Pseudoscalar-Meson Contributions to g-2 via Schwinger's Sum Rule

Franziska Hagelstein (AEC Bern); Vladimir Pascalutsa (JGU Mainz)

arxiv: 1907.06927 · v1 · pith:3MZVATVEnew · submitted 2019-07-16 · ✦ hep-ph · nucl-th

Pseudoscalar-Meson Contributions to g-2 via Schwinger's Sum Rule

Franziska Hagelstein (AEC Bern) , Vladimir Pascalutsa (JGU Mainz) This is my paper

Pith reviewed 2026-05-24 20:59 UTC · model grok-4.3

classification ✦ hep-ph nucl-th

keywords Schwinger sum rulemuon g-2hadronic light-by-light scatteringpseudoscalar mesonsanomalous magnetic moment

0 comments

The pith

Schwinger's sum rule serves as a tool for calculating pseudoscalar meson contributions to the muon anomalous magnetic moment.

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

The paper proposes the Schwinger sum rule as a method to study hadronic contributions to the muon anomalous magnetic moment. It demonstrates the approach through preliminary calculations of light-by-light scattering from pseudoscalar mesons. A sympathetic reader would care because these contributions form part of the theoretical input needed for precision tests of the Standard Model. The method is positioned as potentially reducing reliance on more complex modeling for this specific piece.

Core claim

The Schwinger sum rule is presented as a new promising tool to study the hadronic contributions to the muon anomalous magnetic moment. In particular, preliminary results are shown for the light-by-light scattering contribution of pseudoscalar mesons.

What carries the argument

Schwinger's sum rule, a dispersion relation that integrates a forward scattering amplitude or related structure function to yield a fixed value, applied here to isolate meson effects in light-by-light scattering.

If this is right

The sum rule provides a route to compute specific hadronic light-by-light terms entering the muon g-2.
Preliminary numerical estimates become available for the pseudoscalar meson channel.
The approach opens a pathway to treat other hadronic states through the same integral relation.

Where Pith is reading between the lines

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

If validated, the method could be cross-checked against lattice QCD results for the same quantities.
Extension to vector mesons or other resonances would follow naturally from the same sum-rule structure.
The technique might help isolate theoretical uncertainties in the overall hadronic vacuum polarization contribution.

Load-bearing premise

The Schwinger sum rule applies directly to extract pseudoscalar meson contributions to hadronic light-by-light scattering without additional corrections or unaccounted model dependencies.

What would settle it

A numerical evaluation of the pseudoscalar light-by-light contribution using the sum rule that deviates substantially from independent calculations via dispersion relations or other established techniques.

Figures

Figures reproduced from arXiv: 1907.06927 by Franziska Hagelstein (AEC Bern), Vladimir Pascalutsa (JGU Mainz).

**Figure 1.** Figure 1: Hadronic contributions to the anomalous magnetic moment: (a) hadronic vacuum polarization and (b) hadronic light-by-light scattering. Hadronic excitations are indicated by red blobs. 2. The Schwinger Sum Rule As we recently argued [8], the Schwinger sum rule encompasses the dispersive formula (1.3) for HVP and provides its analogue for HLbL (or, in fact, any other contribution). The Schwinger 1 [PITH_FULL… view at source ↗

**Figure 2.** Figure 2: Tree-level Compton scattering diagrams. 3. Unified Treatment of Hadronic Contributions To evaluate the impact of a given mechanism on aµ via the Schwinger sum rule, we need to measure its effect on the photoabsorption cross section σLT . There are two fundamentally different ways in which hadrons affect the photoabsorption on a lepton: 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Different channels contributing to the photoabsorption process: hadron photoproduction through timelike Compton scattering (a) or the Primakoff mechanism (b), and hadronic light-by-light contributions to the Compton scattering (c, d) (crossed diagrams omitted). The first type of contributions can in principle be measured experimentally. They begin to contribute to σLT at O(α 3 ), and hence are of O(α 2 ) … view at source ↗

**Figure 4.** Figure 4: Single-meson (red dashed line) contribution to the anomalous magnetic moment. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 5.** Figure 5: Diagrams contributing to the process of pseudoscalar-meson production off the muon. The most recent value for the dominant π 0 -pole contribution [15]: a π 0 -pole µ = 6.26+0.30 −0.25 ×10−10 , (4.2) concurs with the first of the two calculations in Eq. (4.1). It is interesting to see whether the Schwinger sum-rule approach can tell us something new. To leading order in α, we need to consider the following … view at source ↗

