Nucleon matrix element of Weinberg's CP-odd gluon operator from the instanton vacuum

We calculate the nucleon matrix element of Weinberg's dimension-6 CP-odd gluon operator $f^{abc}{\left({\tilde{F}}_{\mu \nu }\right)}^a{\left(F^{\mu \rho }\right)}^b{\left(F_{\rho }^{\nu }\right)}^c$ in the instanton vacuum. In leading order of the instanton packing fraction, the dimension-6 operator is effectively proportional to the topological charge density ${\left({\tilde{F}}_{\mu \nu }\right)}^a{\left(F^{\mu \nu }\right)}^a$, whose nucleon matrix element is given by the flavor-singlet axial charge and constrained by the $U{\left(1\right)}_A$ anomaly. The nucleon matrix element of the dimension-6 operator is obtained substantially larger than in other estimates, because of the strong localization of the nonperturbative gluon fields in the instanton vacuum. We comment on the neutron electric dipole moment induced by CP violation proportional to the dimension-6 operator.