Available online at www.sclencedlrect.com

\0 ScienceDirect

Nuclear Physics B 821 (2009) 327-353

www.elsevier.com/locate/nuclphysb

The Cabibbo angle in a supersymmetric D14 model

A. Bluma, C. Hagedornabc*

a Max-Planck-Institut für Kernphysik, Postfach 10 39 80, 69029 Heidelberg, Germany b Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Beirut 4, I-34014 Trieste, Italy c INFN, Sezione di Trieste, Italy

Received 7 March 2009; received in revised form 19 June 2009; accepted 25 June 2009

Available online 30 June 2009

Abstract

We construct a supersymmetric model with the flavor symmetry D14 in which the CKM matrix element \Vud | can take the value \Vud | = cos( n) & 0.97493 implying that the Cabibbo angle 0c is sin(0c) & \Vus\ & sin(n) & 0.2225. These values are very close to those observed in experiments. The value of \ Vud \ (0c ) is based on the fact that different Z2 subgroups of D14 are conserved in the up and down quark sector. In order to achieve this, D14 is accompanied by a Z3 symmetry. The spontaneous breaking of D14 is induced by flavons, which are scalar gauge singlets. The quark mass hierarchy is partly due to the flavor group D\4 and partly due to a Froggatt-Nielsen symmetry ^(1)fn under which only the right-handed quarks transform. The model is natural in the sense that the hierarchies among the quark masses and mixing angles are generated with the help of symmetries. The issue of the vacuum alignment of the flavons is solved up to a small number of degeneracies, leaving four different possible values for \ Vud \. Out of these, only one of them leads to a phenomenological viable model. A study of the Z2 subgroup breaking terms shows that the results achieved in the symmetry limit are only slightly perturbed. At the same time they allow \ Vud \ (0c) to be well inside the small experimental error bars. © 2009 Elsevier B.V. All rights reserved.

1. Introduction

The explanation of the hierarchy among the charged fermion masses and of the peculiar fermion mixings, especially in the lepton sector, is one of the main issues in the field of model building. The prime candidate for the origin of fermion mass hierarchies and mixing patterns

* Corresponding author at: Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Beirut 4, I-34014 Trieste, Italy.

E-mail addresses: alexander.blum@mpi-hd.mpg.de (A. Blum), hagedorn@sissa.it (C. Hagedorn).

0550-3213/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2009.06.028

seems to be a flavor symmetry under which the three generations of Standard Model (SM) particles transform in a certain way. Unlike the majority of studies which concentrate on the leptonic sector we propose a dihedral group,1 Du, as flavor symmetry to predict the CKM matrix element | Vud | or equivalently the Cabibbo angle GC. The crucial aspect in this model is the fact that | Vud | is given in terms of group theoretical quantities, like the index n of the dihedral group Dn, the index j of the representation 2j under which two of the generations of the (left-handed) quarks transform and the indices ku,d of the subgroups which remain unbroken in the up, Z2 = (BAku), and the down quark sector, Z2 = (BAkd). Thereby, A and B are the two generators of the dihedral group. The general formula for | Vud | is [2-4]

|Vud | =

^n(ku - kd)j ^

In particular, the Cabibbo angle neither depends on arbitrarily tunable numbers, nor is it connected to the quark masses as is the case for the Gatto-Sartori-Tonin (GST) relation [5], sin(GC) Vus | ^ *Jmd/ms. The only dependence arises through the fact that the ordering of the mass eigenvalues determines which element in the CKM mixing matrix is fixed by the group theoretical quantities. However, since the hierarchy among the quark masses is also naturally accommodated in our model, partly by the flavor group D14 itself and partly by an additional Froggatt-Nielsen (FN) symmetry ^(1)FN [6], this sort of arbitrariness in the determination of the Cabibbo angle is avoided.2

In this paper we consider as framework the Minimal Supersymmetric SM (MSSM). The construction used in our model is in several aspects analogous to the one used in [7] to generate tri-bimaximal mixing in the lepton sector with the help of the group A4. The flavor group is broken at high energies through vacuum expectation values (VEVs) of gauge singlets, the flavons. The prediction of the actual value of the mixing angle originates from the fact that different subgroups of the flavor symmetry are conserved in different sectors (up and down quark sector) of the theory. The separation of these sectors can be maintained by an additional cyclic symmetry, which is Z3 in our case. The other crucial aspect for preserving different subgroups is the achievement of a certain vacuum alignment. As in [7], an appropriate flavon superpotential can be constructed by introducing a ^(1)« symmetry and adding a specific set of scalar fields, the driving fields, whose F-terms are responsible for aligning the flavon VEVs. As we show, the vacuum can be aligned such that in the up quark sector a Z2 symmetry with an even index ku is preserved, whereas in the down quark sector the residual Z2 symmetry is generated by BAkd with kd being an odd integer. Thus, two different Z2 groups are maintained in the sectors. We can set ku = 0 without loss of generality. However, we are unable to predict the exact value of kd such that our model leads to four possible scenarios with four different possible values of | Vud |. Out of these scenarios only one, namely kd = 1 or kd = 13, results in a phenomenologically viable value of |Vud | (and Gq )3

|Vud| =cos( 14 j « 0.97493 and sin(Gc) ^ 0.2225. (2)

1 Dihedral symmetries have already been frequently used as flavor symmetries, see [1].

2 This cannot, for example, be avoided in the A4 models [7], which successfully predict tri-bimaximal mixing in the lepton sector, since the hierarchy among the light neutrinos, which determines the ordering of the columns in the lepton mixing matrix, is very mild. Actually, a certain fine-tuning is necessary to achieve that the atmospheric mass squared difference is larger than the solar one.

3 Note that the result 8q = 14 has been derived in another context in [8].

Similar to [7], subleading corrections to masses and mixings arise from higher-dimensional operators. In general they are at most of relative order e & X2 & 0.04, so that Eq. (2) holds within

The model presented here surpasses the non-supersymmetric one constructed in [4] in several ways. Since the flavor symmetry is broken only spontaneously at the electroweak scale in the latter model, it contains several copies of the SM Higgs doublet. In contrast to this, the model which we discuss in the following possesses the two MSSM Higgs doublets hu and hd, which are neutral under the flavor group, and gauge singlets, the flavons and driving fields, which transform under flavor. The flavons are responsible for breaking the flavor symmetry. As a consequence, none of the problems usually present in models with an extended Higgs doublet sector, such as too low Higgs masses and large flavor changing neutral currents, is encountered here. Additionally, the problem of the vacuum alignment, which determines the value of the Cabibbo angle, is solved, up to a small number of degeneracies. This is impossible in the case of a multi-Higgs doublet model due to the large number of quartic couplings. Only a numerical fit can show that (at least) one set of parameters exists which leads to the desired vacuum structure. Finally, the breaking of the flavor group at high energies is also advantageous, because then domain walls generated through this breaking [9] might well be diluted in an inflationary era.

In the class of models [7] which extends the flavor group A4, being successful in predicting tri-bimaximal mixing for the leptons, to the quark sector one usually observes that the Cabibbo angle 0C = X & 0.22 produced is generically only of the order of e & X2 and thus too small by a factor of four to five.4 This observation might indicate that it is not possible to treat the Cabibbo angle only as a small perturbation in this class of models.

The paper is organized as follows: in Section 2 we repeat the necessary group theory of D14 and the properties of the subgroups relevant here. Section 3 contains an outline of the model in which the transformation properties of all particles under the flavor group are given. The quark masses and mixings, in the limit of conserved Z2 subgroups in up and down quark sector, are presented in Section 4. In Section 5, corrections to the quark mass matrices are studied in detail and the results of Section 4 are shown to be only slightly changed. The flavon superpotential is discussed in Section 6. We summarize our results and give a short outlook in Section 7. Details of the group theory of D14 such as Kronecker products and Clebsch-Gordan coefficients can be found in Appendix A. In Appendix B the corrections to the flavon superpotential and the shifts of the flavon VEVs are given.

2. Group theory of D14

In this section we briefly review the basic features of the dihedral group D14. Its order is 28, and it has four one-dimensional irreducible representations which we denote as 1;, i = 1,..., 4 and six two-dimensional ones called 2j, j = 1,..., 6. All of them are real and the representations 2j with an odd index j are faithful. The group is generated by the two elements A and B which fulfill the relations [11]

±0.04.

A14 = 1, B2 = 1, ABA = B. The generators A and B of the one-dimensional representations read 11: A = 1, B = 1,

4 This also happens in a recently proposed model using the flavor group S4 [10].

12: A = 1, B = -1, (4b)

13: A =-1, B = 1, (4c)

I4: A =-1, B = -1. (4d)

For the representation 2 j they are two-by-two matrices of the form

A = (T e-0.j ) B = (1 i). (5)

Note that we have chosen A to be complex, although all representations of D14 are real. Due to

this, we find for (a1,a2)T forming the doublet 2j that the combination (a2,,a\)T transforms as

2j rather than (a'2,a,2_)T. The Kronecker products and Clebsch-Gordan coefficients can be found in Appendix A and can also be deduced from the general formulae given in [2,12].

