  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 a previously unnoticed qualitative
dependence of the third order fixed point on an arbitrary dimensionless
parameter, which strongly suggest the spurious nature of the fixed point.
