# [

###### Abstract

We review the application of non-Abelian discrete groups to Tri-Bimaximal (TB) neutrino mixing, which is supported by experiment as a possible good first approximation to the data. After summarizing the motivation and the formalism, we discuss specific models, mainly those based on but also on other finite groups, and their phenomenological implications, including the extension to quarks. The recent measurements of favour versions of these models where a suitable mechanism leads to corrections to that can naturally be larger than those to and . The virtues and the problems of TB mixing models are discussed, also in connection with lepton flavour violating processes, and the different approaches are compared.

theDOIsuffix \Volume55 \Month01 \Year2007 \pagespan1 \ReceiveddateXXXX \ReviseddateXXXX \AccepteddateXXXX \DatepostedXXXX

TB Mixing and Discrete Symmetries]Tri-Bimaximal Neutrino Mixing and Discrete Flavour Symmetries
G. Altarelli]Guido Altarelli^{1}^{1}1E-mail:
F. Feruglio]Ferruccio Feruglio^{2}^{2}2E-mail:
L. Merlo]Luca Merlo^{3}^{3}3Corresponding author E-mail:

## 1 Introduction

Neutrino mixing [1, 2, 3, 4, 5, 6] is important because it could in principle provide new clues for the understanding of the flavour problem. Even more so since neutrino mixing angles show a pattern that is completely different than that of quark mixing. The bulk of the data on neutrino oscillations are well described in terms of three active neutrinos. By now all three mixing angles have been measured, although with different levels of accuracy (see Tab. 1 [7, 8]). In particular, we have experimental evidence for a non vanishing value of the smallest angle (see Tab. 1 [9, 10, 11, 12]): considering the most precise results from DOUBLE CHOOZ, Daya Bay and RENO, we get

(1) |

for both the mass orderings.

[h] \vchcaptionRecent fits to neutrino oscillation data from [7, 8]. In the brackets the IH case. In this case the full is allowed. Quantity Fogli et al. [7] Schwetz et al. [8] () () () () () () () ()

[h!] \vchcaptionThe reactor angle measurements from the recent experiments T2K[9], MINOS[10], DOUBLE CHOOZ[11], Daya Bay [12] and RENO [13], for the normal (inverse) hierarchy. Quantity T2K[9] () () MINOS[10] () () DC[11] DYB[12] RENO[13]

Models of neutrino mixing based on discrete flavour groups have received a lot of attention in recent years [14, 15, 16, 17, 18, 19]. There are a number of special mixing patterns that have been studied in that context. Most of these mixing matrices have , , values that are a good approximation to the data, and differ by the value of the solar angle . The observed , the best measured mixing angle, is very close, from below, to the so called Tri-Bimaximal (TB) value [20, 21, 22, 23, 24] which is (see Fig. 1). Alternatively it is also very close, from above, to the Golden Ratio (GR) value [25, 26, 27, 28] which is

In the following we will concentrate on TB mixing which is perhaps the most plausible and certainly the most studied first approximation to the data. The simplest symmetry that, in leading order (LO), leads to TB is , the group of even permutations of 4 objects, a subgroup of that includes all 4-object permutations. Thus, in the following, we will devote a special attention to models, but alternative theories of TB mixing will also be briefly considered. The plan of the paper is as follows. In Sect. 2 we recall the definitions of TB, GR and BM mixing and the symmetries of the corresponding mass matrices. In Sect. 3 we summarize the group theory of . In Sect. 4 we review the structure of models of lepton masses and mixings and, in two separate subsections, we first describe the baseline models and then those special models [58, 59] where additional dynamical ingredients allow that the angle can naturally be of different (and larger) order of magnitude than the deviations of from the TB value. We also discuss the comparison with present data of the two options. In Sect. 5 we discuss the possible extension of the TB models to include quarks, possibly also in a GUT context. Our speculations on the origin of either as a subgroup of the modular group or as a remnant of an extra dimensional spacetime symmetry are presented in Sect. 6. A number of alternative theories of TB mixing are briefly considered in Sect. 7. Sect. 8 contains a summary on the implications for lepton flavour violation of the different models described in Ref. [58]. Finally in Sect. 9 we derive our conclusions.

The values of for TB or GR or BM mixing are compared with the data at .

