Reduction of a Class of Three-Loop Vacuum Diagrams to Tetrahedron Topologies

TL;DR

This paper develops a method to simplify complex three-loop vacuum diagrams in quantum field theory by reducing them to tetrahedron topologies, enabling analytical computation of their finite parts and application to the effective potential of massive  theory.

Contribution

It introduces a reduction technique for three-loop vacuum diagrams to tetrahedron topologies using integration-by-parts, facilitating analytical calculations.

Findings

Finite parts of three-loop vacuum diagrams obtained.

Reduction to tetrahedron diagrams simplifies calculations.

Application to  theory effective potential demonstrated.

Abstract

We obtain finite parts (as well as $\epsilon$-pole parts) of massive three-loop vacuum diagrams with three-point and/or four-point interaction vertices by reducing them to tetrahedron diagrams with both massive and massless lines, whose finite parts were given analytically in a recent paper by Broadhurst. In the procedure of reduction, the method of integration-by-parts recurrence relations is employed. We use our result to compute the $\bar{\rm MS}$ effective potential of the massive $\phi^4$ theory.