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arxiv: 2605.29902 · v1 · pith:6QGU3DT5new · submitted 2026-05-28 · ✦ hep-lat · hep-ph

Electromagnetic pion mass splitting using a Pauli-Villars-regulated photon propagator

Alessandro De Santis , Dominik Erb , Harvey B. Meyer This is my paper

Pith reviewed 2026-06-28 23:43 UTC · model grok-4.3

classification ✦ hep-lat hep-ph
keywords lattice QCDelectromagnetic correctionspion mass splittingPauli-Villars regulatorCottingham formulafinite volume effects
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The pith

Lattice QCD with a Pauli-Villars photon regulator produces the pion electromagnetic mass splitting of 4.56 MeV after cutoff removal.

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

The paper establishes a lattice QCD result for the charged-neutral pion mass difference at electromagnetic order using a Pauli-Villars regulated photon propagator defined in the continuum and infinite-volume limit. The regulator introduces an auxiliary cutoff Lambda that eliminates power-law finite-volume effects and permits direct extrapolation to infinite volume. Calculations on CLS ensembles are performed at several Lambda values, followed by continuum and physical-point extrapolations and a Cottingham-formula decomposition into elastic and inelastic pieces at fixed Lambda. Removing the cutoff yields a final value in agreement with experiment, confirming that the regulated framework reproduces physical electromagnetic effects in this controlled case.

Core claim

Using the Pauli-Villars regulated photon propagator on CLS ensembles, with extrapolations in volume, lattice spacing, quark mass and the regulator scale Lambda to infinity, the electromagnetic contribution to the pion mass splitting is found to be 4.56(22) MeV.

What carries the argument

The Pauli-Villars regulated photon propagator, which serves as the electromagnetic interaction kernel and removes power-law finite-volume artifacts by construction.

If this is right

  • Electromagnetic corrections to other hadronic observables become accessible without power-law finite-volume corrections.
  • Elastic and inelastic contributions to the mass splitting can be isolated at fixed regulator scale before the final limit.
  • The same regulated propagator can be inserted into calculations of electromagnetic shifts for other quantities on the same ensembles.

Where Pith is reading between the lines

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

  • The approach may simplify electromagnetic corrections for baryon masses or form factors where finite-volume effects have been more severe.
  • Direct numerical comparison of the Cottingham decomposition against dispersion-relation evaluations at the same Lambda would test internal consistency.
  • Extension to non-degenerate quark masses would allow the method to address isospin-breaking effects beyond the pion.

Load-bearing premise

The regulated propagator, once the auxiliary cutoff is removed, reproduces the complete physical electromagnetic effects without residual cutoff dependence or unaccounted biases after all extrapolations.

What would settle it

A repeat of the calculation with an independent photon regulator that produces a splitting differing by more than the quoted uncertainty after all limits are taken would falsify the result.

read the original abstract

We present a lattice QCD calculation of the charged-neutral pion mass splitting $M_{\pi^+} - M_{\pi^0}$ at $\mathcal{O}(\alpha_\mathrm{em})$ using a recently proposed framework based on a Pauli-Villars (PV) regulated photon propagator defined in the continuum and infinite-volume limit, with $\Lambda$ acting as an additional UV cutoff scale. The use of this propagator avoids power-law finite-volume effects, allowing for a straightforward treatment of the infinite-volume limit. We perform the calculation using CLS ensembles, studying finite-volume effects, the continuum limit and the extrapolation to the physical point for several values of the scale $\Lambda$. By means of the Cottingham formula, we further decompose the result into elastic and inelastic contributions at fixed $\Lambda$. Our final result, after removing the cutoff scale $\Lambda$, is $M_{\pi^+} - M_{\pi^0} = 4.56(22)$ MeV, in good agreement with the experimental measurement. This calculation serves as a validation of the formalism in a well-controlled setting and offers useful insights into the application of electromagnetic corrections to other observables.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a lattice QCD calculation of the electromagnetic pion mass splitting ΔM_π = M_π⁺ − M_π⁰ at O(α_em) on CLS ensembles. A Pauli-Villars regulated photon propagator with UV cutoff Λ is employed in the continuum and infinite-volume limit to eliminate power-law finite-volume effects. For several fixed values of Λ the authors perform simultaneous extrapolations in lattice spacing a→0, spatial extent L→∞ and pion mass to the physical point; the resulting ΔM_π(Λ) values are then extrapolated to Λ→∞. The final result is 4.56(22) MeV, stated to be in agreement with experiment. The Cottingham formula is used to separate elastic and inelastic contributions at each Λ.

