CP-violating decays of the pseudoscalars eta and eta and their connection to the electric dipole moment of the neutron

Using the present upper bound on the neutron electric dipole moment, we give an estimate for the upper limit of the CP-violating couplings of the eta(eta) to the nucleon. Using this result, we then derive constraints on the CP-violating eta(eta)-pi-pi couplings, which define the two-pion CP-violating decays of the eta and eta mesons. Our results are relevant for the running and planned measurements of rare decays of the eta and eta mesons by the GlueX Collaboration at JLab and the LHCb Collaboration at CERN.