Abstract
We present a concise review of models for neutrino masses and mixings with particular emphasis on recent developments and current problems. We discuss in detail attempts at reproducing approximate tribimaximal mixing starting from discrete symmetry groups, notably A4. We discuss the problems encountered when trying to extend the symmetry to the quark sector and to construct Grand Unified versions.
RM3TH/0717
CERNPH/2007213
Lectures on Models of Neutrino Masses and Mixings
Guido Altarelli ^{1}^{1}1
Dipartimento di Fisica e Sezione INFN, Universita’ di Roma Tre, 00146 Rome, Italy
Theory Division, CERN, 1211 Geneva 23, Switzerland
1 Introduction
At the Institute I gave two lectures on neutrino masses and mixings. Much of the material covered in my first lecture is written down in a review on the subject that I published not long ago with F. Feruglio [1] (see also [2]). Here, I make a relatively short summary (with updates) of the content of my first lecture, referring to our review for a more detailed presentation, and then I expand on the content of the second lecture which was dedicated to recent developments, in particular models of tribimaximal neutrino mixing, which were not covered in the review.
By now there is convincing evidence for solar and atmospheric neutrino oscillations. The values and mixing angles are known with fair accuracy. A summary of the results, taken from Ref. [3] is shown in Table 1. For the we have: eV and eV. As for the mixing angles, two are large and one is small. The atmospheric angle is large, actually compatible with maximal but not necessarily so: at :
lower limit  best value  upper limit  
()  ()  
7.2  7.9  8.6  
1.8  2.4  2.9  
0.27  0.31  0.37  
0.34  0.44  0.62  
0  0.009  0.032 
A very important recent experimental progress was the result obtained by the MiniBooNE Collaboration [5] that does not confirm the LSND signal (already not seen by the KARMEN experiment). This is very relevant because if the LSND claim had been proven right we would have needed an additional number of sterile neutrinos (i.e. without weak interactions) being involved in neutrino oscillations, or a violation of CPT symmetry (in order to make the spectrum of neutrinos and antineutrinos different). Actually on the MiniBooNE result there is some residual caveat in that their run was with a neutrino beam, while the LSND signal was seen in antineutrinos. But MiniBooNE is now running with antineutrinos, so that we will soon know if there is a difference. Also MiniBooNE has some excess at small neutrino energies which is not understood and their exclusion result is based on data at energies above a corresponding cut. In the following we assume that the LSND signal is not really there and assume that there are only the 3 known light active neutrinos.
In spite of this experimental progress there are still many alternative routes in constructing models of neutrino masses. This variety is mostly due to the considerable ambiguities that remain. First of all, neutrino oscillations only determine mass squared differences and a crucial missing input is the absolute scale of neutrino masses. Also the pattern of the neutrino mass spectrum is not known: it could be approximately degenerate with or of the inverse hierarchy type (with the solar doublet on top) or of the normal hierarchy type (with the solar doublet below).
The following experimental information on the absolute scale of neutrino masses is available. From the endpoint of tritium beta decay spectrum we have an absolute upper limit of 2 eV (at 95% C.L.) on the mass of “” [6], which, combined with the observed oscillation frequencies under the assumption of three CPTinvariant light neutrinos, represents also an upper bound on the masses of all active neutrinos. Less direct information on the mass scale is obtained from neutrinoless double beta decay (). The discovery of decay would be very important because it would directly establish lepton number violation and the Majorana nature of ’s (see section 3). The present limit from is affected by a relatively large uncertainty due to ambiguities on nuclear matrix elements. We quote here two recent limits (c.l.) [7]: eV [NEMO3()] or eV [Cuoricino()], where in terms of the mixing matrix and the mass eigenvalues (see eq.(9)). Complementary information on the sum of neutrino masses is also provided by measurements in cosmology [8], where an extraordinary progress has been made in the last years, in particular data on the cosmic microwave background (CMB) anisotropies (WMAP), on the large scale structure of the mass distribution in the Universe (SDSS, 2dFGRS) and from the Lyman alpha forest. WMAP by itself is not very restrictive: eV (at 95% C.L.). Combining CMB data with those on the large scale structure one obtains eV. Adding also the data from the Lyman alpha forest one has eV [9]. But this last combination is questionable because of some tension (at ’s) between the Lyman alpha forest data and those on the large scale structure. In any case, the cosmological bounds depend on a number of assumptions (or, in fashionable terms, priors) on the cosmological model. In summary, from cosmology for 3 degenerate neutrinos of mass , depending on which data sets we include and on our degree of confidence in cosmological models, we can conclude that eV.
