Neutrino mass and oscillation as probes of physics beyond the Standard Model ^{*}^{*}*Invited article prepared for the Journal of the Egyptian Mathematical Society.
Abstract:
We present a review of the present status of the problem of neutrino masses and mixing including a survey of theoretical motivations and models, experimental searches and implications of recently appeared solar and atmospheric neutrino data, which strongly indicate nonzero neutrino masses a mixing angles.
SUSXTH00023
1 Introduction
The existence of a socalled neutrino, a light, neutral, feebly interacting fermion, was first proposed by W. Pauli in 1930 to save the principle of energy conservation in nuclear beta decay [1]. The idea was promptly adopted by the physics community; in 1933 E. Fermi takes the neutrino hypothesis, gives the neutrino its name and builds his theory of beta decay and weak interactions. But, it was only in 1956 that C. Cowan and F. Reines were able to discover the neutrino, more exactly the antineutrino, experimentally [2]. Danby et al. [3] confirmed in 1962 that there exist, at least, two types of neutrinos, the and . In 1989, the study of the Z boson lifetime allows to show with great certitude that only three light neutrino species do exist. Only in 2000, it has been confirmed by direct means [4] the existence of the third type of neutrino, the in addition to the and . Until here the history, with the present perspective we can say that the neutrino occupies a unique place among all the fundamental particles in many ways and as such it has shed light on many important aspects of our present understanding of nature and is still believed to hold a key role to the physics beyond the Standard Model (SM).
In what respects its mass, Pauli initially expected the mass of the neutrino to be small but not necessary zero: not very much more than the electron mass, F. Perrin in 1934 showed that its mass has to be less than that of the electron. After more than a half a century, the question of whether the neutrino has mass is still one open question, being one of the outstanding issues in particle physics, astrophysics, cosmology and theoretical physics in general. Presently, there are several theoretical, observational and experimental motivations which justify the searching for possible nonzero neutrino masses (see i.e. [5, 6, 7, 8, 9, 10, 11] for excellent older reviews on this matter).
Understanding of fermion masses in general are one of the major problems of the SM and observation of the existence or confirmation of nonexistence of neutrino masses could introduce useful new perspectives on the subject. If they are confirmed as massless they would be the only fermions with this property. A property which is not dictated by any known fundamental underlying principle, such as gauge invariance in the case of the photon. If it is concluded that they are massive then the question is why are their masses so much smaller than those of their charged partners. Although theory alone can not predict neutrino masses, it is certainly true that they are strongly suggested by present theoretical models of elementary particles and most extensions of the SM definitively require neutrinos to be massive. They therefore constitute a powerful probe of new physics at a scale larger than the electroweak scale.
If massive, the superposition postulates of quantum theory predict that neutrinos, particles with identical quantum numbers, could oscillate in flavor space. If the absolute difference of masses among them is small enough then these oscillations could have important phenomenological consequences. Some hints at accelerator experiments as well as the observed indications of spectral distortion and deficit of solar neutrinos and the anomalies on the ratio of atmospheric neutrinos and their zenith distribution are naturally accounted by the oscillations of a massive neutrino. Recent claims of the highstatistics highprecision SuperKamiokande (SK) experiment are unambiguous and left little room for the scepticism as we are going to see along this review.
Moreover, neutrinos are basic ingredients of astrophysics and cosmology. There may be a hot dark matter component (HDM) to the Universe: simulations of structure formation fit the observations only when some significant quantity of HDM is included. If so, neutrinos would be, at least by weight, one of the most important ingredients in the Universe.
Regardless of mass and oscillations, astrophysical interest in the neutrino and their properties arises from the fact that it is copiously produced in high temperature and/or high density environment and it often dominates the physics of those astrophysical objects. The interactions of the neutrino with matter is so weak that it passes freely through any ordinary matter existing in the Universe. This makes neutrinos to be a very efficient carrier of energy drain from optically thick objects becoming very good probes for the interior of such objects. For example, the solar neutrino flux is, together with heliosysmology, one of the two known probes of the solar core. A similar statement applies to objects as the typeII supernovas: the most interesting questions around supernovas, the explosion dynamics itself with the shock revival, and, the synthesis of the heaviest elements by the socalled rprocesses, could be positively affected by changes in the neutrino flux, e.g. by MSW active or sterile conversions [12]. Finally, ultra high energy neutrinos are called to be useful probes of diverse distant astrophysical objects. Active Galactic Nuclei (AGN) should be copious emitters of ’s, providing both detectable point sources and an observable diffuse background which is larger in fact than the atmospheric neutrino background in the very high energy range [13].
This review is organized as follows, in section 2 we discuss the neutrino in the SM. Section 3 is devoted to the possible ways for generating neutrino mass terms and different models for these possibilities are presented. Neutrino oscillation in vacuum and in matter are studied in section 4. The cosmological and the astrophysical constraints on diverse neutrino properties are summarized in section 5. In section 6 we give an introduction to the phenomenological description of neutrino oscillations in vacuum and in matter. In section 7 we give an extensive description of the different neutrino experiments, their results and their interpretation. Finally we present some conclusions and final remarks in section 7.
2 The neutrino in the Standard Model.
The current Standard Model of particles and interactions supposes the existence of three neutrinos. The three neutrinos are represented by twocomponent Weyl spinors each describing a lefthanded fermion. They are the neutral, upper components of doublets with respect the group, the weak interaction group, we have,
They have the third component of the weak isospin and are assigned an unit of the global th lepton number. The three righthanded charged leptons have however no counterparts in the neutrino sector and transform as singlets with respect the weak interaction.
These SM neutrinos are strictly massless, the reason for this can be understood as follows. The only Lorenz scalar made out of them is the Majorana mass, of the form ; it has the quantum number of a weak isotriplet, with as well as two units of total lepton number. Thus to generate a renormalizable Majorana mass term at the tree level one needs a Higgs isotriplet with two units of lepton number. Since in the stricter version of the SM the Higgs sector is only constituted by a weak isodoublet, there are no treelevel neutrino masses. When quantum corrections are introduced we should consider effective terms where a weak isotriplet is made out of two isodoublets and which are not invariant under lepton number symmetry. The conclusion is that in the SM neutrinos are kept massless by a global chiral lepton number symmetry (and more general properties as renormalizability of the theory, see Ref.[8] for an applied version of this argument). However this is a rather formal conclusion, there is no any other independent, compelling theoretical argument in favor of such symmetry, or, with other words, there is no reason why we would like to keep it intact.
