The Two Component Optical Conductivity in the Cuprates: A Necessary Consequence of Preformed Pairs

We address how the finite frequency real conductivity $\sigma(\omega)$ in the underdoped cuprates is affected by the pseudogap, contrasting the behavior above and below $T_c$. The f-sum rule is analytically shown to hold. Here we presume the pseudogap is associated with non-condensed pairs arising from stronger-than-BCS attraction. This leads to both a Drude and a mid infrared (MIR) peak, the latter associated with the energy needed to break pairs. These general characteristics appear consistent with experiment. Importantly, there is no more theoretical flexibility (phenomenology) here than in BCS theory; the origin of the two component conductivity we find is robust.