DFPD-05/TH/14

CERN-PH-TH/2005-067

Tri-Bimaximal Neutrino Mixing from Discrete Symmetry in Extra Dimensions

Guido Altarelli ^{1}^{1}1e-mail address:

CERN, Department of Physics, Theory Division

CH-1211 Geneva 23, Switzerland

and

Dipartimento di Fisica ‘E. Amaldi’, Università di Roma Tre

INFN, Sezione di Roma Tre, I-00146 Rome, Italy

Ferruccio Feruglio ^{2}^{2}2e-mail address:

Dipartimento di Fisica ‘G. Galilei’, Università di Padova

INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padua, Italy

We discuss a particularly symmetric model of neutrino mixings where, with good accuracy, the atmospheric mixing angle is maximal, and the solar angle satisfies (Harrison-Perkins-Scott (HPS) matrix). The discrete symmetry is a suitable symmetry group for the realization of this type of model. We construct a model where the HPS matrix is exactly obtained in a first approximation without imposing ad hoc relations among parameters. The crucial issue of the required VEV alignment in the scalar sector is discussed and we present a natural solution of this problem based on a formulation with extra dimensions. We study the corrections from higher dimensionality operators allowed by the symmetries of the model and discuss the conditions on the cut-off scales and the VEVs in order for these corrections to be completely under control. Finally, the observed hierarchy of charged lepton masses is obtained by assuming a larger flavour symmetry. We also show that, under general conditions, a maximal can never arise from an exact flavour symmetry.

## 1 Introduction

By now there is convincing evidence for solar and atmospheric neutrino oscillations. The values and mixing angles are known with fair accuracy [1]. For 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 : with central value around . The solar angle is large, , but certainly not maximal (by about 5-6 now [2]). The third angle is strongly limited, mainly by the CHOOZ experiment, and has at present a upper limit given by about .

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, it is essential to know whether the LSND signal [3], which has not been confirmed by KARMEN [4] and is currently being double-checked by MiniBoone [5], will be confirmed or will be excluded. If LSND is right we probably need at least four light neutrinos; if not we can do with only the three known ones, as we assume here in the following. As neutrino oscillations only determine mass squared differences a crucial missing input is the absolute scale of neutrino masses (within the existing limits from terrestrial experiments and cosmology [6],[7]). 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 for granted that neutrinos are Majorana particles, their masses can still arise either from the see-saw mechanism or from generic dimension-five non renormalizable operators.

At a more direct level, we do not know how small the mixing angle is and how close to maximal is . One can make a distinction between ”normal” and ”special” models. For normal models is not too close to maximal and is not too small, typically a small power of the self-suggesting order parameter , with . Special 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. We expect that experiment will eventually find that is not too small and that is sizably not maximal. But if, on the contrary, either very small or very close to maximal will emerge from experiment, then theory will need to cope with this fact. Thus it is interesting to conceive and explore dynamical structures that could lead to special models in a natural way.

We want to discuss here some particularly special models where both and exactly vanish
^{1}^{1}1More precisely,
they vanish in a suitable limit, with correction terms that can be made negligibly small.. Then the neutrino mixing matrix (,,, ), in the basis of diagonal charged leptons, is given by, apart from sign convention redefinitions:

(1) |

where and stand for and , respectively. It is much simpler to write natural models of this type with small and thus many such attempts are present in the early literature. More recently, given the experimental value of , the more complicated case of large was also attacked, using non abelian symmetries, either continuous or discrete [8, 9, 10, 12, 13, 14, 11]. In many examples the invoked symmetries are particularly ad hoc and/or no sufficient attention is devoted to corrections from higher dimensional operators that can spoil the pattern arranged at tree level and to the highly non trivial vacuum alignment problems that arise if naturalness is required also at the level of vacuum expectation values (VEVs).

