  We study two-magnon excitations in resonant inelastic x-ray scattering (RIXS)
at the transition-metal $K$-edge. Instead of working with effective Heisenberg
spin models, we work with a Hubbard-type model ($d$-$p$ model) for a typical
insulating cuprate La$_2$CuO$_4$. For the antiferromagnetic ground state within
the spin-density-wave (SDW) mean-field formalism, we calculate the dynamical
correlation function within the random-phase approximation (RPA), and then
obtain two-magnon excitation spectra by calculating the convolution of it.
Coupling between the $K$-shell hole and the magnons in the intermediate state
is calculated by means of diagrammatic perturbation expansion in the Coulomb
interaction. Calculated momentum dependence of RIXS spectra agrees well with
that of experiments. A notable difference from previous calculations based on
the Heisenberg spin models is that RIXS spectra have a large two-magnon weight
near the zone center, which may be confirmed by further careful high-resolution
experiments.
