  QCD Laplace sum rules are used to calculate heavy quarkonium (charmonium and
bottomonium) hybrid masses in several distinct $J^{PC}$ channels. Previous
studies of heavy quarkonium hybrids did not include the effects of
dimension-six condensates, leading to unstable sum rules and unreliable mass
predictions in some channels. We have updated these sum rules to include
dimension-six condensates, providing new mass predictions for the spectra of
heavy quarkonium hybrids. We confirm the finding of other approaches that the
negative-parity $J^{PC}=(0,1,2)^{-+},\,1^{--}$ states form the lightest hybrid
supermultiplet and the positive-parity $J^{PC}=(0,1)^{+-},\,(0,1,2)^{++}$
states are members of a heavier supermultiplet. Our results disfavor a pure
charmonium hybrid interpretation of the $X(3872)$, in agreement with previous
work.
