Infrared resummation for derivative interactions in de Sitter space

TL;DR

This work addresses infrared logarithms that arise for scalar fields in de Sitter space, which threaten perturbation theory, by extending the stochastic IR resummation to derivative-interacting models. Through a covariant Yang-Feldman analysis, it derives Langevin equations with white noise for the nonlinear sigma model and a general noncanonical kinetic term model, along with a covariant Fokker-Planck equation, capturing leading IR dynamics on curved target spaces. The results yield equilibrium distributions, $\rho_\infty(\phi)=Z^{-1}\sqrt{G(\phi)}$ for the sigma model and $\rho_\infty(\phi)=Z^{-1}\sqrt{G(\phi)}\exp\left\{-\frac{2\pi^{(D+1)/2}}{\Gamma((D+1)/2)H^D}V(\phi)\right\}$ for the general case, and connect these nonequilibrium IR effects to the Euclidean zero-mode dynamics on a $D$-sphere. This framework provides a nonperturbative handle on IR effects in curved spacetime and lays groundwork for incorporating gravitational IR dynamics in inflationary contexts.

Abstract

In de Sitter space, scale invariant fluctuations give rise to infrared logarithmic corrections to physical quantities, which eventually spoil perturbation theories. For models without derivative interactions, it has been known that the field equation reduces to a Langevin equation with white noise in the leading logarithm approximation. The stochastic equation allows us to evaluate the infrared effects nonperturbatively. We extend the resummation formula so that it is applicable to models with derivative interactions. We first consider the nonlinear sigma model and next consider a more general model which consists of a noncanonical kinetic term and a potential term. The stochastic equations derived from the infrared resummation in these models can be understood as generalizations of the standard one to curved target spaces.