The finite and large-$N$ behaviors of independent-value matrix models

TL;DR

This paper studies the behavior of independent-value O(N)-invariant matrix models at finite and large N, revealing a continuous connection between interacting and pseudo-free theories, with implications for nonrenormalizable models.

Contribution

It demonstrates that all interaction couplings vanish in the large N limit, connecting interacting models to a pseudo-free theory, and extends this result to finite N with nonperturbative methods.

Findings

Interacting theories approach a pseudo-free state as couplings go to zero.

The behavior is proven for finite N models and partially for N→∞ models.

The models are nonrenormalizable but tractable via nonperturbative techniques.

Abstract

We investigate the finite and large $N$ behaviors of independent-value O(N)-invariant matrix models. These are models defined with matrix-type fields and with no gradient term in their action. They are generically nonrenormalizable but can be handled by nonperturbative techniques. We find that the functional of any O(N) matrix trace invariant may be expressed in terms of an O(N)-invariant measure. Based on this result, we prove that, in the limit that all interaction coupling constants go to zero, any interacting theory is continuously connected to a pseudo-free theory. This theory differs radically from the familiar free theory consisting in putting the coupling constants to zero in the initial action. The proof is given for generic finite-size matrix models, whereas, in the limiting case $N\rightarrow\infty$, we succeed in showing this behavior for restricted types of actions using a particular scaling of the parameters.