An interesting special case of eq. (1) is obtained for , i.e. the so-called tri-bimaximal or Harrison-Perkins-Scott mixing pattern (HPS) [13], with the entries in the second column all equal to in absolute value:

(2) |

This matrix is a good approximation to present data
^{2}^{2}2In the HPS scheme , to be compared with the latest experimental
determination [2]: .. It would be interesting to find a natural and appealing scheme that leads to this matrix with good accuracy. In fact this is a most special model where not only and vanish but also assumes a particular value. Clearly, in a natural realization of this model, a very constraining and predictive dynamics must be underlying. We think 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: at the other extreme, anarchical models have been proposed [15], where no structure at all is assumed in the lepton sector, so that, for example, and are predicted to be in no way special, except that there must be a smallest angle (probably near to the present bound) and a largest angle (expected sizably different from maximal).

Interesting ideas on how to obtain the HPS mixing matrix have been discussed in refs [13]. The most attractive models are based on the discrete symmetry , which appears as particularly suitable for the purpose, and were presented in ref. [10, 11]. In the present paper we start by discussing some general features of HPS models. We then present a new version of an model, with (moderate) normal hierarchy, and discuss in detail all aspects of naturalness in this model, also considering effects beyond tree level and the problem of vacuum alignment. There are a number of substantial improvements in our version with respect to Ma in ref. [11]. First, the HPS matrix is exactly obtained in a first approximation when higher dimensional operators are neglected, without imposing ad hoc relations among parameters (in ref. [11]. the equality of and is not guaranteed by the symmetry). The observed hierarchy of charged lepton masses is obtained by assuming a larger flavour symmetry. The crucial issue of the required VEV alignment in the scalar sector is considered with special attention and we present a natural solution of this problem. We also keep the flavour scalar fields distinct from the normal Higgs bosons (a proliferation of Higgs doublets is disfavoured by coupling unification) and singlets under the Standard Model gauge group. Last not least, we study the corrections from higher dimensionality operators allowed by the symmetries of the model and discuss the conditions on the cut-off scales and the VEVs in order for these corrections to be completely under control.

## 2 General Considerations

The HPS mixing matrix implies that in a basis where charged lepton masses are diagonal the effective neutrino mass matrix is given by :

(3) |

The eigenvalues of are , , with eigenvectors , and , respectively. In general, apart from 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 (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). From the general expression of the eigenvectors one immediately sees that this mass matrix, independent of the values of , leads to the HPS mixing matrix. It is a curiosity that the eigenvectors are the same as in the case of the Fritzsch-Xing (FX) matrix [16] but with the roles of the first and the third ones interchanged (so that for HPS is maximal while , while for FX the two mixing angles keep the same values but are interchanged).

If the atmospheric mixing angle is really maximal as in the HPS ansatz
or close to maximal,
it seems quite natural to interpret this as the effect of a
flavour symmetry. It would be tempting to think of an approximate
flavour symmetry such that arises in the
limit of exact symmetry, that is by neglecting all symmetry breaking
effects.
Here we will show that this is not the case and that, under quite general
conditions, we can never obtain as a result of an
exact flavour symmetry ^{3}^{3}3For related observations see ref.
[17].. We assume that this symmetry is a
meaningful symmetry, that is it is only broken by
small effects, in the real world.
In other words here we exclude symmetries that need
breaking terms of order one to describe the observed fermion masses
and mixing angles. Apart from that the symmetry can be of whatever
type, global or local, continuous or discrete.
Being interested in the limit of exact symmetry, we can
neglect the sector giving rise to flavour symmetry breaking.
We assume that the fields on which such symmetry acts
are the fields of the standard model, plus possibly the right-handed
neutrinos, so that our results will also cover the
see-saw case.
Last, we assume canonical kinetic terms,
so that the symmetry acts on the fields of the standard model
through unitary transformations.

