Renormalization group for one-dimensional Fermi systems: Three-loop scaling equations

The one-dimensional Fermi system with Tomonaga-type (g₂) and back-scattering (g₁) interactions is studied by a weak-coupling renormalization-group method at the three-loop level. When the transfer cutoff procedure is used, the Gell-Mann-Low function for g₁ depends on g₁ alone, dg₁dξ=g₁²+1/2g₁³+1/8g₁⁴. Further, in the absence of umklapp processes the combination g₁-2g₂ is a scale invariant. These conclusions agree with the results of Luther and Emery derived by the boson representation method. Some novel calculational techniques for the derivation of the scaling equations are described and compared to the conventional scheme.