Phase structure of non-commutative scalar field theories

TL;DR

This paper explores the phase transitions in non-commutative scalar field theories, revealing ordered phases that break translation symmetry and discussing their implications for renormalizability.

Contribution

It provides a self-consistent analysis of phase structure, demonstrating the existence of renormalizable non-commutative scalar field theories in dimensions two and above.

Findings

Ordered phases break translation invariance.

Transition into ordered phases is first order.

Explicit renormalization shown in large N limit.

Abstract

We investigate the phase structure of non-commutative scalar field theories and find evidence for ordered phases which break translation invariance. A self-consistent one-loop analysis indicates that the transition into these ordered phases is first order. The phase structure and the existence of scaling limits provides an alternative to the structure of counter-terms in determining the renormalizability of non-commutative field theories. On the basis of the existence of a critical point in the closely related planar theory, we argue that there are renormalizable interacting non-commutative scalar field theories in dimensions two and above. We exhibit this renormalization explicitly in the large $N$ limit of a non-commutative O(N) vector model.