Extracting the chiral anomaly from $e^+e^-\to 3\pi$

Bai-Long Hoid; Bastian Kubis; Martin Hoferichter

REVIEW 3 major objections 5 minor 3 cited by

Extracting the chiral anomaly from $e^+e^-\to 3\pi$

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper claims that a dispersive fit to $e^+e^-\to 3\pi$ data extracts the chiral anomaly strength $F_{3\pi}$ at about 5% precision, in agreement with the Wess–Zumino–Witten prediction.

desk verdict First extraction of the chiral anomaly from e+e−→3π data is a useful consistency check, but the 5% precision claim is softer than it looks because the conformal-polynomial expansion has not demonstrably converged. read the letter →

arxiv 2504.13827 v2 pith:57VUYGSA submitted 2025-04-18 hep-ph hep-exhep-latnucl-th

Martin Hoferichter , Bai-Long Hoid , Bastian Kubis This is my paper · ORCID

classification hep-phhep-exhep-latnucl-th

keywords chiralanomalyWess–Zumino–Wittene+e−→3πdispersiverepresentationKhuri–Treimanformalismpiondecayconstanthadronicvacuumpolarizationmuong−2

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 aims to convert measurements of $e^+e^-\to 3\pi$ into a precision test of the chiral (Wess–Zumino–Witten) anomaly, which predicts the strength $F_{3\pi}$ of the $\gamma^*\to 3\pi$ coupling from the pion decay constant. Using a dispersive representation in which the anomaly enters as a subtraction constant tied through a sum rule to the integral of the amplitude's imaginary part, the authors fit cross-section data from SND, CMD-2, and BaBar and obtain $F_{3\pi}=33.1(1.7)\,\mathrm{GeV}^{-3}$, agreeing with the chiral prediction at the $5\%$ level. If the extraction is sound, it validates the anomaly prediction at a precision comparable to existing low-energy measurements, and it supports the common practice of imposing the anomaly as a constraint in dispersive calculations of hadronic processes.

What carries the argument

The machinery is a dispersive (Khuri–Treiman) representation of the $\gamma^*\to 3\pi$ amplitude, writing the scalar function $F(s,t,u;q^2)$ in terms of partial-wave amplitudes and a subtraction function $a(q^2)$. The anomaly enters through the subtraction constant $\alpha_A=C_A/3$ in $a(q^2)$, and the load-bearing identity is the sum rule (2.11), which is imposed at every fit step and makes the $\omega$, $\phi$, and $\omega'$ resonances plus a truncated conformal polynomial carry the anomaly information. The formalism includes $\rho$–$\omega$ mixing through a correction factor and dispersively improved Breit–Wigner parameterizations for the resonances.

What would settle it

Evaluate the sum rule $\alpha_A=\frac{1}{\pi}\int ds'\,\mathrm{Im}\,a(s')/s'$ directly from the measured cross section over the full energy range, without assuming the fitted parameterization, and compare the resulting $\alpha_A$ with $C_A/3$ from the fit; a disagreement larger than the quoted $5\%$ uncertainty would falsify the extraction, as would a future high-statistics Belle II measurement whose fitted $C_A/C_A^{\mathrm{WZW}}$ excludes $1.028$ by more than the combined errors.

Watch

Extended reading notes

Core claim

The central claim is that the chiral anomaly can be extracted from $e^+e^-\to 3\pi$ data as a global fit parameter rather than imposed as an input. In the dispersive framework the subtraction constant $\alpha_A=C_A/3$ is fixed by the sum rule $\alpha_A=\frac{1}{\pi}\int_{s_{\mathrm{thr}}}^{\infty} ds'\,\mathrm{Im}\,a(s')/s'$, so the anomaly becomes a property of the whole measured cross section rather than only its low-energy threshold. Fitting $C_A$ to SND, CMD-2, and BaBar data yields $C_A/C_A^{\mathrm{WZW}}=1.028(15)(48)(17)[53]$, corresponding to $F_{3\pi}=33.1(1.7)\,\mathrm{GeV}^{-3}$, consistent with the WZW prediction within $5\%$. Fits to recent Belle II data instead show tensions with the dispersive constraints, lower $\omega$ and $\phi$ widths, and an upward pull in both $a_\mu^{3\pi}$ and the extracted anomaly, which the paper interprets as a problem in the interpretation of that data set.

Load-bearing premise

The extraction depends on the assumption that the fitted $\omega$, $\phi$, and $\omega'$ resonances plus a truncated conformal polynomial capture all of the dispersive strength in the measured region, so any missing high-energy contribution or wrong energy dependence of the polynomial would shift the extracted anomaly value.

Editorial extensions

If this is right

The WZW prediction for $\gamma\to 3\pi$ is supported at $5\%$ precision, making $F_{3\pi}$ a reliable input for dispersive hadronic light-by-light and $e^+e^-\to 3\pi$ cross-section calculations entering the muon $g-2$.
The anomaly can be left as a free parameter in future $e^+e^-\to 3\pi$ fits, turning every new data set into an independent consistency check on the chiral prediction.
The global fit determines $\omega$ and $\phi$ masses and widths consistent with literature values, while Belle II data pull the widths down by $3.2\sigma$ and $2.6\sigma$, showing that the dispersive fit can expose tensions inside a data set.
The upward pull in $C_A$ from Belle II data tracks the upward pull in $a_\mu^{3\pi}$, so resolving the Belle II tensions will sharpen both the anomaly test and the hadronic-vacuum-polarization estimate for the muon's anomalous magnetic moment.

