REVIEW 2 major objections 4 minor 30 references
Long-distance effects fixed by data bring the short-distance coefficient C9 in B to K* mu mu to within about 2 sigma of the Standard Model.
Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →
T0 review · grok-4.5
2026-07-13 01:20 UTC pith:H2DY2RE3
load-bearing objection Solid data-driven update that brings the C9 tension down to ~2σ with a transparent dispersive model; the resonance-saturation assumption is the real soft spot, but the low-q^{2} validation and S7,8,9 postdictions are useful. the 2 major comments →
Disentangling short- vs. long-distance dynamics in Bto K^(*)μ^+μ^-
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
After the long-distance contributions are fixed by a dispersive resonance model whose parameters come from the global amplitude fit, a bin-by-bin extraction of the short-distance coefficient yields a constant, helicity-independent C9 = 3.75 +0.13/-0.12. Compared with the Standard Model value 4.27 this produces Delta C9 = 0.52 +/- 0.25, a tension that does not exceed the 2-sigma level. The same fit supplies precise postdictions for the absorptive angular observables S7,8,9 that are already tighter than present measurements.
What carries the argument
A once-subtracted dispersion relation for the non-local four-quark matrix elements, saturated by a finite sum of single-particle vector-meson Breit-Wigner resonances whose couplings and phases are taken from the global amplitude analysis; this converts the long-distance pieces into a q2- and helicity-dependent shift of C9 that can be subtracted bin by bin.
Load-bearing premise
The spectral functions that feed the dispersion relation are assumed to be completely saturated by a handful of single-particle vector-meson resonances with constant widths, ignoring multi-particle continua and coupled-channel dynamics above the open-charm threshold.
What would settle it
A future high-precision measurement of S7, S8 and S9 in the low-q2 region that systematically violates the paper's postdictions (or the approximate sum rule that links the three observables) would falsify the dispersive saturation assumption and reopen the extraction of C9.
If this is right
- The residual ~2-sigma tension in C9 can no longer be attributed solely to unconstrained long-distance effects of the type already included.
- Precise future data on S7,8,9 become direct tests of the resonance-saturation hypothesis rather than free parameters of a polynomial fit.
- The same dispersive subtraction can be applied to related modes such as Bs to phi mu mu, allowing a cross-channel consistency check of short-distance physics.
- If the postdictions hold, analyses that previously floated unconstrained non-local parameters will report a systematically smaller |Delta C9|.
Where Pith is reading between the lines
- The same resonance-saturated dispersion relation could be used as a prior in global SMEFT fits of b to s ell ell data, reducing the present degeneracy between short-distance Wilson coefficients and hadronic pollution.
- If analogous dispersive analyses of B to K mu mu and inclusive b to s ell ell continue to prefer a lower C9, the residual discrepancy would point more cleanly toward new short-distance physics rather than mode-specific long-distance effects.
- The sum rule among S7,8,9 offers an internal experimental consistency check that does not require any theoretical model of rescattering, and could already be applied to existing unbinned datasets.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper presents a data-driven analysis of short- versus long-distance dynamics in B o K*μ+μ- using the recent LHCb full angular distribution [15] and the global amplitude fit [16]. Non-local contributions are parameterized via a once-subtracted dispersion relation for the hadronic functions H_λ(q^{2}), with the spectral density saturated by a finite sum of vector-meson Breit–Wigner resonances (five charmonia plus ρ,ω,φ). Resonance couplings and phases are taken from [16] and held fixed (within errors) while a short-distance C9 is extracted bin-by-bin and under a constant-C9 hypothesis. The global fit yields C9=3.75+0.13-0.12, so that ΔC9=CSM9-Cfit9=0.52±0.25 (Eq. 5.3), a tension with the SM that does not exceed ~2σ. The same framework produces precise postdictions for S7,8,9 and an approximate sum rule among them.
