The effective potential in three - dimensional O ( N ) models

We consider the effective potential in three-dimensional models with O(N) symmetry. For generic values of N , and in particular for the physically interesting cases N = 0, 1, 2, 3, we determine the six-point and eight-point renormalized coupling constants which parametrize its small-field expansion. These estimates are obtained from the analysis of their ǫ-expansion, taking into account the exact results in one and zero dimensions, and, for the Ising model (i.e. N = 1), the accurate high-temperature estimates in two dimensions. They are compared with the available results from other approaches. We also obtain corresponding estimates for the two-dimensional O(N) models.