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$D_N$-Orthogonal Freedom in the Canonical Seesaw: Flavor Invariants and Physical Non-Equivalence of F-Classes

Jianlong Lu

TL;DR

This work establishes a complete group-theoretic classification of the right-handed seesaw freedom that preserves the light neutrino mass matrix $m_\nu$ in the canonical Type-I seesaw with three heavy neutrinos, identifying the invariance group $\mathcal{G}=D_N^{1/2}O(3,\mathbb{C})D_N^{-1/2}$. It decomposes flavor invariants into class-blind and class-sensitive categories, showing that oscillation data fix $U_{\rm PMNS}$ and mass splittings but not the $F$-class; the latter is probed by heavy–light non-unitarity, LFV, and CP-odd leptogenesis invariants such as $\mathcal{I}_{\rm CP}^{(1)}$, with $\det\eta$ being class-independent. Through six benchmark points, the authors demonstrate that class-sensitive invariants can vary across classes by orders of magnitude while class-blind quantities stay fixed, and that heavy-mass degeneracies enlarge the symmetry and suppress unflavored CP probes, motivating flavored invariant probes. The framework provides a practical, basis-invariant map from high-energy completions to low-energy observables, enabling data-driven discrimination of seesaw realizations compatible with the same $m_\nu$. These results offer a roadmap for using precision non-unitarity, LFV, and leptogenesis observables to distinguish among UV completions sharing identical low-energy neutrino phenomenology.

Abstract

We study basis-independent structures in the Type-I seesaw mechanism for light Majorana neutrinos, assuming the canonical scenario with three heavy right-handed (sterile) neutrinos. Let $m_ν$ denote the $3\times3$ mass matrix of light neutrinos, obtained at tree level from heavy Majorana singlets with diagonal mass matrix $D_N = \mathrm{diag}(M_1,M_2,M_3)$ and Dirac matrix $m_D$. We show that all right-actions $F$ on the seesaw matrix that leave $m_ν$ unchanged form the group $G = D_N^{1/2} O(3,\mathbb{C}) D_N^{-1/2}$. While oscillation data determine the PMNS matrix $U_{\rm PMNS}$ and the mass-squared splittings, they do not fix the $F$-class within $G$. We classify basis-invariant quantities into those that are class-blind (e.g.\ $\detη$) and class-sensitive (e.g. $\mathrm{Tr}\,η$, $\mathrm{Tr}\,η^2$, an alignment measure, and CP-odd traces relevant to leptogenesis), where $η$ denotes the non-unitarity matrix of the light sector. We provide explicit formulas and both high-scale and GeV-scale benchmark examples that illustrate these invariant fingerprints and their scaling with $D_N$. This converts the degeneracy at fixed $m_ν$ into measurable, basis-invariant fingerprints.

$D_N$-Orthogonal Freedom in the Canonical Seesaw: Flavor Invariants and Physical Non-Equivalence of F-Classes

TL;DR

This work establishes a complete group-theoretic classification of the right-handed seesaw freedom that preserves the light neutrino mass matrix in the canonical Type-I seesaw with three heavy neutrinos, identifying the invariance group . It decomposes flavor invariants into class-blind and class-sensitive categories, showing that oscillation data fix and mass splittings but not the -class; the latter is probed by heavy–light non-unitarity, LFV, and CP-odd leptogenesis invariants such as , with being class-independent. Through six benchmark points, the authors demonstrate that class-sensitive invariants can vary across classes by orders of magnitude while class-blind quantities stay fixed, and that heavy-mass degeneracies enlarge the symmetry and suppress unflavored CP probes, motivating flavored invariant probes. The framework provides a practical, basis-invariant map from high-energy completions to low-energy observables, enabling data-driven discrimination of seesaw realizations compatible with the same . These results offer a roadmap for using precision non-unitarity, LFV, and leptogenesis observables to distinguish among UV completions sharing identical low-energy neutrino phenomenology.

Abstract

We study basis-independent structures in the Type-I seesaw mechanism for light Majorana neutrinos, assuming the canonical scenario with three heavy right-handed (sterile) neutrinos. Let denote the mass matrix of light neutrinos, obtained at tree level from heavy Majorana singlets with diagonal mass matrix and Dirac matrix . We show that all right-actions on the seesaw matrix that leave unchanged form the group . While oscillation data determine the PMNS matrix and the mass-squared splittings, they do not fix the -class within . We classify basis-invariant quantities into those that are class-blind (e.g.\ ) and class-sensitive (e.g. , , an alignment measure, and CP-odd traces relevant to leptogenesis), where denotes the non-unitarity matrix of the light sector. We provide explicit formulas and both high-scale and GeV-scale benchmark examples that illustrate these invariant fingerprints and their scaling with . This converts the degeneracy at fixed into measurable, basis-invariant fingerprints.

Paper Structure

This paper contains 16 sections, 8 theorems, 69 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Seesaw setup and diagonalization
  3. Classification of the $m_\nu$-preserving freedom
  4. Flavor invariants: class-blind versus class-sensitive
  5. Six benchmark points and numerical fingerprints
  6. Heavy-mass degeneracies and flavor invariants
  7. Phenomenological implications
  8. Conclusions and outlook
  9. Numerical details for Table \ref{['tab:highscale']}
  10. Inputs (fixed for all six benchmark points $A$--$F$)
  11. Square roots and helpful diagonals
  12. Representatives for the six benchmark points (Casas--Ibarra right action)
  13. Exact construction for each benchmark point
  14. Invariant definitions and numerics
  15. Cross-checks and controls
  16. ...and 1 more sections

Key Result

Theorem 1

With $D_N>0$ diagonal as above, where $O(3,\mathbb{C})\equiv\{H\in GL(3,\mathbb{C})\ |\ H H^{T}=\mathbb{I}\}$ is the complex orthogonal group.

Theorems & Definitions (8)

  • Theorem 1: Conjugate complex-orthogonal classification
  • Lemma 1
  • Proposition 1: Class-blind low-energy controls
  • Lemma 2
  • Proposition 2: Non-unitarity invariants
  • Proposition 3: Alignment invariant
  • Theorem 2: A CP-odd leptogenesis invariant
  • Proposition 4: Class-independence of $\det\eta$