UWThPh199821 DFTT 19/98 May 1998
Neutrino Masses and Mixing
in the Light of Experimental Data^{1}^{1}1 Lecture given by W.G. at the 5th Workshop on High Energy Physics Phenomenology, January, 1226, 1998, IUCAA, Pune, India
[5mm] S.M. Bilenky
Joint Institute for Nuclear Research, Dubna, Russia, and
INFN, Sezione di Torino, Via P. Giuria 1, I–10125 Torino, Italy
[3mm] C. Giunti
INFN, Sezione di Torino, and Dipartimento di Fisica Teorica, Università di Torino,
Via P. Giuria 1, I–10125 Torino, Italy
[3mm] and
[3mm] W. Grimus
Institute for Theoretical Physics, University of Vienna,
Boltzmanngasse 5, A–1090 Vienna, Austria
Abstract
[3mm] All the possible schemes of neutrino mixing with four massive neutrinos inspired by the existing experimental indications in favour of neutrino mixing are considered. It is shown that the scheme with a neutrino mass hierarchy is not compatible with the experimental results, likewise all other schemes with the masses of three neutrinos close together and the fourth mass separated by a gap needed to incorporate the LSND neutrino oscillations. Only two schemes with two pairs of neutrinos with close masses separated by this gap of the order of 1 eV are in agreement with the results of all experiments. We carefully examine the arguments leading to this conclusion and also discuss experimental consequences of the two favoured neutrino schemes.
1 Indications in favour of neutrino oscillations
1.1 Notation
Neutrino masses and neutrino mixing are natural phenomena in gauge theories extending the Standard Model (see, for example, Ref.[1]). However, for the time being masses and mixing angles cannot be predicted on theoretical grounds and they are the central subject of the experimental activity in the field of neutrino physics.
In the general discussion, we assume that there are neutrino fields with definite flavours and that neutrino mixing is described by a unitary mixing matrix such that
(1) 
Note that the neutrino fields other than the three active neutrino flavour fields , , must be sterile to comply with the result of the LEP measurement of the number of neutrino flavours. The fields () are the lefthanded components of neutrino fields with definite mass . We assume the ordering for the neutrino masses. In Eq.(1) and in the following discussion of neutrino oscillations it does not matter if the neutrinos are of Dirac or Majorana type. One should only keep in mind that different types cannot mix.
The most striking feature of neutrino masses and mixing is the quantummechanical effect of neutrino oscillations [2]. The probability of the transition is given by
(2) 
where , is the distance between source and detector and is the neutrino momentum. Eq.(2) is valid for ().^{2}^{2}2There are additional conditions depending on the neutrino production and detection processes which must hold for the validity of Eq.(2). See, e.g., Ref.[3] and references therein. Evidently, from neutrino oscillation experiments only differences of squares of neutrino masses can be determined. The probability for transitions is obtaind from Eq.(2) by the substitution .
1.2 Indications in favour of neutrino masses and mixing
At present, indications that neutrinos are massive and mixed have been found in solar neutrino experiments (Homestake [4], Kamiokande [5], GALLEX [6], SAGE [7] and SuperKamiokande [8, 9]), in atmospheric neutrino experiments (Kamiokande [10], IMB [11], Soudan [12] and SuperKamiokande [13, 9]) and in the LSND experiment [14]. From the analyses of the data of these experiments in terms of neutrino oscillations it follows that there are three different scales of neutrino masssquared differences:

Solar neutrino deficit: Interpreted as effect of neutrino oscillations the relevant value of the masssquared difference is determined as
(3) The two possibilities for correspond, respectively, to the MSW [18] and to the vacuum oscillation solutions of the solar neutrino problem.

LSND experiment: The evidence for oscillations in this experiment leads to
(5) where is the neutrino masssquared difference relevant for shortbaseline (SBL) experiments.
Thus, at least four light neutrinos with definite masses must exist in nature in order to accommodate the results of all neutrino oscillation experiments. Denoting by a generic neutrino masssquared difference we can summarize the discussion in the following way:
3 different scales of 4 neutrinos (or more).
Therefore there exists at least one noninteracting sterile neutrino [19, 20, 21, 22, 23, 24, 25].
However, we must also take into account the fact that in several shortbaseline experiments neutrino oscillations were not observed. The results of these experiments allow to exclude large regions in the space of the neutrino oscillation parameters. This will be done in the next section.
The plan of this report is as follows. In section II we extensively discuss SBL neutrino oscillations for an arbitrary number of neutrinos. In section III we argue that a 4neutrino mass hierarchy is disfavoured by the experimental data. Thereby, solar and atmospheric neutrino flux data play a crucial role. In section IV we introduce the two 4neutrino mass and mixing schemes favoured by all neutrino oscillation experiments. We discuss possibilities to check these schemes in longbaseline (LBL) neutrino oscillation experiments in section V. Our conclusions are presented in section VI.
2 SBL experiments
2.1 The oscillation phase
As a guideline, SBL neutrino oscillation experiments are sensitive to masssquared differences eV. A generic oscillation phase is given by
(6) 
Distinguishing reactor and accelerator experiments and assuming that experiments are roughly sensitive to phases (6) around 0.1 or larger we get the following conditions from eV:

Reactors: MeV and therefore m.

