⚛️ phenomenology

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

This paper presents a systematic analysis of relativistic Coulomb corrections in various neutral $B$ -meson decays involving lepton pairs, demonstrating that including these effects significantly improves the agreement between theoretical predictions and experimental data for channels like $B_s^0 \to \mu^+\mu^-$ and $B^0 \to K^{(*)0}\mu^+\mu^-$ , thereby establishing them as a crucial systematic consideration for high-precision $B$ -physics and New Physics searches.

Original authors: S. I. Manukhov, N. V. Nikitin

Published 2026-08-04

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CC BY 4.0

Original authors: S. I. Manukhov, N. V. Nikitin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, bustling dance floor where particles are the dancers. In this high-energy ballroom, heavy particles called B-mesons occasionally decide to break up and transform into lighter, faster dancers. Sometimes, they split into a pair of charged partners: a positive lepton and a negative lepton (like an electron or a muon). Because these partners have opposite electric charges, they don't just drift apart; they feel a magnetic-like pull toward each other, a force physicists call the "Coulomb interaction." Think of it like two dancers who, just as they are about to spin away from each other, feel a sudden, invisible tug that makes them stick together a tiny bit longer or move slightly faster than they would if they were alone.

For decades, scientists have been trying to predict exactly how often these B-mesons break up into these specific pairs. They use a set of rules called the Standard Model, which is like the ultimate choreography manual for the universe. However, when they compare their manual's predictions to what they actually see in giant particle detectors (like the LHCb or CMS experiments), the numbers don't always match perfectly. Sometimes the dancers appear more often than expected; other times, less. To fix these mismatches, physicists have been adding tiny "corrections" to their math, accounting for things like the emission of soft, invisible photons (light particles) that the detectors might miss. But there was one subtle force that the standard calculations were mostly ignoring: the specific, direct pull between the two charged lepton partners right at the moment of their creation.

This paper is a systematic investigation into that missing pull. The authors, S.I. Manukhov and N.V. Nikitin, decided to stop treating the lepton partners as if they were strangers passing in the night and instead calculated exactly how their mutual attraction changes the outcome of the dance. They didn't just guess; they compared three different mathematical ways to describe this attraction. First, they looked at an old, non-relativistic method (the Gamow-Sommerfeld-Sakharov factor), which is like using a simple map for a slow walk. Then, they used a much more complex, modern "relativistic" method (the Crater-Alstine-Sazdjian formalism) that accounts for the fact that these particles are moving at near-light speeds. Finally, they checked their results against standard quantum loop calculations. They found that, surprisingly, the simple old map and the complex modern GPS gave almost the exact same answer, differing by less than 0.3%. This confirmed that they could use a reliable, unified formula to fix their predictions.

When they applied this new "Coulomb correction" to the real-world data, the results were a small but meaningful improvement. For the decay of a neutral B-meson into a pair of muons ( $B^0_s \to \mu^+\mu^-$ ), the correction increased the predicted rate by about 2.3%. This tiny shift brought the theoretical prediction closer to the experimental measurements, reducing the discrepancy between the two to just 2%. While this is still smaller than the current experimental error bars (which are around 11%), it tightens the net. In other decays, like those involving a kaon and a muon pair ( $B^0 \to K^0\mu^+\mu^-$ ), the correction helped shrink the gap between theory and experiment from 2% down to less than 1%. For decays involving the heavy tau-lepton, the correction was even larger, reaching about 4%.

The authors emphasize that while these corrections might seem small compared to other uncertainties (like the difficulty in measuring the "shape" of the particles involved), they are a necessary piece of the puzzle. Currently, tools used by experimentalists to simulate soft light radiation (like a program called photos) do not include this specific Coulomb tug. By adding it, the paper suggests that we are refining our understanding of the Standard Model to a higher level of precision. This is crucial because if we want to spot "New Physics"—signs of particles or forces we haven't discovered yet—we first need to make sure our baseline predictions are as perfect as possible. The authors conclude that in this high-precision era of particle physics, ignoring the Coulomb pull between the final partners is no longer an option; it's a systematic effect that must be accounted for to ensure that any future surprises are truly new discoveries and not just mathematical oversights.

Technical Summary: Coulomb Corrections in Rare Decays of Neutral B Mesons with $\ell^+\ell^-$ Pair in Final State

Problem Statement
Rare semileptonic and ultra-rare leptonic decays of neutral $B$ mesons ( $B^0_{d,s} \to X^0 \ell^+\ell^-$ ) are critical probes for the Standard Model (SM) and potential New Physics. While experimental collaborations (LHCb, CMS, ATLAS, Belle-II) have measured partial widths and differential distributions with increasing precision, discrepancies persist between theoretical predictions and experimental data. Notably, the $B^0_s \to \mu^+\mu^-$ channel shows a central value slightly above the SM prediction, while $B^0 \to K^{(*)}\mu^+\mu^-$ and $B^0_s \to \phi\mu^+\mu^-$ exhibit tensions in Wilson coefficients and differential observables.

