Abstract
Seesawtype and lowscale models of neutrino masses are reviewed, along with the corresponding structure of the lepton mixing matrix. The status of neutrino oscillation parameters as of June 2006 is given, including recent fluxes, as well as latest SNO, K2K and MINOS results. Some prospects for the next generation of experiments are given. This writeup updates the material presented in my lectures at the Corfu Summer Institute on Elementary Particle Physics in September 2005.
Neutrino physics overview
J. W. F. Valle
AHEP Group, Instituto de Física Corpuscular, C.S.I.C. – Universitat de València
Edificio de Institutos de Paterna, Apartado 22085, E–46071 València, Spain
Contents
1 Introduction
The historic discovery of neutrino oscillations [1, 2, 3, 4, 5] marks a turning point in particle and nuclear physics and implies that neutrinos have mass. This possibility has been first suggested by theory since the early eighties, both on general grounds and on the basis of different versions of the seesaw mechanism [6, 7, 8, 9, 10, 11].
The general characterization of neutrino mass theories in terms provided a modelindependent basis to analyse the seesaw [9, 10]. It also indicated a fundamental difference between the lepton and the quark mixing matrices, namely, the appearance of new phases associated to the Majorana nature of neutrinos [9].
Irrespective of what the ultimate origin of neutrino mass may turn out to be, the basic gauge theoretic mechanism to account for the smallness of neutrino mass is in terms of the feebleness of BL violation. The seesaw is one realization of the idea, by far not unique. There are two classes of theories of neutrino mass, that differ by the scale at which Lsymmetry is broken. They are summarized in Sec. 3. The corresponding structure of the lepton mixing matrix that follows from theory is described Sec. 4. This forms the basis for the analysis of the data from current neutrino oscillation experiments [1, 2, 3, 4, 5] [12, 13]. The status of neutrino mass and mixing parameters as determined from the world’s neutrino oscillation data within the simplest CPconserving threeneutrino mixing scheme is summarized in Sec. 5.1 [14]. In addition a determination of the solar angle , the atmospheric angle and the corresponding mass squared splittings and , one gets a constraint on the last angle in the three–neutrino leptonic mixing matrix, . Together with the small ratio the angle holds the key for further progress in neutrino oscillation searches.
Some attempts at predicting neutrino masses and mixing are given in Sec. 5.2. Lepton number violating processes such as neutrinoless double beta decay [15, 16] are briefly discussed in Sec. 5.3. Searching for constitutes a very important goal for the future, as this will probe the fundamental nature of neutrinos, irrespective of the process that induces it, a statement known as the “blackbox” theorem [17]. In addition, will be sensitive to the absolute scale of neutrino mass and to CP violation induced by the socalled Majorana phases [9], inaccessible in conventional oscillations [18, 19, 20]. Finally in Sec. 6 the robustness of the oscillation interpretation and the role of nonstandard neutrino interactions in future precision oscillation studies is briefly mentioned.
2 Dirac and Majorana masses
Electrically charged fermions must be Dirac type. In contrast, electrically neutral fermions, like neutrinos (or supersymmetric “ inos”), are expected to be Majoranatype on general grounds, irrespective of how they acquire their mass. Phenomenological differences between Dirac and Majorana neutrinos are tiny for most processes, such as neutrino oscillations: first because neutrinos are known to be light and, second, because the weak interaction is chiral, well described by the VA form.
The most basic spin fermion corresponding to the lowest representation of the Lorentz group is given in terms of a 2component spinor , with the following free Lagrangean [9]
(1) 
where are the usual Pauli matrices and , being the identity matrix. I use Pauli’s metric conventions, where , . Under a Lorentz transformation, , the spinor transforms as where obeys
(2) 
The kinetic term in Eq. (1) is clearly invariant, and so is the mass term, as a result of unimodular property . However, the mass term is not invariant under a phase transformation
(3) 
The equation of motion following from Eq. (1) is
(4) 
As a result of the conjugation and Clifford properties of the matrices, one can verify that each component of the spinor obeys the KleinGordon waveequation.
Start from the usual Lagrangean describing of a massive spin Dirac fermion, given as
(5) 
where by convenience we use the chiral representation of the Dirac algebra in which is diagonal,
(6) 
In this representation the charge conjugation matrix obeying
(7)  
(8)  
(9) 
is simply given in terms of the basic conjugation matrix as
(10) 
In order to display clearly the relationship between the Majorana theory in Eq. (1) and the familiar Dirac Lagrangean in Eq. (5), one splits a Dirac spinor as
(11) 
so that the corresponding chargeconjugate spinor is the same as but exchanging and , i. e.
(12) 
A 4component spinor is said to be Majorana or selfconjugate if which amounts to setting . Using Eq. (11) we can rewrite Eq. (5) as
(13) 
where
(14) 
are the left handed components of and of the chargeconjugate field , respectively. This way the Dirac fermion is shown to be equivalent to two Majorana fermions of equal mass. The symmetry of the theory described by Eq. (5) under corresponds to continuous rotation symmetry between and
which result from the mass degeneracy between the ’s, showing that, indeed, the concept of fermion number is not basic.
The mass term in Eq. (1) vanishes unless and are anticommuting, so the Majorana fermion is, right from the start, a quantized field. The solutions to Eq. (1) are easily obtained in terms of those of Eq. (5), which are well known:
(15) 
where and is the massshell condition. The creation and annihilation operators obey canonical anticommutation rules and, like the ’s and ’s, depend on the momentum and helicity label . The expression in Eq. (15) describes the basic Fourier expansion of a massive Majorana fermion. It differs from the usual Fourier expansion for the Dirac spinor in Eq. (16) in two ways:

spinors are twocomponent, as there is a chiral projection on the ’s and ’s

there is only one Fock space, particle and antiparticle coincide, showing that a massive Majorana fermion corresponds to one half of a conventional massive Dirac fermion.
The ’s and ’s are the same wave functions that appear in the Fourier decomposition the Dirac field
(16) 
Using the helicity eigenstate wavefunctions,
(17)  
(18) 
one can show that, out of the linearly independent wave functions and , only two survive as the mass approaches zero, namely, and [21]. This way the LeeYang twocomponent massless neutrino theory is recovered as the massless limit of the Majorana theory.
