Cosmological Constant Problem and Renormalized Vacuum Energy Density in Curved Background

Abstract: 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, $m \lesssim M{\rm Pl}$.