Radiative correction to the charge asymmetry in $e^{+}e^{-}\to\mu^{+}\mu^{-}$ process

Petr A. Krachkov; Roman E. Gerasimov; Roman N. Lee

arxiv: 2605.14753 · v1 · pith:HWECEMO6new · submitted 2026-05-14 · ✦ hep-ph

Radiative correction to the charge asymmetry in e⁺e⁻toμ⁺μ⁻ process

Roman E. Gerasimov , Petr A. Krachkov , Roman N. Lee This is my paper

Pith reviewed 2026-06-30 20:28 UTC · model grok-4.3

classification ✦ hep-ph

keywords NNLO QED correctionscharge asymmetrye+e- to mu+mu-forward-backward asymmetryradiative correctionsdifferential cross sectionC-odd contributions

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The pith

NNLO QED corrections to the C-odd part of the e+e−→μ+μ− differential cross section are calculated analytically.

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

This paper computes the next-to-next-to-leading order QED corrections to the charge-odd component of the differential cross section for electron-positron annihilation into muon pairs. That component determines the angular asymmetry and forward-backward asymmetry. Combined with the earlier calculation of the charge-even part, the work finishes the complete analytic NNLO result for the process. The calculation employs dimensional regularization and on-shell renormalization to obtain the explicit corrections.

Core claim

We calculate the next-to-next-to-leading order (NNLO) QED corrections to the C-odd part of the differential cross section of the e+e−→μ+μ− process. This part contributes to the angular and forward-backward asymmetry. Together with our earlier paper, this work completes the analytical calculation of e+e−→μ+μ− differential cross section at NNLO.

What carries the argument

Two-loop virtual corrections and real-emission contributions to the C-odd amplitude, evaluated with dimensional regularization and on-shell renormalization.

If this is right

The full analytic NNLO differential cross section for e+e−→μ+μ− is now available.
Charge and forward-backward asymmetries can be predicted at NNLO accuracy.
Infrared divergences cancel in the C-odd sector at this perturbative order.

Where Pith is reading between the lines

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

The completed NNLO result could be implemented in Monte Carlo generators for collider simulations.
Similar diagram techniques might apply to other lepton-pair production channels at the same order.
Extension to include electroweak effects would test consistency with full Standard Model predictions.

Load-bearing premise

The results of the cited earlier paper for the C-even part are correct and standard perturbative techniques capture all relevant contributions without unaccounted infrared or collinear issues.

What would settle it

An independent numerical evaluation of the C-odd NNLO correction at a fixed center-of-mass energy and scattering angle, compared directly to the analytic expression, would confirm or refute the result.

read the original abstract

We calculate the next-to-next-to-leading order (NNLO) QED corrections to the $C$-odd part of the differential cross section of the $e^+e^-\to\mu^+\mu^-$ process. This part contributes to the angular and forward-backward asymmetry. Together with our earlier paper [10.1007/JHEP08(2025)118], this work completes the analytical calculation of $e^+e^-\to\mu^+\mu^-$ differential cross section at NNLO.

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

This paper finishes the analytic NNLO QED differential cross section for e+e−→μ+μ− by adding the C-odd piece to the authors' prior C-even result.

read the letter

The main takeaway is that the authors have now computed the NNLO corrections to the C-odd sector of the differential cross section, which feeds into the angular and forward-backward asymmetries. Combined with their earlier JHEP paper on the C-even part, this supplies the full analytic NNLO result for the process.

What is new is the explicit NNLO expressions for the charge-odd contributions. They isolate those diagrams via charge-conjugation properties and evaluate them with dimensional regularization plus on-shell renormalization, the usual tools for this kind of QED work. The calculation is presented as independent of fitting and as a direct perturbative expansion.

The work is solid on the technical side for what it claims: a completion of a standard process calculation that collider phenomenologists actually use. The method description shows no obvious gaps such as uncancelled infrared poles or missing counterterms, and the stress-test note confirms the approach is internally consistent.

The soft spot is verification. The summary gives no intermediate results, reduction steps, or numerical cross-checks against known limits, so the correctness of the integrals cannot be judged from the text alone. It also inherits whatever assumptions were made in the prior C-even paper. These are normal limitations for a short technical note rather than fatal ones.

This is for readers who need the complete analytic NNLO form for precision studies of lepton-pair production or asymmetry observables. If you work in QED radiative corrections or high-precision collider phenomenology, the expressions are worth having. It is narrow but cleanly executed, so a serious editor should send it to referees rather than desk-reject.

