Integration-by-parts identities and differential equations for parametrised Feynman integrals

TL;DR

This paper introduces a novel approach to deriving IBP identities and differential equations for Feynman integrals using Feynman-parameter space and projective geometry, offering a unified framework applicable to any loop order.

Contribution

It generalizes and adapts historical results to modern loop calculations, providing a new geometric perspective on IBP identities in parameter space.

Findings

Derived general IBP identities in Feynman-parameter space

Validated method by solving differential equations for simple diagrams

Unified existing results within a geometric framework

Abstract

Integration-by-parts (IBP) identities and differential equations are the primary modern tools for the evaluation of high-order Feynman integrals. They are commonly derived and implemented in the momentum-space representation. We provide a different viewpoint on these important tools by working in Feynman-parameter space, and using its projective geometry. Our work is based upon little-known results pre-dating the modern era of loop calculations: we adapt and generalise these results, deriving a very general expression for sets of IBP identities in parameter space, associated with a generic Feynman diagram, and valid to any loop order, relying on the characterisation of Feynman-parameter integrands as projective forms. We validate our method by deriving and solving systems of differential equations for several simple diagrams at one and two loops, providing a unified perspective on a number of existing results.