Abstract
We obtain predictions for $B_{(s)}\to e^+e^-\gamma$ and $B_{(s)}\to \mu^+\mu^-\gamma$ decays. All the contributions containing long-distance QCD effects are calculated in the framework of relativistic quark model. The contributions of the light vector-meson resonances related to the virtual photon emission from valence quarks of the $B$-meson are included. The highest branching ratios for the radiative leptonic B-decays are ${\cal B}(\bar B^0_s\to e^+e^-\gamma)=18.8\times10^{-9}$ and ${\cal B}(B^0_s\to \mu^+\mu^-\gamma)=12.2\times10^{-9}$. We also give the distribution of the foward-backward asymmetry.
Highlights
Rare radiative leptonic B(s) → + −γ decays are induced by the flavour-changing neutral current transitions b → s, d which are forbidden at tree level in the Standard Model and are described by penguin and box diagrams, leading to small probabilities of the order of 10−8 – 10−10
The penguin form factors FTV,T A(q21, q22) are defined as functions of two variables: q1 is the momentum of the photon emitted from the penguin, and q2 is the momentum of the photon emitted from the valence quark of the B meson
2.2 Emission of the virtual photon from B-meson valence quarks. Another process contributing to the amplitude is that with the real photon emitted from the penguin, whereas one of the valence quarks directly emits the virtual photon which goes into the final + −
Summary
Rare radiative leptonic B(s) → + −γ decays are induced by the flavour-changing neutral current transitions b → s, d which are forbidden at tree level in the Standard Model and are described by penguin and box diagrams, leading to small probabilities of the order of 10−8 – 10−10 (see e.g. [1]). The braching ratio of B+ → K+μ+μ− is itself 30% lower that the SM value at 2σ [9,10,11,12,13]. Another inconsistency is probably related to the decay Bs → μ+μ−. The joint CMS and LHCb measurement of its branching ratio [14] gives the value which is 25% lower than the SM prediction, but here it is 1σ effect only. For the branching ratios of Bs → μ+μ− and Bs → μ+μ−γ decays the following relation takes place.
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