Dispersive analysis of the pion vector form factor without zeros

Peter Stoffer; Thomas P. Leplumey

REVIEW 2 major objections 3 minor 1 cited by

Dispersive analysis of the pion vector form factor without zeros

T0 review · 2 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read By imposing the absence of complex zeros on the pion vector form factor, this paper shows the largest systematic error in dispersive fits disappears, the data do not want zeros, and the CMD-3 versus KLOE tension in the muon g−2 hadronic…

desk verdict A careful, genuinely useful dispersive analysis that sharpens the CMD-3 vs. other-data tension, provided you buy the zero-free assumption it openly defers. read the letter →

arxiv 2501.09643 v1 pith:ZJ5IBDQH submitted 2025-01-16 hep-ph hep-exhep-latnucl-th

Thomas P. Leplumey , Peter Stoffer This is my paper · ORCID

classification hep-phhep-exhep-latnucl-th PACS 13.40.Gp14.60.Ef

keywords pionvectorformfactordispersiverepresentationOmnèsfunctioncomplexzeroshadronicvacuumpolarizationmuonanomalousmagneticmomentchargeradiuse+e-annihilation

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 biggest theoretical uncertainty in dispersive fits of the pion vector form factor comes from unconstrained complex zeros in the inelastic conformal polynomial, and that imposing their absence removes this uncertainty without hurting fit quality. The authors show that the available $e^+e^-\to\pi^+\pi^-$ data are compatible with a zero-free form factor, and that under this assumption the discrepancies between CMD-3 and the other data sets grow, with the CMD-3–KLOE tension reaching $8.9\sigma$ in the two-pion contribution to the muon anomalous magnetic moment. This matters because it turns the pion charge radius into a sharp discriminator between data sets and offers an independent lattice-QCD cross-check of the hadronic vacuum polarization puzzle.

What carries the argument

The machinery is the dispersive representation $F_V^\pi(s)=\Omega_1^1(s)\,G_\omega(s)\,G_{\mathrm{in}}(s)$, where $\Omega_1^1$ is the Omnès function built from the Roy-equation $\pi\pi$ P-wave phase shift, $G_\omega$ describes $\rho$–$\omega$ mixing, and $G_{\mathrm{in}}$ is a conformal polynomial for inelastic states above the $\pi^0\omega$ threshold. Zeros can appear only in $G_{\mathrm{in}}$; the paper excludes them either by factorizing $G_{\mathrm{in}}$ as $Q_{r,n}(z)P_n(z)$ with roots restricted to the unit circle, or by adding a sum-rule penalty $\chi^2_{\mathrm{zeros}}$ derived from the unsubtracted dispersion relation for $\psi(s)=(s_{\mathrm{thr}}-s)^{-3/2}\log[F_V^\pi(s)/F_V^\pi(s_{\mathrm{thr}})]$. A hybrid phase-modulus representation reconstructs $G_{\mathrm{in}}$ from its modulus via a modulus dispersion relation, with sum-rule corrections fixing the asymptotic behavior; both methods agree, and the zero-free constraint is what removes the dominant $N$-variation systematic.

What would settle it

A high-precision experiment or lattice calculation that finds a complex zero of $F_V^\pi(s)$ at a phase not excluded by Refs. [81,90]—for example, by measuring the space-like form factor and comparing it with the zero-free analytic continuation—would falsify the constraint. More directly, if a fit that allows zeros achieved a significantly better p-value than the constrained fit for any single data set when correlations are fully accounted for, the claim that the data do not prefer zeros would fail.

Watch

Extended reading notes

Core claim

The central discovery is that the large systematic uncertainty in earlier dispersive analyses, caused by varying the order of the conformal polynomial describing inelastic effects, is tied to the appearance of complex zeros in the form factor. Using two implementations—an explicit zero-free parametrization and a sum-rule penalty—the fits stabilize, the p-values remain essentially unchanged, and the systematic uncertainty on $a_\mu^{\pi\pi}$ drops dramatically. The data therefore do not prefer zeros. Consequently, the constraints of analyticity and unitarity sharpen rather than dilute the experimental discrepancies: CMD-3 disagrees with KLOE at $8.9\sigma$ in $a_\mu^{\pi\pi}|_{\le 1\,\mathrm{GeV}}$, and the pion charge radius becomes a discriminating observable, with a combined-fit value $\langle r_\pi^2\rangle=0.4290(17)\,\mathrm{fm}^2$ versus $0.4367(24)\,\mathrm{fm}^2$ from CMD-3 alone.

