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Description
We present a systematic analysis of Coulomb corrections for the interaction of charged leptons in the final state in lepton and semi-lepton decays of neutral $B$-mesons. The calculation is performed within the Furry picture. For scalar systems, a comparison is made between the nonrelativistic Gamow-Sommerfeld-Sakharov approximation, the exact relativistic method of Crater-Alstine-Sazdjian, and the Furry approach. Based on this comparison, an assumption is made about the applicability of the Furry method for calculating Coulomb corrections in the considered $B$-meson decays. The corrections are applied to the analysis of lepton ($B^0_{d,s}\to \ell^+\ell^-$) and semi-lepton ($B^0_{d,s}\to h^0\,\ell^+\ell^-$, $B^0_{d,s}\to V^0\ell^+\ell^-$) channels. Differential and angular distributions and partial decay widths are calculated.
For the $B_s^0 \to \mu^+\mu^-$ channel, Coulomb corrections improve the prediction of the partial width by 2%, improving the agreement with the LHCb/CMS experimental results within the current experimental (11%) and theoretical (5% lattice QM) errors. In the decays $B^0\to K^0\mu^+\mu^-$ and $B^0 \to K^{*0}\mu^+\mu^-$, Coulomb effects also reduce the discrepancies between theoretical predictions and experimental data (to less than 1% and from 11% to 4% respectively). Finally, for the decays $B^0_{d,s}\to \{h^0, V^0\}\,\tau^+\tau^-$, the Coulomb correction reaches 7\%, which is comparable to the uncertainties of the non-perturbative contributions of the strong interaction. The latter result suggests that the Coulomb correction may become significant in the search for new physics in rare $B$-meson decays, especially those involving $\tau$-leptons in the final states.
