Asymptotic behavior of quasiparticle damping in a degenerate electron gas

An analytic expression has been derived within the random-phase approximation for the imaginary part of the self-energy ${\Sigma }_2(\overrightarrow{\mathrm{k}}, \omega )$ of a degenerate electron gas, valid in the limit $\frac{\omega }{4E_0}>3+\frac{k}{k_0}+{\left(\frac{k}{2k_0}\right)}^2$, where $E_0$ is the Fermi energy and $k_0$ the Fermi wave number. Our calculation reveals that for such large values of $\omega$, ${\Sigma }_2(\overrightarrow{\mathrm{k}}, \omega )$ drops off as ${\omega }^{-\frac{3}{2}}$. An analytic expression for the self-energy valid for large $\overrightarrow{\mathrm{k}}$ has also been obtained and it is shown that for each value $\omega$ there is a particular $k$ beyond which ${\Sigma }_2(\overrightarrow{\mathrm{k}}, \omega )$ would vanish.