Optimized Rayleigh–Schrödinger expansion of the effective potential

Abstract

An optimized Rayleigh-Schrödinger expansion scheme of solving the functional Schrödinger equation with an external source is proposed to calculate the effective potential beyond the Gaussian approximation. For a scalar field theory whose potential function has a Fourier representation in a sense of tempered distributions, we obtain the effective potential up to the second order, and show that the first-order result is just the Gaussian effective potential. Its application to the λφ 4 field theory yields the same post-Gaussian effective potential as obtained in the functional integral formalism.

Sine-Gordon effective potential beyond Gaussian approximation

“…In fact, the proper functions in Minkowski space coincide with those in Euclidean space [18], and discussions on the λφ 4 model in Refs. [13,14,16] have suggested and supported this point.…”

“…If the series is truncated at any order of δ, then the truncated result will depend on arbitrary parameters µ and Φ. The background field Φ does not affect the EP as long as the wave function renormalization is not involved and so Φ can conveniently and directly be rendered as ϕ r , the vacuum expectation value of the field operator φ [13,14]. As for µ, it should be determined according to the principle of minimal sensitivity [19].…”

“…Under such a circumstance, a better approximated EP of sG field theory beyond the Gaussian approximation will be worth considering and may give a substantial correction to the sG Gaussian EP. Recently, the calculation of the EP beyond the Gaussian approximation has attracted much interest [12,13] and the λφ 4 model as well as some simple fermion models were considered [12,13,14]. Very recently, we also considered a class of scalar field theories with the optimized Rayleigh-Schrödinger expansion [14].…”

“…Recently, the calculation of the EP beyond the Gaussian approximation has attracted much interest [12,13] and the λφ 4 model as well as some simple fermion models were considered [12,13,14]. Very recently, we also considered a class of scalar field theories with the optimized Rayleigh-Schrödinger expansion [14]. Therefore, it is interesting to add an exponential interaction model, for example the sG field theory, to this list.…”

Rayleigh-Schr�dinger-Goldstone variational perturbation theory for many Fermion systems

“…Further improvements beyond the Gaussian approxiamtion have been investigated mostly in two directions. One is to try it with non-Gaussian wavefunctionals [16,17], and the other is to perform appropriate expansions based on Gaussian trial wavefunctional [18][19][20][21][22][23][24].…”