Exact Beta Functions in the Vector Model and Renormalization Group Approach

TL;DR

This paper uses the vector model to clarify the renormalization group approach at large N, deriving exact beta functions, fixed points, and susceptibility exponents through an exact difference equation and reparametrization identities.

Contribution

It provides an exact derivation of beta functions and fixed points in the vector model, demonstrating the effectiveness of the renormalization group approach at large N.

Findings

Exact difference equation relating free energies for different N

Infinite identities reduce coupling space to finite dimensions

Exact fixed points and susceptibility exponents obtained

Abstract

The validity of the renormalization group approach for large $N$ is clarified by using the vector model as an example. An exact difference equation is obtained which relates free energies for neighboring values of $N$. The reparametrization freedom in field space provides infinitely many identities which reduce the infinite dimensional coupling constant space to that of finite dimensions. The effective beta functions give exact values for the fixed points and the susceptibility exponents.