Given that neutrino masses are certainly extremely small, it is really difficult from the theory point of view to avoid the conclusion that L conservation is probably violated. In fact, in terms of lepton number violation the smallness of neutrino masses can be naturally explained as inversely proportional to the very large scale where lepton number L is violated, of order the grand unification scale or even the Planck scale . If neutrinos are Majorana particles, their masses arise from the generic dimensionfive non renormalizable operator of the form:
(1) 
with being the ordinary Higgs doublet, the SU(2) lepton doublets, a matrix in flavour space, a large scale of mass and a charge conjugation matrix between the lepton fields is understood.
Neutrino masses generated by are of the order for , where is the vacuum expectation value of the ordinary Higgs. A particular realization leading to comparable masses is the seesaw mechanism [10], where derives from the exchange of heavy ’s: the resulting neutrino mass matrix reads:
(2) 
that is, the light neutrino masses are quadratic in the Dirac masses and inversely proportional to the large Majorana mass. For eV and with GeV we find GeV which indeed is an impressive indication for . Thus probably neutrino masses are a probe into the physics at .
2 Basic Formulae for ThreeNeutrino Mixing
Neutrino oscillations are due to a misalignment between the flavour basis, , where is the partner of the mass and flavour eigenstate in a lefthanded (LH) weak isospin SU(2) doublet (similarly for and ) and the mass eigenstates :
(3) 
where is the unitary 3 by 3 mixing matrix. Given the definition of and the transformation properties of the effective light neutrino mass matrix :
(4)  
we obtain the general form of (i.e. of the light mass matrix in the basis where the charged lepton mass is a diagonal matrix):
(5) 
The matrix can be parameterized in terms of three mixing angles , and () and one phase () , exactly as for the quark mixing matrix . The following definition of mixing angles can be adopted:
(6) 
where , . In addition, if are Majorana particles, we have the relative phases among the Majorana masses , and . If we choose real and positive, these phases are carried by . Thus, in general, 9 parameters are added to the SM when nonvanishing neutrino masses are included: 3 eigenvalues, 3 mixing angles and 3 CP violating phases.
In our notation the two frequencies, , are parametrized in terms of the mass eigenvalues by
(7) 
where and . The numbering 1,2,3 corresponds to our definition of the frequencies and in principle may not coincide with the ordering from the lightest to the heaviest state. From experiment, see table 1, we know that is small, according to CHOOZ, (3).
If would be exactly zero there would be no CP violations in oscillations. A main target of the new planned oscillation experiments is to measure the actual size of . In the next decade the upper limit on will possibly go down by at least an order of magnitude (T2K, NoA, DoubleCHOOZ…..). Even for three neutrinos the pattern of the neutrino mass spectrum is still undetermined: it can be approximately degenerate, or of the inverse hierarchy type or normally hierarchical. Given the observed frequencies and the notation , with and , the three possible patterns of mass eigenvalues are:
(8) 
The sign of can be measured in the future through matter effects in long baseline experiments. Models based on all these patterns have been proposed and studied and all are in fact viable at present.
3 Importance of Neutrinoless Double Beta Decay
Oscillation experiments do not provide information about the absolute neutrino spectrum and cannot distinguish between pure Dirac and Majorana neutrinos. The detection of neutrinoless double beta decay, besides its enormous intrinsic importance as direct evidence of non conservation, would also offer a way to possibly disentangle the 3 cases. The quantity which is bound by experiments is the 11 entry of the mass matrix, which in general, from , is given by :
(9) 
Starting from this general formula it is simple to derive the following bounds for degenerate, inverse hierarchy or normal hierarchy mass patterns (see fig.1).

