L/E-flatness of the electron-like event ratio, in Super-Kamiokande and a degeneracy, in neutrino masses

doi:10.1016/S0370-2693(99)00811-4

Physics Letters B

Volume 460, Issues 3–4, 12 August 1999, Pages 431-436

https://doi.org/10.1016/S0370-2693(99)00811-4 Get rights and content

Abstract

We show that the L/E-flatness of the electron-like event ratio in the Super-Kamiokande atmospheric neutrino data implies the equality of the expectation values for the muon and tau neutrino masses. We establish this result by obtaining a set of three constraints on the neutrino-oscillation mixing matrix as contained in the indicated flatness. The resulting 3×3 neutrino-oscillation matrix depends only on one angle. A remarkable result that follows directly from this matrix is the consistency between the mixing angles observed by LSND and Super-Kamiokande.

Introduction

One of the most remarkable features of the latest Super-Kamiokande data on the atmospheric neutrino anomaly is the L/E-flatness of the electron-like event ratio [1]. Given the definitions, $R_{e} ≡ Experimentally observed e-like events Theoretically expected e-like events (without ν oscillations),$ $R_{μ} ≡ Experimentally observed μ-like events Theoretically expected μ-like events (without ν oscillations),$ the Super-Kamiokande data reveals a significant L/E dependence for $R_{μ}$ , while $R_{e}$ is consistent with unity with no L/E dependence. In this Letter, following an earlier study [2], we establish that this L/E flatness of $R_{e}$ , consistent with $R_{e} =1$ , constrains the neutrino mixing matrix dramatically. All resulting neutrino-oscillation matrices are equivalent to each other, under appropriate redefinitions of the underlying mass eigenstates. They all carry the property that the expectation values of the muon and tau neutrino masses be equal. Moreover, the mixing matrices derived in this Letter naturally lead to a consistency between the mixing angles observed by LSND and Super-Kamiokande. This circumstance makes us suspect that the solar neutrino anomaly points towards a richer phenomenology, perhaps beyond neutrino masses, in the physics of neutrino oscillations.

Section snippets

Constraints on the neutrino-oscillation matrices

Let us assume that at the top of the atmosphere, at t=0, the number of ν_e and ν_μ produced is N_e and N_μ, respectively. Although both neutrinos and antineutrinos are produced in both flavours, we shall use the terms “electron neutrinos” and “muon neutrinos” loosely, to include both the ν and $ν ̄$ . In general, the ratio of ν_μ to ν_e neutrinos, $r= N_{μ} N_{e}$ is a function of energy. However, for the relevant energy range in Super-Kamiokande, it may be assumed constant (as shall be done in this Letter). Within

The resulting neutrino-oscillation mixing matrices

One possible class of solutions to the system of equations above is given by requiring that the expressions in the square brackets in , , be zero simultaneously. However, this leads to a rather uninteresting class of mixing matrices, namely the unit matrix and others that are basically equivalent to it, under appropriate redefinitions of the mass eigenstates. Hence, the problem of the L/E flatness in $R_{e}$ is trivially solved, but there would be no neutrino oscillations either. This would

The degenerate muon and tau neutrino masses

Referring to the obtained neutrino-oscillation matrices U above, and taking note of the definition for the expectation value of the neutrino masses (recall that for δ=0 all the U_ℓj elements are real) $〈m(ν_{ℓ})〉≡ ∑ j U^{2}_{ℓj} m_{j},$ we immediately come to the general conclusion on the mass degeneracy of the muon and tau neutrinos: $〈m(ν_{μ})〉=〈m(ν_{τ})〉.$ For the three cases enumerated above, one readily sees that the “degenerate mass” carries the values $12 m_{1} +s_{β}^{2} m_{2} +c_{β}^{2} m_{3},$ $12 s_{β}^{2} m_{1} +m_{2} +c_{β}^{2} m_{3},$ $12 s_{θ}^{2} m_{1} +c_{θ}^{2} m_{2} +m_{3},$ respectively. The

Implications for the muon-like event ratio: LSND versus Super-Kamiokande

Apart from the mass degeneracy in the muon and tau neutrino masses, a remarkable result that follows directly from the derived neutrino-oscillation matrices is the consistency between the mixing angles observed by LSND and Super-Kamiokande. We briefly discuss this in the following.

The number of muon neutrinos at the Super-Kamiokande detector, N_μ′, is given by $N_{μ} ′=N_{μ} P_{μμ} +N_{e} P_{eμ},$ and thus, the muon-like event ratio reads $R_{μ} = N_{μ} ′ N_{μ} =P_{μμ} + 1 r P_{eμ} .$ Without loss of generality, let us consider the mixing matrix

The solar neutrino deficit

If the Super-Kamiokande/LSND consistency is firmly established by future experiments, then the physics of neutrino oscillations shall be found not only to contain massive neutrinos, but may also point towards new physics. This arises from the long-standing solar neutrino deficit, as measured by a variety of experiments, with different sensitivities and detection techniques 7, 8, 9, 10, 11. Within the framework of neutrino oscillations, the ratio of measured to predicted solar neutrinos, R_e, is

Conclusions

We conclude that L/E flatness of the electron-like event ratio in the Super-Kamiokande data on atmospheric neutrinos implies a mass degeneracy for the muon and tau neutrino. The obtained results support recent considerations on maximal mixing, bi-maximal mixing, and the degenerate neutrino masses 17, 18, 19, 20, 21. More precise data on the discussed L/E flatness would be most helpful to settle the insights gained in this Letter. If the L/E flatness of $R_{e}$ is firmly established in the future by

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