Neutrino masses and mixing from neutrino oscillation experiments
Abstract
We discuss which information on neutrino masses and mixing can be obtained from the results of all neutrino oscillation experiments in the cases of three and four massive neutrinos. We show that in the three-neutrino case the neutrino oscillation data are not compatible with a hierarchy of couplings. In the case of four neutrinos, a hierarchy of masses is disfavored by the data and only two schemes with two pairs of neutrinos with close masses separated by a gap of the order of 1 eV can accommodate the results of all experiments.
pacs:
Talk presented by C. Giunti at the International Europhysics Conference on High Energy Physics, 19–26 August 1997, Jerusalem, Israel.The search for neutrino oscillations is one of the most active branches of today’s high-energy physics. From the LEP measurements of the invisible width of the -boson we know that there are three light active flavor neutrinos: , and . In general, flavor neutrinos are not mass eigenstates and the left-handed flavor neutrino fields are superpositions of the left-handed components of the fields of neutrinos with a definite mass (): , where is a unitary mixing matrix. The number of massive neutrinos can be three or more, without any experimental upper limit. If , there are sterile flavor neutrino fields, i.e., fields of neutrinos which do not take part in standard weak interactions; in this case . Neutrino oscillations is a direct consequence of neutrino mixing; the probability of transitions is given by (see [1])
(1) |
where is the distance between the neutrino source and detector, is the neutrino energy and .
In this report we discuss which information on the neutrino mass spectrum and mixing parameters can be obtained from the results of neutrino oscillation experiments. Many short-baseline (SBL) neutrino oscillation experiments with reactor and accelerator neutrinos did not find any evidence of neutrino oscillations. Their results can be used in order to constrain the allowed values of the neutrino masses and of the elements of the mixing matrix. In our analysis we use the most stringent exclusion plots obtained in the channel by the Bugey experiment [2], in the channel by the CDHS and CCFR experiments [3], and in the channel by the BNL E734, BNL E776 and CCFR [4] experiments.
There are three experimental indications in favor of neutrino oscillations that come from the anomalies observed by the solar neutrino experiments [5], the atmospheric neutrino experiments [6] and the LSND experiment [7]. The solar neutrino deficit can be explained with oscillations of solar ’s into other states and indicates a mass-squared difference of the order of in the case of resonant MSW transitions or in the case of vacuum oscillations. The atmospheric neutrino anomaly can be explained by oscillations () with a mass-squared difference of the order of . Finally, the LSND experiment found indications in favor of oscillations with a mass-squared difference of the order of .
Hence, three different scales of mass-squared difference are needed in order to explain the three indications in favor of neutrino oscillations. This means that the number of massive neutrinos must be bigger than three. In the following we consider the simplest possibility, i.e. the existence of four massive neutrinos (). In this case, besides the three light flavor neutrinos , , , there is a light sterile neutrino .
However, before considering the case of four neutrinos, we discuss the minimal possibility of the existence of only three massive neutrinos (). In this case one of the experimental anomalies mentioned above cannot be explained with neutrino oscillations (we choose to disregard the atmospheric neutrino anomaly).
In both cases of three and four massive neutrinos the oscillations in the LSND experiment imply that the largest mass-squared difference is relevant for SBL oscillations, whereas the other mass-squared differences are much smaller. Hence, the mass spectrum must be composed of two groups of massive neutrinos with close masses ( and ) separated by a mass difference in the eV range () and in SBL experiments we have , for , for . The formula (1) written as
(2) |
leads to the following expression for the transition () and survival () probabilities of neutrinos and anti-neutrinos in SBL experiments:
(3) |
with and the oscillation amplitudes
(4) |
The formulas (3) have the same form of the standard expressions for the oscillation probabilities in the case of two neutrinos (see [1]) with which the data of all the SBL experiments have been analyzed by the experimental groups. Hence, the results of these analyses can be used in order to constrain the possible values of the oscillation amplitudes and .
First, we consider the scheme 3H of Tab.Neutrino masses and mixing from neutrino oscillation experiments, with three neutrinos and a mass hierarchy. This scheme (as all the schemes with three neutrinos) provides only two independent mass-squared differences, and , which we choose to be relevant for the solution of the solar neutrino problem and for neutrino oscillations in the LSND experiment, respectively.
