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

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 $m_\nu$ 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 $m_\nu$ 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.\ $\det\eta$) and class-sensitive (e.g. $\mathrm{Tr}\,\eta$, $\mathrm{Tr}\,\eta^2$, an alignment measure, and CP-odd traces relevant to leptogenesis), where $\eta$ 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 $m_\nu$ into measurable, basis-invariant fingerprints.