Twisted Riemann bilinear relations and Feynman integrals

TL;DR

This paper explores the application of twisted cohomology to derive relations among multi-loop Feynman integrals, revealing new quadratic relations for maximal cuts and clarifying limitations for non-maximal cuts.

Contribution

It introduces a framework using twisted Riemann bilinear relations to analyze Feynman integrals, establishing new quadratic relations for maximal cuts and explaining their limitations.

Findings

Quadratic relations are derived for maximal cuts.

TRBRs do not produce quadratic relations for non-maximal cuts.

The relations are linear in the period matrix and its dual, not quadratic.

Abstract

Using the framework of twisted cohomology, we study twisted Riemann bilinear relations (TRBRs) satisfied by multi-loop Feynman integrals and their cuts in dimensional regularisation. After showing how to associate to a given family of Feynman integrals a period matrix whose entries are cuts, we investigate the TRBRs satisfied by this period matrix, its dual and the intersection matrices for twisted cycles and co-cycles. For maximal cuts, the non-relative framework is applicable, and the period matrix and its dual are related in a simple manner. We then find that the TRBRs give rise to quadratic relations that generalise quadratic relations that have previously appeared in the literature. However, we find that the TRBRs do not allow us to obtain quadratic relations for non-maximal cuts or completely uncut Feynman integrals. This can be traced back to the fact that the TRBRs are not quadratic in the period matrix, but separately linear in the period matrix and its dual, and the two are not simply related in the case of a relative cohomology theory, which is required for non-maximal cuts.