Conservation of $\zeta$ with radiative corrections from heavy field

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