We derive and solve the renormalisation-group (RG) equation of the shape function g17(ω, ω1; μ), which appears at subleading power in the factorisation of the inclusive decays \( \overline{B}\to {X}_s\gamma \) and \( \overline{B}\to {X}_s{\ell}^{+}{\ell}^{-} \) . 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 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 \( {\overline{B}}_{d,s}\to \gamma \gamma \) decays, and which has recently been discussed in the literature.
