Chiral Phase Transition for SU(N) Gauge Theories via an Effective Lagrangian Approach

TL;DR

This work addresses the zero-temperature chiral phase transition in SU($N$) gauge theories as a function of $N_f$ by constructing an anomaly-induced effective potential that incorporates the full $eta$-function and the anomalous dimension $\gamma$ of the quark mass operator. The authors show that chiral symmetry is restored when $\gamma<1$, deriving a criterion for criticality and estimating $N_f^c$ to be about $4N$ using perturbative $\gamma$. The approach relies on mesonic degrees of freedom and saturates trace and axial anomalies to determine the potential, yielding an infinite-order transition at $\gamma=1$ and a GL-like description near criticality. The results connect the chiral/conformal transition to the conformal window and walking technicolor scenarios, offering a framework that complements gap-equation analyses while highlighting the role of anomaly structure in low-energy dynamics.

Abstract

We study the chiral phase transition for vector-like SU(N) gauge theories as a function of the number of quark flavors N_f by making use of an anomaly-induced effective potential. We modify an effective potential of a previous work, suggested for N_f < N, and apply it to larger values of N_f where the phase transition is expected to occur. The new effective potential depends explicitly on the full β-function and the anomalous dimension γof the quark mass operator. By using this potential we argue that chiral symmetry is restored for γ<1. A perturbative computation of γthen leads to an estimate of the critical value N_f^c for the transition.

Figures (1)

• Figure 1: $N_{f}^{c}/N$ as a function of $\alpha N$ is shown as a solid line. The dashed line represents $N_{f}^{s}/N$, while the dot-dashed line describes the $N_{f}^{\ast }/N$ function.