  We examine the effect of core-hole lifetime on the spectral weight of
resonant inelastic x-ray scattering (RIXS) in hole-doped cuprates. We calculate
the spectral weight by using exact diagonalization technique for a 4x4 doped
Hubbard lattice and find that spin-flip channel detecting single-magnon
excitation is less sensitive to the core-hole lifetime while in non-spin-flip
channel the spectral weight is strongly dependent on the lifetime. In the
latter, charge and two-magnon excitations predominately contribute to RIXS for
short and long core-hole lifetimes, respectively. For a realistic value of the
core-hole lifetime in cuprates, both the charge and two-magnon excitations are
expected to contribute to non-spin-flip channel in RIXS when the
incident-photon energy is tuned to the main peak of x-ray absorption spectrum.
