The contribution of the CP violating three-gluon Weinberg operator, $$ \frac{1}{3!}w{f}^{abc}{\epsilon}^{\nu \rho \alpha \beta}{G}_{\mu \nu}^a{G}_{\alpha \beta}^b{G}_{\rho}^{c\mu} $$ 1 3 ! w f abc ϵ νραβ G μν a G αβ b G ρ cμ , to the atomic and nuclear EDMs is estimated using QCD sum rules. After calculating the transition matrix element between the pion and the vacuum through the Weinberg operator, we obtain the long-range CP-odd nuclear force by determining the isovector CP-odd pion-nucleon vertex, using chiral perturbation theory at NLO. The EDMs of 199Hg, 129Xe, and 225Ra atoms, as well as those of 2H and 3He nuclei are finally given including comprehensive uncertainty analysis. While the leading contribution of the 199Hg EDM is given by the intrinsic nucleon EDM, that of 129Xe atom may be dominated by the one-pion exchange CP-odd nuclear force generated by the Weinberg operator. From current experimental data of the 199Hg atomic EDM, we obtain an upper limit on the Weinberg operator magnitude of |w| < 4 × 10−10GeV−2 if we assume that it is the only source of CP violation at the scale μ = 1 TeV.

When the QCD axion is absent in full theory, the strong CP problem has to be explained by an additional mechanism, e.g., the left-right symmetry. Even though tree-level QCD θ parameter is restricted by the mechanism, radiative corrections to θ are mostly generated, which leads to a dangerous neutron electric dipole moment (EDM). The ordinary method for calculating the radiative θ utilizes an equation θ = arg det m loop q based on the chiral rotations of complex quark masses. In this paper, we point out that when full theory includes extra heavy quarks, the ordinary method is unsettled for the extra quark contributions and does not contain its full radiative corrections. We formulate a novel method to calculate the radiative corrections to θ through a direct loop-diagrammatic approach, which should be more robust than the ordinary one. As an application, we investigate the radiative θ in the minimal left-right symmetric model. We first confirm a seminal result that two-loop level radiative θ completely vanishes (corresponding to oneloop corrections to the quark mass matrices). Furthermore, we estimate the size of a nonvanishing radiative θ at three-loop level. It is found that the resultant induced neutron EDM is comparable to the current experimental bound, and the expected size is restricted by the perturbative unitarity bound in the minimal left-right model.

When the QCD axion is absent in full theory, the strong CP problem has to be explained by an additional mechanism, e.g., the left-right symmetry. Even though tree-level QCD $$ \overline{\theta} $$ θ ¯ parameter is restricted by the mechanism, radiative corrections to $$ \overline{\theta} $$ θ ¯ are mostly generated, which leads to a dangerous neutron electric dipole moment (EDM). The ordinary method for calculating the radiative $$ \overline{\theta} $$ θ ¯ utilizes an equation $$ \overline{\theta}=-\arg\ \det {m}_q^{\textrm{loop}} $$ θ ¯ = − arg det m q loop based on the chiral rotations of complex quark masses. In this paper, we point out that when full theory includes extra heavy quarks, the ordinary method is unsettled for the extra quark contributions and does not contain its full radiative corrections. We formulate a novel method to calculate the radiative corrections to $$ \overline{\theta} $$ θ ¯ through a direct loop-diagrammatic approach, which should be more robust than the ordinary one. As an application, we investigate the radiative $$ \overline{\theta} $$ θ ¯ in the minimal left-right symmetric model. We first confirm a seminal result that two-loop level radiative $$ \overline{\theta} $$ θ ¯ completely vanishes (corresponding to one-loop corrections to the quark mass matrices). Furthermore, we estimate the size of a non-vanishing radiative $$ \overline{\theta} $$ θ ¯ at three-loop level. It is found that the resultant induced neutron EDM is comparable to the current experimental bound, and the expected size is restricted by the perturbative unitarity bound in the minimal left-right model.