Let us emphasize that the mass spectrum 3H with three neutrinos and a mass hierarchy is the simplest and most natural one, being analogous to the mass spectra of charged leptons, up and down quarks. Moreover, a scheme with three neutrinos and a mass hierarchy is predicted by the see-saw mechanism for the generation of neutrino masses, which can explain the smallness of the neutrino masses with respect to the masses of the corresponding charged leptons.
In the case of scheme 3H we have and Eq.(4) implies that
(5) |
Hence, neutrino oscillations in SBL experiments depend on three parameters: , and (the unitarity of implies that ). From the exclusion plots obtained in reactor and accelerator disappearance experiments it follows that at any fixed value of , the oscillation amplitudes and are bounded by the upper values and , respectively, which are small quantities for . From Eq.(5) one can see that small upper bounds for and imply that the parameters and can be either small or large (i.e., close to one):
(6) |
for . Both and are small ( and ) for any value of in the range (see Fig.1 of Ref.[8]).
Since large values of both and are excluded by the unitarity of the mixing matrix (), at any fixed value of there are three regions in the – plane which are allowed by the exclusion plots of SBL disappearance experiments: Region I, with and ; Region II, with and ; Region III, with and .
In region III is large and has a small mixing with and , which is insufficient for the explanation of the solar neutrino problem. Indeed, the survival probability of solar ’s is bounded by (see [8]). If , we have at all neutrino energies, which is a bound that is not compatible with the solar neutrino data. Hence, region III is excluded by solar neutrinos.
In region I , which means that the probability of transitions in SBL experiments is strongly suppressed. The corresponding upper bound obtained from the 90% CL exclusion plots of the Bugey [2] disappearance experiment and of the CDHS and CCFR [3] disappearance experiments is represented in Fig.Neutrino masses and mixing from neutrino oscillation experiments by the curve passing trough the circles. The shadowed regions in Fig.Neutrino masses and mixing from neutrino oscillation experiments are allowed at 90% CL by the results of the LSND experiment. Also shown are the 90% CL exclusion curves found in the BNL E734, BNL E776 and CCFR [4] appearance experiments and in the Bugey experiment. One can see from Fig.Neutrino masses and mixing from neutrino oscillation experiments that in region I, the bounds obtained from the results of , and experiments are not compatible with the allowed regions of the LSND experiment [8]. Therefore, region I is disfavored by the results of SBL experiments. This is an important indication, because region I is the only one in which it is possible to have a hierarchy of the elements of the neutrino mixing matrix analogous to the one of the quark mixing matrix.
Having excluded the regions I and III of the scheme 3H, we are left only with region II, where has a large mixing with , i.e., (not ) is the “heaviest” neutrino.
Let us now consider the possible schemes with four neutrinos, which provide three independent mass-squared differences and allow to accommodate in a natural way all the three experimental indications in favor of neutrino oscillations. We consider first the scheme 4H of Tab.Neutrino masses and mixing from neutrino oscillation experiments with four neutrinos and a mass hierarchy. The three independent mass-squared differences, , and , are taken to be relevant for the oscillations of solar, atmospheric and LSND neutrinos, respectively. In the case of scheme 4H we have and Eq.(4) implies that the oscillation amplitudes are given by
(7) |
In this case, neutrino oscillations in SBL experiments depend on four parameters: , , and . From the similarity of the amplitudes (7) with the corresponding ones given in Eq.(5), it is clear that replacing with we can apply to the scheme 4H the same analysis presented for the scheme 3H. Hence, also the regions III and I of the scheme 4H are excluded, respectively, by the solar neutrino problem and by the results of SBL experiments. Furthermore, the purpose of considering the scheme 4H is to have the possibility to explain the atmospheric neutrino anomaly, but this is not possible if the neutrino mixing parameters lie in region II. Indeed, in region II is large and the muon neutrino has a small mixing with the light neutrinos , and , which is insufficient for the explanation of the atmospheric neutrino anomaly [9].
