Infrared dynamics in de Sitter space from Schwinger-Dyson equations

TL;DR

This paper addresses infrared divergences in a light $O(N)$ scalar field theory in de Sitter space by solving the Schwinger-Dyson equations for the two-point function using the $p$-representation. By computing the nonlocal self-energy at two-loop order and concentrating on superhorizon momenta, the authors reformulate the IR dynamics as a one-dimensional problem that admits an exact analytical solution, effectively resumming infrared logarithms into a superposition of modified power laws. The main result is that the IR behavior approaches that of a free massive field with renormalized mass and field strength, with nonperturbative renormalization factors in the massless limit; the findings are consistent with stochastic and Euclidean approaches within perturbation theory and provide a detailed structure of correlators across horizon scales. The work offers a framework for nonperturbative infrared resummations in expanding spacetimes and suggests extensions to quasi-de Sitter and global de Sitter settings, as well as to nonperturbative schemes like $1/N$ or 2PI methods.

Abstract

We study the two-point correlator of an O(N) scalar field with quartic self-coupling in de Sitter space. For light fields in units of the expansion rate, perturbation theory is plagued by large logarithmic terms for superhorizon momenta. We show that a proper treatment of the infinite series of self-energy insertions through the Schwinger-Dyson equations resums these infrared logarithms into well defined power laws. We provide an exact analytical solution of the Schwinger-Dyson equations for infrared momenta when the self-energy is computed at two-loop order. The obtained correlator exhibits a rich structure with a superposition of free-field-like power laws. We extract mass and field-strength renormalization factors from the asymptotic infrared behavior. The latter are nonperturbative in the coupling in the case of a vanishing tree-level mass.

Figures (3)

• Figure 1: The nonlocal self-energy $\bar{\Sigma}(x,x')$ at two-loop order. The lines denote the propagator $G_M$, see Eq. (\ref{['eq:massive']}).
• Figure 2: The resummed statistical function (\ref{['eq:Fmassive']}) for equal momenta $p'=p\le\mu$, normalized by the free-field expression (\ref{['eq:f_loc_ir']}), on a logarithmic scale. It interpolates between the free-field behavior for $p\lesssim \mu$ and the asymptotic behavior (\ref{['eq:f_loc_ir_eff']}) (dotted line) for deep infrared momenta $p\ll\mu$. The employed parameters are $d=3$ and $\varepsilon=\sigma_\rho=0.1$.
• Figure 3: The self-consistent local self-energy $\sigma$. The line denotes the propagator $G_M$, see Eq. (\ref{['eq:massive']}).