Exact NLO Matching and Analyticity in $b\to s\ell\ell$

Understand

Exclusive rare decays mediated by $b\to s\ell\ell$ 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 kinematic 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 ${\cal O}_{1,2}$, we perform an improved calculation of the NLO matching for the leading dimension-three operators.