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 $\chi$.
