Tails of the dynamical structure factor of 1D spinless fermions beyond the Tomonaga approximation

We consider one-dimensional (1D) interacting spinless fermions with a non-linear spectrum in a clean quantum wire (non-linear bosonization). We compute diagrammatically the 1D dynamical structure factor, $S(\om,q)$, beyond the Tomonaga approximation focusing on it's tails, $|\om| \gg vq$, {\it i.e.} the 2-pair excitation continuum due to forward scattering. Our methodology reveals three classes of diagrams: two "chiral" classes which bring divergent contributions in the limits $\om \to \pm vq$, {\it i.e.} near the single-pair excitation continuum, and a "mixed" class (so-called Aslamasov-Larkin or Altshuler-Shklovskii type diagrams) which is crucial for the f-sum rule to be satisfied. We relate our approach to the T=0 ones present in the literature. We also consider the $T\not=0$ case and show that the 2-pair excitation continuum dominates the single-pair one in the range: $|q|T/k_F \ll \om \mp vq \ll T$ (substantial for $q \ll k_F$). As applications we first derive the small-momentum optical conductivity due to forward scattering: $\sigma \sim 1/\om$ for $T \ll \om$ and $\sigma \sim T/\om^2$ for $T \gg \om$. Next, within the $2-$pair excitation continuum, we show that the attenuation rate of a coherent mode of dispersion $\Omega_q$ crosses over from $\gamma_q \propto \Omega_q (q/k_F)^2$, {\it e.g.} $\gamma_q \sim |q|^3$ for an acoustic mode, to $\gamma_q \propto T (q/k_F)^2$, independent of $\Omega_q$, as temperature increases. Finally, we show that the $2-$pair excitation continuum yields subleading curvature corrections to the electron-electron scattering rate: $\tau^{-1} \propto V^2 T + V^4 T^3/\eps_F^2$, where $V$ is the dimensionless strength of the interaction.