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arxiv: 2511.00437 · v2 · submitted 2025-11-01 · ✦ hep-ph

Optimization of factorization scale in QED Drell-Yan-like processes

Andrej Arbuzov , Uliana Voznaya , Aliaksandr Sadouski This is my paper

Pith reviewed 2026-05-18 02:04 UTC · model grok-4.3

classification ✦ hep-ph
keywords factorization scaleinitial state radiationQED correctionsDrell-Yan processese+e- annihilationlogarithmic approximationstwo-loop calculationselectroweak precision
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The pith

Comparisons with exact two-loop results can optimize the factorization scale choice for initial-state radiation corrections in e+e- annihilation.

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

The paper investigates how different choices of factorization scale affect the size and accuracy of initial-state radiation corrections in electron-positron annihilation processes. It evaluates several scale prescriptions inside leading-logarithmic and next-to-leading-logarithmic approximations. The authors then compare these approximate results against known complete two-loop calculations to identify which scale choices bring the approximations closer to the exact answers. This approach matters for collider physics because radiation corrections enter precision predictions for cross sections and distributions. A scale that works better at two loops may reduce theoretical uncertainty when the same approximations are used at higher orders.

Core claim

The dependence of initial-state QED corrections on the factorization scale is studied within leading and next-to-leading logarithmic approximations for Drell-Yan-like processes; comparisons with complete two-loop results are used to select preferred scale prescriptions that improve agreement with the exact calculation.

What carries the argument

Factorization scale prescriptions applied inside leading-logarithmic and next-to-leading-logarithmic approximations for initial-state radiation.

If this is right

  • The optimized scale reduces the difference between logarithmic approximations and exact two-loop results for total cross sections and distributions in e+e- annihilation.
  • The same scale choice can be adopted in Monte Carlo generators that use leading or next-to-leading logarithmic parton showers for initial-state radiation.
  • Improved scale prescriptions lower the theoretical uncertainty assigned to missing higher-order QED corrections in precision electroweak studies.
  • The method provides a concrete criterion for choosing the scale in any process where only partial higher-order results are known.

Where Pith is reading between the lines

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

  • The same tuning procedure could be repeated once three-loop or four-loop analytic results become available, testing stability of the optimal scale.
  • An analogous optimization might be performed for final-state radiation or for mixed initial-final corrections in the same processes.
  • The approach could be extended to Drell-Yan production at hadron colliders by replacing the QED logarithms with QCD ones and comparing to known two-loop QCD results.

Load-bearing premise

A scale choice that matches two-loop results will continue to be optimal or at least better when the same approximations are applied at higher orders or in resummed calculations.

What would settle it

A complete three-loop calculation of the same initial-state radiation corrections would show whether the two-loop-tuned scale choice still reduces the discrepancy relative to untuned choices.

Figures

Figures reproduced from arXiv: 2511.00437 by Aliaksandr Sadouski, Andrej Arbuzov, Uliana Voznaya.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Corrections and differences for factorization scales [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. NLO corrections and differences for factorization scales [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. NNLO corrections and difference for factorization scales [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Corrections vs. factorization scale for [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Corrections vs. factorization scale for [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
read the original abstract

The dependence of corrections due to the initial state radiation in $e^+ e^-$-annihilation processes on the choice of the factorization scale is investigated. Different prescriptions of the factorization scale choice are analyzed within the leading and next-to-leading logarithmic approximations. Comparisons with the known complete two-loop results are used to optimize the scale choice.

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

1 major / 2 minor

Summary. The manuscript investigates the dependence of initial-state radiation corrections in e⁺e⁻ annihilation on the choice of factorization scale. It analyzes different prescriptions within leading-logarithmic (LL) and next-to-leading-logarithmic (NLL) approximations and optimizes the scale by comparing these approximations to known complete two-loop results.

Significance. If the optimized scale choice is robust and generalizes, the work could provide a practical method for improving the accuracy of LL/NLL approximations in QED Drell-Yan-like processes by leveraging exact higher-order benchmarks. This would be useful for precision phenomenology where full higher-order calculations remain computationally intensive.

major comments (1)
  1. [Optimization procedure (around the comparison to two-loop results)] The central optimization procedure selects the factorization scale by minimizing the difference between LL/NLL approximations and complete O(α²) results. This tuning assumes the dominant discrepancy pattern at two loops persists in a form that the same scale will suppress at O(α³) and in resummed series, but the manuscript provides no explicit decomposition of the two-loop residual into logarithmic versus constant terms nor any test at higher orders to support the extrapolation.
minor comments (2)
  1. [Abstract] The abstract and introduction could more explicitly list the specific kinematic cuts or observables (e.g., invariant mass range) used in the numerical comparisons.
  2. [Section 2 or 3] Notation for the factorization scale (often denoted μ_F or similar) should be introduced with a clear definition before the first numerical results are presented.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for the careful review of our manuscript and the constructive comments. We address the major comment below, providing clarification on our optimization approach while acknowledging its limitations based on currently available results.

read point-by-point responses

  1. Referee: [Optimization procedure (around the comparison to two-loop results)] The central optimization procedure selects the factorization scale by minimizing the difference between LL/NLL approximations and complete O(α²) results. This tuning assumes the dominant discrepancy pattern at two loops persists in a form that the same scale will suppress at O(α³) and in resummed series, but the manuscript provides no explicit decomposition of the two-loop residual into logarithmic versus constant terms nor any test at higher orders to support the extrapolation.

    Authors: We agree that our optimization relies on the available complete O(α²) results, as these represent the highest-order exact calculations for QED initial-state radiation corrections in e⁺e⁻ annihilation. This procedure follows standard practice in resummation techniques, where the scale is tuned to the highest known perturbative order to improve the approximation. The LL and NLL terms capture the dominant logarithmic enhancements, and the scale choice is intended to minimize the overall mismatch, including both logarithmic and constant contributions at two loops. We acknowledge that the original manuscript does not include an explicit decomposition of the two-loop residual. To address this point, we have added a brief discussion in the revised version explaining the structure of the corrections and the rationale for expecting the optimized scale to remain effective at higher orders due to the universality of collinear logarithms in QED. However, complete O(α³) results are not available in the literature, preventing direct numerical tests of the extrapolation at this stage. We have included a short paragraph on the expected behavior based on renormalization-group arguments. revision: partial

standing simulated objections not resolved
  • Direct numerical validation of the scale choice at O(α³) or in fully resummed series is not possible at present due to the absence of complete higher-order exact results for this process.

Circularity Check

0 steps flagged

No circularity: scale choice calibrated to independent external two-loop benchmarks

full rationale

The paper explicitly uses known complete two-loop results as an external benchmark to optimize the factorization scale within LL and NLL approximations for ISR. This is a calibration step against independent higher-order calculations rather than any derivation that reduces to its own inputs by construction. No self-definitional loop, fitted prediction presented as first-principles result, or load-bearing self-citation chain appears in the abstract or described procedure. The two-loop results serve as an external reference, making the optimization self-contained against benchmarks. Any concern about whether the tuned scale remains optimal at yet higher orders is an extrapolation assumption, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence and accuracy of previously published two-loop calculations and on the assumption that logarithmic approximations remain valid when the scale is varied.

axioms (1)
  • domain assumption Complete two-loop results for the processes under study are known and can serve as an unbiased benchmark.
    The abstract states that comparisons with these results are used to optimize the scale.

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Reference graph

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