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REVIEW 2 major objections 4 minor 44 references

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

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Coulomb attraction between final-state leptons changes predicted rates of rare neutral-B decays by 2–4%, bringing B_s→μ⁺μ⁻, B→K μ⁺μ⁻, and B→K* μ⁺μ⁻ into closer agreement with measured branching fractions.

desk verdict Useful catalogue of Coulomb K-factors for B decays, but the B_s→μμ headline rests on an unproven no-double-counting assumption and one table row doesn't add up. read the letter →

arxiv 2602.07934 v2 pith:7TAENKPZ submitted 2026-02-08 hep-ph

S. I. Manukhov , N. V. Nikitin This is my paper
classification hep-ph PACS 13.20.He13.25.-k14.40.Nd
keywords CoulombcorrectionsneutralBmesondecaysleptonicsemileptonicradiativeGamow-Sommerfeld-Sakharovfactorfinal-stateQEDB_s→μ+μ−
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the electromagnetic Coulomb attraction between the positively and negatively charged leptons in the final state of rare neutral-B-meson decays is a genuine, systematic correction of 2–4% that should be included in predictions. It compares three ways to compute this correction — the non-relativistic Gamow–Sommerfeld–Sakharov factor, an exact relativistic two-particle formalism applied to scalar systems, and one-loop QED — and finds they agree to better than 0.3% once the relativistic relative velocity of the lepton pair is used. Multiplying the published 'free' decay widths by this factor brings the B_s→μ⁺μ⁻ prediction to within 2% of the measured average, B→K μ⁺μ⁻ to within 0.5%, and B→K* μ⁺μ⁻ from 11% to 4% discrepancy, while leaving e⁺e⁻ channels within their larger errors. The authors present the correction for differential, angular, and double-differential distributions as well as partial widths, so the factor can be folded into current precision analyses.

What carries the argument

The load-bearing object is the Coulomb K-factor, Eq. (2)/(3): K = |Γ(√(1/4−α_em²)+1/2+iα_em/v)/Γ(√(1−4α_em²)+1)|² e^{πα_em/v}. In the non-relativistic limit it reduces to the Gamow–Sommerfeld–Sakharov factor 2πα_em/v / (1−e^{−2πα_em/v}). The paper inserts the relativistic relative velocity v = √(1−4mℓ²/M_B²)/(1−2mℓ²/M_B²) (for semileptonic channels, v = √(1−4m̂/ŝ)/(1−2m̂/ŝ)) and applies the factor multiplicatively to differential, angular, and double-differential widths. The machinery does the work of turning a known QED Coulomb-singularity correction into a simple, spin-independent, universal rescaling of existing predictions; because it is below 4% in all channels, it acts as a systematic

What would settle it

Recompute the B_s→μ⁺μ⁻ partial width in the same effective Hamiltonian with explicit one-loop QED applied to the lepton pair. If that direct calculation already reproduces 3.75×10⁻⁹ without the separate K-factor, the paper's multiplicative recipe double-counts; if it reproduces 3.66×10⁻⁹ with the Coulomb pole absent, the recipe is correct.

Watch

Extended reading notes

Core claim

Central claim: final-state Coulomb rescattering of the ℓ⁺ℓ⁻ pair is a systematic correction in rare neutral-B decays. The paper shows the standard Gamow–Sommerfeld–Sakharov factor survives relativization: an exact relativistic two-particle treatment and one-loop QED agree with it to better than 0.3% once the relativistic relative velocity v = √(1−4mℓ²/M_B²)/(1−2mℓ²/M_B²) is inserted. The K-factor (2.3% for e, μ; up to 4% for τ) multiplies the published 'free' widths. For B_s→μ⁺μ⁻ it moves the SM prediction from 3.66×10⁻⁹ to 3.75×10⁻⁹, cutting the discrepancy to 2%; for B→K μ⁺μ⁻ and B→K* μ⁺μ⁻ it cuts discrepancies to <1% and 4%. Working conclusion: Coulomb corrections should be included in hi

Load-bearing premise

The free decay widths taken from earlier calculations do not already include the same final-state Coulomb/QED photon exchange described by the K-factor; if they do, multiplying by K double-counts the effect and the improved 2% agreement for B_s→μμ is an artifact.

