Three-loop effective potential of general scalar theory via differential equations

TL;DR

This paper analytically computes the three-loop effective potential for a general scalar theory using differential equations, expressing results in cyclotomic polylogarithms and providing a numerical evaluation algorithm.

Contribution

It introduces a method to evaluate three-loop vacuum diagrams with multiple masses using differential equations and cyclotomic polylogarithms, including a new numerical implementation.

Findings

Analytic expressions for three-loop vacuum diagrams with up to two masses.

Expression of master integrals in terms of cyclotomic polylogarithms up to weight four.

A Mathematica package for numerical evaluation of cyclotomic polylogarithms.

Abstract

We consider the scalar sector of a general renormalizable theory and evaluate the effective potential through three loops analytically. We encounter three-loop vacuum bubble diagrams with up to two masses and six lines, which we solve using differential equations transformed into the favorable $\epsilon$ form of dimensional regularization. The master integrals of the canonical basis thus obtained are expressed in terms of cyclotomic polylogarithms up to weight four. We also introduce an algorithm for the numerical evaluation of cyclotomic polylogarithms with multiple-precision arithmetic, which is implemented in the Mathematica package cyclogpl.m supplied here.