Abstract
Abstract We formulate a perturbative framework for the flavor transformation of the standard active three neutrinos but with a non-unitary flavor mixing matrix, a system which may be relevant for the leptonic unitarity test. We use the $\alpha$ parametrization of the non-unitary matrix and take its elements $\alpha_{\beta \gamma}$ ($\beta,\gamma = e,\mu,\tau$) and the ratio $\epsilon \simeq \Delta m^2_{21} / \Delta m^2_{31}$ as the small expansion parameters. Two qualitatively new features that hold in all the oscillation channels are uncovered in the probability formula obtained to first order in the expansion: (1) The phases of the complex $\alpha$ elements always come into the observable in the particular combination with the $\nu$SM CP phase $\delta$ in the form $[e^{- i \delta } \bar{\alpha}_{\mu e}, ~e^{ - i \delta} \bar{\alpha}_{\tau e}, ~\bar{\alpha}_{\tau \mu}]$ under the Particle Data Group convention of a unitary $\nu$SM mixing matrix. (2) The diagonal $\alpha$ parameters appear in particular combinations $\left( a/b - 1 \right) \alpha_{ee} + \alpha_{\mu \mu}$ and $\alpha_{\mu \mu} - \alpha_{\tau \tau}$, where $a$ and $b$ denote, respectively, the matter potential due to charged current and neutral current reactions. This property holds only in the unitary evolution part of the probability, and there is no such feature in the genuine non-unitary part, while the $\delta$–$\alpha$ parameter phase correlation exists for both. The reason for such remarkable stability of the phase correlation is discussed.
Highlights
It appears that the three flavor lepton mixing [1] is well established after the long term best endeavor by the experimentalists, which are recognized in an honorable way [2, 3].Though we do not know the value of CP phase δ, the lepton Kobayashi-Maskawa (KM)phase [4], and the neutrino mass ordering, there appeared some hints toward identifying these unknowns
It is good to know that a part of our canonical phase combination, e−iδαμe, has been observed in the foregoing studies [30, 32, 33].10. Our result places these observation into more generic setting which includes all the complex α parameters, and clarify its nature using the first order analytic formulas
One can observe that the oscillation probability given P(1) and P(1), including both the EV and unitarity violation (UV) parts are invariant under the transformation π φ→φ+
Summary
It appears that the three flavor lepton mixing [1] is well established after the long term best endeavor by the experimentalists, which are recognized in an honorable way [2, 3]. If we use the α matrix to parametrize the UV effect [30] (see section 3.2 for definition) it takes the form [e−iδαμe, e−iδατe, ατμ] under the Particle Data Group (PDG) convention [31] of the flavor mixing MNS matrix UMNS.3 It generalizes the earlier observation of e−iδαμe correlation done in analyzing the data [32, 33] (see [30]) to all the complex α matrix elements in an analytic way.
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