**Figure 6.** Figure 6: Leading contributions to the pseudoscalar-lepton-lepton interaction. 4 [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗

**Figure 7.** Figure 7: Subleading mechanism accompanying the single-meson photoproduction. 5. Conclusion The Schwinger sum rule provides a dispersive data-driven approach to calculating the hadronic contributions in lepton g−2. It encompasses the simple dispersive formula for the HVP contribution, Eq. (1.3), and allows to treat the others (e.g, the HLbL contribution) on similar footing. The required data on the doubly-polarized… view at source ↗

read the original abstract

The Schwinger sum rule is presented as a new promising tool to study the hadronic contributions to the muon anomalous magnetic moment. In particular, we show preliminary results for the light-by-light scattering contribution of pseudoscalar mesons.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper sketches an application of Schwinger's sum rule to pseudoscalar poles in HLbL for a_mu but the abstract gives no derivation of the needed mapping, so the model-independence claim stays untested.

read the letter

The main point is that Hagelstein and Pascalutsa want to use Schwinger's sum rule as a route to the pseudoscalar-meson piece of hadronic light-by-light scattering in the muon anomaly. They present it as a promising tool and include some early numerical results for the light pseudoscalars. That is the new angle they are advertising: taking an old sum rule and pointing it at this particular contribution instead of the usual dispersion or effective-Lagrangian approaches. If the mapping works without extra subtractions or form-factor assumptions, it could give a cleaner handle on the pion, eta and eta-prime poles. The abstract is too brief to show whether they have actually carried out that mapping or just stated the idea. On the positive side, the authors are not claiming a finished calculation; they label the numbers preliminary, which keeps expectations realistic. The approach is at least a different starting point from the standard ones that dominate the literature. The soft spot is the one flagged in the stress-test note. Schwinger's original relation applies to the forward virtual Compton amplitude. Extending it to the four-point HLbL tensor that contracts with the muon line requires a choice of how to isolate the pseudoscalar channel and what subtraction constants to keep. If those steps rely on the same effective vertices or parametrizations already used in conventional pion-pole evaluations, the advertised reduction in model dependence does not materialize. The abstract gives no indication that the necessary identities have been derived or cross-checked against the known analytic result for the pion pole. Without that step the preliminary numbers cannot be assessed for consistency or improvement. This is a short, idea-oriented note rather than a complete calculation. It would interest specialists already working on sum rules or dispersion relations for g-2 who might want to explore whether the route can be made rigorous. For most readers the lack of the explicit mapping makes it hard to extract usable information. I would not cite it in its current form because no new, validated result is on offer. It does not yet look ready for a serious referee; the central technical step is missing. A fuller version that derives the projection and shows a concrete numerical or formal gain would change that assessment.

Referee Report

2 major / 0 minor

Summary. The manuscript proposes Schwinger's sum rule as a new tool for extracting pseudoscalar-meson contributions to the hadronic light-by-light (HLbL) scattering amplitude that enters the muon anomalous magnetic moment a_μ, and presents preliminary numerical results for these contributions.

Significance. If the sum rule can be shown to map directly onto the HLbL tensor structures relevant for a_μ without introducing new model dependence or unsubtracted dispersion relations, the method could supply an independent cross-check on existing pion-pole and pseudoscalar-exchange calculations. The current preliminary status and absence of explicit derivations limit the immediate impact.

major comments (2)

[Abstract] Abstract: the central claim that the Schwinger sum rule supplies a 'new promising tool' for the pseudoscalar HLbL piece rests on an unshown adaptation from the forward virtual Compton amplitude to the four-point HLbL function; no derivation of the required kinematic projections or subtraction constants is supplied, leaving open whether the mapping is parameter-free or reintroduces the same effective-Lagrangian input used in conventional calculations.
[Abstract] Abstract (preliminary results): without explicit spectral decomposition, form-factor parametrization, or comparison to the known pion-pole formula, it is impossible to verify that the reported numbers isolate the pseudoscalar channel in the tensor structures contracted with the muon vertex.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We respond to the major comments point by point below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the Schwinger sum rule supplies a 'new promising tool' for the pseudoscalar HLbL piece rests on an unshown adaptation from the forward virtual Compton amplitude to the four-point HLbL function; no derivation of the required kinematic projections or subtraction constants is supplied, leaving open whether the mapping is parameter-free or reintroduces the same effective-Lagrangian input used in conventional calculations.

