Minimal mixing of quarks and leptons in the SU(3) theory of flavour

Nuclear Physics B 626 (2002) 307–343

www.elsevier.com/locate/npe

Minimal mixing of quarks and leptonsin theSU(3) theory of flavour

J.L. Chkareulia, C.D. Froggattb, H.B. Nielsenc

a Institute of Physics, Georgian Academy of Sciences, 380077 Tbilisi, Georgiab Department of Physics and Astronomy, Glasgow University, Glasgow G12 8QQ, Scotland, UK

c Niels Bohr Institute, Blegdamsvej 17-21, DK 2100 Copenhagen, Denmark

Received 31 October 2001; accepted 16 January 2002

Abstract

We argue that flavour mixing, both in the quark and lepton sector, follows the minimal mixingpattern, according to which the whole of this mixing is basically determined by the physical massgeneration for the first family of fermions. So, in the chiral symmetry limit when the masses ofthe lightest (u and d) quarks vanish, all the quark mixing angles vanish. This minimal patternis shown to fit extremely well the already established CKM matrix elements and to give fairlydistinctive predictions for the as yet poorly known ones. Remarkably, together with generically smallquark mixing, it also leads to large neutrino mixing, provided that neutrino masses appear throughthe ordinary “see-saw” mechanism. It is natural to think that this minimal flavour mixing patternpresupposes some underlying family symmetry, treating families of quarks and leptons in a specialway. Indeed, we have found a local chiralSU(3)F family symmetry model which leads, through itsdominant symmetry breaking vacuum configuration, to a natural realization of the proposed minimalmechanism. It can also naturally generate the quark and lepton mass hierarchies. Furthermorespontaneous CP violation is possible, leading to a maximal CP violating phaseδ = π/2, in theframework of the MSSM extended by a high-scaleSU(3)F chiral family symmetry. 2002 ElsevierScience B.V. All rights reserved.

PACS:12.15.Ff; 11.30.Hv

1. Introduction

The flavour mixing of quarks and leptons is certainly one of the major problems thatpresently confront particle physics (for a recent review see [1]). Many attempts have been

E-mail address:[email protected] (C.D. Froggatt).

0550-3213/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0550-3213(02)00032-9

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308 J.L. Chkareuli et al. / Nuclear Physics B 626 (2002) 307–343

made to interpret the pattern of this mixing in terms of various family symmetries—discreteor continuous, global or local. Among them, the abelianU(1) [2–4] and/or non-abelianSU(2) [5–7] andSU(3) [8,9] family symmetries seem the most promising. They providesome guidance to the expected hierarchy between the elements of the quark–lepton massmatrices and to the presence of texture zeros [10] in them, leading to relationships betweenthe mass and mixing parameters. In the framework of the supersymmetric Standard Model,such a family symmetry should at the same time provide an almost uniform mass spectrumfor the superpartners, with a high degree of flavour conservation [11], that makes itsexistence even more necessary in the SUSY case.

Despite some progress in understanding the flavour mixing problem, one has the uneasyfeeling that, in many cases, the problem seems just to be transferred from one place toanother. The peculiar quark–lepton mass hierarchy is replaced by a peculiar set ofU(1)flavour charges or a peculiar hierarchy of Higgs field VEVs in the non-abelian symmetrycase. As a result there are not so many distinctive and testable generic predictions, strictlyrelating the flavour mixing angles to the quark–lepton masses.

A commonly accepted framework for discussing the flavour problem is based on thepicture that, in the absence of flavour mixing, only the particles belonging to the thirdgenerationt , b andτ have non-zero masses. All other (physical) masses and the mixingangles then appear as a result of the tree-level mixings of families, related to someunderlying family symmetry breaking. They might be proportional to powers of somesmall parameter, which are determined by the dimensions of the family symmetry allowedoperators that generate the effective (diagonal and off-diagonal) Yukawa couplings for thelighter families in the framework of the (ordinary or supersymmetric) Standard Model.

Recently, a new mechanism of flavour mixing, which we call Lightest Family MassGeneration (LFMG), was proposed [12] (see the papers in Ref. [13] for further discussion).According to LFMG the whole of flavour mixing for quarks is basically determined by themechanism responsible for generating the physical masses of theupanddownquarks,mu

andmd , respectively. So, in the chiral symmetry limit whenmu andmd vanish, all thequark mixing angles vanish. This, in fact minimal, flavour mixing model was found to fitextremely well the already established CKM matrix elements and give fairly distinctivepredictions for the yet poorly known ones.

By its nature, the LFMG mechanism is not dependent on the number of quark–leptonfamilies nor on any “vertical” symmetry structure, unifying quarks and leptons inside afamily as in Grand Unified Theories (GUTs). The basic LFMG leads to mass-matriceswhose non-zero diagonal and off-diagonal elements for theN family case are related asfollows:

(1)M22 :M33 : · · · :MNN = |M12|2 : |M23|2 : · · · : |MN−1 N |2.This shows clearly that the heavier families (4th, 5th,. . . ), had they existed, would be moreand more decoupled from the lighter ones and from each other. Indeed, this behaviour canto some extent actually be seen in the presently observed CKM matrix elements involvingthe third family quarkst andb.

The above condition (1) is suggestive of some underlying flavour symmetry, probablynon-abelian rather than abelian, treating families in a special way. Indeed, for the observedthree-family case, we show that the local familySU(3)F symmetry [8] treating the

J.L. Chkareuli et al. / Nuclear Physics B 626 (2002) 307–343 309

quark and lepton families as fundamental chiral triplets, with an appropriate dominantsymmetry breaking vacuum configuration, could lead to a natural realization of the LFMGmechanism. Other outstanding problems in flavour physics are also addressed withinthe framework of the Minimal Supersymmetric Standard Model (MSSM) appropriatelyextended by a high-scaleSU(3)F chiral family symmetry: the quark and lepton masshierarchies, neutrino masses and oscillations and the possibility of spontaneous CPviolation.

The paper is organized in the following way. In Section 2 we give a general expositionof the LFMG mechanism involving two alternative scenarios. In Section 3 we present theSU(3)F theory of flavour, treating the quark and lepton families as fundamental chiraltriplets. We show that there is a dominant symmetry breaking vacuum configuration relatedwith the basic horizontal Higgs supermultiplets, triplets and sextets ofSU(3)F , whichleads to a natural realization of the LFMG mechanism. While we mainly consider thesupersymmetric case—just the MSSM—most of our argumentation remains valid for theStandard Model (and ordinary GUTs) as well. Finally, in Section 4, we give our summary.

2. Minimal flavour mixing: the model

2.1. Quark mixing

The proposed flavour mixing mechanism, driven solely by the generation of the lightestfamily mass, could actually be realized in two generic ways.

The first basic alternative I is when the lightest family mass (mu or md ) appears as aresult of the complex flavour mixing of all three families. It “runs along the main diagonal”of the corresponding 3× 3 mass matrixM, from the basic dominant elementM33 to theelementM22 (via a rotation in the (2-3) sub-block ofM) and then to the primordiallytexture zero elementM11 (via a rotation in the (1-2) sub-block). The direct flavour mixingof the first and third families of quarks and leptons is supposed to be absent or negligiblysmall inM.

The second alternative II, on the contrary, presupposes direct flavour mixing of just thefirst and third families. There is no involvement of the second family in the mixing. Inthis case, the lightest mass appears in the primordially texture zeroM11 element “walkinground the corner” (via a rotation in the (1-3) sub-block of the mass matrixM). Certainly,this second version of the LFMG mechanism cannot be used for both the up and the downquark families simultaneously, since mixing with the second family members is a basicpart of the CKM phenomenology (Cabibbo mixing, non-zeroVcb element, CP violation).However, the alternative II could work for the up quark family provided that the downquarks follow the alternative I.

So, there are two possible scenarios for the LFMG mechanism to be considered.

2.1.1. Scenario A: “both of the lightest quark massesmu andmd running along thediagonal”

We propose that both mass matrices for the Dirac fermions—the up quarks (U = u, c, t)and the down quarks (D = d, s, b)—in the Standard Model, or supersymmetric Standard


Model, are Hermitian with three texture zeros of the following form:

(2)Mi =( 0 ai 0a∗i Ai bi0 b∗

i Bi

), i =U,D.

It is, of course, necessary to assume some hierarchy between the elements, which wetake to be:Bi � Ai ∼ |bi| � |ai |. We derive this hierarchical structure from ourSU(3)Ffamily symmetry model in Section 3.3. The zeros in the(Mi)11 elements correspond to our,and the commonly accepted, conjecture that the lightest family masses appear as a directresult of flavour mixings. The zeros in(Mi)13 mean that only minimal “nearest neighbour”interactions occur, giving a tridiagonal matrix structure.

