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REVIEW 4 major objections 4 minor 34 references

Angular observables and branching ratio for $B_s\to \phi \ell^+ \ell^-$ decay

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

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Pith's one-line read The covariant confined quark model reproduces LHCb angular data for B_s→φμ+μ− once two-loop corrections to the Wilson coefficients are included.

desk verdict A useful CCQM cross-check undermined by a corrupted comparison table and an overclaimed agreement sentence. read the letter →

arxiv 2510.03739 v2 pith:M63LVINN submitted 2025-10-04 hep-ph hep-ex

Aidos Issadykov This is my paper · ORCID
classification hep-phhep-ex PACS 12.39.Ki13.30.Ce14.40.Nd
keywords B_sdecayrareBdecaysangularobservablescovariantconfinedquarkmodeleffectiveHamiltonianWilsoncoefficientsLHCbflavorphysics
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 rare decay B_s→φℓ+ℓ− can be described within the Standard Model by the covariant confined quark model (CCQM), a relativistic quark model, provided the effective Wilson coefficients C7 and C9 are upgraded with next-to-next-to-leading-logarithmic (NNLL) two-loop corrections. Using form factors computed in the CCQM and the standard effective Hamiltonian, the paper computes the branching fraction, forward-backward asymmetry, longitudinal polarization, and optimized angular observables in q² bins, and compares them to LHCb measurements. The central result is that the CCQM predictions for the branching fraction and F_L agree with the 2021 LHCb data in the bins where the two-loop corrections are valid, indicating that no new physics is required for these observables. If correct, this strengthens the case that a simple quark-model approach can capture the dynamics of b→sℓℓ transitions.

What carries the argument

The central object is the set of dimensionless form factors A0, A+, A−, V, a0, a+, g for the B_s→φ transition, parametrized by a dipole form F(q²)=F(0)/(1−a s + b s²) with s=q²/m_{Bs}². These feed the helicity amplitudes that define the observables. The second key ingredient is the effective Hamiltonian with Wilson coefficients, where the long-distance charm-loop effects and two-loop QCD corrections are absorbed into C7_eff and C9_eff; the two-loop corrections are included only in the q² ranges where their expansions are valid, 1.1–5.5 and 8.8–22 GeV². The combination of these ingredients produces predictions that match LHCb data.

What would settle it

A determination of B_s→φμ+μ− in the q² bins [5,8] and [6,8] GeV² with the same two-loop accuracy (or an experimental measurement with uncertainty below ~5%) would test the one-loop CCQM predictions directly; if the measured branching fraction in these bins deviates by more than the model's ~20% uncertainty while the two-loop-valid bins agree, the assumed treatment of the charm region would be falsified. Alternatively, a lattice QCD calculation of the B_s→φ form factors at q² = 0 that yields A0(0)=0.40 and V(0)=0.31 within errors would support the CCQM input.

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Extended reading notes

Core claim

The paper computes the B_s→φ transition form factors within the CCQM and uses them together with the SM effective Hamiltonian to predict the differential branching ratio and angular observables for B_s→φℓ+ℓ−. It shows that after including the NNLL two-loop corrections to C7_eff and C9_eff, the predicted branching fraction B(B_s→φμ+μ−) and the longitudinal polarization F_L fall within the 2021 LHCb measurements in the bins [11,12.5], [15,17], [17,18.9], and [15,18.9] GeV², improving on the agreement with the earlier 2015 data. The paper concludes that the CCQM provides a consistent Standard Model description of this decay.

Load-bearing premise

The numerical predictions inherit the CCQM form factors and model parameters from an earlier paper without recomputation, so the agreement with LHCb data is conditional on those priors being correct; additionally, the long-distance charm-loop effects are modeled by a Breit-Wigner ansatz rather than computed from first principles.

Editorial extensions

If this is right

  • If the CCQM description holds, the measured B_s→φμ+μ− observables are consistent with the Standard Model, and deviations seen in earlier bins may be attributed to missing higher-order corrections rather than new physics.
  • The inclusion of NNLL corrections is necessary for agreement; one-loop predictions are insufficient, especially at low q².
  • The optimized observable S7, which vanishes at leading order, becomes non-zero with the two-loop corrections, providing a testable SM prediction.
  • For the bins [5,8] and [6,8] GeV², where two-loop corrections are not valid, the one-loop CCQM predictions are the best available and can be improved when full NNLO results for the charm region become available.
  • The same framework yields predictions for the τ+τ− mode and B_s→φνν̄, which can be checked by future experiments.

