{'fraction_non_alphanumeric': 0.037182695745441544, 'fraction_numerical': 0.025622690978429268, 'mean_word_length': 4.022142429682825, 'pattern_counts': {'":': 0, '<': 9, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Possible Revelation of Seesaw Mass Pattern in Solar and Atmospheric Neutrino Data

Assuming the solar and atmospheric neutrino deficits to be due to neutrino oscillations, it is shown that the 3X3 mass matrix spanning the e, mu, and tau neutrinos may have already revealed a seesaw mass pattern. Also, this matrix is the natural reduction of a simple 5X5 seesaw mass matrix with one large scale, the 4X4 reduction of which predicts that a fourth neutrino would mix with the e and mu neutrinos in such a way that oscillations between them may occur just within the detection capability of the LSND (Liquid Scintillator Neutrino Detector) experiment.

Whether or not neutrinos have mass is clearly a fundamental issue in particle physics and astrophysics. There is now a good deal of indirect evidence from measurements of the solar ν e flux [1,2,3,4] and the atmospheric ν µ /ν e ratio [5,6] that neutrinos oscillate from one to another. This means that ν e and ν µ are not mass eigenstates; each is rather an admixture of two or more mass eigenstates. Under the simplifying assumption that only two neutrinos are involved in each case, probable central values for the mass-squared difference and the mixing angle are ∆m 2 ≃ 4 × 10 −6 eV 2 , sin 2 2θ ≃ 0.01, (1) from analyzing [7] solar data, assuming the small-angle, nonadiabatic solution, and ∆m 2 ≃ 10 −2 eV 2 , sin 2 2θ ≃ 0.5, from analyzing [8] atmospheric data.
Given only the above information, it is clearly not possible to reconstruct unambiguously the underlying 3 × 3 neutrino mass matrix. However, if a certain empirical relationship can be found among the four parameters listed above, an important insight may be gained as to the form of this mass matrix. Such an example is already very well-known in the case of the quark mass matrix. It was pointed out many years ago [9] that the empirical relationship may be obtained with a 2 × 2 mass matrix of the form This simple observation has generated over the years an enormous literature on quark mass matrices. It is an especially active field of research in the past two or three years. In the jargon of neutrino physics, the form of M in Eq. (4) is called seesaw [10].
We propose here that Eq. (4) is also applicable to the 2 × 2 matrix spanning ν τ and the linear combination ν µ cos θ + ν e sin θ, where θ is the angle measured in atmospheric neutrino oscillations. The orthogonal linear combination ν e cos θ − ν µ sin θ is assumed to be massless.
Since ∆m 2 12 << ∆m 2 13 and a/b << 1, the atmospheric neutrino deficit [5,6] is explained by a simple two-neutrino oscillation between ν µ and ν e . However the solar neutrino deficit comes from the oscillations of all three neutrinos. If matter effects [11] could be neglected, the disappearance probability of solar neutrinos would be given by where U αi are the elements of the mixing matrix in Eq. (6). We see that the effects of the heaviest mass parameter average out, but a dependence on |U e3 | 2 remains. Only in the limit of vanishing |U e3 | 2 = sin 2 θ is the two-neutrino approximation valid for solar neutrinos. This general observation is also true when matter effects are included [12]. Since a large sin 2 θ is required by the atmospheric data, the naive comparison of Eq. (7) with Eq. (1) will have to be modified.
A three-neutrino analysis of all the solar neutrino data [1,2,3,4] is shown in Figs.
(1a), (1b), and (1c). The method is the same as that of an earlier analysis [13] where a continuous range of |U e3 | 2 values is considered. Three values of ∆m 2 13 and sin 2 2θ which fit the atmospheric neutrino data have been chosen, and their predictions for ∆m 2 12 and (5) and (6) are satisfied in different ways across the parameter region indicated by the atmospheric data. Although we do not include the region allowed by the new Kamiokande multi-GeV data [6] which is in conflict with Frejus [14], solutions exist there as well.
