Resolved Q₁q-Q7γ interference in B̄→ Xₛγ at NLO: Hard matching and radiative charm-loop jet function for mc → 0

We study next-to-leading-order (NLO) QCD corrections to the subleading-power resolved-photon contribution to B̄→ Xₛγ arising from the interference of the current-current operator Q₁q and the electromagnetic dipole operator Q7γ. This contribution is among the main sources of theoretical uncertainty in Standard Model predictions for the decay rate. Using a factorisation theorem in soft-collinear effective theory for the kinematic endpoint region, mb-2E_γ=O(ΛQCD), we determine all previously unknown perturbative ingredients at NLO in the limit of a vanishing charm-quark mass, mc→0. Specifically, we compute the mc-independent one-loop hard-matching coefficients, derive their renormalisation-group equations, and obtain accurate analytic approximations to their solutions by retaining the first two terms in a 1/Nc expansion. Our second main result is the two-loop correction to the radiative charm-loop jet function, including its full dependence on the collinear momentum fraction. Combining these results with the known O(αₛ) correction to the quark jet function and the one-loop renormalisation of the subleading shape function g₁₇, previously computed by some of us, we provide the first complete NLO description of this resolved contribution in the massless-charm limit. This leaves the mc dependence of the radiative charm-loop jet function as the remaining ingredient for a phenomenological analysis within renormalisation-group-improved perturbation theory.