## 2 Special Patterns of Neutrino Mixing

Starting from the PNMS mixing matrix (we refer the reader to Ref. [1, 5] for its general definition and parametrisation), the general form of the neutrino mass matrix, in terms of the (complex^{4}^{4}4We absorb the Majorana phases in the mass eigenvalues , rather than in the mixing matrix .
The dependence on these phases drops in neutrino oscillations.) mass eigenvalues , in the basis where charged leptons are diagonal, is given by

(2) |

We present here some particularly relevant forms of and that are important in the following. We start by the most general mass matrix that corresponds to and maximal, that is to given by (in a particular phase convention)

(3) |

with and . By applying eq. (2) we obtain a matrix of the form [60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75]:

(4) |

with complex coefficients , , and . This matrix is the most general one that is symmetric under 2-3 (or ) exchange or

(5) |

where is given by

(6) |

The solar mixing angle is given by

(7) |

where the second equality applies to real parameters. Since , in this limit there is not no CP violation in neutrino oscillations, and the only physical phases are the Majorana ones, accounted for by the general case of complex parameters. We restrict here our consideration to real parameters. There are four of them in eq. (4) which correspond to the three mass eigenvalues and one remaining mixing angle, . Models with - symmetry have been extensively studied [60, 61, 62, 63, 64, 66, 67, 65, 69, 68, 74, 72, 70, 71, 73, 75, 76, 77].

The particularly important case of TB mixing is obtained when or
^{5}^{5}5The other solution gives rise to TB mixing in another phase convention and is physically equivalent to .. In this case the matrix takes the form

(8) |

In fact, in this case, is given by [20, 21, 22, 23, 24]

(9) |

Note that is a rotation matrix with special, fixed angles: indeed all the entries of are pure numbers. This property is related to the particular pattern of which belongs to the category of the form diagonalizable mass matrices [78]. These matrices are diagonalized by unitary transformations that are independent from the eigenvalues. At the LO discrete flavour models give rise to form diagonalizable mass matrices and the physical mixing angles are thus unrelated to masses. From eq. (2), one obtains

(10) |

where

(11) |

are the respective columns of and are the neutrino mass eigenvalues. It is easy to see that the TB mass matrix in eqs. (10) and (11) is indeed of the form in eq. (8). All patterns for the neutrino spectrum are in principle possible. 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 [79, 80, 81, 42, 82, 83, 84]).

Note that the mass matrix for TB mixing, in the basis where charged leptons are diagonal, as given in eq. (8), can be specified as the most general matrix which is invariant under (or 2-3) symmetry and, in addition, under the action of a unitary symmetric matrix (actually and ):

(12) |

where is given by

(13) |

Similarly, it is useful to consider the product , where is the charged lepton mass matrix (defined as ), because this product transforms as , with the unitary matrix that rotates the left-handed (LH) charged lepton fields. The most general diagonal is invariant under a diagonal phase matrix with 3 different phase factors,

(14) |

and conversely a matrix satisfying the above requirement is diagonal. If the matrix generates a cyclic group . In the simplest case and we get but is equally possible. In the case we have

(15) |

where , so that .

We are now in a position to explain the role of finite groups and to formulate the general strategy to obtain the special mass matrix of TB mixing. We must find a group which, for simplicity, must be as small as possible but large enough to contain the and transformations. A limited number of products of and close a finite group . Hence the group contains the subgroups and generated by monomials in and , respectively. We assume that the theory is invariant under the spontaneously broken symmetry described by . Then we must arrange a breaking of such that, at leading order, is broken down to in the neutrino mass sector and down to in the charged lepton mass sector. In a good model this step must be realized in a natural way as a consequence of the stated basic principles, and not put in by hand. The symmetry under in some cases is also part of (this the case of , the permutation group of 4 objects) and then must be preserved in the neutrino sector along with by the breaking or it could arise as a consequence of a special feature of the breaking (for example, in it is obtained by allowing only some transformation properties for the flavons with non vanishing VEV’s). The explicit example of is discussed in the next section. Note that, along the same line, a model with symmetry can be realized in terms of the group generated by products of and (see, for example, Ref. [85, 86]).

## 3 The Group

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 tetrahedron (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

(16) |

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:

(17) | ||||||

Note that, except for the identity which always forms an equivalence class in itself, the other classes are according to the powers of (in , could as well be seen as ).

[h!] \vchcaptionCharacters of Class 1 1 1 3 1 0 1 0 1 1 1 -1

The characters of a group are defined, for each element , as the trace of the matrix that maps the element in a given representation . From the invariance of traces under similarity transformations it follows that equivalent representations have the same characters and that characters have the same value for all elements in an equivalence class. Characters satisfy , where is the number of transformations in the group ( in ). 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 . In a finite group the squared dimensions of the inequivalent irreducible representations add up to . The character table of is given in Tab. 3. From this table one derives that has four inequivalent representations: three of dimension one, , and and one of dimension .

It is immediate to see that the one-dimensional unitary representations are obtained by:

(18) | ||||||||

Note that is the cubic root of 1 and satisfies , .

