  We estimate the size of the hadronic matrix elements of CP-violating
three-gluon and four-gluon Weinberg operators using sum-rule techniques. In the
three-gluon case, we are able to reproduce the expressions given in earlier
works, while the four-gluon results obtained in this article are new. Our paper
therefore represents the first systematic study of contributions to the
electric dipole moment of the neutron due to CP-violating dimension-six and
dimension-eight operators. We provide many details on both the derivation of
the sum rules as well as the analysis of the uncertainties that plague our
final predictions.