Significance. If the multi-stage extrapolation procedure is shown to be free of uncontrolled systematics, the work supplies a controlled validation of the PV-regulated formalism for electromagnetic corrections. The avoidance of power-law FV effects is a technical advantage that could benefit other observables. The quoted 0.22 MeV uncertainty and the decomposition into elastic/inelastic pieces provide concrete benchmarks. The result is not parameter-free and relies on external ensembles, but the direct numerical approach (no fitting to the target splitting itself) is a positive feature.

major comments (2)
  1. [Section 5 (Λ extrapolation and final result)] The headline result 4.56(22) MeV is obtained only after the final Λ→∞ extrapolation performed on top of per-Λ continuum/volume/physical-point fits. The manuscript must demonstrate that the functional form adopted for this extrapolation (presumably a power series in 1/Λ) reproduces the unregulated Cottingham integral without residual O(1/Λ^n) or mixed-term biases once the other limits have already been taken; any mismatch would shift the central value at a level comparable to the quoted uncertainty.
  2. [Section 4.2 and Table 2] Table 2 (or equivalent) reports the per-Λ extrapolated values and their uncertainties before the Λ→∞ step. The combined statistical and systematic error budget for each Λ must be shown to be stable under reasonable variations of the fit ansätze (e.g., inclusion or exclusion of a^2, 1/L^3, m_π^2 terms) so that the subsequent Λ extrapolation does not absorb unquantified systematics from the earlier stages.
minor comments (2)
  1. [Abstract] The abstract states agreement with experiment but does not quote the experimental central value or uncertainty; a brief comparison sentence would improve readability.
  2. [Section 2] Notation for the PV regulator (definition of the regulated propagator, precise relation between Λ and the Pauli-Villars mass) should be collected in a single equation early in the methods section for easy reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and will revise the manuscript accordingly to strengthen the analysis and presentation.

read point-by-point responses

  1. Referee: [Section 5 (Λ extrapolation and final result)] The headline result 4.56(22) MeV is obtained only after the final Λ→∞ extrapolation performed on top of per-Λ continuum/volume/physical-point fits. The manuscript must demonstrate that the functional form adopted for this extrapolation (presumably a power series in 1/Λ) reproduces the unregulated Cottingham integral without residual O(1/Λ^n) or mixed-term biases once the other limits have already been taken; any mismatch would shift the central value at a level comparable to the quoted uncertainty.

    Authors: We agree that explicit validation of the Λ→∞ extrapolation is required to confirm the absence of residual biases. In the revised manuscript we will add a dedicated subsection in Section 5 that (i) motivates the power-series ansatz from the known large-Λ expansion of the PV-regulated Cottingham integrand, (ii) performs the extrapolation with and without higher-order terms (1/Λ², 1/Λ³) and reports the stability of the central value and uncertainty, and (iii) compares the final extrapolated result against a direct evaluation of the unregulated Cottingham formula on the same ensembles at the largest accessible Λ. These checks will quantify any potential mismatch at the level of the quoted 0.22 MeV uncertainty. revision: yes

  2. Referee: [Section 4.2 and Table 2] Table 2 (or equivalent) reports the per-Λ extrapolated values and their uncertainties before the Λ→∞ step. The combined statistical and systematic error budget for each Λ must be shown to be stable under reasonable variations of the fit ansätze (e.g., inclusion or exclusion of a², 1/L³, m_π² terms) so that the subsequent Λ extrapolation does not absorb unquantified systematics from the earlier stages.

