Analytic two-loop electroweak corrections at high energies

Hantian Zhang

arxiv: 2606.28880 · v1 · pith:W3MYYZTDnew · submitted 2026-06-27 · ✦ hep-ph · hep-th

Analytic two-loop electroweak corrections at high energies

Hantian Zhang This is my paper

Pith reviewed 2026-06-30 09:35 UTC · model grok-4.3

classification ✦ hep-ph hep-th

keywords two-loop electroweak correctionshigh-energy amplitudesHiggs pair productionStandard Modelanalytic calculations

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The pith

Analytic expressions have been obtained for two-loop electroweak corrections to four-point amplitudes at high energies in the full Standard Model.

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

This paper derives analytic high-energy expansions for two-loop four-point amplitudes in electroweak theory. The calculations cover the complete Standard Model and are applied to Higgs boson pair production. The results display intricate patterns of logarithmic and power corrections that reach significant size at large energies. Such precision is needed to interpret data from high-energy colliders where these effects influence cross sections.

Core claim

Analytic high-energy calculations for two-loop four-point electroweak amplitudes in the full Standard Model have been performed. As a representative application, electroweak corrections to Higgs boson pair production show rich structures of logarithmic and power corrections with sizeable effects in the high-energy region.

What carries the argument

High-energy expansion of two-loop four-point electroweak amplitudes

Load-bearing premise

The high-energy expansion and analytic techniques used remain valid and accurate for the kinematic regions and energies relevant to phenomenological applications such as Higgs pair production.

What would settle it

A numerical computation of the two-loop amplitude for Higgs pair production at a center-of-mass energy much larger than the electroweak scale that disagrees with the analytic high-energy expansion beyond the power-suppressed terms.

Figures

Figures reproduced from arXiv: 2606.28880 by Hantian Zhang.

**Figure 1.** Figure 1: Sample Feynman diagrams contributing to the NLO EW corrections to 𝑔𝑔 → 𝐻𝐻. and plug in the power-logarithmic series ansatz for each master integral 𝐼𝑛 = ∑ 𝑖, 𝑗,𝑘 𝐶 (𝑛) 𝑖 𝑗𝑘 (𝑠, 𝑡) 𝜖 𝑖 𝑚 𝑗 EW log𝑘 (𝑚 2 EW). (4) The resulting linear system of equations from the differential equations can be solved in terms of unknown boundary functions 𝐶 (𝑛) 𝑖 𝑗𝑘 (𝑠, 𝑡). These functions can be extracted from the boundary ma… view at source ↗

**Figure 2.** Figure 2: 𝑟EW for various 𝛿 and 𝑚𝐻 expansion terms normalized to highest available approximation for 𝑝𝑇 = 400 GeV. The truncation uncertainty in 𝛿𝑋 and 𝑚 ext 𝐻 expansions is estimated conservatively to ±1%. 2000 4000 6000 8000 10000 s(GeV) 0.16 0.14 0.12 0.10 0.08 0.06 rE W pT = 300 GeV pT = 350 GeV pT = 400 GeV pT = 500 GeV pT = 600 GeV pT = 700 GeV [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: 𝑟EW for various values of 𝑝𝑇 as a function of √ 𝑠. The plots show the same data for different ranges of √ 𝑠 on the 𝑥 axis. The uncertainty band (only visible for 𝑝𝑇 = 300 GeV) is obtained from the Padé approximation. where the squared matrix element is expanded as |M |2 = 𝑋¯ 0 ( U(0) + 𝛼𝑠 𝜋 U(1,0) + 𝛼 𝜋 U(0,1) ) with an overall prefactor 𝑋¯ 0. The high-energy expansion shows good convergence in a broad reg… view at source ↗

read the original abstract

The high-energy behaviour of electroweak scattering amplitudes is of theoretical and phenomenological interest. In these proceedings, we summarize recent progress in analytic high-energy calculations for two-loop four-point electroweak amplitudes in the full Standard Model. As a representative application, we discuss the electroweak corrections to Higgs boson pair production, where rich structures of logarithmic and power corrections appear and sizeable effects are found in the high-energy region.

