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REVIEW 3 major objections 4 minor 55 references

Investigating non-local contributions in $B_{s} \to \phi \bar{\ell} \ell$ including higher-twist effects

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Adding twist-5 and twist-6 $B_s$-meson distribution amplitudes enlarges the non-local charm-loop form factors for $B_s\to\phi\bar{\ell}\ell$ by about an order of magnitude, with a twist-5 term dominating the shift.

desk verdict A clean, well-documented LCSR extension to twist-5/6, but the claimed order-of-magnitude enhancement is a one-parameter effect that is not statistically robust once the λ2_E, λ2_H uncertainties are taken into account. read the letter →

arxiv 2502.08427 v2 pith:BIXFXYDW submitted 2025-02-12 hep-ph hep-exhep-lat

M. S. A. Alam Khan , Rusa Mandal , Praveen S Patil , Ipsita Ray This is my paper · ORCID
classification hep-phhep-exhep-lat
keywords B_s->phil+l-decaycharm-loopeffectlight-conesumrulesB-mesondistributionamplitudeshigher-twisteffectsnon-localformfactorsWilsoncoefficientC9rarebstransitions
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

The paper tries to establish that higher-twist (twist-5 and twist-6) three-particle $B_s$-meson light-cone distribution amplitudes cannot be dropped when estimating the non-local charm-loop contribution to $B_s\to\phi\bar{\ell}\ell$. Including them in the light-cone sum rule changes the non-local form factors by roughly an order of magnitude relative to the previous twist-4 truncation, because the new twist-5 term breaks a cancellation that had suppressed the twist-3 and twist-4 contributions. This matters for rare $b\to s\ell\ell$ decays, where the charm loop is one of the largest theory uncertainties and where measured observables have shown deviations from Standard Model expectations. The paper's corrected charm loop translates into a $q^2$-dependent shift of the effective Wilson coefficient $C_9$ that is larger than the factorizable-only prediction but still consistent with the Standard Model within uncertainties.

What carries the argument

The load-bearing object is the set of three-particle $B_s$-meson light-cone distribution amplitudes (LCDAs) of definite collinear twist, in particular the twist-5 LCDA $\tilde\phi_5$ in the Exponential Model, whose normalization is set by $\lambda_2^E+\lambda_2^H$. Twist labels the degree of suppression in the light-cone expansion, and these functions parametrize the $B_s$-to-vacuum matrix element of the non-local quark-antiquark-gluon operator produced when the charm loop emits a soft gluon. The LCSR in Eq. (28) converts them into the non-local form factors $\tilde V_i$. The second engine is the double-subtracted hadronic dispersion relation in $q^2$, with the $\phi$, $J/\psi$, and $\psi(2S)$ poles plus a fitted continuum, which carries the spacelike LCSR predictions into the decay region. The numerical cancellation among Lorentz-structure contributions of the same twist is what makes the twist-5 term decisive.

What would settle it

Compute the same non-local form factor with an independent determination of the twist-5 normalization, for example from a lattice QCD calculation of the relevant three-particle correlation function or from alternative QCD sum rules for $\lambda_2^E$ and $\lambda_2^H$, and check whether $\tilde V_\perp(-1\,\mathrm{GeV}^2)$ stays near $-13\times 10^{-7}$ rather than returning to the old value near $0.3\times 10^{-7}$. A complementary check is to fit the low-$q^2$ $B_s\to\phi\mu^+\mu^-$ angular data with and without the enhanced charm loop and see which reproduces the measured $S_3$ and $S_7$ bins.