    Authors: We acknowledge the need to demonstrate robustness of the per-Λ extrapolations. The revised manuscript will include an expanded Table 2 (or a new supplementary table) that lists results for each Λ under at least four fit variations: the baseline ansatz, the baseline plus an a² term, the baseline plus a 1/L³ term, and the baseline plus an m_π² term. For each variation we will report χ²/dof, the extracted ΔM_π(Λ), and the combined statistical-plus-systematic uncertainty. We will show that the central values remain consistent within the quoted errors and that the final Λ→∞ extrapolation is insensitive to these choices at the present precision. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct lattice evaluation on external ensembles

full rationale

The paper computes the pion mass splitting via lattice QCD on CLS ensembles using a PV-regulated photon propagator, performing independent extrapolations in a, L, m_π and Λ before taking the final Λ→∞ limit. The quoted result is obtained from these numerical evaluations and compared to experiment rather than fitted to it. No load-bearing step reduces the output to the input by construction, self-definition, or unverified self-citation chain; the framework is applied as an external regulator whose effects are removed by explicit extrapolation. This is a standard self-contained first-principles calculation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted from the provided text. The method relies on standard lattice QCD assumptions plus the new regulator whose detailed properties are not stated here.

pith-pipeline@v0.9.1-grok · 5731 in / 1160 out tokens · 24238 ms · 2026-06-28T23:43:42.901787+00:00 · methodology

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Reference graph

Works this paper leans on

41 extracted references · 34 canonical work pages · 16 internal anchors

  1. [1]

    RCstar collaboration, First results on QCD+QED with C ⋆ boundary conditions, JHEP 03 (2023) 012 [ 2209.13183]

    work page arXiv 2023

  2. [2]

    RC⋆ collaboration, Comparing QCD+QED via full simulation versus the RM123 method: U-spin window contribution to aHVP µ , JHEP 10 (2025) 158 [ 2506.19770]

    work page arXiv 2025

  3. [3]

    RM123 collaboration, Leading isospin breaking effects on the lattice , Phys. Rev. D 87 (2013) 114505 [1303.4896]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  4. [4]

    Cottingham, The neutron proton mass difference and electron scattering experiments , Annals Phys

    W.N. Cottingham, The neutron proton mass difference and electron scattering experiments , Annals Phys. 25 (1963) 424

    1963

  5. [5]

    Gasser and H

    J. Gasser and H. Leutwyler, Implications of Scaling for the Proton - Neutron Mass - Difference, Nucl. Phys. B 94 (1975) 269

    1975

  6. [6]

    Gasser and H

    J. Gasser and H. Leutwyler, Quark Masses, Phys. Rept. 87 (1982) 77

    1982

  7. [7]

    QED in finite volume and finite size scaling effect on electromagnetic properties of hadrons

    M. Hayakawa and S. Uno, QED in finite volume and finite size scaling effect on electromagnetic properties of hadrons, Prog. Theor. Phys. 120 (2008) 413 [ 0804.2044]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  8. [8]

    Borsanyi et al., Leading hadronic contribution to the muon magnetic moment from lattice QCD, Nature 593 (2021) 51 [ 2002.12347]

    S. Borsanyi et al., Leading hadronic contribution to the muon magnetic moment from lattice QCD, Nature 593 (2021) 51 [ 2002.12347]

    work page arXiv 2021

  9. [9]

    Position-space approach to hadronic light-by-light scattering in the muon $g-2$ on the lattice

    N. Asmussen, J. Green, H.B. Meyer and A. Nyffeler, Position-space approach to hadronic light-by-light scattering in the muon g − 2 on the lattice , PoS LATTICE2016 (2016) 164 [1609.08454]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  10. [10]

    Charged hadrons in local finite-volume QED+QCD with C* boundary conditions

    B. Lucini, A. Patella, A. Ramos and N. Tantalo, Charged hadrons in local finite-volume QED+QCD with C ⋆ boundary conditions, JHEP 02 (2016) 076 [ 1509.01636]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  11. [11]

    Massive photons: an infrared regularization scheme for lattice QCD+QED

    M.G. Endres, A. Shindler, B.C. Tiburzi and A. Walker-Loud, Massive photons: an infrared regularization scheme for lattice QCD+QED , Phys. Rev. Lett. 117 (2016) 072002 [1507.08916]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  12. [12]

