Advanced predictions for moments of the B̅ →X_sγ photon spectrum

Based on a new, exact QCD factorization formula for the partial B̅ →X_sγ decay rate with a restriction on large photon energy, improved predictions are presented for the partial moments ⟨E_γ⟩ and ⟨E_γ²⟩-⟨E_γ⟩² of the photon spectrum defined with a cut E_γ≥E₀. In the region where Δ=m_b-2E₀ is large compared with Λ_QCD, a theoretical description without recourse to shape functions can be obtained. However, for Δ≪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 Δ and √m_bΔ 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 μ_π² 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 μ_π^2,SF=(0.11±0.19_exp⁡±0.08_th)  GeV² in the shape-function scheme at a scale μ_f=1.5  GeV, while m_b^kin=(4.54±0.11_exp⁡±0.04_th)  GeV and μ_π^2,kin=(0.49±0.18_exp⁡±0.09_th)  GeV² in the kinetic scheme at a scale μ_f=1  GeV.