Significance. If the resonance-saturation hypothesis holds at the claimed precision, the work supplies a concrete, falsifiable reduction of the long-standing C9 anomaly and a set of sharp postdictions for S7,8,9 that future data can test. The bin-by-bin compatibility of Cλ,bin9 with a constant (Figs. 5.1–5.2, Table 5.1), the explicit error budget for the subtraction point, and the sum-rule relation among S7,8,9 are genuine strengths. The analysis also clarifies how much of the previous tension can be absorbed by long-distance effects once they are constrained by data rather than left free.
major comments (2)
- [Sec. 5.2, Eq. (5.3), Table 5.1] The headline result ΔC9=0.52±0.25 (Eq. 5.3) is obtained from the global constant-C9 fit that includes the high-q^{2} bins ([11–12.5] and [15–19] GeV^{2}). Section 5.2 and Table 5.1 already show that the constant-C9 hypothesis is only marginally acceptable in that region (p=0.07, χ^{2}/dof=1.28), and the paper itself notes that a sum of constant-width Breit–Wigners is inadequate above the open-charm threshold (citing Ref. [38]). Because any residual continuum or coupled-channel contribution not absorbed into the resonance parameters can systematically shift the extracted short-distance coefficient, the all-bin number is less robust than the low-q^{2}-only result (C9=3.78+0.17-0.16). The manuscript should either (i) promote the low-q^{2} determination as the primary result or (ii) quantify the possible bias from uncontrolled continuum contributions above ~4 GeV.
- [Sec. 5.1, Appendix A, comparison with [16]] Resonance parameters η Vλ and δ Vλ are taken from the LHCb global amplitude fit [16] that already employs the same dispersive functional form. While the authors correctly exclude the narrow-resonance peaks from their own fit and treat the parameters as external inputs, residual correlations remain and the procedure is not fully independent. A short quantitative assessment of how much the extracted C9 would move if the resonance parameters were varied within a broader, model-independent envelope (or re-determined from an independent data set) would strengthen the claim that the subtraction is data-driven rather than circular.
minor comments (4)
- [Sec. 5.1] The choice Im(Ceff7)=-0.05±0.02 is motivated a posteriori by the first bin of S7 and by theory estimates. While the impact on Re(C9) is stated to be small, a one-sentence sensitivity study (or a brief table entry) showing ΔC9 for Im(Ceff7)=0 and for the pure theory value -0.027 would make the robustness transparent.
- [Fig. 3.1] Figure 3.1 would be clearer if the three kinematic regions discussed in the text were shaded or labeled directly on the plot rather than only described in the caption.
- [Sec. 4, Eq. (4.4)] The approximate sum rule (4.4) is a useful consistency check; it would help the reader if the paper showed the numerical residual of the left-hand side when evaluated on the experimental S7,8,9 (or on the postdictions) in a few representative bins.
- [Abstract, Refs. [15]] Typographical consistency: the abstract and introduction use both “non-local” and “nonlocal”; pick one. Also, the arXiv identifier of the companion LHCb angular paper [15] appears as 2512.18053, which looks like a future date—please verify.
Circularity Check
Resonance parameters ηλV, δλV taken from LHCb amplitude fit that already assumes the same dispersive Breit–Wigner form are fixed and used to subtract LD; constancy of extracted C9 and S7,8,9 postdictions are then presented as validation of that form. Im(Ceff7) is likewise tuned on low-q2 S7.
specific steps
-
fitted input called prediction
[Sec. 2.2 (hypothesis ii), Eq. (2.17), Appendix A, Sec. 5.1 Eq. (5.1), Sec. 5.2]
"Following the data analysis presented in Ref. [16], we limit this sum to five charmonium resonances and the three lightest vector resonances (ρ, ω, ϕ). The coefficients ηλV and δλV, which are determined by data, are reported in Appendix A. … we have fitted the binned data by means of the amplitude (2.5) with the substitution C9 o Ceff,λ9(q2) = Cλ,bin9 + Y(q20) + ΔCλ9(q2)res … the resonance parameters are included as external inputs … The fact that this does not happen … is a non-trivial consistency check of the approach."
ηλ V, δλ V are extracted by LHCb under precisely the same once-subtracted resonance-saturated dispersion relation that the present paper then holds fixed while claiming to ‘validate’ the model by the resulting constancy of C9. The subtraction and the validation therefore share the same fitted inputs; any residual continuum not absorbed into those parameters is invisible by construction.
