The Koide Relation as an Exact Z3 Bogoliubov Identity

kibrom kidane · · DOI 10.5281/zenodo.19826574

Abstract

For any three–component Bogoliubov–de Gennes (BdG) system whose quasiparticle gaps form a real Z3–equivariant triplet ∆k = A+Bcos(θ+2πk/3) with all ∆k ≥ 0, the Koide ratio Q≡ ∑k mk /(∑k √mk )2 satisfies the exact identity Q= 1 3 + B2/(6A2), so that within this positive–gap region the empirical charged–lepton value Q= 2/3 is equivalent to |B/A|= √2. Outside the positive–gap region the identity fails; a counterexample at (B/A,θ) = (√2,π/3) gives Q≈ 0.4094. The algebraic identity is not new mathematics; it is implicit in Brannen’s parametrisation and the broader Koide literature. The contribution of this note is twofold. First, we separate existence from selection: the Koide puzzle reduces to a closed–form theorem on the existence side, leaving two narrower selection questions (Why |B/A|= √2? and Why θ = 2/9?). Second, we argue that the BdG framing supplies the physical reason for why a Z3 symmetry acts on √mrather than on m: the quasiparticle mass is |∆k |2 for an order–parameter gap on which the symmetry naturally acts. Numerical verification against Particle Data Group charged–lepton masses and a fully runnable reproduction script are provided.

Keywords

Koide relation· Bogoliubov–de Gennes spectrum· Z3 symmetry· Charged-lepton masses· Circulant mass matrix· Quasiparticle gap

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