Reading between the lines

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

A testable extension would be a data-driven evaluation of the sum rule directly from binned cross sections above 1 GeV, bypassing the conformal polynomial; the paper imposes the sum rule but does not compare it with an unparameterized integral.
If the Belle II tensions are resolved by future data with proper bin weighting, a combined fit could push the anomaly extraction below 5%, making $e^+e^-\to 3\pi$ the sharpest single observable for $F_{3\pi}$.
The same formalism applies to $\eta\to 3\pi$ and $\eta'\to 3\pi$, where the anomaly normalization enters analogously, offering a cross-channel check of the WZW prediction not performed in this paper.

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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

Dear colleague,

The headline: this is the first extraction of F3π from e+e−→3π data, and it is a sensible consistency check for the dispersive machinery used in muon g−2 work. The central result, F3π = 33.1(1.7) GeV^-3, agrees with the WZW prediction at about the 5% level, but that precision is softer than it appears. The quoted error is dominated by the truncation of the conformal polynomial, and the sequence CA/CWZW_A = 1.038(11), 1.028(13), 0.980(16) as pconf goes 2→3→4 shows the central value moving by the full quoted truncation error in the last step. The paper itself notes the fits become increasingly unstable for larger pconf, so the expansion has not demonstrably converged. Because the sum rule (2.11) makes the extracted anomaly an integral over the imaginary part, and the pconf contributions enter above 1 GeV^2 where the data only constrain up to 1.8 GeV, the 5% claim does depend on an unverified high-energy extrapolation. The stress-test note lands.

That said, credit where it is due. The authors are upfront about the poor global fit quality (p-value around 1e-5 even after scale-factor inflation) and they include the truncation spread as a systematic error rather than hiding it. The Belle II comparison is genuinely useful: those data pull Γω and Γϕ down by 3.2σ and 2.6σ and push the anomaly upward, which is either a data-set issue or a sign the representation is missing something. The authors frame it as a tension to monitor, which is honest.

The circularity concern is mostly a non-issue. F3π is a fitted parameter, the sum rule is a dispersion identity, and the agreement with WZW is an external benchmark. The same group's earlier papers supply the formalism, but that is not circular.

The main weakness is that the central value's stability is not established. The 0.048 shift from pconf 3 to 4 equals the quoted truncation error, so the 5% total uncertainty is an envelope, not a convergence statement. A serious referee should ask for a more principled truncation criterion, such as a Bayesian model average or a check against an alternative parameterization, rather than just the spread across pconf = 2, 3, 4.

Bottom line: this is a useful, honest paper and it deserves proper peer review, not a desk rejection. Send it out with a request to confront the convergence issue head-on and to soften the language about "5% precision" if the systematic envelope is dominated by an unstable expansion. Conditional acceptance is the right call.

Best, [You]

Referee Report

3 major / 5 minor

Summary. The authors extract the chiral anomaly F3π from e+e−→3π cross-section data using the dispersive Khuri–Treiman representation of the γ*→3π amplitude developed in earlier work. The central idea is to leave the subtraction constant α_A (proportional to the anomaly constant C_A) free in the fit, while imposing the sum rule (2.11) at every fit step. Fitting SND, CMD-2, and BaBar data yields F3π = 33.1(1.7) GeV^{-3}, i.e. C_A/C_WZW_A = 1.028(15)(48)(17)[53], in agreement with the WZW prediction at the 5% level. The paper also fits the recent Belle II data and finds tensions with the dispersive representation, with the omega and phi widths, and with the resulting 3π contribution to the muon anomalous magnetic moment, reflected in an upward pull on C_A.

Significance. If correct, this is a valuable new test of the WZW anomaly: it uses only e+e−→3π data and exploits the dispersive sum rule to tie the anomaly to the omega and phi resonance region, making the extraction substantially more sensitive than low-energy data alone. The error breakdown into statistical, conformal-truncation, and rho-omega phase components is transparent, and the careful treatment of bin reweighting and iterative fitting is a strength. The paper also explicitly identifies tensions with Belle II data instead of forcing a combined result. Importantly, there is no circularity: the sum rule is a dispersion identity, and C_A is a fitted output compared against an external chiral benchmark. The main limitation is that the quoted precision depends on the convergence and completeness of the conformal-polynomial representation, which the fit-quality and truncation tests do not yet fully establish.

major comments (3)