Degenerate case. If is the common mass and we set , which is a safe approximation in this case, because cannot compensate for the smallness of , we have . Here the phase ambiguity has been reduced to a sign ambiguity which is sufficient for deriving bounds. So, depending on the sign we have or . We conclude that in this case could be as large as the present experimental limit but should be at least of order given that the solar angle cannot be too close to maximal (in which case the minus sign option could be arbitrarily small). The experimental 2 range of the solar angle does not favour a cancellation by more than a factor of about 3.

Inverse hierarchy case. In this case the same approximate formula holds because is small and the term in eq.(9) can be neglected. The difference is that here we know that so that eV. At the same time, since a full cancellation between the two contributions cannot take place, we expect eV.

Normal hierarchy case. Here we cannot in general neglect the term. However in this case
In the next few years a new generation of experiments will reach a larger sensitivity on by about an order of magnitude. If these experiments will observe a signal this would indicate that the inverse hierarchy is realized, if not, then the normal hierarchy case remains a possibility.
4 Baryogenesis via Leptogenesis from Heavy Decay
In the Universe we observe an apparent excess of baryons over antibaryons. It is appealing that one can explain the observed baryon asymmetry by dynamical evolution (baryogenesis) starting from an initial state of the Universe with zero baryon number. For baryogenesis one needs the three famous Sakharov conditions: B violation, CP violation and no thermal equilibrium. In the history of the Universe these necessary requirements have possibly occurred at different epochs. Note however that the asymmetry generated by one epoch could be erased at following epochs if not protected by some dynamical reason. In principle these conditions could be verified in the SM at the electroweak phase transition. B is violated by instantons when kT is of the order of the weak scale (but BL is conserved), CP is violated by the CKM phase and sufficiently marked outof equilibrium conditions could be realized during the electroweak phase transition. So the conditions for baryogenesis at the weak scale in the SM superficially appear to be present. However, a more quantitative analysis [12] shows that baryogenesis is not possible in the SM because there is not enough CP violation and the phase transition is not sufficiently strong first order, unless the Higgs mass is below a bound which by now is completely excluded by LEP. In SUSY extensions of the SM, in particular in the MSSM, there are additional sources of CP violation and the bound on is modified by a sufficient amount by the presence of scalars with large couplings to the Higgs sector, typically the stop. What is required is that , a stop not heavier than the top quark and, preferentially, a small . But also this possibility has by now become at best marginal with the results from LEP2.
If baryogenesis at the weak scale is excluded by the data it can occur at or just below the GUT scale, after inflation. But only that part with would survive and not be erased at the weak scale by instanton effects. Thus baryogenesis at needs BL violation at some stage like for if neutrinos are Majorana particles. The two effects could be related if baryogenesis arises from leptogenesis then converted into baryogenesis by instantons [13]. The decays of heavy Majorana neutrinos (the heavy eigenstates of the seesaw mechanism) happen with violation of lepton number L, hence also of BL and can well involve a sufficient amount of CP violation. Recent results on neutrino masses are compatible with this elegant possibility. Thus the case of baryogenesis through leptogenesis has been boosted by the recent results on neutrinos.
5 Models of Neutrino Mixing
After KamLAND, SNO and the upper limits on the absolute value of neutrino masses not too much hierarchy in the spectrum of neutrinos is indicated by experiments:
(10) 
Precisely at : 3]. Thus, for a hierarchical spectrum, , which is comparable to the Cabibbo angle or . This suggests that the same hierarchy parameter (raised to powers with o(1) exponents) may apply for quark, charged lepton and neutrino mass matrices. This in turn indicates that, in absence of some special dynamical reason, we do not expect quantities like or the deviation of from its maximal value to be too small. Indeed it would be very important to know how small the mixing angle is and how close to maximal is. Actually one can make a distinction between ”normal” and ”exceptional” models. For normal models is not too close to maximal and is not too small, typically a small power of the selfsuggesting order parameter , with . Exceptional models are those where some symmetry or dynamical feature assures in a natural way the near vanishing of and/or of . Normal models are conceptually more economical and much simpler to construct. Typical categories of normal models are (we refer to the review in ref.[1] for a detailed discussion of the relevant models and a more complete list of references): [

Anarchy. These are models with approximately degenerate mass spectrum and no ordering principle or approximate symmetry assumed in the neutrino mass sector [14]. The small value of r is accidental, due to random fluctuations of matrix elements in the Dirac and Majorana neutrino mass matrices. Starting from a random input for each matrix element, the seesaw formula, being a product of 3 matrices, generates a broad distribution of r values. All mixing angles are generically large: so in this case one does not expect to be maximal and should probably be found near its upper bound.

