  We report a resonant inelastic x-ray scattering study of the dispersion
relations of charge transfer excitations in insulating La$_2$CuO$_4$. These
data reveal two peaks, both of which show two-dimensional characteristics. The
lowest energy excitation has a gap energy of $\sim 2.2$ eV at the zone center,
and a dispersion of $\sim 1$ eV. The spectral weight of this mode becomes
dramatically smaller around ($\pi$, $\pi$). The second peak shows a smaller
dispersion ($\sim 0.5$ eV) with a zone-center energy of $\sim 3.9$ eV. We argue
that these are both highly dispersive exciton modes damped by the presence of
the electron-hole continuum.
