On the O(1/N^2) Beta-Function of the Nambu-Jona-Lasinio Model with Non-Abelian Chiral Symmetry

TL;DR

This paper develops a formalism to compute the critical exponent of the beta function in the Nambu-Jona-Lasinio model with non-Abelian chiral symmetry at order 1/N^2, revealing solvability only for the case M=2.

Contribution

It provides the first analytic expression for the critical exponent at O(1/N^2) in the NJL model with SU(M) x SU(M) symmetry, highlighting a unique solvable case for M=2.

Findings

Equations solvable only for M=2 case.

Analytic expression derived for M=2.

Contrasting behavior between M=2 and M>2 cases.

Abstract

We present the formalism for computing the critical exponent corresponding to the $\beta$-function of the Nambu--Jona-Lasinio model with $SU(M)$ $\times$ $SU(M)$ continuous chiral symmetry at $O(1/N^2)$ in a large $N$ expansion, where $N$ is the number of fermions. We find that the equations can only be solved for the case $M$ $=$ $2$ and subsequently an analytic expression is then derived. This contrasting behaviour between the $M$ $=$ $2$ and $M$ $>$ $2$ cases, which appears first at $O(1/N^2)$, is related to the fact that the anomalous dimensions of the bosonic fields are only equivalent for $M$ $=$ $2$.