We present a detailed analysis of QED corrections to $$ \overline{B}\to \overline{K}{\mathrm{\ell}}^{+}{\mathrm{\ell}}^{-} $$ B ¯ → K ¯ ℓ + ℓ − decays at the double-differential level. Cancellations of soft and collinear divergences are demonstrated analytically using the phase space slicing method. Whereas soft divergences are found to cancel at the differential level, the cancellation of the hard-collinear logs ln mℓ require, besides photon-inclusiveness, a specific choice of kinematic variables. In particular, hard-collinear logs in the lepton-pair invariant mass distribution (q2), are sizeable and need to be treated with care when comparing with experiment. Virtual and real amplitudes are evaluated using an effective mesonic Lagrangian. Crucially, we show that going beyond this approximation does not introduce any further infrared sensitive terms. All analytic computations are performed for generic charges and are therefore adaptable to semileptonic decays such as $$ \overline{B}\to D\mathrm{\ell}\overline{\nu } $$ B ¯ → D ℓ ν ¯ .

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Using analytic results obtained in a meson effective theory, that includes all infrared sensitive logs, we build a dedicated Monte Carlo framework to describe QED corrections in $$ \overline{B} $$ B ¯ →$$ \overline{K} $$ K ¯ ℓ+ℓ− for a generic form factor. For the neutral mode $$ {\overline{B}}^0 $$ B ¯ 0 →$$ {\overline{K}}^0 $$ K ¯ 0 ℓ+ℓ−, we perform a detailed numerical comparison versus those obtained with the general-purpose photon-shower tool PHOTOS. The comparison indicates a good agreement, at the few per-mil level, when focusing on the rare mode only. In addition, our framework allows us to investigate the impact of the charmonium resonances. Interference effects, not described by PHOTOS in the experimental analysis, are found to be small in the dilepton invariant mass region q2< 6GeV2, which is used to determine $$ {R}_{K^{\left(\ast \right)}} $$ R K ∗ . Using a semi-analytic framework we assess the full, rare and resonant, mode. Based thereupon, we discuss strategies to check the subtraction of the resonant mode, which has a sizeable impact at q2 ≈ 6GeV2 in the electron mode.