Now our main hypothesis, that the second and third family diagonal mass matrixelements are practically the same in the gauge and physical quark–lepton bases, meansthat:

(3)Bi = (mt ,mb)+ δi, Ai = (mc,ms)+ δ′i .

The componentsδi andδ′i are supposed to be much less than the masses of the particles in

the next lightest family, meaning the second and first families, respectively:

(4)|δi | � (mc,ms),∣∣δ′i

∣∣� (mu,md).

Since the trace and determinant of the Hermitian matrixMi gives the sum and product ofits eigenvalues, it follows that

(5)δi −(mu,md),

while theδ′i are vanishingly small and can be neglected in further considerations.

It may easily be shown that our hypothesis and related Eqs. (3)–(5) are entirelyequivalent to the condition that the diagonal elements (Ai,Bi ), of the mass matricesMi ,are proportional to the modulus square of the off-diagonal elements (ai, bi ):

(6)Ai

Bi=∣∣∣∣aibi

∣∣∣∣2

, i =U,D.

For the moment we leave aside the question of deriving this proportionality condition,Eq. (6), from some underlying theory beyond the Standard Model (see Section 3)and proceed to calculate expressions for all the elements of the matricesMi and thecorresponding CKM quark mixing matrix, in terms of the physical masses.

Using the conservation of the trace, determinant and sum of principal minors ofthe Hermitian matricesMi under unitary transformations, we are led to a completedetermination of the moduli of all their elements. The results can be expressed to highaccuracy as follows:

(7)Ai = (mc,ms), Bi = (mt −mu,mb −md),

(8)|ai | =(√

mumc,√mdms

),

(9)|bi | =(√

mumt ,√mdmb

).


As to the CKM matrixV , we must first choose a parameterisation appropriate to our pictureof flavour mixing. Among many possible ones, the original Euler parameterisation recentlyadvocated [1] is most convenient:

(10)V =(

cU sU 0−sU cU 0

0 0 1

)(e−iφ 0 0

0 c s

0 −s c

)(cD −sD 0sD cD 00 0 1

)

(11)=(sU sDc+ cUcDe

−iφ sU cDc− cU sDe−iφ sU s

cU sDc− sUcDe−iφ cUcDc+ sU sDe

−iφ cU s

−sDs −cDs c

).

HeresU,D ≡ sinθU,D andcU,D ≡ cosθU,D parameterise simple rotationsRU,D12 between

the first and second families for the up and down quarks, respectively, whiles ≡ sinθand c ≡ cosθ parameterise a rotation between the second and third families. Thisrepresentation ofV takes into account the observed hierarchical structure of the quarkmasses and mixing angles. The CP violating phase is connected directly to the first andsecond families alone.

The quark mass matricesMU andMD are diagonalised by unitary transformationswhich can be written in the form:

(12)VU =RU12R

U23Φ

U, VD =RD12R

D23Φ

D,

whereΦU,D are phase matrices, depending on the phases of the off-diagonal elementsai = |ai |eiαi andbi = |bi |eiβi :

(13)ΦU =(eiαU 0 0

0 1 00 0 e−iβU

), ΦD =

(eiαD 0 0

0 1 00 0 e−iβD

).

The CKM matrix is defined by

(14)V = VUV†D =RU

12RU23Φ

U(ΦD

)∗(RD

23

)−1(RD

12

)−1

and, after a suitable re-phasing of the quark fields, we can use the representation

(15)RU23Φ

U(ΦD

)∗(RD

23

)−1 =(e−iφ 0 0

0 c s

0 −s c

).

The rotation matricesRU,D23 andRU,D

12 for our mass matrices are readily calculated and theCKM matrix expressed in terms of quark mass ratios

(16)sU =√mu

mc

, sD =√md

ms

,

(17)s =∣∣∣∣√md

mb

− eiγ√mu

mt

∣∣∣∣and two phasesφ = αD − αU andγ = βD − βU .

It follows that the Cabibbo mixing is given by the well-known Fritzsch formula [14]

(18)|Vus | ∣∣sU − sDe

−iφ∣∣= ∣∣∣∣√md

ms

− e−iφ√mu

mc

∣∣∣∣,


which fits the experimental value well, provided that the CP-violating phaseφ is requiredto be close toπ/2. So, in the following, we shall assume maximal CP violation in the formφ = π/2, as is suggested by spontaneous CP violation in the framework ofSU(3)F familysymmetry (see Section 3.3). The other phaseγ appearing inVcb andVub,

(19)|Vcb| s, |Vub| = sU s,

can be rather arbitrary, since the contribution√mu/mt to s is relatively small, even

compared with the uncertainties coming from the light quark masses themselves. Thisleads to our most interesting prediction (with the mass ratios calculated at the electroweakscale [15]):

(20)|Vcb| √md

mb

= 0.038± 0.007

in good agreement with the current data|Vcb| = 0.039± 0.003 [16]. For definiteness weshall assume the phaseγ in s, see Eq. (17), to be aligned with the CP violating phaseφ,again as suggested bySU(3)F family symmetry, and takeγ = π/2. This has the effect ofreducing the uncertainty in our prediction Eq. (20) from 0.007 to 0.004. Another predictionfor the ratio:

(21)

∣∣∣∣VubVcb

∣∣∣∣=√mu

mc

is quite general for models with “nearest-neighbour” mixing [1].

2.1.2. Scenario B: “up quark massmu walking around the corner, while down quarkmassmd runs along the diagonal”

Now the mass matrices for the down quarksMD and charged leptonsME are supposedto have the same form as in Eq. (2), while the Hermitian mass matrix for the up quarks istaken to be:

(22)MU =( 0 0 cU

0 AU 0c∗U 0 BU

).

All the elements ofMU can again be readily determined in terms of the physical massesas:

(23)AU =mc, BU =mt −mu, |cU | = √mumt .

The quark mass matrices are diagonalised again by unitary transformations as in Eq. (12),provided that the matrixVU is changed to

(24)VU =RU13Φ

U,

where the (1-3) plane rotation of theu andt quarks and the phase matrixΦU (dependingon the phase of the elementcU = |cU |eiαU ) are parameterised in the following way:

(25)RU13 =

(c13 0 s130 1 0

−s13 0 c13

), ΦU =

(eiαU 0 0

0 1 00 0 1

).

Heres13 ≡ sinθ13 andc13 ≡ cosθ13.


The structure of the CKM matrix now differs from that of Eq. (14) as it contains thedirect (1-3) plane rotation for the up quarks:

(26)V = VUV†D =RU

13ΦU(ΦD

)∗(RD

23

)−1(RD

12

)−1,

although the phases and rotations associated with the down quarks are left the same asbefore. This natural parameterisation is now quite close to the standard one [16]. Theproper mixing angles and CP violating phase (after a suitable re-phasing of the c quark,c → ce−iβD ) are given by the simple and compact formulae:

(27)|Vus | s12 =√md

ms

, |Vcb| s23 =√md

mb

, |Vub| s13 =√mu

mt

and

(28)δ = αU − αD − βD.

While the values of|Vus | and|Vcb| are practically the same as in scenario A and in goodagreement with experiment, a new prediction for|Vub| (not depending on the value of theCP violating phase) should allow experiment to differentiate between the two scenarios inthe near future.

2.1.3. The CKM matrixOur numerical results for both versions of our model, with a maximal CP violating

phase (see discussion in Section 3.3.6), are summarized in the following CKM matrix:

(29)VCKM =

0.975(5) 0.222(4) 0.0023(5) A0.0036(6) B

0.222(4) 0.975(8) 0.038(4)0.0084(18) 0.038(4) 0.999(5)

.

The uncertainties in brackets are largely given by the uncertainties in the quark masses.There is clearly a real and testable difference between scenarios A and B given by the valueof theVub element. Agreement with the experimental values of the already known CKMmatrix elements [16] looks quite impressive. The distinctive predictions for the presentlyrelatively poorly knownVub andVtd elements1 should be tested in the near future, byexperimental data from the B-factories.

2.2. Lepton sector

The lepton mixing matrix is defined analogously to the CKM matrix:

(30)U =UνUE†,

where the indicesν andE stand forν = (νe, νµ, ντ ) andE = (e,µ, τ ). Our model predictsthe charged lepton mixing angles in the matrixUE with high accuracy to be:

(31)sinθeµ =√me

mµ

, sinθµτ =√me

mτ

, sinθeτ 0,

1 Present data from CLEO [17] and LEP [18] favour scenario B.


provided that the charged lepton masses follow alternative I, along with the down quarksones, or

(32)sinθeµ = 0, sinθµτ = 0, sinθeτ =√me

mτ

,

if they follow alternative II, like the up quarks. We consider only alternative I for thecharged leptons in ourSU(3)F family symmetry model, since the alternative II for themwould lead in general to the same hierarchy between their masses as appears for the upquarks, which is experimentally excluded.