Since the flavour symmetry is broken only by small effects, the mass matrices for charged leptons and neutrinos can be written as:

(4) |

where dots denote symmetry breaking effects and has rank less or equal than one. Rank greater than one, as for instance when both the tau and the muon have non-vanishing masses in the symmetry limit, is clearly an unacceptable starting point, since the difference between the two non-vanishing masses can only be explained by large breaking effects, which we have excluded, or by a fine-tuning, which we wish to avoid. If the rank of vanishes, than all mixing angles in the charged lepton sector are undetermined in the symmetry limit and is also completely undetermined. Therefore we can focus on the case when has rank one. If has rank one, then by a unitary transformations we can always go to a field basis where

(5) |

As in the original basis, the action of the flavour symmetry on the new field basis is perfectly defined. If and are the unitary matrices that diagonalize and , it will be possible to adopt the parametrization [18]

(6) |

where is the orthogonal matrix representing a rotation in the sector, and . Moreover:

(7) |

where the angle is completely undetermined. The physical mixing matrix is and we find:

(8) |

Therefore, in general, the atmospheric mixing angle is always undetermined at the leading order. When small symmetry breaking terms are added to and , it is possible to obtain , provided these breaking terms have suitable orientations in the flavour space. If the breaking terms are produced by a spontaneous symmetry breaking through the minimization of the potential energy of the theory, in general two independent scalar sectors are needed. One of them communicates the breaking to charged fermions and the other one feeds the breaking to neutrinos. In such a framework a maximal atmospheric mixing angle is always the result of a special vacuum alignment.

In the literature there are symmetries predicting large, not necessarily maximal, in the limit of exact symmetry [19]. For instance, this is produced by U(1) flavour symmetries, when the U(1) charges of left-handed leptons and right-handed charged leptons are and , respectively, with and all non-vanishing and different. In the symmetry limit, such an assignment implies (, ):

(9) |

and:

(10) |

with , , , and independent parameters of the same order of magnitude. If there is no conspiracy among these parameters, the resulting mixing is generically large.

In conclusion, a large lepton mixing in the 23 sector is possible as the result of an exact flavour symmetry. But if we want to reproduce in some limit of our theory, necessarily this limit cannot correspond to an exact symmetry in flavour space. A maximal atmospheric mixing angle can only originate from breaking effects as a solution of a vacuum alignment problem.

## 3 Basic Structure of the Model

Our model is based on the discrete group following refs [10, 11], where its structure and representations are described in detail. Here we simply recall that is the discrete symmetry group of the rotations that leave a tethraedron invariant, or the group of the even permutations of 4 objects. It has 12 elements and 4 inequivalent irreducible representations denoted 1, , and 3 in terms of their respective dimensions. Introducing , the cubic root of unity, , so that , the three one-dimensional representations are obtained by dividing the 12 elements of in three classes, which are determined by the multiplication rule, and assigning to (class 1, class 2, class 3) a factor for 1, or for or for . The product of two 3 gives . Also , , etc. For , the irreducible representations obtained from their product are:

(11) |

(12) |

(13) |

(14) |

(15) |

Following ref. [11] we assigns leptons to the four inequivalent
representations of : left-handed lepton doublets transform
as a triplet , while the right-handed charged leptons ,
and transform as , and , respectively.
The flavour symmetry is broken by two real triplets
and and by a real singlet .
At variance with the choice made by [11], these fields
are gauge singlets.
Hence we only need two Higgs doublets (not three generations of
them as in ref. [11]), which we take invariant under .
We assume that some mechanism produces and maintains the hierarchy
where is the
cut-off scale of the theory
^{4}^{4}4This is the well known hierarchy
problem that can be solved, for instance, by realizing a supersymmetric
version of this model..
The Yukawa interactions in the lepton sector read:

(16) |

In our notation, transforms as , transforms as and transforms as . Also, to keep our notation compact, we use a two-component notation for the fermion fields and we set to 1 the Higgs fields and the cut-off 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 in section 6. 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 will be motivated later on.

