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

TL;DR

This work quantifies final-state Coulomb corrections in neutral $B$-meson decays with leptons in the final state by contrasting the GSS and CAS relativistic formalisms and corroborating them with one-loop QED results. The authors apply the derived Coulomb factor ${\rm K}(v)$ to ultra-rare leptonic, rare semileptonic (both pseudoscalar and vector final states), and radiative leptonic $B$ decays, finding typically 2–3\% corrections and up to about 4\% for channels with $\tau$ leptons. The corrections reduce discrepancies between theory and experiment in key modes such as $B_s^0\to\mu^+\mu^-$ and $B^0\to K^{0*}\mu^+\mu^-$, reinforcing the necessity of including final-state Coulomb effects in high-precision SM predictions and NP searches. The results provide a robust, relativistically consistent framework for incorporating Coulomb final-state interactions in precision flavor physics analyses.

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 ($B^0_{d,s}\to γ\ell^+\ell^-$) 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 $B_s^0 \to μ^+μ^-$ channel, Coulomb corrections improve the prediction of the partial width to $δ= |\mathcal{B}^{(exp)} - \mathcal{B}^{(theory)}|/\mathcal{B}^{(exp)} = 2\%$. 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 $B^0\to K^0μ^+μ^-$ and $B^0 \to K^{0*}μ^+μ^-$, Coulomb effects also reduce the discrepancies between theoretical predictions and experimental data (to less than $δ= 1\%$ and from $δ= 11\%$ to $δ= 4\%$ respectively). Finally, for the decays involving $τ$-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.

Figures (11)

• Figure 1: The dependence of the Coulomb $\mathcal{K}$-factor as a function of the relative velocity $v = v_{rel}$ (top), and the ratio $\mathcal{K}^{(CAS)}/\mathcal{K}^{(GSS)}$ of the factors as a function of $v$ (bottom). The smaller the velocity, the larger the Coulomb enhancement. It can be seen that the CAS and GSS methods yield practically identical results over the entire range of significant velocities $v\in[10^{-15}, 1]$. The relative difference across considered range of velocities does not exceed $0.3\%$.
• Figure 2: Dependence of the differential branching fraction $10^7 d\mathcal{B}/d\hat{s}$ for the decay $B^0\rightarrow K^0 \mu^+\mu^-$ on $\hat{s} = (p_B-p_K)^2/M_{B}^2$ -- the squared transferred momentum normalized to the square of the $B$-meson mass. The black band corresponds to predictions without Coulomb interaction, the gray band -- with Coulomb interaction. The overlap region is indicated by black-gray hatching.
• Figure 3: Dependence of the differential branching fraction $10^7 d\mathcal{B}/d\hat{s}$ for the decays $B^0\rightarrow \{K^0, \pi^0\} \ell^+\ell^-$ and $B_s^0\rightarrow \{\eta, \eta', K^0\} \ell^+\ell^-$ on $\hat{s} = (p_{B_{d,s}^0}-p_{h^0})^2/M_{B_{d,s}^0}^2$ — the squared transferred momentum normalized to the square of the $B$-meson mass. The black band corresponds to predictions without Coulomb interaction, the gray band — with Coulomb interaction. The overlap region is indicated by black-gray hatching.
• Figure 4: Angular distributions $10^7 d\mathcal{B}/d\cos\theta$ for the decays $B^0\rightarrow \{K^0, \pi^0\} \ell^+\ell^-$ and $B_s^0\rightarrow \{\eta, \eta', K^0\} \ell^+\ell^-$ as a function of $\cos\theta$, where $\theta = \angle (\mathbf{p} _{h^0}, \mathbf{p}_{\ell^+})$ is the angle between the direction of the neutral hadron $h^0$ and the positive lepton $\ell^+$ in the $\ell^+\ell^-$ rest frame. The black band corresponds to predictions without Coulomb interaction, the gray band — with Coulomb interaction. The overlap region is indicated by black-gray hatching.
• Figure 5: Double differetial distributions $10^7\cdot d\mathcal{B}/d\hat{s}d\cos\theta$ for the decays $B^0\rightarrow \{K^0, \pi^0\} \ell^+\ell^-$ and $B_s^0\rightarrow \{\eta, \eta', K^0\} \ell^+\ell^-$. Here $\hat{s} = (p_{B_{d,s}^0}-p_{h^0})^2/M_{B_{d,s}^0}^2$ and $\theta = \angle (\mathbf{p} _{h^0}, \mathbf{p}_{\ell^+})$ is the angle between the direction of the neutral hadron $h^0$ and the positive lepton $\ell^+$ in the $\ell^+\ell^-$ rest frame.