Effective potential and spontaneous symmetry breaking in the noncommutative φ 6 model

We study the conditions for spontaneous symmetry breaking of the $(2+1)$-dimensional noncommutative ${\ensuremath{\varphi}}^{6}$ model in the small-\ensuremath{\theta} limit. In this regime, considering the model as a cutoff theory, it is reasonable to assume translational invariance as a property of the vacuum state and study the conditions for spontaneous symmetry breaking by an effective potential analysis. An investigation of up to the two-loop level reveals that noncommutative effects can modify drastically the shape of the effective potential. Under reasonable conditions, the nonplanar sector of the theory can become dominant and induce symmetry breaking for values of the mass and coupling constants not reached by the commutative counterpart.