As we will demonstrate in section 5, the fields , and develop a VEV along the directions:

(17) |

Therefore, at the leading order of the expansion, the mass matrices and for charged leptons and neutrinos are given by:

(18) |

(19) |

where

(20) |

Charged leptons are diagonalized by

(21) |

and charged fermion masses are given by:

(22) |

We can easily obtain a natural hierarchy among , and by introducing an additional U(1) flavour symmetry under which only the right-handed lepton sector is charged. We assign F-charges , 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

(23) |

In the flavour basis the neutrino mass matrix reads
^{5}^{5}5Notice that a unitary change of basis like the one in eq. (21)
will in general change the relative phases of the eigenvalues of .:

(24) |

and is diagonalized by the transformation:

(25) |

with

(26) |

The leading order predictions are , and . The neutrino masses are , and , in units of . We can express , in terms of , and , being the phase difference between the complex numbers and :

(27) |

To satisfy these relations a moderate tuning is needed in our model. Due to the absence of in eq. (16) which we will motivate in the next section, and are of the same order in , see eq. (20). Therefore we expect that and are close to each other and, to satisfy eqs. (27), should be negative and of order one. We obtain:

(28) |

If , we have a neutrino spectrum close to hierarchical:

(29) |

In this case the sum of neutrino masses is about eV. If is accidentally small, the neutrino spectrum becomes degenerate. The value of , the parameter characterizing the violation of total lepton number in neutrinoless double beta decay, is given by:

(30) |

For we get eV, at the upper edge of the range allowed for normal hierarchy, but unfortunately too small to be detected in a near future. Independently from the value of the unknown phase we get the relation:

(31) |

which is a prediction of our model.

It is also important to get some constraint on the mass scales involved in our construction. From eqs. (27) and (20), by assuming GeV, we have

(32) |

Since, to have a meaningful expansion, we expect , we have the upper bound

(33) |

Beyond this energy scale, new physics should come into play. The smaller the ratio , the smaller becomes the cut-off scale. For instance, when , should be close to GeV. A complementary information comes from the charged lepton sector, eq. (22). A lower bound on can be derived from the requirement that the Yukawa coupling remains in a perturbative regime. By asking GeV, we get

(34) |

Finally, by assuming that all the VEVs fall in approximately the same range, which will be shown in section 5, we obtain the range

(35) |

that will be useful to estimate the effects of higher-dimensional operators in section 6. Correspondingly the cut-off scale will range between about and GeV.

## 4 Vacuum alignment

In this section we investigate the problem of achieving the vacuum alignment of eq. (17). At the same time we should prevent, at least at some level, the interchange between the fields and to produce the desired mass matrices in the neutrino and charged lepton sectors. As we will see, there are several difficulties to naturally accomplish these requirements. By minimizing the scalar potential of the theory with respect to and we get six equations that we would like to satisfy in terms of the two unknown and . Even though we expect that, due to the symmetry , the six minimum conditions are not necessarily independent, such an expectation turns out to be wrong in the specific case, unless some additional relation is enforced on the parameters of the scalar potential. These additional relations are in general not natural. For instance, even by imposing them at the tree level, they are expected to be violated at the one-loop order. Therefore, as we will now illustrate, the minimum conditions cannot be all satisfied by our vacuum configuration.

As an example here we analyze the most general renormalizable scalar potential invariant under and depending upon the triplets and of the Lagrangian in eq. (16). The term in can be forbidden by an additional symmetry, commuting with . One possibility is just the total lepton number or a discrete subgroup of it. Here we consider a symmetry under which transform into , into , is invariant and changes sign. This symmetry also explains why and cannot be interchanged. The scalar potential contains bilinears , trilinears and quartic terms , invariant under the group . A choice of independent invariants is:

(36) |

The scalar potential reads:

(37) | |||||

We start by analyzing the field configuration:

(38) |

The minimum conditions are:

(39) |

The equations are clearly incompatible unless . Even by forcing to vanish, we are left with three independent equations for the two unknown and , which, for generic values of the coefficients, admit only the trivial solution . This negative results cannot be modified by adding to the terms depending on the singlet . Also by investigating the problem in a slightly more general framework, with real and complex, we reach the same conclusion. Although we have not a no-go theorem, these examples show the difficulty to obtain the desired alignment.

