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 ($B_{d,s}^0\to {\ell }^{+}{\ell }^{-}$), semileptonic ($B_{d,s}^0\to h^0{\ell }^{+}{\ell }^{-}$, $B_{d,s}^0\to V^0{\ell }^{+}{\ell }^{-}$) and radiative leptonic ($B_{d,s}^0\to \gamma {\ell }^{+}{\ell }^{-}$) 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. We also discuss the role of the Coulomb factor among other QED corrections, in particular, the contribution of soft-photon radiation, which is effectively simulated in experiments by tools such as PHOTOS. 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 $B^0\to K^{0\ast }{\mu }^{+}{\mu }^{-}$, Coulomb effects also reduce the discrepancies between theoretical predictions and experimental data (from $\delta =2\%$ to less than $\delta =1\%$ and from $\delta =11\%$ to $\delta =4\%$ respectively). Finally, for the decays involving $\tau$-leptons, the Coulomb correction 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.

Comments: 23 pages, 11 figures, 8 tables