  The current vacuum energy density observed as dark energy ${ \rho }_{ \rm
dark }\simeq 2.5\times10^{-47}\ {\rm GeV^{4}}$ is unacceptably small compared
with any other scales. Therefore, we encounter serious fine-tuning problem and
theoretical difficulty to derive the dark energy. However, the theoretically
attractive scenario has been proposed and discussed in literature: In terms of
the renormalization-group (RG) running of the cosmological constant, the vacuum
energy density can be expressed as ${ \rho }_{ \rm vacuum }\simeq m^{2}H^{2}$
where $m$ is the mass of the scalar field and rather dynamical in curved
spacetime. However, there has been no rigorous proof to derive this expression
and there are some criticisms about the physical interpretation of the RG
running cosmological constant. In the present paper, we revisit the RG running
effects of the cosmological constant and investigate the renormalized vacuum
energy density in curved spacetime. We demonstrate that the vacuum energy
density described by ${ \rho }_{ \rm vacuum }\simeq m^{2}H^{2}$ appears as
quantum effects of the curved background rather than the running effects of
cosmological constant. Comparing to cosmological observational data, we obtain
an upper bound on the mass of the scalar fields to be smaller than the Planck
mass, $m \lesssim M_{\rm Pl}$.
