Nonlocal Corrections to Scalar Field Effective Action in de Sitter spacetime

We investigate the one-loop effective action for a test scalar field in a general Friedmann-Lema\^itre-Robertson-Walker (FLRW) background, specifically focusing on quantum corrections up to the second order in the interaction strength. By employing the Schwinger-Keldysh formalism, we derive the equation of motion for the field expectation value, which incorporates not only the standard local radiative corrections but also novel nonlocal features: a memory term and a stochastic noise term. We identify all ultraviolet divergent structures within these nonlocal terms and provide a consistent renormalization procedure. To analyze the physical impact of these terms, we apply a local approximation under the assumption of slowly-varying fields, by which the memory term acts as a negative contribution to the drift coefficient. As a concrete application, we consider a massive $\phi^4$ theory and show that these one-loop corrections lead to a suppression of the field variance in the infrared regime compared to the tree-level results.