Introduction

Years ago we introduced a calculational technique that was quite successful in describing the critical behavior of the $SO(\mathfrak{N})$-symmetric Euclidean $\lambda\phi^{4}$ model in $d\geq 3$ spacetime dimensions [1]. Some quantities were calculated in $d=4$ and compared to the results of computer simulations, yielding surprisingly good results, and describing reliably the most important qualitative aspects of the model. The observables calculated where the expectation value of the field, which is the order parameter of the critical transition of the model and describes the phenomenon of spontaneous symmetry breaking, and the two-point function, from which one can get the renormalized masses and hence the correlation lengths, in both phases of the model.

Although inspired by and superficially similar to perturbation theory, the technique can handle a phenomenon such as spontaneous symmetry breaking, which is usually considered to be out of reach for plain perturbation theory. The innovative and essential aspect of the technique is the use of certain self-consistency conditions within a framework similar to that of perturbation theory. The technique would be better described as a Gaussian approximation rather than a perturbative expansion. As such, it is able to produce good predictions for the one-point and two-point observables, since these are the moments present in the Gaussian distribution, but should not be expected to go much further than that. For lack of a better name, we shall refer to it as the Gaussian-Perturbative approximation.

The important role that the four-component $\lambda\phi^{4}$ model plays in the Standard Model of high-energy particle physics makes it certainly interesting to learn more about it. In this paper we extend the Gaussian-Perturbative technique introduced in [1] to the same model in the presence of external sources. These external sources are not thought of merely as analytical devices used to extract the Green's functions from the functional generators of the model, and to be put to zero afterwards. Instead, they are thought of as actual physical sources of particles in the model. One important objective is to determine how the introduction of the external sources affects the values of the renormalized masses in either phase of the model.

These are analytical calculations performed on the Euclidean lattice, which therefore allow us to discuss, and to explicitly take, specific continuum limits in the quantum theory. As we will see, there is no need for perturbative renormalization, or for any regulation mechanism other than the lattice where the model is defined. All calculations on finite lattices are ordinary straightforward manipulations. Although there are some quantities that do diverge in the continuum limit, they all cancel off from the observables before the limit is taken.