Independent from mass and charge oddities, in any other respect neutrinos are very well behaved particles within the SM framework and some figures and facts are unambiguously known about them. The LEP Z boson lineshape measurements imply that are only three ordinary (weak interacting) light neutrinos [14, 15]. Big Bang Nucleosynthesis (BBN) constrains the parameters of possible sterile neutrinos, nonweak interacting or those which interact and are produced only by mixing [16]. All the existing data on the weak interaction processes in which neutrinos take part are perfectly described by the SM chargedcurrent (CC) and neutralcurrent (NC) Lagrangians:
(1)  
(2) 
where are the neutral and charged vector bosons intermediaries of the weak interaction. The CC and NC interaction Lagrangians conserve three total additive quantum numbers, the lepton numbers while the structure of the CC interactions is what determine the notion of flavor neutrinos .
There are no indications in favor of the violation of the conservation of these lepton numbers in weak processes and very strong bounds on branching ratios of rare, lepton number violating, processes are obtained, for examples see Table 1.
From the theoretical point of view, in the minimal extension of the SM where righthanded neutrinos are introduced and the neutrino gets a mass, the branching ratio of the decay is given by (2 generations are assumed [17]),
where are the neutrino masses, is the mass of the boson and is the mixing angle in the lepton sector. Using the experimental upper limit on the heaviest neutrino one obtains , a value far from being measurable at present as we can see from table 1 The and similar processes are sensitive to new particles not contained in the SM. The value is highly model dependent and could change by several orders of magnitude if we modify the neutrino sector for example introducing an extra number of heavy neutrinos.
3 Neutrino mass terms and models.
3.1 Model independent neutrino mass terms
Phenomenologically, Lagrangian mass terms can be viewed as terms describing transitions between right (R) and left (L)handed states. For a given minimal, Lorenz invariant, set of four fields: , wouldbe components of a generic Dirac Spinor, the most general mass part of the Lagrangian can be written as:
(3) 
In terms of the newly defined Majorana fields (): , , the Lagrangian can be rewritten as:
(4) 
where is the neutrino mass matrix defined as:
(5) 
We proceed further and diagonalizing the matrix M one finds that the physical particle content is given by two Majorana mass eigenstates: the inclusion of the Majorana mass splits the four degenerate states of the Dirac field into two nondegenerate Majorana pairs.
If we assume that the states are respectively active (belonging to weak doublets) and sterile (weak singlets), the terms corresponding to the ”Majorana masses” and transform as weak triplets and singlets respectively. While the term corresponding to is an standard, weak singlet in most cases, Dirac mass term.
The neutrino mass matrix can easily be generalized to three or more families, in which case the masses become matrices themselves. The complete flavor mixing comes from two different parts, the diagonalization of the charged lepton Yukawa couplings and that of the neutrino masses. In most simple extensions of the SM, this CKMlike leptonic mixing is totally arbitrary with parameters only to be determined by experiment. Their prediction, as for the quark hierarchies and mixing, needs further theoretical assumptions (i.e. Ref.[18, 8] predicting maximal mixing).
We can analyze different cases. In the case of a purely Dirac mass term, in Eq.(4), the states are degenerate with mass and a four component Dirac field can be recovered as . It can be seen that, although violating individual lepton numbers, the Dirac mass term allows a conserved lepton number .
In the general case, pure Majorana mass transition terms, or terms in Lagrangian (4), describe in fact a particleantiparticle transition violating lepton number by two units (). They can be viewed as the creation or annihilation of two neutrinos leading therefore to the possibility of the existence of neutrinoless double beta decay.
In the general case where all classes of terms are allowed, it is interesting to consider the socalled ”seesaw” limit in Eq.(4). In this limit taking , the two Majorana neutrinos acquire respectively masses ,. There is one heavy neutrino and one neutrino much lighter than the typical Dirac fermion mass. One of neutrino mass has been automatically suppressed, balanced up (“seesaw”) by the heavy one. The ”seesaw” mechanism is a natural way of generating two well separated mass scales.
3.2 Neutrino mass models
Any fully satisfactory model that generates neutrino masses must contain a natural mechanism that explains their small value, relative to that of their charged partners. Given the latest experimental indications it would also be desirable that includes any comprehensive justification for light sterile neutrinos and large, near maximal, mixing.
Different models can be distinguished according to the new particle content or according to the scale. According to the particle content, of the different open possibilities, if we want to break lepton number and to generate neutrino masses without introducing new fermions in the SM, we must do it by adding to the SM Higgs sector fields carrying lepton numbers, one can arrange then to break lepton number explicitly or spontaneously through their interactions. But, possibly, the most straightforward approach to generate neutrino masses is to introduce for each neutrino an additional weak neutral singlet.
This happens naturally in the framework of LR symmetric models where the origin of SM parity () violation is ascribed to the spontaneous breaking of a baryonlepton (the quantum number) symmetry.
In the GUT the Majorana neutral particle enters in a natural way in order to complete the matter multiplet, the neutral is a singlet.
According to the scale where the new physics have relevant effects, Unification (i.e. the aforementioned GUT) and weakscale approaches (i.e. radiative models) are usually distinguished [19, 20].
The anomalies observed in the solar neutrino flux, atmospheric flux and low energy accelerator experiments cannot all be explained consistently without introducing a light, then necessarily sterile, neutrino. If all the Majorana masses are small, active neutrinos can oscillate into the sterile right handed fields. Light sterile neutrinos can appear in particular seesaw mechanisms if additional assumptions are considered (“singular seesaw “ models) with some unavoidable fine tuning. The alternative to such fine tuning would be seesawlike suppression for sterile neutrinos involving new unknown interactions, i.e. family symmetries, resulting in substantial additions to the SM, (i.e. some sophisticated superstringinspired models, Ref.[21]).
Finally some example of weak scale models, radiative generated mass models where the neutrino masses are zero at tree level constitute a very different class of models: they explain in principle the smallness of for both active and sterile neutrinos. Different mass scales are generated naturally by different number of loops involved in generating each of them. The actual implementation generally requires however the adhoc introduction of new Higgs particles with nonstandard electroweak quantum numbers and lepton numberviolating couplings [22].
