Full Colour for Loop Amplitudes in Yang-Mills Theory

TL;DR

The paper addresses the challenge of keeping full colour information in loop-level Yang–Mills amplitudes, where colour factors proliferate with loop order. It develops a colour-decomposition framework compatible with multi-loop integrand reduction, preserving colour structures of unitarity cuts and mapping cuts to irreducible numerators. It shows that trace-based decompositions can be refined to a KK-independent DDM basis, enabling cancellations of symmetry factors and simplifying higher-loop calculations, with explicit all-plus examples at two and three loops. The approach generalizes to supersymmetric theories and potentially to QCD with quarks, offering a scalable route to fully colour-dressed loop amplitudes.

Abstract

We present a general method to account for full colour dependence Yang-Mills amplitudes at loop level. The method fits most naturally into the framework of multi-loop integrand reduction and in a nutshell amounts to consistently retaining the colour structures of the unitarity cuts from which the integrand is gradually constructed. This technique has already been used in the recent calculation of the two-loop five-gluon amplitude in pure Yang-Mills theory with all positive helicities, arXiv:1507.08797. In this note, we give a careful exposition of the method and discuss its connection to loop-level Kleiss-Kuijf relations. We also explore its implications for cancellation of nontrivial symmetry factors at two loops. As an example of its generality, we show how it applies to the three-loop case in supersymmetric Yang-Mills case.

Figures (13)

• Figure 1: Graphic version of the definition \ref{['TreeTraceColourFactor']} for $T(1,2,\dots,n)$. On the left-hand side it is depicted as an $n$-point vertex, since in Section \ref{['sec:irreducible']} these combine into trace-based colour factors for irreducible numerators.
• Figure 2: The quartic ordered subtraction vertex contains a maximum of two subgraphs (a). Inside the two-loop butterfly topology (b), only the second subgraph's propagator is loop-dependent and contributes to the hierarchy subtraction.
• Figure 3: The quintic ordered subtraction vertex contains a maximum of five one-propagator and five two-propagator subgraphs. These subgraphs are not Feynman diagrams of Yang-Mills theory; for instance, the six-point subtraction vertex contains even subgraphs involving five-point vertices.
• Figure 4: Graphic representation of the process of obtaining the DDM colour factors from traces via "stretching" of the vertex by any two edges (here chosen to be $1$ and $n$).
• Figure 5: Inserting the DDM tree basis into coloured cuts of a one-loop amplitude. As shown in the inset, "stretching" by the two internal edges gives a sum over permutations of the external legs.