Phase structure of non-commutative scalar field theories

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.