Renormalization Group Study of a Fragile Fermi liquid in $1+\epsilon$ dimensions

We present a calculation of the low energy Greens function in $1+\epsilon$ dimensions using the method of extended poor man's scaling, developed here. We compute the wave function renormalization $Z(\omega)$ and also the decay rate near the Fermi energy. Despite the lack of $\omega^2$ damping characteristic of 3-dimensional Fermi liquids, we show that quasiparticles do exist in $1+\epsilon$ dimensions, in the sense that the quasiparticle weight $Z$ is finite and that the damping rate is smaller than the energy. We explicitly compute the crossover from this behavior to a 1-dimensional type Tomonaga-Luttinger liquid behavior at higher energies.