Watch

Extended reading notes

Core claim

On its own terms, the paper claims that the non-local form factors $\tilde V_\perp$, $\tilde V_\parallel$, and $\tilde V_0$ receive a twist-5-dominated contribution that earlier truncations missed. At the benchmark $q^2=-1\,\mathrm{GeV}^2$, the perpendicular combination is $10^7 \tilde V_\perp(-1)=1.536|_{\mathrm{twist}\,3}-1.235|_{\mathrm{twist}\,4}-14.334|_{\mathrm{twist}\,5}+0.668|_{\mathrm{twist}\,6}$, moving the central value from about $0.3$ (the twist-3 plus twist-4 part) to about $-13.4$ in these units. Within the Exponential Model's complete set of eight three-particle LCDAs, the twist-5 function $\tilde\phi_5$ alone contributes $-139.47\times 10^{-8}$ of the total $-143.32\times 10^{-8}$ for $\tilde V_\perp$ at that point. Continuing these LCSR results into the physical region with a double-subtracted dispersion relation that includes the $\phi$, $J/\psi$, and $\psi(2S)$ resonances, the paper obtains a polarization-dependent, $q^2$-dependent correction $\Delta C_{9,\lambda}(q^2)$ that is positive over most of the physical region and larger than the Standard Model prediction without non-factorizable charm loops, yet compatible with it within uncertainties.

Load-bearing premise

The whole order-of-magnitude result rests on one modeling assumption: the exponential shape and normalization of the twist-5 three-particle distribution amplitude of the $B_s$ meson, whose strength is set by $\lambda_2^E+\lambda_2^H$; if that model is wrong, the enhancement could vanish.

Editorial extensions

If this is right

  • The effective $C_9$ shift becomes polarization-dependent and $q^2$-dependent, so $B_s\to\phi\mu^+\mu^-$ angular observables such as $S_3$ and $S_7$ change most in the low-to-intermediate $q^2$ bins, where current data already show some tension.
  • Precision on $\lambda_2^E$ and $\lambda_2^H$ becomes a bottleneck: across the five allowed parameter regions the non-local form factors vary by up to orders of magnitude, so better determinations of these two constants directly translate into sharper rare-decay predictions.
  • The local $B_s\to\phi$ form factors are affected at the sub-percent level by the higher-twist three-particle LCDAs, so the larger charm-loop effect is not coming from rescaled local inputs.
  • Because a similar cancellation pattern holds for $B\to K^*\bar\ell\ell$, the same twist-5 and twist-6 input likely shifts that mode's non-local estimates as well, though with slightly smaller magnitude.

Reading between the lines

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

  • If this enhancement survives scrutiny, global fits of $b\to s\ell^+\ell^-$ data that were tuned to the older, smaller charm-loop estimates may need to be redone; the enlarged effect could absorb part of the apparent tension with the Standard Model or shift the preferred window for new physics.
  • The dominance of $\tilde\phi_5$ suggests that truncating the twist expansion at an even order (twist-4) is structurally unsafe; a direct test would be to repeat the computation with the alternative Local Duality model of the LCDAs, which the paper mentions but does not present, and check whether the order-of-magnitude enhancement survives the model choice.
  • A lattice QCD determination of the moments $\lambda_2^E$ and $\lambda_2^H$ would be a sharp test, since the paper's sensitivity study shows the prediction can vary by orders of magnitude across the currently allowed parameter regions.
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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

3 major / 4 minor

Summary. This paper extends the LCSR calculation of non-local charm-loop form factors for B_s → φ ℓ+ℓ− by including twist-5 and twist-6 three-particle B_s-meson LCDAs. Using the Exponential Model for the LCDAs, the authors find that the twist-5 contributions break the cancellations seen in the earlier twist-4 analysis of Ref. [12], leading to an approximately order-of-magnitude increase in the non-local form factors. The results are continued to the physical region using double-subtracted dispersion relations with resonance inputs from B_s → φ V data and fitted continuum parameters. The paper also updates local form factors with higher-twist three-particle contributions and studies the impact on C9 and angular observables.

