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

Abstract

We present a systematic analysis of Coulomb corrections for leptonic ($B^0_{d,s}\to \ell^+\ell^-$), semileptonic ($B^0_{d,s}\to h^0\,\ell^+\ell^-$, $B^0_{d,s}\to V^0\ell^+\ell^-$) and radiative leptonic ($B^0_{d,s}\to \gamma \ell^+\ell^-$) 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 $\delta = |\mathcal{B}^{(exp)} - \mathcal{B}^{(theory)}|/\mathcal{B}^{(exp)} = 2\%$. 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 $B^0 \to K^{0*}\mu^+\mu^-$, Coulomb effects also reduce the discrepancies between theoretical predictions and experimental data (to less than $\delta = 1\%$ and from $\delta = 11\%$ to $\delta = 4\%$ respectively). Finally, for the decays involving $\tau$-leptons, the Coulomb correction $\mathcal{K} = \mathcal{B}^{(Coulomb)}/ \mathcal{B}^{(free)}$ 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.