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Resummation of double logarithms in electroweak high energy processes

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V.S. Fadin, L.N. Lipatov, A.D. Martin, M. Melles

High Energy Physics - Phenomenology · Submitted 14 Oct 1999 · 18 references

Pith's verdict: The core exponentiation result is sound and worth citing; the Added Note's one-paragraph dismissal of [18] and a normalization slip in Eq. (28) are the things to fix.

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arXiv IDhep-ph/9910338 (version 1)
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CitationA.D. Martin, L.N. Lipatov, M. Melles, V.S. Fadin. "Resummation of double logarithms in electroweak high energy processes." arXiv:hep-ph/9910338 (1999). https://pith.science/paper/AK27CEJZ
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The paper

The paper claims

The paper argues that the large double logarithms of electroweak physics at TeV energies — the corrections that grow like $g^2$ times the square of a logarithm of energy over mass — can be summed into a single exponential per external pat

The reading

At a future TeV electron–positron collider, the corrections to scattering cross sections that grow like $g^2 L^2$, where $L$ is a logarithm of the energy over the weak-boson mass, become comparable to the Born term itself. The paper's aim is to sum these double logarithms for Standard Model electroweak processes, and to do so in a way that respects the broken gauge symmetry, where the photon is a mixture of the $SU(2)$ and $U(1)$ fields and the $W$ and $Z$ are massive. The authors use an evolution equation in an infrared cutoff $|k_|$ on virtual transverse momenta, starting from the Born amplitude at large cutoff and running down through the weak-boson mass scale to the photon-only regime. The payoff is a closed exponential form: the amplitude equals the Born amplitude times $e^{-1/2}$ times a sum over external particles of the probability of emitting a soft, nearly collinear gauge boson. The paper extends this to processes with real soft emission and to forward and backward Regge kinematics, where the relevant logarithms are of a different, non-Sudakov type.

What carries the argument

The infrared evolution equation: a differential equation for the scattering amplitude with respect to a gauge-invariant cutoff $|k_|$ on the transverse momenta of virtual particles, obtained by factoring off the virtual gauge boson with the smallest transverse momentum and taking the remaining amplitude on the mass shell. This is the non-Abelian extension of Gribov's dispersive theorem on soft Bremsstrahlung. Because the factored-off boson couples to external legs through the same soft-collinear factor regardless of the hard process, the equation exponentiates, and the exponent is the emission probability $W_l$ built from the group Casimir and the logarithm of energy over cutoff. The matching at $|k_| = M$, where the $W$ and $Z$ are integrated out and only the photon kernel survives, is what turns the unbroken-phase exponent into the two-stage exponent of Eqs. (26) and (28).

Extended reading notes

Core claim

The central claim, stated in the abstract as 'In all cases the Sudakov double logarithms are found to exponentiate', is realized concretely. For an unbroken non-Abelian gauge theory the amplitude is $M_{Born}$ times $e^{-1/2}$ times the sum over external legs of $W_l(s,|k_|^2)$, where $W_l$ is the probability to emit a soft, almost collinear gauge boson from leg $l$, computed with the transverse momentum cutoff. For the broken electroweak theory the same structure survives: above the weak scale the exponent is a sum over external particles of $g^2 T_i(T_i+1) + g'^2 (Y_i/2)^2$ times $L^2$, with $T_i$ the weak isospin and $Y_i$ the hypercharge; below the weak scale only the photon piece runs, and the two regimes are matched at $|k_| = M$. The authors stress that the split into photon, $W$ and $Z$ contributions is not gauge invariant, and that dropping the virtual photons from the unbroken-phase calculation would give a gauge-dependent, unphysical answer. For processes with real soft emission the exponent gains an extra term $W_A(k_^2,|k_|^2)$ from the emitted gauge boson's own non-Abelian interactions, so that the photon emission is no longer Poisson-distributed in the double-logarithmic approximation. In Regge kinematics the evolution equations become non-linear in the amplitude, reduce to Riccati form in the angular-momentum plane, and in simple cases to a Schrödinger equation with a harmonic-oscillator potential.

Load-bearing premise

Everything rests on the claim that the virtual gauge boson with the smallest transverse momentum can be factored off the amplitude, with the remaining amplitude taken on the mass shell at the substituted cutoff, in a way that is gauge invariant for the broken electroweak theory.

Editorial extensions

If this is right

  • Every electroweak process with all invariants of order $s$ acquires a leading high-energy correction that is a pure exponential of a group-theoretic coefficient, so the double-logarithmic series can be summed once and for all rather than computed process by process.
  • Because the exponent is a sum over external legs, the leading electroweak correction is a quasi-classical, process-independent factor for large-angle scattering, which makes it a clean target for dedicated resummation in collider simulations.
  • Real soft photon emission in the double-logarithmic approximation deviates from a Poisson distribution, since the photon carries a non-Abelian component, so the usual exponentiation of real and virtual corrections into a single Sudakov factor does not hold at the leading-log level.
  • In Regge kinematics the double logarithms of non-Sudakov type are governed by a Schrödinger-type equation with a harmonic-oscillator potential, giving a power-law high-energy behaviour for positive-signature amplitudes and a logarithmic behaviour for negative-signature ones.
  • The framework is modular in the initial condition: new particles or interactions above the weak scale can be absorbed by changing the matching condition of the evolution equation rather than redoing the resummation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the exponent is a sum over external legs of a group Casimir times $\log^2(s/\mu^2)$, then the leading electroweak correction to any process is fixed by the external-state quantum numbers alone, so a single universal factor could be folded into event generators as a per-particle weight rather than a per-process calculation.
  • The claim that the photon's non-Abelian component breaks Poisson emission suggests a measurable diagnostic: the double-logarithmic coefficient of the one-photon-emission cross section relative to the non-radiative one should differ from the naive QED ratio by a term proportional to $T^3_i$ times the $W$ emission probability, which is in principle separable by comparing $Z$ and photon-tagged sample
  • The matching at the weak scale assumes that below $M$ only photon exchange contributes to the double-logarithmic kernel; if the Higgs sector or a light new state contributed logarithmically there, the two-stage exponent would need a third matching step, which is a concrete place to look for sensitivity to physics beyond the Standard Model.
Original abstract

