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Vacuum and MSW interpretations of solar neutrino data with the LNW mass matrix

The Lehmann-Newton-Wu mass matrix, which was recently applied to neutrinos, is further investigated. The analytic results presented earlier are confirmed numerically for the solar density profile of the Standard Solar Model. The combined analysis of atmospheric and solar neutrino data favors the MSW solution over the vacuum-oscillation solution. The total rates from the solar neutrino detectors and spectrum distortion reported by the Super-Kamiokande collaboration for solar neutrinos favors one heavy (m3\sim 0.05 eV) and two light (m1, m2\lsim 0.003 eV) neutrinos.


Introduction
Neutrino flavor transitions (e.g. ν e → ν µ ) are the most widely accepted explanation of the atmospheric neutrino anomaly and the solar neutrino problem (SNP) [1]. In the most popular scenarios for such flavor conversions, vacuum oscillations and matter enhanced transitions (MSW) [2], the neutrino conversion and survival probabilities depend on two types of neutrino-specific parameters. One of them is the mixing matrix U defined from ν α = U αi ν i , where α = e, µ, or τ and the sum runs over the mass states, ν 1 , ν 2 , ν 3 . The other is the differences of masses squared, ∆m 2 kj = m 2 k − m 2 j , where the latter refer to the neutrino mass states ν i . There is no widely accepted mass matrix, therefore the conventional way of analyzing the results from neutrino oscillation experiments is to consider the mixing matrix as independent of the mass-squared differences. But when we assume a certain mass matrix M in the Lagrangian (in a current basis) L =νMν + h.c., (1.1) then the mixing will explicitly depend on the masses through the diagonalization of M.
The maximal mixing arising from the democratic texture [3] is a widely discussed example of this kind (see also [4]). The mass matrix proposed by Lehmann, Newton and Wu (LNW) for quarks [5], was recently applied to neutrinos in [6], and gross features of the model were explored for solar and atmospheric neutrinos. We shall here examine the implications of that model in more detail, focusing on solar neutrinos, and compare the vacuum-oscillation scenario with the MSW scenario. Also, we will for the different cases discuss the energy spectrum of the recoil electrons induced by solar neutrinos. Constraints from atmospheric neutrino data will be taken into account.

The model
Let us briefly review the model being considered [5,6]. One assumes that there are three neutrinos with a real mass matrix of the form: where b 2 = 8c 2 . This symmetric mass matrix is diagonalized by the matrix U, where M d is a diagonal matrix whose entries are the mass eigenvalues. The diagonalization involves solving a cubic equation for a.
For an arbitrary set of mass eigenvalues, (m 1 , m 2 , m 3 ), the model yields two possible mixing matrices (two physical solutions of the cubic equation), referred to as Solution 1 and Solution 2 [6]. The corresponding mixing matrices U are displayed in Fig. 1 for m 2 ≤ m 3 (the physical region of the model extends a little beyond the triangle given by m 1 ≤ m 2 ≤ m 3 ). As illustrated in those figures, the mixing elements are determined by the two mass ratios m 1 /m 3 and m 2 /m 3 . The neutrino oscillation data give quite strong restrictions on both the mixing and the mass-squared differences, when treated as free parameters. Also, a model for the neutrino masses should not violate the experimental upper bounds on the neutrino masses [7]: m νe < ∼ 10 eV, m νµ < 0.17 MeV, m ντ < 18.2 MeV. Furthermore, it should satisfy the mass constraints from cosmology [8]: m ν f < ∼ 94 eV, be capable of explaining the apparent oscillation of atmospheric neutrinos, with ∆m 2 ∼ 10 −3 eV 2 , and give a plausible explanation of the SNP, with ∆m 2 ∼ 10 −10 eV 2 (vacuum oscillation) or ∆m 2 ∼ 10 −5 eV 2 (MSW). The natural scale implied by these values for the mass-squared differences is far too small to be in conflict with any of the above-mentioned mass bounds.
We denote the mass states in such a way that m 1 ≤ m 3 . Of the two independent ∆m 2 kj , one will be reserved for an explanation of the atmospheric neutrino anomaly, and the other will be assigned to a solution of the solar neutrino problem. Thus we are trying in one model to satisfy the two most serious neutrino problems. According to our notation, the smallest mass-squared difference will be either ∆m 2 21 or ∆m 2 32 . In the present model one can fit directly in terms of masses as opposed to mass-squared differences and mixing factors. When the values of the mixing elements are given by the masses, we do not expect as good a theoretical fit to the observed data as when these parameters are considered as independent. For given values of U and ∆m 2 kj the probability expression for flavor conversion will be the same whether the mixing matrix is dependent on the masses or not. Therefore we know to some extent from the traditional analyses where in the parameter space we can expect acceptable fits for the model we are analyzing. For example, for solar neutrinos it is known that both vacuum-and matter-influenced neutrino transitions require small U e3 in order to give good fits. From Fig. 1 we see that this is achieved when both m 1 /m 3 and m 2 /m 3 are small. And from the same figure it is apparent that in this region the mixing elements U e1 and U e2 are very sensitive to small changes in the masses. This means that by small changes in the mass ratios, the mixing elements can be adjusted close to the best-fit values obtained when U and the masses are considered as independent.

