  Despite significant progress in resonant inelastic x-ray scattering (RIXS)
experiments on cuprates at the Cu L-edge, a theoretical understanding of the
cross-section remains incomplete in terms of elementary excitations and the
connection to both charge and spin structure factors. Here we use
state-of-the-art, unbiased numerical calculations to study the low energy
excitations probed by RIXS in undoped and doped Hubbard model relevant to the
cuprates. The results highlight the importance of scattering geometry, in
particular both the incident and scattered x-ray photon polarization, and
demonstrate that on a qualitative level the RIXS spectral shape in the
cross-polarized channel approximates that of the spin dynamical structure
factor. However, in the parallel-polarized channel the complexity of the RIXS
process beyond a simple two-particle response complicates the analysis, and
demonstrates that approximations and expansions which attempt to relate RIXS to
less complex correlation functions can not reproduce the full diversity of RIXS
spectral features.