[Sec. 3, Table 3, Eq. (3.3)] The quoted truncation uncertainty of 0.048 is essentially the full shift between p_conf=3 and p_conf=4 for delta_epsilon=0, and the sequence 1.038(11), 1.028(13), 0.980(16) is monotonic in p_conf. Figure 1 shows the result continues to move at p_conf=5, and the text states that for larger p_conf the fits become increasingly unstable. A shift equal to the quoted systematic is not by itself a demonstration of convergence. Please provide a sharper justification for choosing p_conf=3 as the central value and for treating the set {2,3,4} as an uncertainty envelope, or an independent stability check such as a prior on the higher coefficients, a data-driven order-selection criterion, or a physics-motivated asymptotic constraint on Im C_p.
[Tables 1 and 3; Eq. (2.11)] Even after scale-factor inflation, the global-fit p-values are between 1e-5 and 4e-5 (chi2/dof about 1.31-1.37), so the fitted representation does not describe the data statistically. Because C_A is extracted as an integral over the fitted imaginary part, including the conformal polynomial up to arbitrarily large s', a model that fails at this level could bias the integral in a way not captured by the quoted uncertainties. Please quantify the impact of the model incompleteness, for example by showing results from single-dataset fits, from omitting the highest-energy BaBar points, or from adding an additional high-energy contribution or an alternative asymptotic form, and report the resulting shift in C_A/C_WZW_A.
[Sec. 4, Tables 4-6] The Belle II fits have p-values around 1e-8 and show significant tensions in the omega and phi widths and in a_mu. This is consistent with the model-incompleteness concern raised above and should be discussed not only as a property of the Belle II data set but also as a systematic check on the reliability of the global-fit extraction, especially since the upward pull in C_A from Belle II is of similar size to the global-fit total uncertainty.

minor comments (5)

[Sec. 2, after Eq. (2.8)] The notation sthr = M^2_pi0 for the omega->pi0 gamma threshold would be clearer if written as M_{pi^0}^2 with an explicit statement that this is the square of the neutral-pion mass.
[Table 2] The error decomposition in the first column would be easier to use if the table footnote stated explicitly that the second error is the maximal variation over p_conf in {2,4} and the third error is the variation between delta_epsilon=0 and delta_epsilon=3.5 degrees.
[Sec. 4] The phrase 'thereof 174 data points' should read 'of which 174 data points'.
[References] Reference [113] is a private communication; if a public Belle II auxiliary note or release description exists, citing it would make the singular-value cut reproducible.
[Fig. 1] Adding a horizontal line at C_A/C_WZW_A = 1 would make the comparison with the WZW prediction more direct.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the chiral anomaly is a fitted subtraction constant compared against an external WZW benchmark, not recycled from the input.

full rationale

The paper's central quantity, F3π (equivalently C_A/C_WZW_A), is extracted as a free parameter in the dispersive representation of Eq. (2.8), with the sum rule (2.11) acting as a dispersion-theoretic identity relating the subtraction constant to the fitted imaginary parts; it does not inject the WZW value. The chiral prediction enters only after the fit as an external benchmark via Eqs. (1.2) and (1.3), and the input data (SND, CMD-2, BaBar, Belle II) are independent measured cross sections. The authors' prior work supplies the formalism and the sum-rule implementation, but no equation in the paper sets C_A or F3π equal to its own input by construction; the fits with the anomaly fixed (Table 1) and with the anomaly free (Table 3) are distinct fits, and the resulting value is compared, not defined, against the WZW prediction. The observed p_conf instability and low global-fits p-values are model-dependence and fit-quality concerns, which belong under soundness rather than circularity. Therefore no circular step is identified.

Assumptions & free parameters 11 free parameters · 6 assumptions · 0 invented entities

The paper's extraction depends on a moderately complex dispersive model with many fitted parameters. The key systematic is the conformal polynomial order. The sum rule is an internal consistency condition of the representation, not an external input. No new particles or forces are introduced.

free parameters (11)

C_A / CWZW_A = 1.028(15)(48)(17)[53] (global fit, pconf=3)
Target parameter extracted from the fit; enters through subtraction constant αA = C_A/3 in Eq. (2.8).
Mω = 782.70(1) MeV
Fitted omega mass in global anomaly fit, from Table 3.
Γω = 8.72(1) MeV
Fitted omega width, constrained by the resonance region of the cross section.
Mφ = 1019.21(1) MeV
Fitted phi mass.
Γφ = 4.27(1) MeV
Fitted phi width.
Mω′ = 1.440(13) GeV
Fitted omega-prime mass, contributes to the high-energy tail.
cω, cφ, cω′, cω′′ = 2.94(1), -0.381(1), -0.19(5), -1.66(5) GeV^-1
Residue couplings of vector resonances in the subtraction function.
c1, c2, c3, c4 = -0.21(7), -1.25(4), -0.62(5) GeV^-3, c4=0 for pconf=3
Coefficients of the conformal polynomial Cp(q^2); truncation order is the dominant systematic.
ξCMD-2, ξBaBar, ξ′BaBar = 1.3(5)e-4, 1.3(1)e-3, -2.3(1)e-3 GeV^-1
Energy rescaling factors of the individual data sets.
Re εω = 1.44(13)e-3
Real part of rho-omega mixing parameter.
δϵ = 0 deg or 3.5 deg
Phase of the rho-omega mixing parameter, chosen by hand; variation is included as a systematic.

