Third-order renormalization group applied to the attractive one-dimensional Fermi gas

We consider a Callan-Symanzik and Wilson renormalization-group (RG) approac h to the infrared problem for interacting fermions in one dimension with ba ckscattering. We compute the third-order (two-loop) approximation of the be ta function using both methods, and compare it with the well-known multipli cative Gell-Mann Low approach. We point out a previously unnoticed strong i nstability of the third-order fixed point with respect to an arbitrary dime nsionless parameter, which suggests a RG flow toward a strong-coupling phas e.