  We develop a theory of resonant inelastic x-ray scattering (RIXS) at the $K$
edge in La$_2$CuO$_4$ on the basis of the Keldysh Green's function formalism.
In our previous analysis (Phys. Rev. B 71, 035110 (2005)), the scattering by
the core-hole potential was treated within the Born approximation, and a
crude-model density of states was used for the $4p$ band. We improve the
analysis by taking account of the multiple scattering in Cu3d-O$2p$ bands and
by using a realistic $4p$ DOS obtained from a band calculation. The multiple
scattering effect is evaluated with the use of the time representation
developed by Nozi\`eres and De Dominicis. It is found that the multiple
scattering effect makes the $K$-edge peak in the absorption coefficient shift
to the lower energy region as a function of photon energy, that is, the photon
energy required to excite the $1s$ electron to the $K$-edge peak reduces. It is
also found that the multiple-scattering effect does not change the two-peak
structure in the RIXS spectra but modifies slightly the shape as a function of
energy loss. These findings suggests that the multiple scattering effect could
mainly be included into a renormalization of the core-level energy and partly
justify the Born approximation, leading to a future application to the RIXS in
three-dimensional systems.