Since we derive the value of the element | Vud | (the Cabibbo angle GC), through a non-trivial breaking of D14 in the up and down quark sector, we briefly comment on the relevant type of Z2 subgroups of Di4. These Z2 groups are generated by an element of the form BAk for k being an integer between 0 and 13. With the help of Eq. (3) one easily sees that (BAk)2 = BAkBAk = BAk-1BAk-1 = ••• = B2 = 1. For k being even, singlets transforming as 13 are allowed to have a non-vanishing VEV, whereas k being odd only allows a non-trivial VEV for singlets which transform as 14 under Di4. Clearly, all singlets transforming in the trivial representation 1i of Di4 are allowed to have a non-vanishing VEV. Note that however the fields in the representation 12 are not allowed a non-vanishing VEV, since BAk = -1 for all possible values of k. In the case of two fields ^1,2 which form a doublet 2j a Z2 group generated by BAk is preserved, if

(V1)\_ le 7

mM 1 (6)

In order to see this note that the vector given in Eq. (6) is an eigenvector of the two-by-two matrix BAk to the eigenvalue +1. Due to the fact that singlets transforming as 13 can only preserve Z2 subgroups generated by BAk with k even and singlets in 14 only those with k odd, it is possible to ensure that the Z2 subgroup conserved in the up quark is different from the one in the down quark sector. Note that for this purpose the dihedral group has to have an even index, since only then the representations 13,4 are present [2]. So, it is not possible to choose D7 as flavor symmetry, as it has been done in [3,4], to predict GC, if distinct values of k in the up quark and down quark sector are supposed to be guaranteed by the choice of representations. One can check that the subgroup preserved by VEVs of the form given in Eq. (6) cannot be larger than Z2, if the index j of the representation 2j is odd, i.e. the representation is faithful. For an even index j the subgroup is a D2 group generated by the two elements A7 and BAk with k being an integer between 0 and 6.5 Obviously, in the case that only flavons residing in representations , i = 1,..., 4, acquire a VEV the conserved subgroup is also generally larger than only Z2.

5 In general, for fields in representations 2j, whose index j has a greatest common divisor with the group index n larger

than one, the preserved subgroup is larger than a Z2 symmetry. In the case under consideration, namely n = 14, this

statement is equivalent to the statement that the preserved subgroup is larger than Z2, if the index j of the representation 2j is even. We note that there is a mistake in the first version of [2] concerning this aspect.

3. Outline of the model

In our model the left-handed quarks Q i and Q2 are unified into the Di 4 doublet 2 1, denoted by Qd, while the third generation of left-handed quarks Q3, the right-handed up-type quark tc, and the right-handed down-type quark sc, transform trivially under D i 4, i.e. as 1 i. The remaining two generations of right-handed fields, i.e. cc and uc in the up quark and dc and bc in the down quark sector, are assigned to the one-dimensional representations 13 and 14.6 The MSSM Higgs doublets hu and hd do not transform under D 4. Therefore, we need to introduce gauge singlets, flavons, to form Di 4-invariant Yukawa couplings. These flavons transform according to the singlets 1i, 13, 14 and the doublets 2i, 22 and 24. All Yukawa operators involving flavon fields are non-renormalizable and suppressed by (powers of) the cutoff scale A which is expected to be of the order of the scale of grand unification or the Planck scale. Additionally, we have to introduce a symmetry which allows us to separate the up and down quark sector. The minimal choice of such a symmetry in this setup is a Z3 group. We assign a trivial Z3 charge to left-handed quarks, right-handed up quarks and to the flavon fields fu 2, x"2, H\ 2 and nu, which ought to couple dominantly to up quarks. The right-handed down quarks transform as w2 under

2ni d d d /7

Z3 with w = e 3 . The flavon fields ff 2, xd 2, Hd 2, nd and a, mainly responsible for down quark masses, acquire a phase w under Z3. The MSSM Higgs fields transform trivially also under the Z3 symmetry. Since the right-handed down quarks have charge w2 under Z3, whereas QD, Q3 and hd are neutral, the bottom quark does not acquire a mass at the renormalizable level, unlike the top quark. As a result, the hierarchy between the top and bottom quark is explained without large tan j = {hu)/(hd). The hierarchy between the charm and top quark mass, mc/mt ~ O(e2) with e & X2 & 0.04, is naturally accommodated in our model. To achieve the correct ratio between strange and bottom quark mass, ms/mb ~ O(e), we apply the FN mechanism. We add the FN field 0 to our model which is only charged under U( 1)fn. Without loss of generality we can assume that its charge is — 1. Note that we distinguish in our discussion between the FN field 0 and the flavon fields f u 2, Xu 2, Hi 2, nu, f f 2, Xf 2, Hf 2, nd and a which transform non-trivially under Di4 x Z3. If we assign a U(1)fn charge +1 to the right-handed down-type quark sc, we arrive at ms/mb ~ O(e). Finally, to reproduce the hierarchy between the first generation and the third one, mu/mt ~ O(e4) and md/mb ~ O(e2), also the right-handed quarks, uc and dc, have to have a non-vanishing U(1)fn charge. The transformation properties of the quarks and flavons under Di4 x Z3 x U(1)fn are summarized in Table 1. Given these we can write down the superpotential w which consists of two parts

w = wq + wf. (7)

wq contains the Yukawa couplings of the quarks and wf the flavon superpotential responsible for the vacuum alignment of the flavons. The mass matrices arising from wq are discussed in Sections 4 and 5, while wf is studied in Section 6.

As already explained in the introduction, the prediction of the CKM matrix element |Vud| or equivalently the Cabibbo angle 0C is based on the fact that the VEVs of the flavons { fu 2 , xiu2 , H™2 , nu} preserve a Z2 subgroup of Di4 which is generated by the element BA^,

6 The fact that the transformation properties of the right-handed down quark fields are permuted compared to those of the right-handed up quark fields is merely due to the desire to arrive at a down quark mass matrix Md which has a large (33) entry, see Eq. (18). However, since this is just a permutation of the right-handed fields it is neither relevant for quark masses nor for mixings.

Table 1

Particle content of the model. Here we display the transformation properties of fermions and scalars under the flavor group D14 x Z3 x U(1)fn. The symmetry Z3 separates the up and down quark sector. The left-handed quark doublets are denoted by Qd = (Ql,Q2)T, Ql = (u,d)T, Q2 = (c,s)T, Q3 = (t,b)T and the right-handed quarks by uc, cc, tc and dc, sc, bc. The flavon fields indexed by a u give masses to the up quarks only, at lowest order. Similarly, the fields which carry an index d (including the field 0) couple only to down quarks at this order. We assume the existence of a field 9 which is a gauge singlet transforming trivially under D14 x Z3. It is responsible for the breaking of the U(1)fn

symmetry. Without loss of generality its charge under U(1)fn can be chosen as -1. Note that m is the third root of unity,

i.e. m = e 3 .

Field Qd Q3 uc cc tc dc sc bc hu,d fU,2 x1,2 tu n,2 nu 2 d 1, -Sf xd X1,2 2 nd 0

D14 21 11 14 13 11 13 11 14 11 21 22 24 13 21 22 24 14 11

z3 1 1 1 1 1 m2 m2 m2 1 1 1 1 1 M m m m m

U(1)fn 0 0 2 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0

whereas the VEVs of {fd2, xd2, ,nd ,0} coupling dominantly to down quarks keep a Z2 group originating from the element BAkd conserved with ku = kd. Due to the fact that nu transforms as 13 and nd as 14 under D14 ku has to be an even integer between 0 and 12 and kd an odd integer between 1 and 13, implying the non-equality of ku and kd. Since ku = kd, it is also evident that D14 is completely broken in the whole theory. As mentioned, the separation of the two symmetry-breaking sectors is maintained by the Z3 symmetry. However, in terms with more than one flavon in the down and more than two flavons in the up quark sector the fields

{fU 2 , X\ 2 ' 2 ' VU} couple to down quarks and {f 1 2 ' 2 ' 2 ' nd, o} to up quarks so that this separation of the two sectors is not rigid anymore. Similarly, non-renormalizable operators in the flavon superpotential mix the two different sectors inducing shifts in the aligned flavon VEVs. This fact is explained in more detail in Sections 5 and 6.2. To elucidate the origin of the prediction of |Vud | (9C) we first consider in Section 4 the mass matrices arising in the case that the two different Z2 subgroups remain unbroken in up and down quark sector. Then we turn in Section 5 to the discussion of the mass matrix structures including the subgroup non-preserving corrections from multi-flavon insertions and VEV shifts and show that the results achieved in the limit of unbroken Z2 subgroups in both sectors still hold, especially the prediction of | Vud | (9C) is valid up to O(e) corrections.

4. Quark masses and mixings in the subgroup conserving case

As mentioned above, all Yukawa terms containing up to two flavons in the up and one flavon in the down quark sector preserve a Z2 group generated by BAku and by BAkd, respectively. In the up quark sector the only renormalizable coupling generates the top quark mass

Q3tchu. (8)

Here and in the following we omit order one couplings in front of the operators. The other elements of the third column and the (32) element of the up quark mass matrix Mu arise at the one-flavon level through the terms

-1 {QDfU)tchu and -1 Q3{ecnu)hu, (9)

respectively. We denote with (■ ■ ■) the contraction to a D14 invariant. The elements belonging to the upper 1-2 subblock of Mu are generated at the level of two-flavon insertions

0 2 0 2 2 0 2

A {QDUCx"HU)hu + (QdUC(HU) ) hu + {QDfUnUUC)hu, (10)

1 1 o 1

A {QDCCxuHu)hu + {Qdcc{hu) 2) hu + (QDfu)(nucC)hu. (11)

Thereby, the (11) and (21) entries stem from the terms in Eq. (10), while the terms in Eq. (11) are responsible for the (12) and (22) elements of Mu. Also the elements of the third column receive contributions from two-flavon insertions which, however, can be absorbed into the existing couplings (if we are in the symmetry preserving limit). Therefore, we do not mention these terms explicitly here. Only the (31) element of Mu vanishes in the limit of an unbroken Z2 subgroup in the up quark sector, since the existence of the residual symmetry forbids a non-zero VEV for a flavon (a combination of flavons) in the Di4 representation 14 for even ku. The VEVs of the fields fU 2, xU 2 and H1U2, which preserve a Z2 symmetry generated by the element BAku, are of the form

niku y / , u. \ / 2nik,

fU1)\_ u e 7 \ {xl

= wue 7

{r2)J \ 1 ) \{xU)J \ 1

/£U\\ 1 1 / 4niku

m v=z e"n 1 ) (12)

together with {nu) = 0. As can be read off from Eq. (1), only the difference between ku and kd is relevant for | Vud |. Thus, we set ku = 0. Obviously, the conserved Z2 group in the up quark sector is then generated by B. The up quark mass matrix has the generic form

(-a'^t 2e2 a2,e2 a3,e\

ap 2e2 a^e2 a^e {hu) (13)

0 au4e yt J

in the Z2 symmetry limit. The couplings af and yt are in general complex. The small expansion parameters e and t are given by

vu wu zu {nu) 2 {0)

— ,—,—, — ~ e & X2 & 0.04 and — = t. (14)

A A A A A

All flavon VEVs are of the order of Mf, which is the mass parameter in the flavon superpotential w f,u in Eq. (41), unless we assume accidental cancellations among the dimensionless parameters of wf,u, compare Eq. (44). Eq. (14) thus holds for Mf & eA. Additionally, we take t and e to be real and positive and assume

t & e & X2 & 0.04 (15)

in the following.

