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The cosmological constant term, $\Lambda$, in Einstein's equations has beenfor three decades a building block of the concordance or standard $\Lambda$CDMmodel of cosmology. Although the latter is not free of fundamental problems, itprovides a good phenomenological description of the overall cosmologicalobservations. However, an interesting improvement in such a phenomenologicaldescription and also a change in the theoretical status of the $\Lambda$-termoccurs upon realizing that the vacuum energy density, $\rho_{\textrm{vac}}$, isactually a "running quantity" in quantum field theory in curved spacetime.Several works have shown that this option can compete with the $\Lambda$CDMwith a rigid $\Lambda$term. The so-called, "running vacuum models" (RVM) arecharacterized indeed by a $\rho_{\textrm{vac}}$ which is evolving with time asa series of even powers of the Hubble parameter and its time derivatives. Thisform has been motivated by renormalization group arguments in previous works.Here we review a recent detailed computation by the authors of the renormalizedenergy-momentum tensor of a non-minimally coupled scalar field with the help ofadiabatic regularization. The final result is noteworthy:$\rho_{\textrm{vac}}(H)$ takes the precise structure of the RVM, namely aconstant term plus a dynamical component $\sim H^2$ (which may be detectable inthe present universe) including also higher order effects $\mathcal{O}(H^4)$which can be of interest during the early stages of the cosmological evolution.Besides, it is most remarkable that such renormalized form of$\rho_{\textrm{vac}}$ does not carry dangerous terms proportional to $m^4$, thequartic powers of the masses of the fields, which are a well-known source ofexceedingly large contributions to the vacuum energy density and are directlyresponsible for fine tuning in the context of the CC problem.

Renormalized $\rho_{\rm vac}$ without $m^4$ terms