At future linear $e^+e^-$ collider experiments in the TeV range, Sudakov double logarithms originating from massive boson exchange can lead to significant corrections to the cross sections of the observable processes. These effects are important for the high precision objectives of the Next Linear Collider. We use the infrared evolution equation, based on a gauge invariant dispersive method, to obtain double logarithmic asymptotics of scattering amplitudes and discuss how it can be applied, in the case of broken gauge symmetry, to the Standard Model of electroweak processes. We discuss the double logarithmic effects to both non-radiative processes and to processes accompanied by soft gauge boson emission. In all cases the Sudakov double logarithms are found to exponentiate. We also discuss double logarithmic effects of a non-Sudakov type which appear in Regge-like processes.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,933 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.09 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

At future linear $e^+e^-$ collider experiments in the TeV range, Sudakov double logarithms originating from massive boson exchange can lead to significant corrections to the cross sections of the observable processes. These effects are important for the high precision objectives of the Next Linear Collider. We use the infrared evolution equation, based on a gauge invariant dispersive method, to obtain double logarithmic asymptotics of scattering amplitudes and discuss how it can be applied, in the case of broken gauge symmetry, to the Standard Model of electroweak processes. We discuss the double logarithmic effects to both non-radiative processes and to processes accompanied by soft gauge boson emission. In all cases the Sudakov double logarithms are found to exponentiate. We also discuss double logarithmic effects of a non-Sudakov type which appear in Regge-like processes.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 At future linear $e^+e^-$ collider experiments in the TeV range, Sudakov double logarithms originating from massive boson exchange can lead to significant corrections to the cross sections of the observable processes. background 1.00 0.09 0.09 hedged possibility 0.97
s1 These effects are important for the high precision objectives of the Next Linear Collider. background 0.88 0.06 0.04 asserted 1.00
s2 We use the infrared evolution equation, based on a gauge invariant dispersive method, to obtain double logarithmic asymptotics of scattering amplitudes and discuss how it can be applied, in the case of broken gauge symmetry, to the Standard Model of electroweak processes. method 0.85 0.94 0.12 asserted 0.97
s3 We discuss the double logarithmic effects to both non-radiative processes and to processes accompanied by soft gauge boson emission. contribution 1.00 0.73 0.07 asserted 0.73
s4 In all cases the Sudakov double logarithms are found to exponentiate. result 1.00 0.95 0.48 asserted 1.00
s5 We also discuss double logarithmic effects of a non-Sudakov type which appear in Regge-like processes. contribution 0.97 0.75 0.06 asserted 0.85

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Feynman diagrams contributing to the infrared evolution e
Figure 2. Schematic Feynman diagrams contributing to the real gau
Figure 3. Two-loop ‘rainbow’ Feynman diagrams contributing to DL co

Figures reproduced from arXiv:hep-ph/9910338 by the authors.

References
How the literature uses it

37 arXiv papers cite this in 63 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States a conflicting result 1 paper

Papers citing it · 39 sentences without a confident label

Builds on it 1 paper
Uses its method or tool 2 papers
Compares with it 1 paper
  • A non-trivial consistency check is provided by the agreement of $δ̂^SL$ with the $𝒪(α)$ expansion of the all-order resummed angular-independent NLL corrections obtained via the IREE [ref] in the symmetric $SU(2)⊗U(1)$ theory, as discussed in Section [ref].

    Logarithmic EW corrections at two-loop · Angular-independent NLL · 0.95

Cites it as background 17 papers
  • Double logarithm (DL) contributions, of the form $(αlog^2E^2/^2)^n$ with arbitrary $n$, have been resummed in fully-inclusive and exclusive cross-sections, using respectively Asymptotic Dynamics [ref] and IREE [ref] .

    Learning from Radiation at a Very High Energy Lepton Collider · Theoretical predictions · 0.89

  • To be more precise, in the Sudakov regime, where all kinematic invariants $r_jk$ are of the same order $s$ and much larger than the electroweak scale, $r_jk∼s≫m_W^2$, EW corrections are dominated by double/leading logarithms (LL) and single/next-to-leading logarithms (NLL) of the ratio of the energy scale to the weak gauge-boson masses [ref] .

    Logarithmic EW corrections at one-loop · Introduction · 0.88

  • In the regime of Eq. [ref], the one- and two-loop EW corrections are dominated by logarithmic terms of the form [ref] [equation] where $δ̂_k^LL$ and $δ̂_k^NLL$ are the $𝒪(α^k)$ leading- and next-to-leading logarithmic correction operators, $a$ and $b$ are constants[These constants are in general complex numbers due to the presence of the additional $iπ$ factors coming from analytical continuation of $log(-r_jk)$.

    Logarithmic EW corrections at two-loop · Notation and Logarithmic Approximation · 0.80

Discussion

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