Solving RG equations with the Lambert W function

TL;DR

This paper demonstrates how the Lambert W function can be used to solve 2-loop RG equations in quantum field theories, enabling explicit construction of RG invariants and proposing a potential for the O(N) model.

Contribution

It introduces a method to solve RG equations using the Lambert W function and applies it to construct RG invariants and speculate on the effective potential for the O(N) model.

Findings

Closed-form solutions for 2-loop RG equations using Lambert W

Construction of RG invariant physical parameters

Speculation on an exact effective potential for the O(N) model

Abstract

It has been known for some time that 2-loop renormalization group (RG) equations of a dimensionless parameter can be solved in a closed form in terms of the Lambert W function. We apply the method to a generic theory with a Gaussian fixed point to construct RG invariant physical parameters such as a coupling constant and a physical squared mass. As a further application, we speculate a possible exact effective potential for the O(N) linear sigma model in four dimensions.