Abstract
With one of the appropriate order of magnitude to solve the atmospheric neutrino problem, we study the resulting threegeneration vacuumoscillation fit to the solar neutrino flux. An explanation of the atmospheric neutrino composition in terms of pure oscillations is easily compatible with the wellknown twoflavour oscillation solution for solar neutrinos. The allowed parameter region in the – plane changes little with increasing values of the mixing element , provided this is less than about 0.4. We find that the threefold maximal mixing is disfavoured.
CERNTH/98200
[2mm] hepph/9806339
Solarneutrino oscillations
[5mm] and thirdflavour admixture
P. Osland and G. Vigdel
Department of Physics, University of Bergen,
Allégaten 55, N5007 Bergen, Norway
Theoretical Physics Division, CERN, CH1211 Geneva 23, Switzerland
CERNTH/98200
[2mm] June 1998
1 Introduction
While neutrinooscillation experiments tend to suffer from systematic errors that are difficult to estimate, there are at least two distinct neutrino problems that deserve serious consideration: the deficit of solar neutrinos and the atmospheric neutrino composition. The former has been around for many years. It started with the pioneering Homestake detector and later gained significance with the waterscattering and gallium experiments [1].
The latter anomaly is an inconsistency between the number of detected neutrinos of the muon family and those of the electron family, assumed to be generated by cosmicray hadrons impinging on the atmosphere. The ratio of ratios
(1) 
where is the number of like events and the number of like events, should be, according to theory, close to 1, but is instead around [2].
If one analyses the solarneutrino deficit (a summary of the chlorine [3], gallium [4, 5] and waterscattering experiments [6, 7] is given in Table 1) in terms of twoflavour mixing, the vacuumoscillation solution leads to a set of disconnected regions in the twodimensional parameter space spanned by and , according to the survival probability
(2) 
where is the mixing angle, the difference of the squared masses, the time since creation and the neutrino energy. The favoured region is –0.9, with , but neighbouring regions that differ from by can yield comparable values of , provided is a small integer, and is of the order of (cf. Fig. 1a)^{1}^{1}1The MSW [8] interpretation of the data leads to values of that are several orders of magnitude higher.. The scale is given by the characteristics of the detectors, the neutrino energy (of the order of a MeV) and the distance from the Sun to the Earth: .
The probability expression (2) and Fig. 1a show the ‘just so’ character of vacuum oscillations, the masssquared difference has to be finetuned in relation to the Sun–Earth distance if this model shall solve the solarneutrino problem [9]. (The allowed region in the upper part of Fig. 1a, where , is a new feature of the most recent data [10].)
With three flavours of neutrinos, one may consider the general threeflavour mixing, in which a state, which is of pure flavour (, , ) at , evolves as
(3) 
where the summation index runs over mass states. We write the unitary mixing matrix as
(4) 
The full parameter space then consists of three angles (we ignore violating effects) and two independent squaredmass differences (the energy can be approximated as ). In particular, the survival probability of an electron neutrino takes the form
(5)  
where . Unitarity and data on other phenomena, in particular on the atmospheric neutrino flux, lead to constraints on these parameters.
We denote the mass states , and , according to their intrinsic mass ratio: their respective mass eigenvalues will be sorted as . For the mixing matrix we use the form advocated for quarks by the Particle Data Group [11]^{2}^{2}2There are nine structurally different ways to express the mixing matrix in terms of mixing angles. A discussion of the parametrization of flavour mixing in the quark sector is presented in [12], where also violation is included.:
(6)  
where , , etc., and the mixing angles are in the interval .
2 Threeparameter accommodation of the solar neutrino flux
A threeflavour analysis of the solarneutrino oscillations will restrict the allowed values of the elements in the mixing matrix. Because of unitarity, the solar electronneutrino survival probability can always be expressed in terms of only two mixing angles, here taken to be and . We shall assume that the dominant mixing (represented in Fig. 1a) is a mixing between the first two families, and ask (i) ‘How will the goodness of fit depend on ’ and (ii) ‘How will the allowed regions of Fig. 1a change as one increases the strength of this mixing?’
Before we address these questions, we want to introduce some further restrictions on the neutrino parameters: we reserve one for the vacuumoscillation solution of the solarneutrino problem, . The suppression of atmospheric muon neutrinos relative to electron neutrinos suggests that also oscillates into some other flavour of neutrinos, but (since the ratio relevant to atmospheric neutrinos is orders of magnitudes smaller than that for solar neutrinos) with a masssquared difference that is many orders of magnitude larger than is required to explain the solarneutrino deficit. It is therefore natural to assume that other squaredmass differences are much larger:
(7) 
such that when Eq. (5) is applied to the case of solar neutrinos, the last two terms can be represented by their average values:
(8) 
Here, the last term represents a nonzero component of the mass state in the electronneutrino state, and vanishes with . For we get the familiar twofamily mixing of Eq. (2).