Accelerators: .
2.2 Basic assumption and formalism
We will make the following basic assumption in the further discussion in this report:
A single is relevant in SBL neutrino experiments.
In accordance with Eq.(5) we denote this by .
As a consequence of this assumption the neutrino mass spectrum consists of two groups of close masses, separated by a mass difference in the eV range. Denoting the neutrinos of the two groups by and , respectively, the mass spectrum looks like
(7) 
such that
(8) 
for the purpose of the SBL formalism. In Eq.(8) we have used the notation . Eq.(2) together with Eq.(8) gives the SBL transition probability
(9) 
For the probability of the transition () we obtain from Eq.(9)
(10) 
where the oscillation amplitude is given by
(11) 
The second equality sign in this equation follows from the unitarity of . Furthermore, the oscillation amplitude fulfills the condition . The second part of this equation is a consequence of the Cauchy–Schwarz inequality and the unitarity of the mixing matrix. The survival probability of is calculated as
(12) 
with the survivial amplitude
(13) 
Conservation of probability gives the important relation
(14) 
The expressions (10) and (12) describe the transitions between all possible neutrino states, whether active or sterile. Let us stress that with the basic assumption in the beginning of this subsection the oscillations in all channels are characterized by the same oscillation length . Furthermore, the substitution in the amplitudes (11) and (13) does not change them and therefore it ensues from the basic SBL assumption that the probabilities (10) and (12) hold for antineutrinos as well and hence there is no CP violation in SBL neutrino oscillations.
The oscillation probabilities (10) and (12) look like 2flavour probabilities. Defining , and for , the resemblance is even more striking. It means that the basic SBL assumption allows to use the 2flavour oscillation formulas in SBL experiments. However, genuine 2flavour neutrino oscillations are characterized by a single mixing angle given by .
2.3 Disappearance experiments
For the two flavours and results of disappearence experiments are available. We will use the 90% exclusion plots of the Bugey reactor experiment [26] for disappearance and the 90% exclusion plots of the CDHS [27] and CCFR [28] accelerator experiments for disappearance. Since no neutrino disappearance has been seen there are upper bounds on the disappearance amplitudes for . These experimental bounds are functions of . It follows that
(15) 
and therefore [29]
(16) 
Eq.(16) shows that . In Fig.2.3 the bounds and are plotted as functions of in the wide range
(17) 
In this range is small () and for eV. This means that in the – unit square for every we can distinguish four allowed regions according to or (see Fig.2.3).
2.4 The transition in SBL experiments
Considering the amplitude , with the help of the Cauchy–Schwarz inequality we obtain from Eq.(11)
(18) 
Therefore, we immediately see that
(19) 
In Fig.3 the result of the LSND experiment [14] for the amplitude is shown with 90% CL boundaries (shaded areas). All other experiments measuring this amplitude have obtained upper bounds [30, 31, 32, 33]. In addition, the upper bound on the survivial amplitude of Bugey [26] is indicated by the solid line in Fig.3 since the unitarity relation (14) gives . Finally, the curve passing through the circles represents the bound (19). Inspecting Fig.3 we come to the following conclusion:
Regions I and III are not compatible with the positive result of LSND indicating oscillations and the negative results of all other SBL experiments.
Furthermore, it can be read off from Fig.3 that
(20) 
is the favoured range for the SBL masssquared difference. In this range holds. Let us further mention that for region III is already ruled out by the unitarity of the mixing matrix. The same is valid for and region I.
3 The 4neutrino mass hierarchy is disfavoured
In the case of a neutrino mass hierarchy, , the masssquared differences and are relevant for the suppression of the flux of solar neutrinos and for the atmospheric neutrino anomaly, respectively. This case corresponds to and (see the formalism in subsection 2.2) with . We only have to consider regions II and IV.
