The differential sum rule for the relaxation rate in dirty superconductors

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