φ4model near the critical point: The Gaussian variational approximation

Using the Gaussian variational approximation, we describe the approach to the continuum limit in ${\phi }^4$ theory in 3+1 space-time dimensions. We study the solutions of the variational equations and their stability and compare our results with those of Monte Carlo calculations on a lattice. The importance of lattice effects is investigated by putting the Gaussian wave functional on a lattice. We find an abrupt decrease of the fourth derivative of the effective potential in the vicinity of the critical point, which is consistent with Monte Carlo calculations. In the continuum limit $b\to 0^{+}$, the renormalized theory shows a broken phase which is degenerate with the symmetric phase. In the asymmetric phase the theory is found to be asymptotically free, in agreement with the conclusions of Branchina et al.