The Two-Particle-Irreducible (2PI) formalism as introduced by Cornwall, Jackiw and Tomboulis provides a systematic analytic approach to consistently describing non-perturbative phenomena in Quantum Field Theory (QFT). In spite of its great success, one major problem of the 2PI approach is that its loopwise expansion gives rise to residual violations of symmetries and hence to massive Goldstone bosons in the spontaneously broken phase of the theory. In my talk I will present a novel Symmetry Improved 2PI formalism (SI2PI) which consistently encodes global symmetries in a loopwise expansion, leading to massless Goldstone bosons within quantum loops and to a proper second order phase transition in O(N) theories. I will also show how the SI2PI effective potential is exactly Renormalization Group (RG) invariant, unlike to what happens in the conventional fixed-order perturbation theory, where the effective potential is RG invariant only up to higher orders. In particular, I will illustrate how the SI2PI effective potential of an O(2) theory can reach a higher level of accuracy, even up to one order of magnitude, with respect to the one obtained in the conventional perturbation theory. Possible new directions towards the development of semi-analytic methods for higher order precision computations in QFT will be briefly discussed.