Authors: The manuscript is a short note focused on the conceptual application and preliminary numerical estimates. The adaptation of Schwinger's sum rule is outlined in the main text via the relevant dispersion relation, but we agree that the abstract does not supply the explicit steps. In revision we will add a concise derivation of the kinematic projections onto the HLbL tensor structures that enter a_μ together with a discussion of the subtraction constants, making clear that the only external input remains the standard pseudoscalar transition form factors. revision: yes
Referee: [Abstract] Abstract (preliminary results): without explicit spectral decomposition, form-factor parametrization, or comparison to the known pion-pole formula, it is impossible to verify that the reported numbers isolate the pseudoscalar channel in the tensor structures contracted with the muon vertex.

Authors: The preliminary numbers are obtained from a spectral representation of the sum rule combined with conventional parametrizations of the pseudoscalar transition form factors. To permit verification we will include in the revised manuscript the explicit form of the spectral function, the parametrization adopted for the form factors, and a direct numerical comparison with the standard pion-pole contribution evaluated in the same tensor structures. revision: yes

Circularity Check

0 steps flagged

No circularity; sum rule invoked as external input for HLbL pseudoscalar channel

full rationale

The abstract and description frame the Schwinger sum rule as an independent tool applied to extract pseudoscalar-meson HLbL contributions to a_μ. No equations, self-citations, or fitted parameters are shown that would reduce any claimed prediction to a redefinition of the input. The central step (applying the sum rule to the four-point function) is presented without evidence of being forced by prior author work or by construction from the target observable. This is the normal case of an external relation used on new kinematics.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or new entities.

pith-pipeline@v0.9.0 · 5560 in / 975 out tokens · 21832 ms · 2026-05-24T20:59:48.471456+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · 1 internal anchor

[1]

G. W. Bennett, et al. [Muon g-2 Collaboration], Phys. Rev. D 73, 072003 (2006)

work page 2006
[2]

Jegerlehner, Acta Phys

F. Jegerlehner, Acta Phys. Polon. B 49, 1157 (2018)

work page 2018
[3]

Variations on Photon Vacuum Polarization

F. Jegerlehner, arXiv:1711.06089 [hep-ph]

work page internal anchor Pith review Pith/arXiv arXiv
[4]

Jegerlehner, Springer Tracts Mod

F. Jegerlehner, Springer Tracts Mod. Phys. 274 (2017)

work page 2017
[5]

Logashenko, et al

I. Logashenko, et al. [Muon g-2 Collaboration], J. Phys. Chem. Ref. Data 44, no. 3, 031211 (2015)

work page 2015
[6]

Venanzoni [Fermilab E989 Collaboration], Nucl

G. Venanzoni [Fermilab E989 Collaboration], Nucl. Part. Phys. Proc. 273-275, 584 (2016)

work page 2016
[7]

Otani [E34 Collaboration], JPS Conf

M. Otani [E34 Collaboration], JPS Conf. Proc. 8, 025008 (2015)

work page 2015
[8]

Hagelstein and V

F. Hagelstein and V . Pascalutsa, Phys. Rev. Lett.120, no. 7, 072002 (2018)

work page 2018
[9]

J. S. Schwinger, Proc. Nat. Acad. Sci. 72, 1 (1975); ibid. 72, 1559 (1975) [Acta Phys. Austriaca Suppl. 14, 471 (1975)]

work page 1975
[10]

A. M. Harun ar-Rashid, Nuovo Cim. A 33, 447 (1976)

work page 1976
[11]

Hagelstein, V

F. Hagelstein, V . Pascalutsa and M. Vanderhaeghen, in preparation

work page
[12]

Pascalutsa, V

V . Pascalutsa, V . Pauk and M. Vanderhaeghen, Phys. Rev. D85, 116001 (2012)

work page 2012
[13]

Knecht and A

M. Knecht and A. Nyffeler, Phys. Rev. D 65, 073034 (2002)

work page 2002
[14]

Melnikov and A

K. Melnikov and A. Vainshtein, Phys. Rev. D 70, 113006 (2004)

work page 2004
[15]

Hoferichter, B.-L

M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold and S. P. Schneider, Phys. Rev. Lett. 121, 112002 (2018)

work page 2018
[16]

Drell, Il Nuovo Cimento 11, 693-697 (1959)

S. Drell, Il Nuovo Cimento 11, 693-697 (1959)

work page 1959
[17]

