Conclusions

It was already known, for some time, that the approximation scheme that we name here the Gaussian-Perturbative approximation gives good results for the $SO(\mathfrak{N})$-symmetric $\lambda\phi^{4}$ model in $d=4$, regarding its critical behavior [1]. It is interesting to speculate that the triviality of the model in $d=4$, which is fairly well established numerically but still lacks rigorous proof, is somehow behind the fact that this approximation works as well as it does in that case. This is so because a trivial model would have a Gaussian effective action, which would allow for a good approximation of its expectation values by a Gaussian measure, which is what we do in the Gaussian-Perturbative approximation.

In this work we extended that technique to the same model in the presence of an external source. This resulted in specific predictions for the values of the expectation value of the field and for the renormalized masses as functions of the external source. Such predictions could motivate future numerical studies with the objective of evaluating their worth by comparing them to the results of appropriate stochastic simulations. In particular, the fact that the renormalized masses do depend on the external sources through the expectation value of the field is important for the very design of some such numerical simulations.

The simulations done in the past to test this technique used what we named back-rotation simulations, which introduce some additional uncertainties into the whole analysis. This was done because neither in the analytical calculations nor in the numerical simulations we were capable at that time to deal appropriately with the external sources. It is now possible to perform simulations in the presence external sources, without the use of the back-rotation idea. With such simulations and the results presented in this paper, it should be possible to do much better comparison of the numerical and analytical results.

We also pointed out a simple and interesting consequence of the results regarding the application of the $\lambda\phi^{4}$ model in the Standard Model of particle physics. The results allow us to determine the critical point $(\alpha,\lambda)$ in the parameter plane of the model that should correspond to the continuum-limit flows leading to the Standard Model. This is a rather unique situation, in which actual experimental data is used to determine the values of bare, dimensionless parameters within the mathematical structure of the $\lambda\phi^{4}$ model.

Although this result in itself may be no more than a curiosity, it would be interesting to determine whether or not this technique and its results could not find a more widespread application for the computation of physical predictions from the Standard Model. This, combined with the use of the very probable fact that the renormalized coupling constant $\lambda_{R}$ is in fact exactly zero in the continuum limits of this model, could very well result in the extraction of new and interesting insights from the Standard Model.

As an example, we may point out that the attribution of a negative value to the bare parameter $\alpha$ is not really a matter of choice, as is implied in the usual treatment of the Standard Model. It is in fact a very strict requirement for the existence of physically meaningful continuum limits of the model, as we have shown in this paper. There are in fact no continuum limits, in which the fundamental action is not Gaussian and the model has finite renormalized masses, such that $\alpha>0$ in the limit.

Is in important to point out that the results presented here for critical behavior and symmetry breaking within the $\lambda\phi^{4}$ model are quite independent of the renormalized coupling constant $\lambda_{R}$. In particular, they are quite independent of whether or not $\lambda_{R}$ is zero in the continuum limit. In other words, the probable triviality of the model in the continuum limit does not disturb the mechanism of phase transition and symmetry breaking, and hence would not void the Higgs mechanism.