On Third Order Renormalization Group applied to the attractive one–dimensional Fermi Gas

We consider a Callan–Symanzik and a Wilson Renormalization Group approach to the infrared problem for interacting fermions in one dimension with backscattering. We compute the third order (two–loop) approximation of the beta function using both methods and compare it with the well known multiplicative Gell–Mann Low approach. We point out an unnoticed qualitative dependence of the third order fixed point on an arbitrary dimensionless parameter, which strongly suggest the spurious nature of the fixed point. Typeset using REVTEX