The differential sum rule for the relaxation rate in the cuprates

Motivated by recent experiments by Basov {\it et al}, we study the differential sum rule for the effective scattering rate $1/\tau (\omega)$. We show that in a dirty BCS superconductor, the area under $1/\tau (\omega)$ does not change between the normal and the superconducting states. For magnetically mediated pairing, similar result holds between $T<T_c$ and $T \geq T_c$, while in the pseudogap phase, $1/\tau (\omega)$ is just suppressed compared to $1/\tau(\omega)$ in the normal state. We argue that this violation of the differential sum rule is due to the fact that low-energy fermions in the pseudogap phase are still incoherent.