Gaussian-Perturbative Calculations with a Homogeneous External Source

We derive the equation of the critical curve and calculate the renormalized masses of the $SO(N)$-symmetric $\lambda\phi^{4}$ model in the presence of a homogeneous external source. We do this using the Gaussian-Perturbative approximation on finite lattices and explicitly taking the continuum limit. No disabling divergences are found in the final results, and no renormalization is necessary. We show that the results give a complete description of the critical behavior of the model and of the phenomenon of spontaneous symmetry breaking, at the quantum-field-theoretical level. We show that the renormalized masses depend on the external source, and point out the consequences of that fact for the design of computer simulations of the model. We point out a simple but interesting consequence of the results, regarding the role of the $\lambda\phi^{4}$ model in the Standard Model of high-energy particle physics. Using the experimentally known values of the mass and of the expectation value of the Higgs field, we determine uniquely the values of the bare dimensionless parameters $\alpha$ and $\lambda$ of the model, which turn out to be small numbers, significantly less that one.