A Complete Two-Loop, Five-Gluon Helicity Amplitude in Yang-Mills Theory

TL;DR

This work advances perturbative QCD by delivering the complete two-loop, full-colour five-gluon all-plus amplitude in Yang-Mills theory. It combines generalized unitarity with tree-level BCJ relations to deduce non-planar information from planar data, and employs a multi-peripheral colour decomposition to manage colour factors. The authors construct irreducible numerators, assemble the full colour integrand, and perform analytic checks of the leading soft divergences, highlighting the infrared structure and connections to N=4 SYM via dimension shifting. The approach demonstrates how on-shell symmetries and tree-level identities can streamline high-loop, high-multiplicity calculations beyond traditional Feynman-diagram methods, with potential extension to other helicities and processes.

Abstract

We compute the integrand of the full-colour, two-loop, five-gluon scattering amplitude in pure Yang-Mills theory with all helicities positive, using generalized unitarity cuts. Tree-level BCJ relations, satisfied by amplitudes appearing in the cuts, allow us to deduce all the necessary non-planar information for the full-colour amplitude from known planar data. We present our result in terms of irreducible numerators, with colour factors derived from the multi-peripheral colour decomposition. Finally, the leading soft divergences are checked to reproduce the expected infrared behaviour.

Figures (4)

• Figure 1: Multi-peripheral diagram for the colour factors in eq. \ref{['DDM']}.
• Figure 2: Inserting the DDM tree basis into the colour dressed cuts of a two-loop amplitude. The upper insert $(a)$ shows the simple case of two loop propagators, while the lower insert $(b)$ shows the case of three loop propagators. The sums run over the permutations of the external legs in the tree-level amplitude.
• Figure 3: Inserting the DDM tree basis into a colour dressed cut of a butterfly topology at two-loops. There are four loop propagators in this case, and the insert $(a)$ shows the result of inserting the DDM tree decomposition fixing the two right legs. The sums run over the permutations of the external legs in the tree-level amplitude.
• Figure 4: The two non-planar maximal topologies.