The origin of the different Dirac and Majorana mass terms appearing above is usually understood by a dynamical mechanism where at some scale or another some symmetry is spontaneously broken as follows.
First we will deal with the Dirac mass term. For the case of interest, and are SU(2) doublets and singlets respectively, the mass term describes then a transition and is generated from SU(2) breaking with a Yukawa coupling:
(6) 
Where are the components of the Higgs doublet. The coefficient is the Yukawa coupling. One has that, after symmetry breaking, where is the vacuum expectation value of the Higgs doublet. A neutrino Dirac mass is qualitatively just like any other fermion masses, but that leads to the question of why it is so small in comparison with the rest of fermion masses: one would require in order to have eV. Or in other words: while for the hadronic sector we have .
Now we will deal with the Majorana mass terms. The term will appear if is a gauge singlet. In this case a renormalizable mass term of the type is allowed by the SM gauge group symmetry. However, it would not be consistent in general with unified symmetries, i.e. with a full SO(10) symmetry and some complicated mechanism should be invocated. A term is usually associated with the breaking of some larger symmetry, the expected scale for it should be in a range covering from TeV (LR models) to GUT scales GeV.
Finally, the term will appear if is active, belongs to some gauge doublet. In this case we have =1 and must be generated by either a) an elementary Higgs triplet or b) by an effective operator involving two Higgs doublets arranged to transform as a triplet. In case a), for an elementary triplet , where is a Yukawa coupling and is the triplet VEV. The simplest implementation (the old GelminiRoncadelli model [23]) is excluded by the LEP data on the Z width: the corresponding Majoron couples to the Z boson increasing significantly its width. Variant models involving explicit lepton number violation or in which the Majoron is mainly a weak singlet ([24], invisible Majoron models) could still be possible. In case b), for an effective operator originated mass, one expects where is the scale of the new physics which generates the operator.
A few words about the range of expected neutrino masses for different types of models depending on the values of . For TeV (LR models) and with typical ’s, one expects masses of order eV, 10 keV, and 1 MeV for the respectively. GUT theories motivates a big range of intermediate scales GeV. In the lower end of this range, for GeV (some superstringinspired models, GUT with multiple breaking stages) one can obtain light neutrino masses of the order eV, eV, 10 eV). At the upper end, for (grand unified seesaw with large Higgs representations) one typically finds smaller masses around , , ) eV somehow more difficult to fit into the present known experimental facts.
3.3 The magnetic dipole moment and neutrino masses
The magnetic dipole moment is another probe of possible new interactions. Majorana neutrinos have identically zero magnetic and electric dipole moments. Flavor transition magnetic moments are allowed however in general for both Dirac and Majorana neutrinos. Limits obtained from laboratory experiments are of the order of a few and those from stellar physics or cosmology are . In the SM electroweak theory, extended to allow for Dirac neutrino masses, the neutrino magnetic dipole moment is nonzero and given, as ([15] and references therein):
(7) 
where is the Bohr magneton. The proportionality of to the neutrino mass is due to the absence of any interaction of other than its Yukawa coupling which generates its mass. In LR symmetric theories is proportional to the charged lepton mass: a value of can be reached still too small to have practical astrophysical consequences.
Magnetic moment interactions arise in any renormalizable gauge theory only as finite radiative corrections. The diagrams which generate a magnetic moment will also contribute to the neutrino mass once the external photon line is removed. In the absence of additional symmetries a large magnetic moment is incompatible with a small neutrino mass. The way out suggested by Voloshin consists in defining a SU(2) symmetry acting on the space , magnetic moment terms are singlets under this symmetry. In the limit of exact SU(2) the neutrino mass is forbidden but is allowed [25]. Diverse concrete models have been proposed where such symmetry is embedded into an extension of the SM (leftright symmetries, SUSY with horizontal gauge symmetries [26]).
4 Aspects of some theoretical models for neutrino mass
4.1 Neutrino masses in LR models
A very natural way to generate neutrino mass is to minimally extend the SM including additional 2spinors as right handed neutrinos and at the same time extend the, nonQCD, SM gauge symmetry group to . The resulting model, initially proposed in 19731974, is known as the leftright (LR) symmetric model [27]. This kind of models were first proposed with the goal of seeking a spontaneous origin for violation in weak interactions: CP and P are conserved at large energies; at low energies, however, the group breaks down spontaneously at a scale . Any new physics correction to the SM would be of order where ; if we choose we obtain only small corrections, compatible with present known physics. We can satisfactorily explain in this case the small quantity of CP violation observed in present experiments and why the neutrino mass is so small, as we will see below.
The quarks () and leptons () in LR models transform as doublets under the group as follows: and . The gauge interactions are symmetric between left and right handed fermions; therefore before symmetry spontaneous breaking, weak interactions, as the others, conserve parity
The breaking of the gauge symmetry is implemented by multiplets of LR symmetric Higgs fields, the concrete choosing of these multiplets is not unique. It has been shown that in order to understand the smallness of the neutrino mass, it is convenient to choose respectively one doublet and two triplets as follows:
The Yukawa couplings of these Higgs fields to the quarks and leptons are given by
(8)  
The gauge symmetry breaking proceeds in two steps. The is broken down to by choosing since this carries both and quantum numbers. It gives mass to charged and neutral right handed gauge bosons, i.e.,
Furthermore, as consequence of term in the Lagrangian above this stage of symmetry breaking also leads to a mass term for the righthanded neutrinos of the order .
Next, as we break the SM symmetry by turning on the vev’s for fields as , with , we give masses to the and the bosons and also to quarks and leptons (). At the end of the process of spontaneous symmetry breaking the two bosons of the model will mix, the lowest physical mass eigenstate is identified as the observed W boson. Current experimental limits set the limit (see Ref.[15], at 90% CL) GeV.
In the neutrino sector the above Yukawa couplings after breaking by leads to the Dirac masses for the neutrino. The full process leads to the following mass matrix for the , , (the matrix M in eq.5)
(9) 
From the structure of this matrix we can see the seesaw mechanism at work. By diagonalizing M, we get a light neutrino corresponding to the eigenvalue and a heavy one with mass .
