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 scaling symmetry, 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 scaling symmetry, which yields a consistency relation for the correlation functions of the massive field $χ$.

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