Second-order corrections to the Gaussian effective potential of λφ4theory

We formulate a systematic, nonperturbative expansion for the effective potential of λ${\mathrm{\phi }}^4$ theory. At first order it gives the Gaussian effective potential (GEP), which itself contains the one-loop and leading-order 1/N results. Here, we compute the second-order terms and carry out the renormalization in the four-dimensional, ‘‘precarious’’ case, using dimensional regularization. (Difficulties with other regularizations are briefly discussed.) Remarkably, the final result takes the same mathematical form as the GEP, with only some numerical coefficients being changed. Indeed, in the most natural parametrization, only a single coefficient is changed, from 1 to 1-1/(N+3${)}^2$.