Load-bearing premise

The pion vector form factor has no complex zeros, so imposing the zero-free constraint does not bias the fits.

Editorial extensions

If this is right

The constraint lowers the systematic uncertainty on $a_\mu^{\pi\pi}|_{\le 1\,\mathrm{GeV}}$ enough that dispersive fits become sharper than direct integration of cross-section data outside the very low-energy region.
The CMD-3 versus KLOE tension in $a_\mu^{\pi\pi}|_{\le 1\,\mathrm{GeV}}$ reaches $8.9\sigma$ in constrained fits; discrepancies persist in Euclidean windows, including the very-long-distance window with $t>2.8\,\mathrm{fm}$.
The pion charge radius becomes a useful discriminator: the combined-fit value $0.4290(17)\,\mathrm{fm}^2$ differs from the CMD-3-only value $0.4367(24)\,\mathrm{fm}^2$, and future lattice-QCD determinations of $\langle r_\pi^2\rangle$ can probe the discrepancy independently of full hadronic vacuum polarization computations.
The hybrid representation shows that CMD-3 data above 1 GeV are in tension with BaBar data above 1.4 GeV, reflected in a hybrid fit p-value of $0.9\%$ compared with roughly $20\%$ for the low-energy Omnès fit.
The combined-fit two-pion contribution $a_\mu^{\pi\pi}=504.7(3.5)\times10^{-10}$ agrees with direct-integration evaluations, while the CMD-3-only value $521.8(1.7)\times10^{-10}$ remains much higher.

Reading between the lines

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

If the zero-free assumption holds and future lattice-QCD charge-radius determinations reach percent-level precision, the pion charge radius could arbitrate between CMD-3 and the older data sets before the full hadronic vacuum polarization discrepancy is resolved.
The same zero-free constraint could be applied to other form factors, such as the kaon form factor, where conformal-polynomial inelastic parametrizations suffer similar order-variation systematics.
A direct test of the zero-free assumption could come from precise space-like measurements: the zero-free fits fix the analytic continuation to negative $s$, so a high-precision NA7-like experiment would confirm or exclude the predicted slope and curvature.
The enhanced $8.9\sigma$ KLOE–CMD-3 tension suggests the underlying systematic difference is smooth and broad in energy rather than localized at the $\rho$ peak, so any radiative-correction model proposed to explain it must reproduce that smooth shape.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The real news here is that imposing a zero-free VFF kills the dominant systematic in the dispersive fits—the N-dependence of the conformal polynomial—and leaves fit quality almost unchanged. That is a concrete, nontrivial payoff. The constrained fits stabilize across N, the hybrid phase-modulus representation stays compatible, and the CMD-3 vs. KLOE tension in a_μ(ππ, ≤1 GeV) climbs to 8.9σ. The charge radius becoming a discriminating observable is also a fair consequence, not an afterthought.

The paper does its homework. Uncertainty budgets are separated into fit, systematic, and BaBar-KLOE components; two independent representations are cross-checked; the inelasticity parameter gets a proper Bayesian treatment; and the CMD-3 correlation structure is stress-tested with a de-correlation toy model. That is solid, reproducible-style phenomenology, though no code or data files are released, which limits independent checks.

The soft spot is exactly the one the paper flags. The zero-free assumption is the load-bearing wall, and it is not proven here—it rests on Leutwyler's old arguments and exclusion regions from Refs. [81,90], with a thorough investigation deferred to future work [101]. The stress-test has it right: unconstrained fits allowing zeros have similar p-values, so the data are consistent with no zeros but do not exclude zeros in all phases. If a zero sits in an allowed region, the constrained fits are biased, the N-variation no longer covers that bias, and the 8.9σ becomes an overstatement. I do not see this as a fatal flaw—the paper is honest about the conditionality—but it is the difference between the results as presented and results that would be fully robust. Also minor: the combined fit excludes SND20 and CMD-3, and the hybrid fits pull low-energy results toward BaBar above 1.4 GeV; these choices are explained and should be kept in mind when quoting the central values.

