  We consider a theory of $N$ self-interacting quantum scalar fields with
quartic $O(N)$-symmetric potential, with a coupling constant $\lambda$, in a
generic curved spacetime. We analyze the renormalization process of the
Semiclassical Einstein Equations at leading order in the $1/N$ expansion for
different renormailzation schemes, namely: the traditional one that sets the
geometry of the spacetime to be Minkowski at the renormalization point, and new
schemes (originally proposed in [1,2]) which set the geometry to be that of a
fixed de Sitter spacetime. In particular, we study the quantum backreaction for
fields in de Sitter spacetimes with masses much smaller than the expansion rate
$H$. We find that the scheme that uses the classical de Sitter background
solution at the renormalization point, stands out as the most appropriate to
study the quantum effects on de Sitter spacetimes. Adopting such scheme we
obtain the backreaction is suppressed by $H^2/M_{pl}^2$ with no logarithmic
enhancement factor of $\ln{\lambda}$, giving only a small screening of the
classical cosmological constant due to the backreaction of such quantum fields.
We point out the use of the new schemes can also be more appropriate than the
traditional one to study quantum effects in other spacetimes relevant for
cosmology.
