Evaluating Feynman Integrals through differential equations and series expansions

TL;DR

The paper addresses the challenge of evaluating multi-loop Feynman integrals by adopting the differential equations method and, crucially, a series expansion approach that allows solving for master integrals order-by-order in $$ and in a kinematic variable. It outlines how to derive the DE system $\frac{\partial}{\partial s}\vec{I}(s;\epsilon)=\boldsymbol{A}(s;\epsilon)\vec{I}(s;\epsilon)$, perform an epsilon expansion, and compute solutions via bottom-to-top triangularization, Frobenius, and variation-of-parameters, including extensions to coupled systems. The work discusses boundary conditions, analytic continuation, and the Complex Mass Scheme to handle unstable particles, with practical guidance on managing convergence and branch cuts. It emphasizes the generality and automation potential of the series-expansion framework for high-precision predictions, along with references to public tools like AMFlow, DiffExp, and SeaSyde.

Abstract

We review the method of the differential equations for the evaluation of multi-loop Feynman integrals. In particular, we focus on the series expansion approach for solving the system of differential equation and we discuss how to perform the analytical continuation of the result to entire (complex) phase-space. This approach allow us to consider arbitrary internal complex masses. This review is based on a lecture given by the author at the "Advanced School and Workshop on Multiloop Scattering Amplitudes" held in NISER, Bhubaneswar (India) in January 2024.

Figures (6)

• Figure 1: Comparison of the series solution for $B_2^{(-1)}$ against the exact solution. The series is centered in $x=0$ and the closest singularity is $x=4$. This implies that the series converges in the interval $(-4,4)$, which can be seen from the rapidly growing of the solution around $x=-4$.
• Figure 2: Radius of convergence and example analytic continuation.
• Figure 3: Effect of a branch cut on the analytic continuation. It shrinks the area in which the series converges to the desired value (left) and it modifies the path for moving from one point to another (right).
• Figure 4: Example of a Logarithmic expansion on the left, against a Taylor expansion on the right.
• Figure 5: Examples of two possible ways to move along a branch cut. In case there is no a singularity between them (left) or there is (right).