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arxiv: 2512.10709 · v3 · submitted 2025-12-11 · ✦ hep-ph · hep-ex· nucl-th

Recognition: 2 theorem links

· Lean Theorem

Disperon QED

Yizhou Fang , Sophie Kollatzsch , Marco Rocco , Adrian Signer , Yannick Ulrich , Max Zoller
Authors on Pith no claims yet

Pith reviewed 2026-05-16 23:07 UTC · model grok-4.3

classification ✦ hep-ph hep-exnucl-th
keywords dispersion relationshadronic vacuum polarisationMonte Carlo event generatorstwo-loop QEDpion vector form factoree to pi pieffective field theorythreshold subtraction
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The pith

Disperon QED inserts experimental hadronic data into automated two-loop calculations via dispersion relations and threshold subtraction.

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

The paper introduces disperon QED as a technique to feed real data into theoretical loop computations inside Monte Carlo generators. It combines dispersion relations that link hadronic vacuum polarization to measured quantities, a threshold subtraction to manage the low-energy region, and effective-field-theory approximations that allow automated tools such as OpenLoops to evaluate the resulting integrals. The method is demonstrated on electron-positron annihilation into two pions, where it accounts for hadronic vacuum-polarisation insertions at two loops and extracts the pion vector form factor under the scalar-QED approximation. A sympathetic reader would care because many high-precision observables depend on these hadronic contributions, and direct first-principles calculations remain intractable at the required accuracy.

Core claim

We present disperon QED, a method to deal with data input in loop processes in Monte Carlo codes. It relies on dispersion relations, automated tools such as OpenLoops, effective field theory methods and a threshold subtraction. We apply it to the process ee to pi pi in McMule to deal with hadronic vacuum polarisation insertions in two-loop contributions as well as the vector form factor of the pion within the form-factor scalar QED approximation. The generality of this method for more complicated processes is emphasised.

What carries the argument

Dispersion relations with threshold subtraction and effective-field-theory approximations that convert hadronic data into integrals evaluable by automated loop tools.

If this is right

  • It treats hadronic vacuum-polarisation insertions inside two-loop contributions to ee to pi pi.
  • It extracts the pion vector form factor inside the scalar-QED approximation.
  • It integrates directly with existing Monte Carlo frameworks such as McMule and OpenLoops.
  • It extends to other processes that require data-driven hadronic inputs at loop level.

Where Pith is reading between the lines

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

  • The same dispersion-plus-subtraction pattern could be applied to processes that feed into the muon anomalous magnetic moment.
  • It offers a route to lower theoretical uncertainties in precision observables by replacing model-dependent hadronic estimates with direct data.
  • The approach could be tested by extending the calculation to three-loop order or to different final states such as four pions.

Load-bearing premise

Dispersion relations combined with threshold subtraction and effective-field-theory approximations can be inserted into automated loop tools without introducing uncontrolled systematic errors when hadronic data are used.

What would settle it

A high-precision measurement of the two-loop cross section for ee to pi pi that deviates from the disperon-QED prediction by more than the quoted theoretical uncertainty would show the method introduces uncontrolled errors.

Figures

Figures reproduced from arXiv: 2512.10709 by Adrian Signer, Marco Rocco, Max Zoller, Sophie Kollatzsch, Yannick Ulrich, Yizhou Fang.

Figure 1
Figure 1. Figure 1: The three tree-level diagrams that need to be calculated for the matching to DET. The [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Single-dispersive contribution to the integrand in ( [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Double-dispersive contribution for fixed kinematics, similar to Figure [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Topologies contributing to the threshold singularity in [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The top panel shows the distribution for [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Same plot as in Figure [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Different versions of the forward-backward charge asymmetry defined in ( [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The contour C for the dispersion relation. γ ∗ (k1)γ ∗ (k2) → π +(p)π −(q) Wµν(p, q; k1, k2) = Z d 4x e−ik1·x ⟨π +(p)π −(q)|j µ em(0)j ν em(x)|0⟩ = π +(p) π −(q) γ ∗ (k1) γ ∗ (k2) . (66) The full tensor is a complicated, non-perturbative object. However, it can be written in terms of five form factors that can in principle be measured or estimated [29,30,103]. Alternatively, we can derive a dispersive deco… view at source ↗
read the original abstract

We present disperon QED, a method to deal with data input in loop processes in Monte Carlo codes. It relies on dispersion relations, automated tools such as OpenLoops, effective field theory methods and a threshold subtraction. We motivate this method and apply it to the process $ee\to\pi\pi$ in McMule to deal with hadronic vacuum polarisation insertions in two-loop contributions as well as the vector form factor of the pion within the form-factor scalar QED approximation. The generality of this method for more complicated processes is emphasised.

