We present high-precision results from lattice QCD for the mass splittings of the low-lying charmonium states. For the valence charm quark, the calculation uses Wilson-clover quarks in the Fermilab interpretation. The gauge-field ensembles are generated in the presence of up, down, and strange sea quarks, based on the improved staggered (asqtad) action, and gluon fields, based on the one-loop, tadpole-improved gauge action. We use five lattice spacings and two values of the light sea quark mass to extrapolate the results to the physical point. An enlarged set of interpolating operators is used for a variational analysis to improve the determination of the energies of the ground states in each channel. We present and implement a continuum extrapolation within the Fermilab interpretation, based on power-counting arguments, and thoroughly discuss all sources of systematic uncertainty. We compare our results for various mass splittings with their experimental values, namely, the 1S hyperfine splitting, the 1P-1S splitting and the P-wave spin-orbit and tensor splittings. Given the uncertainty related to the width of the resonances, we find excellent agreement.
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