Post-Gaussian effective potential in scalar and scalar-fermion theories

The post-Gaussian effective potential (PGEP) of λ${\mathrm{\phi }}^4$ theory is examined in detail for d=1, 2, and 3 space-time dimensions. The self-energy is computed in the post-Gaussian approximation in order to determine the physical mass and the wave-function renormalization. The four-dimensional, ‘‘autonomous’’ version of the theory [and its O(N) generalization] is also investigated, using dimensional regularization. Serious difficulties are found: one can indeed obtain an ‘‘autonomous’’-like potential, but it becomes unbounded below for a certain range of parameters. Finally, a Yukawa fermion-scalar theory, without scalar self-interactions, is considered. The PGEP for the scalar sector turns out, after renormalization, to be identical to the GEP.