Explicit estimate of charm rescattering in $B^0\to K^0\bar{\ell }\ell$

Abstract

We analyze $B^0\to K^0\bar{\ell }\ell$ 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 $B^0\to D^{\ast }{D}_s(D_s^{\ast }D)$ transition in order to reproduce the corresponding discontinuity in the $B^0\to K^0\bar{\ell }\ell$ amplitude. The $D^{\ast }{D}_s(D_s^{\ast }D)K$ vertex is estimated using heavy-hadron chiral perturbation theory, obtaining an accurate description of the whole rescattering process in the low-recoil (or high-$q^2$) limit. We also present an extrapolation to the whole kinematical region introducing hadronic form factors. The explicit estimate of the leading $D^{\ast }{D}_s(D_s^{\ast }D)$ 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 $C_9$ are discussed.

Comments: 8 pages, 3 figures, 2 tables. Minor corrections, results unchanged (published version)