Detector
Observed   The rightmost column shows the fraction between observed and predicted fluxes.
In the analysis of the solar-neutrino data (summarized in Table 1) we will see that, for small values of m 1 and m 2 , the two mixing solutions give approximately the same results.
In the traditional analyses of the data from neutrino-oscillation experiments one often expresses the amount of mixing in terms of angles. We adopt the following, common parameterization of the mixing matrix [7]: where c 12 = cos θ 12 , s 12 = sin θ 12 , etc. This will be particularly useful when we discuss the MSW mechanism. If θ 13 = 0 we get the two-generation case.
For vacuum oscillations as well as for the MSW scenario, we shall discuss four cases, small ∆m 2 21 , small ∆m 2 32 , together with Solutions 1 and 2.

Atmospheric-neutrino constraints
We shall not here perform a detailed χ 2 fit to the atmospheric-neutrino data. Instead, we will use best-fit values for U and ∆m 2 kj from the literature as constraints and compare those with the acceptable regions we find for the solar neutrino data. The experimental data indicate that the "missing" muon neutrinos can be explained by a close to maximal ν µ ↔ ν τ oscillation [15], which requires small U e3 . Also, the result from the CHOOZ experiment [16], interpreted in terms of three generations and our choice for the ∆m 2 kj -values, indicates small U e3 . For masses where our model gives good fits to the solar neutrino data, we actually have U e3 ≪ 1. This criterion allows us to neglect matter effects for atmospheric neutrinos [17].
With three neutrino families the oscillation probability between arbitrary flavor states can be written as where R is the distance traveled since the neutrinos were created, and E is the neutrino energy.
For the case of atmospheric neutrinos, the factor R/E is quite small as compared with the case of solar neutrinos. Thus, the term in Eq. (3.1) with a small ∆m 2 kj (henceforth referred to as the SNP value, ∼ 10 −5 eV 2 or ∼ 10 −10 eV 2 ) can be neglected. When this small SNP term involves ∆m 2 21 we get where we use the approximation ∆m 2 31 ≃ ∆m 2 32 . From [15] we know that the best fit between data and the oscillation hypothesis occurs at ∆m 2 31 ∼ 3.2 × 10 −3 eV 2 . This means that the lowest possible value of the heaviest mass state is m 3 = O(0.05 eV).
In Fig. 2 we have plotted the amplitude in Eq. (3.2) for Solutions 1 and 2. One should keep in mind that the approximation leading to Eq. (3.2) is only satisfied close to the diagonal, m 1 = m 2 . For Solution 1, an unrealistically large m 2 /m 3 would be needed to get an acceptable ν µ − ν τ mixing. For Solution 2, we observe that large ν µ − ν τ mixing could be reached for values of ∆m 2 21 which are marginally compatible with the solar-neutrino constraints. We also note that the ∆m 2 21 which is required for the MSW effect will allow a larger amplitude for atmospheric-neutrino transitions than a mass-squared difference relevant for the vacuum oscillation solution to the SNP, since the MSW effect requires a larger m 2 .
When ∆m 2 32 assumes the SNP value, i.e., m 2 ≃ m 3 , then the atmospheric ν µ → ν τ transition probability can be written as P νµ↔ντ (R) ≃ 4U 2 µ1 U 2 τ 1 sin 2 (∆m 2 31 R/4E) instead of (3.2), where we have used the approximation ∆m 2 31 ≃ ∆m 2 21 . The maximal amplitude for this oscillation is about 0.5 for both mixing solutions. This value is too low to be of real interest.  In summary, the atmospheric-neutrino data favor parameters corresponding to Solution 2 with ∆m 2 21 ≪ ∆m 2 31 ≃ ∆m 2 32 . They also favor the MSW solution for the solarneutrino problem, since this mechanism requires a larger m 2 , and thus a larger amplitude for atmospheric-neutrino oscillations.

