  The properties of charmonium states are or will be intensively studied by the
B-factories Belle II and BESIII, the LHCb and PANDA experiments and at a future
Super-c-tau Factory. Precise lattice calculations provide valuable input and
several results have been obtained by simulating up, down and strange quarks in
the sea. We investigate the impact of a charm quark in the sea on the
charmonium spectrum, the renormalization group invariant charm-quark mass and
the scalar charm-quark content of charmonium. The latter is obtained by the
direct computation of the mass-derivatives of the charmonium masses. We do this
investigation in a model, QCD with two degenerate charm quarks. The absence of
light quarks allows us to reach very small lattice spacings down to 0.023 fm.
By comparing to pure gauge theory we find that charm quarks in the sea affect
the hyperfine splitting at a level below 2%. The most significant effects are
5% in the quark mass and 3% in the value of the charm quark content of the
eta_c meson. Given that we simulate two charm quarks these estimates are upper
bounds for the contribution of a single charm quark. We show that lattice
spacings <0.06 fm are needed for safe continuum extrapolations of the
charmonium spectrum with O(a) improved Wilson quarks. A useful relation for the
projection to the desired parity of operators in two-point functions computed
with twisted mass fermions is proven.