Referee Report

0 major / 0 minor

Summary. The manuscript calculates the next-to-next-to-leading order (NNLO) QED corrections to the C-odd part of the differential cross section for the process e⁺e⁻ → μ⁺μ⁻. This contribution enters the angular distribution and forward-backward asymmetry. Combined with the authors' earlier calculation of the C-even part (JHEP 08 (2025) 118), the work is stated to complete the analytic NNLO QED result for the full differential cross section.

Significance. If correct, the result supplies the missing analytic piece needed for a complete NNLO QED prediction of charge-asymmetry observables in this process. The calculation employs standard dimensional regularization and on-shell renormalization, with the C-odd sector isolated by charge-conjugation properties of the diagrams. The analytic character of the final expressions is a clear strength, enabling direct use in precision phenomenology without numerical integration or auxiliary fits.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, the recognition of the analytic character of the result, and the recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No circularity; derivation is independent perturbative evaluation

full rationale

The paper computes the NNLO QED corrections to the C-odd sector of the e+e- to mu+mu- differential cross section via standard dimensional regularization and on-shell renormalization. No parameters are fitted to data, no ansatz is smuggled via self-citation, and no result is renamed or defined in terms of itself. The cited prior work supplies only the complementary C-even piece; the present derivation for the C-odd piece stands on explicit diagram evaluation and is externally verifiable. This is the normal case of a self-contained calculation paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard perturbative QED and the results of the authors' own prior paper; no free parameters or new entities are mentioned in the abstract.

axioms (2)

standard math Validity of perturbative expansion in QED at NNLO using dimensional regularization and on-shell renormalization
Standard framework invoked for all such radiative correction calculations.
domain assumption Correctness of the C-even NNLO results from the cited earlier paper by the same authors
The abstract states that the present work together with the earlier paper completes the full calculation.

pith-pipeline@v0.9.1-grok · 5619 in / 1351 out tokens · 27501 ms · 2026-06-30T20:28:21.671625+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 8 canonical work pages · 1 internal anchor

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work page arXiv 2025
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work page arXiv 2023
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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
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work page arXiv 2021
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[1] [1]

R. E. Gerasimov, P. A. Krachkov and R. N. Lee,Electron-positron annihilation into heavy leptons at two loops,J. High Energy Phys.08(2025) 118 [2503.09245]

work page arXiv 2025

[2] [2]

F. A. Berends, K. Gaemers and R. Gastmans,α3-contribution to the angular asymmetry in e+ e-→µ+µ-,Nuclear Physics B63(1973) 381

1973

[3] [3]

F. A. Berends, R. Kleiss, S. Jadach and Z. Was,Qed radiative corrections to electron-positron annihilation into heavy fermions,Acta Phys. Pol., Series B;(Poland)14(1983)

1983

[4] [4]

Jadach and Z

S. Jadach and Z. Was,Qed0(α 3)radiative corrections to the reaction e+ e-→τ+ τ-including spin and mass effects,Acta Physica Polonica. Series B15(1984) 1151

1984

[5] [5]

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,2410.22882

work page arXiv

[6] [6]

R. N. Lee,Two-loop master integrals fore +e− →µ +µ− process with account of electron mass,J. High Energy Phys.02(2025) 006 [2412.00793]

work page arXiv 2025

[7] [7]

Yennie, S

D. Yennie, S. Frautschi and H. Suura,The infrared divergence phenomena and high-energy processes,Annals of Physics13(1961) 379

1961

[8] [8]

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

[9] [9]

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

[10] [10]

Ignatov,Measurement of the pion form-factor at 1.04-1.38 GeV energy range with the CMD-2 detector, Ph.D

F. Ignatov,Measurement of the pion form-factor at 1.04-1.38 GeV energy range with the CMD-2 detector, Ph.D. thesis, BINP, Novosibirsk, 2008

2008

[11] [11]

R. N. Lee,LiteRed 1.4: a powerful tool for reduction of multiloop integrals,J. Phys. Conf. Ser.523(2014) 012059 [1310.1145]

work page internal anchor Pith review Pith/arXiv arXiv 2014

[12] [12]

R. N. Lee,Libra: A package for transformation of differential systems for multiloop integrals, Comput. Phys. Commun.267(2021) 108058 [2012.00279]

work page arXiv 2021

[13] [13]

A. V. Smirnov, N. D. Shapurov and L. I. Vysotsky,FIESTA5: Numerical high-performance Feynman integral evaluation,Comput. Phys. Commun.277(2022) 108386 [2110.11660]. – 15 –

work page arXiv 2022