    Di Carlo, M.T

    M. Di Carlo, M.T. Hansen, N. Hermansson-Truedsson and A. Portelli, QEDr: a finite-volume QED action with redistributed spatial zero-momentum modes , JHEP 10 (2025) 026 [2501.07936]

    work page arXiv 2025

  13. [13]

    The anomalous magnetic moment of the muon in the Standard Model: an update

    R. Aliberti et al., The anomalous magnetic moment of the muon in the Standard Model: an update, Phys. Rept. 1143 (2025) 1 [ 2505.21476]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  14. [14]

    Biloshytskyi, E.-H

    V. Biloshytskyi, E.-H. Chao, A. G´ erardin, J.R. Green, F. Hagelstein, H.B. Meyer et al., Forward light-by-light scattering and electromagnetic correction to hadronic vacuum polarization, JHEP 03 (2023) 194 [ 2209.02149]

    work page arXiv 2023

  15. [15]

    Parrino, V

    J. Parrino, V. Biloshytskyi, E.-H. Chao, H.B. Meyer and V. Pascalutsa, Computing the UV-finite electromagnetic corrections to the hadronic vacuum polarization in the muon (g −

  16. [16]

    from lattice QCD , JHEP 07 (2025) 201 [ 2501.03192]

    work page arXiv 2025

  17. [17]

    D. Erb, A. G´ erardin, H.B. Meyer, J. Parrino, V. Biloshytskyi and V. Pascalutsa, Isospin-violating vacuum polarization in the muon (g − 2) with SU(3) flavour symmetry from lattice QCD, JHEP 10 (2025) 157 [ 2505.24344]

    work page arXiv 2025

  18. [18]

    Frezzotti, G

    R. Frezzotti, G. Gagliardi, V. Lubicz, G. Martinelli, F. Sanfilippo and S. Simula, Lattice calculation of the pion mass difference M π+-Mπ0 at order O( αem), Phys. Rev. D 106 (2022) 014502 [ 2202.11970]. – 27 –

    work page arXiv 2022

  19. [19]

    X. Feng, L. Jin and M.J. Riberdy, Lattice QCD Calculation of the Pion Mass Splitting , Phys. Rev. Lett. 128 (2022) 052003 [ 2108.05311]

    work page arXiv 2022

  20. [20]

    Bonanno et al., Lattice determination of the QCD low-energy constant ℓ7, Phys

    C. Bonanno et al., Lattice determination of the QCD low-energy constant ℓ7, Phys. Rev. D 113 (2026) 074510 [ 2512.09880]

    work page arXiv 2026

  21. [21]

    Das, G.S

    T. Das, G.S. Guralnik, V.S. Mathur, F.E. Low and J.E. Young, Electromagnetic mass difference of pions, Phys. Rev. Lett. 18 (1967) 759

    1967

  22. [22]

    Chiral Sum Rules and Their Phenomenology

    J.F. Donoghue and E. Golowich, Chiral sum rules and their phenomenology , Phys. Rev. D 49 (1994) 1513 [ hep-ph/9307262]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  23. [23]

    OPAL collaboration, Measurement of the strong coupling constant alpha(s) and the vector and axial vector spectral functions in hadronic tau decays , Eur. Phys. J. C 7 (1999) 571 [hep-ex/9808019]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  24. [24]

    The Physics of Hadronic Tau Decays

    M. Davier, A. Hocker and Z. Zhang, The Physics of Hadronic Tau Decays , Rev. Mod. Phys. 78 (2006) 1043 [ hep-ph/0507078]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  25. [25]

    Bijnens, N

    J. Bijnens, N. Hermansson-Truedsson and A. Rodr´ ıguez-S´ anchez,The Pion Sum Rule beyond the Chiral Limit , 2602.05672

    work page arXiv

  26. [26]

    Strassberger et al., Scale setting for CLS 2+1 simulations , PoS LATTICE2021 (2022) 135 [2112.06696]

    B. Strassberger et al., Scale setting for CLS 2+1 simulations , PoS LATTICE2021 (2022) 135 [2112.06696]

    work page arXiv 2022

  27. [27]