Consider now the possible origin of the 3 × 3 neutrino mass matrix M ν of Eq. (5). The most natural way is to find it as the seesaw [10] reduction of a larger matrix which includes a higher mass scale. A very simple solution is the following 5 × 5 mass matrix spanning ν e , ν µ , ν τ , ν S , and N, where the last two are singlet Majorana neutrinos. In the limit of very large m 5 , the seesaw reduction of M 5 is Assume now that m 2 4 /m 5 is the dominant term in M 4 , then a second seesaw reduction gives the following 3 × 3 matrix: With the further requirement that m 3 m 4 << m 5 m 2 1 + m 2 2 so that a << b, we then obtain Eq. (6). Since we predict that there should be additional oscillations between ν e and ν µ through this fourth neutrino mass eigenstate with restricted values of ∆m 2 and sin 2 2θ such that their product is the same as that for atmospheric oscillations. Using the numerical results obtained earlier in this paper, we would then predict Part of this region is excluded by the E776 neutrino experiment at Brookhaven National Laboratory [16], but another part (∆m 2 ∼ 1 eV 2 , sin 2 2θ ∼ 5 × 10 −3 ) lies within the detection capability of the LSND (Liquid Scintillator Neutrino Detector) experiment at Los Alamos National Laboratory. We also note that the upper bound in the above comes from reactor experiments [15]. Relaxing it would allow a larger ∆m 2 (∼ 6 eV 2 ) which is preferred by the preliminary LSND results, but in conflict with the published E776 data.
To obtain M 5 , the following softly broken discrete Z 3 symmetry may be considered. Let ν e , ν µ , ν τ , ν S , and N transform as ω, ω, 1, ω 2 , and 1 respectively, where ω 3 = 1. Then m 1 , m 2 , and m 3 come from the vacuum expectation value of the standard Higgs doublet, and m 5 is allowed by Z 3 . However, m 4 breaks Z 3 softly and so does the diagonal mass term for ν S , but as long as the latter is much smaller than m 2 4 /m 5 , the reduction to M ν proceeds as before. With four light neutrinos, the nucleosynthesis bound [17] of N ν < 3.3 is an important constraint. Although ν S is a singlet neutrino, it mixes with other neutrinos and may contribute significantly to N ν through oscillations [18]. In the context of M 4 , this constraint implies that m 2 1 + m 2 2 has to be very much smaller than m 2 4 /m 5 , hence the sin 2 2θ parameter in Eq. (14) is too small to be in the range of the LSND detection capability. In other words, the simplest extension of our basic proposal to four neutrinos cannot accommodate simultaneously the LSND results and nucleosynthesis. The latter constraint can be avoided if we switch ν τ and ν S in M 5 , but then the underlying theory becomes much more involved.
For example, the ν e ν τ and ν µ ν τ mass terms must now come from either a Higgs triplet or a radiative mechanism as recently proposed [19]. Details will be given elsewhere.
In conclusion, we have shown in the above how present data on solar and atmospheric neutrinos may have revealed a seesaw mass pattern for the three known neutrinos. This seesaw scenario requires the solar neutrino deficit to involve the oscillations of all three neutrinos. Hence a much more extensive numerical analysis is needed [13] beyond that of the usual two-neutrino assumption [7]. It is interesting to note that with only two neutrinos, the adiabatic solution is not allowed by the combined solar data, but with three neutrinos, it becomes allowed if the mixing of the third neutrino with ν e is large enough.
We have also shown that a simple 5×5 mass matrix with one large scale reduces naturally    2 12 versus sin 2 2θ e2 ≡ 4|U e2 | 2 (1 − |U e2 | 2 ) for various values of sin 2 2θ. The solid contours surround the parameter region allowed by the solar neutrino data [1,2,3,4] at 90% confidence level. The dashed line gives the prediction of Eq. (5) for a specific value of ∆m 2 13 which explains the atmospheric neutrino deficit.  versus sin 2 2θ. The dashed contours surround the parameter region allowed by sub-GeV atmospheric neutrino measurements [6] but excluded by the Frejus data [14] and below constraints from reactor measurements [15]. The shaded regions are where Eq. (5) can satisfy the solar neutrino data; the left shaded region corresponds to the small-angle nonadiabatic solution and the right shaded region corresponds to the adiabatic solution. In between these two regions is a small area where it is not possible to satisfy the solar neutrino data.