Differential sum rule for the relaxation rate in the cuprates

Motivated by recent experiments by Basov 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, a similar result holds between Tor=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 in the pseudogap phase is due to the absence of the feedback effects from the pairing.