    RQCD collaboration, Scale setting and the light baryon spectrum in N f = 2 + 1 QCD with Wilson fermions, JHEP 05 (2023) 035 [ 2211.03744]

    work page arXiv 2023

  28. [28]

    Djukanovic, G

    D. Djukanovic, G. von Hippel, S. Kuberski, H.B. Meyer, N. Miller, K. Ottnad et al., The hadronic vacuum polarization contribution to the muon g − 2 at long distances , JHEP 04 (2025) 098 [ 2411.07969]

    work page arXiv 2025

  29. [29]

    Simulation of QCD with N_f=2+1 flavors of non-perturbatively improved Wilson fermions

    M. Bruno et al., Simulation of QCD with N f = 2 + 1 flavors of non-perturbatively improved Wilson fermions, JHEP 02 (2015) 043 [ 1411.3982]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  30. [30]

    RQCD collaboration, Lattice simulations with Nf = 2 + 1 improved Wilson fermions at a fixed strange quark mass , Phys. Rev. D 94 (2016) 074501 [ 1606.09039]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  31. [31]

    Non-perturbative renormalization and O$(a)$-improvement of the non-singlet vector current with $N_{\mathrm{f}}=2+1$ Wilson fermions and tree-level Symanzik improved gauge action

    A. Gerardin, T. Harris and H.B. Meyer, Nonperturbative renormalization and O(a)-improvement of the nonsinglet vector current with Nf = 2 + 1 Wilson fermions and tree-level Symanzik improved gauge action , Phys. Rev. D 99 (2019) 014519 [ 1811.08209]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  32. [32]

    Feng and L

    X. Feng and L. Jin, QED self energies from lattice QCD without power-law finite-volume errors, Phys. Rev. D 100 (2019) 094509 [ 1812.09817]

    work page arXiv 2019

  33. [33]

    Electromagnetic corrections in hadronic processes

    J. Gasser, A. Rusetsky and I. Scimemi, Electromagnetic corrections in hadronic processes, Eur. Phys. J. C 32 (2003) 97 [ hep-ph/0305260]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  34. [34]

    Jay and E.T

    W.I. Jay and E.T. Neil, Bayesian model averaging for analysis of lattice field theory results , Phys. Rev. D 103 (2021) 114502 [ 2008.01069]

    work page arXiv 2021

  35. [35]

    Akaike, A new look at the statistical model identification , IEEE Trans

    H. Akaike, A new look at the statistical model identification , IEEE Trans. Automatic Control 19 (1974) 716

    1974

  36. [36]

    Stamen, D

    D. Stamen, D. Hariharan, M. Hoferichter, B. Kubis and P. Stoffer, Kaon electromagnetic form factors in dispersion theory , Eur. Phys. J. C 82 (2022) 432 [ 2202.11106]

    work page arXiv 2022

  37. [37]

    Particle Data Group collaboration, Review of particle physics , Phys. Rev. D 110 (2024) 030001. – 28 –

    2024

  38. [38]

    Lorentz-covariant coordinate-space representation of the leading hadronic contribution to the anomalous magnetic moment of the muon

    H.B. Meyer, Lorentz-covariant coordinate-space representation of the leading hadronic contribution to the anomalous magnetic moment of the muon , Eur. Phys. J. C 77 (2017) 616 [1706.01139]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  39. [39]

    Erb, H.B

    D. Erb, H.B. Meyer and K. Ottnad, Factorizing the position-space photon propagator in QED corrections to lattice QCD correlators , 2603.13086

    work page arXiv

  40. [40]

    Use of stochastic sources for the lattice determination of light quark physics

    P.A. Boyle, A. Juttner, C. Kelly and R.D. Kenway, Use of stochastic sources for the lattice determination of light quark physics , JHEP 08 (2008) 086 [ 0804.1501]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  41. [41]

    Stochastic Estimation with $Z_2$ Noise

    S.-J. Dong and K.-F. Liu, Stochastic estimation with Z(2) noise , Phys. Lett. B 328 (1994) 130 [hep-lat/9308015]. – 29 –

    work page internal anchor Pith review Pith/arXiv arXiv 1994