Linear response theory around a localized impurity in the pseudogap regime of an anisotropic superconductor: Precursor pairing vs the d-density-wave scenario

We derive the polarizability of an electron system in (i) the superconducting phase, with d-wave symmetry, (ii) the pseudogap regime, within the precursor pairing scenario, and (iii) the d-density-wave (dDW) state, characterized by a d-wave hidden order parameter, but no pairing. Such a calculation is motivated by the recent proposals that imaging the effects of an isolated impurity may distinguish between precursor pairing and dDW order in the pseudogap regime of the high-T-c superconductors. In all three cases, the wave-vector dependence of the polarizability is characterized by an azimuthal modulation, consistent with the d-wave symmetry of the underlying state. However, only the dDW result shows the fingerprints of nesting, with nesting wave vector Q=(pi,pi), albeit imperfect, due to a nonzero value of the hopping ratio t(')/t in the band dispersion relation. As a consequence of nesting, the presence of hole pockets is also exhibited by the (q,omega) dependence of the retarded polarizability.