However, in both cases, these small charged lepton mixing angles will not markedlyeffect atmospheric neutrino oscillations [19], which appear to require essentially maximalmixing sin2 2θatm 1. It follows then that the large neutrino mixing responsible foratmospheric neutrino oscillations should mainly come from theUν matrix associatedwith the neutrino mass matrix in (30). This requires a different mass matrix texturefor the neutrinos compared to the charged fermions; in this connection a number ofinteresting models have been suggested [21], using different mechanisms for neutrinomass generation. Remarkably, there appears to be no need in our case for some differentmechanism to generate the observed mixing pattern of neutrinos: they can get physicalmasses and mixings via the usual “see-saw” mechanism [20], using the proposed LFMGmechanism for their primary Dirac and Majorana masses.

Let us consider this pattern in more detail. According to the “see-saw” mechanism theeffective mass-matrixMν for physical neutrinos has the form [20]

(33)Mν = −MTNM

−1NNMN,

whereMN is their Dirac mass matrix, whileMNN is the Majorana mass matrix oftheir right-handed components. Therefore, one must first choose which of the twoalternatives I or II works for the neutrino mass-matrix (33), particularly for its Dirac andMajorana ingredients,MN andMNN , respectively. There are in fact four possible cases:(i) alternative I works for both the matricesMN andMNN ; (ii) alternative I works forMN ,while alternative II works forMNN ; (iii) alternative I works forMNN , while alternative IIworks forMN ; (iv) alternative II works for both the matricesMN andMNN . It is easy toconfirm that the cases (iii) and (iv) are completely excluded, since they do not lead to anysignificantνµ − ντ mixing as is required by the SuperKamiokande data [19]. However, thefirst two cases, (i) and (ii), happen to be of actual experimental interest when an appropriatehierarchy is proposed between the matrix elements in the matricesMN andMNN . So, wealso have two possible scenarios in the lepton sector.

2.2.1. Scenario A∗: “all the lightest lepton Dirac and Majorana masses running alongthe diagonal”

Let us enlarge first on the case (i) considered recently in Ref. [22]. The eigenvalues ofthe neutrino Dirac mass matrixMN are taken to have a hierarchy similar to that for thecharged leptons (and down quarks)

(34)MN3 :MN2 :MN1 1 : y2 : y4, y ≈ 0.1


and the eigenvalues of the Majorana mass matrixMNN are taken to have a strongerhierarchy

(35)MNN3 :MNN2 :MNN1 1 : y4 : y6.

One then readily determines the general LFMG matricesMN andMNN to be of the type

(36)MN MN3

( 0 αy3 0αy3 y2 αy2

0 αy2 1

)

and

(37)MNN MNN3

( 0 βy5 0βy5 y4 βy3

0 βy3 1

),

where, for both the order-one parametersα andβ contained in the matricesMN andMNN ,we take an extra condition of the type

(38)|∆− 1| � y2 (∆≡ α,β)

according to which they are supposed to be equal to unity with a few percent accuracy.Substitution in the see-saw formula (33) generates an effective physical neutrino massmatrixMν of the form:

(39)Mν − M2N3

MNN3

0 y 0y 1+ (

y − y2)2 1− (

y − y2)

0 1− (y − y2

)1

.

The physical neutrino masses are then given with an accuracy ofO(y) taken by:

mν1 (

1

2−

√3

2

)M2

N3

MNN3y,

(40)mν2 (

1

2+

√3

2

)M2

N3

MNN3y, mν3 (2− y)

M2N3

MNN3.

Taking the Dirac mass parameterMN3 to approximately equal the top quark massmt ,the Majorana mass parameter can be determined to beMNN3 1 × 1015 GeV fromthe experimentally observed value of*m2

atm 3 × 10−3 eV2 [19] (one may neglect thechanges in the lepton masses and mixings coming from running from the top mass scale tothe MeV scale).

The mass matrix (39) gives essentially maximalνµ–ντ mixing in agreement with theatmospheric neutrino data [19]. Also it is in agreement, while marginal, with the knownLMA (large mixing angle MSW) fit to the solar neutrino experiments [23]. The standardtwo flavour atmospheric and solar neutrino oscillation parameters are expressed in termsof the leptonic CKM matrixU as follows:

sin2 2θsun= 4|Ue1|2|Ue2|2, sin2 2θatm= 4|Uµ3|2(1− |Uµ3|2

),

(41)*m2sun=m2

ν2 −m2ν1, *m2

atm=m2ν3 −m2

ν2


assuming the approximate decoupling condition|Ue3|2 � 1 (see below). The predictedvalues of these parameters in scenario A∗ are [22]:

sin2 2θatm 1, sin2 2θsun 2

3, Ue3 1

2√

2y,

(42)*m2

sun

*m2atm

√

3

4y2

to be well compared with the experimentally allowed regions [19,23] corresponding to theLMA solution for the solar neutrino problem

0.82� sin2 2θatm� 1, 0.65� sin2 2θsun� 1,

(43)|Ue3|2 � 0.05, 3× 10−3 � *m2sun

*m2atm

� 5× 10−2.

2.2.2. Scenario B∗: “the lightest Majorana mass walking around the corner, while thelightest Dirac masses run along the diagonal”

We now turn to case (ii), which appears to lead to another case of neutrino oscillationswhere together with essentially maximal mixing of atmospheric neutrinos the small mixingangle solution MSW (SMA) naturally appears for solar neutrinos. We again take thealternative I form (36) for the Dirac mass matrixMN , while for the Majorana mass matrixMNN we take the general alternative II form

(44)MNN MNN3

( 0 0 βyq

0 yp 0βyq 0 1

),

with an arbitrary eigenvalue hierarchy of the type

(45)MNN3 :MNN2 :MNN1 1 : yp : y2q,

wherep and q are positive integers. Again, for the parametersα and β in the mass-matricesMN andMNN , the extra condition (38) is assumed. The see-saw formula (33)then generates an effective physical neutrino mass matrixMν which for anyp � 2q − 1,q > 1 has a particularly simple form (inessential higher order terms are omitted)

(46)Mν − M2N3

MNN3y4−p

(y2 y y

y 1− yp−2q+2 1+ yp−q−1

y 1+ yp−q−1 1

).

One can see that the matrix (46) automatically leads to the large (maximal)νµ− ντ mixingand smallνe −νµ mixing for any hierarchy in the Majorana mass matrix (44) satisfying theconditionp � 2q − 1, q > 1 mentioned above. Taking, for an example,p = 5 andq = 3one can find for neutrino masses the values

(47)mν1 M2N3

MNN3

y2

3, mν2 −3

2

M2N3

MNN3, mν3 2

M2N3

MNN3y−1,


while the following predictions for the two flavour oscillation parameters naturally appear

sin2 2θatm 1, sin2 2θsun 2

9y2, Ue3 1√

2y,

(48)*m2

sun

*m2atm

9

16y2.

Again takingy ≈ 0.1, our predictions (48) turn out to be inside of the experimentallyallowed intervals [23] for the SMA solution for solar neutrino oscillations:

0.82� sin2 2θatm� 1, 10−3 � sin2 2θsun� 10−2,

(49)|Ue3|2 � 0.05, 5× 10−4 � *m2sun

*m2atm

� 9× 10−3.

Note that in contrast to the LMA case (42), one must include now even the smallcontributions stemming from the charged lepton sector (see Eq. (31)) into the solar neutrinooscillations.

So, one can see that the LFMG mechanism works quite successfully in the lepton sectoras well as in the quark sector. Remarkably, the same mechanism results simultaneouslyin small quark mixing and large lepton mixing (in scenario B∗ for the second andthird families). On the other hand, there could be different sources of the large neutrinomixing. An attractive alternative method for generating large neutrino mixing within thesupersymmetric Standard Model is viaR-parity violating interactions, which can giveradiatively induced neutrino masses and mixing angles in just the area required by thecurrent observational data [24,25].