Editorial extensions

If this is right

  • B_s→μ⁺μ⁻: the SM prediction moves from 3.66×10⁻⁹ to 3.75×10⁻⁹, reducing the theory–experiment discrepancy to 2% against an 11% experimental error.
  • For B→K μ⁺μ⁻ and B→K* μ⁺μ⁻, the discrepancy drops from 1.7% to 0.5% and from 11% to 4%, respectively, both within current errors.
  • Tau modes carry the largest relative correction (up to K≈4% for B_s→η′τ⁺τ⁻ and B_s→φτ⁺τ⁻), where it may eventually rival form-factor uncertainty.
  • The same factor applies to differential and angular distributions, allowing it to be folded into future binned and angular analyses, not just total rates.
  • In e⁺e⁻ channels (B→K e⁺e⁻, B→K* e⁺e⁻), the correction slightly worsens agreement (33%→36% and 14%→17%) but stays within experimental uncertainty, showing the factor is not a blanket fix.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If future experimental precision on B_s→μ⁺μ⁻ falls below ~2%, omitting the K-factor would create a fake new-physics shift; the paper's recipe should become part of standard SM-prediction pipelines.
  • A direct cross-check is to compute the same channels with full next-to-leading-order QED and verify that the α_em/v term appears explicitly; if the reference widths used here already include that term, Eq. (9) double-counts.
  • For τ modes, where v is small, the 4% size of K sits near where the α_em/v expansion begins to probe higher-order terms; a two-loop estimate would show whether the single-factor recipe holds.
  • The angular distributions plotted suggest shifts in observables like forward–backward asymmetries; the paper does not compute those derived observables, so a natural next step is to project the corrected distributions onto NP-sensitive angular variables.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proposes that the final-state Coulomb interaction between charged leptons in rare neutral-B decays is captured by a multiplicative K-factor, either the Gamow-Sommerfeld-Sakharov factor (Eq. 2) or a relativistically derived Crater-Alstine-Sazdjian formula (Eq. 3), with the relativistic relative velocity v. After benchmarking the two factors against each other and against one-loop QED, the authors apply K to literature 'free' widths for leptonic B^0_{d,s}→ℓ^+ℓ^- (Table I), semileptonic B→h^0ℓ^+ℓ^- and B→V^0ℓ^+ℓ^- (Tables II–III), and radiative B→γℓ^+ℓ^- (Table IV). They report K≈2.3% for e/μ and up to ~4% for τ, and claim improved agreement with experiment for B_s→μ^+μ^- (δ=2%), B→Kμ^+μ^- (δ<1%), and B→K^*μ^+μ^- (δ=4%).

Significance. Strengths: the CAS derivation in Appendix B is explicit and self-contained; the agreement between GSS and CAS to 0.3% and the benchmark against the one-loop QED expression (Eq. 6) are reassuring; the K-factor is parameter-free and the paper provides predictions for a wide set of channels and distributions. If the baseline double-counting issue is resolved, the paper would be a useful reference for the high-precision era, since the 2–4% effect is currently smaller than form-factor and experimental uncertainties. However, the central quantitative claims rest on the definition of the 'free' widths and on one table entry that is internally inconsistent. As it stands, the primary conclusions are not yet supported.

major comments (2)
  1. [§III, Eq. (9), Table I] The B_s→μ^+μ^- result multiplies the SM prediction of Ref. [21] by K^(CAS). Ref. [21] is a complete NNLO QCD + NLO EW calculation; its NLO EW part necessarily contains the one-photon exchange between the final-state muons. In the threshold limit that diagram is the same πα/v term as in Eq. (6), and for v≈1 it is a ~2.3% correction. The paper asserts that Γ_free is 'without accounting for the Coulomb interaction' but gives no evidence that the quoted 3.66×10^{-9} excludes this diagram. If the QED correction is already present, Eq. (9) double-counts it, and the resulting 3.75×10^{-9} and δ=2% are artifacts. Please state explicitly which QED corrections are included in each external input ([21], [22], [38]) and provide a quantitative check against [21]'s NLO EW contribution.
  2. [Table III, row B^0→K^{*0}μ^+μ^-] The table lists B_free=0.94±0.11, B_Coulomb=1.02±0.11, and K=2.4%. The implied ratio is 1.085 (8.5%), not 2.4%, and this row is the only one in the table with such an inconsistency. The paper's highlighted conclusion in §V and §VII — the discrepancy drops from δ=11% to δ=4% — depends on this 8.5% factor. With K=2.4%, the corrected width is 0.96×10^{-6} and δ would be about 9%, so the claimed improvement disappears. Please correct the numerical error and update the abstract, §V, and §VII accordingly.
minor comments (4)
  1. [§II.C, Eq. (6)] The O(...) term in Eq. (6) is typeset in a confusing manner; please correct the argument of O( ) (likely O(√(1−4m²/s))) and define all variables.
  2. [§IV, Eq. (11)] When applying K^(CAS)(v) to angular/double-differential distributions (Figs. 4–11), please state whether v is evaluated at each q² slice or at a representative value; the text is silent on this point.
  3. [Tables I–IV and captions] Please proofread the notation: B^0_{d,s} is used inconsistently; standard B_s^0 and B_d^0 would be clearer. Also fix the typos 'differetial' in the captions of Figs. 5, 8, and 11, and 'we provide the Coulomb interaction' in §VI.
  4. [Abstract and §VII] The abstract and §VII state the τ correction 'reaches 4%', but the largest tabulated value is 3.9% (Table II) / 3.8% (Table III) / 2.7% (Table IV). Please round consistently or quote the maximum table value.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Coulomb K-factor is derived from independent formalisms and applied as a parameter-free multiplicative correction to external free-width predictions.

