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Higher order extensions of the Gaussian effective potential

A variational method is discussed, extending the Gaussian effective potential to higher orders. The single variational parameter is replaced by trial unknown two-point functions, with infinite variational parameters to be optimized by the solution of a set of integral equations. These stationary conditions are derived by the self-energy without having to write the effective potential, making use of a general relation between self-energy and functional derivatives of the potential. This connection is proven to any order and verified up to second order by an explicit calculation for the scalar theory. Among several variational strategies, the methods of minimal sensitivity and of minimal variance are discussed in some detail. For the scalar theory, at variance with other post-Gaussian approaches, the pole of the second-order propagator is shown to satisfy the simple first-order gap equation that seems to be more robust than expected. By the method of minimal variance, nontrivial results are found for gauge theories containing fermions, where the first-order Gaussian approximation is known to be useless.