Does a massless Goldstone boson exist?

Abstract: Classical and quantum complex nonlinear scalar fields are considered. A new approach to the quantization of nonlinear fields and the construction of a perturbation theory with allowance for spontaneous symmetry breaking is proposed, based on the use of the relativistic model of a self-consistent field as the main approximation. The concept of a particle is analyzed within the frame-work of the theory of nonlinear quantum fields. When constructing single-particle states, the contribution of vacuum fluctuations is systematically taken into account. Within the framework of the developed approach, the problem of the existence of massless scalar particles is discussed. It is shown that successive consideration of the vacuum averages of not only one field operator, but also the products of two field operators, leads to the appearance of masses for scalar particles. Various states in which the field can exist for given parameters entering into the Lagrangian are considered, and the vacuum energy densities in these states are calculated. It is shown that, depending on the values of the parameters entering into the Lagrangian, the vacuum energy density can be either positive or negative, which is important for modern cosmology.