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The paper claims that, once form-factor uncertainties are treated conservatively with the Dispersive Matrix method, the latest b→sℓ+ℓ− data—especially the strong-phase-sensitive angular observables—favor long-distance hadronic effects (char

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2026-08-02 08:48 UTC pith:EOLETHJF

load-bearing objection Useful new form-factor machinery and an honest fit, but the flagship claim that the data prefer hadronic effects over a C9 shift only holds if NP C9 is real. the 3 major comments →

arxiv 2607.03531 v2 pith:EOLETHJF submitted 2026-07-03 hep-ph hep-ex

A Dispersive Look at Rare B-meson Semileptonic Decays

Marco Ciuchini , Marco Fedele , Ayan Paul , Josua Scholze , Luca Silvestrini , Silvano Simula , Mauro Valli , Ludovico Vittorio This is my paper
classification hep-ph hep-ex PACS 13.20.He
keywords b→s transitionsrare B decayssemileptonic decaysDispersive Matrix form factorscharming penguinslepton universalityB→K*μμ angular observablesb→sνν
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper asks whether the long-standing anomalies in rare B-meson decays point to new physics or to unappreciated hadronic effects. It finds that, when form-factor uncertainties are enlarged by relying only on lattice QCD inputs, the global fit to b→sℓ+ℓ− data no longer prefers a lepton-universal shift in the Wilson coefficient C9. Instead, the data—particularly the new LHCb and CMS angular measurements that are sensitive to strong phases—favor sizeable non-factorizable hadronic matrix elements, the so-called charming penguins. The paper also updates Standard Model predictions for the invisible modes B→K(∗)νν, which are free of these hadronic contaminations, and finds that the recent Belle II excess in B+→K+νν becomes a 2.7σ effect relative to the new prediction. A sympathetic reader would care because the interpretation of the b→s flavor anomalies hinges on this hadronic-versus-short-distance dichotomy.

Core claim

The central claim is that the pattern of deviations in b→sℓ+ℓ− decays is better described by non-factorizable hadronic dynamics than by a short-distance shift in C9. Two independent pieces of evidence support this. First, the real part of a hadronic parameter, Re(h−(1)), which is exactly degenerate with a lepton-universal C9 shift, is pulled away from zero in the direction opposite to the usually claimed negative C9, while the imaginary parts of the hadronic parameters, which cannot be mimicked by a constant C9 shift, are found to be nonzero at 95% probability in the data-driven approach. Second, the q2-dependence of the B+→K+μ+μ− branching fraction at low q2 is naturally reproduced by a q2-

What carries the argument

The Dispersive Matrix (DM) method is used to determine the B→K(∗) and Bs→φ form factors from lattice QCD inputs, enforcing analyticity and unitarity to extrapolate to the full kinematic range and yielding a controlled, conservative uncertainty band. The second key element is the phenomenological parametrization of the non-local hadronic correlator hλ(q2) in powers of q2, with constant complex coefficients hλ(i); the real parts of h−(0) and h−(1) enter the helicity amplitudes in exactly the same way as shifts in C7 and C9, respectively, while all imaginary parts and the higher-order real parts are unambiguously hadronic. This parametrization makes transparent the degeneracy between charming-p

Load-bearing premise

The fit's conclusion depends on the phenomenological polynomial parametrization of the non-local hadronic correlator hλ(q2), truncated at order q4 (with q2^{3/2} terms for B→K) and with constant complex coefficients; if the true q2 dependence of charm-loop and rescattering effects contains stronger or resonant structure, the inferred nonzero hadronic contributions could be artifacts.

