Can three-flavor oscillations solve the solar neutrino problem?
Abstract
The good agreement of standard solar models with helioseismology and the combined analysis of the solar neutrino experiments suggest that the solution to the solar neutrino problem is located in particle physics rather than in astrophysics. The most promising solution are neutrino oscillations, which usually are analyzed within the reduced 2-flavor scheme, because the solutions found therein reasonably well reproduce the recent data of Super-Kamiokande about the recoil-electron energy spectrum, zenith-angle and seasonal variations, and the event rate data of all the neutrino detectors. In this work, however, a survey of the complete parameter space of 3-flavor oscillations is performed. Basically eight new additional solutions could be identified, where the best one with , , , and (denoted SVO) is slightly more probable than any 2-flavor solution. While the 2-flavor results of the atmospheric neutrino problem () would exclude all the 3-flavor solutions of this work, in the 3-flavor atmospheric-neutrino analysis the SLMA-solution with , , , and is still allowed. The relatively weak improvement of the fit using 3-flavor instead of 2-flavor oscillations, which appears to be due to an inconsistency of the different kind of data, indicates that there are possibly still systematic errors in at least one data set, or that the statistics is not yet sufficient. Besides, the ability of SNO and Borexino to discriminate the various 2- and 3-flavor solutions is investigated. Only with very good statistics in these experiments the correct solution to the solar neutrino problem can be identified unambiguously.
pacs:
PACS numbers: 26.65.+t, 14.60.Pq, 96.60.Jw[
]
I Introduction
From the beginning of the first measurements of the solar neutrino flux on Earth [1] until the present time, the origin of the solar neutrino problem could not yet be resolved totally. While previously inaccurate or unknown physics used in solar-model calculations could have been made responsible for the discrepancy between measured and predicted solar neutrino flux, this kind of solution can presently almost be ruled out, basically of two reasons:
Firstly, the high precision in the measurements of the solar p-mode frequencies and the development of helioseismological inversion techniques enable the determination of the solar sound-speed profile with high accuracy [2]. The comparison with standard solar models containing improved input physics like opacity, equation of state and microscopic diffusion shows an excellent agreement with the seismic models [3, 4, 5]. Predictions for the event rates in the solar neutrino detectors deduced from these standard solar models still are inconsistent with the measurements, confirming the solar neutrino problem (Table 2).
Secondly, the three types of currently operating experiments, the chlorine detector [1], the gallium experiments GALLEX/GNO[6] and SAGE [7], and the Čerenkov-light counter Super-Kamiokande have different neutrino-energy thresholds. This allows to determine the contribution of different parts of the solar neutrino spectrum to the total flux without explicitly taking into account solar-model calculations. From this analysis it has been inferred that the experimental results can be explained only with huge changes in the nuclear fusion rates. The best fit with the data is obtained even with a negative flux of neutrinos created in the electron-capture process of Be [8]. Nevertheless, strong modifications of the reaction cross sections would be difficult to explain experimentally and theoretically. Moreover, even if a presently unknown physical process can account for the demanded changes, the resulting solar models would hardly be consistent with helioseismology [9].
The most promising approach to the solution of the solar neutrino problem is an extension to the particle-physics standard model — neutrino mixing. Analogous to the CKM-mixing in the quark sector, weak and mass eigenstates of the neutrinos are supposed not to be identical but connected by a unitary transformation.
Under this assumption an initial solar electron-neutrino can be converted during its propagation to the Earth into another flavor, a - or -neutrino (just-so oscillations [10]). Furthermore, the neutrinos may coherently scatter forward in solar matter (MSW-effect [11]) altering the conversion probability for a certain set of mixing parameters.
The possible values for the mixing parameters, with which the measured event rates in all detectors can be reproduced simultaneously, have been derived by various authors [12, 13], but only the oscillations between two flavors usually is taken into account.
Recently Super-Kamiokande has published more detailed information about the energy distribution of the recoiled electrons and the zenith-angle dependence of the neutrino-signal [14, 15]. In the analysis of the 825-days data it became clear, that it is not possible to explain satisfactorily these data and the event rates of all detectors by one set of mixing parameters (see [16]): An excess of event rates in the high-energy bins was inconsistent with the other data.
