A conformally coupled massive scalar field in the de Sitter expanding universe with the mass term treated as a perturbation

TL;DR

This paper compares different perturbative formalisms for a conformally coupled massive scalar field in de Sitter space, highlighting issues with infrared divergences in in-out theory and demonstrating the effectiveness of Euclidean and in-in approaches.

Contribution

It clarifies the relation between vacua, identifies the source of infrared divergences, and shows that Euclidean and in-in formalisms avoid these issues in perturbation theory.

Findings

In-out perturbation theory breaks down due to infrared divergences.

Euclidean and in-in formalisms do not have infrared divergences.

Two-point functions are correctly reproduced in Euclidean and in-in approaches.

Abstract

We study conformally-coupled massive scalar field theory with the mass term treated as a perturbation in the expanding half of de Sitter spacetime as a toy model for comparing various perturbative formalisms. We point out that the in-out perturbation theory breaks down due to infrared divergences coming from the infinite future. We then clarify the relation between the free-theory vacuum and the true vacuum using the Bogolubov transformation and show that the discrepancy between the free-theory out-vacuum and the true vacuum causes apparent pair creation of free-theory particles and makes the in-out two-point function differ from the true two-point function. We also identify the infinite Bogolubov coefficients as the cause of infrared divergences. We then examine two alternative approaches: the Euclidean and in-in formalisms. We verify that there are no infrared divergences in perturbation theory with the mass term treated as a perturbation in either of these approaches and that the two-point function of massive scalar field is reproduced correctly by these methods.