Semianarchy. We have seen that anarchy is the absence of structure in the neutrino sector. Here we consider an attenuation of anarchy where the absence of structure is limited to the 23 neutrino sector. The typical structure is in this case [15]:
(11) where and are small and by 1 we mean entries of and also the 23 determinant is of . This texture can be realized, for example, without seesaw from a suitable set of charges for , eg appearing in the dim. 5 operator . Clearly, in general we would expect two mass eigenvalues of order 1, in units of , and one small, of order or . This typical pattern would not fit the observed solar and atmospheric observed frequencies. However, given that is not too small, we can assume that its small value is generated accidentally, as for anarchy. We see that, if by chance the second eigenvalue , we can then obtain the correct value of together with large but in general non maximal and and small . The guaranteed smallness of is the main advantage over anarchy, and the relation with normally keeps not too small. For example, in typical models that provide a very economical but effective realization of this scheme .

Inverse hierarchy. One obtains inverted hierarchy, for example, in the limit of exact symmetry for LH lepton doublets [16]. In this limit and is maximal while is generically large. [1]. Simple forms of symmetry breaking cannot sufficiently displace from the maximal value because typically . Viable normal models can be obtained by arranging large contributions to and from the charged lepton mass diagonalization. But then, it turns out that, in order to obtain the measured value of , the size of must be close to its present upper bound [17]. If indeed the shift from maximal is due to the charged lepton diagonalization, this could offer a possible track to explain the empirical relation [18] (with present data ). While it would not be difficult in this case to arrange that the shift from maximal is of the order of , it is not at all simple to guarantee that it is precisely equal to [19] (for a recent attempt, see [20]). Besides the effect of the charged lepton diagonalization, in a seesaw context, one can assume a strong additional breaking of from soft terms in the Majorana mass matrix [21]. Since ’s are gauge singlets and thus essentially uncoupled, a large breaking in does not feedback in other sectors of the lagrangian. In this way one can obtain realistic values for and for all other masses and mixings, in particular also with a small .

Normal hierarchy. Particularly interesting are models with 23 determinant suppressed by seesaw [1]: in the 23 sector one needs relatively large mass splittings to fit the small value of but nearly maximal mixing. This can be obtained if the 23 subdeterminant is suppressed by some dynamical trick. Typical examples are lopsided models with large off diagonal term in the Dirac matrices of charged leptons and/or neutrinos (in minimal SU(5) the dquark and charged lepton mass matrices are one the transposed of the other, so that large lefthanded mixings for charged leptons correspond to large unobservable righthanded mixings for dquarks). Another class of typical examples is the dominance in the seesaw formula of a small eigenvalue in , the righthanded Majorana neutrino mass matrix. When the 23 determinant suppression is implemented in a 3x3 context, normally is not protected from contributions that vanish with the 23 determinant, hence with .
The fact that some neutrino mixing angles are large and even nearly maximal, while surprising at the start, was soon realised to be well compatible with a unified picture of quark and lepton masses within GUTs. The symmetry group at could be either (SUSY) SU(5) or SO(10) or a larger group [1] (for some more recent models, see [22]). For example, normal models leading to anarchy, semianarchy, inverted hierarchy or normal hierarchy can all be naturally implemented by simple assignments of U(1) horizontal charges in a semiquantitative unified description of all quark and lepton masses in SUSY SU(5) U(1). Actually, in this context, if one adopts a statistical criterium, hierarchical models appear to be preferred over anarchy and among them normal hierarchy with seesaw ends up as being the most likely [23].