Reading between the lines

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

  • The paper's reliance on form factors from a previous calculation without recomputation means the numerical agreement is conditional on those model parameters; a public release or independent recalculation of the CCQM parameters would allow direct testing.
  • Because the two-loop corrections are valid only in specific q² ranges, the 'agreement' claim is strongest in those bins; future high-precision data in the [5,8] and [6,8] bins would sharply discriminate the CCQM's one-loop treatment from other models.
  • The near-equality A0(0)≈V(0) noted in the heavy-quark limit is not exact in the CCQM, which explains the smallness of P1; testing whether P1 remains small at higher q² in other models could probe the model's form-factor structure.
  • If the charm-loop Breit-Wigner ansatz is inadequate, the agreement could be accidental; lattice QCD form factors combined with the same effective Hamiltonian would provide a cross-check.
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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

4 major / 4 minor

Summary. This paper presents a calculation of the B_s→φ transition form factors in the covariant confined quark model (CCQM), and uses them to predict branching fractions and angular observables (A_FB, F_L, A_i, S_i) for B_s→φℓ^+ℓ^- in various q² bins. The results are compared with LHCb data from 2015 and 2021 and with PQCD predictions from ref. [19]. The central claim is that the CCQM predictions agree with the latest LHCb data, particularly in specific high-q² bins, and that NNLL two-loop corrections to C7_eff and C9_eff are essential to this agreement. The manuscript includes tables of binned predictions, figures for differential distributions, and a comparison of integrated observables.

Significance. If the numerical results are correct, this would be a useful independent CCQM benchmark for b→sℓℓ transitions and a cross-check of the LHCb angular measurements. The paper's use of externally computed Wilson coefficients and published two-loop corrections is a strength, as is the direct comparison with two LHCb datasets. However, the main comparison table is substantially misaligned and the text overstates the level of agreement. The quantitative conclusions cannot be assessed until the table is corrected and the internal inconsistencies are resolved.

major comments (4)
  1. [Table VI] Table VI, the sole LHCb-comparison table, contains multiple row/column misalignments that invalidate the printed agreement claim. In the AFB block, row [1,6] lists CCQM = 0.69 ± 0.14 and PQCD = 0.777, identical to the FL values for the same bin, whereas the adjacent [1.1,6] AFB row is 0.034 ± 0.006. In the same AFB block, row [15,18.9] lists 0.394 ± 0.003, again an FL value. A5 [0.1,0.98] = 0.268 ± 0.054 while A5 [0.1,2] = 0.0031 ± 0.0006, an implausible jump for a smooth observable. A8 [1,6] = 0.17 ± 0.03 equals the S4 entry. Many S7 rows lack CCQM entries and instead list PQCD values around 0.45–0.99, inconsistent with the stated magnitude of S7. These errors must be corrected before the agreement claim can be evaluated.
  2. [Section V] The concluding sentence in Section V states that the integrated observables are in 'complete agreement' with PQCD [19]. This is contradicted by Table V: for the τ mode, <F_L> is 0.090 ± 0.02 (CCQM) versus 0.396+0.002/−0.003 (PQCD), a factor-of-4 difference. The μ-mode values are close, but the τ-mode F_L is not in agreement. This overstatement should be removed and the discrepancy discussed.
  3. [Section IV, Eq. (20)] The paper states that the [5,8] and [6,8] GeV² bins are omitted from NNLL corrections and therefore use one-loop Wilson coefficients, yet these bins are included in Table VI without any additional systematic uncertainty. Since the text credits NNLL corrections for the agreement with LHCb, the one-loop bins are not on the same footing as the rest of the table. The authors should either assign an uncertainty for the missing two-loop piece in these bins or explicitly exclude them from the agreement claim.
  4. [Eq. (20)] The manuscript gives two conflicting validity ranges for the low-q² NNLL corrections. Equation (20) states 1.1 ≤ q² ≤ 5.5 GeV², while the following paragraph says the low-q² two-loop results are reliable for 0.1 ≤ q² ≤ 6 GeV². Several bins in Table VI start at 0.1 or 0.98 GeV². Because the paper argues that NNLL corrections are essential to the agreement, the reader must know which bins actually received these corrections. Please reconcile the ranges and state the treatment of each bin.
minor comments (4)
  1. [Figures 1–6] The figure labels contain LaTeX artifacts such as '/LParen1GeV2/RParen1' and '/MiΝus'. The sentence before Fig. 1, 'The behavior of the differential branching B(Bs→φνν) is shown in Figs. 4', is incomplete.
  2. [Tables I–II] Table I gives A0(0)=0.40, while Table II lists the CCQM A0(0)=0.28±0.03. The relation between these quantities (via the BSW form-factor convention from ref. [5]) should be stated explicitly in the caption of Table II, otherwise the reader cannot reconcile the two tables.
  3. [Section IV] The text says 'the optimized observable P1 remains small across a large range of values' and that A0(0)−V(0)=0.09 leads to a 'truly small value of P1'. However, Table V reports <P1> = −0.52 ± 0.1 for the muon mode. This is not small in the conventional normalization of P1 and should be clarified.
  4. [Table IV] The branching ratios in Table IV are quoted without explicit reference to the q² integration range. For Bs→φµ+µ−, the total rate should be specified as the full q² range; otherwise the comparison with LHCb and other models is ambiguous.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the CCQM predictions are benchmarked against LHCb data using independent external inputs, not fitted to the comparison data.