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 paper introduces 'disperon QED', a method that combines dispersion relations, automated one-loop tools such as OpenLoops, effective-field-theory approximations, and a threshold subtraction to incorporate experimental hadronic data into loop calculations inside Monte Carlo frameworks. The method is demonstrated on the process e⁺e⁻ → π⁺π⁻ within the McMule generator, specifically for two-loop hadronic-vacuum-polarisation insertions and for the pion vector form factor evaluated in the scalar-QED approximation. The authors stress that the construction is general and can be extended to more complicated processes.

Significance. If the numerical stability of the subtracted dispersion integrals and the controlled matching to the EFT approximation can be established, the approach would provide a practical route for embedding data-driven hadronic inputs into existing automated loop infrastructures. This would be particularly useful for precision QED observables that receive hadronic corrections, such as those relevant to the muon anomalous magnetic moment.

major comments (2)
  1. [§4] §4 (application to two-loop HVP): the manuscript must supply an explicit error budget or direct numerical comparison against an independent evaluation of the same two-loop HVP contribution; without it the claim that the threshold subtraction renders the integrals stable remains unverified.
  2. [§3.2] §3.2 (scalar-QED form-factor approximation): the matching between the dispersion integral and the EFT form factor is performed at a fixed order; the paper should demonstrate that residual higher-order EFT terms do not exceed the target precision when the hadronic data are inserted into the Monte Carlo infrastructure.
minor comments (2)
  1. [Figure 2] Figure 2: the caption should explicitly state the integration contour and the value of the subtraction point used in the dispersion relation.
  2. [§2] Notation: the symbol for the subtracted dispersion kernel is introduced without a dedicated equation; adding a numbered definition would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major points below and will revise the manuscript accordingly to strengthen the numerical validation.

read point-by-point responses

  1. Referee: [§4] §4 (application to two-loop HVP): the manuscript must supply an explicit error budget or direct numerical comparison against an independent evaluation of the same two-loop HVP contribution; without it the claim that the threshold subtraction renders the integrals stable remains unverified.

    Authors: We agree that an explicit numerical validation strengthens the presentation. In the revised version we will add a direct comparison of the two-loop HVP contribution obtained with the disperon-QED implementation against an independent dispersive evaluation (using the same hadronic data but without Monte-Carlo integration). We will also include a concise error budget that separates the contributions from experimental input uncertainties, the threshold-subtraction procedure, and numerical integration errors, thereby confirming the stability of the subtracted integrals. revision: yes

  2. Referee: [§3.2] §3.2 (scalar-QED form-factor approximation): the matching between the dispersion integral and the EFT form factor is performed at a fixed order; the paper should demonstrate that residual higher-order EFT terms do not exceed the target precision when the hadronic data are inserted into the Monte Carlo infrastructure.

    Authors: The scalar-QED framework is employed as a controlled test-bed to illustrate the method. The matching is performed at the order required by the target precision of the McMule implementation. In the revision we will add a dedicated paragraph that uses power counting and a numerical estimate of the next-to-leading EFT corrections to show that their size remains well below the precision goal for the pion vector form factor. This establishes that the fixed-order matching is sufficient for the present application while leaving open the possibility of higher-order matching for future, more demanding processes. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents disperon QED as a methodological combination of dispersion relations, threshold subtraction, EFT approximations, OpenLoops automation and McMule for inserting hadronic data into loop integrals. The ee→ππ application is described as a concrete demonstration within the scalar-QED form-factor approximation; no derivation chain, equations or self-citations are shown that reduce any claimed prediction or result to a fitted input or prior self-result by construction. The construction is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; the central claim rests on the applicability of dispersion relations and EFT approximations to hadronic vacuum polarisation, but no explicit free parameters, axioms, or invented entities are stated.

axioms (2)
  • domain assumption Dispersion relations can be used to incorporate experimental data into loop amplitudes
    Invoked to handle hadronic vacuum polarisation insertions
  • domain assumption Threshold subtraction removes singularities without affecting physical results
    Used to regularise the data input procedure

pith-pipeline@v0.9.0 · 5387 in / 1355 out tokens · 60417 ms · 2026-05-16T23:07:17.160519+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