We can discuss the down quark mass matrix Md in a similar fashion. Taking into account only terms with one flavon we can generate apart from the (33) entry of the matrix the elements of the second column,

AQ3 {bCnd)hd, A Q3SC°hd and ^ {QDfd)sChd. (16)

Actually, the first term is responsible for the (33) entry, while the second one leads to a non-vanishing (32) entry and the third one gives the dominant contribution to the (12) and (22) elements of Md. The flavon VEVs preserving the subgroups generated by BAk are of the form

l,lrd\\ / -nik \ / lxd\\ ,, / -2ni

d( e M /xiM_ „„d^nk e 7

v ))- 1 y Ui )'- " eT| 1 and

A _ zd J^k ( e

4nik 7

(si)>-zen i )■ ,17)

with (nd) and (a) being non-zero. Since we already set ku to zero, we omitted the subscript d of the parameter k which has to be an odd integer ranging between 1 and 13. As discussed in Section 6, the value of k cannot be uniquely fixed through the superpotential w f. The form of the down quark mass matrix is then

/0 a^te 0 \

Md = 0 ad e-Itik/1te 0 (hd). (18)

\0 a^e ybeJ Again, the couplings ad and yb are complex. The expansion parameter e is given by

vd wd zd (nd) (a) 2

~ e « « 0.04. (19)

A A A A A

Again, the fact that all VEVs are of the same order of magnitude (for no accidental cancellations due to some special parameter choice) can be derived from the flavon superpotential wf,d, see Eqs. (47) and (48). Eq. (19) holds for (a) = x « eA. We assume that e in Eq. (19) is the same as in Eq. (14), i.e. the VEVs of all flavons are expected to be of the same order of magnitude. Eq. (15) thus also holds.

For the quark masses we find

m2u : m2c : m2 ~ e8 : e4 : 1, (20a)

m2d : m2 : mcb - 0 : ec : 1, (20b)

m2 : md - e2 : 1, (20c)

where the third equation holds for small tan j3. As one can see, the hierarchy among the up quark masses and the ratio ms/mb are correctly reproduced. The down quark mass vanishes at this level and is generated by Z2 symmetry non-conserving two-flavon insertions, see Eq. (37). The CKM matrix is of the form

cos(kf )| | sin(kf )| 0\ / 0 O(e4) O(e2)\ |VckmI=|| sin (14 )| | cos(kf )| 0] + O(e2) O(e2) O(e) . (21)

0 0 J V O(e) O(e) O(e2)J

The elements | Vud|, | Vus |, | Vcd | and | VcsI are determined by the group theoretical parameter k. Since k takes odd integer values between 1 and 13 we arrive at four possible scenarios: If k takes the value k = 1 (minimal) or k = 13 (maximal), we arrive at | Vud | = cos( 14) ^ 0.97493. This value is very close to the central one, | Vud |exp = 0.97419+0;00022, [13]. For the other three elements of the CKM matrix, also only determined by k, we then find

I Vud I ^ I Vcs | ^ 0.97493 and |Vus |«|Vcd I ^ 0.2225, (22)

which should be compared with the experimental values [13]

IV I _ 0 o73 34+0.00023 IV I _ 0 2 2 57+0.0010 |V | _ 0 2 2 56+0.0010

1 Vcs |exp = °.9'334_0.00023' 1 Vus |exp = °.225/ -0.0010 ' 1 Vcd |exp = 0.22 5 6 -0.0010.

As the experimental errors are very small, the values predicted for |Vud |, |Vus |, |Vcd | and | VCs| are not within the error bars given in [13]. However, as we show in Section 5 the terms which break the residual Z2 subgroups, change the values of the CKM matrix elements | Vud |, | Vus |, | Vcd | and | VCs | by order e so that the results of the model agree with the experimental data.

3n N 14 '

The three other possible values for | Vud | which can arise are cos (14) & 0.78183 for k = 3 and

k = 11, cos(5f) & 0.43388 if k = 5 or k = 9 and finally |Vud | vanishes for k = 7. Thus, k has to be chosen either minimal or maximal to be in accordance with the experimental observations. The other values cannot be considered to be reasonable, since we cannot expect that the corrections coming from symmetry breaking terms change the element | Vud | by more than e & 0.04. For this reason, we set k = 1 in the following discussion. The CKM matrix elements in the third row and column are reproduced with the correct order of magnitude, apart from | Vub | which is slightly too small, e2 & X4 instead of X3, and from | Vtd | which is slightly too large, e & X2 instead of X3. The value of | Vub| gets enhanced through the inclusion of Z2 symmetry breaking terms. In any case by including Z2 symmetry breaking terms it becomes possible to accommodate all experimental data, if some of the Yukawa couplings are slightly enhanced or suppressed. JCP, the measure of CP violation in the quark sector [14], is of the order e3 & X6 and thus of the correct order of magnitude.

Finally, we briefly compare the form of the mass matrices Mu and Md to the general results we achieved in [2]. According to [2] the most general mass matrix arising from the preservation of a Z2 subgroup generated by the element BAku'd for left-handed fields transforming as 21 + 11 and right-handed fields as three singlets is given by

( _Aq Bq Cq \

Mq = Aqe_nikq/1 Bqe_nikq/7 Cqe_nikq'1 for q = u,d. (24)

V 0 Dq Eq )

The parameters Aq, Bq, Cq, Dq and Eq contain Yukawa couplings and VEVs and are in general complex. Comparing Eq. (24) with Eq. (13) shows that Mu is of this form with ku = 0. The down quark mass matrix Md, given in Eq. (18), equals the matrix in Eq. (24), if k = kd and the parameters Ad and Cd are set to zero. This happens, since our model only contains a restricted number of flavon fields and we do not take into account terms with more than one flavon at this level.

5. Quark masses and mixings including subgroup-breaking effects

In this section we include terms which break the residual Z2 symmetries explicitly. These lead to corrections of the results shown in Section 4. They are generated by multi-flavon insertions in which flavon fields belonging to the set [ff 2 , x( 2 , fd 2 , nd, &} give masses to up quarks and flavons belonging to [ff 2 , xU 2 , f 2 , nu} to down quarks. Additionally, non-renormalizable terms in the flavon superpotential lead to complex shifts in the flavon VEVs in Eqs. (12) and (17). They can be parameterized as in Eq. (50) in Section 6.2 (with k = 1). At the same time, x remains a free parameter. The corrections to the flavon superpotential are discussed in detail in Sections 6.2 and Appendix B. The analysis given in these sections shows that the generic size of

the shifts is

( VEV \

5 VEV ~ ) VEV ~ e VEV, (25)

if all VEVs are of the order eA, see Eqs. (14) and (19). Thus, the VEV shifts inserted in Yukawa terms with p flavons contribute at the same level as Yukawa terms containing p + 1 flavons.

In the up quark sector we find that the (11) and (21) elements receive Z2 symmetry breaking corrections through the following operators

n 2 n 2 n 2

Aï[ {QoucSxu^u) + {Qoucxu&^u)]hu + A4 {QDucïu8ï")hu + a* {QoSfunuu^hu

n 2 n 2 n 2

+ a5 {QDfdxd){nduc)K + a {QDuc{xd)3) hu + a (qduc(*d) 2f)k

b 2 n b 2 o b 2

+ a{QDucxd{f) )hu + a{QDuc{xd)hd)hu + a5{QDucxdïdyh

B 2 2 0 2

+ A(QDuc{Hd) )ohu + A{QDtd){nduc)ahu. (26)

The notation of, for example, &xu indicates that the VEV of the fields x\2, shifted through the non-renormalizable operators correcting the flavon superpotential, is used, when calculating the contribution to the up quark mass matrix. Thus, all contributions from the operators in the first line of Eq. (26) arise from the fact that the VEVs become shifted. Note that we omitted the operator stemming from the shift of the VEV of nu, since this field only transforms as singlet under Di4 and thus does not possess any special vacuum structure. (We also do this in the following equations.) The other operators arise from the insertions of three down-type flavon fields. There exist similar operators containing three up-type flavons. However, these still preserve the Z2 symmetry present in the up quark sector at lowest order and therefore can be absorbed into the existing couplings. Analogously, we find that the following operators give rise to Z2 symmetry breaking contributions to the (12) and (22) elements so that also these are no longer equal

A [(qDcCsxU^U) + {QDccxus^u)]hu + A (qDcCïU^U)k + A {QDwu){nucc )hu

+ A3 {QDccfdxdnd)hu + A {QDOc{xd)3) hu + A (QDcc(fd)2

+ A QDccxd (ïd )2)hu + A (QDcc{xd )¥ )hu + A (QDccxdf )

+ A (Qdcc № f)°hu + -A3 (QDOcfdnd )ahu. (27)

Again, the operators in the first line are associated to the shifted VEVs. The rest of the operators originates from three-flavon insertions of down-type flavons. The contributions from the analogous operators with up-type flavons can again be absorbed into the existing couplings. The dominant contribution to the (13) and (23) elements which breaks the residual Z2 symmetry stems from the VEV shift of the fields ^ 2

A (QD&fu)tchu. (28)

All other contributions up to three flavons are either Z2 symmetry preserving or breaking, but subdominant. The (31) element which has to vanish in the symmetry limit is generated through

the following three-flavon insertions of down-type flavons

G 2 G 2 o G 2

A Q3 {nduc)o 2hu + — Q3{ nducWd) 2hu + — Q3( nduc){xd) hu

G 2 o G 2 o G 2

+ — Q3 {nduc){i;d) 2hu + — Q3(ndu%d) 2hu + — Q3{ucfdXdïd)hu

+ Â Q3 {ucfd{ïd)2) hu. (29)

Note that there are symmetry-conserving couplings, i.e. operators with three up-type flavons, of the same order. These, however, vanish, if the vacuum alignment in Eq. (12) is applied. They can only contribute at the next order, if the VEV shifts are taken into account; however, such effects are subdominant. The (32) and (33) elements of the up quark mass matrix already exist at the lowest order and only receive subdominant contributions from higher-dimensional operators and VEV shifts. The up quark mass matrix can thus be cast into the form n 2(-auxc2 + ^e3) aiie2 + ^e3 a^ + p3ue2\ Mu = ap 2e2 au2e2 au3e (hu). (30)

V p4ut 2e3 a^e yt )

We note that without loss of generality we can define the couplings au 2 3 and pu 2 3 in such a way that the corrections stemming from Z2 subgroup breaking terms only appear in the first row of Mu. Due to this and due to the absorption of subdominant contributions the couplings au only coincide at the leading order with those present in Eq. (13). This also holds for yt. Again, all couplings are in general complex. The matrix in Eq. (30) is the most general one arising in our model, i.e. all contributions from terms including more than three flavons can be absorbed into the couplings a'u, ff and yt.

