Conductivity sum rule, implication for in-plane dynamics and c-axis response

Recently observed $c$-axis optical sum rule violations indicate non-Fermi liquid in-plane behavior. For coherent $c$-axis coupling, the observed flat, nearly frequency independent $c$-axis conductivity ${\sigma }_1(\omega )$ implies a large in-plane scattering rate $\mathrm{\Gamma }$ around $(0,\pi )$ and therefore any pseudogap that might form at low frequency in the normal state will be smeared. On the other hand incoherent $c$-axis coupling places no restriction on the value of $\mathrm{\Gamma }$ and gives a more consistent picture of the observed sum rule violation which, we find in some cases, can be less than half.