1 Introduction

MPP-2011-38

SHEP-11-04

SISSA 15/2011/EP

Right unitarity triangles and tri-bimaximal mixing

[4mm] from discrete symmetries and unification

[9mm] S. Antusch 111E-mail: antusch@mppmu.mpg.de, Stephen F. King 222E-mail: king@soton.ac.uk, Christoph Luhn 333E-mail: christoph.luhn@soton.ac.uk, M. Spinrath 444E-mail: spinrath@sissa.it,

[-2mm]

Max-Planck-Institut für Physik (Werner-Heisenberg-Institut),

Föhringer Ring 6, D-80805 München, Germany

School of Physics and Astronomy, University of Southampton,

SO17 1BJ Southampton, United Kingdom

SISSA/ISAS and INFN,

Via Bonomea 265, I-34136 Trieste, Italy

We propose new classes of models which predict both tri-bimaximal lepton mixing and a right-angled Cabibbo-Kobayashi-Maskawa (CKM) unitarity triangle, . The ingredients of the models include a supersymmetric (SUSY) unified gauge group such as , a discrete family symmetry such as or , a shaping symmetry including products of and groups as well as spontaneous CP violation. We show how the vacuum alignment in such models allows a simple explanation of by a combination of purely real or purely imaginary vacuum expectation values (vevs) of the flavons responsible for family symmetry breaking. This leads to quark mass matrices with 1-3 texture zeros that satisfy the “phase sum rule” and lepton mass matrices that satisfy the “lepton mixing sum rule” together with a new prediction that the leptonic CP violating oscillation phase is close to either , , , or depending on the model, with neutrino masses being purely real (no complex Majorana phases). This leads to the possibility of having right-angled unitarity triangles in both the quark and lepton sectors.

1 Introduction

The flavour puzzle, i.e. the origin of the observed pattern of fermion masses, mixing angles and CP violating phases is one of the most challenging puzzles in particle physics. There are various aspects of the flavour puzzle, such as the hierarchy among the quark masses, the origin of CP violation, and the largeness of the leptonic mixing angles, which have turned out to be close to “tri-bimaximal” [1]. In particular this latter observation has led to increasing interest in non-Abelian discrete family symmetries for flavour model building [2].

Recently it has become increasingly clear that current data is indeed consistent with the hypothesis of a right-angled CKM unitarity triangle, with the best fits giving  [3]. Such a right unitarity triangle was suggested long ago, when the error bar on was much larger, by Fritzsch and collaborators as a natural consequence of having quark mass matrices with zeros in the 1-3 element [4]. In the light of recent data, this observation has gained increased momentum, and there have been several papers that attempt to predict by postulating up-type and down-type quark mass matrices with the elements being either purely real or purely imaginary, with texture zeros in the 1-3 elements, [5, 6, 7] (see also [8]). Under these assumptions, it has been shown that the prediction can be understood from a simple analytic “phase sum rule” relation [7] which relates the angle  to phases arising from the quark mass matrices. To be precise, the “phase sum rule” can be expressed as  [7], where the phases and are the arguments of the complex 1-2 rotation angles in the up-type and down-type quark mass matrices, as defined in [7]. To explain one might therefore simply try to realise , or alternatively , in a model of flavour. For hierarchical mass matrices, this corresponds to the 1-2 and the 2-2 elements of the mass matrix being either purely real or imaginary. Such textures have also been considered previously in [5, 6, 7], and in [5] a SUSY Grand Unified Theory (GUT) with a continuous family symmetry responsible for such textures has been proposed, however the vacuum alignment responsible for the spontaneous breaking of the family symmetry was not studied.