Similarly, we analyze the Z2 symmetry breaking contributions to the down quark mass matrix Md. The (11) and (21) elements of Md are dominantly generated by Z2 symmetry breaking effects from two-flavon insertions involving one down- and one up-type flavon. We find five independent operators

— (Qodcïdxu)hd + —{Qodcxd^u)hd + —{QDdcÇdïu)hd

+ — {QDtd){nuàc)hd + —^ (QDdcndr )hd■ (31)

Since they are Z2 symmetry breaking, the (11) and (21) entries are uncorrelated. We note that Z2 symmetry preserving contributions can only arise, if operators with more than two flavons are considered. However, these are always subdominant compared to the operators in Eq. (31). Similar statements apply to the generation of the (13) and (23) element of Md. The dominant (Z2 symmetry breaking) contributions stem from the operators

-1 (QDbcïdxu)hd + —^ {QDbcxdïu)hd + —^ {QDbcfïu)hd

+ — (QDbcfdnu)hd + — (QDr )(ndbc)hd. (32)

The (12) and (22) elements which are already present at the lowest order are corrected by Z2

symmetry breaking terms from the VEV shift of the fields tyf 2 G ( )

——2 {QD8fd)schd (33)

as well as from two-flavon insertions with one up-type and one down-type flavon

— {QDXdtu)schd + — {QotdXu)schd + A {QBfu)oschd. (34)

The (31) entry, which must vanish in the symmetry limit, is generated dominantly by a single operator

A3 Q3(nudc)ahd.

Similarly to the up quark mass matrix, the (32) and (33) elements of Md also receive contributions from Z2 symmetry breaking effects, which can be absorbed into the leading order term. Eventually, the most general form of the down quark mass matrix Md in our model reads

'fidxte2 t(ade + fidAe2) fi^e2-

Md = fi^te2

■,fidte2

ade-ni/1 a^te

The parameters ad and fi^ have been defined so that Z2 symmetry breaking contributions only appear in the (12) element. Note again that all parameters ad, ¡3d and yb are complex. Also note that ad and yb only coincide at leading order with the corresponding parameters in Eq. (18) due to the absorption of subdominant effects.

Before calculating quark masses and mixings the parameters 34, aU, yt, ¡¡d, ad and yb in the third row of Mu and Md are made real by appropriate rephasing of the right-handed quark fields. The resulting quark masses are then (for t « e)

ml = 2|au|2(hu )2-8

m2 = 2 |aK - ^ (hu)2e4 + ^^^, m2 = 2|ad |2(hd)2e4 + O(e5),

e8 + O(e9), m2d = 2 fi - fi^T |2(hd)2e6 + O(e7),

m2 = yf(hu)2 + O (e2), m\ = y2{hd )2e2 + O (e4).

(37b) (37c)

At the subdominant level thus also the correct order of magnitude of the down quark mass is reproduced. The CKM matrix elements are given by

IVud | =cos(^—J + O(e),

IVuS | = + O(e),

fid + fid 2a3,

| Vcs | = cos )+ O(e),

| Vcd | = si^n )+ O(e),

(38a) (38b)

|Vcb | =V2

|Vts| =

+ O (e

fid + fide T a3u(1 + e T )

|Vub | = ^=

fid - fid

+ O (e2),

+ O (e

|Vtd| = -= V2

fid - fideT

a%(1 - e~)

+ O (e

| Vtb | = 1 + O (e2).

(38d) (38e)

Table 2

Driving fields of the model. The transformation properties of the driving fields under the flavor symmetry D14 x Z3. Similar to the flavons none of the driving fields is charged under U(1)fn. The fields indexed with an u (d) drive the VEVs of the flavons giving masses dominantly to the up (down) quarks. Note that all these fields have a U(1)r charge +2.

Field ff0u u2 <p°u2 ^1u2 ff 0d ^12 #2 <°12

d14 11 21 23 25 11 21 23 25

Z3II 1 1 rn rn rn rn

As one can see, |Vud|, \Vus|, |Vcd| and \ Vcs|, which are determined by the group theoretical

indices of this model, get all corrected by terms of order e, so that they can be in full accordance

with the experimental values [13]. The elements of the third row and column are still of the same order of magnitude in e after the inclusion of Z2 subgroup breaking terms, apart from | VUb | which gets enhanced by 1/e. For this reason, | Vid| and |Vub | are both slightly larger in our model, |Vid|, | Vub| ~ e « X2, than the measured values, which are of order X3. However, only a moderate tuning is necessary in order to also accommodate these values. For the Jarlskog invariant JCP we find

Jcp = ^ sin(y) (23bRe((«3)* № - pd6)) - y (Kl2 - Kl2)) + O(e3). (39)

Similar to | Vub| JCP gets enhanced by 1/e compared to the result in the symmetry limit. Thus, it has to be slightly tuned to match the experimental value, Jcp,exp = (3.05+°'i,") x 10-5 [13], which is around e3 « X6. However, already the factor sin(y)/4 « 0.11 leads to a certain suppression of JCP.

6. Flavon superpotential

6.1. Leading order

Turning to the discussion of the flavon superpotential wf we add—analogously to, for example, [15]—two additional ingredients. First, we introduce a further U(1) symmetry which is an extension of R-parity called U(1)R. Second, a set of so-called driving fields whose F-terms account for the vacuum alignment of the flavon fields is added to the model. Quarks transform with charge +1, flavon fields, hu^d and 0 are neutral and driving fields have a charge +2 under U(1)R. In this way all terms in the superpotential wf are linear in the driving fields, whereas these fields do not appear in the superpotential wq, responsible for the quark masses. Since we expect the flavor symmetry to be broken at high energies around the seesaw scale or the scale of grand unification, soft supersymmetry breaking effects will not affect the alignment so that considering only the F-terms is justified. The driving fields, required in order to construct wf, can be found in Table 2. The flavon superpotential at the renormalizable level consists of two parts

Wf = Wf,u + Wf,d, (40)

where wf,u gives rise to the alignment of the flavons with an index u, and wf,d to the alignment of the flavons coupling mainly to down quarks. wf,u reads

Wfu = Muf{f Uf22u + fUfT) + "u{ r^u^ + fUxUvT)

+ bu(^u/2u^0u + ^2uXiu^20u) + Cu(^u^2u^0u + ^l>0u) + du^u (fu^0u + ^2>2°u) + ^u (£ uf1uP20u + V&P0u) + fu^(xuV0u + X2v20u) + «u^ 0u^u^2u + lu° ^XuXu

+ »u^ 0u^2u + ?ua 0u(^u)2.

The conditions for the vacuum alignment are given by the F-terms

dwf,u = M"fu + bufuxu = 0,

= Mi^u + bu^2ux1u = 0,

= au s2ux2u + cu su^u + du = 0,

df0u dwf,u

d-f = aul ux u + ^ 1u + dunu& = 0, dfu = euS^2u + funuxt = 0,

^ = eul^i + funUx2 = 0,

d W f,u

= guff + lux1 x2 + "uf!uf2u + iu(n")2 = 0.

(42a) (42b) (42c) (42d) (42e) (42f) (42g)

If we assume that none of the parameters in the superpotential vanishes and acquires a nonzero VEV, we arrive at

(£xu)

u 2niku / e 7 = zue 7 '

= wue 7

wu = - v

2du e.

■ (cu fu 4audu eu fu + (cu fu) ) ,

(vu)2 = -

lu(wu)2 + nu(zu)2 niku

gu + qu( fUuWu )2

e 7 and = -

eu v z"

as unique solution. The flavon VEVs are aligned and their alignment only depends on the parameter ku which is an even integer between 0 and 12 (see Section 2). Thus, all vacua conserve a Z2 subgroup of Di4 generated by the element BAku. Since only the difference between ku and kd is relevant for the prediction of the CKM matrix element | Vud |, we set ku = 0, as it has been done in Section 4, when we study quark masses and mixings. The size of the flavon VEVs is determined by the dimensionless parameters au, bu, ... in wf,u and the mass parameter M£. Assuming the dimensionless parameters are order one couplings leads to flavon VEVs which are all of the order

of Mf « еЛ, as done in Sections 4 and 5. Choosing the parameters in wf,u appropriately, we can make all VEVs in Eqs. (43) and (44) positive for ku = 0.

Analogously, the flavon superpotential which drives the vacuum alignment of the fields f d 2, Xd2, Hd 2, nd and a is given by

wf, d = mfa(f df0d + f2df 0d ) + ad{Yxxdvld + fV^ )

+bd fdxdf0d+f2dxdf20d )+cd fd^+) +ddnd (Hdv0d - H2d^20d )+^d (f dHdP20d+fdHdp0d )

+fdnd{xdp0d - x^p^t)+Bd* 0dtdtd+idff 0dxdxd+^ od&>.