Significance. If the enhancement survived scrutiny, it would have important implications for b → s ℓℓ phenomenology, as the charm-loop correction to C9 would be large and polarization-dependent. The paper is thorough in providing breakup tables (Table 3, Appendix D), explicit sensitivity expressions (Eq. 43), and a detailed account of the LCSR framework, and it reproduces the twist-4 limit of Ref. [12] (up to input differences). However, the headline enhancement is driven by a single twist-5 LCDA ψ̃5 whose normalization λ2_H is poorly constrained, and the paper's own Fig. 2 and sensitivity analysis show that alternative determinations can change the prediction by orders of magnitude. Hence the central quantitative claim is not yet established at the 1σ level.

major comments (3)
  1. [Sec. 4.1.1, Eq. (42), Table 3] The claimed order-of-magnitude enhancement is driven by the twist-5 term in Eq. (42), which for eV⊥(−1) equals −14.334×10⁻⁷ and is dominated by the ψ̃5 contribution (−139.47×10⁻⁸ of the −143.32×10⁻⁸ twist-5 total in Table 3). In the Exponential Model of Eq. (70), ψ̃5 is proportional to λ2_H, so the enhancement is controlled by one poorly known input combination. The difference between the twist-4-only value eV⊥(−1)=0.302±2.049 and the full result −13.364±8.171 (Table 2) is only about 1.6σ once uncertainties are added in quadrature. The paper should provide a quantitative significance statement for the enhancement, and should present central predictions under the alternative λ2_E,H determinations (Regions II–V of Fig. 2) to show whether the order-of-magnitude effect persists. As it stands, the headline claim is not statistically robust.
  2. [Sec. 3, Eqs. (38)–(39), Sec. 4.2.2, Table 8] The double-subtracted dispersion relations in Eqs. (38)–(39) contain the subtraction constants Hλ(q0²) and dHλ/dq² at q0² = −1 GeV², but Table 8 lists only the phases φ0_V and continuum parameters aλ, bλ. The manuscript does not state how the subtraction constants and their derivatives are obtained: if they are taken from the LCSR predictions at q0² = −1, the derivative is not directly available from the sum rule and requires an additional modeling assumption; if they are fitted, they should appear as fit parameters with uncertainties. This information is needed to validate the extrapolation used for the C9 predictions in Fig. 3.
  3. [Sec. 4.1.2, Eq. (43), Fig. 2] Eq. (43) and Fig. 2 show that eVλ is linear in λ2_H and R = λ2_E/λ2_H, and the paper acknowledges that along Regions IV and V 'such estimates can even vary by several orders of magnitude'. The uncertainties quoted in Table 2 (e.g., eV⊥(−1) = −13.364±8.171) reflect only the Region I 1σ ranges of Table 1 and do not include the spread across the alternative determinations of λ2_E,H presented in Regions II–V. The paper should either restrict the analysis to a well-justified range of λ2_E,H or propagate the full spread into the quoted errors; otherwise the central values are not representative of the current state of knowledge.
minor comments (4)
  1. [Sec. 4.1.1, Table 2] The comparison with Ref. [12] for eV0 at q² = −1 GeV² shows a difference of about 2.5σ between the central values (0.101±0.065 vs −0.15±0.08), which is more than the 'slight difference' described in the text; the authors should clarify the input choices that produce this shift.
  2. [Sec. 4.2.2, Table 8] The fitted phases φ0_V are quoted without uncertainties, although they are fit parameters; the authors should report their errors or explain why they are fixed.
  3. [Throughout] There are numerous typographical artifacts, e.g., 'Ge V2' instead of 'GeV²' in Sec. 4.1.1 and elsewhere, and inconsistent spacing in 'Bs → ϕ¯ℓℓ'; the manuscript would benefit from a careful proofreading pass.
  4. [Sec. 3, Eq. (40)] The paper notes that anomalous thresholds from multiparticle states are ignored and a simple linear continuum model is used (Eq. (40)); a brief comment on the expected size of this approximation would be useful, especially given the recent literature cited as Refs. [26,27].

