One-dimensional scattering of two-dimensional fermions near quantum criticality

Forward and backscattering play an exceptional role in the physics of two-dimensional interacting fermions. In a Fermi liquid, both give rise to a non-analytic $\omega^2 \log(\omega)$ form of the fermionic scattering rate at second order in the interaction. Here we argue that higher powers of $\log(\omega)$ appearin the backscattering contribution at higher orders. We show that these terms come from "planar" processes, which are effectively one-dimensional. This is explicitly demonstated by extending a Fermi liquid to the limit of $N \gg 1$ fermionic flavors, when only planar processes survive. We sum the leading logarithms for the case of a 2D Fermi liquid near a nematic transition and obtain an expression for the scattering rate at $T=0$ to all orders in the interaction. For a repulsive interaction, the resulting rate is logarithmically suppressed, and the result is valid down to $\omega = 0$. For an attractive interaction, the ground state is an $s$-wave superconductor with a gap $\Delta_0$. We show that in this case the scattering rate increases as $\omega$ is reduced towards $\Delta_0$. At $\omega \geq \Delta_0$, the behavior of the scattering rate is rather unconventional as many pairing channels compete near a nematic critical point, and $s$-wave wins only by a narrow margin. We take superconductivity into consideration and obtain the scattering rate also at smaller $\omega \simeq \Delta_0$.