+ qdff 0d (nd)2 + Pdff 0dff 2. Setting the F-terms of the driving fields ^d,, ^0d>, p<0d2 and a0d to zero we find

df0 dw

f,d = mfaf + bdf dxdd = 0,

df2 dw

f,d = mfafd + bdfdx d = 0,

f,d = adfdxd + Cdf dH2d + ddndH d = 0,

= adf dxd + CdfdHd - ddnaHd = 0

dV~2 dw

f,d = edfdHd + fdndX d = 0,

= edf dHd - fd^xd = 0,

dp! dw

= gdf d f 2 + kx idX2d + «dHidH2d + qd{nd )2 + Pda2 = 0.

(46a) (46b) (46c) (46d) (46e) (46f) (46g)

These equations lead to the same VEV structure as shown in Eq. (43), if we assume that again none of the parameters in the flavon superpotential vanishes and the two fields f d and a get a non-vanishing VEV. Thus, (f d2), (x !2) and (H d2) have the same form as (f U2), (x U2) and (HU2) with obvious replacements {vu, wu, zu} ^ {vd, wd, zd}, ku ^ kd and kd being an odd integer. wd, zd and vd are given by

and zd =

(cdfd ±У4adadedfd + (cdfd)2),

d)2 ld(wd)2 + nd(zd)2 + pdx2 ПЦ

(vd)2 = ■

qd(efiwd) '

and the VEVs of the two singlets nd and a read

I d\ ed VdZd -4£kL , . . n/ = —--e 7 and {a )=x.

We find that the parameter x remains as free parameter in the model. However, taking the di-mensionless parameters of wf,d to be of similar size all flavon VEVs are of the same order of magnitude, i.e. x « eA. Note further that the two possible signs appearing in Eqs. (44) and (47) are uncorrelated. We can choose the parameters such that vd, wd, zd and x are positive. The parameter kd is an odd integer in the range {1,..., 13}. Similar to ku being even, kd is required to be odd by the transformation property of the flavon nd under Di4. Especially, kd is different from ku so that we preserve different Z2 subgroups in both sectors. As a consequence, the derived mixing angle is non-trivial. However, we cannot uniquely fix the parameter kd and thus the mixing angle by the vacuum alignment deduced from wf .As discussed in Section 4, we are left with a small number (four) of different possible values for | Vud |. Due to the different subgroups preserved in up and down quark sector D14 is eventually completely broken in the whole theory. As we already set ku to zero, we omit the index of the parameter kd from now on also in the discussion of the flavon superpotential.

We end with a few remarks about the VEVs of the driving fields, the absence of a ^,-term and the mass spectrum of the gauge singlets transforming under Di4. The VEVs of the driving fields are determined by the F-terms of the flavon fields. We find that the VEVs of all driving fields can vanish at the minimum, if we plug in the solutions for the VEVs of the flavons found in Eqs. (43), (44), (47) and (48). The term \xhuhd is forbidden by the symmetry. It cannot be generated through terms including one driving field, hu and hd and an appropriate number of flavon fields (to make it invariant under the symmetry Di4 x Z3), as long as the driving fields do not acquire non-vanishing VEVs. Thus, the ^,-term should originate from another mechanism, see also [15]. Massless modes which might exist in the spectrum of flavons and driving fields are expected to become massive, if soft supersymmetry breaking masses are included into the potential.

6.2. Corrections to the leading order

In the flavon superpotential, terms containing three flavons and one driving field lead to corrections of the vacuum alignment achieved through the superpotential wf, i.e. they induce (small) shifts in the VEVs of the flavons. Such terms are suppressed by the cutoff scale A. Due to the Z3 symmetry two types of three-flavon combinations can couple to a driving field with an index u, namely either all three flavons also carry an index u or all three of them belong to the set {fd 2 ,xf2 ,^d2,Vd,a}. If the driving field has an index d, two of the three flavons have to be down-type flavons, while the third one necessarily has to carry an index u. These corrections to the flavon superpotential can be written as

Awf = Awf, u + Awf, d, (49)

where the terms of Awf u (Awf d) are responsible for the shifts in the VEVs of the flavons uncharged (charged) under the Z3 symmetry. The exact form of the terms is given in Appendix B. We choose the following convention for the shifts of the VEVs

[fu) = vu + Svu, (x?) = wu + 8wu, (g) = zu + 8zu,

[nu) = -j w- + Snu> M) = e- ^ V + V), f) = vd + 8vd,

[xd) = e- ^ wd + Swd), [xd) = e ^ wd + Swd), $) = e- ^ {zd + Szd),

fä) = e2^ zd + Sz2) and [nd) = e-^ + (50)

{ff) = x

remains as free parameter. As can be read off from the equations given in Appendix B, x is also not fixed by the corrections to the superpotential. We do not fix the parameter k in Eq. (50), although we showed in Section 4 that only k = 1 and k = 13 lead to a phenomenologically viable model. This is done, because the complexity of the calculation of the shifts does not depend on the actual value of k. (k still has to be an odd integer.) One finds that also the inclusion of the corrections to the flavon superpotential does not fix the value of k. The detailed calculations given in Appendix B show that the generic size of the shifts is

for all VEVs being of the order eA. The shifts are expected to be in general complex, without having a particular phase.

7. Summary and outlook

We presented an extension of the MSSM in which the value of the CKM matrix element |Vud | or equivalently the Cabibbo angle 0C is fixed by group theoretical quantities of the flavor symmetry D14, up to the choice among four different possible values. The determination of |Vud| originates from the fact that residual Z2 symmetries of D14 exist in the up and down quark sector. We have shown that these can be maintained by the vacuum alignment resulting from a properly constructed flavon superpotential. Furthermore, it is ensured through the choice of flavon representations that the Z2 symmetries of the up and down quark sector do not coincide so that the quark mixing cannot be trivial. It turns out that the vacua of Z2 symmetries generated by BAk with k being either even or odd are degenerate so that we arrive at the mentioned four possible values for |Vud| . Out of these only one is phenomenologically viable, namely | Vud |= cos( 14) « 0.97493. The CKM matrix elements | Vus|, | Vcd| and | V„| are as well predicted to be | Vu4^| Vcd| « 0.2225 and |Vcs| Vud| ^ 0.97493. For the other elements we find the following orders of magnitude in e « X2 (including Z2 subgroup breaking effects): | Vcb |, | V„|, | Vub|, | V,d|~ e « X2 and | Vi6| = 1 + O(e2) = 1 + O(X4). Thus, | V,d| and | Vu^ turn out to be slightly too large. The same is true for JCP which is of the order of e2 « X4 instead of X6. However, it only requires a moderate tuning of the parameters of the model to accommodate the experimentally measured values. All quark masses are appropriately reproduced. The large top quark mass results from the fact that the top quark is the only fermion acquiring a mass at the renormalizable level. Since the bottom quark mass stems from an operator involving one flavon, the correct ratio mb/mt ~ e is produced without large tan j. The hierarchy mu : mc : mt ~ e4 : e2 : 1 in the up quark sector is accommodated in the Z2 subgroup conserving limit. Thereby, the suppression of the up quark mass is (partly) due to the non-vanishing FN charge of the right-handed up quark. The correct order of magnitude of the strange quark mass can as well be achieved through the FN mechanism. The down quark mass which vanishes at the lowest order is generated by operators with two-flavon insertions. Also its correct size is guaranteed by the FN mechanism. The main problem which cannot be solved in this model is the fact that the parameter k(d)—and therefore also | Vud |—is not uniquely fixed, but can take a certain number of different values. We presume that a new type of mechanism for the vacuum alignment is necessary which also fixes the (absolute) phase of the VEVs of the flavons so that

the parameter k(d) is determined. One possibility might arise in models with extra dimensions. For a recent discussion of the breaking of a flavor symmetry with extra dimensions see [16].

As a next step, it is interesting to discuss the extension of our model to the leptonic sector. In the literature models with the dihedral flavor group D3 (= S3) [17] or D4 [18] can found which also use the fact that different subgroups of the flavor symmetry are conserved in the charged lepton and neutrino (Dirac and right-handed Majorana neutrino) sector to predict the leptonic mixing angle 023 to be maximal and 0i3 to be zero. These models are non-supersymmetric and contain Higgs doublets transforming non-trivially under the flavor group in their original form. However, recently variants of [18] have been discussed whose framework is the MSSM and in which only gauge singlets break the flavor group spontaneously at high energies [19]. A possibility to combine such a variant with the model presented here by using a (possibly larger) dihedral group is worth studying.

As has been discussed in [2], the assignment 2 + 1 for the left-handed and 1 + 1 + 1 for the right-handed fields is not the only possible one in order to predict one element of the mixing matrix in terms of group theoretical quantities only. Alternatively, we can consider a model in which both, left- and right-handed fields, are assigned to 2 + 1. Such an assignment usually emerges when we consider grand unified theories (GUTs), e.g. in SU(5) where the left- and right-handed up quarks both reside in the representation 10.7 However, the following problem might occur: the product 2 x 2 contains an invariant of the dihedral group, if left- and right-handed fields transform as the same doublet. The group theoretical reason is the fact that all two-dimensional representations of dihedral groups are real. The existence of the invariant leads to a degenerate mass spectrum among the first two generations, e.g. in an SU(5) GUT to the prediction that up quark and charm quark mass are degenerate. One possibility to circumvent this difficulty might be to resort to a double-valued dihedral group. Such a group additionally possesses pseudo-real (two-dimensional) representations. One of their properties is that the product of a representation with itself contains the invariant/trivial representation 11 in its anti-symmetric part. In an SU(5) model one can then use the fact that the contribution of a Higgs field in the GUT representation 5 to the up quark mass matrix leads to a symmetric mass matrix, in order to avoid the invariant coupling. However, it is still not obvious whether the mass hierarchy among the up quarks can be generated (through the FN mechanism) without tuning the parameters. Even in non-unified models in which the two-dimensional representations under which left- and right-handed fields transform do not have to be equivalent, it might not be obvious that the fermion mass hierarchy can be appropriately accommodated (with an additional FN symmetry).