It was the initial motivation of the present work to examine, whether an expansion of the neutrino analysis to the more general 3-flavor case could resolve this discrepancy between the different types of data. However, after the Super-Kamiokande group reanalyzed their data and included new data (1117 days in total), in particular the excess in the energy-bin data could be diminished and now all kind of data can be explained simultaneously by 2-flavor oscillations. Furthermore, the neutrino spectrum in the B-decay has been measured recently in the laboratory [17]. Although within the errors the spectrum is in agreement with the one predicted theoretically in [18], the number of high-energetic neutrinos is overall higher than previously thought. Thus, the excess in the high-energy bins is further reduced which yields a yet slightly better reproduction of the data by the 2-flavor solutions (see section IV.1). Nevertheless, it is presently still not clear, which solution to the solar neutrino problem is the correct one, and hence all possible solution should be deduced.
Therefore, in this work the most general case of 3-flavor oscillations is investigated without making any assumptions about the mass scale as, for instance, inspired by the atmospheric neutrino problem. The latter is taken in various publications [20, 21, 22] as a constraint to investigate 3-flavor oscillations. But here, the aim is to examine whether the expansion to three flavors leads to new solutions with which the fits to all kind of data can be improved compared to the usual 2-flavor analysis. The implications for the atmospheric neutrino puzzle are discussed afterwards.
In section II the equations for 3-flavor oscillations are derived from the solution to the Klein-Gordon equation and the size of the parameter space is deduced. After describing the underlying solar model and the neutrino analysis in section III, the results for 2-flavor and 3-flavor oscillations are shown (section IV). Finally the ability of forthcoming experiments like SNO and Borexino to discriminate the various solutions are discussed (section V).
Ii Theory of neutrino oscillations
In the following an overview of the basic equations for neutrino oscillations are provided with particular emphasis to the 3-flavor case. A more thorough description can be found e.g. in [23] or [9].
ii.1 Vacuum oscillations
If neutrinos have mass, a mixing matrix similar to the Kobayashi-Maskawa matrix in the quark sector can be developed
(1) |
where denotes the weak and the mass eigenstates. The unitary matrix can be parameterized by
(2) | |||||
where and are abbreviations for resp. () and is a CP-violating phase [24], which is neglected in the following^{2}^{2}2The effect of the CP-violating phase on the analysis of neutrino oscillation data has been elaborated in [25]. The equation of motion for a neutrino beam in vacuum obeys the Klein-Gordon equation for free particles ()
(3) |
where the mass matrix is defined as
with being the mass of the neutrino mass eigenstate . Generally the solution is given by a superposition of plane waves with the dispersion relation
In the case of the Sun with an almost stationary neutrino flux, can be expanded in components of fixed frequency . For a spherically symmetric flux of relativistic neutrinos () one finally gets
(4) |
The constant “potential” can be removed by shifting the energy scale, and by using Eq. (4) can be formally written as a more familiar Schrödinger-type equation
(5) |
with . The general solution to this equation is
(6) |
The probability to detect a neutrino with energy as a neutrino of type at distance from the source is therefore given by ()
(7) |
Using the matrix writes as
With the mass eigenstate of an electron-neutrino being
(8) |
Eq. (7) yields for the survival probability of ’s in vacuum
(9) | |||||
The superscript denotes the case of 3-flavor mixing. depends on four quantities, two mass-squared differences, and (), and two mixing angles and . The third mixing angle does not appear in Eq. (8) and hence is independent of this quantity. The survival probabilities for and depend on , but its value cannot be determined by solar-neutrino experiments, as in the energy range of the solar neutrinos and interact equally with the detector material via NC-interactions. Thus, only the total number of - plus -neutrinos, given by 1, influences the event rates in detectors like Super-Kamiokande, SNO or Borexino.
For oscillations between two neutrino flavors where no mixing into the third flavor occurs (e.g. , ), Eq. (9) simplifies to the well known formula
(10) |
where or for -– resp. -–oscillations ( defined analogously).
ii.2 Matter effect
During the propagation of the neutrinos through the Sun they coherently scatter forward on the particles of the solar plasma. Unlike - and -neutrinos, which only interact via NC-reactions, electron-neutrinos can additionally couple via -bosons to the electrons. Thus, the scattering cross section of a is altered as against the one of the other two neutrino-flavors. This can lead to a resonant flavor transition, which may create a pure - or -beam from the originally created electron-neutrinos, first theoretically postulated and described in [11] (MSW-effect).
