Advanced predictions for moments of theB¯→Xsγphoton spectrum

Based on a new, exact QCD factorization formula for the partial $\overline{B}\to X_s\gamma$ 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 $E_{\gamma }\ge E_0$. In the region where $\Delta =m_b-2E_0$ is large compared with ${\Lambda }_{\mathrm{QCD}}$, a theoretical description without recourse to shape functions can be obtained. However, for $\Delta \ll m_b$ 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 $m_b^{\mathrm{SF}}=(4.62\pm {0.10}_{\exp }\pm {0.03}_{\mathrm{th}})\mathrm{GeV}$ and ${\mu }_{\pi }^{2,\mathrm{SF}}=(0.11\pm {0.19}_{\exp }\pm {0.08}_{\mathrm{th}}){\mathrm{GeV}}^2$ in the shape-function scheme at a scale ${\mu }_f=1.5\mathrm{GeV}$, while $m_b^{\mathrm{kin}}=(4.54\pm {0.11}_{\exp }\pm {0.04}_{\mathrm{th}})\mathrm{GeV}$ and ${\mu }_{\pi }^{2,\mathrm{kin}}=(0.49\pm {0.18}_{\exp }\pm {0.09}_{\mathrm{th}}){\mathrm{GeV}}^2$ in the kinetic scheme at a scale ${\mu }_f=1\mathrm{GeV}$.