Variants of the basic model include the possibility of having Dirac neutrinos as the expense of enlarging the particle content. The introduction of two new singlet fermions and a new set of carefullychosen Higgs bosons, allows us to write the mass matrix [28]:
(10) 
Matrix 10 leads to two Dirac neutrinos, one heavy with mass and another light with mass . This light four component spinor has the correct weak interaction properties to be identified as the neutrino. A variant of this model can be constructed by addition of singlet quarks and leptons. One can arrange these new particles in order that the Dirac mass of the neutrino vanishes at the tree level and arises at the oneloop level via mixing.
Leftright symmetric models can be embedded in grand unification groups. The simplest GUT model that leads by successive stages of symmetry breaking to leftright symmetric models at low energies is the based model. A example of LR embedding GUT Supersymmetric theories will be discussed below in the context of Superstringinspired models.
4.2 SUSY models: Neutrino masses without righthanded neutrinos
Supersymmetry (SUSY) models with explicit broken parity provide an interesting example of how we can generate neutrino masses without using a righthanded neutrino but incorporating new particles and enlarging the Higgs sector.
In a generic SUSY model, due to the Higgs and lepton doublet superfields have the same quantum numbers, we have in the superpotential terms, bilinear or trilinear in the superfields, that violate baryon and lepton number explicitly. They lead to a mass for the neutrino but also to to proton decay with unacceptable high rates. One radical possibility is to introduce by hand a symmetry that rule out these terms, this is the role of the symmetry introduced in the MSSM.
A less radical possibility is to allow for the existence in the of superpotential of a bilinear term, i.e. . This is simplest way to illustrate the idea of generating neutrino mass without spoiling current limits on proton decay. The bilinear violation of parity implied by the term leads [29] by a minimization condition to a nonzero sneutrino vev, . In such a model the neutrino acquire a mass, due to the mixing between neutrinos and neutralinos. The and neutrinos remain massless in this model, it is supposed that they get masses from scalar loop contributions. The model is phenomenologically equivalent to a three Higgs doublet model where one of these doublets (the sneutrino) carry a lepton number which is broken spontaneously. We have the following mass matrix for the neutralinoneutrino sector, in block form the matrix reads:
(11) 
where corresponding to the two gauginos masses. The is a matrix containing the vevs of , and the sneutrino. The next two rows are Higgsinos and the last one denotes the tau neutrino. Let us remind that gauginos and Higgsinos are the supersymmetric fermionic counterparts of the gauge and Higgs fields.
In diagonalizing the mass matrix , a “seesaw” mechanism is again at work, in which the role of scale masses are easily recognized. It turns out that mass is given by (),
where is the largest gaugino mass. However, in an arbitrary SUSY model this mechanism leads to (although relatively small if is large) still too large masses. To obtain a realistically small mass we have to assume universality among the soft SUSY breaking terms at GUT scale. In this case the mass is predicted to be small due to a cancellation between the two terms which makes negligible the .
We consider now the properties of neutrinos in superstring models. In a number of these models, the effective theory imply a supersymmetric grand unified model, with matter fields belonging to the dimensional representations of group plus additional singlet fields. The model contains additional neutral leptons in each generation and neutral singlets, gauginos and Higgsinos. As before but with a larger number of them, all of these neutral particles can mix, making the understanding of neutrino masses quite difficult if no simplifying assumptions are employed.
Several of these mechanisms have been proposed to understand neutrino masses [28]. In some of these mechanisms the huge neutral mixing mass matrix is reduced drastically down to a neutrino mass matrix result of the mixing of the with an additional neutral field whose nature depends on the particular mechanism. In the basis the mass matrix is of the form (with possibly being zero):
(12) 
We distinguish two important cases, the Rparity violating case and the mixing with a singlet, where the sneutrinos, superpartners of , are assumed to acquire a v.e.v. of order .
In the first case the field corresponds to a gaugino with a Majorana mass that can arise at twoloop order. Usually GeV, if we assume TeV additional dangerous mixing with the Higgsinos can be neglected and we are lead to a neutrino mass eV. Thus, smallness of neutrino mass is understood without any fine tuning of parameters.
In the second case the field corresponds to one of the singlets presents in the model [30, 31]. One has to rely on symmetries that may arise in superstring models on specific CalabiYau space to conveniently restrict the Yukawa couplings. If we have in matrix 12, this leads to a massless neutrino and a massive Dirac neutrino. There would be neutrino mixing even if the light neutrino remains strictly massless. If we include a possible Majorana mass term for the fermion of order GeV we get similar values of the neutrino mass as in the previous case.
It is worthy to mention that mass matrices as that one appearing in expression 12 have been proposed without embedding in a supersymmetric or any other deeper theoretical framework. In this case small tree level neutrino masses are obtained without making use of large scales. For example, the model proposed by Ref.[32] (see also Ref.[33]) which incorporates by hand additional isosinglet neutral fermions. The smallness of neutrino masses is explained directly from the, otherwise left unexplained, smallness of the parameter in such a model.
4.3 Neutrino masses and extra dimensions
Recently, models where spacetime is endowed with extra dimensions (4+) have received some interest [34]. It has been realized that the fundamental scale of gravity need not be the 4dimensional “effective” Planck scale but a new scale , as low as TeV. The observed Planck scale is then related to in dimensions, by
where is the typical length of the extra dimensions. i.e., the coupling is for TeV. For , the radii of the extra dimensions are of the order of the millimeter, which could be hidden from many, extremely precise, measurements that exist at present but it would give hope to probe the concept of hidden space dimensions (and gravity itself) by experiment in the near future.
According to current theoretical frameworks (see for example Ref. [34]), all the SM groupcharged particles are localized on a dimensional hypersurface ‘brane’ embedded in the bulk of the extra dimensions. All the particles split in two categories, those that live on the brane and those which exist every where, as ‘bulk modes’. In general, any coupling between the brane and the bulk modes are suppressed by the geometrical factor . Graviton and possible other neutral states belongs to the second category. The observed weakness of gravity can be then interpreted as a result of the new space dimensions in which gravity can propagate.
The small coupling above can also be used to explain the smallness of the neutrino mass [35]. The left handed neutrino having weak isospin and hypercharge must reside on the brane. Thus it can get a naturally small Dirac mass through the mixing with some bulk fermion which can be interpreted as right handed neutrinos :
Here are the Higgs doublet fields and a Yukawa coupling. After EW breaking this interaction will generate the Dirac mass . The right handed neutrino has a whole tower of KaluzaKlein relatives . The masses of these states are given by , the couples with all with the same mixing mass. We can write the mass Lagrangian as where , and the resulting mass matrix being:
(13) 
The eigenvalues of the matrix are given by a transcendental equation. In the limit, , or , the eigenvalues are , with a doublydegenerated zero eigenvalue.
