Abstract
In the low-energy effective theory of neutrinos, the Haar measure for unitary matrices is very likely to give rise to something similar to the observed Pontecorvo-Maki-Nakagawa-Sakata matrix. Assuming the Haar measure, we determine the probability density functions for all quadratic, quartic Majorana, and quartic Dirac rephasing invariants for an arbitrary number of neutrino generations. We show that for a fixed number of neutrinos, all rephasing invariants of the same type have the same probability density function under the Haar measure. We then compute the moments of the rephasing invariants to determine, with the help of the Mellin transform, the three probability density functions. We finally investigate the physical implications of our results in function of the number of neutrinos.
Highlights
In flavor physics, the passage from gauge eigenstates to mass eigenstates encodes flavor mixing
In the Standard Model of particle physics, there is no equivalent mixing for the lepton sector
After reviewing the Haar measure, we demonstrate that all rephasing invariants of the same type have the same probability density function (PDF) with respect to the Haar measure
Summary
The passage from gauge eigenstates to mass eigenstates encodes flavor mixing. It was proven there that the PDF for arbitrary neutrino numbers factorizes into a PDF for the light neutrino mass eigenvalues and a PDF for the mixing angles and phases of the PMNS matrix The former is given by a complicated multidimensional integral, while the latter is the Haar measure (independently of the seesaw mechanism, as foreseen on physical grounds in [10]). We introduce another technique relying on the knowledge of the moments and the Mellin transform This method leads to direct expressions for all rephasing invariant PDFs for arbitrary neutrino numbers in terms of Meijer G functions.
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