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A Two-Loop Four-Gluon Helicity Amplitude in QCD

Z. Bern, L. Dixon, D. A. Kosower

TL;DR

This paper computes the pure-glue two-loop four-gluon amplitude in QCD for the all-plus helicity configuration using a unitarity-based approach in $D=4-2\epsilon$, avoiding direct Feynman diagrams. The authors reconstruct the amplitude from its two- and three-particle cuts in the HV scheme, guided by auxiliary scalar-loop amplitudes, and express the result in terms of planar and non-planar double-box and bow-tie integrals expanded to ${\cal O}(\epsilon^0)$. They provide explicit leading- and subleading-color decompositions, verify the infrared and ultraviolet divergence structure against Catani’s predictions, and demonstrate numerical cross-checks of the cuts. The work illustrates the viability of dimensional unitarity for non-supersymmetric two-loop QCD amplitudes and offers a pathway toward NNLO jet cross-section calculations.

Abstract

We present the two-loop pure gauge contribution to the gluon-gluon scattering amplitude with maximal helicity violation. Our construction of the amplitude does not rely directly on Feynman diagrams, but instead uses its analytic properties in 4 - 2 epsilon dimensions. We evaluate the loop integrals appearing in the amplitude through O(epsilon^0) in terms of polylogarithms.

A Two-Loop Four-Gluon Helicity Amplitude in QCD

TL;DR

This paper computes the pure-glue two-loop four-gluon amplitude in QCD for the all-plus helicity configuration using a unitarity-based approach in , avoiding direct Feynman diagrams. The authors reconstruct the amplitude from its two- and three-particle cuts in the HV scheme, guided by auxiliary scalar-loop amplitudes, and express the result in terms of planar and non-planar double-box and bow-tie integrals expanded to . They provide explicit leading- and subleading-color decompositions, verify the infrared and ultraviolet divergence structure against Catani’s predictions, and demonstrate numerical cross-checks of the cuts. The work illustrates the viability of dimensional unitarity for non-supersymmetric two-loop QCD amplitudes and offers a pathway toward NNLO jet cross-section calculations.

Abstract

We present the two-loop pure gauge contribution to the gluon-gluon scattering amplitude with maximal helicity violation. Our construction of the amplitude does not rely directly on Feynman diagrams, but instead uses its analytic properties in 4 - 2 epsilon dimensions. We evaluate the loop integrals appearing in the amplitude through O(epsilon^0) in terms of polylogarithms.

Paper Structure

This paper contains 9 sections, 72 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Overview of method
  3. Two-loop amplitudes with an internal scalar loop
  4. The pure gluon amplitude
  5. Leading color
  6. Subleading color
  7. Divergences
  8. Conclusions
  9. Integrals

Figures (12)

  • Figure 1: The $s$-channel two-particle cuts of a two-loop amplitude as products of tree and one-loop amplitudes. We take all external momenta to be outgoing. The dashed lines represent the cuts.
  • Figure 2: The three-particle cut of a two-loop amplitude.
  • Figure 3: The $s$-channel double two-particle cut of a two-loop amplitude separates it into a product of three tree amplitudes. The dashed lines represent the generalized cuts.
  • Figure 4: Representative diagrams for the contributions with a single scalar loop. The class (a) diagrams can be drawn on the plane with all external legs on the outside, and one internal gluon line on the inside of the scalar loop. The class (b) diagrams have all the gluon lines on the outside of the scalar loop.
  • Figure 5: The planar and non-planar double box integrals.
  • ...and 7 more figures