107 extracted references · 107 canonical work pages · 47 internal anchors

  1. [1]

    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

  2. [2]

    Borsanyi et al.,Leading hadronic contribution to the muon magnetic moment from lattice QCD,Nature593(2021) 51 [2002.12347]

    S. Borsanyi et al.,Leading hadronic contribution to the muon magnetic moment from lattice QCD,Nature593(2021) 51 [2002.12347]. [3]RBC, UKQCDcollaboration, T. Blum et al.,Update of Euclidean windows of the hadronic vacuum polarization,Phys. Rev. D108(2023) 054507 [2301.08696]

    work page arXiv 2021

  3. [3]

    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 muong−2at long distances,JHEP04 (2025) 098 [2411.07969]

    work page arXiv 2025

  4. [4]

    Afanasev et al.,Radiative corrections: from medium to high energy experiments,Eur

    A. Afanasev et al.,Radiative corrections: from medium to high energy experiments,Eur. Phys. J. A60(2024) 91 [2306.14578]

    work page arXiv 2024

  5. [5]

    Pohl et al.,The size of the proton,Nature466(2010) 213

    R. Pohl et al.,The size of the proton,Nature466(2010) 213

    work page 2010

  6. [6]

    Aliberti et al.,Radiative corrections and Monte Carlo tools for low-energy hadronic cross sections in𝑒+𝑒− collisions,2410.22882

    R. Aliberti et al.,Radiative corrections and Monte Carlo tools for low-energy hadronic cross sections ine +e− collisions,SciPost Phys. Comm. Rep.2025(2025) 9 [2410.22882]

    work page arXiv 2025

  7. [7]

    R. D. Bucoveanu and H. Spiesberger,Second-Order Leptonic Radiative Corrections for Lepton-Proton Scattering,Eur. Phys. J. A55(2019) 57 [1811.04970]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  8. [8]

    Banerjee, T

    P. Banerjee, T. Engel, A. Signer and Y. Ulrich,QED at NNLO with McMule,SciPost Phys.9(2020) 027 [2007.01654]

    work page arXiv 2020

  9. [9]

    C. M. Carloni Calame, M. Chiesa, S. M. Hasan, G. Montagna, O. Nicrosini and F. Piccinini,Towards muon-electron scattering at NNLO,JHEP11(2020) 028 [2007.01586]

    work page arXiv 2020

  10. [10]

    Banerjee, T

    P. Banerjee, T. Engel, N. Schalch, A. Signer and Y. Ulrich,Bhabha scattering at NNLO with next-to-soft stabilisation,Phys. Lett. B820(2021) 136547 [2106.07469]

    work page arXiv 2021

  11. [11]

    Banerjee, T

    P. Banerjee, T. Engel, N. Schalch, A. Signer and Y. Ulrich,Møller scattering at NNLO, Phys. Rev. D105(2022) L031904 [2107.12311]

    work page arXiv 2022

  12. [12]

    Budassi, C

    E. Budassi, C. M. Carloni Calame, M. Chiesa, C. L. Del Pio, S. M. Hasan, G. Montagna et al.,NNLO virtual and real leptonic corrections to muon-electron scattering,JHEP11 (2021) 098 [2109.14606]. 28

    work page arXiv 2021

  13. [13]

    Broggio et al.,Muon-electron scattering at NNLO,JHEP01(2023) 112 [2212.06481]

    A. Broggio et al.,Muon-electron scattering at NNLO,JHEP01(2023) 112 [2212.06481]

    work page arXiv 2023

  14. [14]

    Engel, F

    T. Engel, F. Hagelstein, M. Rocco, V. Sharkovska, A. Signer and Y. Ulrich,Impact of NNLO QED corrections on lepton-proton scattering at MUSE,Eur. Phys. J. A59(2023) 253 [2307.16831]

    work page arXiv 2023

  15. [15]

    Matching perturbative and Parton Shower corrections to Bhabha process at flavour factories

    G. Balossini, C. M. Carloni Calame, G. Montagna, O. Nicrosini and F. Piccinini,Matching perturbative and parton shower corrections to Bhabha process at flavour factories,Nucl. Phys. B758(2006) 227 [hep-ph/0607181]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  16. [16]

