On the proper treatment of massless fields in Euclidean de Sitter space
Arvind Rajaraman
TL;DR
The paper shows that infrared divergences in massless scalar field theory on Euclidean de Sitter space arise from an improper treatment of the zero-mode; by quantizing the zero-mode exactly while treating nonzero modes perturbatively, the massless limit becomes finite and perturbation theory is well defined, with loop corrections scaling as $(\sqrt{\lambda})^k$ for $m^2 \ll \lambda H^2$. It provides explicit expressions for the two-point function, including a finite zero-mode contribution controlled by $\lambda_{\rm eff}$ and constants $c_2$ and $c_4$, demonstrating a mass gap generated by quartic interactions. The results establish a controlled framework for perturbative QFT in de Sitter backgrounds and discuss implications for the in-in formalism and inflationary contexts, highlighting where nonperturbative methods may be needed. Overall, the work clarifies how proper zero-mode treatment yields finite, computable predictions for massless fields in curved spacetime and paves the way for applying these techniques to cosmological perturbation theory.
Abstract
We analyze infrared divergences arising in calculations involving light and massless fields in de Sitter space. We show that these arise from an incorrect treatment of the constant mode of the field, and show that a correct quantization leads to a well-defined and calculable perturbation expansion. We illustrate this by computing the first nontrivial loop correction in a theory of a massless scalar field with a quartic interaction.