Ametller, A

L. Ametller, A. Bramon, E. Masso, Phys. Rev. D 30 251 (1984)

work page 1984
[18]

A. E. Dorokhov and M. A. Ivanov, Phys. Rev. D 75 114007 (2007)

work page 2007
[19]

Nyffeler, Phys

A. Nyffeler, Phys. Rev. D 79, 073012 (2009)

work page 2009
[20]

Tanabashi, et al

M. Tanabashi, et al. (PDG), Phys. Rev. D 98, 030001 (2018)

work page 2018
[21]

Grifﬁths, Introduction to Elementary Particles, WILEY-VCH Verlag, 2008

D. Grifﬁths, Introduction to Elementary Particles, WILEY-VCH Verlag, 2008. 7

work page 2008

[1] [1]

G. W. Bennett, et al. [Muon g-2 Collaboration], Phys. Rev. D 73, 072003 (2006)

work page 2006

[2] [2]

Jegerlehner, Acta Phys

F. Jegerlehner, Acta Phys. Polon. B 49, 1157 (2018)

work page 2018

[3] [3]

Variations on Photon Vacuum Polarization

F. Jegerlehner, arXiv:1711.06089 [hep-ph]

work page internal anchor Pith review Pith/arXiv arXiv

[4] [4]

Jegerlehner, Springer Tracts Mod

F. Jegerlehner, Springer Tracts Mod. Phys. 274 (2017)

work page 2017

[5] [5]

Logashenko, et al

I. Logashenko, et al. [Muon g-2 Collaboration], J. Phys. Chem. Ref. Data 44, no. 3, 031211 (2015)

work page 2015

[6] [6]

Venanzoni [Fermilab E989 Collaboration], Nucl

G. Venanzoni [Fermilab E989 Collaboration], Nucl. Part. Phys. Proc. 273-275, 584 (2016)

work page 2016

[7] [7]

Otani [E34 Collaboration], JPS Conf

M. Otani [E34 Collaboration], JPS Conf. Proc. 8, 025008 (2015)

work page 2015

[8] [8]

Hagelstein and V

F. Hagelstein and V . Pascalutsa, Phys. Rev. Lett.120, no. 7, 072002 (2018)

work page 2018

[9] [9]

J. S. Schwinger, Proc. Nat. Acad. Sci. 72, 1 (1975); ibid. 72, 1559 (1975) [Acta Phys. Austriaca Suppl. 14, 471 (1975)]

work page 1975

[10] [10]

A. M. Harun ar-Rashid, Nuovo Cim. A 33, 447 (1976)

work page 1976

[11] [11]

Hagelstein, V

F. Hagelstein, V . Pascalutsa and M. Vanderhaeghen, in preparation

work page

[12] [12]

Pascalutsa, V

V . Pascalutsa, V . Pauk and M. Vanderhaeghen, Phys. Rev. D85, 116001 (2012)

work page 2012

[13] [13]

Knecht and A

M. Knecht and A. Nyffeler, Phys. Rev. D 65, 073034 (2002)

work page 2002

[14] [14]

Melnikov and A

K. Melnikov and A. Vainshtein, Phys. Rev. D 70, 113006 (2004)

work page 2004

[15] [15]

Hoferichter, B.-L

M. Hoferichter, B.-L. Hoid, B. Kubis, S. Leupold and S. P. Schneider, Phys. Rev. Lett. 121, 112002 (2018)

work page 2018

[16] [16]

Drell, Il Nuovo Cimento 11, 693-697 (1959)

S. Drell, Il Nuovo Cimento 11, 693-697 (1959)

work page 1959

[17] [17]

Ametller, A

L. Ametller, A. Bramon, E. Masso, Phys. Rev. D 30 251 (1984)

work page 1984

[18] [18]

A. E. Dorokhov and M. A. Ivanov, Phys. Rev. D 75 114007 (2007)

work page 2007

[19] [19]

Nyffeler, Phys

A. Nyffeler, Phys. Rev. D 79, 073012 (2009)

work page 2009

[20] [20]

Tanabashi, et al

M. Tanabashi, et al. (PDG), Phys. Rev. D 98, 030001 (2018)

work page 2018

[21] [21]

Grifﬁths, Introduction to Elementary Particles, WILEY-VCH Verlag, 2008

D. Grifﬁths, Introduction to Elementary Particles, WILEY-VCH Verlag, 2008. 7

work page 2008