Revisiting the $\phi^6$ Theory in Three Dimensions at Large $N$

TL;DR

This paper analyzes the three-dimensional $O(N)$-symmetric $$ theory at large $N$, computing the effective potential and examining the stability and fixed points, revealing that certain phenomena are not physically realized and that the UV fixed point persists.

Contribution

It provides the first explicit computation of the effective potential at next-to-leading order in $1/N$ for this theory and clarifies the nature of the tricritical line and Bardeen-Moshe-Bander phenomenon.

Findings

The tricritical line is not a physical manifestation in the strict $N o $ limit.

The effective potential at next-to-leading order is computed and its stability analyzed.

The Bardeen-Moshe-Bander phenomenon disappears without spontaneous scale symmetry breaking.

Abstract

We investigate the $O(N)$--symmetric $\phi^6$ theory in three spacetime dimensions using dimensional regularisation and minimal subtraction. The predictions of other methods are scrutinised in a large-$N$ expansion. We show how the tricritical line of fixed point emerges in a strict $N\to\infty$ limit but argue that it is not a physical manifestation. For the first time in this explicit manner, we compute the effective potential at next-to-leading order in the $1/N$-expansion and discuss its stability. The Bardeen-Moshe-Bander phenomenon is also analysed at next-to-leading order, and we demonstrate that it disappears without breaking the scale invariance spontaneously. Our findings indicate that the UV fixed point found by Pisarski persists at large $N$.