An explicit estimate of charm rescattering in $B^0 → K^0 ̄ $

We analyze B⁰ → K⁰ ̄ long-distance contributions induced by the rescattering of a pair of charmed and charmed-strange mesons. We present an explicit estimate of these contributions using an effective description in terms of hadronic degrees of freedom, supplemented by data on the B⁰ → D^*Dₛ (D^*ₛD) transition in order to reproduce the corresponding discontinuity in the B⁰ → K⁰ ̄ amplitude. The D^* Dₛ(D^*ₛ D) K vertex is estimated using heavy-hadron chiral perturbation theory, obtaining an accurate description of the whole rescattering process in the low-recoil (or high-q²) limit. We also present an extrapolation to the whole kinematical region introducing appropriate form factors. The explicit estimate of the leading D^*Dₛ (D^*ₛD) intermediate state leads to a long-distance amplitude which does not exceed a few percent relative to the short-distance one. The consequences of this result for the extraction of the short-distance coefficient C₉ are discussed.