A try for dark energy in quantum field theory: The vacuum energy of neutrino field

The quartic-divergent vacuum energy poses an ultraviolet (UV) challenge (the cosmological constant problem) in probing the nature of dark energy. Here we try to evaluate the contribution of the vacuum energy to dark energy with a method of the UV-free scheme. The result indicates that it is not a problem in the UV region but a question of the contributions of heavy fields being suppressed. Then, we explore an effective description via scale decoupling. The parameter spaces suggest that the vacuum energy of active neutrino fields can naturally meet the observation of dark energy density, and a neutrino with a typical mass $\sim$ 10 meV $({10}^{-3}$ eV) is expected. The normal ordering neutrinos are preferred by naturalness, and the neutrino mass window set by dark energy is 6.3 meV $\lesssim m_1\lesssim$ 16.3 meV, 10.7 meV $\lesssim m_2\lesssim$ 18.4 meV, 50.5 meV $\lesssim m_3\lesssim$ 52.7 meV.