Two independent propagators follow from Eq. (1),
(19)  
(20) 
where is the usual Feynman function. The first one is the “normal” propagator that intervenes in total lepton number conserving () processes, while the one in Eq. (20) describes the virtual propagation of Majorana neutrinos in processes such as neutrinoless doublebeta decay.
The Lagrangean in Eq. (1) can easily be generalized to an arbitrary number of Majorana neutrinos, giving
(21) 
where the sum runs over the “neutrinotype” indices and . By Fermi statistics the mass coefficients must form a symmetric matrix, in general complex. This matrix can always be diagonalized by a complex unitary matrix (See [9] for the proof)
(22) 
When is real its diagonalizing matrix may be chosen to be orthogonal and, in general, the mass eigenvalues can have different signs. These may be assembled as a signature matrix
(23) 
For two neutrino types there are two classes of models, one with and another characterized by . The class with contains as a limit the case where the two fermions make up a Dirac neutrino. Note that one can always make all masses positive by introducing appropriate phase factors in the wave functions, such as the factors of in Eq. (14). However, when interactions are added these signs become physical. As emphasized by Wolfenstein, they play an important role in the discussion of (neutrinoless double beta decay) [22].
3 The origin of neutrino mass
Table 1 gives the fifteen basic building blocks of matter. They are all 2component sequential “lefthanded” chiral fermions, one set for each generation. Parity violation in the weak interaction is incorporated “effectively” by having “left” and “right” fermions transform differently with respect to the gauge group. In contrast to charged fermions, neutrinos come only in one chiral species.
It has been long noted by Weinberg [8] that one can add to the Standard Model (SM) an effective dimensionfive operator where denotes a lepton doublet for each generation and is the SM scalar doublet.
Once the electroweak symmetry breaks through the nonzero vacuum expectation value (vev) , Majorana neutrino masses are induced, in contrast to the masses of the charged fermions which arise from basic renormalizable interactions, and are linear in . Moreover, the dimensionfive operator violates lepton number by two units (), whereas the charged fermion masses arise from renormalizable Lconserving Yukawa interactions. This naturally accounts for the smallness of neutrino masses irrespective of the specific origin of neutrino mass. From such general point of view the emergence of Dirac neutrinos would be a surprise, justified only in the presence of an “accidental” lepton number symmetry. For example, neutrinos could naturally get very small Dirac masses via mixing with a bulk fermion in models involving extra dimensions [23, 24, 25]. Barring such very special circumstances, gauge theories give rise to Majorana neutrinos.
Little more can be said from first principles about the mechanism giving rise to the operator in Fig. 1, its associated mass scale or its flavour structure. For example, the strength of the operator may be suppressed by a large scale in the denominator (topdown) scenario, leading to
where is some unknown dimensionless constant. Gravity, which in a sense ”belongs” to the SM, could induce the dimensionfive operator , providing the first example of a topdown scenario with , the Planck scale [26]. In this case the magnitude of the resulting Majorana neutrino masses are too small.
Alternatively, the strength of the operator may be suppressed by small parameters (e.g. scales, Yukawa couplings) in the numerator and/or loopfactors (bottomup scenario). Both classes of scenarios are viable and have many natural realizations. While models of the topdown type are closer to the idea of unification, bottomup schemes are closer to experimental test.
Models of neutrino mass may also be classified according to whether or not additional neutral heavy states are present, in addition to the three isodoublet neutrinos. As an example, such leptons could be singlet “righthanded” neutrinos. In what follows we first consider topdown, then bottomup scenarios.
3.1 Seesaw and related models
The most popular topdown scenario is the seesaw [6]. The idea is to generate the operator by the exchange of heavy states, both fermions (typeI) and scalars (typeII), as shown in Fig. 2.
This can be implemented in many ways, with different gauge groups and multiplet contents. The main point is that, as the masses of the intermediate states go to infinity, neutrinos naturally become light [7]. The seesaw provides a simple realization of Weinberg’s operator [8]. Note that the seesaw idea does not require the gauging of BL, nor does it require it to be broken spontaneously. In fact, most of the physics it encodes, and which has been brilliantly confirmed by the recent oscillation experiments, lies in its “effective” lowenergy form [9]. Only in lowscale schemes like the inverseseesaw discussed in Sec. 3.2.3 effects associated with “seesawdynamics” may be observable (see Secs. 5.4.1 and 5.4.2).
3.1.1 “Effective” seesaw
Much of the low energy phenomenology, such as that of neutrino oscillations is blind to the details of the underlying seesaw theory at high energies, e. g. its gauge group, multiplet content or the nature of BL. For this purpose the most general way to describe the physics of the seesaw is to characterize it, effectively, in terms of the SM gauge structure [9]. In the basis , , corresponding to the three “left” and three “right” neutrinos, respectively, the seesaw mass matrix has SU(2) triplet, doublet and singlet terms described as [9]
(24) 
Here we use the original notation of reference [9], where the “Dirac” entry is proportional to , the comes from a triplet vev, and may be added by hand, as it is a gauge singlet. The particular case was first mentioned in Ref. [6].
Note that, though symmetric, by the Pauli principle, the matrix is complex, so that its Yukawa coupling submatrices as well as and are also complex matrices, the last two symmetric. It is diagonalized by performing a unitary transformation ,
(25) 
so that
(26) 
This yields 6 mass eigenstates, including the three light neutrinos with masses , and three twocomponent heavy leptons of masses . The light neutrino mass states are given in terms of the flavour eigenstates via eq. (25). The effective light neutrino mass, obtained this way is of the form
(27) 
The smallness of light neutrino masses is understood by assuming . The above general structure forms the basis for the description of the seesaw lepton mixing matrix [9], given in Sec. 4.3.
While it constitutes the most general description, and also the common denominator of all seesaw schemes, such an “effective” seesaw does not give a dynamical insight on the origin of neutrino mass. For this reason we now turn to schemes where lepton number symmetry is broken spontaneously.
