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.