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found: the twist-5 enhancement is an openly parameter-sensitive model prediction, not a fitted reproduction of the target observables.

full rationale

The paper's central claim—that including twist-5 and twist-6 three-particle B_s-meson LCDAs enhances the non-local charm-loop form factors by about an order of magnitude—is a genuine LCSR computation. The enhancement is traced to the twist-5 LCDA \psi_tilde5, whose normalization is proportional to \lambda_2^E + \lambda_2^H, a non-perturbative input taken from the literature (Ref. [36]), not fitted to the B_s -> \phi \mu^+\mu^- data shown in Fig. 4. The authors explicitly expose the linear dependence on these parameters in Eq. (43) and quantify the sensitivity across five literature regions in Fig. 2, so the result is a model-dependent prediction rather than a hidden restatement of an input. The dispersion-relation parameters (phases \phi^0_V, continuum coefficients a_\lambda, b_\lambda, Table 8) are fitted to the LCSR spacelike predictions and then used for analytic continuation into the timelike region; this is a standard matching procedure, and the final comparison with experimental data is only a comparison, not a fit. The only self-citations (Refs. [18] and [35]) supply the input \lambda_{B_s}, which has independent QCD sum-rule and lattice support and does not carry the central claim. No definitional equivalence, fitted-input-renamed-as-prediction, or ansatz-smuggling-via-citation was identified.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central calculation rests on the LCSR framework and the Exponential Model parametrization of the B_s-meson three-particle LCDAs, with normalization controlled by the poorly constrained parameters lambda2_E and lambda2_H. All parameters listed as free are either inputs from external QCD sum rules or fit parameters introduced in this paper for the z-expansion and dispersion-relation extrapolation. No new physical entities are postulated.

free parameters (6)
  • lambda2_E = 0.03 +/- 0.02 GeV^2 (input from Ref. [36])
    Sets the normalization of several three-particle LCDAs in the Exponential Model (Eq. 70); the twist-5 contribution to eV_lambda is proportional to it (Eq. 43). The sensitivity study in Sec. 4.1.2 and Fig. 2 shows the non-local form factors vary by factors of several across allowed determinations.
  • lambda2_H = 0.06 +/- 0.03 GeV^2 (input from Ref. [36])
    Same role as lambda2_E; enters the twist-5 and twist-6 DAs. The paper shows order-of-magnitude variation in eV_lambda across the allowed Regions IV and V in Fig. 2.
  • lambda_Bs = 480 +/- 92 MeV (input from Ref. [35])
    Inverse moment of the leading-twist B_s-meson LCDA; fixes the scale omega0 in the Exponential Model and therefore the overall size of the three-particle LCDA contributions.
  • phi0_V (V = phi, J/psi, psi(2S)) = -0.301, 0.034, 0.018 (Table 8)
    Longitudinal resonance phases left free in the double-subtracted dispersion relation (Eqs. 38-39) and fitted to the LCSR predictions at spacelike q^2.
  • a_lambda and b_lambda (lambda = perp, par, 0) = Complex values in Table 8
    Coefficients of the linear continuum model in Eq. (40), fitted to the LCSR predictions. They carry the extrapolation uncertainty into the timelike region.
  • z-expansion coefficients aF_i for the 7 local form factors = Table 5 (21 coefficients)
    Fitted to the LCSR local form factor points via chi-square minimization (Eq. 46); used in the analytic continuation of F_lambda into the timelike region for Delta C9,lambda.
assumptions (5)
  • standard math Semi-local quark-hadron duality identifies the continuum threshold s0 with the LCSR effective threshold (Sec. 2.1, Eq. 28).
    Standard LCSR assumption used to eliminate excited states; taken over from Refs. [8,12].
  • domain assumption Exponential Model for the three-particle B_s-meson LCDAs, Eq. (70), including the twist-5 forms psi5, ~psi5, ~phi5 and twist-6 phi6.
    The shape and normalization of the higher-twist DAs are not derived in this paper; they are taken from Refs. [14,39]. The dominant twist-5 contribution to eV_lambda comes from ~phi5, proportional to lambda2_E plus lambda2_H, so the central enhancement inherits this model dependence.
  • domain assumption Higher-order terms in the light-cone OPE are small and neglected (Sec. 3, after Eq. 33).
    The paper explicitly neglects subleading power corrections such as the two-particle twist-5 off-light-cone effect that Refs. [16,17] estimate at the roughly 25% level in similar modes.
  • domain assumption The continuum in the dispersion relation is modeled as a linear function of q^2, Eq. (40).
    Adopted instead of a more detailed treatment of anomalous thresholds and higher resonances; the authors state the numerical effect is small compared to the single-pole alternative.
  • domain assumption The resonance amplitudes A_lambda_V and phases phi_perp and phi_par are extracted from LHCb two-body data [30-32], with longitudinal phases phi0_V free in the fit.
    These inputs anchor the dispersion relation in the timelike region; the free phases are fitted to the LCSR predictions at spacelike q^2 (Table 8).

