IR finite correlation functions in de Sitter space, a smooth massless limit, and an autonomous equation

TL;DR

This paper develops a vacuum-independent method to compute infrared correlation functions of a massive scalar field in de Sitter space, ensuring a smooth massless limit and connecting quantum field theory with stochastic processes.

Contribution

It introduces a novel vacuum-independent approach for correlation functions in de Sitter space that yields a smooth massless limit and constructs an autonomous equation inspired by renormalization group ideas.

Findings

Correlation functions match Ornstein-Uhlenbeck process predictions.

Results agree with Schwinger-Keldysh, stochastic, and Hartree-Fock methods.

Autonomous equation reproduces perturbative series and late-time behavior.

Abstract

We explore two-point and four-point correlation functions of a massive scalar field on the flat de Sitter background in the long-wavelength approximation. By employing the Yang-Feldman-type equation, we compute the two-point correlation function up to the $\lambda^3$ order and the four-point correlation function up to the $\lambda^2$ one. In contrast to the standard theory of a massive scalar field based on the de Sitter-invariant vacuum, we develop the vacuum-independent reasoning that may not possess de Sitter invariance but results in a smooth massless limit of the correlation function's infrared part. Our elaboration affords to calculate correlation functions of a free massive scalar field and to proceed with quantum corrections, relying only on the known two-point correlation function's infrared part of a free massless one. Remarkably, the two-point correlation function of a free massive scalar field coincides with the Ornstein-Uhlenbeck stochastic process's one and has a clear physical interpretation. We compared our results with those obtained with the Schwinger-Keldysh diagrammatic technique, Starobinsky's stochastic approach, and the Hartree-Fock approximation. At last, we have constructed a renormalization group-inspired autonomous equation for the two-point correlation function. Integrating its approximate version, one obtains the non-analytic expression with respect to a self-interaction coupling constant $\lambda$. That solution reproduces the correct perturbative series up to the two-loop level. At the late-time limit, it almost coincides with the result of Starobinsky's stochastic approach in the whole interval of a new dimensionless parameter $0 \leq \tfrac{\pi^2 m^4}{3\lambda H^4} < \infty$.