We will now take into account information from the solar neutrino anomaly assuming that it is solved by neutrino oscillations. From the fact that the 4th column vector in pertaining to is not affected by solar neutrino oscillations we obtain a lower bound on the average survival probability of solar neutrinos given by (see Refs.[34, 20])
(21) 
In region IV we have or and therefore holds for all solar neutrino energies. Such a large lower bound is not compatible with the solar neutrino data and we conclude:
For a 4neutrino mass hierarchy region IV is not compatible with the solar neutrino data.
Let us mention that inequality (21) is not completely exact. In the solar neutrino problem the matter background is important and it enters the total Hamiltonian for neutrino propagation. Nevertheless, to very good accuracy the largest eigenvalue of the Hamiltonian is given by with eigenvector and corrections to this are of order where , denotes the electron number density in the sun and in the solar core eV. Furthermore, the evolution of in solar matter is adiabatic to an even better accuracy. Thus Eq.(21) is accurate for our purpose.
It remains to discuss region II. To this end we consider the atmospheric neutrino anomaly which is expressed through the deviation of the double ratio
(22) 
from 1. In Eq.(22) is the ratio of muon and electron events without neutrino oscillations. It is obtained by a Monte Carlo calculation which gives for subGeV events. For atmospheric neutrinos matter effects are nonnegligible. Analogously to Eq.(21) we have the lower bound
(23) 
Let us assume for the moment that . This is the case if CP is conserved or if the oscillating parts in the probabilities occurring in Eq.(23) drop out because of averaging processes involving neutrino energy and distance between source and detector. Then it is easily shown by Eqs.(22) and (23) that [21]
(24) 
for all energy ranges and zenith angle bins. In this case in region II we obtain
(25) 
The assumption is not fully satisfactory because it is not clear if or how well it is fulfilled. Let us therefore dispense with it now. The evolution of oscillation probabilities with a matter background has the general form [34, 20]
(26) 
where is a unitary matrix and diagonalizes the Hamiltonian for neutrino propagation in matter at the location . Note that Eq.(26) is the generalization of Eq.(2) referring to vacuum oscillations where has only diagonal elements given by and . Because , the matrix decomposes approximately into a 3 3 and a 1 1 block and therefore (see the discussion after Eq.(21))
(27) 
This consideration leads to
(28) 
For , the central expression of Eq.(28) has the minimum with respect to at . This explains the second part of the inequality. Eq.(28) represents a general bound valid for all energy ranges and zenith angles, whether assumption is fulfilled or not. Its righthand side is a decreasing function in and therefore in region II we arrive at
(29) 
Let us take advantage of the bin ( is the zenith angle) of the subGeV Superkamiokande events where (90% CL) [35]. Here is particularly small. In Fig.3 the horizontal lines indicate with its 90% CL interval taken from Ref.[35], the dashed line represents the bound (25) and the solid line the general bound (29). Taking into account that the SBL experiments and, in particular, LSND restrict to the range (20) ( eV) we see that the bound (25) rules out region II. However, the general bound (29) it is not tight enough around eV to fully exclude region II with a neutrino mass hierarchy because gets too large there.
There is a possiblity to improve the bound around 0.3 eV in the following way. For a mass hierarchy we have or where is the minimum measured by LSND. Thus we get
(30) 
The dashdotted curve in Fig.3 which branches off from the solid curve corresponds to the part of the lower bound (30) originating from . Therefore, comparing the lower bounds on obtained by using 90% CL data, namely the solid and the dashdotted lines, with the uppermost horizontal line which corresponds to the 90% CL experimental upper bound on we see that only a tiny allowed triangle is left in Fig.3. Thus we arrive at the conclusion:
With a 4neutrino mass hierarchy region II is strongly disfavoured by the atmospheric neutrino data and the results of all SBL neutrino oscillation experiments.
Let us summarize our findings for a 4neutrino mass hierarchy:

Region I: Excluded by the unitarity of .

Region II: Strongly disfavoured by atmospheric neutrino data.

Region III: Ruled out by LSND.

Region IV: Ruled out by solar neutrino data.
It is easy to show that with the arguments presented here all neutrino mass schemes where three masses are clustered and the fourth one is separated by the “LSND gap” are disfavoured by the present data [21, 22].
4 The favoured nonhierarchial 4neutrino mass spectra
Now we are left with only two possible neutrino mass spectra in which the four neutrino masses appear in two pairs separated by :
(31) 
We have to check that these mass spectra are compatible with the results of all neutrino oscillation experiments.
In schemes A and B the quantities (15) are defined with . Clearly, regions I and III (see Fig.2.3) are ruled out by LSND (see subsection 2.4). Let us first consider scheme A. For the survival probability of solar ’s have [34, 20]
(32) 
where is the survival probability involving , only. If , it follows from Eq.(32) that the survival probability of solar ’s practically does not depend on the neutrino energy and . This is disfavoured by the solar neutrino data [36]. Consequently, regions II and III are ruled out by the solar neutrino data. This argument does not apply to region IV and one can easily convince oneself that also the atmospheric neutrino anomaly is compatible with this region. Furthermore, looking at Eq.(18) we see that this upper bound on is linear in the small quantity in region IV. Since for all values of , in the case of scheme A the bound (18) is compatible with the result of the LSND experiment. For scheme B the analogous arguments lead to region II. Therefore we come to the conclusion that [21, 22]
(33) 
Schemes A and B have different consequences for the mesurement of the neutrino mass through the investigation of the endpoint part of the H spectrum. From Eq.(33) it follows that in the case of scheme A the neutrino mass that enters in the usual expression for the spectrum of H decay is approximately equal to the “LSND mass”, i.e., . If scheme B is realized in nature and , are very small, the mass measured in H experiments is at least two order of magnitude smaller than [21, 22].
5 Checks of the favoured neutrino schemes in LBL experiments
LBL neutrino oscillation experiments are sensitive to the socalled “atmospheric range” of – eV. For reactor experiments with MeV this amounts to km [37, 38] whereas in accelerator experiments with –10 GeV the length of the baseline is of order km [39, 40, 41] (see Eq.(6)). Let us consider scheme A for definiteness. Then in vacuum the probabilities of transitions in LBL experiments are given by
(34) 
This formula has been obtained from Eq.(2) taking into account the fact that in LBL experiments and dropping the terms proportional to the cosines of phases much larger than ( for and ). Such terms do not contribute to the oscillation probabilities averaged over the neutrino energy spectrum.
To obtain limits on the LBL oscillation probability (34) from the results of the SBL oscillation experiments, we employ the Cauchy–Schwarz inequality on the term with the summation over and use (15) with to find the inequalites
(35) 
and
(36) 
It can easily be shown [23] that Eq.(36) is schemeindependent and that both equations also hold for antineutrinos. Considering reactor experiments and taking into account Eq.(33) we obtain the bound
(37) 
which holds for both schemes. Inserting the numerical values of the function (see Fig.2.3) it turns out that the upper bound (37) is below the sensitivity of the CHOOZ experiment in the preferred range (20) of . For the accelerator experiments matter effects have to be taken into account. We have shown [23] that the mattercorrected version of Eq.(36) leads to stringent bounds on and LBL transition probabilities of the order of to depending on the value of and on the energy of the neutrino beam (for a study of LBL CP violation in schemes A and B see Ref.[24]).
6 Conclusions
In this report we have discussed the possible form of the neutrino mass spectrum that can be inferred from the results of all neutrino oscillation experiments, including the solar and atmospheric neutrino experiments. The crucial input are the three indications in favour of neutrino oscillations given by the solar neutrino data, the atmospheric neutrino anomaly and the result of the LSND experiment. These indications, which all pertain to different scales of neutrino masssquared differences, require that apart from the three wellknow neutrino flavours at least one additional sterile neutrino (without couplings to the and bosons) must exist. In our investigation we have assumed that there is one sterile neutrino and that the 4neutrino mixing matrix (1) is unitary. We have considered all possible schemes with four massive neutrinos which provide three scales of . We have argued that a neutrino mass hierarchy is not compatible with the abovementioned indications in favour of neutrino oscillations together with the negative results of all other SBL neutrino oscillation experiments other than LSND. The same holds for all mass spectra with three squares of neutrino masses clustered together, such that the gap between the cluster and the remaining masssquared determines relevant in SBL experiments.
Thus only two possible spectra of neutrino masses, denoted by A and B (see Eq.(31)), with two pairs of close masses separated by a mass difference of the order of 1 eV are compatible with the results of all neutrino oscillation experiments. The positive result of the LSND experiment confines the SBL masssquared to the interval (see Fig.3). If, of the two neutrino schemes defined by Eqs.(31) and (33), scheme A is realized in nature, the neutrino mass that is measured in H decay experiments coincides with the “LSND mass”. If the massive neutrinos are Majorana particles, in the case of scheme A, the experiments on the search for decay have good chances to obtain a positive result. Furthermore, schemes A and B have severe consequences for longbaseline neutrino oscillations: the survivial probability is close to one and the and transitions are strongly constrained.
Finally, we can ask ourselves what happens if not all experimental input data leading to schemes A and B are confirmed in future experiments. Among the many questions in this context, the two most burning ones concern LSND and the zenith angle variation in the atmospheric neutrino flux. Clearly, if LSND is not confirmed, three neutrinos are sufficient. If one nevertheless requires a 4th neutrino with a mass in the eV range for cosmological reasons then the neutrino spectrum is likely to be hierarchial because region III (see Fig.2.3) cannot be excluded in this case. If, on the other hand, the zenith angle variation in the atmospheric neutrino flux is not confirmed, a 3neutrino mixing scheme with eV and other definite predictions is possible [42]. We have to wait for future experimental results to see if the present interesting and puzzling situation concerning the neutrino mass and mixing pattern persists.
Acknowledgement
W.G. would like to thank the organizers of the workshop for their great hospitality and the stimulating and pleasant atmosphere.
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