assumptions (6)

domain assumption Reconstruction theorem (2.3): F(s,t,u;q2) = F(s;q2)+F(t;q2)+F(u;q2), valid when F- and higher partial waves are negligible below the ρ3(1690) resonance.
Used to simplify the partial-wave expansion in Eq. (2.4); this is a standard approximation in the Khuri-Treiman framework.
standard math Sum rule (2.11) αA = (1/π)∫ ds' Im a(s')/s' is imposed at each step in the fit.
Dispersive identity connecting the subtraction constant to the imaginary parts; its validity requires the assumed analytic structure of a(q2).
domain assumption Imaginary part of a(q2) is saturated by ω, φ, ω′ resonances and a conformal polynomial, with thresholds sthr=Mπ0^2 and sinel=1 GeV^2.
Model for the spectral function; incomplete saturation would bias the extracted anomaly.
ad hoc to paper Conformal polynomial truncated at order pconf=3, with pconf=2 and 4 used to estimate systematic uncertainties.
The truncation order is a choice; higher orders make the fit unstable, so the error is derived from the spread rather than from a convergence criterion.
ad hoc to paper The ρ-ω mixing phase δϵ is set to 0 or 3.5 degrees.
The data cannot determine the phase; the two choices bracket the uncertainty.
domain assumption Chiral correction relation C_A = F3π × 1.066(10) from Ref. [19] is used to translate C_A into F3π.
Quark-mass corrections from chiral perturbation theory; uncertainty is negligible compared to the fit uncertainty.

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Pith. "Pith review of Extracting the chiral anomaly from $e^+e^-\to 3\pi$." pith.science (2026). https://pith.science/paper/57VUYGSA

@misc{pith2026250413827,
  author       = {Pith},
  title        = {Pith review of: Extracting the chiral anomaly from $e^+e^-\to 3\pi$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/57VUYGSA}},
  note         = {Machine review of arXiv:2504.13827}
}

abstract

The strength of the interaction of three pions and a photon, $F_{3\pi}$ is predicted by the axial anomaly in terms of the pion decay constant, a relation that is frequently used to constrain low-energy radiative processes involving pions, but only tested experimentally at the $10\%$ level. Here, we present a new avenue to test this prediction, via a fit of a dispersive description of the $\gamma^*\to3\pi$ amplitude to data for $e^+e^-\to 3\pi$. From the global fit to SND, CMD-2, and BaBar data we obtain $F_{3\pi}=33.1(1.7)\,\text{GeV}^{-3}$, in agreement with the chiral prediction at the level of $5\%$. We also consider dispersive fits to the recent data by Belle II, in which case we observe tensions with the dispersive constraints, the width parameters of $\omega$ and $\phi$, and the chiral anomaly.

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Works this paper leans on

115 extracted references · 8 canonical work pages · cited by 3 Pith papers

[53]

Colangelo, M

G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, JHEP09, 091 (2014), arXiv:1402.7081 [hep-ph]

arXiv 2014
[1]

Wess and B

J. Wess and B. Zumino, Phys. Lett. B37, 95 (1971)

1971
[2]

Witten, Nucl

E. Witten, Nucl. Phys. B223, 422 (1983)

1983
[3]

S. L. Adler, B. W. Lee, S. B. Treiman, and A. Zee, Phys. Rev. D4, 3497 (1971)

1971
[4]

M. V. Terent’ev, Phys. Lett. B38, 419 (1972)

1972
[5]

Aviv and A

R. Aviv and A. Zee, Phys. Rev. D5, 2372 (1972)

1972
[6]

Navas et al

S. Navas et al. (Particle Data Group), Phys. Rev. D110, 030001 (2024)

2024
[7]

Larin et al

I. Larin et al. (PrimEx-II), Science 368, 506 (2020)

2020

Show all 115 references

[8]

Bijnens, A

J. Bijnens, A. Bramon, and F. Cornet, Phys. Rev. Lett.61, 1453 (1988)

1988
[9]

J. L. Goity, A. M. Bernstein, and B. R. Holstein, Phys. Rev. D66, 076014 (2002), arXiv:hep-ph/0206007

2002 arXiv
[10]

Ananthanarayan and B

B. Ananthanarayan and B. Moussallam, JHEP05, 052 (2002), arXiv:hep-ph/0205232

2002 arXiv
[11]

Kampf and B

K. Kampf and B. Moussallam, Phys. Rev. D79, 076005 (2009), arXiv:0901.4688 [hep-ph]

2009 arXiv
[12]

Gérardin, H

A. Gérardin, H. B. Meyer, and A. Nyffeler, Phys. Rev. D100, 034520 (2019), arXiv:1903.09471 [hep-lat]

2019 arXiv
[13]

Bijnens, L

J. Bijnens, L. Girlanda, and P. Talavera, Eur. Phys. J. C23, 539 (2002), arXiv:hep-ph/0110400

2002 arXiv
[14]

Grilli di Cortona, E

G. Grilli di Cortona, E. Hardy, J. Pardo Vega, and G. Villadoro, JHEP01, 034 (2016), arXiv:1511.02867 [hep-ph]

2016 arXiv
[15]

Gao, Z.-H

R. Gao, Z.-H. Guo, J. A. Oller, and H.-Q. Zhou, JHEP04, 022 (2023), arXiv:2211.02867 [hep-ph]

2023 arXiv
[16]

R. Gao, J. Hao, C.-G. Duan, Z.-H. Guo, J. A. Oller, and H.-Q. Zhou, Eur. Phys. J. C85, 97 (2025), arXiv:2411.06737 [hep-ph]