Other examples can be considered which incorporates a LR symmetry (see for example Ref. [36]), a right handed neutrino is assumed to live on the brane together with the standard one. In this class of models, it has been shown that the left handed neutrino is exactly massless whereas assumed bulk sterile neutrinos have masses related to the size of the extra dimensions. They are of order eV, if there is at least one extra dimension with size in the micrometer range.
4.4 Family symmetries and neutrino masses
The observed mass and mixing interfamily hierarchy in the quark and, presumably in the lepton sector might be a consequence of the existence of a number of family symmetries [37]. The observed intrafamily hierarchy, the fact that for each family , seem to require one of these to be anomalous [38, 39].
A simple model with one familydependent anomalous beyond the SM was first proposed in Ref.[38] to produce the observed Yukawa hierarchies, the anomalies being canceled by the GreenSchwartz mechanism which as a byproduct is able to fix the Weinberg angle (see also Ref.[39]).
Recent developments includes the model proposed in Ref.[18], which is inspired by models generated by the heterotic string. The gauge structure of the model is that of the SM augmented by three Abelian symmetries , the first one is anomalous and family independent. Two of the them, the nonanomalous ones, have specific dependences on the three chiral families designed to reproduce the Yukawa hierarchies. There are right handed neutrinos which trigger neutrino masses by the seesaw mechanism.
The three symmetries are spontaneously broken at a high scale M by stringy effects. It is assumed that three fields acquire a vacuum value. The fields are singlets under the SM symmetry but not under the and symmetries. In this way, the Yukawa couplings appear as the effective operators after spontaneous symmetry breaking.
For neutrinos we have [40] the mass Lagrangian
where . The parameter determine the mass and mixing hierarchy, where is the Cabibbo angle. The are the charges assigned respectively to left handed leptons and right handed neutrinos .
These coupling generate the following mass matrices for neutrinos:
(14) 
From these matrices, the seesaw mechanism gives the formula for light neutrinos:
The neutrino mass mixing matrix depends only on the charges assigned to the left handed neutrinos, by a cancellation of right handed neutrino charges by virtue of the seesaw mechanism. There is freedom in assigning charges . If the charges of the second and the third generations of leptons are equal (), then one is lead to a mass matrix which have the following structure:
(15) 
where . This matrix can be diagonalized by a large rotation, it is consistent with a large mixing. In this theory, explanation of the large neutrino mixing is reduced to a theory of prefactors in front of powers of the parameter .
5 Cosmological Constraints
5.1 Cosmological mass limits and Dark Matter
There are some indirect constraints on neutrino masses provided by cosmology. The most relevant is the constraint which follows from demanding that the energy density in neutrinos should not be too high. At the end of this section we will deal with some other limits as the lower mass limit obtained from galactic phase space requirements or limits on the abundance of additional weakly interacting light particles.
Stable neutrinos with low masses ( MeV) make a contribution to the total energy density of the universe which is given by:
(16) 
where the total mass , with the number of degrees of freedom for Dirac (Majorana) neutrinos. The number density of the neutrino sea is related to that one of photons by entropy conservation in the adiabatic expansion of the universe, , and this last one is very accurately obtained from the CMBR measurements, cm (for a Planck spectrum with eV). Writing , where is the critical energy density of the universe (), we have ()
(17) 
where is the reduced Hubble constant, recent analysis [41] give the favored value: .
Constrained by requirements from BBN Nucleosynthesis, galactic structure formation and large scale observations, increasing evidence (luminositydensity relations, galactic rotation curves,large scale flows) suggests that [42]
(18) 
where is the total mass density of the universe, as a fraction of the critical density . This includes contributions from a variety of sources: photons, baryons, nonbaryonic Cold Dark Matter (CDM) and Hot Dark Matter (HDM).
The two first components are rather well known. The photon density is very well known to be quite small: . The deuterium abundance BBN constraints [43] on the baryonic matter density () of the universe
The hot component, HDM is constituted by relativistic longlived particles with masses much less than keV, in this category would enter the neutrinos. Detailed simulations of structure formation fit the observations only when one has some 20 % of HDM (plus CDM), the best fit being two neutrinos with a total mass of 4.7 eV. There seems to be however some kind of conflict within cosmology itself: observations of distant objects favor a large cosmological constant instead of HDM (see Ref.[44] and references therein). One may conclude that the HDM part of does not exceed 0.2.
Requiring that , we obtain . From here and from Eq.17, we obtain the cosmological upper bound on the neutrino mass
Mass limits, in this case lower limits, for heavy neutrinos ( GeV) can also be obtained along the same lines. The situation gets very different if the neutrinos are unstable, one gets then joint bounds on mass and lifetime, then mass limits above can be avoided.
There is a limit to the density of neutrinos (or weak interacting dark matter in general) which can be accumulated in the halos of astronomical objects (the TremaineGunn limit): if neutrinos form part of the galactic bulges phasespace restrictions from the FermiDirac distribution implies a lower limit on the neutrino mass [45]:
The abundance of additional weakly interacting light particles, such as a light sterile , is constrained by BBN since it would enter into equilibrium with the active neutrinos via neutrino oscillations. A limit on the mass differences and mixing angle with another active neutrino of the type eV should be fulfilled in principle. From here is deduced that the effective number of neutrino species is
However systematical uncertainties in the derivation of the BBN bound make it too unreliable to be taken at face value and can eventually be avoided [46].
5.2 Neutrino masses and lepton asymmetry
In supersymmetric LR symmetric models, inflation, baryogenesis (or leptogenesis) and neutrino oscillations can become closely linked.
Baryosinthesis in GUT theories is in general inconsistent with an inflationary universe. The exponential expansion during inflation will wash out any baryon asymmetry generated previously at GUT scale. One way out of this difficulty is to generate the baryon or lepton asymmetry during the process of reheating at the end of the inflation. In this case the physics of the scalar field that drives the inflation, the inflaton, would have to violate CP (see Ref.[45] and references therein).
