The $\bar B \to \bar K \pi \ell \ell$ and $\bar B_s \to \bar K K \ell \ell$ distributions at low hadronic recoil

Abstract

The rare multi-body decays B->K pi ll and B_s->K K ll are both important as backgrounds to precision analyses in the benchmark modes B->K* ll and B_s->phi ll as well as sensitive probes of flavor physics in and beyond the standard model. We work out non-resonant contributions to B->K pi ll and B_s->K K ll amplitudes, where l=e,mu, at low hadronic recoil in a model-independent way. Using the operator product expansion in 1/m_b, we present expressions for the full angular distribution. The latter allows to probe new combinations of |Delta B|=|Delta S|=1 couplings and gives access to strong phases between non-resonant and resonant contributions. Exact endpoint relations between transversity amplitudes based on Lorentz invariance are obtained. Several phenomenological distributions including those from the angular projections to the S-, P-, D-waves are given. Standard model branching ratios for non-resonant B->K pi ll and B_s->K K ll decays are found to be in the few 10^{-8} region, but drop significantly if cuts around the K* or phi mass are employed. Nevertheless, the non-resonant contributions to B->K pi ll provide the dominant background in the B->K* ll signal region with respect to the low mass scalars. In B_s->K K ll, the narrowness of the phi allows for more efficient background control. We briefly discuss lepton-flavor non-universal effects, also in view of the recent data on R_K.