Running vacuum in QFT in FLRW spacetime: The dynamics of $\rho_{\rm vac}(H)$ from the quantized matter fields

Phenomenological work in the last few years has provided significant support to the idea that the vacuum energy density (VED) is a running quantity with the cosmological evolution and that this running helps to alleviate the cosmological tensions afflicting the $\Lambda$CDM. On the theoretical side, recent devoted studies have shown that the properly renormalized $\rho_{\rm vac}$ in QFT in FLRW spacetime adopts the"running vacuum model"(RVM) form. While in three previous studies by two of us (CMP and JSP) such computations focused solely on scalar fields non-minimally coupled to gravity, in the present work we compute the spin-$1/2$ fermionic contributions and combine them both. The calculation is performed using a new version of the adiabatic renormalization procedure based on subtracting the UV divergences at an off-shell renormalization point $M$. The quantum scaling of $\rho_{\rm vac}$ with $M$ turns into cosmic evolution with the Hubble rate, $H$. As a result the"cosmological constant"$\Lambda$ appears in our framework as the nearly sustained value of $8\pi G(H)\rho_{\rm vac}(H)$ around (any) given epoch $H$, where $G(H)$ is the gravitational coupling, which is also running, although very mildly (logarithmically). We find that the VED evolution at present reads $\delta \rho_{\rm vac}(H)\sim \nu_{\rm eff}\, m_{\rm Pl}^2 \left(H^2-H_0^2 \right)\ (|\nu_{\rm eff}|\ll 1)$. The coefficient $\nu_{\rm eff}$ receives contributions from all the quantized fields, bosons and fermions. Remarkably, there also exist higher powers ${\cal O}(H^{6})$ which can trigger inflation in the early universe. Finally, the equation of state (EoS) of the vacuum receives also quantum corrections from bosons and fermion fields, shifting its value from -1. The striking consequence is that the EoS of the quantum vacuum may nowadays effectively appears as quintessence.