Plasmon excitations and their attenuation in dirty superconductors

We develop a theory for the plasmon spectrum in dirty superconductors across the entire temperature range. Starting with the microscopic Keldysh sigma model description, we link the plasmon dispersion $\omega (q)$ to the optical conductivity $\sigma (\omega ,T)$ of a superconductor, which requires analytical continuation to the lower half-plane of complex frequency. This approach reveals a discontinuity at the superconducting transition: a jump in both the real and imaginary parts of $\omega (q)$ at $T_c$. For any temperature below $T_c$, the plasmon dispersion terminates at a critical wave vector $q_c(T)$ where plasmons remains undamped, with weakly $T$-dependent $\omega [q_c(T)]\approx 2\mathrm{\Delta }(0)$. Plasmons significantly attenuate only within a narrow 5% temperature window near $T_c$, with the propagating mode recovering at large $q$.