Rescaled Leptonic Unitarity Triangles and Rephasing Invariants

Technical Summary: Rescaled Leptonic Unitarity Triangles and Rephasing Invariants

Problem Statement
While the field of neutrino physics has made significant progress in measuring oscillation strengths and frequencies over the past two decades, a direct geometric link between these observables and the fundamental parameters of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix remains a challenge. Specifically, the amplitudes governing neutrino oscillation probabilities are rephasing invariants of quartet forms of the PMNS matrix elements. However, the standard Unitarity Triangles (UTs), defined by the orthogonality relations of the $3\times3$ unitary matrix, are not directly observable in their standard form. Furthermore, the evolution of these invariants in the presence of matter effects (crucial for long- and medium-baseline experiments) is often described through effective mixing angles and mass-squared differences, obscuring the direct relationship between vacuum invariants and their matter counterparts. The paper aims to bridge this gap by establishing a systematic framework linking quartet observables directly to rescaled UTs and deriving precise relations for their evolution in matter.

Methodology
The authors employ a systematic analysis of the nine CP-conserving rephasing invariants, denoted as $R_{\gamma k} \equiv \text{Re}[V_{\alpha i}V_{\beta j}V^*_{\alpha j}V^*_{\beta i}]$, and the unique CP-violating Jarlskog invariant, $J$.

1. Rescaling Unitarity Triangles: The authors rescale the six standard UTs defined by the orthogonality of the PMNS matrix rows and columns. This rescaling produces 18 "rescaled UTs" in the complex plane whose sides are phase-rephasing invariant. These rescaled triangles share a common height equal to the Jarlskog invariant $J$.
2. Matrix Representation: The nine real components $R_{\gamma k}$ are arranged into a $3\times3$ matrix $R$. The authors analyze the symmetries of this matrix, particularly the approximate $\mu-\tau$ symmetry, by defining average ($R_i$) and difference ($\Delta R_i$) components.
3. Matter Effect Analysis: The study investigates the propagation of neutrinos in matter using the effective Hamiltonian. Instead of relying solely on effective mixing parameters, the authors derive "Naumov-like" relations connecting the vacuum invariants ($R_{\alpha i}, J$) to their effective matter counterparts ($\tilde{R}_{\alpha i}, \tilde{J}$).
4. Composition Matrices: The effective invariants are expressed as linear combinations of vacuum invariants via composition matrices ($X_e$ and $X_{\mu\tau}$). The evolution of these matrices and the invariants is analyzed across three regimes: vacuum-dominated, resonance, and matter-dominated regions, using the latest global fit data for neutrino masses and mixing angles.

Key Contributions

• Direct Link to Observables: The paper establishes that the rescaled UTs are directly determined by the quartet observables ($R_{\gamma k}$ and $J$) found in oscillation probabilities. This allows for the experimental reconstruction of specific UTs based on measurements from appearance and disappearance channels.
• Naumov-like Relations for CP-Conserving Invariants: Extending the well-known Naumov relation for the Jarlskog invariant ($J$), the authors derive explicit relations connecting the CP-conserving invariants $R_{\alpha i}$ in vacuum to $\tilde{R}_{\alpha i}$ in matter.
• Composition Matrices: The authors demonstrate that effective CP-conserving invariants in matter are linear combinations of their vacuum counterparts. The composition matrices governing these transformations are primarily determined by the matter potential ($A_{CC}$), neutrino mass-squared differences, and the first row of the PMNS matrix ($|V_{e1}|^2, |V_{e2}|^2, |V_{e3}|^2$).
• Symmetry Analysis: The paper provides a detailed analysis of $\mu-\tau$ symmetry within the $R$ matrix and its preservation or breaking under matter effects. It defines specific conditions under which UTs become right-angled or isosceles.

Results

• Vacuum Properties: Using global fit data, the authors provide best-fit values and ranges for $J$ and the nine $R_{\alpha i}$ invariants for both Normal (NMO) and Inverted (IMO) Mass Orderings. They find that at the electroweak scale, $\triangle_e$ and $\triangle_3$ are obtuse triangles, while others can be acute, right, or obtuse depending on the specific parameter values.
• Matter Evolution:
  • Vacuum-Dominated Region: The composition matrices are close to the identity matrix with small perturbations, allowing for accurate prediction of effective invariants given precise measurements of the first row of the PMNS matrix.
  • Resonance Regions: Significant changes occur in the invariants near solar ($A_{CC} \sim \Delta_{21}$) and atmospheric ($A_{CC} \sim \Delta_{31}$) resonances. The paper details which specific invariants undergo significant resonance (e.g., $\tilde{R}_{e3}, \tilde{R}_{\mu3}, \tilde{R}_{\tau3}$ at solar resonance).
  • Matter-Dominated Region: As $|A_{CC}| \to \infty$, electron neutrinos decouple. The system reduces to a two-flavor mixing structure between $\nu_\mu$ and $\nu_\tau$. In this limit, all CP-violating invariants vanish, and eight of the nine CP-conserving invariants vanish, leaving only one non-zero invariant $\tilde{R}_f \approx -1/4 \sin^2 2\tilde{\theta}$. The mass ordering (NMO vs. IMO) can be distinguished by the specific fixed values of the remaining invariants and the composition matrix elements.
• $\mu-\tau$ Symmetry: The study finds that if the vacuum $R$ matrix exhibits exact $\mu-\tau$ symmetry, the effective matrix in matter preserves this symmetry. The matter effect itself does not introduce additional symmetry breaking.

Significance
The paper claims that this framework provides a robust method for exploring unknown neutrino oscillation parameters using existing data. Because the composition matrices are determined by parameters already measured with high precision (the first row of the PMNS matrix and mass-squared differences), the evolution of invariants in matter can be calculated with high accuracy. This allows for the conversion of data from diverse experiments (accelerator, reactor, atmospheric) probing different energy ranges and baselines into a unified vacuum basis ($R_{\alpha i}$ and $J$). This synergy facilitates a global analysis to reduce uncertainties in the $\theta_{23}$ octant, the CP phase $\delta$, and the neutrino mass ordering. Furthermore, the relations serve as a foundation for testing the unitarity of the PMNS matrix and identifying potential deviations caused by new physics, such as sterile neutrinos or non-standard interactions.