  We determine $J^{PC}=0^{++}$ and $2^{++}$ hadron-hadron scattering amplitudes
in the charmonium energy region up to 4100 MeV using lattice QCD, a
first-principles approach to QCD. Working at $m_\pi\approx 391$ MeV, more than
200 finite-volume energy levels are computed and these are used in extensions
of the L\"uscher formalism to determine infinite-volume coupled-channel
scattering amplitudes. We find that this energy region contains a single
$\chi_{c0}$ and a single $\chi_{c2}$ resonance. Both are found as pole
singularities on the closest unphysical Riemann sheet, just below 4000 MeV with
widths around 70 MeV. The largest couplings are to kinematically-closed $D^*
\bar{D}^*$ channels in $S$-wave, and couplings to several decay channels
consisting of pairs of open-charm mesons are found to be large and significant
in both cases. Above the ground state $\chi_{c0}$, no other scalar bound-states
or near-$D\bar{D}$ threshold resonances are found, in contrast to several
theoretical and experimental studies.
