  We perform an exploratory lattice QCD simulation of $D \bar D$ scattering,
aimed at determining the masses as well as the decay widths of charmonium
resonances above open charm threshold. Neglecting coupling to other channels,
the resulting phase shift for $D \bar D$ scattering in p-wave yields the
well-known vector resonance $\psi(3770)$. For $m_\pi = 156$ MeV, the extracted
resonance mass and the decay width agree with experiment within large
statistical uncertainty. The scalar charmonium resonances present a puzzle,
since only the ground state $\chi_{c0}(1P)$ is well understood, while there is
no commonly accepted candidate for its first excitation. We simulate $D \bar D$
scattering in s-wave in order to shed light on this puzzle. The resulting phase
shift supports the existence of a yet-unobserved narrow resonance with a mass
slightly below 4 GeV. A scenario with this narrow resonance and a pole at
$\chi_{c0}(1P)$ agrees with the energy-dependence of our phase shift. Further
lattice QCD simulations and experimental efforts are needed to resolve the
puzzle of the excited scalar charmonia.
