High-precision calculation of multi-loop Feynman integrals by difference equations

TL;DR

The paper introduces a comprehensive, automated framework for computing multi-loop Feynman integrals by converting IBP identities into finite systems of linear relations and, crucially, into one-variable difference equations for master integrals. It develops two complementary solution strategies: factorial-series expansions and Laplace-transform-based differential equations, both yielding high-precision results and ε-expansions, even for diagrams with hundreds of masters. The SYS program implements the entire pipeline—from IBP-based reduction to numerical evaluation of master integrals—demonstrating applications up to three loops for vacuum and self-energy diagrams and two loops for vertex and box diagrams, with extensive cross-checks and discussion of stability and deformation issues. The approach generalizes to differential equations in masses and momenta, enabling reconstruction of integrals across kinematic regimes, including non-Euclidean regions and zero-mass cases, thereby facilitating precise, automated multi-loop calculations with broad applicability to phenomenology, such as higher-order contributions in gauge theories.

Abstract

We describe a new method of calculation of generic multi-loop master integrals based on the numerical solution of systems of difference equations in one variable. We show algorithms for the construction of the systems using integration-by-parts identities and methods of solutions by means of expansions in factorial series and Laplace's transformation. We also describe new algorithms for the identification of master integrals and the reduction of generic Feynman integrals to master integrals, and procedures for generating and solving systems of differential equations in masses and momenta for master integrals. We apply our method to the calculation of the master integrals of massive vacuum and self-energy diagrams up to three loops and of massive vertex and box diagrams up to two loops. Implementation in a computer program of our approach is described. Important features of the implementation are: the ability to deal with hundreds of master integrals and the ability to obtain very high precision results expanded at will in the number of dimensions.

Figures (5)

• Figure 1: Vacuum diagrams up to three loops.
• Figure 2: Self-energy diagrams up to three loops.
• Figure 3: Vertex diagrams up to two loops.
• Figure 4: Box diagrams up to two loops.
• Figure 5: Diagrams with massless lines (dashed).