  The Symmetry Improved Two-Particle-Irreducible (SI2PI) formalism is a
powerful tool to calculate the effective potential beyond perturbation theory,
whereby infinite sets of selective loop-graph topologies can be resummed in a
systematic and consistent manner. In this paper we study the
Renormalization-Group (RG) properties of this formalism, by proving for the
first time a number of new field-theoretic results. First, the RG runnings of
all proper 2PI couplings are found to be UV finite, in the Hartree-Fock and
sunset approximations of the 2PI effective action. Second, the SI2PI effective
potential is exactly RG invariant, in contrast to what happens in the ordinary
One-Particle-Irreducible (1PI) perturbation theory, where the effective
potential is RG invariant only up to higher orders. Finally, we show how the
effective potential of an O(2) theory evaluated in the SI2PI framework,
appropriately RG improved, can reach a higher level of accuracy, even up to one
order of magnitude, with respect to the corresponding one obtained in the 1PI
formalism.