In this paper we show that large classes of models involving discrete family symmetry and supersymmetric unification (so called SUSY GUTs of Flavour) that have previously been proposed to account for tri-bimaximal lepton mixing are quite capable of providing an explanation of the right-angled unitarity triangle, subject to a constraint on the “shaping symmetry” that helps to shape the vacuum alignment superpotential. Such a “shaping symmetry” is always necessary in realistic models, but here we constrain the nature of the symmetry to be a discrete symmetry of a particular type. The main technical accomplishment in this paper is to propose a mechanism for vacuum alignment based on discrete symmetries, which can give rise to purely real or purely imaginary vacuum alignments for the flavon fields responsible for spontaneously breaking the discrete family symmetry.111In the context of continuous family symmetries, flavon alignments with real or imaginary values in the multiplet components have been discussed in [9]. There are four different aspects to these models which are important for our approach, as follows.

(i) Supersymmetric Unification: We impose gauge coupling unification, which severely restricts the choice of available models in the literature, since many of the existing models are not unified. The role of unification is to relate the lepton sector to the quark sector, since we want to make the connection between tri-bimaximal mixing and quark mixing and CP violation. Here we shall consider the minimal SUSY gauge group. It is, however, worth emphasising that our method of obtaining a right-angled CKM unitarity triangle can also be applied to models without grand unification.

(ii) Discrete Family Symmetry: As already stated we are also concerned with discrete family symmetries that have been proposed to account for tri-bimaximal lepton mixing. The approach is applicable for all types of discrete family symmetries and does not depend on whether the neutrino flavour (Klein) symmetry associated with tri-bimaximal mixing is identified as a subgroup of the family symmetry (as in the so called direct models) or as an accidental symmetry which results from having flavons aligned along the columns of the tri-bimaximal mixing matrix (as in the so called indirect models). Recall that in the latter case, the flavons break the Klein symmetry only due to an overall minus sign, and bilinears of flavons appearing in the neutrino sector respect the Klein symmetry (for a full discussion of this see [10]). We shall consider an example model of both the direct and the indirect kind.

(iii) Discrete Shaping Symmetry: We assume an extra shaping symmetry based on products of symmetries, where is an even number. All realistic models involve some extra shaping symmetry that can help to control the presence of operators in the sectors responsible for Yukawa couplings and vacuum alignment, so the idea of an extra shaping symmetry is not new. What is new is that our mechanism restricts the shaping symmetries to be strings of discrete symmetries such as and symmetries in order that the vevs of the flavon fields be forced to be purely real or purely imaginary. In particular, this prohibits the use of continuous shaping symmetries such as for example a symmetry; discrete symmetries like, e.g.  or symmetries are also not suitable as they would not lead to purely real or purely imaginary vevs. This means that many of the existing models which have been proposed to describe tri-bimaximal lepton mixing are not viable for explaining the right-angled unitarity triangle, and we are forced to invent new models.

(iv) Spontaneous CP violation: Another requirement of our mechanism is that CP is conserved in the theory at the high energy scale, and is only broken spontaneously by the (complex) vevs of flavons. Such a scenario has been proposed previously in order to account for the smallness of CP violation in the soft SUSY sector [11]. Here it will be an essential ingredient in obtaining the prediction of . Thus we envisage models with family symmetries and spontaneous CP violation, in which the flavour structure as well as CP violation are generated from family symmetry breaking.

In section 2 we explain our mechanism in general terms. In sections 3 and 4 we then turn to two realistic examples of GUT models with and family symmetries, respectively, plus extra shaping symmetries. The model in section 3 is an example of an indirect model, similar in nature to the model proposed in [12], while the model in section 4 is an example of a direct model, similar in nature to the model proposed in [13]. The models are modified here to take account of the restriction on the shaping symmetry in (iii). Both models illustrate that the right-angled CKM unitarity triangle can indeed be understood from an underlying flavour model with discrete symmetries which were introduced previously for the purpose of providing as an explanation of tri-bimaximal lepton mixing. In both models the quark mass matrices have 1-3 texture zeros and satisfy the “phase sum rule” derived in [7], while the lepton mass matrices lead to the “lepton mixing sum rule” [14, 15] together with a new prediction that the leptonic CP violating oscillation phase is close to either , , , or depending on the model, with neutrino masses being purely real (no complex Majorana phases). We summarise and conclude the paper in section 5, providing appendices on the more technical aspects of the models including their ultraviolet completion.

