The Physical Range of Majorana Neutrino Mixing Parameters

TL;DR

This paper clarifies the physical parameter ranges for Majorana neutrino mixing angles and phases, resolving confusion and providing a consistent framework for understanding neutrino mixing scenarios, including sterile neutrinos and potential new interactions.

Contribution

It establishes unambiguous ranges for Majorana phases and mixing angles, and discusses how these ranges extend or change with sterile neutrinos and new interactions.

Findings

Majorana phases can be constrained between 0 and pi.

Mixing angles can be positive and up to pi/2 with Dirac phases between -pi and pi.

Parameter space may need to be enlarged for new flavor-changing interactions.

Abstract

If neutrinos are Majorana fermions, the lepton mixing parameter space consists of six mixing parameters: three mixing angles and three CP-odd phases. A related issue concerns the physical range of the mixing parameters. What values should these take so that all physically distinguishable mixing scenarios are realized? We present a detailed discussion of the lepton mixing parameter space in the case of two and three active neutrinos, and in the case of three active and N sterile neutrinos. We emphasize that this question, which has been a source of confusion even among "neutrino" physicists, is connected to an unambiguous definition of the neutrino mass eigenstates. We find that all Majorana phases can always be constrained to lie between 0 and pi, and that all mixing angles can be chosen positive and at most less than or equal to pi/2 provided the Dirac phases are allowed to vary between -pi and pi. We illustrate our results with several examples. Finally, we point out that, in the case of new flavor-changing neutrino interactions, the lepton mixing parameter space may need to be enlarged. We properly qualify this statement, and offer concrete examples.