Nucleon electric dipole moment and gluonic content of light hadrons

We discuss the contributions to the nucleon electric dipole moment $d_N$ associated with the flavor-diagonal purely gluonic and mixed quark-gluon $P$- and $\mathrm{CP}$-violating $J^{\mathrm{PC}}=0^{-+}$ operators. These include the dimension-5 and -6 operators, corresponding to the color-electric dipole moment of quarks and gluons, and the set of dimension-8 purely gluonic operators. The large-$N_c$ current-algebra ${\eta }_0$-meson dominance) approach relates the hadronic matrix elements of these higher-dimension operators to the amplitudes involving the dimension-4 topological field operator $Q\left(x\right)\mathrm{}\sim G\tilde{G}$. The basic parameters which appear in these relations correspond to the ${\eta }_0$-meson decay coupling constants induced by the higher-dimension operators. These parameters determine the long-distance behavior of the mixed vacuum correlation function between $Q\left(x\right)\mathrm{}$ and the higher-dimension operators. We evaluate these parameters by an application of the QCD sum-rule method. The treatment of dimension-8 operators is incomplete, due to a restriction to a subset of contributions to coefficient functions and to use of vacuum factorization. Estimates for $d_N$ are presented for the case where the $\mathrm{CP}$ violation is induced by the neutral Higgs sector of the two-Higgs-doublet extension of the standard model.