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 $\omega \left(q\right)$ 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 $\omega \left(q\right)$ at $T_c$. For any temperature below $T_c$, the plasmon dispersion terminates at a critical wave vector $q_c\left(T\right)$ where plasmons remain undamped, with weakly $T$-dependent $\omega \left[q_c\left(T\right)\right]\approx 2\mathrm{\Delta }\left(0\right)$. Plasmons significantly attenuate only within a narrow 5% window near $T_c$, with the propagating mode recovering at large $q$.

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• Plasmonics
• Plasmons
• Superconductivity
• Superconductors
• Nonlinear sigma model
• Usadel equation