  The microwave surface impedance measurements on cuprate superconductors
provide the crucial information of the effect of the impurity scattering on the
quasiparticle transport, however, the full understanding of the effect of the
impurity scattering on the quasiparticle transport is still a challenging
issue. Here based on the microscopic octet scattering model, the effect of the
impurity scattering on the low-temperature microwave conductivity in cuprate
superconductors is investigated in the self-consistent $T$-matrix approach. The
impurity-dressed electron propagator obtained in the Fermi-arc-tip
approximation of the quasiparticle excitations and scattering processes is
employed to derive the electron current-current correlation function by taking
into account the impurity-induced vertex correction. It is shown that the
microwave conductivity spectrum is a non-Drude-like, with a sharp cusp-like
peak extending to zero-energy and a high-energy tail falling slowly with
energy. Moreover, the microwave conductivity decreases with the increase of the
impurity concentration or with the increase of the strength of the impurity
scattering potential. In a striking contrast to the dome-like shape of the
doping dependence of the superconducting transition temperature, the microwave
conductivity exhibits a reverse dome-like shape of the doping dependence. The
theory also show that the highly unconventional features of the microwave
conductivity are generated by both the strong electron correlation and
impurity-scattering effects.