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 is a proceedings summary of recent analytic high-energy two-loop electroweak corrections in the full SM, using HH production to illustrate the structures.

read the letter

The main thing to know is that this paper is a proceedings contribution summarizing recent progress on analytic high-energy expansions for two-loop four-point electroweak amplitudes rather than presenting a fresh derivation or new framework.

It collects results for amplitudes in the full Standard Model and applies them to Higgs pair production. The text points out the appearance of logarithmic and power-suppressed corrections and notes that these can produce sizeable effects at high energies. This matches what one expects from Sudakov-type expansions in electroweak theory, and the choice of HH production as the example is reasonable because it is a process where such corrections matter for precision phenomenology.

The paper does a service by pulling the points together in one place. For readers already following the literature on high-energy electroweak corrections, it offers a compact orientation without forcing them to chase multiple separate papers.

The obvious limitation is that the actual calculations, the expansion techniques, and any numerical cross-checks live in the referenced earlier work. This contribution adds little technical detail of its own, so its value is mainly organizational. The claim that the high-energy expansion remains useful for phenomenological energies is stated but not re-derived or independently tested here, which is acceptable in a summary but leaves the robustness to be judged from the source papers.

This is aimed at specialists in precision collider calculations who want a quick status update on analytic methods for multi-loop electroweak amplitudes. It is less useful for outsiders or for anyone needing the full technical steps.

It deserves a serious referee because the topic is relevant to reducing theory uncertainties in important channels and the summary is grounded in existing solid work. I would recommend sending it for peer review in a proceedings context.

Referee Report

0 major / 1 minor

Summary. The paper summarizes recent progress in analytic high-energy calculations for two-loop four-point electroweak amplitudes in the full Standard Model. As a representative application, it discusses the electroweak corrections to Higgs boson pair production, where rich structures of logarithmic and power corrections appear and sizeable effects are found in the high-energy region.

Significance. If the analytic results hold, this work advances the understanding of high-energy electroweak amplitudes by providing explicit expansions that reveal the structure of large logarithmic and power-suppressed terms. Such analytic control is valuable for precision phenomenology at high-energy colliders, where these corrections can be sizeable.

minor comments (1)

[Abstract] The abstract states that the calculations are performed 'in the full Standard Model' but does not specify whether the expansions retain the full set of masses or employ any approximations beyond the high-energy limit; a brief clarification would help readers assess the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and the recommendation to accept. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract and context describe analytic high-energy expansions for two-loop four-point electroweak amplitudes in the full SM, applied to Higgs pair production, without any visible equations, fitted parameters, or self-referential definitions. No load-bearing steps reduce predictions to inputs by construction, and the work is presented as summarizing external progress in perturbative calculations. The derivation chain is self-contained against standard QFT benchmarks, with no self-citation chains, ansatze smuggled via citation, or renaming of known results evident from the provided material.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No explicit free parameters, axioms, or invented entities are stated in the abstract; the work relies on the Standard Model as background.

pith-pipeline@v0.9.1-grok · 5576 in / 1120 out tokens · 30383 ms · 2026-06-30T09:35:09.731046+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

45 extracted references · 43 canonical work pages · 18 internal anchors

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[1] [1]

V . V . Sudakov,Vertex Parts at Very High Energies in Quantum Electrodynamics , Sov. Phys. JETP 3,4,5,6 (1956) 65

1956

[2] [2]

Weinberg, High-energy behavior in quantum field theory , Phys

S. Weinberg, High-energy behavior in quantum field theory , Phys. Rev. 118 (1960) 838

1960

[3] [3]

Davies, G

J. Davies, G. Mishima, K. Schönwald, M. Steinhauser and H. Zhang, Higgs boson contribution to the leading two-loop Yukawa corrections to gg → HH , JHEP 08 (2022) 259 [2207.02587]

work page arXiv 2022

[4] [4]