full rationale

The central correction factor K is not fitted to the decays it is applied to. In Sec. II the GSS factor (Eq. 2) is a standard Coulomb enhancement; the CAS factor (Eq. 3) is derived in Appendix B from the two-body CAS equations with only alpha_em and the relative velocity v as inputs; and Eq. (6) is an independent one-loop QED benchmark. The relativistic velocity v is a kinematic function of masses, not a parameter adjusted to branching-fraction data. The corrected widths (Eqs. 9 and 11) are multiplicative products of published free widths ([21], [22], [38]) and K, so no 'prediction' is obtained by inverting an experimental number or by renaming a fitted quantity. The paper cites the authors' own earlier work ([22], [35], [38]) for form factors, Wilson coefficients, and free differential widths, but those are external published inputs, not the source of the Coulomb effect; the K-factor derivation does not reduce to those citations. A legitimate physics concern, noted by a reader, is that the SM prediction from [21] may already include final-state QED/Coulomb corrections, in which case Eq. (9) would double-count. However, that is a possible baseline-overlap mistake, not a circular derivation: the paper's definition of Gamma_free as 'without accounting for the Coulomb interaction' is asserted but not verified against the content of [21]. This caveat affects the reliability of the numerical improvement claim, not the internal logical structure of the derivation.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted free parameters; the K factor depends only on α and masses. However, the paper borrows baseline decay widths from earlier work (including the authors' own [22,38]) and does not prove the absence of double-counting with respect to QED corrections in those baselines.

assumptions (4)
  • domain assumption The Coulomb factor from nonrelativistic/relativistic two-body wave functions can be factored as a multiplicative K(v) modifying the full differential and total decay width.
    Used throughout (Eqs. 9 and 11); assumes FSI factorization independent of production mechanism and spin structure.
  • ad hoc to paper B_free values from [21], [22], and [38] represent the width without Coulomb final-state interaction.
    Necessary for Eq. (9); [21] is a full SM/QED calculation, so the overlap is not established.
  • domain assumption The CAS relativistic two-particle equation with constant coupling α and Coulomb potential applies to the decay production process.
    Appendix A-B; used to derive Eq. (3); relies on constant α and regularization at r~1/m being negligible.
  • domain assumption The one-loop QED correction from [31] is universal and spin-independent.
    Section IIC; used to justify applying a scalar-derived K factor to leptons.

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Pith. "Pith review of Coulomb corrections in rare decays of neutral $B$ mesons with $\ell^+\ell^-$-pair in final state." pith.science (2026). https://pith.science/paper/7TAENKPZ

@misc{pith2026260207934,
  author       = {Pith},
  title        = {Pith review of: Coulomb corrections in rare decays of neutral $B$ mesons with $\ell^+\ell^-$-pair in final state},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7TAENKPZ}},
  note         = {Machine review of arXiv:2602.07934}
}
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. 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 $B^0 \to K^{0*}\mu^+\mu^-$, 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.

Figures

Figures reproduced from arXiv: 2602.07934 by the authors.

Figure 1
Figure 1. FIG. 1: The dependence of the Coulomb [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Dependence of the differential branching fraction [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Dependence of the differential branching fraction 10 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Angular distributions 10 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Double differetial distributions 10 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Dependence of the differential branching fraction 10 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Angular distributions for [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Double differetial distributions 10 [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Dependence of the differential branching fraction 10 [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Angular distributions for [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: Double differetial distributions 10 [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]

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