Generalized Neutrino Mixing from the Atmospheric Anomaly

We determine the neutrino mixing and mass parameters that are allowed by the Super–Kamiokande atmospheric neutrino data in a three–neutrino model with one mass–squared difference contributing to the oscillations. We find that although νμ → ντ oscillations are favored, νμ → νe oscillations with amplitude as large as 0.18 are allowed even after accounting for the limit from the CHOOZ reactor experiment. The range of allowed parameters permit observable νμ ↔ νe and νe → ντ oscillations in future long–baseline experiments. Introduction. It was suggested long ago [1] that the atmospheric neutrino anomaly [2] could be explained by the oscillation of muon neutrinos and antineutrinos into another neutrino species. This interpretation has been confirmed by the zenith angle dependence measured by the Super–Kamiokande (SuperK) experiment [3]. Neutrino oscillations can also be invoked to separately explain the solar neutrino deficit [4, 5] and the results of the LSND experiment [6]. Because confirmation of the LSND results awaits future experiments and recent measurements in the KARMEN detector exclude part of the LSND allowed region [7], a conservative approach is to assume that oscillations need only account for the solar and atmospheric data. Then the two mass–squared difference scales in a three–neutrino model are sufficient to describe the data. Interest in the implications for models of the atmospheric neutrino anomaly has recently intensified [8, 9, 10]. An attractive possibility is that both the atmospheric νμ and solar νe oscillate maximally or near–maximally at the δm 2 atm and δm 2 sun scales, respectively [9, 10]. In this letter we use the recent Super–Kamiokande atmospheric neutrino data [3] to determine the allowed values for the general three–neutrino mixing matrix under the assumption that one mass–squared difference, δm2atm, explains the atmospheric neutrino oscillations. In our scenario the other mass–squared difference, δm2sun ≪ δm 2 atm, can explain the solar neutrino oscillations via either an MSW [11] or vacuum long–wavelength scenario [12]. We find that although pure νμ → ντ oscillations of atmospheric neutrinos are favored, there exist three–neutrino solutions with non–negligible νμ ↔ νe oscillations, even after applying the constraints from the CHOOZ reactor experiment [13]. Consequently νμ ↔ νe and νe → ντ oscillations may be observable in future long–baseline experiments. Oscillation probabilities. We begin our analysis with the survival probability for a given neutrino flavor in a vacuum [14] P (να → να) = 1− 4 ∑ k<j PαjPαk sin 2 ∆jk , (1) where Pαj ≡ |Uαj | 2 , (2) U is the neutrino mixing matrix (in the basis where the charged–lepton mass matrix is diagonal), ∆jk ≡ δm 2 jk L/4E = 1.27(δm 2 jk/eV )(L/km)/(E/GeV), δm2jk ≡ m 2 j −m 2 k, and the sum is over all j and k, subject to k < j. The matrix elements Uαj are the mixings between the flavor (α = e, μ, τ) and the mass (j = 1, 2, 3) eigenstates, and we assume without loss of generality that m1 < m2 < m3. The solar oscillations are driven by |∆21| ≡ ∆sun and the atmospheric oscillations are driven by |∆31| ≃ |∆32| ≡ ∆atm ≫ ∆sun. The off-diagonal vacuum oscillation probabilities of this three-neutrino model are P (νe → νμ) = 4Pe3Pμ3 sin 2 ∆atm − 4Re{Ue1U ∗ e2U ∗ μ1Uμ2} sin 2 ∆sun − 2 J sin 2∆sun , (3) P (νe → ντ ) = 4Pe3Pτ3 sin 2 ∆atm − 4Re{Ue1U ∗ e2U ∗ τ1Uτ2} sin 2 ∆sun + 2 J sin 2∆sun , (4) P (νμ → ντ ) = 4Pμ3Pτ3 sin 2 ∆atm − 4Re{Uμ1U ∗ μ2U ∗ τ1Uτ2} sin 2 ∆sun − 2 J sin 2∆sun , (5) where the CP–violating “Jarlskog invariant” [15] is J = ∑ k,γ ǫijkǫαβγIm{UαiU ∗ αjU ∗ βiUβj} for any α, β, i, and j (e.g., J = Im{Ue2U ∗ e3U ∗ μ2Uμ3} for α = e, β = μ, i = 2, and j = 3). The CP– odd term changes sign under reversal of the oscillating flavors. We note that the CP–violating