Vacuum oscillations
Vacuum oscillations provide a plausible solution to the solar-neutrino problem [18,19]. In this scenario, the length of the relevant oscillation cycle (see Eq. (3.1)) is about one Sun-Earth distance, i.e. ∼ 1.5×10 11 m. The corresponding ∆m 2 kj is several orders of magnitude smaller than the one which is relevant for atmospheric neutrinos. As mentioned, it will be either ∆m 2 21 or ∆m 2 32 which is relevant for solar neutrinos. Thus, the allowed values of ∆m 2 32 will be of the order relevant for either atmospheric or solar neutrinos. One necessary condition for acceptable fit between data and the vacuum oscillation hypothesis is that the smallest ∆m 2 kj ∼ 10 −10 eV 2 . The other independent mass-squared difference is held fixed at 3.2 × 10 −3 eV 2 , in order to be able to accommodate the atmospheric neutrino data. To find the optimal mixing matrix we have to search through a large number of values for the mass ratios m 1 /m 3 and m 2 /m 3 , which we do by varying m 1 and ∆m 2 21 .

Small ∆m 2 21
The interesting region is here close to the diagonal in the m 1 -m 2 plane (see Fig. 1). The best fit is achieved at m 1 = 3.1 × 10 −6 eV, ∆m 2 21 = 8.0 × 10 −11 eV 2 which gives χ 2 = 4.1. This χ 2 -value is about 0.1% higher than the value obtained in an analysis where U and ∆m 2 kj are treated as independent. This good agreement is explained by the high sensitivity of the mixing matrix to the mass values. Fig. 3 shows how the relevant mixing elements depend on m 1 when ∆m 2 21 and ∆m 32 are held fixed at 8.0×10 −11 eV 2 and 3.2×10 −3 eV 2 , respectively. As can be seen from this figure, the mixing solutions are equal for m 1 10 −3 eV. For Solution 1, U µ3 ≪ 1, which means no ν µ ↔ ν τ for atmospheric neutrinos. For Solution 2 this amplitude is ∼ 0.4. This value is unchanged over a broad range of mass values, including the best fit value for the solar neutrino solution. In Fig. 4 we have plotted the χ 2 values corresponding to Fig. 3. When m 1 is reduced beyond the best-fit value, the mixing matrix approaches the identity matrix, hence the fit becomes poor.

Small ∆m
In this case we can not get an equally good fit for solar neutrinos as we had for small ∆m 2 21 . It turns out that the best fit is achieved when we, in the expression (3.1), have vanishing amplitude for the oscillation factors containing the largest ∆m 2 kj [18]. This can not be achieved within the model, neither for Solution 1 nor for Solution 2, see Fig. 1. For Solution 1 the best fit gives χ 2 ≃ 6.1, while for Solution 2 one gets U e2 ≃ U e3 ≃ 1/ √ 2, with the best fit result χ 2 ≃ 5.0. Also, the relevant amplitude for atmospheric neutrino oscillations would be rather low in this case.

Summary on vacuum oscillations
If we consider only the SNP, none of the best fits can be excluded. The best-fit point is achieved with small ∆m 2 21 and at practically the same mass values for Solutions 1 and 2. Including atmospheric neutrino data, the preferred solution would be small ∆m 2 21 and Solution 2. But the maximum possible oscillation amplitude for ν µ ↔ ν τ is only 0.4, so the vacuum oscillation solution to the SNP is disfavored by the model in [6]. On the other hand, the recoil spectrum has an enhancement at high energies, as suggested by the data (see Fig. 7).

The MSW interpretation
For an exponential solar density, the three-flavor MSW equations have recently been solved analytically for arbitrary masses [20]. Here, since we assume two widely separated values of ∆m 2 kj , we use the more conventional approach [21]. This allows for a numerical comparison of the two approaches. The agreement was found to be very good. For example, for (m 1 , m 2 , m 3 ) = (0.002, 0.01, 0.057) eV and E = 1 MeV, the two approaches give ν e survival probabilities of 0.668 and 0.660. For some of the mass points we also check the exponential solar density approximation against the solar density profile calculated in a standard solar model [14,22], by performing numerical integrations on the latter. The agreement is within a few percent.
In the two-generation MSW case where the angle and mass-squared difference are varied independently, one arrives at two possible solutions in the parameter space spanned by sin 2 (2θ) and ∆m 2 . There is one region with low χ 2 at small angles, and another minimum at large angles [23]. As we will see, this is also the case for the model in [6].
For solar neutrinos, Solutions 1 and 2 are nearly identical for the masses relevant at the best-fit point. Therefore we will divide the presentation of our results into two parts, small ∆m 2 21 and small ∆m 2 32 .

