A Systematic and Efficient Method to Compute Multi-loop Master Integrals

TL;DR

A novel method to compute multi-loop master integrals by constructing and numerically solving a system of ordinary differential equations, with almost trivial boundary conditions, which can be systematically applied to problems with arbitrary kinematic configurations is proposed.

Abstract

We propose a novel method to compute multi-loop master integrals by constructing and numerically solving a system of ordinary differential equations, with almost trivial boundary conditions. Thus it can be systematically applied to problems with arbitrary kinematic configurations. Numerical tests show that our method can not only achieve results with high precision, but also be much faster than the only existing systematic method sector decomposition. As a by product, we find a new strategy to compute scalar one-loop integrals without reducing them to master integrals.

Figures (3)

• Figure 1: Diagrams of nonfactorizable vacuum master integrals up to 3 loops.
• Figure 2: An example distribution of singularities of ODEs. Singularities are labeled as crosses. Solid dots are points where to expand the ODEs.
• Figure 3: (a) A 1-loop box diagram; (b) A 2-loop non-planar box diagram.