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The rare multi-body decays B ¯ → K ¯ πℓℓ $$ \overline{B}\to \overline{K}\pi \ell \ell $$ and B ¯ s → K ¯ Kℓℓ $$ {\overline{B}}_s\ \to \overline{K}K\ell \ell $$ are both important as backgrounds to precision analyses in the benchmark modes B ¯ → K ¯ * ℓℓ $$ \overline{B}\to {\overline{K}}^{*}\ell \ell $$ and B ¯ s → ϕℓℓ $$ {\overline{B}}_s\to \phi \ell \ell $$ as well as sensitive probes of flavor physics in and beyond the standard model. We work out non-resonant contributions to B ¯ → K ¯ πℓℓ $$ \overline{B}\to \overline{K}\pi \ell \ell $$ and B ¯ s → K ¯ Kℓℓ $$ {\overline{B}}_s\ \to \overline{K}K\ell \ell $$ 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 |Δ B | = |Δ 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 distri- butions including those from the angular projections to the S-, P-, D-waves are given. Standard model branching ratios for non-resonant B ¯ → K ¯ πℓℓ $$ \overline{B}\to \overline{K}\pi \ell \ell $$ and B ¯ s → K ¯ Kℓℓ $$ {\overline{B}}_s\ \to \overline{K}K\ell \ell $$ 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 B ¯ → K ¯ πℓℓ $$ \overline{B}\to \overline{K}\pi \ell \ell $$ provide the dominant background in the B ¯ → K ¯ * ℓℓ $$ \overline{B}\to {\overline{K}}^{*}\ell \ell $$ signal region with respect to the low mass scalars. In B ¯ s → K ¯ Kℓℓ $$ {\overline{B}}_s\ \to \overline{K}K\ell \ell $$  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

the journal of high energy physics/the journal of high energy physics

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