    Photon pair production at flavour factories with per mille accuracy

    G. Balossini, C. Bignamini, C. M. C. Calame, G. Montagna, O. Nicrosini and F. Piccinini, Photon pair production at flavour factories with per mille accuracy,Phys. Lett. B663 (2008) 209 [0801.3360]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  17. [17]

    Budassi, C

    E. Budassi, C. M. Carloni Calame, M. Ghilardi, A. Gurgone, G. Montagna, M. Moretti et al.,Pion pair production ine +e− annihilation at next-to-leading order matched to Parton Shower,JHEP05(2025) 196 [2409.03469]

    work page arXiv 2025

  18. [18]

    Towards a Fully Automated Differential $\text{NNLO}_\text{EW}$ Generator for Lepton Colliders

    A. Price and F. Krauss,Towards a Fully Automated Differential NNLO EW Generator for Lepton Colliders,2512.04959

    work page internal anchor Pith review Pith/arXiv arXiv

  19. [19]

    Muon-electron scattering at NLO

    M. Alacevich, C. M. Carloni Calame, M. Chiesa, G. Montagna, O. Nicrosini and F. Piccinini,Muon-electron scattering at NLO,JHEP02(2019) 155 [1811.06743]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  20. [20]

    Arbuzov, S

    A. Arbuzov, S. Bondarenko, Y. Dydyshka, L. Kalinovskaya, L. Rumyantsev, R. Sadykov et al.,Effects of Electroweak Radiative Corrections in Polarized Low-Energy Electron–Positron Annihilation into Lepton Pairs,JETP Lett.116(2022) 199 [2206.09469]

    work page arXiv 2022

  21. [21]

    Kollatzsch and Y

    S. Kollatzsch and Y. Ulrich,Lepton pair production at NNLO in QED with EW effects, SciPost Phys.15(2023) 104 [2210.17172]

    work page arXiv 2023

  22. [22]

    Kollatzsch, D

    S. Kollatzsch, D. Moreno, D. Radic and A. Signer,Parity violation in Møller scattering within low-energy effective field theory,JHEP09(2025) 196 [2507.17652]

    work page arXiv 2025

  23. [23]

    V. S. Fadin and R. N. Lee,Two-loop radiative corrections toe +e− →γγ ∗ cross section, JHEP11(2023) 148 [2308.09479]

    work page arXiv 2023

  24. [24]

    Badger, J

    S. Badger, J. Kry´ s, R. Moodie and S. Zoia,Lepton-pair scattering with an off-shell and an on-shell photon at two loops in massless QED,JHEP11(2023) 041 [2307.03098]

    work page arXiv 2023

  25. [25]

    Badger, M

    S. Badger, M. Czakon, H. B. Hartanto, R. Moodie, T. Peraro, R. Poncelet et al.,Isolated photon production in association with a jet pair through next-to-next-to-leading order in QCD,JHEP10(2023) 071 [2304.06682]

    work page arXiv 2023

  26. [26]

    Petit Ros` as and W

    P. Petit Ros` as and W. J. Torres Bobadilla,Fast evaluation of Feynman integrals for Monte Carlo generators,JHEP09(2025) 210 [2507.12548]

    work page arXiv 2025

  27. [27]

    Virtual photon-photon scattering

    M. Hoferichter, G. Colangelo, M. Procura and P. Stoffer,Virtual photon-photon scattering, Int. J. Mod. Phys. Conf. Ser.35(2014) 1460400 [1309.6877]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  28. [28]

    Dispersion relation for hadronic light-by-light scattering: theoretical foundations

    G. Colangelo, M. Hoferichter, M. Procura and P. Stoffer,Dispersion relation for hadronic light-by-light scattering: theoretical foundations,JHEP09(2015) 074 [1506.01386]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  29. [29]

    Dispersion relations for $\gamma^*\gamma^*\to\pi\pi$: helicity amplitudes, subtractions, and anomalous thresholds

    M. Hoferichter and P. Stoffer,Dispersion relations forγ ∗γ∗ →ππ: helicity amplitudes, subtractions, and anomalous thresholds,JHEP07(2019) 073 [1905.13198]. 29

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  30. [30]

    L. W. Mo and Y.-S. Tsai,Radiative Corrections to Elastic and Inelasticepandµp Scattering,Rev. Mod. Phys.41(1969) 205

    work page 1969

  31. [31]

    L. C. Maximon and J. A. Tjon,Radiative corrections to electron proton scattering,Phys. Rev. C62(2000) 054320

    work page 2000

  32. [32]