While a simultaneous fit to both solar and atmosphericneutrino fluxes also would involve the angle as well as two masssquared differences, under the assumption (7) the above formula (8) provides a threeparameter accommodation of the solar flux.
We show in Fig. 1b the results of twoparameter fits to the solarneutrino data. (The high region in , cf. Fig. 1a, is not displayed here.) In our calculations we use the solarneutrino flux as given by the 1998 Standard Solar Model (SSM) by Bahcall, Basu and Pinsonneault [10], and for the detection rates the values given in Table 1. The second mixing angle, , is held fixed at different values, and the fit is performed in the variables and . It is seen that regions that are separated in the twofamily case, can be smoothed out to one big region when the more general case of three flavours is considered. However, there is a slight worsening of the quality of the fit as the coupling to the third state is increased. For , 0.2, and 0.4, the values of are 3.6, 4.2 and 5.6, respectively. For two degrees of freedom, 4 experiments (SAGE and GALLEX are combined) minus 2 parameters, the probability of getting larger than those values by chance, are 17, 12 and 6%, respectively. None of these is particularly good, but they are not excluded.
Since there is some uncertainty associated with the important boron flux [10], we have checked how the fit changes under 10% rescalings of the flux. In Table 2 we show the resulting and goodness of fit for three different fractions of the SSM boron flux (keeping the other components of the flux fixed), for two different values of . For the lowest of these fractions, the goodness of fit is appreciably better than for the standard flux. It is also seen that, for the lowest of our tabulated boron flux values, the fit deteriorates less when is nonzero.
A different representation, for several values of , is given in Fig. 2, including also the higher region. As increases, the allowed regions get broader, but also the fit gets poorer.
It has been argued that the data suggest ‘maximal mixing’ [13]. In our variables, maximal mixing corresponds to and . We find that the fit gets rather poor for such parameters, with .
3 Longbaseline experiments
The future longbaseline experiments [14] will provide new restrictions on these mixing elements. For example, if we assume that a longbaseline experiment is such that oscillations other than those involving
(9) 
can be neglected, then the condition (7) leads to the following transition probabilities:
(10) 
(11) 
and the survival probability:
(12) 
Note that the coefficient in Eq. (10) can be written as . The transition can thus be given a simple physical interpretation, valid for small values of the mixing angle . When , the transition can be interpreted as an ‘indirect’ transition, involving the admixtures of as given by and . This is different from the case of , where the ‘direct’ transition is effective via the states and .
The atmosphericneutrino data are usually explained in terms of or oscillations with a large amplitude [15, 16]. When applying Eq. (8) to explain the solarneutrino deficit, the transition is preferable for the atmospheric neutrinos, because it allows for a small or vanishing value of , which gives the best fit of the observed solar neutrino rate. An early threeflavour analysis (not including data from SuperKamiokande) gave for the best fit a value for (or ) in the range 0.06–0.14 [17]. More recent studies (including SuperKamiokande data) prefer values for of 0.1 [15] and [16]. As we have seen, these low values of also lead to acceptable fits to the solarneutrino data, with –5.6.
In the longbaseline experiments [14, 18], neutrinos from either an accelerator or a reactor will be counted after travelling a ‘long’ distance, from 1 km (for CHOOZ) to 730 km (for the MINOS experiment). In fact, the recent data from the CHOOZ experiment, where the survival probability is measured, does not exclude any of the cases studied in Fig. 2, provided . Indeed, no evidence for oscillations was found for and . This means that has to be either small () or quite big () (or ). With the largeangle solution, Eq (8) gives a poor fit to the solarneutrino data, this alternative should therefore be disregarded. The remaining lowangle solution is also preferred in the analysis of the atmosphericneutrino data [15]. In this case (with ) the lower two boxes of Fig. 2 would be excluded. The Bugey reactor experiment [19], which is less sensitive, excludes down to about for , consistent with CHOOZ.
A measurement of a nonzero value for will imply a nonzero value for . Such information, which would be a major result from the planned longbaseline experiments, combined with Eq. (8), can help us to further restrict the allowed parameter space relevant to the solar neutrinos.
