Lectures on differential equations for Feynman integrals

TL;DR

This paper reviews recent advances in solving Feynman integrals with differential equations, highlighting algorithms, canonical forms, and new methods for simplifying and computing these integrals in quantum field theory.

Contribution

It introduces new algorithms and approaches for simplifying Feynman integrals using differential equations and properties of loop integrands, including canonical forms and basis optimization.

Findings

Algorithms for simplifying differential equations of Feynman integrals.

Connection between canonical forms and recent conjectures.

Application of the Drinfeld associator to bootstrap single-scale integrals.

Abstract

Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations. These lectures give a review of these developments, while not assuming any prior knowledge of the subject. After an introduction to differential equations for Feynman integrals, we point out how they can be simplified using algorithms available in the mathematical literature. We discuss how this is related to a recent conjecture for a canonical form of the equations. We also discuss a complementary approach that allows based on properties of the space-time loop integrands, and explain how the ideas of leading singularities and d-log representations can be used to find an optimal basis for the differential equations. Finally, as an application of the differential equations method we show how single-scale integrals can be bootstrapped using the Drinfeld associator of a differential equation.