Long-distance weak annihilation contribution to the $B^{\pm}\to (π^{\pm},K^{\pm}) \ell^+\ell^-$ decays

Abstract

We propose an alternative evaluation of the long-distance weak annihilation (WA, also called one-photon exchange in this paper) contribution to the rare semileptonic $B^{\pm}\to (π^{\pm},K^{\pm})\ell^+\ell^-$ ($\ell=e,\,μ$) decays. This hadronic description at low energies is matched at intermediate energies to its short-distance counterpart in terms of quark and gluon degrees of freedom. Although the WA contribution does not contribute to solve the possible breaking of lepton-universality observed by LHCb in the $B^{\pm}\to K^{\pm}(μ^+μ^-/e^+e^-)$ ratio, nor provides an important hadronic contamination to their decay rates, its contribution to the branching ratios (and direct CP asymmetry) of the $B^{\pm}\to π^{\pm}\ell^+\ell^-$ transitions turns out to be significant. This hadronic pollution should be taken into account when looking for new physics effects in decays into pions, which suggests to restrict these searches to squared lepton-pair invariant mass in the $(1,8)$ GeV$^2$ range. The interference of the one-photon exchange contribution with the dominant short-distance one-loop amplitude induces a sizable CP asymmetry in these rare decays, which calls for dedicated measurements.

Matching of the long-distance and short-distance descriptions of the WA contributions added and CP violation results changed accordingly in Table II. Extended discussion of the (subleading) neglected non-factorizable corrections to WA at large distances. Title changed, three figures and several references added, and minor improvements in the text. Version accepted by PRD