Existing theoretical calculations of QED corrections, particularly those based on Soft-Collinear Effective Theory (SCET) or slicing methods (e.g., Refs. [21, 28]), successfully account for structure-dependent effects and soft-photon radiation (simulated by tools like photos). However, these analyses systematically omit the Coulomb interaction between the two charged final-state leptons ( $\ell^+\ell^-$ ). This omission leaves a gap in the precision of theoretical predictions, as the Coulomb correction represents a non-negligible systematic effect, particularly near kinematic thresholds or for heavy leptons ( $\tau$ ).

Methodology
The authors present a systematic analysis of Coulomb corrections across 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.

Validation of Relativistic Treatment: To ensure accuracy in the relativistic regime of $B$ -meson decays, the authors compare three approaches for calculating the Coulomb factor ( $K_C$ ):
- The non-relativistic Gamow-Sommerfeld-Sakharov (GSS) factor.
- The exact relativistic Crater-Alstine-Sazdjian (CAS) formalism, derived from two-particle relativistic equations for scalar systems.
- One-loop QED calculations.
- Finding: The CAS and GSS methods yield practically identical results (difference $<0.3\%$ ) across the entire physically significant velocity range ( $v \in [10^{-13}, 1]$ ). The authors adopt the CAS result for subsequent calculations, denoting the correction simply as $K_C$ .
Factorization: The total QED correction is factorized into a soft-photon term ( $K_{soft}$ ), which is effectively captured by Monte Carlo tools like photos, and the Coulomb term ( $K_C$ ), which is not included in such simulations. The paper explicitly calculates $K_C$ for differential, angular, and double-differential distributions, as well as partial decay widths.
Application: The Coulomb correction is applied to the effective Hamiltonian for $b \to q \ell^+\ell^-$ transitions. The correction is applied as a multiplicative factor to the "free" decay width (calculated without Coulomb interaction):
$\Gamma^{(Coulomb)} = \Gamma^{(free)} \cdot K_C(v)$
where $v$ is the relative velocity of the lepton pair.

Key Results
The inclusion of Coulomb corrections modifies theoretical predictions for branching fractions and reduces discrepancies with experimental data in several channels:

Leptonic Decays ( $B^0_{d,s} \to \ell^+\ell^-$ ):
- For the $B^0_s \to \mu^+\mu^-$ channel, the Coulomb correction increases the theoretical branching fraction by approximately 2.3%. This brings the prediction closer to the LHCb/CMS experimental results, reducing the relative discrepancy ( $\delta$ ) to 2%. This improvement remains within current experimental (11%) and theoretical (5%) uncertainties.
- Corrections for electron and tau channels are similar in magnitude (~2.3% for $e, \mu$ ; ~2.4% for $\tau$ ).
Semileptonic Decays ( $B^0_{d,s} \to h^0 \ell^+\ell^-$ and $V^0 \ell^+\ell^-$ ):
- $B^0 \to K^0 \mu^+\mu^-$ : The discrepancy between theory and experiment is reduced from $\delta = 1.7\%$ to 0.5%.
- $B^0 \to K^{*0} \mu^+\mu^-$ : The discrepancy is significantly reduced from $\delta = 11\%$ to 4%.
- $B^0 \to K^0 e^+e^-$ : The correction formally increases the discrepancy from 33% to 36%, though this remains within the large (45%) experimental uncertainty.
- $\tau$ -lepton channels: Coulomb corrections reach approximately 3–4%. While currently smaller than dominant form-factor uncertainties, they constitute a systematic effect.
Radiative Leptonic Decays ( $B^0_{d,s} \to \gamma \ell^+\ell^-$ ):
- Corrections are calculated for $q^2 \in [1, 6]$ GeV $^2$ (for $e, \mu$ ) and the full kinematic range (for $\tau$ ). The relative correction is approximately 2.3–2.7%.

Significance and Claims
The paper claims that while the Coulomb correction is currently smaller than the dominant uncertainties arising from hadronic form factors and experimental errors, it represents a non-negligible systematic effect.

Precision Era: In the "high-precision era of B-physics," where experimental and theoretical errors are shrinking, these corrections become significant for the accurate interpretation of potential New Physics signals.
Completeness: The work fills a gap in the literature by systematically accounting for the Coulomb interaction between final-state leptons, an effect previously missing from comprehensive QED analyses of neutral $B$ -meson decays.
Relativistic Validity: The study confirms that the non-relativistic GSS approximation is numerically equivalent to the exact relativistic CAS formalism for these decays, validating the use of the simpler GSS form with relativistic velocity substitution for future calculations.

The authors conclude that these corrections should be accounted for in future high-precision analyses to ensure that observed deviations from the SM are not artifacts of incomplete QED modeling.

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