Small ∆m 2 21
For small values of m 2 /m 3 and very small values of m 1 /m 3 we get U e1 ≃ 1 for both mixing solutions. These are practically equivalent for solar neutrinos, both of them have a best fit with χ 2 = 1.8 at m 1 = 2.8 × 10 −6 eV, ∆m 2 21 = 7.0 × 10 −6 eV 2 , where we have imposed ∆m 2 31 = 3.2 × 10 −3 eV 2 . The latter quantity is held fixed at this value, which is compatible with the atmospheric neutrino data [15]. Because it is large compared to the SNP-relevant ∆m 2 21 , its exact value is, within the uncertainty, unimportant for the solar-neutrino analysis. These mass values correspond to small mixing angle, see Eq. (2.3). With the above-mentioned values of the mass-squared differences held fixed, we show in Fig. 5 (left panel) how U e1 and χ 2 vary when m 1 is slightly perturbed around its best-fit value.
As previously noted, for atmospheric neutrinos we consider only vacuum oscillations, due to the smallness of U e3 . In the parameter region shown in Fig. 5 (left panel), U e3 < 10 −3 . From Eq. (3.2) it follows that the ν µ ↔ ν τ oscillation amplitude is 0.05 and 0.6 for Solutions 1 and 2, respectively. Thus, Solution 1 is unacceptable in this parameter region. We also note that the χ 2 value is quite sensitive to the mass values m 1 and m 2 .
In Fig. 5 (right panel) we show how the model compares with the data in the region known as the "large-mixing-angle solution". In this region, which requires somewhat larger values of m 1 , the relevant mixing elements are very sensitive to changes in the mass ratios  kinds of detectors and two parameters) and the global minimum has χ 2 min = 1.8, from the relation χ 2 = χ 2 min + 5.99, these contours corresponds to 95% C.L. In the same plot we also show the contours for atmospheric ν µ ↔ ν τ amplitudes equal to 0.7, 0.8 and 0.9. No point in this figure gives a mass-squared difference ∆m 2 31 outside of the 68% C.L. region given by [15].

Small ∆m
In this case the mixing elements are far less sensitive to small changes in the mass ratios. And for this reason the mixing elements are less adjustable within the relevant values of the mass-squared differences. For Solution 1 we get a best fit χ 2 ≃ 12. For Solution 2 it is unavoidable to get U e2 = U e3 = 1/ √ 2, see Fig. 1. The predicted fit to the observed values corresponds to χ 2 ≃ 15, which is quite poor. Small ∆m 2 32 is also unfavored in view of the atmospheric neutrino data.

Spectrum distortion
The Super-Kamiokande detector measures the recoil energy of the scattered electrons coming from the process ν + e − → ν + e − [24]. For each of the solar sources, the form of the neutrino energy spectrum is measured in laboratories, and is independent of the total flux. Thus, the Super-Kamiokande measurements of the upper part of the 8 B energy spectrum gives restrictions on the neutrino parameters, independent of solar models. If an energy dependent suppression of the flux should be established by such a method, it would be a very strong hint for neutrino oscillations because it would in principle not involve the uncertainties associated with solar models. However, the hep neutrino flux is poorly known [25]. This fact tends to reduce the significance of these measurements. In Fig. 7 we have plotted the ratio of the measured counting rate against the expectation in the absence of oscillations [24]. If there is no neutrino flavor transition, this ratio should be a straight horizontal line. However, it seems to increase for E 13 MeV. In the same figure we have tried to reproduce the observed spectrum within three scenarios: the vacuum, SMA, and LMA solution. These are actually all acceptable, but, as concerns the spectrum, the vacuum and SMA solutions are favored.

Conclusion
We have extended earlier work on the application of the Lehmann-Newton-Wu mass matrix [5] to neutrinos by investigating also the vacuum oscillation case, and by studying in more detail the MSW solutions. Two new fits to the solar neutrino counting rates have been found, beyond those of [6], one at m 1 ∼ 3 × 10 −6 eV, with m 2 ∼ 3 × 10 −3 eV and m 3 ∼ 0.05 eV, corresponding to a small mixing angle solution in the MSW interpretation, and a vacuum-oscillation solution with m 1 and m 2 of the order 10 −6 -10 −5 eV. Both of these  [24,14].
solutions lead to electron energy recoil spectra that are in better agreement with the data than the LMA solutions, but they are only in marginal agreement with the atmospheric neutrino data. For both the vacuum and MSW solutions of the solar neutrino problem, only hierarchical mass values gives viable fits. In both cases, best fits are obtained when the largest mass value is m 3 ≃ 0.05. When atmospheric neutrino constraints are taken into account, only the large-angle part of the MSW effect gives an acceptable fit to the observed data. The result from the combined analyses of the atmospheric and solar neutrino data will be further constrained with the forthcoming long-baseline experiments and SNO data [26].