We note that, for oscillations, the K2K experiment [20] will have a sensitivity of and the MINOS experiment could probe values as low as . In our parametrization, this means that they would be sensitive to values of as low as 0.16 and 0.05. If this mixing element should turn out to be significantly larger, the vacuumoscillation solution to the solarneutrino problem would have serious difficulties.
4 Concluding remarks
The data from the solarneutrino detectors can very well be reconciled with the atmosphericneutrino data. If we assume that the solution of the latter anomaly is the transition (which is induced by the direct mixing and gives the best fit), then the vacuumoscillation solution for the atmospheric and the solarneutrino problems can be regarded as two separate twogeneration oscillations. Even if the transition , which for is induced by the twostep process of and mixing (with angles and , respectively) should turn out to take place, the rate has to be quite large before it can severely alter the fit of the vacuumoscillation solution to the solarneutrino problem.
When the mixing angle is increased from zero, the allowed ranges of merge to a wider one, but beyond , which is well within the sensitivity that will be reached by the K2K experiment, the fit deteriorates rapidly.
References

[1]
Solar Neutrinos: The First 30 Years,
edited by: J. N. Bahcall et al. (Addison Wesley, New York, 1994);
J. N. Bahcall, Phys. Lett. B 338, 276 (1994).  [2] Y. Fukuda et al. (SuperKamiokande Collaboration), ICRR411987 (1998), hepex/9803006 and hepex/9805006.

[3]
B. T. Cleveland, Nucl. Phys. (Proc. Suppl.) B 38, 47 (1995);
K. Lande et al., in Neutrino ’96, Proceedings of the 17th International Conference on Neutrino Physics and Astrophysics, Helsinki, 1996, edited by K. Huitu, K. Enqvist and J. Maalampi (World Scientific, Singapore, 1997). 
[4]
P. Anselmann et al., Phys. Lett. B 342, 440 (1995);
W. Hampel et al., Phys. Lett. B 388, 384 (1996).  [5] J. N. Abdurashitov et al., Phys. Lett. B 328, 234 (1994). V. Gavrin et al., in Neutrino ’96, op. cit.
 [6] Y. Fukuda et al., Phys. Rev. Lett. 77, 1683 (1996).

[7]
Y. Fukuda et al.,
hepex/9805021, submitted to Phys. Rev. Lett.
See also
Y. Itow (contains S.K. data until 20 October 1997)
http://wwwsk.icrr.utokyo.ac.jp/doc/sk/pub/pub sk.html 
[8]
S. P. Mikheyev and A. Yu. Smirnov, Yad. Fiz. 42, 1441 (1985)
[Sov. J. Nucl. Phys. 42, 913 (1985)],
Nuovo Cimento 9C, 17 (1986);
L. Wolfenstein, Phys. Rev. D 17, 2369 (1978), ibid. 20, 2634 (1979).  [9] V. Barger, R. J. N. Phillips and K. Whisnant, Phys. Rev. D 24, 538 (1981); S. L. Glashow and L. M. Krauss, Phys. Lett. 190 B, 199 (1987).
 [10] J. N. Bahcall, S. Basu and M. H. Pinsonneault, astroph/9805135.
 [11] Particle Data Group (R. M. Barnett et al.), Phys. Rev. D 54, 1 (1996).
 [12] H. Fritzsch and Z.z. Xing, Phys. Rev. D 57, 594 (1998); A. Rasin, hepph/9708216.
 [13] P. F. Harrison and W.G. Scott, Phys. Lett. B333, 471 (1994); P. F. Harrison, D. H. Perkins and W. G. Scott, Phys. Lett. B349, 137 (1995), ibid. B374, 111 (1996) and B396, 186 (1997).
 [14] For a short description of some of these experiments, see Neutrino ’96, op. cit.
 [15] O. Yasuda, hepph/9804400.
 [16] T. Teshima and T. Sakai, preprint CUTP/9805, hepph/9805386.
 [17] S. M. Bilenky, C. Giunti and C. W. Kim, Astropart. Phys. 4, 241 (1996), hepph/9505301.
 [18] M. Apollonio et al. (CHOOZ Collaboration), Phys. Lett. B 420, 397 (1998).
 [19] B. Achkar et al., Nucl. Phys. B 434, 503 (1995).
 [20] Y. Oyama, talk at YITP Workshop on Flavour Physics, Kyoto, January 1998, hepex/9803014.
Experiment  Counting rate  SSM [10]  Counted/SSM 

Homestake [3]  
GALLEX [4]  
SAGE [5]  
Kamiokande [6]  
SuperKamiokande [7] 
2.0  3.6  5.8  2.9  4.8  7.2  
38  17  5.5  23  9  3 