Finally, further interesting aspects to analyze are the anomaly conditions holding for the flavor symmetry D14 which in general lead to additional constraints [20] as well as the origin of such a flavor symmetry, see for instance [21].

Acknowledgements

We would like to thank Lorenzo Calibbi, Ferruccio Feruglio and Andrea Romanino for discussions. We also acknowledge the referee's comments on the number of free parameters among the flavon VEVs and flat directions present in the flavon potential. A.B. acknowledges support from the Studienstiftung des Deutschen Volkes. C.H. was partly supported by the "Sonderforschungsbereich" TR27.

7 The case in which all three generations transform as singlets is not very appealing, since the up quark sector would be merely determined by an Abelian flavor symmetry rather than a non-Abelian one.

Appendix A. Kronecker products and Clebsch-Gordan coefficients

Here we list the explicit form of the Kronecker products as well as the Clebsch-Gordan coefficients. More general results for dihedral groups with an arbitrary index n can be found in [2,12].

A.1. Kronecker products

The products 1; x 1j are

1; X 1; = 1i, 1i X 1i = 1i for i = 1.....4,

12 x I3 = I4, I2 x I4 = I3 and I3 x I4 = I2. For 1; x 2j we find

11,2 x 2j = 2j and 13,4 x 2j = 27-j for all j. The products of 2; x 2; decompose into

[2; x 2;] = 1i + 2j and {2; x 2;} = 12,

where the index j equals j = 2;' for ; < 3 and j = 14 - 2;' holds for ; > 4. [v x v] denotes thereby the symmetric part of the product v x v, while {v x v} is the anti-symmetric one. For the mixed products 2; x 2j with ; = j two structures are possible either

2; x 2j = 2k + 2i,

with k = |; - j | and l being ; + j for ; + j < 6 and 14 - (; + j) for ; + j > 8. For ; + j = 7 we find instead

2; x 2j = 13 +14 + 2k, where k is again |; - j |.

A.2. Clebsch-Gordan coefficients

For si — 1i and (a1,a2)T — 2j we find

SiaM - 2j, (S2M - 2j, - 27-j and fs4a2 .-

s1a2j J \-S2a2J \S3a1J \-S4a1

The Clebsch-Gordan coefficients of the product of (a1,a2)T, (b1,b2)T ~ 2; read

(«1^1 \ f a2b2 \

, 2j or 2j

«2b2j j \a1b1J j

depending on whether j = 2; as it is for ; < 3 or j = 14 - 2; which holds if; > 4. For the two doublets (a1,a2)T ~ 2; and (b1,b2)T ~ 2j we find for ; + j = 7

a1b2) ~ 2k (k = ; - j) or (a2M ~ 2k (k = j -;), ab1 J \ a^J

abO ~ 2i (l = ; + j) or (£) ~ 2i (l = 14 - (; + j)).

If i + j = 7 holds the covariants read

aibi + a2b2 ~ 13, a\b\ — 02^2 ~ 14, I^M ^ 2k or (aibl

\a2b1J \a1b2

Again, the first case is relevant for k = i — j, while the second form for k = j — i

Appendix B. Corrections to the flavon superpotential

In this appendix we discuss the form of the VEV shifts induced by the corrections of the flavon superpotential. These corrections can be written as

Aw/ = Awf,u + Awf,d.

We can parameterize the shifted VEVs as shown in Eq. (50). x remains unchanged, since it is a free parameter. As mentioned, since the complexity of the calculation is not increased, if k is not fixed, it is kept as parameter in the VEVs. For the actual calculation of the shifts we choose a plus sign in zu and zd in front of the square root as well as the positive root for vu and vd, see Eqs. (44) and (47). The corrections to the flavon superpotential, which induce shifts in the VEVs of the fields with an index u, are given by

i / 16 11 12 9 N

1 I \ ' U rR,U . \ - u jS,U . \ 1 u rT,u . \ ^ u jP,U

> = II +su II +tl II +^ Pl II

The invariants I,R'U read

1 = a 2 (f^ + fW),

R,u__Li,.d „d„i,0u I ,.i,.d „d,,i„0u\

IRR,U = a(fd/2df?U + f2 X d f 2 U)'

R = fdf 0U + fdf°U(nd\2, R = (f?f°°u + f2Ufi°U\(nU\2

R = (f df2°U + fdf°U) (fdfi), R = {fdlXd1^2f°U + f2dX2d^1df2°U)>

= (f»f°°u + ff

iRR,u = (fdf°°u + fdf°U(xdxd),

= (ff + f2Uf1°1

iR,U = (fW + f°dfx^)(f1df:

jR,u TS.U

=(fi f°u+f2uf°^(^2u:

For I^ we find

R = (fM^fi + f°ux°uf1uf°°u)

\(fUf°u\,

RU = nd(Xldfldf°U - X2df2df2°U\,

)(X!X2U), I14,U = nU(XlUflUfl°U + X2Uf2Uf2°U\,

R = nd ((tf\2f°U - (^2d\2f°U\,

I1R6U = 2f2°u + № 2f°u)

\2 /,°u\

h = a \f1 X1 ^2 + f2X2^1 j, I7 = f X2 f1 ^2 + f2 X1 f2 j

^ = af!^ + f2df1d^2°^,

I3S,U = and(Hdv°u - f2d^2°u\,

ISS,U = (\X1d) + \X2d) 2f №),

IS,U = ((X^f!^ + (X2U\2fUV?u\,

IS,U = (f^ + (f2d) 3<p°u), IS°U = /((xD 2^°u - (X2d)2 ^°u)

„ d\3

,d\2 °u\

I5,u=a f D ^+(fDv?

I6S,U = (f + f2dXldf2d^?U\,

\3, „°u\

IS1,U = 2^?U + (X2U) 2^2U)

\2 „°u\

and for I,

jT , u , d 0u I 0u\ jT , u i d s-d > d „ 0u , d ^d > d „0u\

V = a(flflP2 + f2f2Pl )' I7 = Ul^1^2P2 + X2f2fPl )'

I2 = an XPl - X2P2 )' 78 = Ul f2 P2 + X2 f2 Pl j

^ = /((f d)2P0u - (f2d)2P20^' lT'u = ((fld)2f dP0u + ($2d)2f2dP20^'

'd \2 0u

d\2 0u

^ = ^ ((f D 2p0u + (f2u) 2P20^' I T0u = ( (f D 2f "P0u + 2f2uP20u)

\ 2 u 0u

iTu = (f d(x d) 2P20u+fdixD 2p0u), iTiu=dp 0u - x fe^),

■d d\2 _0u

•d (,,d\2 _0u

iT'u = f Kx D 2P20u + fKxD 2P0

Finally, IP,u read

i T2 u = n^flV 0u + x ?f2uP20u)-

IpP'u = a 0u((ff) 2X2u + f) 2X D'

jP,u „0u {{,,u\2u 1 i„u\2i-u \

Ip = a ((x 1)f2+ (X2) f 1)'

I3P'u = a 0uaf ff!,

P'u 0u d d

14 = a axl X2 ,

P,u 0u d d

15 = a af l f 2 '

P = a °u((f0 2X2d + (f2d) 2X0,

jP,u 0 Ip = a

P,u 0u 3 I7 = a a ,

d\2 d\

P = a 0u((x 0 2fl + (X2d) 2f l

d\2 d\

The shifts in the VEVs of the set of fields [f f 2, x 12, H1 2 , nd, a} originate from non-renormalizable terms which are of the form

, /2 l l 4 l 6 7

d = ji^I?'d+E sdISd +E tdITd +E pth

The invariants IR,d are the following lR,f = a 2(f lf°2d + f^f i0d ),

I R = (f lf20d - f2df ?Vnd,

I? = a (f lxuf 0d + f2dX rf20^, I?3,d = (f uf20d + f2uf i0d)(nd)

R = a(f l X2f 0d + f2uX lf20^,

Ifr = (f ux Ulf 0d+f2x2f df20d),

I4/,d = ((f d)2f2 f2d + (f2d)2f !f H, I/5,d = (f lxuflf 0d + f2dX2ufldf20^,

d\2 u 0d

= (f !f20d + f2uf Hf df

I/,d = (f df20dXdx2u + f2f 0dX2dXlu\, I/7'd = nd (x luf df 0d - X2uf2df20d\,

I7/,d = (fdf20dX2dXr + f2df 0dxdx2u\, If8d = nd (X df ff 0d - X2f2 f ^\,

jR,d 11 6

= (f dx df!f 0d + f2dX2df luf20^'

IR,d = (f !f20d + f!f ?d)(x dX2d),

I/R,d = (f df20df df2u + f2f 0dfd2f ?), IR' = n^fldf u f 0 d - f2df2uf

I R,d=nu (x df df r+x2f2f2d^,

IR,d = (f df20df2df lu + flf 00df ldf!), I2Rl,d = nu{{f d) 2f20d + (f2d) 2f 00d)

d\2 0d

i.d\2 , 0d\

I R, = f Uf20d + f2Uf 1d^2d)-

The second set reads

I'S,d = a {f^xd^ + f2X2y^d^,

IS'd = (f uX2dfld^20d + fu2xdflv0d),

lS,d = a tyUUv™ + , = (fhUfxV2 + fdxUHivV),

rs,d _ umU-M I .kuU-M\ jS,d = (^1d/2dfl"^2d + fdxd^vV)>

Ifid = ((/ld + (*2d W Up)1),

4d = (x fx i"^2d^0d + xfxUf (v0d),

^ = a{f USfp 0d + f;^1),

^ = a (f f^P0d + fU!<P2 ),

I5s'd = and{HIpO1 - ^), /6S,d = anu(fdp0d + ^), lS3d = nd (xIxIpO1 - X2dxlP0d),

iS,d = ( fd) 2f I P0d + (f2d) fVO1), Iis4d = nU((xf) 2p0d + (x2d) 2P0d)

d)2 i 0d

d)2 0d

d Od )

and lT,d are given by

jT,d i i Od I ,.i..i yd „ 0d\ jT,d Od i t-d„rd ^ 0d\

A = a Vf 1 S1 P2 + f2$2Pl )' I9 = (x 1 S1 f2 P2 + x2 S2 f 1 P 1 j,

i2 = af fs IpO1 + fdS21p'0^, iTod = (x 1s if2dp0d + xIs; f IpO11), l3T'd = an1 (x fp 0d + xfpO11), T = (x 1f2P0d + xfS^ff 1P 0d ),

l4r'd = and(x1p0d - x1P20^, IT;1 = ((Si1)2f IPO1 + (S2d)2f1P20^,

-dOd )