2 The real/imaginary vacuum alignment mechanism

The goal of this paper is to show how the nearly right-angled CKM unitarity triangle (i.e. ) can be explained in flavour models. In this section we describe in general terms how this can be achieved in theories with spontaneous CP violation via purely real or imaginary alignments for the flavon fields. The flavon fields break the family symmetry, give rise to the flavour structure, and have to induce the observed CP violation via their vevs.

2.1 Motivation: The phase sum rule

The motivation for this approach is provided by the phase sum rule of [7], which states that if the 1-3 mixing in both, the up-type quark mass matrix as well as the down-type mass matrix vanish (approximately), then there holds the following relation for the angle of the CKM unitarity triangle:

(2.1)

The phases and are the arguments of the complex 1-2 rotation angles in the up-type and down-type quark mass matrices, as defined in [7].

To explain one might therefore simply try to realise , or alternatively , in a model of flavour. For hierarchical mass matrices, this corresponds to the 1-2 and the 2-2 elements of the mass matrix being either purely real or imaginary. When the Yukawa matrices are generated after the breaking of some (non-Abelian) family symmetry, we thus need vevs of the flavons which have either purely real or purely imaginary components.

We now discuss in general terms how this might be achieved in scenarios with discrete symmetries in addition to non-Abelian family symmetries and later on we will also give two concrete examples.

2.2 Method: Discrete vacuum alignment

As mentioned above, we assume spontaneous CP violation, i.e. that CP violation is induced via the vevs of the flavons only, whereas the fundamental theory conserves CP. More specifically, we will assume that in the phase of unbroken family symmetry, there exists a basis where all parameters are real.

Furthermore, we will consider the case that in addition to a non-Abelian discrete family symmetry, the flavour model features extra shaping symmetries. When a flavon carries single charge under the symmetry (), then typical terms in the flavon superpotential, which “drive” the flavon vev non-zero, have the form

(2.2)

The field is the so-called “driving superfield”, meaning that the -term generates the potential for the scalar component of which enforces a non-zero vev. Here and in the following we will use the same letters for the superfields and its component fields. is the (real and positive) suppression scale of the effective operator, typically associated with the mass(es) of the messenger field(s) involved in its generation, and here is simply a (real) mass scale. From the potential for ,

(2.3)

we see that the vev of has to satisfy

(2.4)

The final step to explain our method is to argue that, whenever the flavon vev depends on just one single parameter, Eq. (2.2) forces the phase of the flavon vev to take only certain discrete values. For instance, in the simplest case where is a singlet under the non-Abelian family symmetry, it is clear that the phase is determined to be:

(2.5)

For example with a symmetry and a “”-sign in Eq. (2.2) the phase satisfies and thus the vev is real. For the “”-sign we have and the vev is purely imaginary. Similarly, with a symmetry, we see that for the “”-sign the phase can take the possible values and for the “”-sign it can take the values . So only for the “”-sign the flavon vev is either purely real or purely imaginary.222We note that with symmetries other than or one obtains different discrete possibilities for the phases which may also be interesting for model building. In this paper, however, we will focus on and symmetries since we are interested in purely real or purely imaginary flavon vevs. As we argued in the previous subsection, such either purely real or purely imaginary aligned flavons will be the building blocks for the proposed explanation of the right-angled CKM unitarity triangle.

The above arguments continue to hold true if the flavons are, for example, triplets of the non-Abelian family symmetry. In fact, typically, in explicit models we will deal with flavons which (by means of other terms in the flavon potential) are forced to point in specific directions in flavour space and thus depend only on one continuous parameter, say . Examples for such flavons may be

(2.6)

When the vevs of such flavons are driven by terms as in Eq. (2.2), the phases are again forced to take only values as specified in Eq. (2.5).

We have argued in this section that flavons which have either purely real or purely imaginary vevs could be the building blocks for flavour models capable of explaining the nearly right-angled CKM unitarity triangle (i.e. ). A model-independent discussion and a derivation of the phase sum rule can be found in [7]. We have outlined a possible method of how such purely real or purely imaginary flavons can be realised in models. The next step will be to apply the method to construct two example models featuring .