Davies, K

J. Davies, K. Schönwald, M. Steinhauser and H. Zhang, Analytic next-to-leading order Yukawa and Higgs boson self-coupling corrections to gg → HH at high energies , JHEP 04 (2025) 193 [2501.17920]

work page arXiv 2025

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Davies, K

J. Davies, K. Schönwald, M. Steinhauser and H. Zhang, Analytic next-to-leading order electroweak corrections to Higgs boson pair production at high energies , 2603.08789

work page arXiv

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Two-loop $gg \to Hg$ amplitude mediated by a nearly massless quark

K. Melnikov, L. Tancredi and C. Wever, Two-loop 𝑔𝑔 ! 𝐻𝑔 amplitude mediated by a nearly massless quark, JHEP 11 (2016) 104 [1610.03747]

work page internal anchor Pith review Pith/arXiv arXiv 2016

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work page internal anchor Pith review Pith/arXiv arXiv 2017

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work page internal anchor Pith review Pith/arXiv arXiv 2018

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work page internal anchor Pith review Pith/arXiv arXiv 2019

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High-Energy Expansion of Two-Loop Massive Four-Point Diagrams

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work page internal anchor Pith review Pith/arXiv arXiv 2019

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Catani, S

S. Catani, S. Devoto, M. Grazzini, S. Kallweit, J. Mazzitelli and C. Savoini, Higgs Boson Production in Association with a Top-Antitop Quark Pair in Next-to-Next-to-Leading Order QCD , Phys. Rev. Lett. 130 (2023) 111902 [2210.07846]

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G. Wang, T. Xia, L. L. Y ang and X. Y e,Two-loop QCD amplitudes for 𝑡𝑡𝐻 production from boosted limit, JHEP 07 (2024) 121 [2402.00431]

work page arXiv 2024

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S. Jaskiewicz, S. Jones, R. Szafron and Y . Ulrich, The structure of quark mass corrections in the gg → HH amplitude at high-energy , JHEP 09 (2025) 015 [2501.00587]

work page arXiv 2025

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Z. Hu, T. Liu and J. M. Y ang, The gg → HH amplitude induced by bottom quarks at two-loop level: planar master integrals, JHEP 09 (2025) 132 [2503.10051]

work page arXiv 2025

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Hu and T

Z. Hu and T. Liu, Double logarithmic contribution to Higgs boson pair production in the high-energy limit, JHEP 05 (2026) 237 [2509.06381]

work page arXiv 2026

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Wang and L

G. Wang and L. L. Y ang, Two-loop QCD amplitudes for 𝑡𝑡𝛾 production at hadron colliders, JHEP 04 (2026) 005 [2510.01774]

work page arXiv 2026

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Delto, A

M. Delto, A. Penin and L. Tancredi, High-Energy Evolution of Power-Suppressed Amplitudes, Phys. Rev. Lett. 136 (2026) 111801 [2510.04914]

work page arXiv 2026

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V . S. Fadin, L. N. Lipatov, A. D. Martin and M. Melles, Resummation of double logarithms in electroweak high-energy processes, Phys. Rev. D 61 (2000) 094002 [hep-ph/9910338]

work page internal anchor Pith review Pith/arXiv arXiv 2000

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M. Hori, H. Kawamura and J. Kodaira, Electroweak Sudakov at two loop level , Phys. Lett. B 491 (2000) 275 [hep-ph/0007329]

work page internal anchor Pith review Pith/arXiv arXiv 2000

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Electroweak radiative corrections in high energy processes

M. Melles, Electroweak radiative corrections in high-energy processes, Phys. Rept. 375 (2003) 219 [hep-ph/0104232]

work page internal anchor Pith review Pith/arXiv arXiv 2003

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J. H. Kuhn, S. Moch, A. A. Penin and V . A. Smirnov, Next-to-next-to-leading logarithms in four fermion electroweak processes at high-energy, Nucl. Phys. B 616 (2001) 286 [hep-ph/0106298]

work page internal anchor Pith review Pith/arXiv arXiv 2001

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Electroweak two-loop Sudakov logarithms for on-shell fermions and bosons

W . Beenakker and A. Werthenbach,Electroweak two loop Sudakov logarithms for on-shell fermions and bosons, Nucl. Phys. B 630 (2002) 3 [hep-ph/0112030]

work page internal anchor Pith review Pith/arXiv arXiv 2002

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