Measuring Dirac CP-violating phase with intermediate energy beta beam facility

We find that spontaneously broken parity ( $P$ ) or left–right symmetry stabilizes dark matter in a beautiful way. If dark matter has a non-real intrinsic parity $\pm i$ (e.g. if it entails Majorana fermions), parity can ensure that it cannot decay to all normal particles with real intrinsic parities. However, if Majorana couplings are absent either in the lepton or the dark sector, $P$ symmetry can be redefined to remove relative non-real intrinsic phases. It is therefore predicted that neutrinos and dark matter fermions must have Majorana masses if dark matter is stable due to parity. The strong CP problem is solved by additionally imposing CP and including vectorlike fermions that help generate CP violation. If leptonlike heavy fermions are provided purely imaginary intrinsic parity phase, they do not couple to the usual leptons, and leptonic CP phases are not generated, which is a testable prediction. Experimentally if leptonic CP phases are not found (if they are consistent with $0$ or $\pi $ ) it can be evidence for the type of models in this work where CP is spontaneously or softly broken and there is also a second hidden or softly broken symmetry such as $P$ , $Z_2$ or $Z_4$ . However, leptonic CP violation can be present in closely related or some non-minimal versions of these models, such as by also including vectorlike leptons with real intrinsic parities.