What would settle it

Compute the charm-loop correlator directly in lattice QCD for B→K and B→K∗, following the first-principles framework the paper cites as future work; if the resulting real and imaginary parts across q2 are small and smooth, the data-driven fit's preference for sizeable penguin matrix elements would disappear and the C9-shift interpretation would revive.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the hadronic explanation is correct, the b→sℓ+ℓ− anomalies do not require new physics in the Wilson coefficient C9; the apparent tensions are absorbed by penguin matrix elements, and strong-phase-sensitive observables such as S7 become key discriminants.
  • The choice of whether to include light-cone sum-rule inputs in the form-factor determination materially changes the room left for hadronic effects: using only lattice QCD inflates large-recoil uncertainties and enlarges the allowed space for non-factorizable contributions.
  • Under the model-dependent treatment of power corrections, a sizeable negative shift in C9 (about −1.2 to −1.7 units, lepton-flavor dependent) is still preferred, so the new-physics interpretation survives only if one imposes that the hadronic correlator is small and smooth.
  • The updated Standard Model prediction for B(B+→K+νν) = (3.95±0.14)×10−6 is slightly lower than earlier estimates, increasing the significance of the Belle II excess to 2.7σ, while the B→K∗νν predictions are roughly twice as uncertain in the lattice-only approach.
  • In the b→sνν sector, current data show a mild preference for a negative right-handed quark-current coefficient δCRν at more than 2σ, a result that is stable across both form-factor determinations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The conclusion hinges on the polynomial truncation of the non-local correlator hλ(q2) at O(q4) with constant coefficients; a first-principles lattice computation of the charm-loop correlator would directly test whether the inferred nonzero imaginary parts are physical or artifacts of missing higher-order and resonant structures.
  • The paper's dichotomy—hadronic effects versus a real, lepton-universal C9 shift—would collapse if new physics had absorptive phases or non-universal lepton couplings; future data on lepton-flavor ratios such as RK(∗) and on CP asymmetries could therefore probe this hidden assumption.
  • The same data-driven framework could be extended to other b→s modes and to b→d transitions, where the hierarchy of CKM factors and penguin contributions differs, offering a consistency check of the 'sizeable penguin matrix elements' interpretation across channels.
  • The authors' emphasis on B→K∗νν as a chiral-sensitive observable suggests that a future Belle II measurement of that mode, combined with the B→Kνν excess, could distinguish left- and right-handed new physics more cleanly than currently possible.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper presents a global analysis of b→s ℓ+ℓ− and b→s νν transitions using form factors obtained with the Dispersive Matrix method. Two hadronic-input setups are compared: one using only lattice QCD (LQCD DM) and one adding LCSR inputs for vector final states (LQCD+LCSR DM). Non-local charm-loop contributions are treated either in a Data Driven way, with complex polynomial coefficients fitted to data, or in a Model Dependent way. The central claim is that the latest LHCb and CMS angular data favour sizeable non-factorizable hadronic effects — in particular non-zero imaginary parts of the non-local correlator — rather than a short-distance shift in C9. The paper also provides updated SM predictions for B→K(*)νν and NP fits for left- and right-handed neutrino Wilson coefficients.

Significance. If the central claim holds, this is an important update: it would mean that the long-standing b→s ℓ+ℓ− tensions can be accommodated within the SM once form-factor uncertainties and non-local hadronic matrix elements are treated conservatively. The paper is technically solid in several respects: the DM form-factor framework is clearly described; full BGL posterior means, standard deviations and correlation matrices are provided in Appendices B–D, which supports reproducibility; the fits are implemented in HEPfit; and the new B→K(*)νν SM predictions, which depend only on local form factors, are a useful benchmark for Belle II. The comparison between LQCD-only and LQCD+LCSR inputs is also a valuable sensitivity study. However, the headline interpretation — hadronic effects rather than C9 new physics — is conditional on assumptions about the reality and flavour structure of the NP Wilson coefficients, and the evidence is partly degenerate with a complex C9 shift. The analysis is therefore best read as evidence against a real, lepton-universal C9 shift under the adopted phenomenological parametrization of non-local effects, rather than as a model-independent discrimination.

