Rescaled Leptonic Unitary Triangles and Rephasing Invariants

TL;DR

The paper develops a framework that links leptonic CP-violating and CP-conserving rephasing invariants, ${\cal J}$ and ${\cal R}_{\gamma k}$, to neutrino-oscillation observables via rescaled unitarity triangles. It introduces Naumov-like relations and composition matrices $X^{e}$ and $X^{\mu\tau}$ that map vacuum invariants ${\cal R}_{\alpha i}$ to their matter-modified counterparts $\widetilde{\cal R}_{\alpha i}$ and relate $\widetilde{\cal J}$ to ${\cal J}$; it analyzes vacuum-dominated, resonance, and matter-dominated regimes and provides numerical illustrations with the latest global fits. The work shows that the effective invariants in matter are linear combinations of vacuum invariants, preserving $\mu$-$\tau$ symmetry patterns and enabling a transparent, geometry-based interpretation of neutrino oscillations in matter. This approach supports unitarity tests of the PMNS matrix with multi-experiment data and offers a foundation for exploring new physics such as NSI or sterile neutrinos.

Abstract

The field of neutrino physics has made significant progress in measuring the strength and frequency of neutrino and antineutrino oscillations in the past two decades. It is clear that the amplitudes involved in the neutrino oscillation probabilities are all phase-reshaping invariants of the quartet forms of the elements of the PMNS mixing matrix. We show in this paper how these quartet observables can be directly linked to the rescaled leptonic unitarity triangles within the framework of three active neutrinos. We provide a systematic discussion of the nine CP-conserving quartets ${\cal R}^{}_{γk} \equiv {\rm Re} \left [ V^{}_{αi} V^{}_{βj} V^{*}_{αj} V^{*}_{βi}\right ] $ along with the universal Jarlskog invariant of CP violation ${\cal J} \equiv \sum_γε^{}_{αβγ} \sum_k ε^{}_{ijk} \; {\rm Im} \left [ V^{}_{αi} V^{}_{βj} V^{*}_{αj} V^{*}_{βi} \right ]$, and place particular emphasis on the matter effect on these quartets. In addition to the well-known Naumov relation for the Jarlskog invariant ${\cal J}$, similar relations connecting ${\cal R}$ in vacuum and its effective counterparts $\widetilde{\cal R}$ in matter are introduced and examined in detail. We find that the effective CP-conserving invariants $\widetilde{\cal R}^{}_{αi}$ in matter can be regarded as linear combinations of their vacuum counterparts. With the latest global fit data of neutrino masses and mixing elements, numerical analyses are carried out to give an intuitive understanding of how these phase-rephasing invariants evolve as the matter density increases.

Figures (13)

• Figure 1: A numerical illustration of the rescaled UTs originating from $\triangle^{}_{e}$, $\triangle^{}_{\mu}$ and $\triangle^{}_{\tau}$ respectively in the complex plane, where the best-fit values of the relevant flavor mixing and CP-violating parameters Esteban:2024eliNufit for NMO have been input.
• Figure 2: A numerical illustration of the rescaled UTs originating from $\triangle^{}_{1}$, $\triangle^{}_{2}$ and $\triangle^{}_{3}$ respectively in the complex plane, where the best-fit values of the relevant flavor mixing and CP-violating parameters Esteban:2024eliNufit for NMO have been input.
• Figure 3: A numerical illustration of the rescaled UTs originating from $\triangle^{}_{e}$, $\triangle^{}_{\mu}$ and $\triangle^{}_{\tau}$ respectively in the complex plane, where the best-fit values of the relevant flavor mixing and CP-violating parameters Esteban:2024eliNufit for IMO have been input.
• Figure 4: A numerical illustration of the rescaled UTs originating from $\triangle^{}_{1}$, $\triangle^{}_{2}$ and $\triangle^{}_{3}$ respectively in the complex plane, where the best-fit values of the relevant flavor mixing and CP-violating parameters Esteban:2024eliNufit for IMO have been input.
• Figure 5: The evolution of the effective CP-violating Jarlskog invariant $\widetilde{\cal J}$ in matter with respect to the dimensionless ratio $A^{}_{\rm CC} / 2.5 \times 10^{-3}_{} {\rm eV}^{2}_{}$ in both the MNO case (left) and the IMO case (right), and for both neutrinos (in blue) and antineutrinos (in red), where the best-fit value, $1\sigma$ and $3\sigma$ ranges of $\widetilde{\cal J}$ are present with solid line, dark and light shadows respectively in the plots.