3.1.2 The “123” seesaw mechanism
The simplest possibility for the seesaw is to have ungauged lepton number. It is also the most general, as it can be studied in the framework of just the gauge group. The mass terms in eq. (24) are given by triplet, doublet and singlet vevs, respectively, as [10]
(28) 
As already mentioned, the Yukawa coupling submatrices as well as and are complex matrices, the last two symmetric.
The new dynamical insight provided by such “123” seesaw containing singlet, doublet and triplet scalar multiplets, is that they obey a simple vev seesaw relation of the type
(29) 
where denotes the SM Higgs doublet vev, fixed by the Wboson mass. This hierarchy implies that the triplet vev as the singlet vev grows. This is consistent with the minimization of the corresponding invariant scalar potential, and implies that the triplet vev is “induced”.
Small neutrino masses arise either by heavy singlet “righthanded” neutrino exchange (type I) or by the small effective triplet vev (type II), as illustrated in Fig. 2. The effective light neutrino mass becomes,
(30) 
The corresponding seesaw diagonalization matrices can be given explicitly as a systematic matrix perturbation series expansion in , given in Ref. [10].
3.1.3 Leftright symmetric and SO(10)
A more symmetric setting for the seesaw is a gauge theory containing BL as a generator, such as or the unified models based on or [6, 7, 11]. For example in each matter generation is naturally assigned to a 16 (spinorial in ) so that the 16 . 16 . 10 and 16 . 126 . 16 couplings generate all entries of the seesaw neutrino mass matrix,
(31) 
Here the basis is , , as before, and denote the Yukawas of the 126 of , whose vevs give rise to the Majorana terms. They correspond to and of the simplest “123” model. On the other hand denotes the 16 . 16 . 10 Dirac Yukawa coupling. In one has a discrete parity symmetry which implies and as recently emphasized in Ref. [28]. Since this may get broken, we prefer to keep, for generality, as independent.
Small neutrino masses are induced either by heavy singlet “righthanded” neutrino exchange (type I) or heavy scalar boson exchange (type II) as illustrated in Fig. 2. The matrix is diagonalized by a unitary mixing matrix as before. The diagonalization matrices can be worked out explicitly as a perturbation series, using the same method of Ref. [10]. This means that the explicit formulas for the unitary diagonalizing matrix given in Ref. [10] also hold in the leftright case, provided one takes into account that and .
The effective light neutrino mass, obtained this way is of the form
(32) 
We have the new vev seesaw relation
(33) 
which naturally follows from minimization of the leftright symmetric scalar potential, together with the vev hierarchy
(34) 
This implies, as before, that both type I and type II contributions vanish as . Notice that, strictly speaking, this version of leftsymmetry is inconsistent with typeI seesaw [7] and requires the full form of the seesaw mass matrix [9, 10]. There are, however, tripletless leftright seesaw variants where a similar vevseesaw formula holds, see Refs. [31, 32, 33, 34] and Sec. 3.1.5.
If one can arrange for the breakdown of parity invariance to be spontaneous, then the smallness of neutrino masses gets correlated to the observed maximality of parity violation in lowenergy weak interactions, as stressed by Mohapatra and Senjanovic [7]. However elegant this connection may be, it is phenomenologically not relevant, in view of the large value of the BL scale. The latter is required both to fit the neutrino masses, as well as to unify the gauge couplings. Another important difference with respect to the simplest “123” seesaw is the absence of the majoron, which is now absorbed as the longitudinal mode of the gauge boson corresponding to the BL generator which picks up a huge mass.
3.1.4 “Double” seesaw mechanism
One can add any number of (anomalyfree) gauge singlet leptons to the SM, or any other gauge theory [9]. For example, in and one may add leptons outside the 16 or the 27, respectively. New important features may emerge when the seesaw is realized with nonminimal lepton content. Here we mention the seesaw scheme suggested in Ref. [29] motivated by string theories [30]. The model extends minimally the particle content of the SM by the sequential addition of a pair of twocomponent singlet leptons, , with a generation index running over . In the basis, the neutral leptons mass matrix is given as
(35) 
in the basis , , . Again is an arbitrary complex Yukawa matrix, and are SU(2) singlet complex mass matrices, being symmetric. Notice that it has zeros in the  and  entries, a feature of several string models [30].
For one has to first approximation that the decouple leaving the simpler seesaw at scales below that. In such a “double” seesaw scheme the three light neutrino masses are determined from
(36) 
The mass generation is illustrated in Fig. 3.
A new feature is that there are two independent scales, and , of which only breaks the BL symmetry. Both scales can be large but, as we will see in Sec. 3.2.3, it is natural for to be small [29], instead of large, , see Sec. 3.2.3. For the case one formally recovers the usual seesaw form, this is useful to present some results in simplified form, as used in Fig. 14.
Irrespective of what sets its scale, the entry may be proportional to the vev of an singlet scalar, in which case the model contains a singlet majoron, see Sec. 5.4.1.
3.1.5 “Novel” seesaw mechanism
The seesaw is a mechanism which allows for many possible realizations. Schemes leading to the same pattern of neutrino masses may differ in many other respects. Correspondingly, there are many types of seesaw. In addition to type I [6, 7] [11], type II [9, 10], there are extended seesaw schemes, like typeIII [31, 32, 33] and the double/inverse seesaw described above.
Here I turn to yet another seesaw that has recently been suggested in Ref. [34]. It belongs to the class of supersymmetric models with broken Dparity. In addition to the states in the 16, it contains three sequential gauge singlets with the following mass matrix
(37) 
in the same basis , , as previously. Notice that it has zeros along the diagonal, specially in the  and  entries, thanks to the fact that there is no 126, a feature of several string models [30, 29]. The resulting light neutrino mass matrix is
(38) 
where is the unification scale, and denote independent combinations of Yukawa couplings of the . One can see that the neutrino mass is suppressed by the unification scale independently of the BL breaking scale. In contrast to all previous seesaws, this one is linear in the Dirac Yukawa couplings , as illustrated in Fig. 4.
