On “dynamical mass” generation in Euclidean de Sitter space

Abstract

We consider the perturbative treatment of the minimally coupled, massless, self-interacting scalar field in Euclidean de Sitter space. Generalizing work of Rajaraman, we obtain the dynamical mass of the scalar for nonvanishing Lagrangian masses and the first perturbative quantum correction in the massless case. We develop the rules of a systematic perturbative expansion, which treats the zero-mode nonperturbatively and goes in powers of $\sqrt{\ensuremath{\lambda}}$. The infrared divergences are self-regulated by the zero-mode dynamics. Thus, in Euclidean de Sitter space the interacting, massless scalar field is just as well defined as the massive field. We then show that its dynamical mass ${m}^{2}\ensuremath{\propto}\sqrt{\ensuremath{\lambda}}{H}^{2}$ can be recovered from the diagrammatic expansion of the self-energy and a consistent solution of the Schwinger-Dyson equation, but it requires the summation of a divergent series of loop diagrams of arbitrarily high order. Finally, we note that the value of the long-wavelength mode two-point function in Euclidean de Sitter space agrees at leading order with the stochastic treatment in Lorentzian de Sitter space, in any number of dimensions.