3

As a first example we will now discuss a model based on an GUT with family symmetry (plus extra discrete symmetries and an -symmetry), broken by the vevs of five flavon fields and . This may be regarded as a variation of the indirect model in [12]. We start with the discussion of the flavon potential. Following the method described in the previous section, we use as discrete symmetries only and symmetries, such that the vevs will be either purely real or imaginary. The field content and the symmetries are listed in Tabs. 1 and 2 for the flavon sector and the matter sector, respectively. The complete messenger sector of the model will be presented in App. B.

3.1 Flavon sector

Flavons
3 0 0 0 0 0 0
3 3 0 0 0 0 0
0 0 0 0 1 0 0
0 0 3 0 0 0 0
0 0 3 3 0 0 0
0 0 1 0 0 0 0
Driving Fields
0 0 0 0 0 0 2
2 0 0 0 0 0 2
2 2 0 0 0 0 2
0 0 0 0 0 0 2
0 0 2 0 0 0 2
2 1 0 0 0 0 2
1 0 0 0 1 0 2
1 1 0 0 1 0 2
1 0 1 1 0 0 2
0 0 2 1 0 0 2
Table 1: Flavon and driving fields of the model. Note that depending on which option for the alignment of is chosen, the fields and are present or not, as described in the main text. The symmetry is not necessary for the alignment itself, but it will be required for the matter (cf. Tab. 2) and the messenger sectors (cf. App. B).

In Sec. 2 we have discussed in general terms how the phases of flavon vevs may be predicted from the flavon potential. In this section we will apply this method to construct a simple model capable of predicting a CKM unitarity triangle with . As we will see, the alignment we produce here enables us to predict the CKM phase correctly and also give predictions for all the phases in the lepton sector which are not yet experimentally determined. The flavons and driving fields and their charges under the imposed symmetries are given in Tab. 1.

In addition to the flavons and responsible for the flavour structure, the table also contains the “auxiliary flavon” , which will be used to align the vev of . The “driving fields” will be called , and , and we use a notation that via their -term contributions to the flavon potential, the singlet fields fix the phase as discussed in Sec. 2, the triplets force flavons to point into certain specific directions in flavour space, and the singlet fields align the vev of the flavon to be orthogonal to the one of . Notice that the driving fields and are all distinguished from one another by their charge assignments, whereas the driving fields are completely neutral under all shaping symmetries. Hence, in a generic basis, each of these fields couples to the same set of terms with different coupling constants. In order to apply the method outlined in Sec. 2 it is necessary to disentangle the equations by a suitable basis transformation, the details of which are presented in App. A. For the sake of simplicity we will sometimes suppress (real and positive) order one coupling constants where they are not relevant for the model predictions. In dealing with we will use the standard “ basis” for which the singlet of is given by the -type inner product. The two triplets of are constructed from the usual (antisymmetric) cross product ’’ and the symmetric star product ’’ (see, for example, [16]). The symmetric product is defined analogous to the cross product but with a relative plus sign instead of a minus sign.

Let us now start our discussion of the alignment by specifying the required form of the vevs of , , , , for the construction of the model with :

(3.1)

For predicting , one possibility will be to have an imaginary and real , , , . This is explicitly incorporated in Eq. (3.1) by assuming real proportionality constants here and in the following. The vevs of are familiar from various flavour models and lead to tri-bimaximal neutrino mixing.

For realising the required vacuum alignment of , we discuss two options:

  • Option A: The (super-)potential for the first option is

    (3.2)

    where displays a SUSY-breaking soft mass term for with positive . The brackets and mean that the fields are contracted to and representations of . stands for . The (real) constants , , and denote messenger masses, see App. B.