The three-dimensional unitary representation, in a basis where the element is diagonal, is built up from:

(19) |

The multiplication rules are as follows: the product of two 3 gives and , , etc. If is a triplet transforming by the matrices in eq. (19) 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:

(20) |

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. (19) we have the representation 3 in a basis where is diagonal. We shall see that for our purposes it is convenient to go to a basis where instead it is that is diagonal. This is obtained through the unitary transformation:

(21) | |||||

(22) |

where:

(23) |

The matrix is special in that it is a 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 [87, 88]. We shall see in the following that in the diagonal basis the charged lepton mass matrix (to be precise the matrix ) is diagonal. Notice that the matrices of eqs. (21) and (22) coincide with the matrices of the previous section.

In this basis the product rules of two triplets, () and () of , according to the multiplication rule are different than in the diagonal basis (because for Majorana mass matrices the relevant scalar product is and not )and are given by:

(24) |

An obvious representation of is obtained by considering the matrices that directly realize each permutation. For and we have

(25) |

The matrices and satisfy the relations in eq. (16), thus providing a representation of . Since the only irreducible representations of are a triplet and three singlets, the representation described by and is not irreducible. It decomposes into the sum of the invariant singlet plus the triplet representation. In fact the vector is clearly invariant under permutations and similarly the 3-dimensional space orthogonal to it. In matrix terms this decomposition is realized by the unitary matrix [89] given by

(26) |

This matrix maps and into matrices that are block-diagonal:

(27) |

where and are the generators of the three-dimensional representation in eq. (19).

In the following we will work in the diagonal basis, unless otherwise stated. In this basis the 12 matrices of the 3-dimensional representation of are given as follows:

: | ||
---|---|---|

: | ||

: | ||

: | ||

We can now see why works for TB mixing. In Sec. 2 we have already mentioned that the most general mass matrix for TB mixing in eq. (8), in the basis where charged leptons are diagonal, can be specified as one which is invariant under the 2-3 (or ) symmetry and under the unitary transformation, as stated in eq. (12). This observation plays a key role in leading to as a candidate group for TB mixing, because is a matrix of . Instead the matrix is not an element of (because the 2-3 exchange is an odd permutation). We shall see that in models the 2-3 symmetry of the neutrino mass matrix arises as an accidental symmetry of the LO Lagrangian by imposing that there are no flavons transforming as or that break with two different VEV’s (in particular one can assume that there are no flavons in the model transforming as or ). It is also clear that a generic diagonal charged lepton matrix is characterized by the invariance under , or .

The group has two obvious subgroups: , which is a reflection subgroup generated by , and , which is the group generated by , which is isomorphic to . If the flavour symmetry associated to is broken by the VEV of a triplet of scalar fields, there are two interesting breaking pattern. The VEV

(28) |

breaks down to , while

(29) |

breaks down to . As we will see, and are the relevant low-energy symmetries of the neutrino and the charged-lepton sectors, respectively. Indeed we have already seen that the TB mass matrix is invariant under and a diagonal charged lepton mass is invariant under .

## 4 Applying to Lepton Masses and Mixings

In the lepton sector a typical model works as follows [90]. One assigns leptons to the four inequivalent representations of : LH lepton doublets transform as a triplet , while the RH charged leptons , and transform as , and , respectively. These models can be realized both with and without a see-saw mechanism. In the first case there are three right-handed neutrinos transforming as a triplet of , while in the second case the source of neutrino masses is a set of higher dimensional operators violating the total lepton number. Here we consider a see-saw realization, so we also introduce conjugate neutrino fields transforming as a triplet of . The fact that LH lepton doublets and, in the see-saw case, also the RH neutrinos , transform as triplets is crucial to realize the fixed ratios of mass matrix elements needed to obtain TB mixing. A drawback is that for the ratio , defined by , one would expect to be compared with the experimental value is , which implies a moderate fine-tuning.

One adopts a supersymmetric (SUSY) context also to make contact
with Grand Unification (flavour symmetries are supposed to act near the GUT scale^{6}^{6}6When the flavour symmetry is broken contextually with the electroweak one, such as in Refs. [91, 92, 93], strong constraints from FCNC transitions are usually present [94, 95], that can eventually rule out the model.). In fact, as well known, SUSY is important in GUT’s for offering a solution to the hierarchy problem, for improving coupling unification and for making the theory compatible with bounds on proton decay. But, in models of lepton mixing, SUSY also helps for obtaining the vacuum alignment, because the SUSY constraints are very strong and limit the form of the superpotential very much. Thus SUSY is not necessary but it is a plausible and useful ingredient. The flavour symmetry is broken by two sets of flavons and , invariant under the SM gauge symmetry, that at the LO break down to and , respectively. At this order couples only to the charge lepton sector and to the neutrino sector. Typically and include triplets and invariant singlets under , but models with flavons transforming as and have also been considered [96, 97]. For example can consist of the triplet with the vacuum alignment in eq. (29) and can include the triplet with the vacuum alignment in eq. (28) and two invariant singlets , .
Two Higgs doublets , invariant under , are also introduced. One can obtain the observed hierarchy among , and
by introducing an additional U(1) flavour symmetry [98] under
which only the RH lepton sector is charged (recently some models were proposed with a different VEV alignment such that the charged lepton hierarchies are obtained without introducing a symmetry [99, 100]).
We recall that is a simple flavour symmetry where particles in different generations are assigned (in general) different values of an Abelian charge. Also Higgs fields may get a non zero charge. When the symmetry is spontaneously broken the entries of mass matrices are suppressed if there is a charge mismatch and more so if the corresponding mismatch is larger.
We assign FN-charges , and to , and
, respectively. There is some freedom in the choice of .
Here we take .
By assuming that a flavon , carrying
a negative unit of FN charge, acquires a VEV
, where , the Yukawa couplings
become field dependent quantities
and we have