The challenge of any baryosinthesis model is to predict the observed asymmetry which is usually written as a baryon to photon (number or entropy) ratio. The baryon asymmetry is defined as
(19) 
At present there is only matter and not known antimatter, . The entropy density is completely dominated by the contribution of relativistic particles so is proportional to the photon number density which is very well known from CMBR measurements, at present . Thus, . From BBN we know that so we arrive to and from here we obtain equally the lepton asymmetry ratio.
It was shown in Ref. [47] that hybrid inflation can be successfully realized in a SUSY LR symmetric model with gauge group . The inflaton sector of this model consists of the two complex scalar fields and which at the end of inflation oscillate about the SUSY minimum and respectively decay into a pair of righthanded sneutrinos () and neutrinos. In this model, a primordial lepton asymmetry is generated [48] by the decay of the superfield which emerges as the decay product of the inflaton. The superfield decays into electroweak Higgs and (anti)lepton superfields. This lepton asymmetry is subsequently partially converted into baryon asymmetry by nonperturbative EW sphalerons.
The resulting lepton asymmetry [49] can be written as a function of a number of parameters among them the neutrino masses and mixing angles and compared with the observational constraints above.
It is highly nontrivial that solutions satisfying the constraints above and other physical requirements can be found with natural values of the model parameters. In particular, it is shown that the values of the neutrino masses and mixing angles which predict sensible values for the baryon or lepton asymmetry turn out to be also consistent with values required to solve the solar neutrino problem.
6 Phenomenology of Neutrino Oscillations
6.1 Neutrino Oscillation in Vacuum
If the neutrinos have nonzero mass, by the basic postulates of the quantum theory there will be in general mixing among them as in the case of quarks. This mixing will be observable at macroscopic distances from the production point and therefore will have practical consequences only if the difference of masses of the different neutrinos is very small, typically eV.
In presence of masses, weak () and mass () basis of eigenstates are differentiated. To transform between them we need an unitary matrix . Neutrinos can only be created and detected as a result of weak processes, at origin we have a weak eigenstate:
We can easily construct an heuristic theory of neutrino oscillations if we ignore spin effects as follows. After a certain time the system has evolved into
where is the Hamiltonian of the system, free evolution in vacuum is characterized by where . In most cases of interest (MeV, eV), it is appropriated the ultrarelativistic limit: in this limit and . The effective neutrino Hamiltonian can then be written and
In the last expression we have written the effective Hamiltonians in the weak basis with . This derivation can be put in a firm basis and one finds again the same expressions as the first terms of rigorous expansions in , see for example the treatment using FoldyWoythusen transformations in Ref.[104].
The results of the neutrino oscillation experiments are usually analyzed under the simplest assumption of oscillations between two neutrino types, in this case the mixing matrix is the well known 2dimension orthogonal rotation matrix depending on a single parameter . If we repeat all the computation above for this particular case, we find for example that the probability that a weak interaction eigenstate neutrino has oscillated to other weak interaction eigenstate neutrino after traversing a distance is
(20) 
where the oscillation length is defined by and . Numerically, in practical units, it turns out that
These probabilities depend on two factors: a mixing angle factor and a kinematical factor which depends on the distance traveled, on the momentum of the neutrinos, as well as on the difference in the squared mass of the two neutrinos. Both, the mixing factor and the kinematical factor should be of to have a significant oscillations.
6.2 Neutrino Oscillations in Matter
When neutrinos propagate in matter, a subtle but potentially very important effect, the MSW effect, takes place which alters the way in which neutrinos oscillate into one another.
In matter the neutrino experiences scattering and absorption, this last one is always negligible. At very low energies, coherent elastic forward scattering is the most important process. As in optics, the net effect is the appearance of a phase difference, refractive index or equivalently a neutrino effective mass.
This effective mass can considerable change depending on the densities and composition of the medium, it depends also on the nature of the neutrino. In the neutrino case the medium is flavordispersive: the matter is usually nonsymmetric with respect and and the effective mass is different for the different weak eigenstates [50].
This is explained as follows for the simpler and most important case, the solar electron plasma. The electrons in the solar medium have charged current interactions with but not with or . The resulting interaction energy is given by , where and are the Fermi coupling and the electron density. The corresponding neutral current interactions are identical for all neutrino species and hence have no net effect on their propagation. Hypothetical sterile neutrinos would have no interaction at all. The effective global Hamiltonian in flavor space is now the sum of two terms, the vacuum part we have seen previously and the new interaction energy:
The practical consequence of this effect is that the oscillation probabilities of the neutrino in matter could largely increase due to resonance phenomena [51]. In matter, for the two dimensional case and in analogy with vacuum oscillation, one defines an effective mixing angle as
(21) 
The presence of the term proportional to the electron density can give rise to a resonance. There is a critical density , given by
for which the matter mixing angle becomes maximal , irrespective of the value of mixing angle . The probability that oscillates into a after traversing a distance in this medium is given by Eq.(20), with two differences. First . Second, the kinematical factor differ by the replacement of . Hence it follows that, at the critical density,
(22) 
This formula shows that one can get full conversion of a weak interaction eigenstate into a weak interaction eigenstate, provided that the length and the energy satisfy the relations
There is a second interesting limit to consider. This is when the electron density is so large such that or . In this limit, there are no oscillations in matter because vanishes and we have
7 Experimental evidence and phenomenological analysis
In the second part of this review, we will consider the existing experimental situation. It is fair to say that at present there are at least an equal number of positive as negative (or better ”nonpositive”) indications in favor of neutrino masses and oscillations.
7.1 Laboratory, reactor and accelerator results.
No indications in favor of a nonzero neutrino masses have been found in direct kinematical searches for a neutrino mass.
From the measurement of the high energy part of the tritium decay spectrum, upper limits on the electron neutrino mass are obtained. The two more sensitive experiments in this field, Troitsk [52] and Mainz [53], obtain results which are plagued by interpretation problems: apparition of negative mass squared and bumps at the end of the spectrum.
In the Troistk experiment, the shape of the observed spectrum proves to be in accordance with classical shape besides a region 15 eV below the endpoint, where a small bump is observed; there are indications of a periodic shift of the position of this bump with a period of “exactly” year [54]. After accounting for the bump, they derive the limit eV, or eV (95[54].
The latest published results by the Mainz group leads to eV (1998 “Mainz data 1”), From which an upper limit of eV [53] (95% C.L., unified approach) is obtained. Preliminary data (1998 and 1999 measurements) provide a limit eV [55]. Some indication for the anomaly, reported by the Troitsk group, was found, but its postulated half year period is not supported by their data.
