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

TL;DR

This paper investigates how radiative corrections from a heavy scalar field affect the conservation of the curvature perturbation  in inflation, showing that symmetry preservation ensures  remains constant at super Hubble scales.

Contribution

It derives conditions under which  remains conserved despite loop corrections from a heavy field, emphasizing the role of scaling symmetry at the quantum level.

Findings

When the heavy field mass is much larger than H, corrections are local and  is conserved.

For comparable mass to H, non-local corrections can break  conservation.

Preservation of scaling symmetry ensures  remains constant at super Hubble scales.

Abstract

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 $\chi$.