Neutrino Masses from Planck to Planck

Koide parameters for both neutrino and mirror neutrino triplets are examined in light of the CMB mirror neutrino correspondence. This note is just to record slightly improved rest mass predictions for the neutrinos. In the 1980s, Koide predicted the τ lepton rest mass using the simple relation √ memμ + √ mμmτ + √ mτme me + mμ + mτ = 1 4 (1) involving the more accurately known electron and muon masses [1]. Later it was observed that 3 × 3 circulant matrices naturally parameterize such mass triplets [2]. The square roots of the three masses form an eigenvalue triplet for the Hermitian matrix √ M √ 2   r eiφ e−iφ e−iφ r eiφ eiφ e−iφ r   , (2) where √ M determines a unit scale. The charged lepton mass triplet is closely fitted by the values φ = 2/9 and r = √ 2, with M the dynamical quark mass [3]. In [3], Brannen extended the Koide relation to the unknown neutrino rest mass triplet by allowing minus signs in (1). Using the known experimental constraints on ∆m2 values, noted below, a neutrino triplet is fitted with parameters φν = 2/9 + π/12 rν = √ 2 (3) and a suitably small value of Mν . This is a normal hierarchy triplet, amenable to testing with current generation neutrino experiments. Koide triplets also exist for quarks [4][5], strengthening the argument for a universal 2/9 phase. In any theory governing the parameters of (2) the phase φ, for a general triplet, may have several components. Although the phase conjugation φ 7→