The renormalization group functions and the effective potential

Abstract

It is first demonstrated how the effective potential Veff in a self-interacting scalar theory can be computed using operator regularization. We examine [Formula: see text] and [Formula: see text] theories, recovering the usual results in the former case and showing how Veff is a power series in the square root of the coupling λ in the latter. Scheme dependence of Veff is considered. Since no explicit divergences occur when one uses operator regularization, the renormalization group functions (β and γ) associated with the dependence of λ and [Formula: see text] on the radiatively induced scale parameter μ must be determined by considering the finite effective potential. It is shown that one must in fact compute Vefff to a higher power in the perturbative expansion than if β and γ were to be computed using Green's functions. The usual results to lowest order are recovered in the [Formula: see text] model. Finally, a nonperturbative β function is determined by requiring that the mass generated by radiative effects be independent of μ2; it is found that both [Formula: see text] and [Formula: see text] are asymptoticly free with this β function. In the appendix we explicitly compute a two-loop integral encountered in the evaluation of Veff.