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A Two-Loop Five-Gluon Helicity Amplitude in QCD

Simon Badger, Hjalte Frellesvig, Yang Zhang

TL;DR

This work extends integrand reduction to D dimensions and employs D-dimensional generalized unitarity with six-dimensional tree amplitudes to compute the planar two-loop, five-gluon all-plus amplitude in QCD. The authors derive a compact integrand form requiring only topologies with six or more propagators, and they provide explicit analytic results for butterfly-type topologies alongside a full numerical validation of the infrared structure. By leveraging FDH regularization and momentum-twistor parametrization, they obtain a clean decomposition into eight integral families with precise coefficients, establishing a first five-point two-loop result in a non-supersymmetric theory. The findings reveal a close structural relation to N=4 results, emphasize the importance of choosing ISP bases for simplification, and set the stage for future extensions to non-planar and other helicity configurations.

Abstract

We compute the planar part of the two-loop five gluon amplitude with all helicities positive. To perform the calculation we develop a D-dimensional generalized unitarity procedure allowing us to reconstruct the amplitude by cutting into products of six-dimensional trees. We find a compact form for the integrand which only requires topologies with six or more propagators. We perform cross checks of the universal infra-red structure using numerical integration techniques.

A Two-Loop Five-Gluon Helicity Amplitude in QCD

TL;DR

This work extends integrand reduction to D dimensions and employs D-dimensional generalized unitarity with six-dimensional tree amplitudes to compute the planar two-loop, five-gluon all-plus amplitude in QCD. The authors derive a compact integrand form requiring only topologies with six or more propagators, and they provide explicit analytic results for butterfly-type topologies alongside a full numerical validation of the infrared structure. By leveraging FDH regularization and momentum-twistor parametrization, they obtain a clean decomposition into eight integral families with precise coefficients, establishing a first five-point two-loop result in a non-supersymmetric theory. The findings reveal a close structural relation to N=4 results, emphasize the importance of choosing ISP bases for simplification, and set the stage for future extensions to non-planar and other helicity configurations.

Abstract

We compute the planar part of the two-loop five gluon amplitude with all helicities positive. To perform the calculation we develop a D-dimensional generalized unitarity procedure allowing us to reconstruct the amplitude by cutting into products of six-dimensional trees. We find a compact form for the integrand which only requires topologies with six or more propagators. We perform cross checks of the universal infra-red structure using numerical integration techniques.

Paper Structure

This paper contains 23 sections, 109 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Integrand reduction and generalized unitarity in $D$ dimensions
  4. $D$-dimensional integrand reduction
  5. Tree-level amplitudes with the six-dimensional spinor-helicity formalism
  6. The six-dimensional spinor-helicity formalism
  7. Generalized unitarity in $D$ dimensions
  8. Feynman diagram set-up for the FDH scheme
  9. Integrand reduction for the five gluon amplitude
  10. The pentagon-box: $(431)$
  11. The massive double-box: $(331;M_2)$
  12. The five-legged double-box: $(331;5L)$
  13. The box-triangle butterfly: $(430)$
  14. The massive double-triangle butterfly: $(330;M_2)$
  15. The five-leg double-triangle butterfly: $(330;5L)$
  16. ...and 8 more sections

Figures (7)

  • Figure 1: A pictorial representation of the planar two-loop topology denoted $I^{[D]}_{n_1 n_2 1;\mathcal{P}}$
  • Figure 2: Conventions for the momentum flow in the four-point double box, $(331)$, and the butterfly topology, $(330)$.
  • Figure 3: The flavour contributions to $\Delta_{s,\;331}^{[6]}$.
  • Figure 4: The flavour contributions to $\Delta_{s,\;330}^{[6]}$.
  • Figure 5: The flavour contribution to $\Delta_{2s,\;330}^{[6]}$.
  • ...and 2 more figures