Invariant formulation of the Functional Renormalisation Group method for $U(n)\times U(n)$ symmetric matrix models

TL;DR

This paper develops an invariant formulation of the Functional Renormalisation Group for $U(n) imes U(n)$ symmetric matrix models, enabling systematic analysis of symmetry effects and anomalies in these theories.

Contribution

It provides explicit invariant formulations of the LPA for $U(n) imes U(n)$ models, including formulas for two- and three-flavor cases, and introduces a method to study the $U_A(1)$ anomaly separately.

Findings

Formulated invariant LPA for $U(n) imes U(n)$ models.

Derived explicit formulas for $U(2) imes U(2)$ and $U(3) imes U(3)$ cases.

Proposed a RG equation to analyze the $U_A(1)$ anomaly effects.

Abstract

The Local Potential Approximation (LPA) to the Wetterich-equation is formulated explicitly in terms of operators, which are invariant under the $U(n)\times U(n)$ symmetry group. Complete formulas are presented for the two-flavor ($U(2)\times U(2)$) case. The same approach leads to a unique natural truncation of the functional driving the renormalisation flow of the potential of the three-flavor case ($U(3)\times U(3)$). The procedure applied to the $SU(3)\times SU(3)$ symmetric theory, results in an equation, which potentially allows an RG-investigation of the effect of the 't Hooft term representing the $U_A(1)$ anomaly, disentangled from the other operators.