Why quasilocalized modes in glasses scale as $\omega$ to the fourth

Simulations of quenched glasses show non-phononic quasilocalized modes with density $D_{\mathrm{loc}}(\omega) = A_g\,\omega^4$ at low $\omega$, with the same exponent in $d = 3\ \text{and}\ 4$ and, with stronger finite-size effects, in $d = 2$, and with the prefactor $A_g$ decreasing for better-annealed glasses. Derive the $\omega^4$ law and the dependence of $A_g$ on the preparation (parent) temperature from a controlled microscopic theory in finite d.

A derivation of $D_{\mathrm{loc}}(\omega) \sim \omega^{4}$ and of $A_{g}(T_{\mathrm{parent}})$ in finite d that matches simulation data quantitatively.