IT5,d = ndfifUiPid - f2df2UP20^, 1T311 = (SlSfffpO1 + ^^P;01),

IT,d = n^(fl)2P0d + (f2d) p)1),

T = (fU(xf )2P20d + fU(x2d )2p0d), I8['d = (ffxUxfp;01 + f2dx2Ux2dp0d),

1 4 = nd(xUS 1p0d - x IS2 p1)1), T = nd (xfS upO1 - x feV;01), T = nU(xfS IpO1 + x fS2dP20d )•

For IlP,d we find

P = a 0^((ff) 2x1 + (f;d) 2xU), P = a 0da (flfl + fff U)

'„d )2

fP,i1 „Od// d)2f. U , /„d\ 2?-i\ fP,d „Od / d„U 1 „d „u\

Ip = a ((x 1 j S2 + U2) Slj, I6 = a a(x ^2 + x;x J, P = a 01 (f 1 xf f U + ffx if 2), P = a 01 a (S f^ + & U), Ip1 = a 01 (x fSfx U + x;dS idx2U)^

(B. 10)

To actually calculate the shifts of the VEVs we take the parameterization given in Eq. (50) and plug this into the F-terms arising from the corrected superpotential. We then linearize the equations in 8 VEV and 1 ¡A and can derive the following for the shifts of the flavons with index u from the F-terms of the driving fields ^0 2' ^0 2' P02 and a0u

K^W + w^Sv1} - 8v2i)) + - rUx2vd + r^w1

+ {vd) 3e-

+ r5U (w^ 2 vd + r6! (wU)2 vU + vd (zd )2

r U3 fd r 1 51 fiwi

+ v^zU)

r8 - r 4

+ r unvdwdzd + r U2v1w1z1

bu (vusw¡ H wu (5v2 - S^U)) H 1 j r¡e-^x2vd H r2Ue-~

H (vd)3e

d)3 -2n-

eiz fdw"

H r6U(wU)2vu H vd(zd)2e-^Г-H vu(zu)

'7_ '13 fä~'15\fdw"

H r,e- ^ (wd )2vd

'8 - '14

e¡zu f¡w¡

H r"e- — vdwdzd H r¡2vuwuz^ = G.

au(vu8w¡ H wuàv2) H cu{vu8zu2 Hz28vU) H d¡z¡

W - ( ^ )áz¡

и____d

H--je 7 s 2 xv w He 7 xv z'

s2 e"z so —

H s4U(v^ 3 H s5U(vU)3

H vdwdzd e ^ ( s¡ - sUG — ) H vu wuzu ( s7U - s¡1 — 6 1G fd 11 fu

H e^s. vd (wd)2 H s9U vu(wu)2\ = G.

au(vU8wU H wU8vU) H c¡ (v28zU H zu8v2) H "¡z1

11 2ni- u i i "--il: i i

H--ie 7 slxv" w" H e 7 xv"z"

H s5 (vu)3 H vdwdzde-^ (s6U - s¡G

eiz" fdw1

H s4¡e-^ (vd )3

H vuwuzu[ s¡ - s) H s¡e- — vd(wd)2 H s9UvU(wU)2 = G.

fuwu8nu H e^ vu8z¡ - w^U H zu8vu^j H ^xvdzde^ (^t2 - tlf- t2e- (v^3 ( ^^ï - t¡ (vu)3 f ^) H t^ vd (wd)2

H tlvu(wu)2 H t7¡e—vdwdzd H t8¡vuwuzu

„d(„d )2 Ьл-I и ¡ .и ed \ . ,,u(„u)2/ .и .

- v (z) e 7 H J ) H v (z j UK» - t

12 y)} = G. fu

fuwu8nu H e^ v28zU - H zu8v¡ J H j\xvdzde-^ - tlf-

H t^H3( e*!) -1-2 (v2)3 ) H t¡e-^

d { d\2 v" [w"j

H t6U vu(wu)2 H t7Ue- —vdwdzd H t8U vu wuz!

,,d (_ d )2 -1™-( и , tu e"\ i ,K(U2l fu . - v (z) e 7 H hl f) H v(z) ( ^0 -1

(B.12)

(B.13)

(B.14)

(B.15)

(B.16)

ru _ ru

guvu(Sv1 + 5«2) + luWu{SwU + 8wu2) + nuzu{Sz\ + 8z1^ - 2quvu( fW ^

+ ^2(Pu(v")2wu + 2zu) + e-"(v^ x ( P3 - P6

+ pu (wd)2x + pu (z^2x + p^x3 + 2(pue-^ (v^2wd + p"u (wd)2zd) \ = 0. (B.17)

- (vd)2wd

eaz faw" d)2 d

Note that we replaced the mass parameter Mf by the VEV wu. Analogously, we replace the dimensionless coupling mf with the VEV wd. We also frequently use the fact that k is an odd integer in order to simplify the phase factors appearing in the formulae.

Similarly, we can deduce another set of equations from the F-terms of the driving fields f0d2,

v0 2, p0 2 and a0L which gives rise to the shifts in the VEVs of the flavons f d2, Xd2, Hd2, nL and a

bd(vd8wad + wd(8vd - 8vd)) + 1 \rdvux2 + rfe-nrkvdwux + r3denrvuwdx

+ vu(vd)

r4 + e-

d eaeuzdzu

+ va w" wu (e ^Trf" + e-'^Trf) + rfv^w")2

,d,„d,„u(a^„d , ^t „d

fd fu Wf W

d u( d)2

a — u a 1 vuwa

w i - r13e

niW eaz

+ vdzdzu

+ vu(zL) 2

"rQa + e-

r " 1 + e 7 r

a u a nii

+ vawuza e 7

T 1 5 - ' 17

r10 - e

fu wu d\

2nk a ed -a / eaz

+ vuwdzd

a nUk a 3nik euz rf4e 7 - rf9e 7 '

+ r "6e- ^ vawazM = 0 ,

(B. 8)

e- Srbd{vd8wd + wd(8vd - 8vf)) + - j r dvux2 + rdvdwux + r3d e- 27ivuwdx

+ vu(vd)

a 2nUk a a 2nik I eaeuz z

rf e 7 + e 7 r| + rf2e 7 1

fd fu wd wu

d -ntt / e"z - T 13e

+ rdvu(wd )2 + vdzdzu

+ vdwdwu( e-^r6d + r7d)

3nik a ea zUk a I eaz

e"7 Tf + e7 T "0 - T "8 fa - 67 T2% fdwd

+ vu(zd)

a Mk a euz" Ta1 + e 7 T2a1

+ vuwdzd

„d a-„d a3nk I euz

r14e 7 - Ta9e 7

.a „,,u „a

+ vawuza e 7

a / edz

r15 - r17

+ rf6e —vdwdzu\ = 0 ,

(B.19)

ad(vd8wd + u/8vd) + ca{vd8zd2 + zd8v^ - daz'

eav" faw"

8zf + 8nd

■.drJ^r „.,u..,d i „d„.,d.„u

+ — js"e 7 xvuwa + ^dLxv"wu + xvuzd

d„-euz

s3e 7 - e 7 s6

+ xvdzu

d щк( e"z

,d „d e_± fd

fdWd d

+ sd{vd)2 vu + s8d e n]kvuwdzd

+ vdwuzde^f sd - sd3 ed ) + sd0e-nkvdwdzu + s"^ vu (wd)2

+ sd2vdwdwu - sd4e-27]kvu(wd)2' e"z

(B.20)

ad(vdSw" + wdSv") + Cd(vdSz" + zdSvd) - "dz'

e"v fdwa

Szd2 + Snd

+ — js"e 7 xvuwd + s"e 7 xvdwu + xvuz'

da - 2nrk arnrk^( e"z

+ xvdzu

d , d -*ßi e"z S4 + S5

+ s7d e-

2vu + s0" e

+ vdwuzde-^ (s9d - sd3 ^ + s'"0vdwdzu + sd1e-^ vu (wd)2 + sd2e-^ vd wd wu

d , d)U e"z

- s"4e—vu(wd)

- t"e i xvuz'

Vf-w u

Sv2d - z < wd

e-wd> !-1"

f-wu )

(B.21)

- (v")

d e"zd \ d _dп]kí e-z"

n i _ i + t"e i '

+ wuzdvde i

.d .d / e"z t9 - t14

+ td0vdwdzu

+ — jt " e~ xvuzd + t^xvdzu

+ td e^ vu(wd )2 + t8d e nkvdwdw

+ wdzdvu

id „3nk ^d / e"z

zuzdvd e-"V( t " +115

,d e± fd

f-w" = 0,

-1 d2e^vu(zd )2

(B.22)

e^ vdSz" + zdSvd - ^TSwi) - fdwdSnd

+ -1 \t" e-22TLxvUzd

+ t2de-^rxvdzu - t"e^xvV'

e„wd\ d -nk d d[ e"wu +14e i xv"z

+ (v")

vd)2vu

d -si / e"z'

+ wuzdvde-^

dd t9 - t 4

edz" fdw"

d( e"z

+ tie- 2jiJr vu (wd)2 + t8de-П

vdwdwu

+1 "0e-n~vdwdzu

+ w z v

j 3nik j nik ( e-z

td e- — -1 ""e"

td e--nrvu(„d)2 - 11 2e ' v

(zd)2 + z

uzdvdp^itd,td ed z v e ' I 11 3 + 11 5 — fd

(B.23)

2 ( (nk — i cos — A\ V 7

+ cosf^)(pd2 (wd\2zu + p7xzdzu

+ ldwd (Swd + W \

e-(5u2 + V) - 2?d

+ ndzd (5zd + Z\ + J{cos(y) (p2e-^ (vd\2wu + p2w2wuz2 + pd^wdwux\

u\ + cos^n^je-^ vdvu (plwd + p2*\} = 0.