major comments (3)
  1. [§III B, Eqs. (23)–(25), Fig. 4(right)] The statement that the imaginary parts of h parameters 'can be identified as genuine hadronic contributions without any ambiguity' is not supported by the amplitude structure. In Eq. (23), and analogously in Eq. (25), h_-^(1) enters only through the combination C9^SM − h_-^(1) multiplying the same form-factor function as C9. A lepton-universal NP shift δC9 = δC9^R + i δC9^I is exactly degenerate with h_-^(1) → h_-^(1) − δC9, leaving every observable invariant. Thus the nonzero Im(h_-^(1)) in Fig. 4(right) can equally be read as an absorptive (weak) phase in C9. The paper’s own NP fits in Sec. IV B restrict δC9 to be real; if complex NP is admitted, the hadronic-vs-NP dichotomy collapses. Please restrict the abstract and conclusions to 'a real lepton-universal C9 shift', or extend the NP fits to complex δC9 and present the combined posterior for the degenerate direction.
  2. [§III B, Eqs. (22), (24), Table XIII] The q^2-dependence argument used in Sec. V relies on the polynomial truncation of hλ(q^2) at O(q^4) (and O((q^2)^{3/2}) for B→K) being a faithful representation of the non-local correlator. The fit attributes the low-q^2 B→K ℓℓ behaviour to Re(h_K^(2)) at about 3σ, but if the true correlator has anomalous-threshold or branch-cut structures not reproducible by a low-order polynomial, the fitted coefficients could absorb those long-distance effects in a way that biases the conclusion toward 'sizeable penguin matrix elements'. A stability test is needed, for example adding O(q^6) terms or comparing with a dispersion-relation-based parametrization (e.g. Refs. [66,98,99]), and verifying that the inference of non-zero imaginary parts and q^2 dependence survives. Without such a test, the 'strengthened evidence' in the abstract is conditional on a phenomenological assumption that is itself part
  3. [§IV B (Information Criterion) and Fig. 4(left)] The claim that 'within the Data Driven approach the NP fits all show larger IC compared to the SM case, while the situation is the opposite in the Model Dependent approach' is not quantified: no IC values, ΔIC, or posterior uncertainties are reported. Since this comparison is used to argue that hadronic effects are preferred, the IC values should be given at least for the four combinations of FF set and hadronic model. In addition, Fig. 4(left) shows that in the more conservative LQCD DM case the 95% contour for Re(h_-^(1))–Re(h_-^(2)) includes the origin; the evidence for non-zero hadronic parameters is thus largely carried either by the LCSR-informed FF choice or by the imaginary parts, which are degenerate with complex NP. The robustness of the central claim is therefore weaker than the abstract suggests.
minor comments (4)
  1. [Table XIII] Typo: 'LCQD+LCSR DM' should read 'LQCD+LCSR DM'. The colour-coded highlighting of non-zero ranges is not visible in monochrome print; consider adding a symbol or footnote.
  2. [Figs. 2–3 captions] The captions state that LCSR points at q^2 = −15, −10, −5 GeV^2 are included in the fit but not shown; this is useful, but the same sentence should make clear whether the displayed black points at q^2 = 0, 5 GeV^2 are the only ones entering the fit. Also, the vertical scale makes the low-recoil differences between blue and green bands hard to judge.
  3. [§IV B, Eq. (IC)] The definition IC_M ≡ −2⟨logL⟩ + 4σ^2_{logL} is unconventional; the relation to the standard DIC/WAIC definitions from Ref. [108] should be stated, together with the numerical values used for the comparisons.
  4. [Appendices B–D] The dense correlation matrices are valuable for reproducibility, but they are hard to read in print. Consider providing an electronic supplement with machine-readable tables, or restricting the printed matrices to correlations above a threshold.

Circularity Check

2 steps flagged

Hadronic-vs-NP conclusion relies on restricting NP C9 to real values, making the fitted complex h parameters 'genuinely hadronic' by definition.

specific steps
  1. self definitional [Sec. III B, after Eqs. (23)-(25)]
    "Notice that imaginary parts of h^(i)_λ parameters as well as real parts of h^(2)_-, h^(0,1,2)_+, h^(0,1)_0 and h^(1,2)_K do not enter the helicity amplitudes in the same way as NP contributions to C7 and C9, so they can be identified as genuine hadronic contributions without any ambiguity."