    It is not obvious, that this potential gives the desired alignment of or . Let us assume that , then the invariant gives a contribution to the -term conditions of the form

    This combination obviously vanishes for , which is already the desired alignment. Nevertheless, having only this invariant coupling to a driving field is not sufficient since the scale of is completely arbitrary up to this stage. This is fixed by the term and the soft mass. Indeed, we have checked numerically that for we end up in a vacuum, where or , if we assume to be real.

  • Option B: Alternatively one can achieve the alignment of the vev of by:

    (3.3)

    The -term equations for and have three distinct solutions. With the potential reduces to the case in Eq. (3.4) giving, e.g. . For there exist two possibilities for : the vev of could vanish or, alternatively, point into the directions . The latter can be understood in the following way. The -term conditions of the first term in Eq. (3.3) give three relations between the vev and the components of which enter the equations cyclically preferring already the solution . We only need a term, which drives the vevs to non-zero values which is done by the second term in Eq. (3.3).

    In the following we assume, that the latter option is realised. Compared to option A, the “auxiliary flavon” and the additional driving field are introduced, however no soft terms are involved in the alignment.

Now that we have this alignment at hand the alignment of the other flavons is comparatively straightforward. The vevs of the other flavons are determined by the following additional superpotential terms:

(3.4)
(3.5)

Note that in Eq. (3.4) only the symmetric coupling appears, since the cross product vanishes for two identical fields (which is of course also allowed by the symmetries). As discussed in Sec. 2, and assuming spontaneous CP violation, we obtain that the phases of the vevs of , , , , can only take the values , as desired. Among these possible vacua, we will concentrate in the following on the solution where is purely imaginary and the other flavon vevs are real.333We note that there are other combinations of the flavon vevs’ phases leading to the same results, whereas others are phenomenologically invalid. In principle, higher-dimensional (Planck scale suppressed) operators may violate the discrete symmetries and favour one vacuum over the others. This preferred vacuum may then expand and finally be the only one in our observable universe. In a more fundamental theory one may even attempt to calculate which vacua are preferred, but for the present work this is clearly beyond the scope.

Together, the flavon superpotential in Eqs. (3.2)-(3.5) can result in the desired flavon alignment with imaginary and real as specified in Eq. (3.1).

3.2 Matter sector and predictions

Matter Fields
0 0 0 0 0 1 1
1 0 0 0 0 0 1
0 0 1 0 0 0 1
0 0 0 0 1 0 1
0 0 3 3 0 0 1
0 0 3 0 0 0 1
Higgs Fields
0 1 0 0 0 0 0
0 0 0 1 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 1 0
Table 2: The matter and Higgs fields of the model.

With the breaking flavon sector and the alignment of the flavon vevs at hand we will now turn to the fermion masses and mixings within the GUT model. The matter content of the Standard Model fits into the five-dimensional representation of

(3.6)

which we assume to be a triplet under , and the ten-dimensional representations of

(3.7)

which we assume to be singlets under . We also add two right-handed neutrinos and , being singlets under and , to generate masses for two of the light neutrinos via the seesaw mechanism [17]. The Higgs sector consists of , and , . is the GUT symmetry breaking Higgs field while contains the MSSM up-type Higgs doublet, and the down-type Higgs doublet is a linear combination of the doublet components of the fields.

The model will predict the GUT scale ratios and . Instead of the commonly encountered - Yukawa unification and the Georgi–Jarlskog relation [18] for , which are phenomenologically somewhat challenged in CMSSM scenarios [19, 20, 21], our model predicts the GUT scale relations [21]

(3.8)

which differ from the Georgi-Jarlskog predictions by an overall factor of 3/2 giving better agreement with phenomenology. The operators yielding these predictions contain with its vev given by the diagonal matrix

(3.9)

The non-renormalisable superpotential which is generated after integrating out the messenger fields, cf. App. B, is given by the following terms:

(3.10)
(3.11)
(3.12)
(3.13)

As before, order one coefficients are dropped where they have no influence on the model predictions. The new masses are the masses of the messenger fields, as will be discussed in App. B.