Who gets value: anyone working on the e+e− → π+π− cross-section puzzle, muon g−2, or dispersive form-factor methods. This deserves a serious referee and likely publication after the zero-free question is either addressed or explicitly framed as an assumption with a bounded systematic error. I would engage with it, and I would cite it once the assumption is either proven or its domain of validity is mapped.

Referee Report

2 major / 3 minor

Summary. The paper presents an updated dispersive analysis of the pion vector form factor using high-statistics e+e- -> pi+pi- data. Two representations are used: a low-energy Omnes representation with a conformal-polynomial inelastic factor, and a hybrid phase-modulus representation extending up to 3 GeV. The key new ingredient is the imposition of the absence of complex zeros of the form factor, implemented either through an explicit zero-free parametrization (Eq. 2.8) or through a sum-rule penalty (Eq. 2.17). The authors find that, under this assumption, the dominant systematic uncertainty associated with varying the conformal-polynomial order is drastically reduced, while fit quality and central values remain largely unchanged. Based on the constrained fits, the dispersive constraints sharpen the discrepancies between different e+e- data sets in the hadronic vacuum polarization contribution to the muon anomalous magnetic moment, reaching 8.9 sigma between KLOE and CMD-3. Tensions are also seen in Euclidean windows, including a very-long-distance window, and in the pion charge radius, which the authors propose as a future discriminant using lattice QCD.

Significance. If the zero-free assumption is valid, the paper represents a genuine methodological advance: it identifies the appearance of complex zeros as the source of the previously dominant systematic uncertainty and demonstrates a stable, data-compatible way to remove that uncertainty. The two independent representations, the detailed uncertainty budget separating fit, systematic, and BaBar-KLOE tension components, and the explicit N-dependence tables are strengths of the manuscript. However, the central quantitative claims are conditional on an assumption that the paper itself defers to future work, and the unconstrained fits do not exclude zeros at phases or positions not covered by existing bounds. The paper is careful in most places to say 'under this assumption', but the headline results, including the sharpened discrepancy claims, inherit that conditionality.

major comments (2)

[Sec. 2.2, Eqs. (2.8), (2.17); Tables 1-3] The absence of complex zeros is imposed, not established. The p-value comparisons in Table 1 show only that the data are compatible with the zero-free constraint; they do not exclude zeros in regions of the complex plane not already covered by Refs. [81,90]. Since the largest systematic uncertainty is declared eliminated under this assumption and the discrepancy claims in Table 3 are correspondingly sharpened, a quantitative stress test is needed: for example, scan zero positions over the currently allowed region, recompute the central values and uncertainties for the derived observables, and show that the constrained results are stable or quantify the resulting bias. Alternatively, the constrained results should be presented as a scenario and the unconstrained results kept as the primary uncertainty estimate. As written, the central claim is load-bearing on a conjecture that is explicitly deferred to Ref. [101].
[Sec. 5.3, Table 5] The very-long-distance window is presented as evidence that the tensions 'persist even at very long distances', but the authors state that this window is correlated a posteriori by more than 80% with the intermediate window in their fits. The VLD result is therefore a derived consequence of the same low-energy fit rather than an independent confirmation. The correlation is disclosed, but the wording in the abstract and conclusions should be qualified so that the VLD window is not read as an independent probe of the discrepancy.

minor comments (3)

[Sec. 3, Eq. (3.4)] The notation sa in Eq. (3.4) conflicts with the earlier definition sa = (1 GeV)^2 in Eq. (2.7); in the modulus representation sa is used as a generic threshold. Please rename one of them to avoid confusion.
[Sec. 3 and Sec. 5.1] There are typos: 'under the asumption' in Sec. 3 and 'Is is important' in Sec. 5.1 should be corrected.
[Sec. 5.4, Table 9] The charge-radius discrepancy significances depend strongly on whether symmetric or asymmetric N-variation errors are used, as shown by the curly-bracket entries in Table 9. The main text should state more prominently that this choice, rather than the data themselves, drives the reported discriminating power of the charge radius in the low-energy fits.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's outputs are data-driven determinations, and the zero-free assumption is transparently conditional.

