Renormalisation group evolution of the shape function g 17 in B ¯ → X s γ $$ \overline{B}→ X_sγ $$ and B ¯ → X s ℓ + ℓ − $$ \overline{B}→ X_s{\ell}^{+}{\ell}^{-} $$ at subleading power - Riccardo Bartocci; Philipp Böer; Tobias Hurth (Journal of High Energy Physics)
Abstract We derive and solve the renormalisation-group (RG) equation of the shape function g 17(ω, ω 1; μ), which appears at subleading power in the factorisation of the inclusive decays B ¯ → X s γ $$ \overline{B}→ X_sγ $$ and B ¯ → X s ℓ + ℓ − $$ \overline{B}→ X_s{\ell}^{+}{\ell}^{-} $$ . Our results provide the first ingredient for a next-to-leading order analysis of the respective resolved-photon Q 1 c − Q 7 γ $$ Q_1^c-Q_{7γ } $$ interference contribution, whose current uncertainties are among the largest ones in both inclusive penguin modes. As a by-product of our study, we find that the analytic properties of the soft anomalous dimension as well as the jet functions in a factorisation theorem allow for a simplified renormalisation of operators in Heavy-Quark Effective Theory that are composed of fields smeared along distinct light-cone directions. Their hadronic matrix elements become relevant in various inclusive and exclusive B-decays beyond leading power in the heavy-quark and large-energy expansion. Using these insights, we derive and solve a simpler “reduced” RG equation of an amplitude-level soft function that describes the long-distance QCD dynamics for penguin contributions to exclusive B ¯ d , s → γγ $$ {\overline{B}}_{d,s}→ γ γ $$ decays, and which has recently been discussed in the literature.