Figure 1

The inelastic scattering rate as a function of energy due to (a) Coulomb and (b) Hubbard interactions. Red dashed lines correspond to the asymptotic analytical expressions given by Eqs. (8) and (10). At exponentially small energies, the scattering rate logarithmically approaches zero in both cases (not displayed in this figure). (a) The scattering rate exhibits nonmonotonic behavior with a sharp peak at $\varepsilon ={\omega }_{\text{pl}}/2$ owing to thermal plasmons. The coupling constant equals $\alpha N=0.1$. (b) Coupling constant $\lambda$ is such that $\lambda N{T}^2/v_F^3=0.03.$ The inset shows that the scattering rate is well described within the second-order perturbation theory (dashed line) in a wide range of energies around $\varepsilon \approx T$. At higher energies, however, the perturbative result smoothly crosses over to a constant, which can only be obtained after the RPA summation. This crossover occurs only for extremely large values of the ultraviolet cutoff $\mathrm{\Lambda }$ satisfying $\mathrm{\Lambda }\gg \sqrt{v_F/\lambda Nln(v_F\mathrm{\Lambda }/T)}$.