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Cite this review

Pith. "Pith review of Investigating non-local contributions in $B_{s} \to \phi \bar{\ell} \ell$ including higher-twist effects." pith.science (2026). https://pith.science/paper/BIXFXYDW

@misc{pith2026250208427,
  author       = {Pith},
  title        = {Pith review of: Investigating non-local contributions in $B_s \to \phi \bar\ell \ell$ including higher-twist effects},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BIXFXYDW}},
  note         = {Machine review of arXiv:2502.08427}
}
abstract

We analyze the impact of higher-twist three-particle $B_s$-meson light-cone distribution amplitudes (LCDAs) on the non-local form factors for the $B_s\to \phi \bar{\ell} \ell$ transition focusing on the `charm-loop' contribution within the light-cone sum rule (LCSR) framework. To analytically continue these charm-loop contributions into the kinematically allowed region of the decay, we employ a hadronic dispersion relation that incorporates intermediate resonant states such as the $\phi,\,J/\Psi$ and $\psi(2S)$ mesons. Here, the LCSR predictions serve as inputs, supplemented by experimental data from two-body decays $B_s \to \phi ~+$ resonance states. Our results indicate that the inclusion of twist-5 and twist-6 LCDAs enhances the non-local form factors by approximately an order of magnitude compared to previous estimates, due to partial disruption of cancellation among different twist contributions. This leads to a dilepton invariant mass-squared ($q^2$)-dependent correction to the Wilson coefficient $C_9$, which is higher than, but still consistent with the Standard Model prediction without the non-factorizable charm-loop corrections within uncertainties. Additionally, we update the local form factors to include contributions from higher-twist three-particle $B_s$-meson LCDAs. The phenomenological implications, particularly for the differential branching fraction and angular observables, are also discussed.

Figures

Figures reproduced from arXiv: 2502.08427 by the authors.

Figure 1
Figure 1. Charm-loop contributions to the mode Bs → ϕ¯ℓℓ with (a)-leading-order factorizable contribution, (b)-next-to-leading-order factorizable contribution (hard gluon exchange) and (c)- non-factorizable soft-gluon correction. The dot denotes the effective four-quark vertex. where the ellipses represent the higher powers in the light-cone OPE. The matching coefficients ∆C9 and ∆C7 have been computed to next-to-leading orde… view at source ↗
Figure 2
Figure 2. Contour plots illustrates the dependence of the non-local form factors [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. The real and imaginary parts of ∆C9,λ are obtained from the analytic continuation of Hλ(q 2 ) and Fλ using Eq. (51), along with the fitted values from Table (8) in the dispersion relations (38)-(39). Local form factors are included from Eq. (44), incorporating inputs from Table (5). The shaded band represents 1σ uncertainties at different q 2 values, while the error bars at q 2 < 0 correspond to the LCSR predictions… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Predictions of the differential branching fraction, longitudinal polarization fraction [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]

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