Scalar field effective potentials in de Sitter spacetime

TL;DR

This work analyzes two definitions of the scalar-field effective potential in de Sitter spacetime—the standard effective potential and the constraint effective potential—and computes them at one-loop order for real and O(N)-multicomponent scalars. It shows that the standard potential suffers infrared divergences in the light-field regime ($M^2\ll H^2$), causing perturbative breakdown, while the constraint potential remains perturbatively well-defined and finite in the same regime. The authors derive explicit power-series expansions in heavy, near-conformal, and light mass limits, highlighting the different infrared behavior and the role of the homogeneous ($n=0$) mode in the standard case. They provide evidence that the constrained potential is the appropriate object for the stochastic Starobinsky-Yokoyama description of inflationary dynamics, linking quantum field theory to stochastic cosmology and suggesting practical avenues for Monte Carlo implementation and SM Higgs applications, with the caveat that $\bar{\phi}$ corresponds to a field averaged on the Euclidean four-sphere. $p_{stoch}(\phi)\propto \exp(-\tfrac{8\pi^2}{3H^4}\mathcal{V}_{\mathcal{C}}(\phi))$ and the long-distance correlator agree with QFT at one loop when the constraint potential is used, supporting the conjecture that $\mathcal{V}_{\mathcal{C}}$ governs stochastic inflation.

Abstract

We investigate two different definitions of a scalar field effective potential in quantum field theory in de Sitter spacetime: the standard textbook definition, and the constraint effective potential proposed by O'Raifeartaigh et al. in 1986. While these definitions are equivalent in Minkowski spacetime, they differ significantly in de Sitter. We demonstrate this by computing them both explicitly at one-loop order in perturbation theory. It is well known that the perturbative expansion of the standard effective potential fails converge for light fields. In contrast, the constraint effective potential does not suffer from this infrared problem, and it can therefore be computed using perturbation theory. We discuss the physical interpretation of the two effective potentials. In particular, we provide evidence supporting an earlier conjecture that the constraint effective potential is the correct one to use in the stochastic Starobinsky-Yokoyama theory.