    Emilis Kaziuk˙ enas,Dispersive definition ofγ ∗γ∗γ→π +π− for muong−2applications, in RadioMonteCarLow 2 Satellite 2025, (Liverpool), 2025, https://indico.ph.liv.ac.uk/event/2169/contributions/10215/

    work page 2025

  33. [33]

    Muon-electron scattering at NNLO: the hadronic corrections

    M. Fael and M. Passera,Muon-Electron Scattering at Next-To-Next-To-Leading Order: The Hadronic Corrections,Phys. Rev. Lett.122(2019) 192001 [1901.03106]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  34. [34]

    Hadronic corrections to $\mu$-$e$ scattering at NNLO with space-like data

    M. Fael,Hadronic corrections toµ-escattering at NNLO with space-like data,JHEP02 (2019) 027 [1808.08233]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  35. [35]

    Balzani, S

    E. Balzani, S. Laporta and M. Passera,Hadronic vacuum polarization contributions to the muong−2in the space-like region,Phys. Lett. B834(2022) 137462 [2112.05704]

    work page arXiv 2022

  36. [36]

    J. J. Sakurai and D. Schildknecht,Generalized vector dominance and inelastic electron - proton scattering,Phys. Lett. B40(1972) 121

    work page 1972

  37. [37]

    Ignatov and R

    F. Ignatov and R. N. Lee,Charge asymmetry ine +e− →π +π− process,Phys. Lett. B833 (2022) 137283 [2204.12235]

    work page arXiv 2022

  38. [38]

    A. V. Gramolin, V. S. Fadin, A. L. Feldman, R. E. Gerasimov, D. M. Nikolenko, I. A. Rachek et al.,A new event generator for the elastic scattering of charged leptons on protons,J. Phys.G41(2014) 115001

    work page 2014

  39. [39]

    R. E. Gerasimov and V. S. Fadin,Analysis of approximations used in calculations of radiative corrections to electron-proton scattering cross section,Phys. Atom. Nucl.78 (2015) 69

    work page 2015

  40. [40]

    Bystritskiy, E

    Y. Bystritskiy, E. Kuraev and E. Tomasi-Gustafsson,Structure function method applied to polarized and unpolarized electron-proton scattering: A solution of theG E(p)/GM(p) discrepancy,Phys. Rev.C75(2007) 015207

    work page 2007

  41. [41]

    Choudhary, U

    P. Choudhary, U. Raha, F. Myhrer and D. Chakrabarti,Analytical evaluation of elastic lepton-proton two-photon exchange in chiral perturbation theory,Eur. Phys. J. A60(2024) 69 [2306.09454]

    work page arXiv 2024

  42. [42]

    S. P. Dye, M. Gonderinger and G. Paz,Elements of QED-NRQED effective field theory: NLO scattering at leading power,Phys. Rev. D94(2016) 013006 [1602.07770]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  43. [43]

    S. P. Dye, M. Gonderinger and G. Paz,Elements of QED-NRQED Effective Field Theory: II. Matching of Contact Interactions,Phys. Rev. D100(2019) 054010 [1812.05056]

    work page arXiv 2019

  44. [44]

    Two-photon exchange corrections to elastic $e^-$-proton scattering: Full dispersive treatment of $\pi N$ states at low momentum transfers

    O. Tomalak, B. Pasquini and M. Vanderhaeghen,Two-photon exchange corrections to elastice −-proton scattering: Full dispersive treatment ofπNstates at low momentum transfers,Phys. Rev. D95(2017) 096001 [1612.07726]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  45. [45]

    Two-photon exchange contribution to elastic $e^-$-proton scattering: Full dispersive treatment of $\pi N$ states and comparison with data

    O. Tomalak, B. Pasquini and M. Vanderhaeghen,Two-photon exchange contribution to elastice − -proton scattering: Full dispersive treatment ofπNstates and comparison with data,Phys. Rev. D96(2017) 096001 [1708.03303]. 30

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  46. [46]

    Ahmed, P

    J. Ahmed, P. G. Blunden and W. Melnitchouk,Two-photon exchange from intermediate state resonances in elastic electron-proton scattering,Phys. Rev. C102(2020) 045205 [2006.12543]

    work page arXiv 2020

  47. [47]