Hence, the scheme 4H is disfavored by the results of neutrino oscillation experiments. For the same reasons, all possible schemes with four neutrinos and a mass spectrum in which three masses are clustered and one mass is separated from the others by a gap of about 1 eV (needed for the explanation of the LSND data) are disfavored by the results of neutrino oscillation experiments. Therefore, there are only two possible schemes with four neutrinos which are compatible with the results of all neutrino oscillation experiments: the schemes 4A and 4B of Tab.Neutrino masses and mixing from neutrino oscillation experiments. In these two schemes the four neutrino masses are divided in two pairs of close masses separated by a gap of about 1 eV. In scheme A, is relevant for the explanation of the atmospheric neutrino anomaly and is relevant for the suppression of solar ’s. In scheme B, the roles of and are reversed.
From Eq.(4) and using the unitarity of the mixing matrix, the oscillation amplitudes in the schemes 4A and 4B (, ) are given by , with in the scheme 4A and in the scheme 4B. This expression for has the same form as the one in Eq.(7), with replaced by . Therefore, we can apply the same analysis presented for the scheme 4H and we obtain four allowed regions in the – plane (now the region with large and is not excluded by the unitarity of the mixing matrix, which gives the constraint ): Region I, with and ; Region II, with and ; Region III, with and ; Region IV, with and . Following the same reasoning as in the case of scheme 4H, one can see that the regions III and IV are excluded by the solar neutrino data and the regions I and III are excluded by the results of the atmospheric neutrino experiments [9]. Hence, only region II is allowed by the results of all experiments.
If the neutrino mixing parameters lie in region II, in the scheme 4A (4B) the electron (muon) neutrino is “heavy”, because it has a large mixing with and , and the muon (electron) neutrino is “light”. Thus, the schemes 4A and 4B give different predictions for the effective Majorana mass in neutrinoless double-beta decay experiments: since , we have in the scheme 4A and in the scheme 4B. Hence, if the scheme 4A is realized in nature, the experiments on the search for neutrinoless double-beta decay can reveal the effects of the heavy neutrino masses . Furthermore, the smallness of in both schemes 4A and 4B implies that the electron neutrino has a small mixing with the neutrinos whose mass-squared difference is responsible for the oscillations of atmospheric neutrinos (i.e., , in scheme 4A and , in scheme 4B). Hence, the transition probability of electron neutrinos and antineutrinos into other states in atmospheric and long-baseline experiments is suppressed [10]^{1}^{1}1After we finished this paper the results of the first long-baseline reactor experiment CHOOZ appeared (M. Apollonio et al., preprint hep-ex/9711002). No indications in favor of transitions were found in this experiment. The upper bound for the transition probability of electron antineutrinos into other states found in the CHOOZ experiment is in agreement with the limit obtained in Ref.[10]..
References
- [1] S.M. Bilenky and B. Pontecorvo, Phys. Rep. 41, 225 (1978).
- [2] B. Achkar et al., Nucl. Phys. B 434, 503 (1995).
- [3] F. Dydak et al., Phys. Lett. B 134, 281 (1984); I.E. Stockdale et al., Phys. Rev. Lett. 52, 1384 (1984).
- [4] L.A.Ahrens et al., Phys. Rev. D 36, 702 (1987); L.Borodovsky et al., Phys. Rev. Lett. 68, 274 (1992); A.Romosan et al., ibid 78, 2912 (1997).
- [5] B.T. Cleveland et al., Nucl. Phys. B (P.S.) 38, 47 (1995); K.S. Hirata et al., Phys. Rev. D 44, 2241 (1991); GALLEX Coll., Phys. Lett. B 388, 384 (1996); SAGE Coll., Phys. Rev. Lett. 77, 4708 (1996).
- [6] Y. Fukuda et al., Phys. Lett. B 335, 237 (1994); R. Becker-Szendy et al., Nucl. Phys. B (P.S.) 38, 331 (1995); W.W.M. Allison et al., Phys. Lett. B 391, 491 (1997).
- [7] C. Athanassopoulos et al., Phys. Rev. Lett. 77, 3082 (1996).
- [8] S.M. Bilenky, A. Bottino, C. Giunti and C.W. Kim, Phys. Rev. D 54, 1881 (1996).
- [9] S.M. Bilenky, C. Giunti and W. Grimus, preprint hep-ph/9607372.
- [10] S.M. Bilenky, C. Giunti and W. Grimus, preprint hep-ph/9710209.
3H: | |
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4H: | |
4A: | |
4B: |
Table Neutrino masses and mixing from neutrino oscillation experiments
Figure Neutrino masses and mixing from neutrino oscillation experiments