Spontaneous symmetry breaking in noncommutative field theories

The spontaneous symmetry breaking of rotational $O\left(N\right)\mathrm{}$ 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 $\overrightarrow{N}\infty$ 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 the Goldstone theorem holds true. We also find a new purely noncommutative coupling which gives contributions starting at order $N-2$ loops.