-next-to-leading order in renormalization-group improved perturbation theory, including two-loop matching contributions and three-loop running. The results presented here can be used to extract the b-quark mass and the quantity mu_pi^2 with excellent theoretical precision. A fit to experimental data reported by the Belle Collaboration yields m_b^{SF}=(4.62+-0.10_{exp}+-0.03_{th})GeV and mu_pi^{2,SF}=(0.11+-0.19_{exp}+-0.08_{th})GeV^2 in the shape-function scheme at a scale mu_f=1.5GeV, while m_b^{kin}=(4.54+-0.11_{exp}+-0.04_{th})GeV and mu_pi^{2,kin}=(0.49+-0.18_{exp}+-0.09_{th})GeV^2 in the kinetic scheme at a scale mu_f=1GeV.