It is rather remarkable that one can indeed take the BL scale as low as TeV without generating inconsistencies, neither with neutrino masses, nor with gauge coupling unification [34].
3.2 Lowscale models
There are many models of neutrino mass where the operator is induced from physics at accessible scales, TeV or less. The smallness of its strength may be naturally achieved due to loop and Yukawa couplings suppression. Moreover, the strength of the operator may be suppressed by small lepton number violating parameters that appear in its numerator, instead of its denominator, as commonly assumed. The latter correspond to the seesawtype schemes previously discussed. The former correspond to the class we are about to describe. Before I do so, let me emphasize that such models are also “natural” in t’Hooft’s sense [35]: “an otherwise arbitrary parameter may be taken as small when the symmetry of the Lagrangean increases by having it vanish”.
Since all particles are at the TeV scale, these models naturally lead to possibly testable phenomenological implications, including lepton flavour violation and modifications in muon and tau decays.
Moreover, when the breaking of lepton number entailed in these models takes place spontaneously, the corresponding Goldstone boson has a remarkable property: it may couple substantially to the SM Higgs boson, which can therefore have a sizeable invisible decay branching ratio [36, 37, 38, 39]
(39) 
where is the majoron. This show that, although neutrino masses are small, the neutrino mass generation may have very important implications for the mechanism of electroweak symmetry breaking. One must therefore take into account the existence of the invisible channel in designing Higgs boson search strategies at future collider experiments [40, 41]. Further discussion in Sec. 5.4.1.
3.2.1 Radiative models
Neutrino masses may be induced by calculable radiative corrections [42]. For example, they can arise at the twoloop level [43] as illustrated in Fig. 5. Up to a logarithmic factor one has, schematically,
(40) 
in the limit where the doublycharged scalar is much heavier than the singly charged one. Here denotes a charged lepton, and are their Yukawa coupling matrices and denotes the SM Higgs Yukawa couplings to charged leptons. Here denotes an singlet vev used in Ref. [44]. Clearly, even if the proportionality factor is large, the neutrino mass can be made naturally small by the presence of a product of five small Yukawas and the appearance of the twoloop factor.
3.2.2 Supersymmetry and neutrino mass
Low energy supersymmetry can be the origin of neutrino mass [45]. The intrinsically supersymmetric way to break lepton number is to break the socalled R parity. This could happen spontaneously, driven by a nonzero vev of an singlet sneutrino [46, 47, 48], and leads to an effective model characterized by purely bilinear R parity violation [49]. This also serves as reference model, as it provides the minimal way to add neutrino masses to the MSSM, we call it RMSSM. Neutrino mass generation takes place in a hybrid scenario, with one scale generated at tree level and the other induced by “calculable” radiative corrections [50].
Here the two blobs in the graph Fig. 6 denote insertions, while the crossed blob accounts for chirality flipping, and denotes the trilinear soft supersymmetry breaking coupling. The general form of the expression is quite involved but the approximation
(41) 
holds in some regions of parameters. The neutrino mass spectrum naturally follows a normal hierarchy, with the atmospheric scale generated at the tree level and the solar mass scale arising from calculable loops.
3.2.3 “Inverse” seesaw mechanism
Here we mention that there are also lowscale treelevel neutrino mass schemes with naturally light neutrinos. One is the inverse seesaw scheme suggested in [29] and already described in Sec. 3.1.4. The mass matrix is the same as that of the double seesaw model, given in eq. (35). The only difference is that now the entry is taken very small, e. g. [29]. Notice that for small neutrino masses vanish with ,
as illustrated in Fig. 3.
The fact that the neutrino mass vanishes as is just the opposite of the behaviour of the seesaw formulas in Eqs. (27) (30) and (32); thus this is sometimes called inverse seesaw model of neutrino masses. The entry may be proportional to the vev of an SU(2) singlet scalar, in which case spontaneous BL violation leads to the existence of a majoron [51]. This would be characterized by a relatively low scale, so that the corresponding phase transition could take place after the electroweak transition. The naturalness of the model stems from the fact that in the limit when lepton number is recovered, increasing the symmetry of the theory.
4 The lepton mixing matrix
In any gauge theory in order to identify physical particles one must diagonalize all relevant mass matrices, which typically result from gauge symmetry breaking. Mechanisms giving mass to neutrinos generally imply the need for new interactions whose Yukawas (like in seesawtype schemes) will coexist with that of the charged leptons, . Since in general these are independent one has that, like quarks, massive neutrinos generally mix. The structure of this mixing is not generally predicted from first principles. Whatever the ultimate high energy gauge theory may be it must be broken to the SM at low scales, so one should characterize the structure of the lepton mixing matrix in terms of the structure. The procedure is the familiar one from the quark sector.
4.1 Dirac case
Here we derive the structure of the lepton mixing matrix of massive Dirac neutrinos and its parametrization, as presented in Ref. [9]. From the start can be parametrized as
(42) 
where
is a diagonal unitary “Cartan” matrix, described by real parameters ^{1}^{1}1By choosing an overall relative phase between charged leptons and Dirac neutrinos we can take as unimodular, i. e. det , so that .. On the other hand each factor
is a complex rotation in with parameter . For example,
(43) 
Once the charged leptons and Dirac neutrino mass matrices are diagonal, one can still rephase the corresponding fields by and , respectively, keeping invariant the form of the free Lagrangean. This results in the form
(44) 
where we are still free to choose the values associated to Dirac neutrino phase redefinitions. Using the conjugation property
(45) 
we arrive at the final Dirac lepton mixing matrix which is, of course, identical in form to that describing quark mixing. It involves a set of
(46) 
where phases were eliminated. This is the parametrization as originally given in Ref. [9], with unspecified factor ordering. From Eq. (46) one sees that for there are 3 angles and precisely one leptonic CP violating phase, just as in the KobayashiMaskawa matrix describing quark mixing. Two of the three angles are involved in solar and atmospheric oscillations, so we set and . The last angle in the three–neutrino leptonic mixing matrix is ,
(47) 
A convenient ordering prescription is to take 23 13 12, or “atmospheric” “reactor” “solar”, with the phase being associated to . In summary, if neutrinos masses are added a la Dirac their charged current weak interaction has exactly the same structure as that of quarks.
