Updated limits on the $CP$ violating $\eta \pi \pi$ and $\eta \prime \pi \pi$ couplings derived from the neutron EDM

We complete our derivation of upper limits on the $CP$ violating $\eta \pi \pi$ and ${\eta }^{\prime }\pi \pi$ couplings from an analysis of their two-loop contributions to the neutron electric dipole moment (nEDM). We use a phenomenological Lagrangian approach which is formulated in terms of hadronic degrees of freedom—nucleons and pseudoscalar mesons. The essential part of the Lagrangian contains the $CP$ violating couplings between $\eta ({\eta }^{\prime })$ and pions. Previously, we included photons using minimal substitution in case of the proton and charged pions. Now we extend our Lagrangian by adding the nonminimal couplings, i.e., anomalous magnetic couplings of nucleons with the photon. The obtained numerical upper limits for the $\eta \pi \pi$ and ${\eta }^{\prime }\pi \pi$ couplings $|f_{\eta \pi \pi }(M_{\eta }^2)|<4.4\times {10}^{-11}$ and $|f_{{\eta }^{\prime }\pi \pi }(M_{{\eta }^{\prime }}^2)|<3.8\times {10}^{-11}$ can be useful for the related, planned experiments at the JLab Eta Factory. Using present experimental limits on the nEDM, we derive upper limits on the $CP$ violating $\overline{\theta }$ parameter of $\overline{\theta }<4.7\times {10}^{-10}$.

Figure 1

Diagrams contributing to the nEDM which are induced by the minimal electromagnetic couplings of proton and charged pions. The solid square denotes the $CP$ violating $\eta {\pi }^{+}{\pi }^{-}$ vertex.

Figure 2

Diagrams contributing to the nEDM which are induced by the nonminimal electromagnetic couplings (anomalous magnetic moments) of nucleons. The solid square denotes the $CP$ violating $\eta ({\eta }^{\prime })\pi \pi$ vertex. Empty and shaded circles correspond to the nonminimal electromagnetic couplings of neutron and proton, respectively.