Effective potentials for de Sitter and anti de Sitter quantum fields

We derive a systematic treatment of one-loop effective potentials for interacting scalar fields in curved spacetimes, providing a general formula valid in arbitrary geometries and explicit results for de Sitter and anti-de Sitter backgrounds. We then compute the effective potential for a scalar $O(N)$ theory on a de Sitter space in any integer dimension. In $d=3$ and dimensional regularization, we extend the calculation up to two loops and compute the $\beta$-function and the anomalous mass dimension. They coincide exactly with flat-space results, despite dramatic curvature modifications to physical masses/couplings. The flat limit $R\to\infty$ recovers Coleman-Weinberg, confirming consistency. Working in $d=3$ dimensions, we repeat the calculation for $AdS_3$ by using point-splitting regularization, obtaining analogous results for the $\beta$-function and anomalous mass dimension.