  We demonstrate two properties of the trace of the energy-momentum tensor
$T^{\mu}_{~ \mu}$ in the flat spacetime. One is the decoupling of heavy degrees
of freedom; i.e., heavy degrees of freedom leave no effect for low-energy
$T^{\mu}_{~ \mu}$-inserted amplitudes. This is intuitively apparent from the
effective field theory point of view, but one has to take into account the
so-called trace anomaly to explicitly demonstrate the decoupling. As a result,
for example, in the $R^{2}$ inflation model, scalaron decay is insensitive to
heavy degrees of freedom when a matter sector ${\it minimally}$ couples to
gravity (up to a non-minimal coupling of a matter scalar field other than the
scalaron). The other property is a quantum contribution to a non-minimal
coupling of a scalar field. The non-minimal coupling disappears from the action
in the flat spacetime, but leaves the so-called improvement term in $T^{\mu}_{~
\mu}$. We study the renormalization group equation of the non-minimal coupling
to discuss its quantum-induced value and implications for inflation dynamics.
We work it out in the two-scalar theory and Yukawa theory.