In conclusion we expect that experiment will eventually find that is not too small and that is sizably not maximal. But if, on the contrary, either is found from experiment to be very small or to be very close to maximal or both, then theory will need to cope with this fact. Normal models have been extensively discussed in the literature [1], so we concentrate here in more detail on a particularly interesting class of exceptional models.
6 Approximate Tribimaximal Mixing
Here we want to discuss particular exceptional models where both and exactly vanish (more precisely, they vanish in a suitable limit, with correction terms that can be made negligibly small) and, in addition, , a value which is in very good agreement with present data (as already noted in the Introduction, the angle is the best measured at present). This is the socalled tribimaximal or HarrisonPerkinsScott mixing pattern (HPS) [24], with the entries in the second column all equal to in absolute value. Here we adopt the following phase convention:
(12) 
In the HPS scheme , to be compared with the latest experimental determination [4]: (at ). Thus the HPS mixing matrix is a good representation of the present data within one . The challenge is to find natural and appealing schemes that lead to this matrix with good accuracy. Clearly, in a natural realization of this model, a very constraining and predictive dynamics must be underlying. It is interesting to explore particular structures giving rise to this very special set of models in a natural way. In this case we have a maximum of ”order” implying special values for all mixing angles. Interesting ideas on how to obtain the HPS mixing matrix have been discussed in refs. [24, 25, 26]. Some attractive models are based on the discrete symmetry A4, which appears as particularly suitable for the purpose, and were presented in ref. [27, 28, 29, 30, 31, 32].
The HPS mixing matrix suggests that mixing angles are independent of mass ratios (while for quark mixings relations like are typical). In fact in the basis where charged lepton masses are diagonal, the effective neutrino mass matrix in the HPS case is given by :
(13) 
where:
(14) 
The eigenvalues of are , , with eigenvectors , and , respectively. In general, disregarding possible Majorana phases, there are six parameters in a real symmetric matrix like : here only three are left after the values of the three mixing angles have been fixed à la HPS. For a hierarchical spectrum , , and could be negligible. But also degenerate masses and inverse hierarchy can be reproduced: for example, by taking we have a degenerate model, while for and an inverse hierarchy case is realized (stability under renormalization group running strongly prefers opposite signs for the first and the second eigenvalue which are related to solar oscillations and have the smallest mass squared splitting).
It is interesting to recall that the most general mass matrix, in the basis where charged leptons are diagonal, that corresponds to and maximal is of the form [35]:
(15) 
Note that this matrix is symmetric under 23 or exchange [36]. For there is no CP violation, so that, disregarding Majorana phases, we can restrict our consideration to real parameters. There are four of them in eq.(15) which correspond to three mass eigenvalues and one remaining mixing angle, . In particular, is given by:
(16) 
In the HPS case is also fixed and an additional parameter can be eliminated, leading to:
(17) 
It is easy to see that the HPS mass matrix in eqs.(1314) is indeed of the form in eq.(17).
Different models have been formulated that lead or can accomodate approximate tribimaximal mixing. There are models where the assumed symmetries or textures lead to a mass matrix expressed in terms of a number of parameters. Then those parameters are fixed in such a way as to reproduce the desired result for mixings. Other models are more predictive in that approximate tribimaximal mixing is obtained in the most general case as a natural consequence of the assumptions made (parameter fitting is then only present to fix the observed mass eigenvalues for charged leptons or for the neutrino values, within the desired mixing pattern). In the next sections we will present models of tribimaximal mixing based on the A4 group that belong to the latter class of more ambitious models. We first introduce A4 and its representations and then we show that this group is particularly suited to the problem.
7 The A4 Group
A4 is the group of the even permutations of 4 objects. It has 4!/2=12 elements. Geometrically, it can be seen as the invariance group of a tethraedron (the odd permutations, for example the exchange of two vertices, cannot be obtained by moving a rigid solid). Let us denote a generic permutation simply by . can be generated by two basic permutations and given by and . One checks immediately that:
(18) 
This is called a ”presentation” of the group. The 12 even permutations belong to 4 equivalence classes ( and belong to the same class if there is a in the group such that ) and are generated from and as follows:
Note that, except for the identity which always forms an equivalence class in itself, the other classes are according to the powers of (in C4 could as well be seen as ).