full rationale

No circular step is present. The paper fits dispersive representations of the pion vector form factor to e+e- cross-section data; the derived quantities (a_mu^pi pi and the pion charge radius) are integrals or derivatives of the fitted function. These are determinations from the same data, not independent predictions, and the paper consistently frames them as such ('allows us to determine', 'direct evaluation'), so there is no fitted input renamed as a prediction. The central zero-free assumption is imposed as a constraint, motivated by external references [79,81,90], and the paper explicitly states that a thorough proof is left to future work [101]. The claim that the data are compatible with the assumption is supported by the fits themselves: constrained and unconstrained fits have similar p-values, while constrained results are more stable in N. This is an empirical finding, not a tautology. Self-citations [39,49,55,70] provide the previously developed dispersive framework, but the current results are obtained from fits performed here and are compared against external benchmarks (DHMZ, KNT, lattice, PDG). Thus the derivation chain is self-contained and the limitation is openly conditional rather than circular.

Assumptions & free parameters 8 free parameters · 6 assumptions · 0 invented entities

The analysis rests on standard dispersive machinery and openly fitted parameters. The paper's own new assumptions are the zero-free condition and the GS-model-dependent modulus extension. No new physical entities are introduced.

free parameters (8)

Conformal polynomial coefficients c_k = not tabulated; N-1=3 central constrained fits
Parametrize inelastic effects in G_in; fitted to e+e- data; order N and coefficients are varied for systematics.
Roy-solution phase-shift parameters delta_1^1(s0) and delta_1^1(s1) = e.g. 110.3(1)(5) deg and 165.8(0.1)(2.0) deg in combined fit
Free parameters in each fit that adjust the elastic pi pi phase shift input.
Omega meson mass M_omega = 782.07(12)(1) MeV combined fit
Fitted in the rho-omega mixing factor; correlated with the mixing phase.
Rho-omega mixing parameter epsilon_omega = 10^3 Re eps = 1.99(2)(1), delta_eps = 3.8(9)(4) deg combined fit
Complex parameter fitted to reproduce the omega lineshape and radiative channels.
Per-experiment energy rescaling factors = not reported
One factor per data set, constrained by experimental calibration uncertainty, absorbing normalization and scale shifts.
Order N of the conformal polynomial (model choice) = N-1=3 central for constrained fits; varied N-1=2..5 for systematics
Higher orders introduce complex zeros in unconstrained fits; the selection strongly affects systematic error bars.
Inelasticity parameter iota_1 (Bayesian nuisance) = prior 0.05(5), posterior sampled
Treated with a Gaussian prior and integrated out via the posterior distribution to avoid fixed-value bias.
Gounaris-Sakurai parameters in hybrid fits = not tabulated; M_rho and Gamma_rho use priors from a pure GS fit
Used to interpolate the modulus above the inelastic threshold; model-dependent parameters.

assumptions (6)

domain assumption The I=1 elastic pi pi P-wave phase shift is accurately described by Roy-equation solutions.
Used for the Omnes factor; relies on Refs. [94,96]; any deficiency propagates to all fits.
standard math Unitarity and analyticity justify the product decomposition Eq. (2.1) and the Omnes representation Eq. (2.2).
Standard dispersive factorization of the vector form factor.
standard math Watson theorem fixes the VFF phase below the inelastic threshold to the elastic pi pi phase shift.
Used to identify the phase of the Omnes factor; standard result of unitarity.
ad hoc to paper The pion vector form factor has no complex zeros.
Imposed as a constraint following Ref. [79] and Refs. [81,90]; a full investigation is deferred to future work [101].
domain assumption The Eidelman-Lukaszuk bound constrains the inelastic phase as in Eq. (2.7).
Implemented as a chi^2 penalty; the bound is model-independent but its specific form is an input.
ad hoc to paper The modulus dispersion relation Eq. (3.3) reconstructs the function from its modulus, and the GS parametrization Eq. (3.2) describes the modulus above the inelastic threshold.
Used only for the hybrid representation; any GS model bias affects the high-energy continuation and charge radius.