3. The SU(3)F theory of flavour mixing

3.1. Motivation

The choice of a local chiralSU(3)F symmetry [8] as the underlying flavour symmetrybeyond the Standard Model is based on the following three “pillars”:

(i) It provides a natural explanation of the number three of observed quark–leptonfamilies, correlated with three species of massless or light (mν < MZ/2) neutrinoscontributing to the invisibleZ boson partial decay width [16];

(ii) Its local nature could be expected by analogy with the other fundamental symmetriesof the Standard Model, such as weak isospin symmetrySU(2)W or colour symmetrySU(3)C . A stronger motivation follows from superstrings (at least it is one of the mostimportant model-independent results derived from perturbative string theory); there are noglobal non-abelian symmetries in the 4D string models (for a review see [26]);

(iii) Any family symmetry should be completely broken at low energies in order toconform with reality. This symmetry should be chiral, rather than a vectorlike one underwhich left-handed and right-handed fermions have the same group-theoretical features.A vectorlike symmetry would not provide any mass protection and would give degeneraterather than hierarchical quark and lepton family mass spectra. Interestingly, both known


examples of local vectorlike symmetries, electromagneticU(1)EM and colourSU(3)C ,appear to be exact symmetries. Also, in standard GUT models fermions and antifermionscan lie in the same irreducible representation; so a GUT⊗ SU(3)H model necessarily hasa chiral family symmetry.

So, if one takes these naturality criteria seriously, all the candidates for flavour symmetrycan be excluded except for local chiralSU(3)F symmetry. The chiralU(1) symmetries[2–4] do not satisfy the criterion (i) and are in fact applicable to any number of quark–lepton families. Also, the chiralSU(2) symmetry can contain, besides two light familiestreated as its doublets, any number of additional (singlets or new doublets ofSU(2))families. All the global non-abelian symmetries are excluded by criterion (ii), while thevectorlike ones are excluded by the last criterion (iii).

Among the applications of theSU(3)F symmetry, the most interesting ones are thedescription of the quark and lepton masses and mixings in the Standard Model andGUTs [8], neutrino masses and oscillations [27] and rare processes [28]. Remarkably,the SU(3)F invariant Yukawa coupling are always accompanied by an accidental globalchiralU(1) symmetry [28,29], which can be identified with the Peccei–Quinn symmetry[30] provided it is not explicitly broken in the Higgs sector, thus giving a solution to thestrong CP problem. In the SUSY context [31], theSU(3)F model leads to a special relationbetween fermion masses and the soft SUSY breaking terms at the GUT scale, so that allthe dangerous supersymmetric flavour-changing processes are naturally suppressed [32].

Another sector of applications is related with a new type of topological defect—non-abelian cosmic strings appearing during the spontaneous breaking of theSU(3)F—considered as a possible candidate for the cold dark matter in the Universe [33]. And thelast point worthy of note is that the local chiralSU(3)F symmetry has been applied toGUTs not only in a direct product form, such asSU(5)⊗ SU(3)F [8], SO(10)⊗ SU(3)F[34] or E(6) ⊗ SU(3)F [34], but also as a subgroup of the family unifiedSU(8) GUTmodel [35].

3.2. The matter and Higgs supermultiplets

In the MSSM extended by the local chiralSU(3)F symmetry the quark and leptonsuperfields are supposed to beSU(3)F chiral triplets, so that their left-handed (weak-doublet) components

Lαf =[(

UL

DL

)α

,

(NL

EL

)α

]

(50)=[((

u

d

)L

,

(c

s

)L

,

(t

b

)L

),

((νee

)L

,

(νµµ

)L

,

(νττ

)L

)],

are triplets ofSU(3)F , while their right-handed (weak-singlet) components

Rαuf = [

UαR,N

αR

]= [(u, c, t)R, (Ne,Nµ,Nτ )R

],

(51)Rαdf = [

DαR,E

αR

]= [(d, s, b)R, (e,µ, τ )R

],

are anti-triplets (or vice versa). For completeness we have included the right-handedneutrino superfieldsNα

R as well. Hereα is a family symmetry index (α = 1,2,3), whilethe indexf simply stands for the type of basic fermion, quark (f = q) or lepton (f = l).


The matter sector also includes some set of right-handed states, which are neededto cancel all theSU(3)F triangle anomalies. While being singlets under the StandardModelSU(3)C ⊗ SU(2)W ⊗U(1)Y gauge symmetry, these states can be chosen as sixteenright-handed tripletsrnα (n= 1, . . . ,16) or some other combination ofSU(3)F multipletsproperly compensating the triangle anomalies. All of them receive heavy masses of orderthe family symmetry breaking scaleMF and are practically decoupled from the low-energyparticle spectrum. In fact, they look like right-handed heavy neutrinos and, as such, cancontribute to the ordinary light neutrino masses and mixings via the “see-saw” mechanism[20] (see below).

In addition to the standard weak-doublet scalar supermultipletsH and �H of the MSSM,the Higgs sector contains theSU(3)F symmetry-breaking horizontal scalars. These scalarsare anti-triplets̄ηi and sextetsχj of SU(3)F (indicesi andj number the scalar multiplets,i = 1,2, . . . , j = 1,2, . . .), which transform underSU(3)F like matter bi-linears so as togive fermion masses through the symmetry-allowed general effective Yukawa couplings:

WY = L̄αf R

βufHu

[Aiuf

χi{αβ}MF

+Biuf

η̄i[αβ]MF

]+ L̄α

f Rβdf

�Hd

[Aidf

χi{αβ}MF

+Bidf

η̄i[αβ]MF

]

(52)+NαRN

βR

[AiNNχ

i{αβ}],

for the up and down fermions, quarks (f = q) and leptons (f = l). Also Majoranacouplings are included for the right-handed neutrinos, where only the symmetrical partsare allowed to appear. The constantsAi

uf,df,N and Biuf,df,N (Ai

uq ≡ AiU , A

iul ≡ Ai

N

and so on) are the dimensionless Yukawa constants associated with the symmetric andanti-symmetric couplings in family space, respectively (we have here used the anti-symmetric feature of triplets inSU(3): η̄[αβ] ≡ εαβγ η̄

γ ). These couplings normallyappear via the well-known see-saw type mechanism [2,8] due to the exchange of aspecial set of heavy (of order the flavour scaleMF ) vectorlike fermions or directlythrough the gravitational interactions, being suppressed in the latter case by inversepowers of the Planck scaleMPl [36]. One can even make the Yukawa couplings (52)renormalizable, by introducing additional scalars with both electroweak and horizontalindices. The VEVs of the horizontal scalarsη̄i andχi , taken in general as large asMF

(or MPl), are supposed to be hierarchically arranged along the different directions infamily space, so as to properly imitate the observed quark and lepton mass and mixinghierarchies even with coupling constantsAi

uf,df,N and Biuf,df,N taken of the order

O(1). The Majorana couplings of the right-handed neutrinosNαR do not, of course,

involve the Weinberg–Salam Higgs fieldHu and are thus naturally of orderO(MF).As was mentioned above, the Yukawa couplings (52) admit an extra global chiralsymmetryU(1) identified with the Peccei–Quinn symmetry2 U(1)PQ [30]. However,in general, one must also consider the Higgs superpotential (see below), in orderto determine whether there exists globalU(1) or discreteZN symmetry(ies) in themodel.

2 Even in the non-supersymmetric Standard Model which contains only one Higgs doublet this symmetry canbe successfully implemented [8,29], by assigning theU(1)PQ charges directly to the horizontal scalarsηi andχi .


There is a more economic way of generating effective Yukawa couplings, by using justhorizontal tripletsηiα and anti-tripletsη̄i[αβ] and imitating sextets by the pairing of tripletscalars. Instead of the couplings (52), in this case [28] one has:

WY = L̄αf R

βufHu

[Aijuf · ηiαηjβIk

ij

uf +Biuf · η̄i[αβ]I

niuf]

+ L̄αf R

βdf

�Hd

[Aijdf · ηiαηjβIk

ijdf +Bi

df · η̄i[αβ]Inidf]

(53)+NαRN

β

R

[Aij

NN · ηiαηjβIkijNN],

where all the horizontal “bold” scalar fieldsηi are taken now to be scaled by theSU(3)F symmetry breaking scaleMF , ηiα ≡ ηiα/MF . The hierarchy in the correspondingmass matrices (including the up–down hierarchy) depends solely on the powerskiju,d,Nandniu,d,N of the specially introduced hierarchy-making singlet scalar fieldI (I ≡ I/MF )in (53). This description turns out to be well fitted to the supersymmetric case where bothtypes of scalar,ηiα and its SUSY counter-partη̄iα ≡ εαβγ η̄i[βγ ], appear automatically.

The basic Yukawa couplings (52) or (53) can lead to various types of fermion massmatrices, depending on the vacuum configurations developed by the horizontal tripletsηi ,anti-triplets η̄i and sextetsχi involved. Remarkably, a texture-zero structure emergesas one of the generic features of a typicalSU(3)F symmetry breaking pattern. This iscontrolled by the antisymmetric (anti-triplet) VEVs〈η̄i〉 giving off-diagonal elements inthe mass matrices, while the symmetric (sextet) ones〈χi〉 generate the dominant diagonalelements. Therefore, any of the texture-zero models catalogued in the literature (see [10])are readily available in the framework of theSU(3)F model.