(B.24)

One can infer the generic size of the shifts of the VEVs from these equations. In the case of no accidental cancellation among the various terms present here we expect all of them to be of the order VEV2/A which is eVEV « e2 A for all VEVs being of the order eA with e « X2 « 0.04.

References

[1] Papers in which dihedral groups are used as flavor symmetry, e.g., J. Kubo, A. Mondragon, M. Mondragon, E. Rodriguez-Jauregui, Prog. Theor. Phys. l09 (2003) 795, arXiv:hep-ph/0302l96;

J. Kubo, A. Mondragon, M. Mondragon, E. Rodriguez-Jauregui, Prog. Theor. Phys. ll4 (2005) 287, Erratum; T. Kobayashi, J. Kubo, H. Terao, Phys. Lett. B 568 (2003) 83, arXiv:hep-ph/0303084;

E. Ma, Phys. Rev. D 6l (2000) 0330l2, arXiv:hep-ph/9909249;

S.L. Chen, M. Frigerio, E. Ma, Phys. Rev. D 70 (2004) 073008, arXiv:hep-ph/0404084; S.L. Chen, M. Frigerio, E. Ma, Phys. Rev. D 70 (2004) 079905, Erratum; S. Morisi, M. Picariello, Int. J. Theor. Phys. 45 (2006) l267, arXiv:hep-ph/0505ll3;

F. Caravaglios, S. Morisi, arXiv:hep-ph/05l032l;

F. Caravaglios, S. Morisi, arXiv:hep-ph/0503234;

N. Haba, K. Yoshioka, Nucl. Phys. B 739 (2006) 254, arXiv:hep-ph/05lll08; Y. Koide, Phys. Rev. D 73 (2006) 05790l, arXiv:hep-ph/05092l4; Y. Koide, Eur. Phys. J. C 50 (2007) 809, arXiv:hep-ph/06l2058;

R.N. Mohapatra, S. Nasri, H.B. Yu, Phys. Lett. B 636 (2006) ll4, arXiv:hep-ph/0603020; R.N. Mohapatra, S. Nasri, H.B. Yu, Phys. Lett. B 639 (2006) 3l8, arXiv:hep-ph/0605020;

G. Seidl, arXiv:hep-ph/030l044;

P.H. Frampton, T.W. Kephart, Phys. Rev. D 64 (200l) 086007, arXiv:hep-th/00lll86; C.D. Carone, R.F. Lebed, Phys. Rev. D 60 (l999) 096002, arXiv:hep-ph/9905275; E. Ma, Fizika B l4 (2005) 35, arXiv:hep-ph/0409288; S.L. Chen, E. Ma, Phys. Lett. B 620 (2005) l5l, arXiv:hep-ph/0505064;

C. Hagedorn, M. Lindner, F. Plentinger, Phys. Rev. D 74 (2006) 025007, arXiv:hep-ph/0604265; P. Ko, T. Kobayashi, J.h. Park, S. Raby, Phys. Rev. D 76 (2007) 035005, arXiv:0704.2807 [hep-ph]; P. Ko, T. Kobayashi, J.h. Park, S. Raby, Phys. Rev. D 76 (2007) 05990l, Erratum; J. Kubo, Phys. Lett. B 622 (2005) 303, arXiv:hep-ph/0506043;

Y. Kajiyama, J. Kubo, H. Okada, Phys. Rev. D 75 (2007) 03300l, arXiv:hep-ph/06l0072;

M. Frigerio, S. Kaneko, E. Ma, M. Tanimoto, Phys. Rev. D 7l (2005) 0ll90l, arXiv:hep-ph/0409l87;

P.H. Frampton, A. Rasin, Phys. Lett. B 478 (2000) 424, arXiv:hep-ph/99l0522;

K.S. Babu, J. Kubo, Phys. Rev. D 7l (2005) 056006, arXiv:hep-ph/04ll226;

Y. Kajiyama, E. Itou, J. Kubo, Nucl. Phys. B 743 (2006) 74, arXiv:hep-ph/05ll268;

P.H. Frampton, T.W. Kephart, Phys. Rev. D 5l (l995) l, arXiv:hep-ph/9409324;

P.H. Frampton, O.C.W. Kong, Phys. Rev. Lett. 75 (l995) 78l, arXiv:hep-ph/9502395;

P.H. Frampton, O.C.W. Kong, Phys. Rev. D 53 (l996) 2293, arXiv:hep-ph/95ll343;

P.H. Frampton, O.C.W. Kong, Phys. Rev. Lett. 77 (l996) l699, arXiv:hep-ph/9603372;

M. Frigerio, E. Ma, Phys. Rev. D 76 (2007) 096007, arXiv:0708.0l66 [hep-ph];

M. Mitra, S. Choubey, Phys. Rev. D 78 (2008) ll50l4, arXiv:0806.3254 [hep-ph].

[2] A. Blum, C. Hagedorn, M. Lindner, Phys. Rev. D 77 (2008) 076004, arXiv:0709.3450 [hep-ph].

[3] C.S. Lam, Phys. Lett. B 656 (2007) l93, arXiv:0708.3665 [hep-ph].

[4] A. Blum, C. Hagedorn, A. Hohenegger, JHEP 0803 (2008) 070, arXiv:07l0.506l [hep-ph].

[5] R. Gatto, G. Sartori, M. Tonin, Phys. Lett. B 28 (l968) l28.

[6] C.D. Froggatt, H.B. Nielsen, Nucl. Phys. B l47 (l979) 277.

[7] G. Altarelli, F. Feruglio, Nucl. Phys. B 720 (2005) 64, arXiv:hep-ph/0504l65; G. Altarelli, F. Feruglio, Nucl. Phys. B 74l (2006) 2l5, arXiv:hep-ph/05l2l03;

G. Altarelli, F. Feruglio, Y. Lin, Nucl. Phys. B 775 (2007) 3l, arXiv:hep-ph/06l0l65;

F. Feruglio, C. Hagedorn, Y. Lin, L. Merlo, Nucl. Phys. B 775 (2007) l20, arXiv:hep-ph/0702l94;

G. Altarelli, F. Feruglio, C. Hagedorn, JHEP 0803 (2008) 052, arXiv:0802.0090 [hep-ph]; Y. Lin, arXiv:0804.2867 [hep-ph].

[8] A.M. Gavrilik, Nucl. Phys. B (Proc. Suppl.) l02 (200l) 298, arXiv:hep-ph/0l03325.

[9] Y.B. Zeldovich, I.Y. Kobzarev, L.B. Okun, Zh. Eksp. Teor. Fiz. 67 (l974) 3, Sov. Phys. JETP 40 (l974) l.

[10] F. Bazzocchi, L. Merlo, S. Morisi, arXiv:090l.2086 [hep-ph].

[11] J.S. Lomont, Applications of Finite Groups, Academic Press, l959, p. 346; P.E. Desmier, R.T. Sharp, J. Math. Phys. 20 (l979) 74;

J. Patera, R.T. Sharp, P. Winternitz, J. Math. Phys. l9 (l978) 2362.

[12] P.H. Frampton, T.W. Kephart, Int. J. Mod. Phys. A l0 (l995) 4689, arXiv:hep-ph/9409330.

[13] C. Amsler, et al., Particle Data Group, Phys. Lett. B 667 (2008) l.

[14] C. Jarlskog, Phys. Rev. Lett. 55 (l985) l039.

[15] See second reference in [7].

[16] N. Haba, A. Watanabe, K. Yoshioka, Phys. Rev. Lett. 97 (2006) 04l60l, arXiv:hep-ph/0603ll6; T. Kobayashi, Y. Omura, K. Yoshioka, Phys. Rev. D 78 (2008) ll5006, arXiv:0809.3064 [hep-ph];

G. Seidl, arXiv:08ll.3775 [hep-ph].

[17] W. Grimus, L. Lavoura, JHEP 0508 (2005) 0l3, arXiv:hep-ph/0504l53.

[18] W. Grimus, L. Lavoura, Phys. Lett. B 572 (2003) l89, arXiv:hep-ph/0305046.

[19] H. Ishimori, T. Kobayashi, H. Ohki, Y. Omura, R. Takahashi, M. Tanimoto, Phys. Lett. B 662 (2008) l78, arXiv:0802.23l0 [hep-ph];

H. Ishimori, T. Kobayashi, H. Ohki, Y. Omura, R. Takahashi, M. Tanimoto, Phys. Rev. D 77 (2008) ll5005, arXiv:0803.0796 [hep-ph];

A. Adulpravitchai, A. Blum, C. Hagedorn, arXiv:08l2.3799 [hep-ph].

[20] T. Araki, Prog. Theor. Phys. ll7 (2007) lll9, arXiv:hep-ph/06l2306;

T. Araki, T. Kobayashi, J. Kubo, S. Ramos-Sanchez, M. Ratz, P.K.S. Vaudrevange, Nucl. Phys. B 805 (2008) l24, arXiv:0805.0207 [hep-th];

C. Luhn, P. Ramond, JHEP 0807 (2008) 085, arXiv:0805.l736 [hep-ph]; C. Luhn, Phys. Lett. B 670 (2009) 390, arXiv:0807.l749 [hep-ph].

[21] T. Kobayashi, S. Raby, R.J. Zhang, Nucl. Phys. B 704 (2005) 3, arXiv:hep-ph/0409098;

T. Kobayashi, H.P. Nilles, F. Ploger, S. Raby, M. Ratz, Nucl. Phys. B 768 (2007) l35, arXiv:hep-ph/06ll020; G. Altarelli, F. Feruglio, Y. Lin, Nucl. Phys. B 775 (2007) 3l, arXiv:hep-ph/06l0l65.