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The current vacuum energy density observed as dark energy ${ \rho }_{ \rmdark }\simeq 2.5\times10^{-47}\ {\rm GeV^{4}}$ is unacceptably small comparedwith any other scales. Therefore, we encounter serious fine-tuning problem andtheoretical difficulty to derive the dark energy. However, the theoreticallyattractive scenario has been proposed and discussed in literature: In terms ofthe renormalization-group (RG) running of the cosmological constant, the vacuumenergy 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 curvedspacetime. However, there has been no rigorous proof to derive this expressionand there are some criticisms about the physical interpretation of the RGrunning cosmological constant. In the present paper, we revisit the RG runningeffects of the cosmological constant and investigate the renormalized vacuumenergy density in curved spacetime. We demonstrate that the vacuum energydensity described by ${ \rho }_{ \rm vacuum }\simeq m^{2}H^{2}$ appears asquantum effects of the curved background rather than the running effects ofcosmological constant. Comparing to cosmological observational data, we obtainan upper bound on the mass of the scalar fields to be smaller than the Planckmass, $m \lesssim M_{\rm Pl}$.

Cosmological Constant Problem and Renormalized Vacuum Energy Density in Curved Background