Towards a Basis for Planar Two-Loop Integrals

TL;DR

This paper develops a method to reduce planar two-loop integrals to a finite basis, enhancing the efficiency of multi-loop amplitude computations in quantum field theories.

Contribution

It provides an explicit construction for reducing planar two-loop integrals to finite bases, applicable to all orders in the dimensional regulator and truncated cases.

Findings

Constructed a finite basis for planar two-loop integrals.

Reorganized integration-by-parts equations for efficient basis element extraction.

Used Gram determinants to further reduce the basis size.

Abstract

The existence of a finite basis of algebraically independent one-loop integrals has underpinned important developments in the computation of one-loop amplitudes in field theories and gauge theories in particular. We give an explicit construction reducing integrals to a finite basis for planar integrals at two loops, both to all orders in the dimensional regulator e, and also when all integrals are truncated to O(e). We show how to reorganize integration-by-parts equations to obtain elements of the first basis efficiently, and how to use Gram determinants to obtain additional linear relations reducing this all-orders basis to the second one. The techniques we present should apply to non-planar integrals, to integrals with massive propagators, and beyond two loops as well.