4.2 Majorana case
Here we consider the form of the lepton mixing matrix in models where neutrino masses arise in the absence of righthanded neutrinos, such as those in Sec. 3.2.1. The unitary form also holds, to a good approximation, to models where SU(2) doublet neutrinos mix only slightly with other states, like highscale seesaw models.
For generations of Majorana neutrinos the lepton mixing matrix has exactly the same form given in Eq. (44). The difference is that in the case of Majorana neutrinos their mass term is manifestly not invariant under rephasings of the neutrino fields. As a result, the parameters in Eq. (44) can not be used to eliminate Majorana phases as we just did in Sec. 4.1. Consequently these are additional phases [9] which show up in Lviolating processes [19, 20]. Such new sources of CP violation in gauge theories with Majorana neutrinos are called “Majorana phases”. They already exist in a theory with just two generations of Majorana neutrinos [9], , whose mixing matrix is described by
(48) 
where is the Majorana phase (recall that Cabibbo mixing has no CP phase). Such “Majorana” CP phases are, in a sense, mathematically more “fundamental” than the Dirac phase, whose existence, as we just saw, requires at least three generations.
For the case of three neutrinos the lepton mixing matrix can be parametrized as [9]
(49) 
where each factor in the product of the ’s is effectively , characterized by an angle and a CP phase. Such symmetrical parameterization of the lepton mixing matrix, can be written as:
All three CP phases are physical [19]: and . The “invariant” combination corresponds to the “Dirac phase”. If neutrinos are of Dirac type, only a single phase (say ) may be taken to be nonzero. This phase corresponds to the phase present in the KobayashiMaskawa matrix, and this is the one that affects neutrino oscillations. The other two phases are associated to the Majorana nature of neutrinos and show up only in leptonnumber violating processes, like neutrinoless double beta decay [19, 20].
An important subtlety arises regarding the conditions for CP conservation in gauge theories of massive Majorana neutrinos. Unlike the case of Dirac fermions, where CP invariance implies that the mixing matrix should be real, in the Majorana case the condition is [21]
where is the signature matrix describing the relative signs of the neutrino mass eigenvalues that follow from diagonalizing the relevant Majorana mass matrix, if one chooses to use real diagonalizing matrices [22]. Consequently say, for , both and correspond to CP conservation, as emphasized by Wolfenstein. These important signs determine the CP properties of the neutrinos and play a crucial role in .
4.3 Seesawtype mixing
The most general theory of neutrino mass is effectively described in SM terms by , being the number of isodoublets and the number of isosinglet twocomponent leptons (the case was given above). Here we assume an arbitrary number of gauge singlets, since they carry no anomaly. The usual seesaw in Secs. 3.1.1, 3.1.2 and 3.1.3 has , while the extended seesaw in Secs. 3.1.4, 3.1.5 and 3.2.3 have . Isosinglets have in general a gauge and Lorentz invariant Majorana mass term. The procedure holds in any scheme of Majorana neutrino masses where isosinglet and isodoublet mass terms coexist [9].
The effective form of such a “seesaw” lepton mixing matrix has, in addition to Majorana phases, many doubletsinglet mixing parameters, in general complex [9]. Its general structure is substantially more complex than “usual”, being described by a rectangular matrix, called . As a result one finds that leptonic mixing as well as CP violation may take place even in the massless neutrino limit [52, 53].
The existence of these neutral heavy leptons could be inferred from low energy weak decay processes, where the neutrinos that can be kinematically produced are only the light ones. The mixing matrix describing the charged weak interactions of the light (masseigenstate) neutrinos is effectively nonunitary, since the coupling of a given light neutrino to the corresponding charged lepton is decreased by a certain factor associated with the heavy neutrino coupling. There are constraints on the strength of the such mixing matrix elements that follow from weak universality and low energy weak decay measurements, as well as from LEP.
The full weak charged current mixing matrix of the general models involves
(50) 
For the explicit parametrization the reader is referred to the original paper, Ref. [9]. One sees that, for example, the usual seesaw model [labeled in our language] is characterized by 12 mixing angles and 12 CP phases (both Dirac and Majoranatype) [9].
This number far exceeds the corresponding number of parameters describing the charged current weak interaction of quarks. As already mentioned, the reason is twofold: (i) neutrinos are Majorana particles, their mass terms are not invariant under rephasings, and (ii) the isodoublet neutrinos in general mix with the singlets. As a result, there are far more physical CP phases that may play a role in neutrino oscillations and/or leptogenesis (see below).
Another important feature which arises in any theory based on where isosinglet and isodoublet lepton mass terms coexist is that the leptonic neutral current is nontrivial [9]: there are nondiagonal couplings of the Z to the masseigenstate neutrinos. They are expressed as a projective hermitian matrix
This contrasts with the neutral current couplings of masseigenstate neutrinos in schemes where lepton number is conserved (Sec. 4.1) or where no isosinglet leptons are present, i.e., (Sec. 4.2). In both cases, just as for quarks, the neutral current couplings are diagonal (GlashowIliopoulosMaiani mechanism).
Before we close, note that, in a scheme with , neutrinos will remain massless, while neutrinos will acquire Majorana masses, light and heavy [9]. For example, in a model with and one has one light and one heavy Majorana neutrino, in addition to the two massless ones. In this case clearly there will be less parameters than present in a model with .
5 Phenomenology
Obviously the first phenomenological implication of neutrino mass models is the phenomenon of neutrino oscillations, required to account for the current solar and atmospheric neutrino data. The interpretation of the data relies on good calculations of the corresponding fluxes [54, 55], neutrino cross sections and response functions, as well as on an accurate description of neutrino propagation in the Sun and the Earth, taking into account matter effects [56, 57].
