Getting Around the Nielsen-Ninomiya Theorem, towards the Rome Approach

The no-go'' theorem of Nielsen and Ninomiya has been the most tenacious obstacle against the construction of a chiral gauge theory with reasonable low energy spectrum, couplings and anomaly. In this paper we construct a model which supplements the usual (bilinear in the Fermi fields) lagrangian with quadrilinear fermionic terms. We show that in a certain region of the parameter space the difficulties of the no-go’’ theorem may be overcome, and a ``renormalized’’ perturbative strategy can be carried out, akin to the one followed in the Rome Approach (RA), whose counterterms are forced to be gauge invariant.