Transcendental equations of the running coupling

TL;DR

This paper analyzes the structure of transcendental equations arising from the running coupling in field theories at higher loop orders, proposing methods to approximate solutions using Pade approximants and power series expansions.

Contribution

It introduces a systematic study of transcendental equations in the context of running couplings, including simplification techniques and solution representations at various loop orders.

Findings

Transcendental equations at higher loops can be simplified with Pade approximants.

Solutions are expressed as power series generalizations of Lambert's equation.

Analytical methods for solving these equations are discussed and developed.

Abstract

The running coupling of a generic field theory can be described through a separable differential equation involving the corresponding $\beta$-function. Only the first loop order can be solved analytically in terms of well-known functions, all further loop orders lead to transcendental equations. While obscure nowadays, many analytical methods have been devised to study them, most specifically the Lagrange-B\"urmann formula. In this article we discuss the structure of transcendental equations that take place at various loop orders. Beyond the first two loop orders, these equations are simplified by applying an optimal Pade approximant on the $\beta$-function. In general, these lead to generalizations of Lambert's equation, the solutions of which are presented in terms of a power series.