Coulomb corrections in rare decays of neutral $B$ mesons with ${\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$-pair in final state

We present a systematic analysis of Coulomb corrections for leptonic ($B_{d,s}^0\to {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$), semileptonic ($B_{d,s}^0\to h^0{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$, $B_{d,s}^0\to V^0{\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$) and radiative leptonic ($B_{d,s}^0\to \gamma {\mathrm{\ell }}^{+}{\mathrm{\ell }}^{-}$) decays of neutral $B$-mesons. The relativization of the Coulomb factor was performed by comparing the Gamow-Sommerfeld-Sakharov factor, the exact relativistic approach of Crater-Alstine-Sazdjian applied by us to scalar systems, and well-known one-loop QED calculations. Coulomb corrections are calculated for differential, angular, and double-differential distributions, as well as for partial decay widths. For the $B_s^0\to {\mu }^{+}{\mu }^{-}$ channel, Coulomb corrections improve the prediction of the partial width to $\delta =\mathrm{\mid }{\mathcal{B}}^{(exp)}-{\mathcal{B}}^{(theory)}\mathrm{\mid }\mathrm{/}{\mathcal{B}}^{(exp)}=2\mathrm{\%}$. This improvement brings the prediction closer to the LHCb/CMS experimental results within the current experimental (11%) and theoretical (5% lattice QCD) errors. In the decays $B^0\to K^0{\mu }^{+}{\mu }^{-}$ and $B^0\to K^{0\ast }{\mu }^{+}{\mu }^{-}$, Coulomb effects also reduce the discrepancies between theoretical predictions and experimental data (to less than $\delta =1\mathrm{\%}$ and from $\delta =11\mathrm{\%}$ to $\delta =4\mathrm{\%}$ respectively). Finally, for the decays involving $\tau$-leptons, the Coulomb correction $\mathcal{K}={\mathcal{B}}^{(Coulomb)}\mathrm{/}{\mathcal{B}}^{(free)}$ reaches 4%. While currently smaller than the dominant form-factor uncertainties and experimental errors, the Coulomb correction represents a non-negligible systematic effect. It should be accounted for in the high-precision era of $B$-physics, where such effects may become significant for the interpretation of potential New Physics signals.