Conservation of ζ with radiative corrections from heavy field

In this paper, we address a possible impact of radiative corrections from a heavy scalar field χ on the curvature perturbation ζ. Integrating out χ, we derive the effective action for ζ, which includes the loop corrections of the heavy field χ. When the mass of χ is much larger than the Hubble scale H, the loop corrections of χ only yield a local contribution to the effective action and hence the effective action simply gives an action for ζ in a single field model, where, as is widely known, ζ is conserved in time after the Hubble crossing time. Meanwhile, when the mass of χ is comparable to H, the loop corrections of χ can give a non-local contribution to the effective action. Because of the non-local contribution from χ, in general, ζ may not be conserved, even if the classical background trajectory is determined only by the evolution of the inflaton. In this paper, we derive the condition that ζ is conserved in time in the presence of the radiative corrections from χ. Namely, we show that when the dilatation invariance, which is a part of the diffeomorphism invariance, is preserved at the quantum level, the loop corrections of the massive field χ do not disturb the constant evolution of ζ at super Hubble scales. In this discussion, we show the Ward-Takahashi identity for the dilatation invariance, which yields a consistency relation for the correlation functions of the massive field χ.