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We consider the differential sum rule for the effective scattering rate $%1/\tau (\omega)$ and optical conductivity $\sigma_{1}(\omega) $ in a dirty BCSsuperconductor, for arbitrary ratio of the superconducting gap $% \Delta$ andthe normal state constant damping rate $1/\tau$. We show that if $\tau$ isindependent of $T$, the area under $1/\tau (\omega)$ does not change betweenthe normal and the superconducting states, i.e., there exists an exactdifferential sum rule for the scattering rate. For \textit{any} value of thedimensionless parameter $\Delta\tau $, the sum rule is exhausted at frequenciescontrolled by $\Delta$. %but the numerical convergence is weak. We show that inthe dirty limit the convergence of the differential sum rule for the scatteringrate is much faster then the convergence of the $f-$sum rule, but slower thenthe convergence of the differential sum rule for conductivity.

Differential sum rule for the relaxation rate in dirty superconductors