Running vacuum in quantum field theory in curved spacetime: renormalizing $\rho_{vac}$ without $\sim m^4$ terms

The $\Lambda$-term in Einstein's equations is a fundamental building block of the `concordance' $\Lambda$CDM model of cosmology. Even though the model is not free of fundamental problems, they have not been circumvented by any alternative dark energy proposal either. Here we stick to the $\Lambda$-term, but we contend that it can be a `running quantity' in quantum field theory (QFT) in curved spacetime. A plethora of phenomenological works have shown that this option can be highly competitive with the $\Lambda$CDM with a rigid cosmological term. The, so-called, `running vacuum models' (RVM's) are characterized by the vacuum energy density, $\rho_{vac}$, being a series of (even) powers of the Hubble parameter and its time derivatives. Such theoretical form has been motivated by general renormalization group arguments, which look plausible. Here we dwell further upon the origin of the RVM structure within QFT in FLRW spacetime. We compute the renormalized energy-momentum tensor with the help of the adiabatic regularization procedure and find that it leads essentially to the RVM form. This means that $\rho_{vac}(H)$ evolves as a constant term plus dynamical components ${\cal O}(H^2)$ and ${\cal O}(H^4)$, the latter being relevant for the early universe only. However, the renormalized $\rho_{vac}(H)$ does not carry dangerous terms proportional to the quartic power of the masses ($\sim m^4$) of the fields, these terms being a well-known source of exceedingly large contributions. At present, $\rho_{vac}(H)$ is dominated by the additive constant term accompanied by a mild dynamical component $\sim \nu H^2$ ($|\nu|\ll1$), which mimics quintessence.