Simple Connection Between Atmospheric and Solar Neutrino Vacuum Oscillations

Extending the minimal standard model of particle interactions (without righthanded singlet neutrinos) to include a heavy scalar triplet ξ to obtain nonzero Majorana neutrino masses, I derive the following simple realistic connection between atmospheric and solar neutrino vacuum oscillations: (∆m)sol(∆m)atm/mν (sin 2 2θ)atm = 2I2, where mν is the assumed common approximate mass of each neutrino (which may be suitable for hot dark matter) and I = (3/16π)(GF / √ 2)mτ ln(m 2 ξ/m 2 W ) comes from the radiative splitting of the degeneracy due to the charged leptons. There is now a vast literature on models of neutrino oscillations [1]. Most try to understand why atmospheric neutrino oscillations [2] of νμ(ν̄μ) to ντ (ν̄τ ) require near-maximal mixing [3]. Many also suggest that solar neutrino oscillations [4] of νe to a linear combination of νμ and ντ should have near-maximal mixing as well [5]. Both are possible in the context of three nearly mass-degenerate neutrinos [6, 7] which could then be considered as candidates for hot dark matter [8]. Recently it has been pointed out [9] that if all three neutrinos obtain equal Majorana masses of order 1 eV from the canonical seesaw mechanism [10], then their splitting due to the different charged-lepton masses from the two-loop exchange of two W bosons [11] is of the right magnitude for solar neutrino vacuum oscillations. However, the inclusion of atmospheric neutrino oscillations has to be rather ad hoc in this case. In fact, it is rare indeed that any bona fide model of neutrino masses even gets a relationship between the mass difference of one oscillation and that of another. [One exception is the recently proposed model [12] of radiative masses for νe, νμ, ντ , plus a singlet (sterile) neutrino νs, which explains atmospheric and solar neutrino oscillations as well as the ν̄μ(νμ) to ν̄e(νe) data of the LSND (Liquid Scintillator Neutrino Detector) experiment [13]. It has the successful relationship (∆m)atm ' 2[(∆m)sol(∆m)LSND], where (∆m)sol refers to the matterenhanced solution [14] of the solar neutrino deficit.] In this note I will present the most economical model to date of neutrino masses which has the following simple realistic connection between atmospheric and solar neutrino vacuum oscillations: (∆m)sol(∆m )atm mν (sin 2 2θ)atm = 2I = 4.9× 10−13 ( ln mξ mW )2 , (1) where mν is the assumed common approximate mass of each neutrino, mξ is the mass of a heavy scalar triplet, and I = 3GF m 2 τ 16π2 √ 2 ln mξ mW (2)