For example, the original Fritzsch ansatz [14] with the zero mass-matrix elements,M11 = M22 = M13 = 0, while presently excluded by experiment, appears from a basicvacuum configuration of the Higgs potential [8] (or superpotential [31]) with a simple setof horizontal scalars—two anti-triplets̄η[αβ] and ξ̄[αβ] and one sextetχ{αβ} developingVEVs along the following directions in family space:3

(54)η̄[12] = −η̄[21], ξ̄[23] = −ξ̄[32], χ{33}.

The “improved” Fritzsch ansatz (see [1]) with a non-zeroM22 element appears, when asecond sextetω{αβ} is included,4 which develops a VEVω{22}. Unfortunately, this ansatzis not completely predictive of the CKM matrix in terms of quark mass ratios (whichwould require no more than three independent parameters for each of the mass matrices).In particular, it gives no possibility of predicting theVcb element in the CKM matrix.

As a matter of fact the only ansatz left, which successfully predicts all the CKM matrixelements in terms of quark mass ratios (not to mention here some eclectic approaches usingad hoc GUT and string arguments, which are already largely excluded by experiment) is

3 Interestingly, the VEVsχ33 andη̄[12] in the coherent basis correspond to the flavour democracy picture [1]for the mass-matrices of quarks and leptons with symmetrical and antisymmetrical “pairing forces” respectively,while the VEVξ[23] breaks this picture appropriately.

4 Both of these ansatzes, original and “improved”, readily appear in a pure triplet realization (53) as well, withthree and four scalar triplets, respectively.


that suggested by the LFMG mechanism. In the next subsection, we show how the LFMGcan be derived within theSU(3)F model.

3.3. Minimal flavour mixing from family symmetry

3.3.1. Basic vacuum configurationsWe start with an essential question: how does one get the proportionality condition (1)

underlying the LFMG in the framework of theSU(3)F model? In terms of the Yukawasuperpotential (52), this condition (1) means that not only the VEVs of the horizontalsupermultiplets, sextetsχi{αβ} and anti-tripletsη̄i[αβ] of SU(3)F , but also their Yukawa

coupling constantsAiu,d,N and Bi

u,d,N must be properly arranged. Since the couplingconstants are in general quite independent even in the framework ofSU(3)F , this in turnmeans that the number of horizontal supermultiplets involved in the Yukawa couplingsshould be restricted. At the same time, some of these scalars have to develop VEVs alongseveral directions in the family space in order to satisfy the proportionality condition (1).

We take the set of scalar supermultiplets in the model to consist of three pairs of tripletsand anti-triplets and two pairs of sextets and anti-sextets ofSU(3)F :

(55)ηα(η̄α), ξα

(ξ̄ α), ζα

(ζ̄ α), χ{αβ}

(χ̄ {αβ}), ω{αβ}

(ω̄{αβ}).

In Section 3.3.2 we will see that just the triplet scalars, to a large extent, determine themasses and mixings of the quarks and leptons, while the sextets govern the basic vacuumconfiguration in family space leading to the LFMG picture of flavour mixing.

We take the Higgs superpotential to have the form

W =Mηηη̄+Mξξξ̄ +Mζζ ζ̄ +Mχχχ̄ +Mωωω̄+ f εαβγ ηαξβζγ + f̄ εαβγ η̄αξ̄β ζ̄ γ

(56)+ Fεαβγ εα′β ′γ ′

χ{αα′}χ{ββ ′}ω{γ γ ′} + �Fεαβγ εα′β ′γ ′ χ̄ {αα′}χ̄ {ββ ′}ω̄{γ γ ′},with massesMη,ξ,ζ andMχ,ω and coupling constantsf , f̄ , F and �F . One can readilynotice that all the trilinear couplings of triplets and sextets in the superpotentialW arevery specific just to theSU(3) symmetry. These couplings by themselves induce, as wewill see later, strictly orthogonal (to each other) VEVs for the triplets and sextets involved.However, as was argued above, some of these scalars also have to develop hierarchicallysmall parallel VEVs in the family space, so as to get the proportionality condition (1)satisfied. For this purpose we include in the superpotentialW the additional terms

(57)*W1 = a

(S

MF

)rMF η̄ζ

with a particular coupling of triplets̄η andζ which, through their mixing, will cause someextra small parallel VEVs for them whose magnitudes will be determined by the powerr of the VEV of the singlet scalarS. This hierarchy-making scalarS also appears in theYukawa couplings (see below). We emphasize here that, besides theSU(3)F symmetry,the superpotentialW has a globalU(1) symmetry. This globalU(1) is in substance thecustodial symmetry strictly protecting the form of the superpotentialW (56) considered.5

5 Let us note that thisU(1) may be identified with a superstring-inherited anomalousU(1)A gauge symmetry[37] broken at a high scale through the Fayet–Iliopoulos (FI)D-term [38]. One can take just this scale as the


Let us come now to a general analysis of the non-trivial supersymmetric vacuumconfigurations of the superpotentialW . One can quickly find from the vanishing F-termsof the triplet supermultiplets involved,

Mηηα + f̄ εαβγ ξ̄β ζ̄ γ + a

(S

MF

)rMF ζα = 0, Mηη̄

α + f εαβγ ξβζγ = 0,

Mξξβ + f̄ εαβγ η̄αζ̄ γ = 0, Mξ ξ̄

β + f εαβγ ηαζγ = 0,

Mζ ζγ + f̄ εαβγ η̄αξ̄β = 0,

(58)Mζ ζ̄γ + f εαβγ ηαξβ + a

(S

MF

)rMF η̄

γ = 0,

that they develop VEVs satisfying the equations

ηαη̄α = MξMζ

f f̄, ξαξ̄

α = MηMζ

f f̄, ζαζ̄

α = MηMξ

f f̄,

(59)εαβγ ηαξβζγ = −MηMξMζ

f f̄ 2, εαβγ η̄

αξ̄β ζ̄ γ = −MηMξMζ

f̄ 2f.

All but one of the scalar products between pairs of triplets vanish

ηαξ̄α = ξαη̄

α = ξαζ̄α = ζαη̄

α = ζαξ̄α = 0,

(60)ηαζ̄α = − a

f f̄

(S

MF

)rMFMξ ,

and their non-zero components can be chosen as follows:

ηα = (X,0,0), η̄α = (�X,0, x̄),ξα = (0, Y,0), ξ̄α = (0,�Y,0),

(61)ζα = (z,0,Z), ζ̄ α = (z̄,0,�Z).The components̄x, z and z̄ which appear in them are small and just arise due to theη̄ζ

mixing term in the superpotentialW mentioned above. We propose here that the VEV ofthe singlet scalarS responsible for this mixing is typically somewhat smaller than the basicVEVs of the triplets (59), so as to have some input hierarchy parameterS/MF in the model(see below).

The vanishing of the D-terms

(62)(Φ,T AΦ

)= 0,

where all the scalars fields (triplets and sextets) are grouped in the vectorΦ (T A aregenerators ofSU(3)F acting on the reducible representation given by the vectorΦ), resultsin further limitations on the possible supersymmetric vacuum configurations. Particularly,for the generatorT 4+i5, Eq. (62) leads to an additional non-trivial condition on the VEVs

family symmetry scaleMF . While for the heterotic strings this scale normally lies only a few orders of magnitudebelow the Planck massMPl, in the orientifold case, where the FI term appears moduli-dependent [39], it can bemade to locate much lower.


of the tripletsηα(η̄α) andζα(ζ̄ α)

(63)−�X∗x̄ +Z∗z− �Z∗z̄= 0.

When this condition is combined with the basic F-term equations (58), the values of thesmall VEV components̄x, z and z̄ can be expressed in terms of the VEV of the scalarsingletS (the bold letterS stands forS/MF ) and the ratios of the big VEV components asfollows:

x̄ = −aSrMF

�XMζ

Y

Mη

�Z∗

Z∗1

1+ ∣∣f YMη

∣∣2 ,z= −aSrMF

�XMζ

�Z∗

Z∗1

1+ ∣∣f YMη

∣∣2 ,(64)z̄= −aSrMF

�XMζ

.

The VEVs slide by themselves into the valleys determined by the F- and D-term conditions(58), (62) conserving supersymmetry.

