Monte Carlo and Renormalization Group Effective Potentials in Scalar Field Theories

We study constraint effective potentials for various strongly interacting ${\varphi }^4$ theories. Renormalization group (RG) equations for these quantities are discussed and a heuristic development of a commonly used RG approximation is presented which stresses the relationships among the loop expansion, the Schwinger-Dyson method and the renormalization group approach. We extend the standard RG treatment to account explicitly for finite lattice effects. Constraint effective potentials are then evaluated using Monte Carlo (MC) techniques and careful comparisons are made with RG calculations. Explicit treatment of finite lattice effects is found to be essential in achieving quantitative agreement with the MC effective potentials. Excellent agreement is demonstrated for $d=3$ and $d=4$, O(1) and O(2) cases in both symmetric and broken phases.