    C. E. Carlson and M. Vanderhaeghen,Two-Photon Physics in Hadronic Processes,Ann. Rev. Nucl. Part. Sci.57(2007) 171 [hep-ph/0701272]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  48. [48]

    Review of two-photon exchange in electron scattering

    J. Arrington, P. G. Blunden and W. Melnitchouk,Review of two-photon exchange in electron scattering,Prog. Part. Nucl. Phys.66(2011) 782 [1105.0951]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  49. [49]

    Two-photon exchange in elastic electron-proton scattering

    A. Afanasev, P. G. Blunden, D. Hasell and B. A. Raue,Two-photon exchange in elastic electron–proton scattering,Prog. Part. Nucl. Phys.95(2017) 245 [1703.03874]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  50. [50]

    Borisyuk and A

    D. Borisyuk and A. Kobushkin,Two-Photon Exchange in Elastic Electron Scattering on Hadronic Systems,Ukr. J. Phys.66(2021) 3 [1911.10956]

    work page arXiv 2021

  51. [51]

    Cabibbo and R

    N. Cabibbo and R. Gatto,Electron Positron Colliding Beam Experiments,Phys. Rev.124 (1961) 1577

    work page 1961

  52. [52]

    Colangelo, M

    G. Colangelo, M. Hoferichter, J. Monnard and J. R. de Elvira,Radiative corrections to the forward-backward asymmetry ine +e− →π +π−,JHEP08(2022) 295 [2207.03495]. [54]CMD-3collaboration, F. V. Ignatov et al.,Measurement of thee +e− →π +π− cross section from threshold to 1.2 GeV with the CMD-3 detector,Phys. Rev. D109(2024) 112002 [2302.08834]. [55]CMD-3co...

    work page arXiv 2022

  53. [53]

    Pion Pair Production with Higher Order Radiative Corrections in Low Energy e+e- Collisions

    A. Hoefer, J. Gluza and F. Jegerlehner,Pion pair production with higher order radiative corrections in low energye +e− collisions,Eur. Phys. J. C24(2002) 51 [hep-ph/0107154]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  54. [54]

    Tracz,Radiative corrections to hadrons-photons interactions, Ph.D

    S. Tracz,Radiative corrections to hadrons-photons interactions, Ph.D. thesis, University of Silesia, 2018

    work page 2018

  55. [55]

    A. B. Arbuzov, T. V. Kopylova and G. A. Seilkhanova,Forward–backward asymmetry in electron–positron annihilation into pion or kaon pairs revisited,Mod. Phys. Lett. A35 (2020) 2050210 [2003.14054]

    work page arXiv 2020

  56. [56]

    Dispersive approach to hadronic light-by-light scattering

    G. Colangelo, M. Hoferichter, M. Procura and P. Stoffer,Dispersive approach to hadronic light-by-light scattering,JHEP09(2014) 091 [1402.7081]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  57. [57]

    The radiative return at phi- and B-factories: FSR at next-to-leading order

    H. Czy˙ z, A. Grzelinska, J. H. Kuhn and G. Rodrigo,The Radiative return atΦand B factories: FSR at next-to-leading order,Eur. Phys. J. C33(2004) 333 [hep-ph/0308312]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  58. [58]

    McMulemanual

    F. Campanario, H. Czy˙ z, J. Gluza, T. Jeli´ nski, G. Rodrigo, S. Tracz et al.,Standard model radiative corrections in the pion form factor measurements do not explain thea µ anomaly, Phys. Rev. D100(2019) 076004 [1903.10197]. [62]McMuleTeam, “McMulemanual.”https://doi.org/10.5281/zenodo.6046769

    work page doi:10.5281/zenodo.6046769 2019

  59. [59]

    Engel, A

    T. Engel, A. Signer and Y. Ulrich,A subtraction scheme for massive QED,JHEP01 (2020) 085 [1909.10244]. 31

    work page arXiv 2020

  60. [60]

    On-the-fly reduction of open loops

    F. Buccioni, S. Pozzorini and M. Zoller,On-the-fly reduction of open loops,Eur. Phys. J. C 78(2018) 70 [1710.11452]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  61. [61]

    Buccioni et al.,OpenLoops 2, Eur

    F. Buccioni, J.-N. Lang, J. M. Lindert, P. Maierh¨ ofer, S. Pozzorini, H. Zhang et al., OpenLoops 2,Eur. Phys. J. C79(2019) 866 [1907.13071]

    work page arXiv 2019

  62. [62]