5.1 Status of neutrino oscillations
Current neutrino oscillation data have no sensitivity to CP violation. Thus we neglect all phases in the analysis and take, moreover, the simplest unitary 3dimensional form of the lepton mixing matrix in Eq. (49) with the three phases set to zero. In this approximation oscillations depend on the three mixing parameters and on the two masssquared splittings and characterizing solar and atmospheric neutrinos. The hierarchy implies that one can set , to a good approximation, in the analysis of atmospheric and accelerator data. Similarly, one can set to infinity in the analysis of solar and reactor data.
The world’s neutrino oscillation data and their analysis, as of June 2006, are given in Ref. [14] and will not be repeated here. The new developments are: new Standard Solar Model [58], new SNO salt data [59], latest K2K [60] and MINOS [61] data. These are briefly described in Appendix C of hepph/0405172 (v5). In what follows we summarize the updated results of the analysis which takes into account all these new data. Apart from the “positive” data already mentioned, the analysis also includes the constraints from “negative” oscillation searches at reactor experiments, CHOOZ and Palo Verde.
The three–neutrino oscillation parameters that follow from the global oscillation analysis in Ref. [14] are summarized in Fig. 7. In the upper panels of the figure the is shown as a function of the three mixing parameters , minimized with respect to the undisplayed parameters. The lower panels show twodimensional projections of the allowed regions in the fivedimensional parameter space. In addition to a confirmation of oscillations with , accelerator neutrinos provide a better determination of . For example, comparing dashed and solid lines in Fig. 7 one sees that the inclusion of the new data (mainly MINOS [61]) leads to a slight increase in and an improvement on its determination (see [14] for details). On the other hand reactors [3] have played a crucial role in selecting largemixingangle (LMA) oscillations [62] out of the previous “zoo” of solutions [63]. and two mass squared splittings
The best fit values and the allowed 3 ranges of the oscillation parameters from the global data are summarized in Table 2.
parameter  best fit  3 range 

7.9  7.1–8.9  
2.6  2.0–3.2  
0.30  0.24–0.40  
0.50  0.34–0.68  
0.00  0.040 
Note that in a three–neutrino scheme CP violation disappears when two neutrinos become degenerate or when one of the angles vanishes [64]. As a result CP violation is doubly suppressed, first by and also by the small mixing angle .
The left panel in Fig. 8 gives the parameter , as determined from the global analysis. The right panel shows the impact of different data samples on constraining . One sees that, although for larger values the bound on is dominated by CHOOZ, this bound deteriorates quickly as decreases (see Fig. 8), so that the solar and KamLAND data become relevant.
There is now a strong ongoing effort aimed at probing and CP violation in future neutrino oscillation searches at reactors and accelerators [65, 66, 67]. As we saw, the basic parameters and characterizing the strength of CP violation in neutrino oscillations are small. Prospects for probing at long baseline reactor and accelerator neutrino oscillation experiments are given in Ref. [68],
5.2 Predicting neutrino masses and mixing
Gauge symmetry alone is not sufficient to predict particle mixings, neither for the quarks, nor for the leptons: such “flavour problem” has remained with us for a while.
As we saw in Sec. 5.1 five of the basic parameters of the lepton sector are currently probed in neutrino oscillation studies. These point towards a well defined pattern of neutrino mixing angles, quite distinct from that of quarks. Such pattern is not easy to account for in the context of unified schemes where quarks and leptons are connected. The data seem to indicate an intriguing complementarity between quark and lepton mixing angles [71, 72, 73, 74].
There has been a rush of papers attempting to understand the values of the leptonic mixing angles from underlying symmetries at a fundamental level. For example the following form of the neutrino mixing angles has been proposed [75]
(51)  
Such HarrisonPerkinsScott pattern [76] could result from some kind of flavour symmetry, valid at a very high energy scale where the dimensionfive neutrino mass operator arises.
One approach to predict neutrino masses and mixing angles was the idea that neutrino masses arise from a common seed at some “neutrino mass unification” scale [77], very similar the merging of the SM gauge coupling constants at high energies due to supersymmetry [78]. However, in its simplest form this very simple theoretical ansatz is now inconsistent (at least if CP is conserved) with the current observed value of the solar mixing angle inferred from current data.
A more satisfactory and fully viable alternative realization of the “neutrino mass unification” idea employs an flavour symmetry introduced by Ernest Ma, in the context of a seesaw scheme [79]. Starting from threefold degeneracy of the neutrino masses at a high energy scale, a viable low energy neutrino mass matrix can indeed be obtained in agreement with neutrino data as well as constraints on lepton flavour violation in and decays. The model predicts maximal atmospheric angle and vanishing ,
Although the solar angle is unpredicted, one expects ^{2}^{2}2There have been realizations of the symmetry that also predict the solar angle, e. g. Ref. [80].
When CP is violated becomes arbitrary and the Dirac phase is maximal [81].
Within such flavour symmetric seesaw scheme one can show that the lepton and slepton mixings are intimately related. The resulting slepton spectrum must necessarily include at least one mass eigenstate below 200 GeV, which can be produced at the LHC. The prediction for the absolute Majorana neutrino mass scale eV ensures that the model will be tested by future cosmological tests and searches. Rates for lepton flavour violating processes typically lie in the range of sensitivity of coming experiments, with BR and BR.
5.3 Absolute scale of neutrino mass and
Neutrino oscillations are blind to whether neutrinos are Dirac or Majorana. As we have seen, on general grounds, neutrinos are expected to be Majorana [9]. Neutrinoless double beta decay and other lepton number violation processes, such as neutrino transition electromagnetic moments [21, 22] [85, 86] are able to probe the basic nature of neutrinos.
The significance of neutrinoless double beta decay stems from the fact that, in a gauge theory, irrespective of the mechanism that induces , it necessarily implies a Majorana neutrino mass [17], as illustrated in Fig. 9.
Thus it is a basic issue. Quantitative implications of the “blackbox” argument are modeldependent, but the theorem itself holds in any “natural” gauge theory.