For the low energy charged lepton and down-type quark Yukawa couplings we define

(3.14)

where we have dropped order one coefficients and Higgs mixing angles. Similarly for the up-type quark sector we define

(3.15)

The top Yukawa coupling is generated at tree-level and the would-be vanishes due to the orthogonality of and . For the neutrino Yukawa couplings we define

(3.16)

and for the right-handed neutrino masses

(3.17)

With these definitions at hand we can express the Yukawa couplings in a simple form using the PDG convention [3], namely

(3.18)

Regarding the quark Yukawa matrices, from Eqs. (3.10) and (3.11) and using the above definitions as well as the alignments of Eq. (3.1) we obtain:

(3.19)
(3.20)

Note that due to the complex conjugation in the definition of the Yukawa couplings a factor of appears now in the 1-2 element of (instead of ). We see that the “phase sum rule” of Ref. [7] applies since in both the up and the down quark sector we have zero 1-3 mixing. As discussed in [7], the structures of and give the correct quark masses and mixings including a CKM matrix that features a right-angled unitarity triangle with . In order to obtain the positive sign of we need to require a relative sign difference between the omitted real order one coefficients of the 1-2 and 2-2 elements of either or (but not both). Note that the moduli of the parameters are not predicted in our model, since these Yukawa couplings stem from effective operators generated by messenger fields with (in general) different masses. They will be fixed by the fit to the up-type quark masses and the quark mixing angles.

For the neutrino and charged lepton sector we obtain:

(3.21)
(3.22)
(3.23)

The size of the neutrino Yukawa couplings is given by the two parameters and , which are of the order and . The right-handed neutrino masses and can be chosen such that the two observed neutrino mass squared differences are obtained, with one of the light neutrinos being massless (by construction since we have assumed only two right-handed neutrinos).

The mixing in the neutrino sector is “tri-bimaximal” to a good approximation, since the neutrino Yukawa matrix above satisfies the conditions of constrained sequential dominance [14]. In the considered GUT framework, is connected to and we obtain predictions for the lepton mixing parameters due to “charged lepton” corrections.

The alignment of Eq. (3.1) fixes all the phases in the lepton sector as well, leading to a Maki-Nakagawa-Sakata (MNS) mixing matrix with or , depending on the relative sign of and , and vanishing CP violating Majorana phases.444We remark that there exist other similar vacuum alignment possibilities for leading to other discrete predictions for , i.e. . A scenario with maximal leptonic CP violation is discussed in Sec. 4. Strictly speaking, our approach is in general only predicting one out of these discrete possibilities. With a specific alignment chosen, here with the one in Eq. (3.1), the values of all phases (including also the two Majorana phases) can be calculated. The would-be leptonic unitarity triangle thus collapses to a line. Combining tri-bimaximal neutrino mixing with the charged lepton corrections the predictions satisfy the lepton mixing sum rule [14, 15]. With , the model therefore predicts a shift of the solar mixing angle from its tri-bimaximal value of .

In summary, we have constructed a simple model based on symmetry, plus discrete and shaping symmetries, which is capable of predicting a right-angled CKM unitarity triangle following the method of “discrete vacuum alignment” described in Sec. 2.

4

In this section we present a variation of the direct model in [13]. We adopt the same basis as well as the same notation; for the sake of brevity we refer the reader to [13] whenever appropriate. As the neutrino sector remains unaltered we do not delve into an in-depth discussion thereof. With the quark sector being our primary focus we wish to accommodate the right-angled CKM unitary triangle by means of real and imaginary entries in the quark mass matrices. Thus we are led to drop the flavon field and introduce two new ones, and . This entails slight changes in the choice of superfields which drive the flavon vevs. We begin by briefly stating the leading Yukawa superpotential terms, the assumed vacuum configurations and the resulting quark mass matrices. We proceed by discussing the required flavon potential in detail. The shaping symmetry which was introduced in [13] in order to control the allowed terms must be replaced by a set of symmetries as discussed in Sec. 2. We construct all possible such symmetries that can help constrain our model at the effective level and determine all allowed terms which - if present - would spoil the desired structure. This investigation shows that even the most general set of allowed symmetries is insufficient to yield a viable model. However, we argue that all additional dangerous terms can be forbidden by invoking a set of messenger fields which gives rise to the required terms but disallows the dangerous ones.