2025 arXiv
[17]

Hoid, PoSCD2024, 028 (2025)

B.-L. Hoid, PoSCD2024, 028 (2025)

2025
[18]

Moinester, (2024), arXiv:2412.03669 [hep-ph]

M. Moinester, (2024), arXiv:2412.03669 [hep-ph]

2024
[19]

Bijnens, A

J. Bijnens, A. Bramon, and F. Cornet, Phys. Lett. B237, 488 (1990)

1990
[20]

Alkofer and C

R. Alkofer and C. D. Roberts, Phys. Lett. B369, 101 (1996), arXiv:hep-ph/9510284

1996 arXiv
[21]

B. R. Holstein, Phys. Rev. D53, 4099 (1996), arXiv:hep-ph/9512338

1996 arXiv
[22]

Hannah, Nucl

T. Hannah, Nucl. Phys. B593, 577 (2001), arXiv:hep-ph/0102213

2001 arXiv
[23]

T. N. Truong, Phys. Rev. D65, 056004 (2002), arXiv:hep-ph/0105123

2002 arXiv
[24]

Ametller, M

L. Ametller, M. Knecht, and P. Talavera, Phys. Rev. D64, 094009 (2001), arXiv:hep-ph/0107127

2001 arXiv
[25]

Benić and D

S. Benić and D. Klabučar, Phys. Rev. D85, 034042 (2012), arXiv:1109.3140 [hep-ph]

2012 arXiv
[26]

Bijnens, K

J. Bijnens, K. Kampf, and S. Lanz, Nucl. Phys. B860, 245 (2012), arXiv:1201.2608 [hep-ph]

2012 arXiv
[27]

Hoferichter, B

M. Hoferichter, B. Kubis, and D. Sakkas, Phys. Rev. D86, 116009 (2012), arXiv:1210.6793 [hep-ph]. – 12 –

2012 arXiv
[28]

Hoferichter, B

M. Hoferichter, B. Kubis, and M. Zanke, Phys. Rev. D96, 114016 (2017), arXiv:1710.00824 [hep-ph]

2017 arXiv
[29]

Y. M. Antipovet al., Phys. Rev. D36, 21 (1987)

1987
[30]

Giller, A

I. Giller, A. Ocherashvili, T. Ebertshäuser, M. A. Moinester, and S. Scherer, Eur. Phys. J. A 25, 229 (2005), arXiv:hep-ph/0503207

2005 arXiv
[31]

Ecker (COMPASS), Nuovo Cim

D. Ecker (COMPASS), Nuovo Cim. C47, 217 (2024), arXiv:2310.09138 [hep-ph]

2024 arXiv
[32]

Maltsev, PoSCD2024, 002 (2025)

A. Maltsev, PoSCD2024, 002 (2025)

2025
[33]

R. A. Briceño, J. J. Dudek, R. G. Edwards, C. J. Shultz, C. E. Thomas, and D. J. Wilson, Phys. Rev. Lett.115, 242001 (2015), arXiv:1507.06622 [hep-ph]

2015 arXiv
[34]

R. A. Briceño, J. J. Dudek, R. G. Edwards, C. J. Shultz, C. E. Thomas, and D. J. Wilson, Phys. Rev. D93, 114508 (2016), [Erratum: Phys. Rev. D105, 079902 (2022)], arXiv:1604.03530 [hep-ph]

2016 arXiv
[35]

Alexandrou, L

C. Alexandrou, L. Leskovec, S. Meinel, J. Negele, S. Paul, M. Petschlies, A. Pochinsky, G. Rendon, and S. Syritsyn, Phys. Rev. D98, 074502 (2018), [Erratum: Phys. Rev. D105, 019902 (2022)], arXiv:1807.08357 [hep-lat]

2018 arXiv
[36]

Niehus, M

M. Niehus, M. Hoferichter, and B. Kubis, JHEP12, 038 (2021), arXiv:2110.11372 [hep-ph]

2021 arXiv
[37]

Z. Xing, H. Dang, M. A. Sultan, K. Raya, and L. Chang, Phys. Rev. D109, 054028 (2024), arXiv:2401.03169 [hep-ph]

2024 arXiv
[38]

A. S. Miramontes, G. Eichmann, and R. Alkofer, Phys. Lett. B868, 139659 (2025), arXiv:2504.20899 [hep-ph]

2025 arXiv
[39]

Aoyama et al., Phys

T. Aoyama et al., Phys. Rept.887, 1 (2020), arXiv:2006.04822 [hep-ph]

2020 arXiv
[40]

Aoyama, M

T. Aoyama, M. Hayakawa, T. Kinoshita, and M. Nio, Phys. Rev. Lett.109, 111808 (2012), arXiv:1205.5370 [hep-ph]

2012 arXiv
[41]

Aoyama, T

T. Aoyama, T. Kinoshita, and M. Nio, Atoms7, 28 (2019)

2019
[42]

Czarnecki, W

A. Czarnecki, W. J. Marciano, and A. Vainshtein, Phys. Rev. D67, 073006 (2003), [Erratum: Phys. Rev. D73, 119901 (2006)], arXiv:hep-ph/0212229

2003 arXiv
[43]