    In Eq. (23) H_-^V is proportional to (C9^SM - h^(1)_-) \tilde V_{L-}; similarly H_+^V and H_0^V contain the same combination. A complex NP shift δC9 = δC9^R + i δC9^I is therefore exactly equivalent to h^(1)_- → h^(1)_- - δC9, with all observables unchanged. Hence Im(h^(1)_-) is degenerate with Im(δC9). The assertion that imaginary parts do not enter 'in the same way as NP contributions' holds only if NP Wilson coefficients are assumed real, which is precisely the restriction used in the fits. Calling the fitted Im(h^(1)_-) 'genuine hadronic without any ambiguity' is a definitional convention, not an empirical result.

  2. fitted input called prediction [Sec. III B, Fig. 4 discussion]
    "Notice that even if Re(h^(2)_-) = 0 and Im(h^(0)_-) = 0, then data hint at Re(h^(1)_-)∼Im(h^(1)_-)≠0, disfavouring the interpretation in terms of NP only."

    In the Data Driven fit the h-parameters are complex and fitted directly to the b→sℓℓ data; the NP side of the comparison is restricted to real, lepton-universal C9 shifts, with the LFU direction C^NP_9,+ explicitly degenerate with Re(h^(1)_-) and left flat. When the fit returns Re(h^(1)_-) ~ Im(h^(1)_-) ≠ 0, this is a property of the fitted hadronic parameters. Presenting it as 'disfavouring the interpretation in terms of NP only' assumes NP cannot have the same complex phase structure. The preference for hadronic effects over a short-distance C9 shift is therefore built into the restricted NP space rather than derived from the data.

full rationale

The paper is largely transparent about its main degeneracy: it states that the Data Driven treatment 'comes at the price of giving up the possibility of identifying eventual lepton-universal NP contributions to C9' and that Re(h^(1)_-) 'can be reinterpreted as a lepton universal NP contribution, C^NP_9,U'. It also explicitly labels the non-local parametrization as phenomenological. The Dispersive Matrix form-factor determination is benchmarked against external lattice points, LCSR inputs, and an independent BGL band, and the b→sνν SM predictions depend only on local form factors, so those parts are not circular. The circularity is concentrated in the central 'hadronic rather than short-distance C9' claim. By Eq. (23), the hadronic coefficient h^(1)_- enters only in the combination C9^SM - h^(1)_-, so a complex NP shift is exactly equivalent to a shift in h^(1)_-. The paper's statement that imaginary parts are 'genuine hadronic contributions without any ambiguity' is therefore valid only under the unstated assumption of real NP C9. The NP fits in Sec. IV B indeed restrict C9 to real values, and the LFU direction is degenerate with Re(h^(1)_-) and not constrained. Consequently the strong-phase evidence, the nonzero Im(h^(1)_-), and the resulting 'disfavouring the interpretation in terms of NP only' reduce in part to the chosen definition of which parameters count as hadronic and which as NP. This is partial circularity: the genuine empirical content is that a real, q^2-independent C9 shift cannot fit the data, but the stronger abstract claim against a short-distance C9 shift with arbitrary phase is not supported and is forced by the restricted NP hypothesis. Self-citations to the authors' prior Data Driven framework are not treated as load-bearing here because the paper identifies the parametrization as phenomenological and does not invoke an external uniqueness theorem.

Axiom & Free-Parameter Ledger

3 free parameters · 5 axioms · 0 invented entities

The central claim rests on a flexible hadronic parametrization with many fitted parameters and on external lattice/LCSR inputs. There are no invented particles or forces.