This effect can be included in Eq. (5) by substituting with = , where
influences solely the -contribution of the neutrino beam. is the Fermi coupling-constant and the electron number density. The new Hamiltonian is no longer diagonal in the mass basis. To evaluate the survival probability and thus it is therefore necessary to diagonalize by a unitary transformation
Similar to the vacuum oscillations the constant phase can be removed from as it does not change the survival probability. The complicated analytical expressions for have been evaluated in [26]. Recently, has been calculated in [27] by using Cayley-Hamilton’s theorem without explicitly deriving . In the present work, however, and are computed with a fast numerical algorithm using none of the analytic expressions.
ii.3 The parameter space
Recently various publications came up about the actual size of the necessary parameter space covering all possible solutions of the solar neutrino problem[28, 29, 30]. In this section the sometimes confusing statements about this topic are summarized and clarified.
The mixing angles can be defined to lie in the first quadrant by appropriately adjusting the neutrino field phases similar as in the quark sector [31]. This can also be verified from the final formulae, as, for instance, depends solely on the square of resp. (). Moreover, it has been shown in [28] that is is also sufficient to consider , if resp. are to be measured. In the case of the matter-enhanced oscillations the situation is not as trivial, but using an analytical formula for the evolution of a neutrino state in matter as derived in [27] (Eq. 44 therein), one can also show that the evolution of an initial electron-neutrino is determined only by the squares of and .
ii.3.1 Two flavors
First, the case of 2-flavor oscillations is examined. Exchanging the first and second row in the definition of the mass eigenstates (Eq. 1) implies that and ()^{3}^{3}3The indices 12 or 13 of the mass-squared difference resp. mixing angle are omitted in the 2-flavor case.. Since the assignment of the masses to the respective mass eigenstate must not change the results, e.g. the case , is equivalent to , for any possible form of the Hamiltonian. Thus, without loss of generality can be chosen to be within .
For pure vacuum oscillations Eq. (10) can be applied which yields that is independent of the sign of and under the transformation . Hence, in this case it is even sufficient to consider .
The situation is different for oscillations in matter, where the resonance condition for MSW-transition is given by
(11) |
A resonance may occur, if is positive, thus (). Although no resonance occurs for negative values of , the matter still influences the evolution of the neutrino flavor composition in the solar interior (for a more detailed study see e.g. [26]).
In Fig. 1 the effect of solar matter on the energy dependence of is demonstrated. For vacuum oscillations would yield an energy-independent survival probability for electron-neutrinos, as the vacuum oscillation length is small and therefore only a mean value of (Eq. 10)
(12) |
would be measurable on Earth. For this leads to . In contrast to the case with (solid line), where the resonant flavor conversion diminishes the -contribution of the solar neutrino current, for (dashed line) the -portion is even enhanced compared to the pure vacuum-oscillation case. Thus, the minimum value of the -contribution for is obtained for pure vacuum oscillations. However, since vacuum oscillations in this mass range yield at most a suppression of the -flux of 50% (Eq. 12) and the solution to the solar neutrino problem demands, at least to explain the Homestake experiment, a stronger suppression of about 60%, for no reasonable solution can be obtained. Hence it is sufficient to consider in the 2-flavor MSW case similar to the 2-flavor vacuum oscillations solely ().
Recently, it has been pointed out in [30, 32] that the -ranges of the LMA- and LOW-solutions (see below) extend outside this region, but additional solutions for the solar neutrino problems cannot be found there. In their analysis, however, was fixed to be positive and thus has been examined. The region of their parameter space (termed “the dark side”) is equivalent to the region and discussed above.
ii.3.2 Three flavors
The considerations of the 2-flavor case are now extended to three flavors. In Fig. 2 the flavor space for the electron-neutrino is illustrated, where is represented as a (yet unknown) point on the surface of an eight of an unit sphere. While in the 2-flavor case the ordering of the masses enables two cases to be distinguished ( resp. ), in the 3-flavor scenario six cases can be identified. But, since exchanging any axes in Fig. 2 maps the flavor space onto itself, each mass hierarchy can of course be obtained from the “canonical” one () simply by exchanging the respective assignment of to in Eq. (1).
Unlike the 2-flavor case, where for pure vacuum oscillations the parameter space could be further decreased, it is not possible in the general 3-flavor scenario, as e.g. , is not invariant under the transformation .
For 3-flavor neutrino oscillations in matter an exact analytic resonance condition can hardly be obtained because of the complicated formula for the mass eigenvalues . If the masses are well separated, the 2-flavor resonance condition (Eq. 11) can be applied for both systems by substituting the quantities () by () resp. () [34]. Hence, in this case considering (sector I) extended to the dashed borders in sector III of Fig. 2) with is sufficient to obtain all possible solutions where two resonances may occur. However, the solution to the solar neutrino problem may also be a combination of a non-resonant and a resonant oscillation and the masses can be in principle very similar, too. Thus, the whole parameter space must be taken into account.
