NON PERTURBATIVE RENORMALIZATION OF FERMION THEORIES

The present work is devoted to the study of the non-perturbative renormalization of quantum field theories (QFT) of self-interacting fermions, which is investigated with the help of a fermion Wilsonian RG method. By starting with the appropriate scale dependent action S_k for fermion models and performing a non-trivial extension of previous techniques, a new renormalization group (RG) equation for a generalized N flavors Gross-Neveu model in 2 ¿ d ¿ 4 dimensions is established. The renormalizability of such a model is related to the existence of a non-Gaussian fixed point of the RG equations, which turns out also to be related to the critical point of the chiral symmetry breaking. Within the Wilson RG approach to critical phenomena, the critical exponents of the Gross-Neveu model are computed by considering the behaviour of the running mass m(k) and Fermi constant G(k) around such a non-Gaussian fixed point (with divergent correlation length). By means our approach, this model can be studied even far from the applicability domain of other traditional analytical tools (as epsilon or 1/N expansion) relying on the bosonization of the model. The RG equations for the running mass m(k) and Fermi constant G(k) are numerically studied in the whole (G, m) plane. From this analysis, in turns out that the physics of the chiral phase transition can be described well in terms of a cross-over phenomenon triggered by the presence of an infinitesimal bare mass. The impact of higher powers operators (¿¿)^4 in the Wilsonian potential is also considered. In the large N limit, the failure of the hyperscaling and the presence of a logarithmic behaviour at d=4 dimensions with the scale of the quartic operator (¿¿)^4 are thus recovered in our RG fermion language. By considering the impact of odd interaction terms in the potential, the marginality of the cubic operator in d = 3 dimensions is also recovered. The anomalous dimension of this operator is computed and found to be in agreement with results given in the literature at the next to leading order in the 1/N expansion. Finally the Wilsonian RG equation is extended to theories which involve both fermions and bosons. For N=1 and d=4 this equation coincides with that found in literature. A non-trivial FP which should describe the chiral transition for d<4 is then found. If, however, a four Fermi interaction term is added to the Yukawa theory, an intriguing and unexpected result is found. Namely, the non-perturbative scaling of the running Fermi constant G(k) triggers the appearence of a non-Gaussian fixed point which heals the triviality of the Yukawa coupling in d=4 dimensions.