Non-perturbative techniques in strongly coupled field theories

Non-perturbative techniques are needed to study strongly coupled systems. One powerful approach is the n-particle irreducible effective action. The method provides a systematic expansion for which the truncation occurs at the level of the action. The standard renormalization procedure can be modified at the 2PI level by introducing multiple sets of counterterms, but counterterms cannot be used at higher orders. We have developed a method based on a renormalization group approach which allows us to construct a renormalized theory at any order in the nPI approximation. We present results from a calculation using a scalar theory with quartic coupling in 4 dimensions, at the 4 loop level. Both the 2PI and 4PI theories can be renormalized using one bare coupling constant and one bare mass, which are introduced at the level of the Lagrangian.