In [20], for instance, 3-flavor oscillations were investigated assuming canonical mass hierarchy and using a fixed value for in agreement with the atmospheric-neutrino results. They examined the remaining parameter space applying an analytical formula for the survival probability, which approximates the solar electron-density profile by an exponential function and is valid for and significantly smaller than . Under these conditions the parameter space could be reduced considerably.
In the present work, the most general case of 3-flavor oscillations of solar neutrinos is investigated, and thus these restrictions are not applicable here. Instead of performing a survey over the whole flavor space () with canonical mass hierarchy, I prefer to consider the case which covers two possible mass hierarchies. Thus, in this case only half of the total flavor space must be overviewed. In the illustration provided in Fig. 2 this reduced area is given by sector II and the lower half of sector I defined by the dotted line.
To simplify the numerical survey and the analysis a variable transformation to the mixing angles is applied in each sector: Obviously as defined in Eq. (9) is not symmetric under the exchange of the indices . However, sector I is symmetric under , and it would be useful to have two quantities and which fulfill
Defining in sector I (, ) the quantities (= ) and by
~s12
=
s12
~s13 = s131-s122c132, |
(13) |
yields such a pair. With these quantities Eq. (9) writes as ()
which is unaltered under the exchange of the indices .
Moreover, it can be proven that the evolution matrix of neutrinos in matter is not changed when exchanging the indices [9]. Hence, using and in sector I yields a survival probability, which is constant under this transformation in vacuum as well as in matter.
Similar to sector I also sector II (, ) can be brought in a quadratic shape by using the transformation
(14) |
But this does not yield a survival probability invariant under the exchange of the indices , as under this operation sector II would be mapped on sector III.
Iii Calculations
iii.1 Standard solar model
For the following calculations my standard solar model (“GARSOM4”) as described in [35] is used. It has been calculated using the latest input physics, equation of state and opacity from the OPAL-group [36, 37], and nuclear reaction rates as proposed by [38]. In addition, microscopic diffusion of H, He, He, the CNO-isotopes, and 4 heavier elements (among them Fe) is included using the diffusion constants of [39]. By treating convection as a fast diffusive process the chemical changes due to nuclear burning and diffusion (mixing) are evaluated in a common scheme.
A peculiarity of GARSOM4 is the inclusion of realistic 2d-hydrodynamical model atmospheres [40] until an optical depth of 1000. The improvement of the high-degree p-mode frequencies due to a better reproduction of the superadiabatic layers just below the photosphere is similar to the one obtained by using 1d-model atmospheres like in [3]. The advantage of using the 2d- instead of the 1d-model atmospheres is the extension of the former to greater optical depths, where the stratification is already adiabatic and thus the solar model gets nearly independent of the applied convection theory.
0.975 | 0.275 | 0.020 | 0.245 | 0.018 | 1.57 | 152 | 0.713 | 0.188 |
In Table 1 typical quantities of GARSOM4 using the convection theory developed in [41] are summarized. The predicted event rates of GARSOM4 for the three presently operating types of neutrino experiments GALLEX/GNO/SAGE, Homestake, and Super-Kamiokande are summarized in Table 2 together with the values obtained by [4]. Since the input physics is very similar in both models, the predicted rates agree very well within the errors as quoted by [4]. In addition, the influence of the assumed B-neutrino spectrum to the predicted rates is shown. Using the recently measured spectrum [17] yields slightly higher rates for all three experiments than including the theoretically predicted one [18], which is caused by the somewhat larger number of high-energetic B-neutrinos in the former spectrum and the strongly inclining detection probability toward higher energies.
Figure 3 shows the sound-speed profile of GARSOM4 compared to the seismic model inferred by [45]. The deviations of GARSOM4 from the latter are of the same size as standard solar models from other groups [4, 46].
Ga [SNU] | Cl [SNU] | Super-K [] | |
128.7 | 7.79 | 5.18 | |
GARSOM4 | 128.4 | 7.58 | 5.06 |
BP98 | |||
Experiment | ^{4}^{4}4Reference [42] | ^{5}^{5}5Reference [43] | ^{6}^{6}6Reference [44] |
iii.2 Neutrino-oscillation analysis
Using the neutrino flux as provided by GARSOM4 the evolution of the initial electron neutrinos through Sun, space and Earth is computed taking into account oscillations between the flavors (see Appendix). The electron-density profile and the radial distribution of the neutrinos is taken from the solar model, too. For the electron-density profile of the Earth the spherically symmetric PREM-model [48] is applied. For each set of mixing parameters the neutrino energy spectrum observed on Earth is evaluated and folded with the detector response functions. The combinations of mixing parameters which reproduce the measured data are found by applying a -analysis.
