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

Forward scattering and backscattering play an exceptional role in the physics of two-dimensional interacting fermions. In a Fermi liquid, both give rise to nonanalytic ${\ensuremath{\omega}}^{2}ln(\ensuremath{\omega})$ form fermionic rate at second order interaction. Here we argue that higher powers $ln(\ensuremath{\omega})$ appear contribution orders. We show these terms come from ``planar'' processes, which are effectively one-dimensional. This is explicitly demonstrated by extending liquid limit $N\ensuremath{\gg}1$ flavors, when only planar processes survive. sum leading logarithms for case near nematic transition, obtain expression $T=0$ all orders For repulsive interaction, resulting logarithmically suppressed, result valid down $\ensuremath{\omega}=0$. attractive ground state $s$-wave superconductor with gap ${\mathrm{\ensuremath{\Delta}}}_{0}$. this increases as $\ensuremath{\omega}$ reduced toward At $\ensuremath{\omega}\ensuremath{\ge}{\mathrm{\ensuremath{\Delta}}}_{0}$, behavior rather unconventional many pairing channels compete critical point, wins narrow margin. take superconductivity into consideration also smaller $\ensuremath{\omega}\ensuremath{\simeq}{\mathrm{\ensuremath{\Delta}}}_{0}$.