Explicit solutions for effective four- and five-loop QCD running coupling

TL;DR

This paper derives explicit solutions for the QCD running coupling at four- and five-loop levels using Lambert W functions, providing a new analytical approach to understanding gauge coupling evolution.

Contribution

It introduces a class of beta functions solvable via Lambert W functions that match high-loop QCD beta functions, extending analytical solutions in gauge theories.

Findings

Explicit four- and five-loop solutions for QCD coupling

Construction of beta functions matching high-loop coefficients

Use of Lambert W function for analytical solutions

Abstract

We start with the explicit solution, in terms of the Lambert W function, of the renormalization group equation (RGE) for the gauge coupling in the supersymmetric Yang-Mills theory described by the well-known beta function of Novikov et al.(NSVZ). We then construct a class of beta functions for which the RGE can be solved in terms of the Lambert W function. These beta functions are expressed in terms of a function which is a truncated Laurent series in the inverse of the gauge coupling. The parameters in the Laurent series can be adjusted so that the first coefficients of the Taylor expansion of the beta function in the gauge coupling reproduce the four-loop or five-loop QCD (or SQCD) beta function.