There are four contributions to the total value of originating from the four different available data sets, the event rates, the recoil-electron energy spectrum, the zenith-angle distribution, and the annual variation. The latter three are available only from the Super-Kamiokande detector, while to the first one all three types of neutrino experiments contribute.
For the event-rate portion the commonly used formula holds
(15) |
where denotes GALLEX/GNO/SAGE, Homestake or Super-Kamiokande. and are the experimental resp. theoretical 1-errors (Table 2). Since the input physics in GARSOM4 is similar to the one used in [4], the theoretical errors derived therein are taken for . are the measured event rates, which are quoted together with the uncertainties in Table 2. Note, that the Super-Kamiokande data are usually reported as B-neutrino flux relatively to a standard solar-model prediction. Actually this number has to be understood as an event-rate ratio. The total number of measured events are divided by the theoretically expected value (e.g. from GARSOM4). This ratio is then often falsely taken to be the suppression rate of the total B-neutrino flux. However, with the energy window of the recoiled electrons being between 5.5^{7}^{7}7There are already data for the energy window from 5.0 to 5.5 MeV available, but the systematic errors are still to be derived. and 20 MeV no statement about the total number of B-neutrinos below this window is possible. Moreover, neutrino oscillations may alter the energy spectrum of the B electron-neutrino flux and as the scattering cross section of the neutrinos in Super-Kamiokande is energy-dependent, the same number of event rates can be obtained with different B-neutrino fluxes. Thus in the present analysis the event rate in Super-Kamiokande following Eq. (21) are used and not the total B-neutrino flux.
The recoil-electron energy spectrum is examined by
(16) |
where the sum extends over all 18 energy bins (Fig. 6(a)) and is the quadratic sum of statistical and systematic errors taken from [44]. Since the absolute value of the event rate in Super-Kamiokande has already been used in the parameter is introduced, by which the spectrum can be normalized adequately, independent of the total rates.
Iv Results
iv.1 Two flavors
In a first step the solar neutrino problem is analyzed taking into account oscillations only between two flavors. Figure 4 shows the allowed oscillation parameters using the experimentally derived B-neutrino spectrum [17], if solely the event rates of the three experiments are fitted. Clearly the four commonly known solution-islands (see e.g. in [12, 13]) can be identified, the small-mixing (SMA) and large-mixing angle (LMA), the low mass-squared difference (LOW), and the vacuum-oscillation (VO) solutions. The -values for the best-fit parameters in these solutions are quoted in Table 3. Apart from LOW all solutions have a ratio of to the number of degrees-of-freedom (d.o.f.) which is less than one and therefore these solutions are acceptable candidates as correct solution for the solar neutrino puzzle.
In the last four columns in Table 3 the best-fit values using the theoretically derived B-neutrino spectrum of reference[18] are provided. The somewhat higher expected event rates (Table 2) result in slightly different mixing parameters compared to the case using the measured B-neutrino spectrum [17] (first four rows). While the -values for the SMA-, LMA-, and VO-solutions are marginally worse with the theoretical spectrum, the LOW-solution give a slightly better fit to the experiments. However the changes are in all cases relatively small.
(eV) | |||
---|---|---|---|
SMA | |||
LMA | 0.79 | ||
LOW | 1.00 | ||
VO | 0.95 | ||
SMA | |||
LMA | 0.78 | ||
LOW | 1.00 | ||
VO | 0.73 |
While the event-rates alone favor the SMA-solution, including into the analysis the recoil-electron energy spectrum, the zenith-angle and annual variations recorded by Super-Kamiokande, yields the LMA-solution as the best fit (Table 4). This result could also be found with the previous 825-day Super-Kamiokande data (see e.g. in[19]). In contrast to the earlier analyses, where no set of parameter could be found, which reproduces all sets of data at the same time [9], with the 1117-days data such simultaneous fits can be performed. In Table 4 the best-fit values of these solutions, the respective -contributions and the ratio of to the available d.o.f. are provided.
