Intersection theory, relative cohomology and the Feynman parametrization

TL;DR

This paper introduces a new method for reducing loop integrals in quantum field theory by applying intersection theory and relative cohomology, providing a mathematically rigorous framework for Feynman integral computation.

Contribution

It develops a novel approach to Feynman integral reduction using intersection theory and relative cohomology, offering a new perspective and computational technique.

Findings

Successfully applied to several examples demonstrating correctness

Provides a mathematically rigorous framework for integral reduction

Discusses subtleties in degenerate limits

Abstract

We present a novel approach for loop integral reduction in the Feynman parametrization using intersection theory and relative cohomology. In this framework, Feynman integrals correspond to boundary-supported differential forms in the language of relative cohomology. The integral reduction can then be achieved by computing intersection numbers. We apply our method in several examples to demonstrate its correctness, and discuss the subtleties in certain degenerate limits.