Abstract
A generalized connection between the quantum mechanical Bargmann invariants and the geometric phases was established for the Dirac fermions. We extend that formalism for the Majorana fermions by defining proper quantum mechanical ray and Hilbert spaces. We then relate both the Dirac and Majorana type Bargmann invariants to the rephasing invariant measures of CP violation with the Majorana neutrinos, assuming that the neutrinos have lepton number violating Majorana masses. We then generalize the recursive parametrization for studying any unitary matrices to include the Majorana fermions, which could be useful for studying the neutrino mixing matrix.
Highlights
One of the most interesting problems of the standard model is to understand the origin of CP violation
The state vectors represent Dirac fermions, and the BIs may be identified with rephasing invariant measures of CP violation. We shall generalize these results to the case when the state vectors represent both Dirac and Majorana fermions by defining the ray and the Hilbert spaces properly. This will introduce additional BIs representing CP violation arising from the Majorana phases and this, in turn, will relate the rephasing invariant measures of CP violation for both Dirac and Majorana fermions to the complete sets of Bargmann invariants
We shall first demonstrate how one can define the quantum mechanical Bargmann variables for the Majorana fermions and relate them with the geometric phases after defining the proper quantum mechanical ray and Hilbert spaces for the Majorana fermions
Summary
One of the most interesting problems of the standard model is to understand the origin of CP violation. We shall generalize these results to the case when the state vectors represent both Dirac and Majorana fermions by defining the ray and the Hilbert spaces properly This will introduce additional BIs representing CP violation arising from the Majorana phases and this, in turn, will relate the rephasing invariant measures of CP violation for both Dirac and Majorana fermions to the complete sets of Bargmann invariants. One can study the CP phases in the neutrino mixing matrix and the Majorana phases through the recursive parametrization of the unitary matrices We shall extend these analyses and present explicit forms of the rephasing invariant quantities for a few examples when Majorana fermions are included
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