The 1, 2, and 3-regions (63.7, 95.4, and 99.7% C.L.) including all available data (Fig. 5) are much bigger than in the case of taking solely the event rates (Fig. 4). Most probably neglecting the correlations between the different data sets of Super-Kamiokande has caused this growth. More detailed data published from the Super-Kamiokande collaboration are desirable to be able to perform a more accurate analysis of the solar neutrino data. In addition, theoretical correlations between e.g. the annual and zenith-angle data should be taken into consideration. In order to show, that the most probable solution regions have indeed not changed drastically compared with the pure-rate analysis, the 10% C.L. areas are plotted in Fig. 5, too. The same region would result as the 1-area, if the minimum -value would be zero instead of 12.5 (=, Table 4).
Independent of whether the theoretically derived or measured B-neutrino spectrum is used, the LMA-solution can reproduce each single kind of data, the event rates, the recoil-electron energy spectrum, the zenith-angle and the seasonal variation acceptably. Although, of the LOW- and VO-solutions are less than one, these solution do not lead to an acceptable fit of the event-rate data. Whether the SMA-solution can already be ruled out by the new Super-Kamiokande data depends on the B-neutrino spectrum included in the analysis. While with the theoretically derived spectrum [18], the event rates can just be reproduced (), including the measured one [17] hardly yields a reasonable fit.
(eV) | /d.o.f. | ||||||
---|---|---|---|---|---|---|---|
SMA | |||||||
LMA | 0.79 | ||||||
LOW | 0.89 | ||||||
VO | 0.95 | ||||||
SMA | |||||||
LMA | 0.88 | ||||||
LOW | 1.00 | ||||||
VO | 0.99 | ||||||
d.o.f. |
By assuming that the Chlorine rate is due to unknown systematic errors 30% higher than quoted and by reducing the reaction rate for the Be-proton capture by about 15–20%, a VO-solution could be obtained which would be able to reproduce the event rates and the energy spectrum of Super-Kamiokande fairly well [49]. However, presently there is no evidence for any hidden systematic uncertainties in Homestake, which would increase the event rate to the 3.5-level.
The parameters of the best-fit LMA- and SMA-solutions are weakly modified when including all the data in the analysis (Tables 3 and 4). The recoil-electron energy spectrum mainly influences the SMA-solution, while the zenith-angle data causes a slight shift of the best-fit values of the LMA-solution (Fig. 5).
For – eV the seasonal variations are caused by the zenith-angle variation (on the northern hemisphere more night data are recorded during winter than during summer), and thus no additional constraints can be obtained from the former data set in this parameter space. For smaller the eccentricity of the Earth orbit leads to a “real” annual dependent signal, which can be used to constrain the mixing parameters. Note, that in contrast to the –region the seasonal dependence in is now producing tiny day-night variations.
Anyway, for deviations of less than 2% from an annually constant neutrino flux are predicted for the Super-Kamiokande data^{8}^{8}8In the analysis of the seasonal data the neutrino signal has been corrected for the -dependence of the flux., which is consistent within the errors with the recorded value. Thus, for the VO-solutions only very weak constraints can be obtained from the present seasonal variation data, while for the region of the LOW-solution these data provide important information. In fact, the position of the best-fit value of the LOW-solution has been changed by including the annual-variation data in the analysis (Fig. 5).
In the regime of the VO-solution, the recoil-electron energy spectrum provides very stringent constraints on the allowed mixing parameters, excluding great part of the region favored by the rates. Hence, no good solution in the VO-region could be found, which reproduces the recoil-electron energy spectrum as well as the rates recorded in GALLEX/SAGE/GNO, Homestake, and Super-Kamiokande.
Nevertheless, the LMA-solution is presently the favored solution to the solar neutrino problem, whereas the earlier favored SMA-solution seem to be almost ruled out. But still, improved statistics in the recoil-electron data of Super-Kamiokande is required to identify more reliably the correct solution to the neutrino problem.
iv.2 Three flavors
In spite of the LMA-solution being able to reproduce the recoil-electron energy spectrum, the zenith-angle and seasonal variations, and the event rates acceptably, the analysis is extended to all three families to deduce whether a better fit to the data can be achieved. Besides, this is the physically correct treatment, which contains the 2-flavor case as a limiting one.