-
fitted input called prediction
[Sec. 5.1 ‘Determination of Im(Ceff7)’, Sec. 5.3, Fig. 5.4]
"we adopt the reference value Im(Ceff7) = -0.05 ± 0.02. This figure is chosen such that … it covers both the central value of the theoretical estimate … and the central value … obtained by LHCb [16] (-0.07). The consistency of this choice is supported by the quality of the fit in the low q2 region, whose p-value increases from p = 0.25 for Im(Ceff7) = 0 to p = 0.70 for Im(Ceff7) = -0.05 ± 0.02. As expected, this improvement is driven primarily by the different χ2 contribution associated with S7 … we obtain precise postdictions for these quantities [S7,8,9]"
Im(Ceff7) is adjusted so that the low-q2 S7 data are better described; the same S7 (and the related S8,9) are then presented as precise postdictions of the dispersive framework. The low-q2 absorptive prediction is therefore partly a re-statement of the tuned input.
full rationale
The paper’s central claim (ΔC9 ≈ 0.52 ± 0.25, tension ≲ 2σ) rests on subtracting non-local contributions whose free parameters are taken from an external but same-ansatz global fit (LHCb [16]). Because the functional form of Yλ(q2) is identical, the subsequent observation that Cλ,bin9 is approximately constant (and that S7,8,9 are well postdicted) is partly forced by construction rather than an independent first-principles test. The low-q2 region still supplies a non-trivial consistency check (resonance peaks are excluded and free C9 per bin/helicity remains flat), so the circularity is only partial; the high-q2 continuum limitation is a separate correctness issue, not circularity. No uniqueness theorem or pure self-definition is involved. Score 5 reflects one clear fitted-input-as-prediction loop plus a secondary tuning of Im(Ceff7) on the same observable later postdicted.
Axiom & Free-Parameter Ledger
free parameters (4)
- C9 (constant short-distance Wilson coefficient) =
3.75^{+0.13}_{-0.12}
- Im(C7eff) =
-0.05 ± 0.02
- Y(q0^{2}) subtraction constant and its theory uncertainty =
0.13 ± 0.21
- Resonance couplings η Vλ and phases δ Vλ (8 resonances imes 3 helicities) =
see Tables A.1–A.3
axioms (4)
- domain assumption Analyticity of the non-local hadronic functions Hλ(q^{2}) with a single cut for q^{2} ≥ 4mπ^{2}, allowing a once-subtracted dispersion relation.
- ad hoc to paper Spectral density ρλ(s) is dominated by single-particle vector-meson intermediate states (finite sum of Breit–Wigner resonances with constant widths).
- domain assumption Yλ(q0^{2}) at the subtraction point q0^{2} = -4.6 GeV^{2} can be computed in heavy-quark perturbation theory (NLO) and is helicity-independent.
- domain assumption Local B o K* form factors are taken from light-cone sum-rule determinations (Bharucha–Straub–Zwicky) and treated as fixed central values.
read the original abstract
We present an updated data-driven analysis of short- vs. long-distance dynamics in $B\to K^{*}\mu^+\mu^-$. The analysis is based on the recent LHCb measurement of the full angular distribution of this process, taking advantage of the previously published global amplitude fit. Thanks to the precise di-lepton invariant-mass binning and angular observables directly sensitive to absorptive phases, current data provide novel insights into non-local contributions. We analyze the latter using a dispersive description, whose free parameters are determined via the amplitude fit. The dispersive model provides a satisfactory overall description of current data, without the need to introduce additional ad hoc hadronic parameters. The extracted value of the short-distance coefficient $C_9$ is lower than its Standard Model prediction, but significantly closer to it than in analyses where long-distance contributions are not constrained by data. The overall tension with the SM does not exceed the $2\sigma$ level. We also obtain precise postdictions for the angular observables $S_{7,8,9}$, and envisage novel consistency tests among them. In view of future, more precise data, these could provide valuable tests of the description of non-local effects in this process that, if further validated, would enable an even more robust extraction of short-distance information.
Figures
Reference graph
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discussion (0)
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