The B ¯ → K ¯ πℓℓ $$ \overline{B}\to \overline{K}\pi \ell \ell $$ and B ¯ s → K ¯ Kℓℓ $$ {\overline{B}}_s\ \to \overline{K}K\ell \ell $$ distributions at low hadronic recoil

Abstract

The rare multi-body decays $$ \overline{B}\to \overline{K}\pi \ell \ell $$ and $$ {\overline{B}}_s\ \to \overline{K}K\ell \ell $$ are both important as backgrounds to precision analyses in the benchmark modes $$ \overline{B}\to {\overline{K}}^{*}\ell \ell $$ and $$ {\overline{B}}_s\to \phi \ell \ell $$ as well as sensitive probes of flavor physics in and beyond the standard model. We work out non-resonant contributions to $$ \overline{B}\to \overline{K}\pi \ell \ell $$ and $$ {\overline{B}}_s\ \to \overline{K}K\ell \ell $$ amplitudes, where ℓ = e,μ, 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 |ΔB| = |ΔS| = 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 distri- butions including those from the angular projections to the S-, P-, D-waves are given. Standard model branching ratios for non-resonant $$ \overline{B}\to \overline{K}\pi \ell \ell $$ and $$ {\overline{B}}_s\ \to \overline{K}K\ell \ell $$ decays are found to be in the few 10−8 region, but drop significantly if cuts around the K * or ϕ mass are employed. Nevertheless, the non-resonant contributions to $$ \overline{B}\to \overline{K}\pi \ell \ell $$ provide the dominant background in the $$ \overline{B}\to {\overline{K}}^{*}\ell \ell $$ signal region with respect to the low mass scalars. In $$ {\overline{B}}_s\ \to \overline{K}K\ell \ell $$ , the narrowness of the ϕ allows for more efficient background control. We briefly discuss lepton-flavor non-universal effects, also in view of the recent data on R K .