Diverse exotic explanations have been proposed to explain the Troitsk bump and their seasonal dependence. The main feature of the effect might be “phenomenologically” interpreted, not without problems, as He capture of relic neutrinos present in a high density cloud around the Sun [52, 56].
The Mainz and Troitsk ultimate sensitivity expected to be limited by systematics lies at the eV level. In the near future, it is planned a new large tritium experiment with sensitivity eV [55].
Regarding the heavier neutrinos, other kinematical limits are the following:

Limits for the muon neutrino mass have been derived using the decay channel at intermediate energy accelerators (PSI, LANL). The present limits are keV [57].

A tau neutrino mass of less than 30 MeV is well established and confirmed by several experiments: limits of 28, 30 and 31 MeV have also been obtained by the OPAL, CLEO and ARGUS experiments respectively (see Ref.[58] and references therein). The best upper limit for the neutrino mass has been derived using the decay mode by the ALEPH collaboration [59]: 18 MeV (95% CL).
Many experiments on the search for neutrinoless doublebeta decay [],
have been performed. This process is possible only if neutrinos are massive and Majorana particles. The matrix element of the process is proportional to the effective Majorana mass . Uncertainties in the precise value of upper limits are relatively large since they depend on theoretical calculations of nuclear matrix elements. From the nonobservation of , the HeidelbergMoscow experiment gives the most stringent limit on the Majorana neutrino mass. After 24 kg/year of data [60] (see also earlier results in Ref.[61]), they set a lower limit on the halflife of the neutrinoless double beta decay in Ge of yr at 90% CL, thus excluding an effective Majorana neutrino mass eV (90% CL). This result allows to set strong constraints on degenerate neutrino mass models. In the next years it is expected an increase in sensitivity allowing limits down to the eV levels (GENIUS I and II experiments, [62]).
Many shortbaseline (SBL) neutrino oscillation experiments with reactor and accelerator neutrinos did not find any evidence of neutrino oscillations. For example experiments looking for be or dissaperance (Bugey, CCFR [63, 64]) or oscillations (CCFR,E776[64, 65]).
The first reactor longbaseline (L 9981115 m) neutrino oscillation experiment CHOOZ found no evidence for neutrino oscillations in the disappearance mode [66, 67]. CHOOZ results are important for the atmospheric deficit problem: as is seen in Fig.(1) they are incompatible with an oscillation hypothesis for the solution of the atmospheric problem. Their latest results [67] imply an exclusion region in the plane of the twogeneration mixing parameters (with normal or sterile neutrinos) given approximately by for maximum mixing and for large (as shown approximately in Fig.(1) (left) which corresponds to early results). Lower sensitivity results, based only on the comparison of the positron spectra from the two differentdistance nuclear reactors, has also been presented, they are shown in Fig.(1) (right). These are independent of the absolute normalization of the antineutrino flux, the cross section and the target and detector characteristics and are able alone to almost completely exclude the SK allowed oscillation region [67].
The Palo Verde Neutrino Detector searches for neutrino oscillations via the disappearance of electron antineutrinos produced by a nuclear reactor at a distance m. The experiment has been taking neutrino data since October 1998 and will continue taking data until the end of 2000 reaching its ultimate sensitivity. The analysis of the 19981999 data (first 147 days of operation) [68] yielded no evidence for the existence of neutrino oscillations. The ratio of observed to expected number of events:
The resulting exclusion plot is very similar to the CHOOZ one. Together with results from CHOOZ and SK, concludes that the atmospheric neutrino anomaly is very unlikely to be due to oscillation.
Los Alamos LSND experiment has reported indications of possible oscillations [69]. They search for ’s in excess of the number expected from conventional sources at a liquid scintillator detector located 30 m from a proton beam dump at LAMPF. It has been claimed that a signal has been detected via the reaction with energy between 36 and 60 MeV, followed by a from (2.2 MeV).
The LSND experiment took its last beam on December, 1998. The analysis of the complete 19931998 data set (see Refs.[70, 71, 72]) yields a fittedestimated excess of of . If this excess is attributed to neutrino oscillations of the type , it corresponds to an oscillation probability of . The results of a similar search for oscillations where the (high energy, MeV) are detected via the CC reaction provide a value for the corresponding oscillation probability of (19931997 data).
There are other exotic physics explanations of the observed antineutrino excess. One example is the leptonnumber violating decay , which can explain these observations with a branching ratio , a value which is lower but not very far from the respective existing upper limits (, [15]).
The surprisingly positive LSND result has not been confirmed by the KARMEN experiment (Rutherford Karlsruhe Laboratories). This experiment, following a similar experimental setup as LSND, searches for produced by oscillations at a mean distance of 17.6 m. The time structure of the neutrino beam is important for the identification of the neutrino induced reactions and for the suppression of the cosmic ray background. Systematic time anomalies not completely understood has been reported which rest credibility to any further KARMEN claim. They see an excess of events above the typical muon decay curve, which is sigmas off (19901999 data, see Ref.[73]) and which could represent an unknown instrumental effect.
Exotic explanations as the existence of a weakly interacting particle “X”, for example a mixing of active and sterile neutrinos, of a mass 33.9 MeV have been proposed as an alternative solution to these anomalies and their consequences extensively studied [73, 74]. This particle might be produced in reactions the and decay as . KARMEN set upper limits on the visible branching ratio lifetime . From their results [73] one obtains the relation ()
More concretely, the results are as it follows. About antineutrino signal, the 19901995 and early 19971998 KARMEN data showed inconclusive results: They found no events, with an expected background of events, for oscillations [75]. The results of the search Feb. 1997 Dec. 1999 which include a 40fold improvement in suppression of cosmic induced background has been presented in a preliminary way [73, 76]. They find this time 9.5 oscillation candidates in agreement with the, claimed, well known background expectation of events. An upper limit for the mixing angle is deduced: (90% C.I.) for large eV). The positive LSND result in this channel could not be completely excluded but they are able to exclude the entire LSND favored regions above 2 eV and most of the rest of its favored parameter space.