Gnendiger, D

C. Gnendiger, D. Stöckinger, and H. Stöckinger-Kim, Phys. Rev. D88, 053005 (2013), arXiv:1306.5546 [hep-ph]

2013 arXiv
[44]

Davier, A

M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, Eur. Phys. J. C77, 827 (2017), arXiv:1706.09436 [hep-ph]

2017 arXiv
[45]

Keshavarzi, D

A. Keshavarzi, D. Nomura, and T. Teubner, Phys. Rev. D97, 114025 (2018), arXiv:1802.02995 [hep-ph]

2018 arXiv
[46]

Colangelo, M

G. Colangelo, M. Hoferichter, and P. Stoffer, JHEP02, 006 (2019), arXiv:1810.00007 [hep-ph]

2019 arXiv
[47]

Hoferichter, B.-L

M. Hoferichter, B.-L. Hoid, and B. Kubis, JHEP08, 137 (2019), arXiv:1907.01556 [hep-ph]

2019 arXiv
[48]

Davier, A

M. Davier, A. Hoecker, B. Malaescu, and Z. Zhang, Eur. Phys. J. C80, 241 (2020), [Erratum: Eur. Phys. J. C80, 410 (2020)], arXiv:1908.00921 [hep-ph]

2020 arXiv
[49]

Keshavarzi, D

A. Keshavarzi, D. Nomura, and T. Teubner, Phys. Rev. D101, 014029 (2020), arXiv:1911.00367 [hep-ph]. – 13 –

2020 arXiv
[50]

B.-L. Hoid, M. Hoferichter, and B. Kubis, Eur. Phys. J. C80, 988 (2020), arXiv:2007.12696 [hep-ph]

2020 arXiv
[51]

A. Kurz, T. Liu, P. Marquard, and M. Steinhauser, Phys. Lett. B734, 144 (2014), arXiv:1403.6400 [hep-ph]

2014 arXiv
[52]

Melnikov and A

K. Melnikov and A. Vainshtein, Phys. Rev. D70, 113006 (2004), arXiv:hep-ph/0312226

2004 arXiv
[54]

Colangelo, M

G. Colangelo, M. Hoferichter, B. Kubis, M. Procura, and P. Stoffer, Phys. Lett. B738, 6 (2014), arXiv:1408.2517 [hep-ph]

2014 arXiv
[55]

Colangelo, M

G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, JHEP09, 074 (2015), arXiv:1506.01386 [hep-ph]

2015 arXiv
[56]

Masjuan and P

P. Masjuan and P. Sánchez-Puertas, Phys. Rev. D95, 054026 (2017), arXiv:1701.05829 [hep-ph]

2017 arXiv
[57]

Colangelo, M

G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, Phys. Rev. Lett.118, 232001 (2017), arXiv:1701.06554 [hep-ph]

2017 arXiv
[58]

Colangelo, M

G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, JHEP04, 161 (2017), arXiv:1702.07347 [hep-ph]

2017 arXiv
[59]

Hoferichter, B.-L

M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold, and S. P. Schneider, Phys. Rev. Lett. 121, 112002 (2018), arXiv:1805.01471 [hep-ph]

2018 arXiv
[60]

Hoferichter, B.-L

M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold, and S. P. Schneider, JHEP10, 141 (2018), arXiv:1808.04823 [hep-ph]

2018 arXiv
[61]

Bijnens, N

J. Bijnens, N. Hermansson-Truedsson, and A. Rodríguez-Sánchez, Phys. Lett. B798, 134994 (2019), arXiv:1908.03331 [hep-ph]

2019 arXiv
[62]

Colangelo, F

G. Colangelo, F. Hagelstein, M. Hoferichter, L. Laub, and P. Stoffer, Phys. Rev. D101, 051501 (2020), arXiv:1910.11881 [hep-ph]

2020 arXiv
[63]

Colangelo, F

G. Colangelo, F. Hagelstein, M. Hoferichter, L. Laub, and P. Stoffer, JHEP03, 101 (2020), arXiv:1910.13432 [hep-ph]

2020 arXiv
[64]

T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin, C. Jung, and C. Lehner (RBC, UKQCD), Phys. Rev. Lett.124, 132002 (2020), arXiv:1911.08123 [hep-lat]

2020 arXiv
[65]

Colangelo, M

G. Colangelo, M. Hoferichter, A. Nyffeler, M. Passera, and P. Stoffer, Phys. Lett. B735, 90 (2014), arXiv:1403.7512 [hep-ph]

2014 arXiv
[66]

Aliberti et al., (2025), arXiv:2505.21476 [hep-ph]

R. Aliberti et al., (2025), arXiv:2505.21476 [hep-ph]

2025 arXiv
[67]

S. P. Schneider, B. Kubis, and F. Niecknig, Phys. Rev. D86, 054013 (2012), arXiv:1206.3098 [hep-ph]

2012 arXiv
[68]

Hoferichter, B

M. Hoferichter, B. Kubis, S. Leupold, F. Niecknig, and S. P. Schneider, Eur. Phys. J. C74, 3180 (2014), arXiv:1410.4691 [hep-ph]

2014 arXiv
[69]

Hoferichter, B.-L

M. Hoferichter, B.-L. Hoid, B. Kubis, and J. Lüdtke, Phys. Rev. Lett.128, 172004 (2022), arXiv:2105.04563 [hep-ph]

2022 arXiv
[70]