In a finite group the squared dimensions of the inequivalent irreducible representations add up to , the number of transformations in the group ( in ). has four inequivalent representations: three of dimension one, , and and one of dimension . It is immediate to see that the onedimensional unitary representations are obtained by:
(20)  
Note that is the cubic root of 1 and satisfies , .
Class  
1  1  1  3  
1  0  
1  0  
1  1  1  1 
The threedimensional unitary representation, in a basis where the element is diagonal, is built up from:
(21) 
The characters of a group are defined, for each element , as the trace of the matrix that maps the element in a given representation . It is easy to see that equivalent representations have the same characters and that characters have the same value for all elements in an equivalence class. Characters satisfy . Also, for each element , the character of in a direct product of representations is the product of the characters: and also is equal to the sum of the characters in each representation that appears in the decomposition of . The character table of A4 is given in Table II [27]. From this Table one derives that indeed there are no more inequivalent irreducible representations other than , , and . Also, the multiplication rules are clear: the product of two 3 gives and , , etc. If is a triplet transforming by the matrices in eq.(21) we have that under : (here the upper index indicates transposition) and under : . Then, from two such triplets , the irreducible representations obtained from their product are:
(22) 
(23) 
(24) 
(25) 
(26) 
In fact, take for example the expression for . Under it is invariant and under it goes into which is exactly the transformation corresponding to .
In eq.(21) we have the representation 3 in a basis where is diagonal. It is interesting to go to a basis where instead it is which is diagonal. This is obtained through the unitary transformation:
(27)  
(28) 
where:
(29) 
The matrix is special in that it is a 3x3 unitary matrix with all entries of unit absolute value. It is interesting that this matrix was proposed long ago as a possible mixing matrix for neutrinos [37]. We shall see in the following that the matrix appears in models as the unitary transformation that diagonalizes the charged lepton mass matrix. In the , basis the product composition rule is different [31]:
(30) 
(31) 
(32) 
(33) 
(34) 
There is an interesting relation [31] between the model considered so far and the modular group. This relation could possibly be relevant to understand the origin of the A4 symmetry from a more fundamental layer of the theory. The modular group is the group of linear fractional transformations acting on a complex variable :
(35) 
where are integers. There are infinite elements in , but all of them can be generated by the two transformations:
(36) 
The transformations and in (36) satisfy the relations
(37) 
and, conversely, these relations provide an abstract characterization of the modular group. Since the relations (18) are a particular case of the more general constraint (37), it is clear that A4 is a very small subgroup of the modular group and that the A4 representations discussed above are also representations of the modular group. In string theory the transformations (36) operate in many different contexts. For instance the role of the complex variable can be played by a field, whose VEV can be related to a physical quantity like a compactification radius or a coupling constant. In that case in eq. (36) represents a duality transformation and in eq. (36) represent the transformation associated to an ”axionic” symmetry.
A different way to understand the dynamical origin of was recently presented in ref. [32] where it is shown that the symmetry can be simply obtained by orbifolding starting from a model in 6 dimensions (6D) (see also [33], [34]). In this approach appears as the remnant of the reduction from 6D to 4D spacetime symmetry induced by the special orbifolding adopted. There are 4D branes at the four fixed points of the orbifolding and the tetrahedral symmetry of connects these branes. The standard model fields have components on the fixed point branes while the scalar fields necessary for the breaking are in the bulk. Each brane field, either a triplet or a singlet, has components on all of the four fixed points (in particular all components are equal for a singlet) but the interactions are local, i.e. all vertices involve products of field components at the same spacetime point. This approach suggests a deep relation between flavour symmetry in 4D and spacetime symmetry in extra dimensions. However, the specific classification of the fields under A4 which is adopted in our model does not follow from the compactification and is separately assumed.
