Illustrations of Integrand-Basis Building at Two Loops

TL;DR

This paper presents a detailed method for constructing integrand bases for two-loop scattering amplitudes with six particles, emphasizing contour choices and subdividing bases by finiteness and transcendental weight.

Contribution

It introduces a systematic approach for building prescriptive integrand bases beyond the planar limit, including contour-based subdivisions and explicit basis construction for six particles.

Findings

Constructed a complete two-loop integrand basis for six particles.

Demonstrated how contour choices affect basis finiteness and divergence.

Showed how to subdivide bases by transcendental weight using double-poles.

Abstract

We outline the concrete steps involved in building prescriptive master integrand bases for scattering amplitudes beyond the planar limit. We highlight the role of contour choices in such bases, and illustrate the full process by constructing a complete, triangle power-counting basis at two loops for six particles. We show how collinear contour choices can be used to divide integrand bases into separately finite and divergent subspaces, and how double-poles can be used to further subdivide these spaces according to (transcendental) weight. Complete details of the basis constructed for six particles is provided in the ancillary files for this work's submission to the arXiv.