The electron-neutrino survival probability for 3-flavor neutrino oscillations is determined by four quantities , , , and , where the appropriate pairs describe each the mixing of two flavors. Hence a 4-dimensional parameter space has to be examined to deduce all possible solutions. In the Appendix the numerical realization is described with which the 4-dimensional parameter survey can be performed efficiently. As worked out in section II.3, all possible solutions for MSW-solutions are obtained by considering sector I and II (Fig. 2) with , . Using in sector I the quantity as defined in Eq. (13) instead of , allows to describe the pure 2-neutrino -–oscillations equivalently to the -–case, which has been investigated thoroughly in various publications [12, 13]. In addition, with the survival probability being symmetric in the exchange of the indices 2 and 3 unnecessary computations can be avoided. Nevertheless, the computations are very extensive due to the 4-dimensional parameter space. Therefore, the grid in the 3-flavor oscillation-survey has to be chosen less dense than in the 2-flavor case, where only a 2-dimensional grid had to be overviewed. However, the grid must still be fine enough that those solutions are not missed which might be confined to small regions in the parameter space. Furthermore, the number of neutrino paths from the Sun to the detector to cover the whole year of data recording was reduced compared to the pure 2-flavor neutrino oscillations. This leads to slightly different -values for effectively pure 2-flavor solutions, which are also found in the full analysis.
Subspaces of the entire possible parameter space have been investigated e.g. in [20] or [50]. In both publications an analytical expression for the survival probability derived in [24] has been used, which is valid for large mass separations resp. small mixing angles. In this description is determined by two 2-flavor probabilities for each mass splitting and . By approximating the electron-density profile in the solar interior with an exponential function, can also be evaluated analytically [24].
The results obtained in [20] or [50] could always been reproduced in the respective mass ranges. However, in those investigations of the 3-flavor scenario, only the event-rate data were available. Using the new type of data (recoil-electron energy spectrum, zenith-angle and seasonal variations) almost all solutions found in [20, 50] are disfavored. Furthermore, new solutions are identified since the respective parts of the parameter space were not covered in the analyses therein.
In the -analysis, applied to constrain the mixing parameters, always the whole available experimental data set was used for the present study. Furthermore, solely the measured B-neutrino spectrum [17] has been included. The -–plane is divided according to the 2-flavor case in three subregions where the two oscillation-branches and are either both of matter type, or both of vacuum type, or one of matter and the second of vacuum nature.
iv.2.1 Oscillations in the MSW mass regime
First sector I is investigated, where and both may undergo resonant MSW-transition, i.e. . In Fig. 7 the projection of the region with on the 3-dimensional subspace -- is shown. The allowed parameters are located within the grey-shaded surroundings.
The pure 2-flavor oscillations where the mixing into is negligible are given by vertical structures. For instance, the LMA- and LOW-solutions are represented by the half-pillars at and and . The horizontal planes provide the solutions independent of , and thus are 2-flavor oscillations in the system, where the influence from is not visible. However, the symmetry of sector I in the exchange of the indices 2 and 3 implies that they must be equivalent to solutions independent of . Therefore, the plane around is equivalent to the pillar-like structures at and can thus be identified as an overlap of mainly LMA- and LOW-solutions. Similarly the plane at represents the SMA-solution. The 3-flavor solutions with the smallest -values are sited near the intersection regions of the horizontal planes and vertical objects, which means that they are at least slightly more probable than the pure 2-flavor solutions and involve indeed all three flavors.
In Fig. 8 the regions with of the present survey are shown. Due to the symmetry in and the overlap of solution islands in this projection the 6 basic regions in this figure belong only to four distinct solutions. Two of them can be identified as “double” SMA- resp. LMA-solutions, i.e. the same kind of solution in and (DSMA resp. DLMA). In addition, two solutions are combinations of a SMA-solution in () and a LMA resp. LOW in ( and resp. .), therefore denoted SLMA and SLOW. The extension of the SLOW-solution in the --plane at is very small and thus its position is marked by a small circle. The assignment of to SMA and to LMA resp. LOW is ambiguous and could also be chosen vice versa. This is reflected by the second appearance of the SLOW- and SLMA-solutions in Fig. 8 at and . The respective solution islands merge in this projection.
In Fig. 9 the projection of the DSMA-, DLMA-, SLMA-, and SLOW-solutions into the typical 2-flavor planes are shown. One of the areas labelled, for instance, SLMA has to be identified with oscillations in and the other with -–mixing. The oscillation parameters are provided in Table 5 together with the -values. Since two more parameters are adjusted in the 3-neutrino as compared to the 2-neutrino case, the number of degrees-of-freedom reduces from 26 to 24.
