Conservation of superhorizon curvature perturbations at one loop: Backreaction in the in-in formalism and Renormalization

We show that the superhorizon-limit curvature perturbations are conserved at one-loop level in single-field inflation models with a transient non-slow-roll period. We take the spatially-flat gauge, where the backreaction plays a crucial role for the conservation of superhorizon curvature perturbations unless the counter terms are tuned. We calculate the backreaction with the in-in formalism. In addition, we explicitly show the renormalization of the UV divergences with the counter terms.