Conventional neutrino oscillations are also insensitive to the absolute scale of neutrino masses [18, 19, 20]. Although the latter will be tested directly in high sensitivity tritium beta decay studies [87], as well as by its effect on the cosmic microwave background and the large scale structure of the Universe [88, 89, 90] may give valuable complementary information. For example, as seen above, the model [79] gives a lower bound on the absolute Majorana neutrino mass eV and may therefore be tested in searches.
Now that oscillations are experimentally confirmed we know that must be induced by the exchange of light Majorana neutrinos, the socalled ”massmechanism”. The corresponding amplitude is sensitive [19, 20] both to the absolute scale of neutrino mass, as well as to Majorana phases [9], neither of which can be probed in oscillations [18, 19].
Fig. 10 shows the estimated average mass parameter characterizing the neutrino exchange contribution to versus the lightest and heaviest neutrino masses. The calculation takes into account the current neutrino oscillation parameters in [14] and stateoftheart nuclear matrix elements [91]. The upper (lower) panel corresponds to the cases of normal (inverted) neutrino mass spectra. In these plots the “diagonals” correspond to the case of quasidegenerate neutrinos [79] [92] [93] In the normal hierarchy case there is in general no lower bound on the rate since there can be a destructive interference amongst the neutrino amplitudes. In contrast, the inverted neutrino mass hierarchy implies a “lower” bound for the amplitude.
A specific normal hierarchy model for which a lower bound on can be placed has been given in Ref. [80]. An interesting feature is that such lower bound depends, as expected, on the value of the Majorana violating phase , as indicated in Fig. 11.
The best current limit on comes from the HeidelbergMoscow experiment. There is also a claim made in Ref. [94] (see also Ref. [95]) which will be important to confirm or refute in future experiments. GERDA will provide an independent check of this claim [96]. SuperNEMO, CUORE, EXO, MAJORANA and possibly other experiments will further extend the sensitivity of current searches [97].
5.4 Other phenomena
If neutrino masses arise a la seesaw the dynamics responsible for generating the small neutrino masses seems most likely untestable. In other words, beyond neutrino masses and oscillations, theories can not be probed phenomenologically at low energies, due to the large scale involved. However when the breaking of lepton number symmetry is spontaneous there is a dynamical “tracer” of the massgeneration mechanism which might be probed experimentally.
5.4.1 Majoron physics
If neutrino masses follow from spontaneous violation of global lepton number, the existence of the Goldstone boson brings new interactions for neutrinos and Higgs boson(s) which may lead to new phenomena. While neither of these is expected within the usual highscale seesaw schemes, both could lead to detectable signals in lowscale models of neutrino mass.
As already mentioned, the majoron may couple substantially to the SM Higgs boson, which can therefore have a sizeable decay branching ratio [36, 37, 38, 39] into the channel in eq. (39). Such “invisible” channel is experimentally detectable as missing energy or transverse momentum associated to the Higgs [40, 41]. Therefore in lowscale models of neutrino mass the neutrino massgiving mechanism may have a strong impact in the electroweak sector.
Majoronemitting neutrino decays can also be lead to detectable signals in lowscale models of neutrino mass,
For example, if the neutrinos decay in high density media, like supernovae, characterized by huge matter densities, then the “matterassisted” decays would lead to detectable signals in underground water Cerenkov experiments [98].
5.4.2 New gauge boson
Although in the usual largescale seesaw with gauged BL symmetry, there are new gauge bosons associated to the neutrino mass generation these are too heavy to give any detectable effect. Only when the BL scale is low, as in the model discussed in Sec. 3.1.5 or the model considered in Ref. [99], there will exist a light new neutral gauge boson, that could be detected in searches for DrellYan processes at the LHC.
5.4.3 Lepton flavour violation
In the presence of supersymmetry, seesaw phenomenology is richer. A generic feature of supersymmetric seesaw models is the existence of processes with lepton flavour violation such as . Supersymmetry contributes through the exchange of charginos (neutralinos) and sneutrinos (charged sleptons) [100, 101, 102], as illustrated in Fig. 12.
Similarly the nuclear conversion arises, as indicated in Fig. 13.
The rates for these process can both be sizeable. As an example, consider the rates for the decay, given in Fig. 14.
The calculation leading to Fig. 14 is done in the framework of the supersymmetric double/inverse seesaw model in Secs. 3.1.4 and 3.2.3 [103]. This allows one to analyse the interplay of neutral heavy lepton [104] ^{3}^{3}3Since in this model flavor and CP violation can occur in the massless neutrino limit, the allowed rates are unsuppressed by the smallness of neutrino masses [104, 52, 53, 105, 106]. and supersymmetric contributions [102]. The figure shows the contours in the plane (logarithmic scales) for hierarchical light neutrinos with eV (left panel) and for degenerate light neutrinos with eV (right panel). The shaded contours correspond to (from left to right): . For large the estimates are similar to those of the standard supersymmetric seesaw. However, if the neutral heavy leptons are in the TeV range (a situation not realizable in the minimal seesaw mechanism), the rate can be enhanced even in the absence of supersymmetry. This is indicated by the contours in the lower left, which depict the contribution from neutral heavy leptons only. For such values around TeV or so, the quasiDirac neutral heavy leptons may be directly produced at accelerators [107]. Note that in order to have low then should be rather small, to keep neutrinos light. This is indicated by the diagonal lines, which indicate contours (top to bottom) of constant GeV. The vertical lines are contours of in the standard supersymmetric seesaw.
5.4.4 Reconstructing neutrino mixing at accelerators
Lowscale models of neutrino mass, considered in Sec. 3.2, offer the tantalizing possibility of reconstructing neutrino mixing at high energy accelerators, like the ”Large Hadron Collider” (LHC) and the ”International Linear Collider” (ILC).
A remarkable example is provided by the models where supersymmetry is the origin of neutrino mass [45], considered in Sec. 3.2.2. A general feature of these models is that, unprotected by any symmetry, the lightest supersymmetric particle (LSP) is unstable. In order to reproduce the masses indicated by current neutrino oscillation data, the LSP is expected to decay inside the detector [50] [110].
