Exact Low-Energy Solution for Critical Fermi Surfaces

We derive multidimensional bosonization directly from the electron gas in a low-energy, low momentum regime where $\omega\gg \frac{k^2}{k_F}$, such that the dispersion can be linearized. To reach this limit, the Fermi momentum and the number of patches are scaled simultaneously keeping the width of each patch finite. We apply this to obtain an exact low-energy solution of the problem of a Fermi surface coupled to a gapless boson, free of disorder and electron-electron scattering. Contrary to claims in the literature, we show that the bosonized theory exactly reproduces the $\omega^{2/3}$ of electrons, previously obtained in large-$N$ theories. We argue that correction to the self-energy due to tangential dispersion are subdominant at sufficiently low energies such that $v_F k\gg \left(\frac{g^4 v_F}{k_F}\right)^{1/3} \omega^{2/3}$, where $g$ is the coupling constant.