Spontaneous Symmetry Breaking in Noncommutative Field Theory

TL;DR

This paper investigates spontaneous symmetry breaking in noncommutative scalar field theories, demonstrating conditions for Goldstone bosons, analyzing supersymmetric extensions, and exploring renormalizability and loop corrections in various models.

Contribution

It provides a detailed analysis of symmetry breaking, Goldstone bosons, and renormalizability in noncommutative field theories, including supersymmetric and higher-dimensional models.

Findings

Certain field orderings eliminate IR/UV mixing.

Goldstone bosons appear under specific conditions.

Model shows potential for all-order renormalizability at large N.

Abstract

The spontaneous symmetry breaking of rotational O(N) symmetry in noncommutative field theory is investigated in a 2+1 dimensional model of scalar fields coupled through a combination of quartic and sextuple self-interactions. There are five possible orderings of the fields in the sextuple interaction and two for the quartic interaction. At one loop, we prove that for some choices of these orderings there is the absence of IR/UV mixing and the appearance of massless excitations. A supersymmetric extension of the model is also studied. Supersymmetry puts additional constraints on the couplings but for any given N there is a Moyal ordering of the superfields for which the requirement for the existence of Goldstone bosons is satisfied. For some ordering and when N goes to infinity we find evidence that the model is renormalizable to all orders in perturbation theory. We also consider a generic chiral model in 3+1 dimensions whose superpotential is invariant under local gauge transformations. We find that for any value of N there is no one loop correction to the pion mass and that, at two loops, there are no pion mass corrections for slowly varying superfields so that Goldstone theorem holds true. We also find a new purely noncommutative coupling which gives contributions starting at order N-2 loops.