Lüdtke, M

J. Lüdtke, M. Procura, and P. Stoffer, JHEP04, 130 (2025), arXiv:2410.11946 [hep-ph]. – 14 –

2025 arXiv
[71]

Hoferichter, P

M. Hoferichter, P. Stoffer, and M. Zillinger, Phys. Rev. Lett.134, 061902 (2025), arXiv:2412.00190 [hep-ph]

2025 arXiv
[72]

Hoferichter, P

M. Hoferichter, P. Stoffer, and M. Zillinger, JHEP02, 121 (2025), arXiv:2412.00178 [hep-ph]

2025 arXiv
[73]

Hoferichter, J

M. Hoferichter, J. Lüdtke, L. Naterop, M. Procura, and P. Stoffer, Phys. Rev. Lett.134, 201801 (2025), arXiv:2503.04883 [hep-ph]

2025 arXiv
[74]

Hoferichter, P

M. Hoferichter, P. Stoffer, and M. Zillinger, Phys. Lett. B866, 139565 (2025), arXiv:2504.10582 [hep-ph]

2025 arXiv
[75]

Hoferichter, B.-L

M. Hoferichter, B.-L. Hoid, B. Kubis, and D. Schuh, JHEP08, 208 (2023), arXiv:2307.02546 [hep-ph]

2023 arXiv
[76]

Hoferichter, G

M. Hoferichter, G. Colangelo, B.-L. Hoid, B. Kubis, J. Ruiz de Elvira, D. Schuh, D. Stamen, and P. Stoffer, Phys. Rev. Lett.131, 161905 (2023), arXiv:2307.02532 [hep-ph]

2023 arXiv
[77]

Stollenwerk, C

F. Stollenwerk, C. Hanhart, A. Kupść, U.-G. Meißner, and A. Wirzba, Phys. Lett. B707, 184 (2012), arXiv:1108.2419 [nucl-th]

2012 arXiv
[78]

Hanhart, A

C. Hanhart, A. Kupść, U.-G. Meißner, F. Stollenwerk, and A. Wirzba, Eur. Phys. J. C73, 2668 (2013), [Erratum: Eur. Phys. J. C75, 242 (2015)], arXiv:1307.5654 [hep-ph]

2013 arXiv
[79]

Kubis and J

B. Kubis and J. Plenter, Eur. Phys. J. C75, 283 (2015), arXiv:1504.02588 [hep-ph]

2015 arXiv
[80]

S. Holz, J. Plenter, C.-W. Xiao, T. Dato, C. Hanhart, B. Kubis, U.-G. Meißner, and A. Wirzba, Eur. Phys. J. C81, 1002 (2021), arXiv:1509.02194 [hep-ph]

2021 arXiv
[81]

S. Holz, C. Hanhart, M. Hoferichter, and B. Kubis, Eur. Phys. J. C82, 434 (2022), [Addendum: Eur. Phys. J. C82, 1159 (2022)], arXiv:2202.05846 [hep-ph]

2022 arXiv
[82]

Holz, The Quest for the η and η′ Transition Form Factors:A Stroll on the Precision Frontier, Ph.D

S. Holz, The Quest for the η and η′ Transition Form Factors:A Stroll on the Precision Frontier, Ph.D. thesis, University of Bonn (2022)

2022
[83]

S. Holz, M. Hoferichter, B.-L. Hoid, and B. Kubis, Phys. Rev. Lett.134, 171902 (2025), arXiv:2411.08098 [hep-ph]

2025 arXiv
[84]

S. Holz, M. Hoferichter, B.-L. Hoid, and B. Kubis, JHEP04, 147 (2025), arXiv:2412.16281 [hep-ph]

2025 arXiv
[85]

M. I. Vysotsky and E. V. Zhemchugov, Phys. Rev. D93, 094029 (2016), [Erratum: Phys. Rev. D 94, 019901 (2016)], arXiv:1512.04438 [hep-ph]

2016 arXiv
[86]

M. Dax, D. Stamen, and B. Kubis, Eur. Phys. J. C81, 221 (2021), arXiv:2012.04655 [hep-ph]

2021 arXiv
[87]

Stamen, J

D. Stamen, J. L. Dammann, Y. Korte, and B. Kubis, Eur. Phys. J. C84, 1267 (2024), arXiv:2409.05955 [hep-ph]

2024 arXiv
[88]

M. N. Achasovet al. (SND), Phys. Rev. D63, 072002 (2001), arXiv:hep-ex/0009036

2001 arXiv
[89]

M. N. Achasovet al. (SND), Phys. Rev. D66, 032001 (2002), arXiv:hep-ex/0201040

2002 arXiv
[90]

M. N. Achasovet al. (SND), Phys. Rev. D68, 052006 (2003), arXiv:hep-ex/0305049

2003 arXiv
[91]

M. N. Achasovet al. (SND), Eur. Phys. J. C80, 993 (2020), arXiv:2007.14595 [hep-ex]

2020 arXiv
[92]

R. R. Akhmetshinet al. (CMD-2), Phys. Lett. B364, 199 (1995)

1995
[93]