(30) |

Had we chosen , we would have needed of order , to reproduce the above result. The superpotential term for lepton masses, is given by

(31) |

with dots denoting higher dimensional operators that lead to corrections to the LO approximation. In our notation, the product of 2 triplets transforms as , transforms as and transforms as . To keep our formulae compact, we omit to write the Higgs and flavon fields , and the cut-off scale . For instance stands for . The parameters of the superpotential are complex, in particular those responsible for the heavy neutrino Majorana masses, . Some terms allowed by the symmetry, such as the terms obtained by the exchange , (or the term ) are missing in . Their absence is crucial and, in each version of models, is motivated by additional symmetries.

The LO superpotential in eq. (31) leads to a diagonal mass matrix for the charged leptons^{7}^{7}7We absorbed in the appropriate factor of .

(32) |

and to a neutrino mass matrix of the same form as that of eq. (8). As for the neutrino spectrum both normal and inverted hierarchies can be realized. It is interesting that models with the see-saw mechanism typically lead to a light neutrino spectrum which satisfies the sum rule (among complex masses):

(33) |

A detailed discussion of a spectrum of this type can be found in Refs. [90, 101, 100, 102]. The above sum rule gives rise to bounds on the lightest neutrino mass. As a consequence, for example, the possible values of are restricted. For normal hierarchy we have

(34) |

while for inverted hierarchy

(35) |

In a completely general framework, without the restrictions imposed by the flavour symmetry, could vanish in the case of normal hierarchy. In this model is always different from zero, though its value for normal hierarchy is probably too small to be detected in the next generation of experiments.

In the leading approximation models lead to exact TB mixing. In these models TB mixing is implied by the symmetry at the leading order approximation which is corrected by non-leading effects. Given the set of flavour symmetries and having specified the field content, the non leading corrections to TB mixing, arising from higher dimensional effective operators, can be evaluated in a well defined expansion.

The departure from the LO approximation depends on the subleading contributions , , to the charged lepton and the neutrino mass matrices, respectively:

(36) |

which can vary according to the model considered. In all models considered here [103, 90, 59, 100] the NLO corrections to the charged lepton mass matrix are of the following type:

where is small adimensional parameter given by the ratio between the flavon VEVs and . The transformation needed to diagonalize is where

(37) |

To discuss the NLO contribution to we distinguish two cases.

### 4.1 Typical Models

In “typical” models [90, 103, 100], the NLO contribution in eq. (36) is a generic symmetric matrix with entries suppressed, compared to the corresponding entries in , by a relative factor , of the order of the ratio between a flavon VEV and . This occurs both with and without the see-saw mechanism. The generic transformation that diagonalizes is where

(38) |

where , and are complex parameters of order one in absolute value. Barring a fine-tuning of the Lagrangian parameters, in these models the suppression factors
and are expected to be of the same order of magnitude. For example, beyond the LO the equations satisfied by and are no longer decoupled and the corrections
to the LO flavon VEVs turn out to be of the same size, for both and . All the elements of the mixing matrix get corrections of the same size . We expect^{8}^{8}8Eq. (40) is a particular case of the general parametrization presented Ref. [104]:

(40) | ||||

According to these expressions, in order to reach the central value for the reactor angle in agreement with eq. (1), the parameter is expected to be . A precise value can be found by studying the success rate to reproduce all the three mixing angles inside the corresponding ranges, depending on the value of . As shown in ref. [58], in a scan with the parameters that multiply treated as random complex numbers with absolute values following a Gaussian distribution around 1 with variance 0.5, the value of that maximizes the success rate is found to be for the NH (IH). The corresponding success rate is , which is not large but not hopelessly small either. For this value of in Fig. 1 we quantitatively analyze the expressions in eq. (40) and their correlations: in the plots on the left (right), we show the correlation between and (