The orbifolding is defined as follows. We consider a quantum field theory in 6 dimensions, with two extra dimensions compactified on an orbifold . We denote by the complex coordinate describing the extra space. The torus is defined by identifying in the complex plane the points related by
(38) 
where our length unit, , has been set to 1 for the time being. The parity is defined by
(39) 
and the orbifold can be represented by the fundamental region given by the triangle with vertices , see Fig. 1. The orbifold has four fixed points, . The fixed point is also represented by the vertices and . In the orbifold, the segments labelled by in Fig. 1, and , are identified and similarly for those labelled by , and , and those labelled by , , . Therefore the orbifold is a regular tetrahedron with vertices at the four fixed points.
The symmetry of the uncompactified 6D space time is broken by compactification. Here we assume that, before compactification, the spacetime symmetry coincides with the product of 6D translations and 6D proper Lorentz transformations. The compactification breaks part of this symmetry. However, due to the special geometry of our orbifold, a discrete subgroup of rotations and translations in the extra space is left unbroken. This group can be generated by two transformations:
(40) 
Indeed and induce even permutations of the four fixed points:
(41) 
thus generating the group . From the previous equations we immediately verify that and satisfy the characteristic relations obeyed by the generators of : . These relations are actually satisfied not only at the fixed points, but on the whole orbifold, as can be easily checked from the general definitions of and in eq. (40), with the help of the orbifold defining rules in eqs. (38) and (39).
8 Applying A4 to Lepton Masses and Mixings
A typical A4 model works as follows [30], [31]. One assigns leptons to the four inequivalent representations of A4: lefthanded lepton doublets transform as a triplet , while the righthanded charged leptons , and transform as , and , respectively. At this stage we do not introduce RH neutrinos, but later we will discuss a seesaw realization. The flavour symmetry is broken by two real triplets and and by a real singlet . These flavon fields are all gauge singlets. We also need one or two ordinary SM Higgs doublets , which we take invariant under A4. The Yukawa interactions in the lepton sector read:
In our notation, transforms as , transforms as and transforms as . Also, to keep our notation compact, we use a twocomponent notation for the fermion fields and we set to 1 the Higgs fields and the cutoff scale . For instance stands for , stands for and so on. The Lagrangian contains the lowest order operators in an expansion in powers of . Dots stand for higher dimensional operators that will be discussed later. Some terms allowed by the flavour symmetry, such as the terms obtained by the exchange , or the term are missing in . Their absence is crucial and, in each version of A4 models, is motivated by additional symmetries. For example , being of lower dimension with respect to , would be the dominant component, proportional to the identity, of the neutrino mass matrix. In addition to that, the presence of the singlet flavon plays an important role in making the VEV directions of and different.
For the model to work it is essential that the fields , and develop a VEV along the directions (in the , basis, i.e. with diagonal, eq.(21):
(43) 
A crucial part of all serious A4 models is the dynamical generation of this alignment in a natural way. If the alignment is realized, at the leading order of the expansion, the mass matrices and for charged leptons and neutrinos are given by:
(44) 
(45) 
where
(46) 
Charged leptons are diagonalized by the matrix
(47) 
This matrix was already introduced in eq.(29) as the unitary transformation between the diagonal to the diagonal 3x3 representation of . In fact, in this model, the diagonal basis is the Lagrangian basis and the diagonal basis is that of diagonal charged leptons. The great virtue of is to immediately produce the special unitary matrix as the diagonalizing matrix of charged leptons and also to allow a singlet made up of three triplets, which leads, for the alignment in eq. (43), to the right neutrino mass matrix to finally obtain the HPS mixing matrix.
The charged fermion masses are given by:
(48) 
We can easily obtain in a a natural way the observed hierarchy among , and by introducing an additional U(1) flavour symmetry under which only the righthanded lepton sector is charged. We assign Fcharges , and to , and , respectively. By assuming that a flavon , carrying a negative unit of F, acquires a VEV , the Yukawa couplings become field dependent quantities and we have
(49) 
In the flavour basis the neutrino mass matrix reads [notice that the change of basis induced by , because of the Majorana nature of neutrinos, will in general change the relative phases of the eigenvalues of