Let us turn now to the sextet superfields in the superpotentialW (56). From theirvanishing F-term equations

Mχχ{αβ} + �Fεαγσ εβδρ ω̄{γ δ}χ̄ {σρ} = 0,

Mχ χ̄{αβ} + Fεαγσ εβδρω{γ δ}χ{σρ} = 0,

Mωω{αβ} + �Fεαγσ εβδρ χ̄ {γ δ}χ̄ {σρ} = 0,

(65)Mχω̄{αβ} + Fεαγσ εβδρχ{γ δ}χ{σρ} = 0,

and the corresponding D-term Eq. (62), one can confirm that the sextets develop strictlyorthogonal vacuum configurations of the type:

χ{αβ} = χ diag(1,1,0)αβ, χ̄ {αβ} = χ̄ diag(1,1,0)αβ,

(66)ω{αβ} = ωdiag(0,0,1)αβ, ω̄{αβ} = ω̄diag(0,0,1)αβ.

In order to show this explicitly, one must first rotate away all the components of thesextetχ{αβ}, except forχ11 andχ22, by the appropriateSU(3)F transformations. Then,successively using the F-term Eq. (65), one can see that the sextetω{αβ} and the anti-sextetω̄{αβ} inescapably develop the VEVs given in Eq. (66) with

(67)ωω̄ = M2χ

�FF ,

while the anti-sextet̄χ {αβ} also develops VEVs on the same components, (11) and (22), asthe sextetχ{αβ} does. Furthermore, a relation appears between them of the form

(68)χ11χ̄11 = χ22χ̄

22 = MχMω

�FF .

Finally, using the D-term Eq. (62) with generatorsT 3 andT 8 and assuming that theχαβandχ̄αβ VEVs are large compared with all the other scalar VEVs contributing to (62), one


unavoidably comes to the equalities

(69)|χ11| =∣∣χ̄11

∣∣= |χ22| =∣∣χ̄22

∣∣.Thus we are led to the VEVs for the scalarsχαβ andχ̄αβ of the form given in Eq. (66)

with |χ | = |χ̄ |. These VEVs by themselves break the familySU(3)F symmetry to the planeSO(2)F symmetry, acting in the family subspace of the first and second families of quarksand leptons. We will see later that just this vacuum configuration turns out to be essentialfor generating Yukawa couplings, which follow our minimal mixing pattern for quarks andleptons. As mentioned above, we assume the VEVs of the sextetsχαβ and χ̄αβ are thelargest in the model, so that this vacuum configuration of the sextets is not disturbed by theVEVs of the triplets. However, it should be noted that this assumption in no way influencesthe hierarchy in masses of the quarks and leptons, which is completely determined by thetriplet scalar VEVs given above. The relative alignment assigned to the VEVs of the triplet(61) and the sextet (66) fields is arranged by introducing a coupling between the triplet andsextet fields in the superpotential W of the form

(70)*W2 = bωζ̄ ζ̄

(S

MF

)p,

so as not to disturb the above vacuum configurations. This term also excludes an accidentalglobal symmetryU(3)triplets⊗U(3)sextetsin the superpotentialW which might, otherwise,induce the appearance of extra goldstones (familons) after symmetry breaking. Finally, onehas the total symmetrySU(3)F ⊗U(1) for the superpotentialW+*W1+*W2 considered.

3.3.2. Yukawa couplingsOne can write down a plethora ofSU(3)F invariant Yukawa couplings. However, by the

use of the additional protecting symmetryU(1), one can choose them in a special formwhich leads to (at least) one of the four experimentally allowed scenarios for the minimalflavour mixing of quarks and leptons. These four scenarios consist of the following fourpairs of combined quark and lepton flavour mixings (see Section 2): A+A∗, A+B∗, B+A∗and/or B+B∗.

Since we have all the triplet superfields needed in our chosen set (55), the mosteconomic choice of Yukawa couplings would be the pure triplet realization of the form(53) discussed in Section 3.2. However, it does not lead to the desired strict proportionalitycondition (1) or (6) between mass-matrix elements of the basic (alternative I) LFMG. Sothe pure triplet Yukawa couplings can only be used for the up quarks in scenario B (withits direct (1-3) mixing) for quark mixing. The up quark Yukawa couplings then take theform:6

(71)WY1 = �UαLU

βRHu

[AU1 · ζαζ β +AU2 · ξαξβIk +BU · ξ̄ [αβ]In

],

which contains two symmetrical couplings related with the bi-linears for the triplet scalarsζα andξα and one antisymmetric coupling related with the anti-tripletξ̄ [αβ] from our scalar

6 We do not consider here the mass matrix of right-handed neutrinos to which the scenario B+B∗ is applicableas well. This will be done later in Section 3.3.5.


set (55). We recall here that the “bold” horizontal scalar fieldsζ , ξ , ξ̄ andI are scaled bythe SU(3)F symmetry mass scale parameterMF (which can generally be as large as thePlank massMPl), so that the coupling constantsAU,NN andBU,NN are all dimensionlessand assumed to be of order unity. The hierarchy in the corresponding mass matrix of the upquarks depends, as in the general case (53), solely on the powersk andn of the hierarchy-making singlet scalar fieldI whose values are determined by theU(1) symmetry imposed.According to the triplet VEVs (61), the first symmetrical coupling inWY1 giving masses tothe heaviest third family are not suppressed, while all other couplings (including the direct(1-3) mixing term) are suppressed by powers of the scalarI. As in the general case (53),by introducing additional scalars with electroweak and horizontal indices combined onecould make the Yukawa couplings (71) renormalizable.

The other possible Yukawa couplings, which can be written with the triplets (55), arethose having the form

WY2 = L̄αf R

βuf Hu

[Auf · η̄[ασ ]η̄[βρ] +Buf · εαβσηρSl

]χ̄ {σρ}

+ L̄αf R

βdf

�Hd

[Adf · η̄[ασ ]η̄[βρ] +Bdf · εαβσ ηρSl

]χ̄ {σρ}

(72)+NαRN

βR

[ANN · η̄[ασ ]η̄[βρ]

]χ̄ {σρ}

for the up and down fermions, quarks (f = q) and leptons (f = l), and right-handedneutrinos. They contain only one triplet scalarηα (and its SUSY counterpart̄η[αβ] ≡εαβγ η̄γ ) developing VEVs (61) along both the first and third directions at the same time.They also include one of the sextetsχ̄ {αβ}, so as to be properly arranged in the familyspace. We remark that the couplingsWY1 (71), like the general Yukawa couplings (53),use horizontal triplets for their symmetric parts and anti-triplets for the antisymmetricparts. However, the couplingsWY2 (72), on the contrary, prefer to use anti-triplets forthe symmetric parts and triplets for the antisymmetric parts. Just those couplings collectedin WY2 turn out to correspond, as we will see later, to the minimal mixing of quarks andleptons with the proportionality condition (1) between their mass matrix elements. Notethat the couplings inWY2 include their own hierarchy making singlet scalar fieldS. ThisscalarS determines (see Eq. (64)) the small components in the VEVs of the tripletsη

(η̄) and ζ (ζ̄ ) and thereby underlie the basic hierarchy between the diagonal elementsM22 andM33 in the quark and lepton mass matrices (see later). Thus, it looks naturalthat the hierarchy in their off-diagonal elements is also dependent solely on the power(l) of the same scalar fieldS and not on the other scalarI introduced in the couplingsWY1 (71). Here again all the horizontal scalar fieldsη, η̄, χ̄ and S are scaled by theSU(3)F symmetry mass scale parameterMF , so that the coupling constantsAuf,df,NN

andBuf,df are all dimensionless and assumed to be of order unity. According to theU(1)symmetry, this hierarchy should be the same for all the fermions involved inWY2. Thesymmetric couplings containing the bi-linears of theη scalar give masses to the fermionsof the heaviest third family and have no suppression, apart from the contributions comingfrom the small components in its VEV configuration (61). For simplicity, we also proposethat the up–down hierarchy in the Yukawa couplings (72) is entirely given by the ratioof the VEVs of the ordinary Higgs doubletsHu and �Hd of the MSSM, tanβ = vu/vd ,thus considering the large tanβ case. One can, of course, include instead extra powers of


the singlet scalar fieldsS in the Yukawa couplings of the down fermions to generate theup–down mass hierarchy.

The Yukawa couplings (72), when taken for all the quarks and leptons involved,correspond in fact to the combined scenario A+A∗ for flavour mixing. In this case thereis only one (natural) hierarchy parameteryS = 〈S〉/MF for all the fermion mass-matrices.However, in the combined scenario B+B∗ where the masses and mixings of the up quarksare determined by the Yukawa couplingsWY1 (71), while those of the down quarks andleptons are determined by the Yukawa couplingsWY2 (72), generally two independenthierarchy parametersyI = 〈I 〉/MF andyS = 〈S〉/MF appear in the model. In general,one can hardly expect that the different Yukawa superpotentials,WY1 andWY2, shouldlead to a similar hierarchy pattern for the corresponding fermions. We shall assume (seeSection 3.3.4) that the hierarchy making scalarsI andS develop VEVs which are close invalue, leading to close values of the hierarchy parametersyI andyS . However, they musthave quite differentU(1) charges, in order to provide the required power-like hierarchy inthe quark and lepton mass matrices.

