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arxiv: 2604.14320 · v1 · submitted 2026-04-15 · ✦ hep-ph

Recognition: unknown

Logarithmic EW corrections at two-loop

J. M. Lindert , L. Mai
Authors on Pith no claims yet

Pith reviewed 2026-05-10 12:30 UTC · model grok-4.3

classification ✦ hep-ph
keywords electroweak correctionstwo-loop orderlogarithmic accuracyvirtual correctionshigh-energy processesnext-to-leading logarithmicphenomenology
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The pith

Two-loop electroweak virtual corrections at next-to-leading logarithmic accuracy are now implemented for automated computation.

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

The paper establishes an implementation for the automated computation of next-to-next-to-leading order electroweak virtual corrections at next-to-leading logarithmic accuracy. This is relevant for processes involving massless fermions and transversely polarised vector bosons. At high energies, the logarithmic corrections are enhanced and can reach several percent at this order in the tails of distributions. Including these terms allows for a reduction in the theoretical uncertainties associated with missing higher-order contributions. The implementation has been validated by comparison to analytical results in the literature.

Core claim

The central claim is that next-to-next-to-leading order electroweak virtual corrections can be computed at next-to-leading logarithmic accuracy in an automated manner for the specified class of processes. This has been validated against analytical results from the literature, and phenomenological results show the impact on high-energy processes by reducing uncertainties from unknown higher orders.

What carries the argument

The automated implementation of two-loop next-to-leading logarithmic electroweak virtual corrections for processes with massless fermions and transversely polarised vector bosons.

If this is right

  • Logarithmic electroweak corrections reach several percent in the high-energy tails of kinematic distributions.
  • Theoretical uncertainties from missing higher-order contributions are reduced.
  • The method applies to processes involving massless fermions and transversely polarised vector bosons.
  • Validation against analytical results confirms the accuracy of the implementation.

Where Pith is reading between the lines

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

  • Future work could extend this to higher logarithmic orders or additional process classes.
  • Improved precision may aid in identifying potential new physics effects in extreme kinematic regions.
  • The logarithmic structure could inform resummation techniques for even higher accuracy.

Load-bearing premise

The implementation accurately captures the next-to-leading logarithmic terms at two-loop order without missing significant contributions for the covered processes.

What would settle it

A calculation of the full two-loop electroweak corrections for a specific process at high energy that deviates substantially from the NLL result would falsify the sufficiency of the approximation.

Figures

Figures reproduced from arXiv: 2604.14320 by J. M. Lindert, L. Mai.

Figure 1
Figure 1. Figure 1: Energy scan for u¯RuR → e + Re − R and u¯LuL → e + L e − L . Shown are the relative two-loop EW corrections with respect to LO: LL (solid red), NLL a.i. (solid green), NLL a.d. (solid orange), and their sum (solid blue). Born couplings are renormalised at µR = √ s; IR divergences are regularised in MR with a fictitious photon mass λ = mW . Analytical results appear as solid lines; numerical OpenLoops predi… view at source ↗
Figure 2
Figure 2. Figure 2: Energy scan for u¯RuR → Z+1g and ¯dLuL → W+ +1 ¯dLdL. Born couplings are renormalised at µR = mW . For the 2 → 3 process we have adopted the following scattering angles: θd¯ = 58.85◦ , ϕd¯ = 114, 05◦ , θd = 85, 13◦ , ϕd = 38, 82◦ , θW+ = 40, 90◦ , ϕW+ = 99, 19◦ . Here θφ and ϕφ are, respectively, the polar and azimuthal scattering angles of the momenta pφ of the final state particles with respect to the di… view at source ↗
Figure 3
Figure 3. Figure 3: Differential distributions in the transverse momentum of the [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Differential distributions in the transverse momentum of the [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Differential distributions in the invariant mass [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Differential distributions in the transverse momentum of the [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Differential distributions in the transverse momentum of the harder, [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Differential distributions in the transverse momentum of the jet [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Differential distributions in the transverse momentum of the hardest [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Differential distributions in the invariant mass of the [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
read the original abstract

We present the implementation of next-to-next-to-leading order (NNLO) electroweak (EW) virtual corrections at next-to-leading logarithmic (NLL) accuracy in the amplitude generator OpenLoops. The implementation covers the automated computation of processes involving massless fermions and transversely polarised vector bosons. For energies above the EW scale, logarithmic EW corrections are strongly enhanced in the tails of kinematic distributions of key LHC processes, reaching several tens of percent at NLO and several percent at NNLO. The two-loop implementation is validated against analytical results from the literature. We present phenomenological results for representative LHC processes and discuss the role of two-loop EW corrections in reducing theoretical uncertainties from missing higher-order contributions.

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

0 major / 2 minor

Summary. The manuscript presents the implementation of NNLO electroweak virtual corrections at NLL accuracy within the OpenLoops amplitude generator. The scope is restricted to processes involving massless fermions and transversely polarised vector bosons. The two-loop implementation is validated against existing analytical results in the literature, and phenomenological applications to representative LHC processes are provided to illustrate the impact on theoretical uncertainties.

Significance. If correct, the automated NLL-accurate NNLO EW implementation supplies a practical tool for evaluating the dominant high-energy logarithmic corrections that reach several percent in LHC tails. Explicit validation against analytical literature results is a clear strength, as is the restriction to a well-defined process class that avoids mass and longitudinal-polarization complications. This work directly supports efforts to reduce missing-higher-order uncertainties in precision phenomenology.

minor comments (2)
  1. [Abstract] Abstract: the statement that corrections reach 'several percent at NNLO' would be more informative if accompanied by a brief indication of the energy scale or process class for which this estimate holds.
  2. [Validation section] Validation discussion: while agreement with analytical results is stated, a compact table or set of relative-difference values for the tested amplitudes would make the numerical precision of the NLL implementation immediately verifiable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the clear summary of its scope and strengths, and the recommendation for minor revision. We note that the report does not list any specific major comments.

Circularity Check

0 steps flagged

No significant circularity in the derivation or validation chain

full rationale

The paper presents an implementation of NNLO electroweak virtual corrections at NLL accuracy in OpenLoops for massless fermions and transversely polarized vector bosons. The central claim rests on automated computation of logarithmic terms and validation against external analytical results from the literature. No equations, predictions, or results are shown to reduce by construction to quantities defined via the authors' own prior fits, self-definitions, or load-bearing self-citations. The restricted process class and external validation benchmarks ensure the derivation chain is self-contained and independent of internal circular reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The implementation rests on standard perturbative quantum field theory and the existing OpenLoops infrastructure; no new free parameters, ad-hoc axioms, or postulated entities are introduced in the abstract.

axioms (1)
  • domain assumption Perturbative expansion of electroweak corrections remains valid and logarithmically enhanced at LHC energies above the electroweak scale.
    Invoked implicitly when stating that logarithmic corrections reach several tens of percent at NLO and several percent at NNLO.

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discussion (0)

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Reference graph

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