More strikingly, LSP decay properties correlate with neutrino mixing angles. For example, if the LSP is the lightest neutralino, it should have the same decay rate into muons and taus, since the observed atmospheric angle is close to [111, 112, 113]. Similar correlations hold irrespective of which supersymmetric particle is the LSP [114] and constitute a smoking gun signature of this proposal that will be tested at upcoming accelerators.
There are other examples of lowscale models that nicely illustrate the possibility of probing neutrino properties at accelerators [115].
5.5 Thermal leptogenesis
Now we briefly discuss one of the cosmological implications of neutrino masses and mixing, in the context of seesaw schemes. It has long been noted [116] that seesaw models open an attractive possibility of accounting for the observed cosmological matterantimatter asymmetry in the Universe through the leptogenesis mechanism [117]. In this picture the decays of the heavy “righthanded” neutrinos present in the seesaw play a crucial role. These take place through diagrams in Fig. 16. In order to induce successful leptogenesis the decay must happen before the electroweak phase transition [118] and must also take place outofequilibrium, i. e. the decay rate must be less than the Hubble expansion rate at that epoch. Another crucial ingredient is CP violation in the lepton sector.
The lepton (or BL) asymmetry thus produced then gets converted, through sphaleron processes, into the observed baryon asymmetry.
In the framework of a supersymmetric seesaw scheme the high temperature needed for leptogenesis leads to an overproduction of gravitinos, which destroys the standard predictions of Big Bang Nucleosynthesis (BBN). In minimal supergravity models, with 100 GeV to 10 TeV gravitinos are not stable, decaying during or after BBN. Their rate of production can be so large that subsequent gravitino decays completely change the standard BBN scenario. To prevent such “gravitino crisis” one requires an upper bound on the reheating temperature after inflation, since the abundance of gravitinos is proportional to . A recent detailed analysis derived a stringent upper bound GeV when the gravitino decay has hadronic modes [119].
This upper bound is in conflict with the temperature required for leptogenesis, GeV [120]. Therefore, thermal leptogenesis seems difficult to reconcile with low energy supersymmetry if gravitino masses lie in the range suggested by the simplest minimal supergravity models. Their required mass is typically too large in order for them to be produced after inflation, implying that the minimal type I supersymmetric seesaw schemes may be in trouble. Two recent suggestions have been made to cure this inconsistency.
One proposal [121] was to add a small Rparity violating term in the superpotential, where are righthanded neutrino supermultiplets. One can show that in the presence of this term, the produced leptonantilepton asymmetry can be enhanced. An alternative suggestion [122] was made in the context of the extended supersymmetric seesaw scheme considered in Sec. 3.1.5. It was shown in this case that leptogenesis can occur at the TeV scale through the decay of a new singlet, thereby avoiding the gravitino crisis. Washout of the asymmetry is effectively suppressed by the absence of direct couplings of the singlet to leptons.
6 Nonstandard neutrino interactions
Most neutrino mass generation mechanisms imply the existence of dimension6 subweak strength nonstandard neutrino interaction (NSI) operators, as illustrated in Fig. 17. These NSI can be of two types: flavourchanging (FC) and nonuniversal (NU). They are conceptually interesting for neutrino propagation since their presence leads to the possibility of resonant neutrino conversions even in the absence of masses [123].
NSI may arise from the nontrivial structure of charged and neutral current weak interactions characterizing seesawtype schemes [9]. While their expected magnitude is rather modeldependent, it may well fall within the range that will be tested in future precision studies [68]. For example, in inverse seesaw model of Sec. 3.2.3 the nonunitary piece of the lepton mixing matrix can be sizeable and hence the induced nonstandard interactions may be phenomenologically important. With neutrino physics entering the precision age it becomes a challenge to scrutinize the validity of the unitary approximation to the lepton mixing matrix, given its theoretical fragility [9].
Relatively sizable NSI strengths may also be induced in supersymmetric unified models [100] and models with radiatively induced neutrino masses, discussed in Sec. 3.2.1.
6.1 Atmospheric neutrinos
The Hamiltonian describing atmospheric neutrino propagation in the presence of NSI has, in addition to the usual oscillation part, another term ,
(52) 
Here holds for neutrinos (antineutrinos) and and parameterize the NSI: is the forward scattering amplitude for the FC process and represents the difference between and elastic forward scattering (NU). is the number density of the fermion along the neutrino path.
6.2 Solar neutrinos
Are solar oscillations robust? Do we understand the Sun, neutrino propagation and neutrino interactions well enough to trust current oscillation parameter determinations? Reactors have played a crucial role in identifying oscillations as “the” solution to the solar neutrino problem [62].
Thanks to KamLAND we have ruled out an otherwise excellent solution of the solar neutrino problem based on spinflavour precession due to convective zone magnetic fields [126, 127]. The absence of solar antineutrinos in KamLAND [128] has been used to establish robustness of the LMA solution with respect to spinflavour precession due to convective zone magnetic fields.
Thanks to KamLAND one could also establish robustness of the LMA solution with respect to small density fluctuations in the solar interior [129, 130], as could arise, say, from solar radiative zone magnetic fields [131].
Finally, again thanks to KamLAND we have ruled out an otherwise excellent solution of the solar neutrino problem based on nonstandard neutrino interactions [132]. However, in contrast to the atmospheric case, nonstandard physics may still affect neutrino propagation properties and detection cross sections in ways that can affect current determinations [133]. This implies that the oscillation interpretation of solar neutrino data is still “fragile” with respect to the presence of nonstandard interactions in the sector, though the required NSI strength for nonrobustness to set in is quite large. In contrast, one can show that even a small residual nonstandard interaction of neutrinos in this channel can have dramatic consequences for the sensitivity to at a neutrino factory [134]. It is therefore important to improve the sensitivities on NSI, another window of opportunity for neutrino physics in the precision age.
Acknowledgements
I thank the organizers for hospitality at Corfu. This work was supported by a Humboldt research award at the Institut für Theoretische Physik of the Universität Tübingen, and also by Spanish grants FPA200501269/BFM200200345, European commission RTN Contract MRTNCT2004503369 and ILIAS/N6 Contract RII3CT2004506222. I thank T. Rashba for proofreading.
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