Analytic and numerical approaches for depictive 3-loop integrals using sector decomposition

TL;DR

This paper combines analytical and numerical methods to evaluate complex 3-loop integrals in quantum field theory, providing detailed insights into their divergent and finite parts and energy dependence.

Contribution

It introduces a simplified sector decomposition approach for 3-loop integrals, enabling analytical determination of divergences and numerical computation of finite parts.

Findings

Analytical coefficients for ultraviolet divergences are obtained.

Finite parts are accurately computed numerically.

Energy dependence of the integrals is explicitly demonstrated.

Abstract

Four 3-loop two-point functions are studied analytically and numerically using a simplified sector decomposition method. The coefficients of the ultraviolet divergent part are determined analytically, and those of the finite part are computed numerically. The energy dependence of the integrals is shown explicitly, and a discussion of its behavior is presented.