Rare FCNC radiative leptonic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:mi>γ</mml:mi><mml:msup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math> decays in the standard model

We revisit rare radiative leptonic decays ${B}_{s,d}\ensuremath{\rightarrow}\ensuremath{\gamma}{e}^{+}{e}^{\ensuremath{-}}$ and ${B}_{s,d}\ensuremath{\rightarrow}\ensuremath{\gamma}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ in the standard model and provide the updated estimates for various differential distributions (the branching ratios, the forward-backward asymmetry, and ${R}_{\ensuremath{\mu}/e}$, the ratio of the differential distribution for muons over electrons in the final state). The new ingredients of this work compared to the existing theoretical analyses are the following: (i) we calculate all ${B}_{d}\ensuremath{\rightarrow}\ensuremath{\gamma}$ and ${B}_{s}\ensuremath{\rightarrow}\ensuremath{\gamma}$ form factors induced by the vector, axial-vector, tensor and pseudotensor quark currents within the relativistic dispersion approach based on the constituent quark picture; (ii) we perform a detailed analysis of the charm-loop contributions to radiative leptonic decays: we obtain constraints imposed by electromagnetic gauge invariance and discuss the existing ambiguities in the charmonia contributions.

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