R. R. Akhmetshinet al. (CMD-2), Phys. Lett. B434, 426 (1998)

1998
[94]

R. R. Akhmetshinet al. (CMD-2), Phys. Lett. B578, 285 (2004), arXiv:hep-ex/0308008. – 15 –

2004 arXiv
[95]

R. R. Akhmetshinet al. (CMD-2), Phys. Lett. B642, 203 (2006)

2006
[96]

J. P. Leeset al. (BaBar), Phys. Rev. D104, 112003 (2021), arXiv:2110.00520 [hep-ex]

2021 arXiv
[97]

Adachi et al

I. Adachi et al. (Belle-II), Phys. Rev. D110, 112005 (2024), arXiv:2404.04915 [hep-ex]

2024 arXiv
[98]

Jacob and G

M. Jacob and G. C. Wick, Annals Phys.7, 404 (1959), [Annals Phys.281, 774 (2000)]

1959
[99]

Niecknig, B

F. Niecknig, B. Kubis, and S. P. Schneider, Eur. Phys. J. C72, 2014 (2012), arXiv:1203.2501 [hep-ph]

2012 arXiv
[100]

N. N. Khuri and S. B. Treiman, Phys. Rev.119, 1115 (1960)

1960
[101]

E. L. Lomon and S. Pacetti, Phys. Rev. D85, 113004 (2012), [Erratum: Phys. Rev. D86, 039901 (2012)], arXiv:1201.6126 [hep-ph]

2012 arXiv
[102]

Moussallam, Eur

B. Moussallam, Eur. Phys. J. C73, 2539 (2013), arXiv:1305.3143 [hep-ph]

2013 arXiv
[103]

Zanke, M

M. Zanke, M. Hoferichter, and B. Kubis, JHEP07, 106 (2021), arXiv:2103.09829 [hep-ph]

2021 arXiv
[104]

Crivellin and M

A. Crivellin and M. Hoferichter, Phys. Rev. D108, 013005 (2023), arXiv:2211.12516 [hep-ph]

2023 arXiv
[105]

Hanhart, Phys

C. Hanhart, Phys. Lett. B715, 170 (2012), arXiv:1203.6839 [hep-ph]

2012 arXiv
[106]

Ropertz, C

S. Ropertz, C. Hanhart, and B. Kubis, Eur. Phys. J. C78, 1000 (2018), arXiv:1809.06867 [hep-ph]

2018 arXiv
[107]

von Detten, F

L. von Detten, F. Noël, C. Hanhart, M. Hoferichter, and B. Kubis, Eur. Phys. J. C81, 420 (2021), arXiv:2103.01966 [hep-ph]

2021 arXiv
[108]

L. A. Heuser, G. Chanturia, F.-K. Guo, C. Hanhart, M. Hoferichter, and B. Kubis, Eur. Phys. J. C84, 599 (2024), arXiv:2403.15539 [hep-ph]

2024 arXiv
[109]

Colangelo, M

G. Colangelo, M. Hoferichter, B. Kubis, and P. Stoffer, JHEP10, 032 (2022), arXiv:2208.08993 [hep-ph]

2022 arXiv
[110]

Stamen, D

D. Stamen, D. Hariharan, M. Hoferichter, B. Kubis, and P. Stoffer, Eur. Phys. J. C82, 432 (2022), arXiv:2202.11106 [hep-ph]

2022 arXiv
[111]

R. D. Ball, L. Del Debbio, S. Forte, A. Guffanti, J. I. Latorre, J. Rojo, and M. Ubiali (NNPDF), JHEP 05, 075 (2010), arXiv:0912.2276 [hep-ph]

2010 arXiv
[112]

D’Agostini, Nucl

G. D’Agostini, Nucl. Instrum. Meth. A346, 306 (1994)

1994
[113]

Sue, (2025), private communication

Y. Sue, (2025), private communication

2025
[114]

Vacuum polarization,

F. Ignatov, “Vacuum polarization,” (2022)

2022
[115]

Aliberti et al., SciPost Phys

R. Aliberti et al., SciPost Phys. Comm. Rep.9, 1 (2025), arXiv:2410.22882 [hep-ph]. – 16 –

2025 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [53]

Colangelo, M

G. Colangelo, M. Hoferichter, M. Procura, and P. Stoffer, JHEP09, 091 (2014), arXiv:1402.7081 [hep-ph]

arXiv 2014

[2] [1]

Wess and B

J. Wess and B. Zumino, Phys. Lett. B37, 95 (1971)

1971

[3] [2]

Witten, Nucl

E. Witten, Nucl. Phys. B223, 422 (1983)

1983

[4] [3]

S. L. Adler, B. W. Lee, S. B. Treiman, and A. Zee, Phys. Rev. D4, 3497 (1971)

1971

[5] [4]

M. V. Terent’ev, Phys. Lett. B38, 419 (1972)

1972

[6] [5]

Aviv and A

R. Aviv and A. Zee, Phys. Rev. D5, 2372 (1972)

1972

[7] [6]

Navas et al

S. Navas et al. (Particle Data Group), Phys. Rev. D110, 030001 (2024)

2024

[8] [7]

Larin et al

I. Larin et al. (PrimEx-II), Science 368, 506 (2020)

2020