The couplingsWY1 (71) andWY2 (72) are the only Yukawa couplings which givemasses to the quarks and leptons. Other possibleSU(3)F couplings containing the samefermion and scalar superfields are, in fact, disallowed by the extraU(1) symmetrymentioned above (see Section 3.3.4 for more details).

3.3.3. Mass-matrices of quarks and leptonsWe show now that, from the four presently (experimentally) allowed LFMG scenarios

for the combined flavour mixing of quarks and leptons, theSU(3)F theory admits onlyone; namely, the B+B∗ scenario which, fortunately, seems to be the most preferable withregard to the experimental situation in quark mixing (the present value of theVub element,see Section 2.1).

Towards this end let us first consider the Yukawa couplingsWY2 (72) to understandwhy they, while being good for all the other fermions, cannot work for the up quarks andright-handed neutrinos. Substituting all the VEV values of the horizontal triplets (61) andsextets (66), as well as those of the ordinary MSSM doubletsHu and �Hd , into the YukawacouplingsWY2 (72), one obtains the following mass matrices for the quarks and leptons:

(73)Muf =m0u

Auf x̄2 0 Auf x̄�X

0 Auf x̄2 Buf XSl

Auf x̄�X −BufXSl Auf�X2

,

(74)Mdf =m0d

Adf x̄2 0 Adf x̄�X

0 Adf x̄2 Bdf XSl

Adf x̄�X −Bdf XSl Adf�X2

,

(75)MNN =m0NN

(ANN x̄2 0 ANN x̄�X

0 ANN x̄2 0ANN x̄�X 0 ANN

�X2

).

Here the mass parametersm0u, m

0d andm0

NN include all the other VEV factors appearingwith the Yukawa couplings inWY2:

(76)m0u = 〈Hu〉χ̄ , m0

d = 〈 �Hd〉χ̄ , m0NN =MF χ̄


(the bold letters stand everywhere for the properly scaled VEVs, for exampleX ≡X/MF

etc.). Now, after diagonalization of the (1-3) blocks in the mass matricesMuf , Mdf andMNN , one immediately comes to the LFMG form (2) for the mass-matrices of all thefermions involved (specifying those for the up and down quarks, neutrinos, charged leptonsand right-handed neutrinos, respectively):

(77)MU =m0u

0 BU

x̄�X XSl 0

−BU x̄�X XSl AU x̄2 BUXSl

0 −BUXSl AU�X2

,

(78)MD =m0d

0 BD

x̄�X XSl 0

−BD x̄�X XSl AD x̄2 BDXSl

0 −BDXSl AD�X2

,

(79)MN =m0u

0 BN

x̄�X XSl 0

−BN x̄�X XSl AN x̄2 BNXSl

0 −BNXSl AN�X2

,

(80)ME =m0d

0 BE

x̄�X XSl 0

−BE x̄�X XSl AE x̄2 BEXSl

0 −BEXSl AE�X2

,

(81)MNN =m0NN

(0 0 00 ANN x̄2 00 0 ANN

�X2

),

with the desired proportionality condition (1) or (6) between their diagonal and off-diagonal elements. This diagonalization, as one can readily see, is reached by a physicallyirrelevant rotation (being the same for the up and down fermions) with a hierarchicallysmall angleΘu,d

13 x̄/�X = x̄/�X ≈ (S/MF )r (see (64)).7

This is, in fact, a crucial point for deriving the LFMG picture within theSU(3)Fsymmetry framework. On the one hand, the scalarη in the Yukawa couplingsWY2 (72)must develop VEVs along the first and third directions in family space, in order that theproportionality condition (6) be satisfied. On the other, these VEVs are allowed to induceat most only non-physical extra elements in the mass-matricesMuf andMdf (73), (74)that could be rotated away by some rotation withΘu

13 =Θd13. Otherwise, one would have

large physical mixing of the order ofO(y) (wherey ≈ 0.1 is the hierarchy parameterintroduced in our discussion of neutrino oscillations in Section 2.2.1) between the first andthird families of fermions that is actually excluded, at least for quarks where this mixingangle is in fact of order ofO(y3), as follows from theVub element value in the observedCKM matrix (see (29)).

There seems to be no other way of getting the LFMG in theSU(3)F model, when thesame formWY2 (72) is taken for the Yukawa couplings of both the up and down fermions.

7 We also neglected the small addition of the order ofO(S2) to the new (33)-elements appearing in all thesemass-matrices after the diagonalization of the (1-3) blocks.


If so, however, all the matricesMuf , Mdf andMNN appear, as one can readily see fromtheir explicit forms (77)–(81), to be largely proportional to each other. This results in anequality of the mass ratios for the heavier leptons and quarks

(82)mc

mt

ms

mb

mµ

mτ

MN2

MN3 MNN2

MNN3,

which should hold with an accuracy of a few percent (given in fact by the ratios of themasses of the first and second families). While such an equality approximately worksfor the down quarks and charged leptons and also might be taken for the Dirac massesof neutrinos, it is certainly unacceptable for the up quarks [16]. This means that weare left with just one possible scenario B for the minimal quark mixing in theSU(3)Fcontext, according to which the up quarks follow the pattern of direct (1-3) mixing (thealternative II, see general discussion in Section 2.2), while the down quark mixing is givenby a mass matrix of the above type (78) corresponding to the alternative I.

As to the lepton mixing, both of the experimentally allowed scenarios A∗ and B∗(Section 2.2) follow the alternative I for the Dirac mass-matrices of leptons. ThusMN andME should have the form given above (79), (80), resulting in the proportionality condition(82) between the Dirac masses of neutrinos and charged leptons. This condition shouldbe necessarily kept into account when analyzing neutrino oscillations (as we have donein Section 2.2), since it allows us to predict the hierarchy parametery appearing in theneutrino Dirac mass matrix:y =√

mµ/mτ . If we take (as we actually did) that the neutrinoMajorana mass matrix also depends on the same parametery then all the characteristicfeature of the neutrino oscillations can be predicted according to Eq. (42) in scenario A∗and Eq. (48) in scenario B∗.

However, scenario A∗ presupposes the stronger hierarchy (35) between the second andthird eigenvalues of the neutrino Majorana mass matrixMNN to get the observed largemixings (see Section 2.2) than the hierarchy (34) appearing in the Dirac mass matrixMN

of neutrinos. The requirement for such a hierarchy to be in the matrixMNN is certainly inconflict with the mass proportionality condition (82), which appears in scenario A∗ withintheSU(3)F framework. Another drawback of this scenario A∗ in theSU(3)F frameworkis the absence of the antisymmetric mixing term in the Yukawa couplings for the right-handed neutrinos (see (72)) that leads to one massless Majorana state inMNN (81). Thus,scenario A∗ of lepton flavour mixing should be dropped in favour of scenario B∗, withdirect (1-3) mixing in the matrixMNN just as it appears for the up quarks in scenario B(see Section 3.3.5 for further discussion of the matrixMNN ). So, we can conclude that thecombined scenario B+B∗ is in substance the only one admitted in theSU(3)F theory. Forthis case the mass proportionality condition (82) takes the reduced form:

(83)ms

mb

mµ

mτ

MN2

MN3,

which seem to work better than the whole Eq. (82), however, not yet in a quite satisfactoryway. Actually, this relation between the masses of the down quarks and leptons is similarto the usual GUT relation, while arising in a different way. However, by enlarging the setof horizontal scalars, one can rearrange or completely decouple the masses of the quarksand leptons.


Now in scenario B, after family symmetry breaking according to the VEVs of thehorizontal triplets (61), one is immediately led from the Yukawa couplings collected inWY1 to the mass matrix for the up quarks:

(84)MU =m0U

(AU1z2 0 AU1zZ −BU�YIn

0 AU2Y2Ik 0AU1zZ +BU�YIn 0 AU1Z2

),

where the mass factorm0U is given by the electroweak symmetry scale involved

(85)m0U = 〈Hu〉.

Further, after diagonalization of the symmetric terms in the (1-3) block, one comes to thefamiliar form

(86)MU =m0U

( 0 0 −BU�YIn

0 AU2Y2Ik 0BU�YIn 0 AU1Z2

),

discussed in Section 2 (see (22)).Let us note that the above partial diagonalization of the mass matrix (84) was in fact

reached by a rotation for the up quarks with the hierarchically small angleΘU13 −z/Z =

z/Z ≈ (S/MF )r (see (64)). It is crucial for the scenario B+B∗ considered that this rotation

turns out to be physically irrelevant: it is in fact the same as for the down quarks andleptons, despite the latter having mass-matrices of another type (74) or (78). However, theorthogonality (60) of the VEVs of the scalars involved keeps the rotations in both sectorsstrictly equivalent to one another

(87)η̄αζα = 0 −→ x̄�X + z

Z= 0 −→ Θ

D,N,E13 =ΘU

13.

