Two-cutoff scaling for the one-dimensional electron gas

We discuss a variety of renormalization-group methods for treating the one-dimensional interacting electron gas when the cutoff for the energy transfer is a phonon frequency ${\omega }_D$, while the electron bandwidth is of order $E_F$, with ${\omega }_D<E_F$. Our results, consistent only in the region of weak coupling, give the scaling trajectories for the coupling constants and cutoffs. The relationship to the conventional one-cutoff renormalization-group approach is discussed.