Large N and the renormalization group

TL;DR

This paper demonstrates that in the large N limit, the local potential approximation becomes exact for the Legendre effective action, and provides insights into the solvability of various renormalization group equations for O(N) scalar theories.

Contribution

It establishes the exactness of the local potential approximation in the large N limit and derives relations between different flow equations for the first time.

Findings

Local potential approximation becomes exact at large N

Derived an exact relation between Polchinski and Legendre potentials

All forms of the renormalization group are exactly solvable at large N

Abstract

In the large N limit, we show that the Local Potential Approximation to the flow equation for the Legendre effective action, is in effect no longer an approximation, but exact - in a sense, and under conditions, that we determine precisely. We explain why the same is not true for the Polchinski or Wilson flow equations and, by deriving an exact relation between the Polchinski and Legendre effective potentials (that holds for all N), we find the correct large N limit of these flow equations. We also show that all forms (and all parts) of the renormalization group are exactly soluble in the large N limit, choosing as an example, D dimensional O(N) invariant N-component scalar field theory.