    Jegerlehner,The Anomalous Magnetic Moment of the Muon, vol

    F. Jegerlehner,The Anomalous Magnetic Moment of the Muon, vol. 274. Springer, Cham, 2017, 10.1007/978-3-319-63577-4

    work page doi:10.1007/978-3-319-63577-4 2017

  63. [63]

    H. H. Patel,Package-X: A Mathematica package for the analytic calculation of one-loop integrals,Comput. Phys. Commun.197(2015) 276 [1503.01469]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  64. [64]

    H. H. Patel,Package-X 2.0: A Mathematica package for the analytic calculation of one-loop integrals,Comput. Phys. Commun.218(2017) 66 [1612.00009]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  65. [65]

    Martin Hoferichter,Rescattering corrections to the pion Compton tensor, in RadioMonteCarLow 2 Satellite 2025, (Liverpool), 2025, https://indico.ph.liv.ac.uk/event/2169/contributions/10214/

    work page 2025

  66. [66]

    Monnard,Radiative corrections for the two-pion contribution to the hadronic vacuum polarization contribution to the muong−2, Ph.D

    J. Monnard,Radiative corrections for the two-pion contribution to the hadronic vacuum polarization contribution to the muong−2, Ph.D. thesis, Bern U., 2021

    work page 2021

  67. [67]

    G. R. Farrar and D. R. Jackson,The Pion Form-Factor,Phys. Rev. Lett.43(1979) 246

    work page 1979

  68. [68]

    ’t Hooft and M

    G. ’t Hooft and M. J. G. Veltman,Regularization and Renormalization of Gauge Fields, Nucl. Phys. B44(1972) 189

    work page 1972

  69. [69]

    On the Rational Terms of the one-loop amplitudes

    G. Ossola, C. G. Papadopoulos and R. Pittau,On the Rational Terms of the one-loop amplitudes,JHEP05(2008) 004 [0802.1876]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  70. [70]

    M. V. Garzelli, I. Malamos and R. Pittau,Feynman rules for the rational part of the Electroweak 1-loop amplitudes,JHEP01(2010) 040 [0910.3130]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  71. [71]

    Z. Bern, A. De Freitas, L. J. Dixon and H. L. Wong,Supersymmetric regularization, two loop QCD amplitudes and coupling shifts,Phys. Rev. D66(2002) 085002 [hep-ph/0202271]

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  72. [72]

    W. B. Kilgore,The Four Dimensional Helicity Scheme Beyond One Loop,Phys. Rev. D86 (2012) 014019 [1205.4015]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  73. [73]

    The Infrared Structure of QCD Amplitudes and $H\to g g$ in FDH and DRED

    C. Gnendiger, A. Signer and D. St¨ ockinger,The infrared structure of QCD amplitudes and H→ggin FDH and DRED,Phys. Lett. B733(2014) 296 [1404.2171]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  74. [74]

    To $d$, or not to $d$: Recent developments and comparisons of regularization schemes

    C. Gnendiger et al.,Tod, or not tod: recent developments and comparisons of regularization schemes,Eur. Phys. J. C77(2017) 471 [1705.01827]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  75. [75]

    Asymptotic expansion of Feynman integrals near threshold

    M. Beneke and V. A. Smirnov,Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B522(1998) 321 [hep-ph/9711391]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  76. [76]

    McMuledataset

    P. Nogueira,Automatic Feynman Graph Generation,J. Comput. Phys.105(1993) 279. [81]McMuleTeam, “McMuledataset.”https://doi.org/10.5281/zenodo.8188752

    work page doi:10.5281/zenodo.8188752 1993

  77. [77]

    J. H. K¨ uhn and S. Uccirati,Two-loop QED hadronic corrections to Bhabha scattering, Nucl. Phys. B806(2009) 300 [0807.1284]. 32

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  78. [78]

    R. E. Cutkosky,Singularities and discontinuities of Feynman amplitudes,J. Math. Phys.1 (1960) 429

    work page 1960

  79. [79]

    L. D. Landau,On the Analytic Properties of Vertex Parts in Quantum Field Theory,Zh. Eksp. Teor. Fiz.37(1960) 62

    work page 1960

  80. [80]

    R. J. Eden, P. V. Landshoff, D. I. Olive and J. C. Polkinghorne,The analytic S-matrix. Cambridge Univ. Press, Cambridge, 1966

    work page 1966

Showing first 80 references.