free parameters (3)
  • Hadronic parameters Re/Im h_λ^(i) for B→K* and Bs→φ (h^(0)_-, h^(1)_-, h^(2)_-, h^(0)_+, h^(1)_+, h^(2)_+, h^(0)_0, h^(1 = 68% HPDIs in Table XIII; e.g. Im(h^(0)_-) = [0.02,0.12] for LQCD+LCSR
    These complex coefficients are fitted to the b→s ℓ+ℓ− data with broad priors; they are the main mechanism by which the Data Driven approach can accommodate the anomalies.
  • Hadronic parameters Re/Im h^(1)_K, h^(2)_K for B→K = Re(h^(2)_K) ≈ [4.47,7.99]×10^4 GeV^-2 (68% HPDI, LQCD+LCSR)
    Fitted to the differential branching fraction of B→K ℓ+ℓ−; the low-q^2 bin drives the nonzero value.
  • NP Wilson coefficients C9_e, C9_µ, C10_e, C10_µ (and C9'_..., C10'_... in extended fits) = e.g., C9_e in Model Dependent LQCD DM: [-1.74,-1.05] at 68% HPDI; C9_µ: [-1.24,-0.93]
    Fitted in NP scenarios; in the Data Driven approach C9_+ is flat/degenerate with Re(h^(1)_-).
axioms (5)
  • ad hoc to paper The non-local hadronic correlator hλ(q2) is adequately described by the polynomial in Eqs. (22) and (24).
    No derivation; truncation and functional form are assumed, and the paper does not quantify the truncation error.
  • ad hoc to paper NP contributions are assumed to be lepton-universal and real in the comparison (δC9, δC10 real; C9_+ degeneracy).
    This assumption is needed to conclude that non-zero imaginary parts imply hadronic, not NP, effects.
  • domain assumption The lattice QCD form-factor inputs from Refs. [70-73] and the exclusion of HPQCD 2013 [75] are reliable.
    The DM bands are built from these inputs; the paper invokes FLAG 2024 to justify the exclusion but does not test its impact.
  • domain assumption The dispersive matrix/unitarity filter applied to combined lattice data does not bias the FF bands (e.g., f0 below HPQCD point at q^2=0).
    The method is from Ref. [35]; the paper notes the f0 deviation but does not assess whether it is a selection artefact.
  • standard math Standard analyticity and unitarity for B-meson form factors (dispersion relations, BGL expansion) hold.
    Invoked throughout Sec. II.

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read the original abstract

Rare semileptonic $b \to s$ flavour-changing neutral current transitions provide stringent tests of the Standard Model. Their interpretation is limited by hadronic uncertainties, notably the $B \to K^{(*)}$ and $B_s \to \phi$ form factors (FFs) and the matrix elements of four-quark operators. We perform a global analysis of $b \to s \ell^+\ell^-$ transitions taking these uncertainties fully into account, determining the FFs through the Dispersive Matrix method and comparing a setup based solely on lattice QCD (LQCD) with one that also includes light-cone sum-rule (LCSR) inputs at low $q^2$. Compared to the case where both input are taken into account, using only LQCD substantially enlarges the FF uncertainties at large recoil. Combined with the latest LHCb and CMS angular measurements sensitive to strong phases, our global fit yields strengthened evidence in favour of long-distance hadronic effects rather than a short-distance shift in $C_9$. We further present new SM predictions for the theoretically clean $b \to s \nu\bar\nu$ modes, which depend only on local FFs, and a New Physics analysis of these transitions in the Weak Effective Theory, discussing their impact on the interpretation of the recent Belle~II measurement and on the available experimental upper bounds.

Figures

Figures reproduced from arXiv: 2607.03531 by Ayan Paul, Josua Scholze, Luca Silvestrini, Ludovico Vittorio, Marco Ciuchini, Marco Fedele, Mauro Valli, Silvano Simula.

Figure 1
Figure 1. Figure 1: As a comparison, in the same figure we report also as orange bands the results obtained in Ref. [65]. The [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Comparison of the FFs entering the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Same as Figure 2 for the [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Left Panel: Joint posterior p.d.f. for Re( [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Two-dimensional marginalized posterior distribution in the ( [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Two- and one-dimensional marginalized posterior distribution in the ( [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Two- and one-dimensional marginalized joint p.d.f. for the set of WCs [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Two- and one-dimensional marginalized joint p.d.f. for the set of 4 WCs [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Two- and one-dimensional marginalized joint p.d.f. for the set of 6 WCs [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Two- and one-dimensional marginalized joint p.d.f. for the set of 8 WCs [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Disentangling short- vs. long-distance dynamics in $B\to K^{*}\mu^+\mu^-$

    hep-ph 2026-07 conditional novelty 5.0

    Dispersive resonance modeling of non-local effects in B o K*μ+μ- reduces the C9 anomaly to ≤2σ while yielding precise postdictions for S7,8,9.

Reference graph

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