New CP Phase and Exact Oscillation Probabilities of Dirac Neutrino derived from Relativistic Equation

Abstract

We present a new formulation deriving the neutrino oscillation probabilities relativistically based on not the Schrödinger equation but the Dirac equation. In two generations, we calculate the oscillation probabilities exactly in the case that there exists only the Dirac mass term. We find that two kinds of new terms appear in the oscillation probabilities derived from the Dirac equation. One is the term dependent on the absolute value of neutrino mass. Although it has been considered that the oscillation probabilities depend only on the mass squared differences until now, we could observe the absolute value of mass through neutrino oscillations in principle. The other is the term including a new CP phase. If there are some interactions to distinguish the flavors of right-handed neutrinos beyond the Standard Model, we could also observe this new CP phase in principle even in the framework of two generations. We discuss the possibility to observe the contribution of these terms by the neutrino oscillations of atomic size. On the other hand, it is negligible in the usual short and long-baseline experiments, and there is no contradiction with previous experiments.

Unification of Neutrino-Neutrino and Neutrino-Antineutrino Oscillations

“…In this paper, we would like to understand the oscillations between the neutrinos and the anti-neutrinos in a unified framework by applying the relativistic formulation used in our previous paper. We show that the Majorana CP phase is a kind of this new CP phase with chirality-flip as in the case of the Dirac neutrino in the previous paper [35]. Namely, the Majorana CP phases arise with chirality-flip, not the lepton number violation.…”

“…In our previous paper [35], we derived the neutrino oscillation probabilities in the case of the Dirac neutrinos by using a relativistic equation, the Dirac equation. In this section, we review the results of our previous paper [35] in order to clarify the similarity and the difference between the case of the Dirac neutrinos and the Majorana neutrinos.…”

“…where we take equal momentum assumption for neutrinos with different flavors, see ref. [35] for details. In the above equation, the bottom-right part and the top-left part are completely separated and therefore they do not mix each other even if the time has passed.…”

“…We cannot observe the CP phase accompanying the unitary matrix V if ν R do not have weak interactions and we can choose the weak eigenstates as the same as the mass eigenstates. Otherwise, the effects may be measurable in the oscillations of atomic size [35].…”

“…In our previous paper [35], we derive the exact oscillation probabilities of the Dirac neutrinos in the framework of two generations relativistically. As a result, we can understand the neutrino oscillations with and without chirality-flip in a unified way.…”