  We investigate CP-violating interactions involving the Higgs boson and gluons
within an effective field theory approach, focusing on the specific class of
new-physics scenarios where the Yukawa couplings of light quarks are zero or
strongly suppressed compared to the standard-model expectations. We compute the
contributions of the most relevant higher-dimensional operators of Weinberg
type to the electric dipole moment of the neutron (nEDM), which are induced by
Feynman diagrams that involve an effective CP-violating Higgs-gluon coupling
and top-quark loops. The resulting nEDM sensitivities and prospects are
discussed and compared to the existing and expected LHC bounds. We find that
future nEDM searches can set non-trivial constraints on CP-violating
Higgs-gluon interactions even if the Higgs only couples to the third generation
of quarks.