/d.o.f. | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
DSMA | I | |||||||||
DLMA | I | 0.67 | 0.38 | |||||||
SLMA | I | 0.40 | ||||||||
SLOW | I | 1.00 | ||||||||
DVO | I | 0.64 | 0.60 | |||||||
DVO’ | II | 0.58 | 0.79 | |||||||
SVO | I | 0.51 | ||||||||
LVO | I | 0.70 | 0.40 | |||||||
d.o.f. |
The mixing angle of the second small mixing-angle branch in the DSMA-solution at is about two order of magnitude smaller than the usual 2-flavor SMA-solution (compare Figs. 9 and 4). This second SMA-branch causes an additional resonance at (see Fig. 10), which enables a better fit to the recoil-electron energy spectrum and the event rates (compare Tables 4 and 5). Thus a solution almost as good as the LMA-solution could be obtained ().
The DLMA- and SLMA-solutions have almost the same values for , , and as the LMA-solution (cp. Tables 4 and 5). Since thus no improvement in explaining the solar neutrino measurements could be achieved by combining the LMA-solution with an additional oscillation in the -system, the 2-flavor LMA-solution itself is a favored solution in the 3-flavor analysis, too
In the SLOW-solution the advantages of the 2-flavor SMA- and LOW-solutions are united. While with the SMA-solution the rates can be reproduced very well (see Table 3), the LOW-solution provides good fits to the energy spectrum, the annual and the zenith angle variations (Table 4).
The best-fit value of the SLOW-solution for is equal to 1, and thus at the border of sector I to sector III. Exchanging the assignment of the indices 2 and 3 transfers the SLOW-solution to the border between sector I and II. In the latter sector, where and a survey has been performed, too. However no new solutions could be identified; solely, the “foothills” of the SLOW-solution into this section have been found. Hence, in the mass regime with the region with does not provide new solutions, which is similar to the findings in the 2-flavor case, where also no additional solutions have been detected in this parameter range.
iv.2.2 Three-flavor vacuum oscillations
In the case when both mass-squared differences are in the vacuum-oscillation regime (,) two minima could be found, denoted DVO and DVO’. While DVO is located in sector I (Fig. 11), i.e. , DVO’ has been found in sector II (Fig. 12), where . Thus although they seem to have very similar mixing parameters (Table 5), there are really distinct solutions with different properties (see Fig. 10). Note that, while in sector I the assignment of the indices can be exchanged, this is not possible in sector II. If an exchange of and in DVO’ is desired, the mixing angles have to be transformed appropriately to lie finally in sector III.
With the DVO- and DVO’-solutions a slightly better fit to the recoil-electron energy spectrum can be achieved compared to the DSMA-, DLMA-, and SLMA-solutions (Table 5). However, because of the event rates being reproduced worse, the -values of DVO and DVO’ are almost equal to the values of DSMA, DLMA, and SLMA.
Compared to the 2-flavor VO-solution an improved fit to the event rates has been obtained with the 3-flavor vacuum-oscillation solutions, but still the event rates are fitted barely acceptably. The DVO- and DVO’-solutions show a compromise between the VO-solution ( in Fig. 11) obtained by fitting solely the event-rate data and the VO-solution of the complete analysis ( in same figure). The mixing angles for the 3-flavor vacuum solution are only about one half of the usual 2-flavor VO-solution, as is now oscillating nearly equally strong into two other flavors.
iv.2.3 Mixed vacuum and MSW oscillations
The combination of oscillation between two neutrino flavors in the MSW mass-regime and an additional vacuum oscillation into the third flavor is investigated completing the mass ranges which have not been covered by the previous sections. This case has been examined in [50] for and . However, in that analysis only the event rates have been included, but not the recoil-electron energy spectrum, the zenith-angle dependence nor the annual variation data recorded by Super-Kamiokande. Furthermore, the Earth-regeneration effect has been neglected. Therefore, this case is reinvestigated including all the available data and calculating the electron-neutrino survival probabilities fully consistently including the Earth effect like in the previous sections.
In this survey is taken to be in the mass range of the MSW-solutions () and in the vacuum oscillation area (). With these conventions about , two possibilities are conceivable for the system, a resonant and a non-resonant oscillation. Since the masses are well separated, the condition for a resonance obtained in the 2-flavor case can be applied (Eq. 11)
Thus, an MSW flavor transition is only possible, if the mixing angle is less than . In the system only pure vacuum oscillations occur, and thus . Since the solution may be a combination of a vacuum oscillation in the and a resonant resp. non-resonant oscillation in the branch, the complete essential parameter space for and