Renormalisation group evolution of the shape function $g_{17}$ in $\bar{B}\to X_s\gamma$ and $\bar{B}\to X_s{\ell }^{+}{\ell }^{-}$ at subleading power

Abstract

We derive and solve the renormalisation-group (RG) equation of the shape function $g_{17}(\omega ,{\omega }_1;\mu )$, which appears at subleading power in the factorization of the inclusive decays $\bar{B}\to X_s\gamma$ and $\bar{B}\to X_s{\ell }^{+}{\ell }^{-}$. Our results provide the first ingredient for a next-to-leading order analysis of the respective resolved-photon $Q_1^c-Q_{7\gamma }$ 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 factorization 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 ${\bar{B}}_{d,s}\to \gamma \gamma$ decays, and which has recently been discussed in the literature.

Comments: v2: 37 pages, 7 Figures, matches published version