In the present phase, the KARMEN experiment will take data until spring 2001. At the end of this period, the KARMEN sensitivity is expected to be able to exclude the whole parameter region of evidence suggested by LSND if no oscillation signal were found (Fig.2). The first phase of a third pion beam dump experiment designed to set the LSNDKARMEN controversy has been approved to run at Fermilab. Phase I of ”BooNe” ( MiniBooNe) expects a 10 signal ( events) and thus will make a decisive statement either proving or ruling it out. Plans are to run early 2001. Additionally, there is a letter of intent of a similar experiment to be carried out at the CERN PS [77, 78].
The K2K experiment started in 1999 the era of very longbaseline neutrinooscillation experiment using a welldefined neutrino beam.
In the K2K experiment ( 250 km), the neutrino beam generated by the KEK proton synchrotron accelerator is aimed at the near and far detectors, which are carefully aligned in a straight line. Then, by comparing the neutrino events recorded in these detectors, they are able to examine the neutrino oscillation phenomenon. SuperKamiokande detector itself acts as the far detector. The K2K near detector complex essentially consists of a one kiloton water Cerenkov detector (a miniature SuperKamiokande detector).
A total intensity of protons on target, which is about 7% of the goal of the experiment, was accumulated in 39.4 days of datataking in 1999 [79]. They obtained 3 neutrino events in the fiducial volume of the SuperKamiokande detector, whereas the expectation based on observations in the front detectors was neutrino events. It corresponds to a ratio of data versus theory . Although the preliminary results are rather consistent with squared mass difference eV and maximal mixing, it is too early to draw any reliable conclusions about neutrino mixing. An complete analysis of oscillation searches from the view points of absolute event numbers, distortion of neutrino energy spectrum, and ratio is still in progress.
7.2 Solar neutrinos
Indications in the favor of neutrino oscillations were found in ”all” solar neutrino experiments (along this section and the following ones, we will make reference to results appeared in Refs. [80, 81, 82, 83, 84, 85]): The Homestake Cl radiochemical experiment with sensitivity down to the lower energy parts of the B neutrino spectrum and to the higher Be line [82]. The two radiochemical Ga experiments, SAGE and GALLEX, which are sensitive to the low energy pp neutrinos and above [81, 80] and the water Cerenkov experiments Kamiokande and SuperKamiokande (SK) which can observe only the highest energy B neutrinos. Water Cerenkov experiments in addition demonstrate directly that the neutrinos come from the Sun showing that recoil electrons are scattered in the direction along the sunearth axis [83, 84, 85].
Two important points to remark are: a) The prediction of the existence of a global neutrino deficit is hard to modify due to the constraint of the solar luminosity on pp neutrinos detected at SAGEGALLEX. b) The different experiments are sensitive to neutrinos with different energy ranges and combined yield spectroscopic information on the neutrino flux. Intermediate energy neutrinos arise from intermediate steps of the thermonuclear solar cycle. It may not be impossible to reduce the flux from the last step (B), for example by reducing temperature of the center of the Sun, but it seems extremely hard to reduce neutrinos from Be to a large extent, while keeping a reduction of B neutrinos production to a modest amount. If minimal standard electroweak theory is correct, the shape of the B neutrino energy spectrum is independent of all solar influences to very high accuracy.
Unless the experiments are seriously in error, there must be some problems with either our understanding of the Sun or neutrinos. Clearly, the SSM cannot account for the data (see Fig.3) and possible highly nonstandard solar models are strongly constrained by heliosysmology studies [see Fig.(4)].
There are at least two reasonable versions of the neutrino oscillation phenomena which could account for the suppression of intermediate energy neutrinos. The first one, neutrino oscillations in vacuum, requires a large mixing angle and a seemingly unnatural fine tuning of neutrino oscillation length with the SunEarth distance for intermediate energy neutrinos. The second possibility, levelcrossing effect oscillations in presence of solar matter and/or magnetic fields of regular and/or chaotic nature (MSW, RSFP), requires no fine tuning either for mixing parameter or neutrino mass difference to cause a selective large reduction of the neutrino flux. This mechanism explains naturally the suppression of intermediate energy neutrinos, leaving the low energy pp neutrino flux intact and high energy B neutrinos only loosely suppressed. Concrete range of parameters obtained including the latest SK (SuperKamiokande) data will be showed in the next section.
7.3 The SK detector and Results.
The high precision and high statistics SuperKamiokande (SK) experiment initiated operation in April 1996. A few words about the detector itself. SK is a 50kiloton water Cerenkov detector located near the old Kamiokande detector under a mean overburden of 2700 meterwaterequivalent. The effective fiducial volume is kt. It is a well understood, well calibrated detector. The accuracy of the absolute energy scale is estimated to be based on several independent calibration sources: cosmic ray throughgoing and stopping muons, muon decay electrons, the invariant mass of ’s produced by neutrino interactions, radioactive source calibration, and, as a novelty in neutrino experiments, a 516 MeV electron LINAC. In addition to the ability of recording higher statistics in less time, due to the much larger dimensions of the detector, SK can contain multiGeV muon events making possible for the first time a measurement of the spectrum of like events up to GeV.
The results from SK, to be summarized below, combined with data from earlier experiments provide important constraints on the MSW and vacuum oscillation solutions for the solar neutrino problem (SNP), [90, 91, 89]:
Total rates. The most robust results of the solar neutrino experiments so far are the total observed rates. Preliminary results corresponding to the first 825 days of operation of SK (presented in spring’2000, [89]) with a total number of events in the energy range MeV. predict the following flux of solar B neutrinos:
a flux which is clearly below the SSM expectations. The most recent data on rates on all existing experiments are summarized in Table (2). Total rates alone indicate that the energy spectrum from the Sun is distorted. The SSM flux predictions are inconsistent with the observed rates in solar neutrino experiments at approximately the 20 level. Furtherly, there is no linear combination of neutrino fluxes that can fit the available data at the 3 level [Fig.(3].
Experiment  Target  E. Th. (MeV)  

SK825d  HO  6.520  
Homestake  Cl  0.8  
Kamiokande  HO  
SAGE  Ga  0.2  
GALLEX  Ga  0.2 
Zenith angle: daynight effect. If MSW oscillations are effective, for a certain range of neutrino parameters the observed event rate will depend upon the zenith angle of the Sun (through a Earth matter regeneration effect). Win present statistics, the most robust estimator of zenith angle dependence is the daynight (or updown) asymmetry, A. The experimental estimation is [89]: