The Gauged Vector Model in Four-Dimensions: Resolution of an Old Problem?

TL;DR

This paper analyzes the phase structure of a four-dimensional gauged U(N) vector model with fermions, revealing conditions for asymptotic freedom and showing the effective potential's behavior in different phases.

Contribution

It provides a leading-order calculation of the effective potential in the gauged vector model, identifying the conditions for asymptotic freedom and phase behavior.

Findings

Two distinct phases identified: asymptotically free and non-free.

Asymptotic freedom occurs when 0 < λ/g^2 < 4/3 (N_f/N - 1) and N_f/N < 11/2.

Effective potential behaves like the tree-level potential in the free phase.

Abstract

A calculation of the renormalization group improved effective potential for the gauged U(N) vector model, coupled to $N_f$ fermions in the fundamental representation, computed to leading order in 1/N, all orders in the scalar self-coupling $\lambda$, and lowest order in gauge coupling $g^2$, with $N_f$ of order $N$, is presented. It is shown that the theory has two phases, one of which is asymptotically free, and the other not, where the asymptotically free phase occurs if $0 < \lambda /g^2 < {4/3} (\frac{N_f}{N} - 1)$, and $\frac{N_f}{N} < {11/2}$. In the asymptotically free phase, the effective potential behaves qualitatively like the tree-level potential. In the other phase, the theory exhibits all the difficulties of the ungauged $(g^2 = 0)$ vector model. Therefore the theory appears to be consistent (only) in the asymptotically free phase.