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Free parameters in quantum field theories: an analysis of the variational approach

The usual renormalization procedure for the variational approximation with a trial Gaussian ansatz for the $\lap$ model in 3+1 dimensions is re-analysed as a departing framework for the investigation of the parameters of the model. The so-called asymmetric phase of the model (where $<vac|ϕ|vac> \neq 0$) is considered for the search of privileged values of these parameters (mass and coupling constant) and possible conditions they may be expected to satisfy. This also may yield a suitable approach for the investigation of the reliability and stability of the approximation. The extremization of the renormalized energy density with relation to the renormalized mass, coupling and $\bphi$ is done. The minimizations of the renormalized energy with relation to the mass and $\bphi$ provide different expressions from the ones obtained by the usual variational procedure for the regularized theory. Sort of ``energy scale'' invariances in expressions for the renormalized mass and coupling constant are found. A different view on the restoration of symmetry issue is discussed. The transcendental character of the GAP equation may be reduced or even eliminated by placing some variables in the complex plane.