full rationale

The derivation chain is not circular in any constructional sense. The form factors in Table I are explicitly inherited from the earlier CCQM paper [5] ('same as in our previous work'), but that is prior independent model output, not a quantity fitted to the LHCb data being compared; the same holds for the Wilson coefficients taken from [30], the two-loop corrections from [28,29], and the Breit-Wigner charm-loop ansatz from [27]. The dipole parametrization in Eq. (4) is a fit to the model's own numerical form factors with a stated <1% relative error, not a fit to the experimental target, so it does not turn the prediction into an input. The comparison with LHCb [1,2] and PQCD [19] is an external benchmark. The paper also honestly flags its own limitation that the [5,8] and [6,8] GeV^2 bins are one-loop because the two-loop validity intervals exclude them. The suspected Table VI row/column misalignments are a presentation and reproducibility concern about the claimed agreement, but they do not amount to a derivation that reduces by construction to its own inputs. The self-citation [5] is load-bearing for the numerical estimates but qualifies as independent prior support rather than circularity because its parameters were not tuned to the LHCb observables with which the paper compares.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The calculation rests on a standard effective Hamiltonian plus two model layers: CCQM hadronic form factors inherited from [5] and long-distance charm-loop parameterization inherited from [27-29]. The paper adds NNLL corrections but does not reduce the number of implicit parameters and does not quantify the impact of the CCQM parameter choice on the quoted uncertainties.

free parameters (3)
  • CCQM model parameters (constituent quark masses, quark-meson couplings g_Bs, g_φ, vertex width, infrared cutoff) = not given; fixed in [5]
    The form factors in Eqs. (2)-(3) and Table I are taken verbatim from ref. [5]; all central observables depend on these parameters, and the paper does not propagate their uncertainty.
  • Dipole fit parameters F(0), a, b for each of the seven form factors = Table I
    Eq. (4) approximates the CCQM numerical form factors with F(0)/(1 - a s + b s^2); the <1% fit error is quoted, but the fit parameters are model-output summaries rather than independently measured inputs, and they enter every subsequent observable.
  • Breit-Wigner parameters for c-cbar resonances absorbed in C9_eff = not specified in this paper
    Sec. III states that long-range c-cbar contributions are parameterized via the Breit-Wigner ansatz [27], but the resonance masses, widths, and relative coefficients used are not listed.
assumptions (4)
  • domain assumption The Standard Model effective Hamiltonian (Eq. 5) with the ten-operator basis and Wilson coefficients taken from [30] is valid at the hadronic scale.
    All predictions are computed in this operator basis; missing operators or scale errors would shift the observables. The Wilson coefficients are external inputs, not derived here.
  • domain assumption The CCQM confining vertex ansatz in Eqs. (2)-(3) reliably approximates the QCD matrix elements for the B_s→φ transition.
    The entire hadronic input is the CCQM replacement of quark propagators and meson vertices by model vertex functions; this is a phenomenological model assumption carried over from [5].
  • domain assumption Charm-loop long-distance effects are represented by C9_eff with the Breit-Wigner ansatz of [27] plus the two-loop corrections of [28,29], and bins [5,8]/[6,8] may use one-loop coefficients.
    Secs. III-IV explicitly restrict NNLL validity to 1.1≤q^2≤5.5 and 8.8≤q^2≤22 GeV^2 and fall back to one-loop in the intermediate bins; no uncertainty from this order-mixing is included.
  • standard math The angular-observable definitions and helicity amplitudes from [31] and the optimized-observable relations from [33] are the correct ones for the B_s→φ(→K+K−)ℓ+ℓ− cascade.
    Formulas (9)-(19) depend on this external formalism; the definitions of A5, A6, A8, A9 are not written explicitly in this paper, so the reader must trust the cited sources.

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Pith. "Pith review of Angular observables and branching ratio for $B_s\to \phi \ell^+ \ell^-$ decay." pith.science (2026). https://pith.science/paper/M63LVINN

@misc{pith2026251003739,
  author       = {Pith},
  title        = {Pith review of: Angular observables and branching ratio for $B_s\to \phi \ell^+ \ell^-$ decay},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M63LVINN}},
  note         = {Machine review of arXiv:2510.03739}
}
abstract

In this paper, an analysis of the $B_s\to \phi \ell^+ \ell^-$ rare decay is presented within the framework of the covariant confined quark model. The $B_s\to \phi$ transition form factors are calculated and then used to compute the branching fractions and angular observables in various $q^2$ bins, including the forward-backward asymmetry $A_{FB}$, the longitudinal polarization $F_L$, and the optimized observables $A_i$ and $S_i$. The results show agreement with the latest experimental data given by LHCb collaboration and compared with available theoretical predictions.

Figures

Figures reproduced from arXiv: 2510.03739 by the authors.

Figure 1
Figure 1. FIG. 1: The behavior of the differential branching fraction [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Forward-backward asymmetry [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Longitudinal polarization [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The behavior of the differential branching fraction [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Observables [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Observables [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

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