Bi - Maximal Mixing of Three Neutrinos

We show that if the solar and atmospheric data are both described by maximal vacuum oscillations at the relevant mass scales then there exists a unique mixing matrix for three neutrino flavors. The solution necessarily conserves CP and automatically implies that there is no disappearance of atmospheric νe, consistent with indications from the Super-Kamiokande experiment. We also investigate the consequences for three-neutrino mixing if the solar and atmospheric oscillations exhibit mixing that is large but not maximal. For non-maximal mixing νe ↔ ντ and νe ↔ νμ oscillations are predicted that may be observable in future long-baseline experiments. Introduction. It is now understood that neutrino oscillations can describe the data on the solar neutrino deficit [1, 2, 3], the atmospheric neutrino anomaly [4, 5, 6], and the results from the LSND experiment [7], only if a sterile neutrino is introduced [8, 9, 10, 11, 12]. However, because confirmation of the LSND results awaits future experiments [13], a conservative approach is to assume that oscillations need only account for the solar and atmospheric data; also, it is possible to introduce new lepton-flavor changing operators with coefficients small enough to evade present exclusion limits, but large enough to explain the small LSND amplitude [14]. Since a comparison of the theoretical expectations [15] with the SuperKamiokande (SuperK) atmospheric data suggests maximal νμ oscillations with mass-squared difference δm2atm ≈ 5× 10−3 eV [5], and in light of the recent data from Super-Kamiokande [3] favoring vacuum long-wavelength oscillations [16, 17, 18] with δm2sun ≈ 10−10 eV and large mixing as a description of the solar neutrino deficit, it is instructive to determine how maximal or near-maximal solar and atmospheric vacuum oscillations may be accommodated within a three-neutrino universe. In this letter we use unitarity constraints to derive conditions on the three-neutrino mixing matrix under the assumption that both solar and atmospheric neutrinos undergo maximal mixing in vacuum. By maximal mixing we mean that the disappearance probabilities are equivalent to those for maximal two-neutrino mixing at the relevant mass scales. We find that there exists a unique three-neutrino mixing solution, up to trivial sign ambiguities, which conserves CP and has no oscillations of atmospheric νe, consistent with indications from the Super-Kamiokande experiment [5]. This mixing matrix predicts that solar νe oscillate maximally into equal numbers of νμ and ντ . We also investigate the consequences for threeneutrino mixing if the solar and atmospheric oscillations are not maximal. We find that for near-maximal solar and atmospheric neutrino mixing the νμ → νe and νe → ντ oscillation amplitudes for atmospheric and long-baseline experiments are approximately equal, and are limited by the deviation of the solar νe disappearance amplitude from unity. Also, for any non-maximal solar and atmospheric neutrino mixing there must exist νe → ντ oscillations that may be visible in future long-baseline experiments. Oscillation probabilities. We begin our analysis with the disappearance probability for neutrino oscillations in a vacuum [19] P (να → να) = 1− 4 ∑ k<j PαjPαk sin 2 ∆jk , (1) where Pαj ≡ |Uαj |2 is the probability to find the α-flavor state in the jth mass state (or vice versa), U is the neutrino mixing matrix (in the basis where the charged-lepton mass matrix is diagonal), ∆jk ≡ δm2jk L/4E = 1.27(δm2jk/eV2)(L/km)/(E/GeV), δm2jk ≡ m2j −mk, 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. We assume a neutrino mass spectrum with mass eigenvalues m1, m2 ≪ m3, where the solar oscillations are driven by ∆21 ≡ ∆sun and the atmospheric oscillations are driven by ∆31 ≃ ∆32 ≡ ∆atm. However, only the form of the neutrino mass matrix depends on the mass hierarchy assumption. All of our other conclusions require only the more general assumption that |∆sun| = |∆21| ≪ |∆31| ≃ |∆32| = |∆atm|, which also could apply, e.g., to the case m3 < m1 ≃ m2, i.e., the solar neutrino vacuum oscillations are driven by a small mass splitting between the two larger masses. 2 The off-diagonal oscillation probabilites of this model are P (νe → νμ) = 4Pe3Pμ3 sin ∆atm − 4Re{Ue1U e2U μ1Uμ2} sin ∆sun − 2 J sin 2∆sun , (2) P (νe → ντ ) = 4Pe3Pτ3 sin ∆atm − 4Re{Ue1U e2U τ1Uτ2} sin ∆sun + 2 J sin 2∆sun , (3) P (νμ → ντ ) = 4Pμ3Pτ3 sin ∆atm − 4Re{Uμ1U μ2U τ1Uτ2} sin ∆sun − 2 J sin 2∆sun , (4) where the CP -violating “Jarlskog invariant” [20] is J = Im{Ue2U e3U μ2Uμ3}. The CP -odd term changes sign under reversal of the oscillating flavors. The solar amplitudes, quartic in the Uαj ’s, may themselves be expressed in terms of the Pαj [21, 22], as may the CP -odd amplitude J [22, 23]. We note that the CP -violating probability at the atmospheric scale is suppressed to order δm2sun/δm 2 atm, the leading term having canceled in the sum over the two light-mass states. Thus, P (να → νβ) = P (νβ → να) at the atmospheric scale. With MSW-enhanced solutions to the solar deficit, sin 2∆sun is effectively replaced by its average value of zero since ∆sun ≫ 1 in this case. Only for the vacuum long-wavelength solution to the solar neutrino anomaly is there hope to measure the CP -odd term. We will use “amplitude” to denote the coefficients of the oscillating factors. The amplitudes of interest are: A 6e sun = 4Pe1Pe2 ≤ (1− Pe3)2 , (5) Aμ6μ atm = 4Pμ3(Pμ1 + Pμ2) = 4Pμ3(1− Pμ3) , (6) A 6e atm = 4Pe3(1− Pe3) , (7) A atm = 4Pe3Pμ3 ≤ (1− Pτ3)2 , (8) A atm = 4Pμ3Pτ3 ≤ (1− Pe3)2 , (9) A atm = 4Pe3Pτ3 ≤ (1− Pμ3)2 . (10) In a three-neutrino model Aμ6μ atm = A μe atm + A μτ atm , and (11) A 6e atm = A μe atm + A eτ atm . (12) The equality in Eq. (5) results from the unitarity condition U∗ μ1Uμ2 + U ∗ τ1Uτ2 = −U∗ e1Ue2. The inequalities in Eqs. (5), (8)–(10) are obtained by maximizing the amplitude under the unitarity constraints Pe3 + Pμ3 + Pτ3 = 1 , and (13) Pe1 + Pe2 + Pe3 = 1 . (14) Thus, the amplitudes depend on just three independent Pαj ’s, and the atmospheric amplitudes depend on just two of the probabilities chosen from Pe3, Pμ3, and Pτ3. Maximal solar and atmospheric oscillations. We first explore a solution for which both solar νe and atmospheric νμ data can be described by maximal oscillations at the relevant δm scales, as suggested by the SuperK data [3, 5]. Maximal atmospheric oscillations require