The difficulty illustrated above is not common to all vacua. For instance the other possible alignment:

(40) |

leads to the minimum conditions:

(41) |

In a non-vanishing portion of the parameter space, these equations have non-trivial solution with non-vanishing and .

It is possible to show that, by sufficiently restricting the form of the most general scalar potential invariant under , the desired alignment can be obtained. Restrictions that are unnatural in a generic model becomes technically natural in a supersymmetric (SUSY) model. The well-known non-renormalization properties of the superpotential allow to accept, at least from a technical viewpoint, a restricted number of terms, compared to what the symmetry would permit. Undesired terms of the superpotential that are set to zero at the tree level are not generated at any order in perturbation theory. Indeed we have produced a SUSY example of this type, where the alignment problem is solved and this example is discussed in detail in the Appendix. However our real aim is to build a fully natural model, where all the terms allowed by the symmetries are present and where the only deviations from the symmetry limit are provided by higher-dimensional operators, rather than by small violations of ad-hoc imposed relations. As we will now see, there exist a simple and economic solution in the context of theories with one extra spatial dimension.

## 5 model in an extra dimension

One of the problems we should overcome in the search for the correct alignment is that of keeping neutrino and charged lepton sectors separate, including the respective symmetry breaking sectors. Here we show that such a separation can be achieved by means of an extra spatial dimension. The space-time is assumed to be five-dimensional, the product of the four-dimensional Minkowski space-time times an interval going from to . At and the space-time has two four-dimensional boundaries, which we will call branes. Our idea is that matter SU(2) singlets such as are localized at , while SU(2) doublets, such as are localized at (see Fig.1). Neutrino masses arise from local operators at . Charged lepton masses are produced by non-local effects involving both branes. Later on we will see how such non-local effects can arise in this theory. The simplest possibility is to introduce a bulk fermion, depending on all space-time coordinates, that interacts with at and with at . The exchange of such a fermion can provide the desired non-local coupling between right-handed and left-handed ordinary fermions. Finally, assuming that and are localized respectively at and , we obtain a natural separation between the two sectors.

### 5.1 Alignment in an extra dimension

Such a separation also greatly simplify the vacuum alignment problem. We can determine the minima of two scalar potentials and , depending only, respectively, on and . Indeed, as we shall see, there are whole regions of the parameter space where and have the minima given in eq. (17). Notice that in the present setup dealing with a discrete symmetry such as provides a great advantage as far as the alignment problem is concerned. A continuous flavour symmetry such as, for instance, SO(3) would need some extra structure to achieve the desired alignment. Indeed the potential energy would be invariant under a much bigger symmetry, SO(3) SO(3), with the SO(3) acting on and leaving invariant and vice-versa for SO(3). This symmetry would remove any alignment between the VEVs of and those of . If, for instance, (17) is minimum of the potential energy, then any other configuration obtained by acting on (17) with SO(3) SO(3) would also be a minimum and the relative orientation between the two sets of VEVs would be completely undetermined. A discrete symmetry such as has not this problem, as we will show now.

Consider first the scalar potential :

(42) |

where , , are defined in eq. (36). The minimum conditions at are:

(43) |

while the minimum condition at is:

(44) |

since in this case are automatically satisfied. Both and can be local minima of , depending on the parameters. The constants should be positive, to have bounded from below. We can look at the region where . When and , the minimum at is the absolute one, while for and is minimized by . Therefore we have a large portion of the parameter space where the minimum is of the desired form: . To be precise, in this region, there are four degenerate minima: , , related by transformations.

Now we turn to . As we did in section 4.1, we assume both and real and odd under the action of a discrete symmetry. The most general renormalizable invariant potential is a combination of , in eq. (36) and the following invariants:

(45) |

We have:

(46) |

We search for a minimum at and :