A mass matrix for atmospheric, solar, and LSND neutrino oscillations

Neutrinos produced by the interaction of cosmic rays with the Earth’s upper atmosphere provide the strongest evidence for neutrino oscillations [1], with νμ → ντ as the favored flavor transition [2]. If the additional evidence from solar [3] and LSND [4] experiments is also confirmed, then it will be necessary to introduce a fourth light neutrino, a so-called “sterile neutrino” νs in addition to the standard electron-, muon-, and tau-neutrinos to account for all the data [5]. The question then arises as to the mass spectrum and mixing scheme for these four particles. In a two-flavor oscillation scenario, the atmospheric data suggest maximal mixing with mass difference ∆m ≈ 3 × 10−3 eV [6]. Of the three types of solution for the solar neutrino data, there are two, namely the large angle MSW (LMSW) and the “just-so” in vacuo ones, which require close to maximal mixing [7]; while the third, small angle MSW (SMSW) requires small mixing [8]. In all three cases, the mass difference ∆m is much smaller than in the atmospheric case. By contrast, the LSND data require small mixing, but with a relatively large ∆m as compared with the atmospheric case [4]. To account for these experimental results, we would like to propose a mass spectrum consisting of two doublets, with the splitting within each doublet being much smaller than the separation between them. The members of the upper doublet are identified as maximal superpositions of νμ and ντ , and the members of the lower doublet are maximal superpositions of νe and νs. Atmospheric neutrino data can then be described as maximal oscillations between the levels of the upper doublet, and solar neutrino data as maximal oscillations between the levels of the lower doublet. LSND is then a weak transition from one doublet to the other. Our approach to the development of a mass matrix for a two-doublet model can be illustrated with the following two-dimensional model: