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Phenomenological work in the last few years has provided significant supportto the idea that the vacuum energy density (VED) is a running quantity with thecosmological evolution and that this running helps to alleviate thecosmological tensions afflicting the $\Lambda$CDM. On the theoretical side,recent devoted studies have shown that the properly renormalized $\rho_{\rmvac}$ 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 computationsfocused solely on scalar fields non-minimally coupled to gravity, in thepresent work we compute the spin-$1/2$ fermionic contributions and combine themboth. The calculation is performed using a new version of the adiabaticrenormalization procedure based on subtracting the UV divergences at anoff-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 nearlysustained 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, althoughvery 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 $\nu_{\rm eff}$ receivescontributions from all the quantized fields, bosons and fermions. Remarkably,there also exist higher powers ${\cal O}(H^{6})$ which can trigger inflation inthe early universe. Finally, the equation of state (EoS) of the vacuum receivesalso quantum corrections from bosons and fermion fields, shifting its valuefrom -1. The striking consequence is that the EoS of the quantum vacuum maynowadays effectively appears as quintessence.

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