Exact RG Invariance and Symmetry Improved 2PI Effective Potential

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.