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Pith. "Pith review of Dispersive analysis of the pion vector form factor without zeros." pith.science (2026). https://pith.science/paper/ZJ5IBDQH

@misc{pith2026250109643,
  author       = {Pith},
  title        = {Pith review of: Dispersive analysis of the pion vector form factor without zeros},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZJ5IBDQH}},
  note         = {Machine review of arXiv:2501.09643}
}

abstract

We perform an updated analysis of $e^+e^-\to\pi^+\pi^-$ cross-section data using a dispersive representation of the pion vector form factor. We show that the available data are compatible with the assumption that the form factor is free of complex zeros and that under this assumption the largest systematic uncertainty in a previous analysis can be eliminated. We investigate both a constrained Omn\`es representation as well as a hybrid phase-modulus representation and we quantify the discrepancies in the hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon based on different $e^+e^-$ data sets. We find that the dispersive constraints exacerbate these discrepancies. Together with the assumption of the absence of zeros, the pion charge radius becomes a useful observable to discriminate between the different data sets. This provides an opportunity for future improved lattice-QCD determinations to probe the discrepancies independently of full computations of hadronic vacuum polarization. We also reevaluate the two-pion contribution to Euclidean windows and we observe that systematic discrepancies between the data sets persist even at very long distances.

Figures

Figures reproduced from arXiv: 2501.09643 by the authors.

**Figure 1.** Values of a ππ µ |≤1GeV from the unconstrained (above) and constrained (below) low-energy fits to single experiments. The smaller error bars show the fit uncertainties while the larger error bars show the full uncertainties including both fit uncertainties and systematic uncertainties. The gray bands correspond to the combined fit to all e +e − data sets (apart from SND20 and CMD-3) and the NA7 data set, with the la… view at source ↗

**Figure 2.** Values of a ππ µ |≤1GeV from the hybrid fits to single experiments and BaBar data above 1.4 GeV. The results of the low-energy constrained fits are shown for comparison as transparent points with dotted error bars. See also [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Values of a ππ µ |[0.6;0.9] GeV from the constrained low-energy fits to single experiments (circles with plain error bars), compared to the evaluations of DHMZ [51] (diamonds with dotted error bars) and KNT [52] (squares with dashed error bars). The direct-scan result in our fits includes SND06, CMD-2, and the NA7 data sets. See also [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Values of a ππ µ |≤1.8 GeV from the hybrid fits to single experiments and BaBar data above 1.4 GeV. Error bars and combination are defined as in [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

**Figure 5.** Figure 5: Two-pion contribution to the VLD window in Euclidean time (t > 2.8 fm), obtained from the constrained low-energy fits. We observe that the constraints of unitarity and analyticity imply that discrepancies between the different e +e − experiments persist even in the VLD…

**Figure 6.** Figure 6: Values of ⟨r 2 π⟩ (in fm2 ) from the low-energy unconstrained and constrained fits and from the hybrid fits to single experiments. Error bars and combination are defined as in [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]

**Figure 7.** Figure 7: Result for a ππ µ |≤1 GeV from the constrained low-energy fits: the filled dots with solid error bars show our central result, while the transparent squares with dotted error bars are the results obtained by taking into account only diagonal systematic errors. α = 0 α …

**Figure 8.** Figure 8: Modified systematic covariance matrices for CMD-3 used to check the effect of reducing correlations. 6 Conclusions In this work, we have presented an updated dispersive analysis of the pion VFF F V π (s), building on top of previous work [39, 49, 55, 70]. We have ident…

**Figure 9.** Figure 9: Result for a ππ µ |≤1 GeV in the constrained low-energy fit to CMD-3 including fit uncertainty, inflated by p χ2/dof, as a function of the decorrelation parameter α. dispersion relation that combines the knowledge about the elastic phase from solutions of the Roy equat…

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