4.1 Outline of the model

To make the following self-contained, we remark that the of is denoted by while the is written as . They furnish the following representations:

The desired Yukawa superpotential terms are

(4.1)
(4.2)
(4.3)

where denotes contractions to the representations, respectively. Here and in the following we assume all order one coefficients to be real and suppress them in our notation. Furthermore, the non-renormalisable terms are suppressed by a common mass scale .

With the vacuum configuration of the flavon fields given as

(4.4)
(4.5)
(4.6)

we are led to the following quark mass matrices

(4.7)

where denotes the sine of the Cabibbo angle. In the up quark mass matrix, the 2-2 element arises from the second term of Eq. (4.1), the 1-2 and 2-1 elements originate from the third term, and the 1-1 element from the fourth. As the vevs of all the flavon fields occurring in Eq. (4.1) are real, the matrix is real as well. Turning to the down quark mass matrix, we obtain the 3-3 entry from the first term of Eq. (4.2), while the 2-2 and 2-3 entries are derived from the second term. Notice the relative factor of 2 in the 2-3 element which is related to the alignment of and serves to improve the fit of the 2-3 CKM mixing. From Eq. (4.7) we get ; evaluating this at the scale of, e.g. the top mass yields a value of around which is to be compared to the measured 2-3 CKM mixing of , see, e.g. [22]. Without the factor of 2 one would be far off. Finally the 1-2 and 2-1 elements of originate from the third term of Eq. (4.2). Due to the vev configurations of the down-type flavons of Eq. (4.5), we find purely imaginary entries for the 1-2 and 2-1 elements while the other elements of the down quark mass matrix are all real. As discussed in [7], the structures of and of Eq. (4.7), with zero 1-3 mixings and a non-vanishing contribution in the 1-1 element of , give the correct quark masses and mixings including a CKM matrix that features a right-angled unitarity triangle with . The positive sign of is again obtained by requiring a relative sign difference between the omitted real order one coefficients of the 1-2 and 2-2 elements of either or (but not both).

Turning to the lepton sector, we point out that the charged lepton mass matrix is related to by transposition and an additional Georgi-Jarlskog factor of in the entries that arise from the term of Eq. (4.2), i.e.

(4.8)

Notice that, unlike in the above model, the left-handed 1-2 charged lepton mixing involves a maximal phase. In the parametrisation of [14] we get for the left-handed charged lepton mixing 

(4.9)

where the sign ambiguity is related to the relative sign difference between the coefficients of the 1-2 and 2-2 elements of in Eq. (4.7): the solution corresponds to identical signs while the solution corresponds to opposite signs. In the neutrino sector, defined by the superpotential of Eq. (4.3) as well as the flavon alignments of Eq. (4.6), we obtain the Dirac and Majorana mass matrices

(4.10)

where , , are independent order one coefficients. The effective light neutrino mass matrix after applying the seesaw mechanism is of exact tri-bimaximal form [13]. This can be easily understood as the superpotential of Eq. (4.3) remains symmetric under the Klein symmetry [10] after the flavons acquire their vevs. With these vevs taking real values, the light neutrino mass matrix ends up being purely real as well. Therefore, it is diagonalised by the tri-bimaximal mixing matrix without any phases. The resulting entries on the diagonal can in general be positive or negative. The latter case would require a Majorana phase which, however, does not violate CP. The neutrino mixing matrix is thus parametrised by555The phase has been introduced because the conventional tri-bimaximal mixing matrix would otherwise be incompatible with the standard PDG parametrisation where mixing angles take values between and .

(4.11)

Using the general relations of [14] it is then straightforward to calculate the resulting MNS matrix in terms of the two distinct left-handed mixing matrices, yielding

(4.12)
(4.13)
(4.14)

This leads to a Dirac CP phase