Exact NLO matching and analyticity in b → sℓℓ

Exclusive rare decays mediated by b → sll transitions receive contributions from four-quark operators that cannot be naively expressed in terms of local form factors. Instead, one needs to calculate a matrix element of a bilocal operator. In certain kine- matic regions, this bilocal operator obeys some type of Operator Product Expansion, with coefficients that can be calculated in perturbation theory. We review the formalism and, focusing on the dominant SM operators O1,2, we perform an improved calculation of the NLO matching for the leading dimension-three operators. This calculation is performed completely analytically in the two relevant mass scales (charm-quark mass mc and dilep- ton squared mass q2), and we pay particular attention to the analytic continuation in the complex q2 plane. This allows for the first time to study the analytic structure of the non-local form factors at NLO, and to calculate the OPE coefficients far below q2 = 0, say q2 < −10 GeV2. We also provide explicitly the contributions proportional to different charge factors, which obey separate dispersion relations.