As a result, one can simultaneously transform the primary (“unrotated”) matrices for the upquarks (84), on the one hand, and those of the down quarks (74) and leptons (neutrinos (73)and charged leptons (74)), on the other, into the final (“rotated”) matrices (86) and (78)–(80), respectively. These matrices are just those which exactly correspond to the combinedscenario B+B∗ for flavour mixing of quarks and leptons. These matrices follow, as wehave seen above, from the Yukawa couplingsWY1 (71) andWY2 (72) when the familysymmetrySU(3)F spontaneously breaks.

3.3.4. Hierarchies in masses and mixingsWe saw in the previous sections that the extraU(1) symmetry, requiring the same

powersk, n and l of the hierarchy making scalarsI andS in similar Yukawa couplingsof quarks and leptons, leads in the considered scenario B+B∗ to similar hierarchical massmatrices for the down quarks (78) and the leptons (79), (80). As a result, we obtained themass relations (83) for the quarks and leptons in the scenario B+B∗.

However, besides this general conclusion theU(1) symmetry turns out, as we nowshow, to generate the required hierarchy in the mass matrices of the quarks and leptons. Inorder to show this in the most general form, we start with a scale-invariant superpotentialwhich, in its only difference with the above-used Higgs superpotentialW (56), contains


no mass parameter for any of the scalar superfields, triplets and sextets, involved. Instead,we assume that their masses are forbidden by theU(1) symmetry and are generated by theVEV of some singlet scalar fieldT from the following allowed couplings:

(88)WT = T(Aηηη̄+Aξξ ξ̄ +Aζ ζ ζ̄ +Aχχχ̄ +Aωωω̄

)+ P(I,S,T ).

So the masses of the scalars, which were directly introduced above (see Section 3.3.1), arenow given by

(89)Mη,ξ,ζ,χ,ω =Aη,ξ,ζ,χ,ω〈T 〉,through the VEV of the scalar fieldT and the coupling constantsAη,ξ,ζ,χ,ω. Thus, the VEVof the scalar fieldT gives the basic flavour scaleMF in the model, which is proposed nowto appear dynamically rather than being introduced by hand. This, in substance, shouldstem from the polynomialP(I,S,T ) which includes all three singlet scalars; one of which(T ) gives the basic mass scale in the model, while the other two (I andS) determinethe mass and mixing hierarchy of the quarks and leptons. The hierarchy parameters arebasically given by the characteristic VEV ratiosI/T andS/T determined by the properminimum of the polynomialP(I,S,T ) (see below).

Based on the above comments, we can now write the new total Higgs superpotential(see Eqs. (56), (57), (70), (88)) in the form

Wtot = T(Aηηη̄+Aξξ ξ̄ +Aζζ ζ̄ +Aχχχ̄ +Aωωω̄

)+ f εαβγ ηαξβζγ + f̄ εαβγ η̄

αξ̄β ζ̄ γ

+Fεαβγ εα′β ′γ ′

χ{αα′}χ{ββ ′}ω{γ γ ′} + �Fεαβγ εα′β ′γ ′ χ̄ {αα′}χ̄ {ββ ′}ω̄{γ γ ′}

(90)+ aη̄ζSrMF + bωζ̄ ζ̄Sp,

with I andS standing forI/MF andS/MF and the polynomialP(I,S,T ) being as yetomitted. One can now quickly derive, from the new total Higgs superpotentialWtot andthe Yukawa couplingsWY1 (71) andWY2 (72), that theU(1) charges of all the horizontalscalar fields involved are given as follows in terms of the charges of the singlet scalarsS

andI :

Qη = −5l

3QS + 8n− 2k

3QI , Qξ = −2l

3QS + 5n− 2k

3QI ,

Qζ = −2l

3QS + 10n− k

6QI , Qχ = −4l + 3p

3QS + 14n− 5k

6QI ,

(91)Qω = − l + 3p

3QS + 8n+ k

6QI , QT = −lQS + 4n− k

2QI .

TheU(1) charges for their supersymmetric counterpartsη̄, ξ̄ , ζ̄ , χ̄ andω̄ are given by

(92)Qη̄,ξ̄,ζ̄ ,χ̄,ω̄ = −Qη,ξ,ζ,χ,ω + 2QT .

In fact, theU(1) charges of all the horizontal scalars can be expressed in terms of oneindependent chargeQS , as one can readily see from the relation

(93)QI = 2l + r

3n− kQS,


arising from theη̄ζ mixing term in the total superpotentialWtot. Here the numbersk, l andn are just the powers of the singlet scalarsI andS in the Yukawa couplingsWY1 andWY2,while the numbersr andp appear as the powers ofS in the triplet–triplet and triplet–sextetmixing terms respectively in the superpotentialWtot (90).

When the above horizontal superfields undergo aU(1) transformation, according to thecharges (91), the total superpotentialWtot acquires a physically inessential overall phase8

(94)Wtot → ei3αT Wtot,

whereαT is the transformation phase corresponding to the singlet scalar superfieldT .Under the same transformation, the Yukawa couplingsWY1 (71) andWY2 (72) are leftinvariant.

As to the matter superfields, theU(1) charges of the quarks

(95)

(UL

DL

)α

, UαR, Dα

R,

leptons

(96)

(NL

EL

)α

, NαR, Eα

R

(see (50), (51)) and the ordinary Higgs doublets,Hu and �Hd , are in substance arbitrary9

and can be chosen in many possible ways.We will now investigate which powersk, l, n and r are required to get the observed

mass and mixing hierarchy from the quark and lepton mass matrices. First of all, oneshould make sure that no other matrix elements are allowed, by theU(1) symmetry andthe chosen powers, than those required by the scenario B+B∗ in all four mass matricesconsidered; namely, for the up quarks (MU (86)), down quarks (MD (78)), neutrinos (MN

(79)) and charged leptons (ME (80)). One can readily check that for an even value ofr andan odd value ofk with

(97)n > k (k is odd),

the undesired horizontal scalar combinations of typeηη, ηξ , ηζ andξζ (accompanied byany powers of the singlet scalarsI andS) in the Yukawa couplingsWY1 (71), as well as thecombinations̄ζ ζ̄ χ̄ , ξ̄ ξ̄ χ̄ andζ χ̄ in the Yukawa couplingsWY2 (72) are strictly prohibitedto appear. This is important since their presence might crucially destroy scenario B+B∗.The only allowed extra combination isξχ̄ which appears inWY2, thus giving an additionalYukawa coupling

(98)L̄αf R

βdf

�Hd

[B ′df · εαβσ ξρIn

]χ̄ {σρ}.

8 This means that theU(1) considered is in fact the globalR-symmetryU(1)R of the superpotentialWtot(90).

9 One can immediately check that, in the B+B∗ scenario, the Yukawa couplingsWY1 (71) andWY2 (72) onlygive five equations for the eight charges (or phases) of the six independent fermion (95), (96) and two Higgs(Hu, �Hd ) multiplets involved.


This actually turns out to be negligibly small as compared to the other couplings for thedown quarks and leptons (f = D,N,E) involved. So, one can see that the form of allthe mass matrices considered are highly protected by theU(1) symmetry from any scalarcontributions, other than those allowed by the B+B∗ scenario of the minimal mixing ofquarks and leptons.

Let us consider now the mass matricesMD , MN andME (78)–(80). All of them havea similar structure with the proportionality (1) between their diagonal and off-diagonalmatrix elements. The natural orders of magnitude of these mass matrix elements are givenby powers of the VEV of the singlet scalarS scaled by the basic massMF (y = c〈S〉/MF ):

(Mf )33 ≈m0, (Mf )22 ≈m0x̄2 ≈m0y2r ,

(99)(Mf )23 ≈m0yl, (Mf )12 ≈m0yl+r

(f =D,N,E) as directly follows from Eqs. (78)–(80) and (64). Since for the down quarksthe approximate equality(MD)22 ≈ (MD)23 (or equivalentlyms ≈ √

mdmb) is known towork well phenomenologically, one is led to the following relation between the powersl

andr:

(100)l = 2r.

Thus, according to the scenario B+B∗ considered, all three matricesMD , MN andME

should have a similar hierarchical structure of the above type (99) withl = 2r, wherer isas yet an arbitrary even number.

We turn now to the mass matrixMU (86) for the up quarks. The natural orders ofmagnitude of its elements are given by the appropriate powers of the VEV of anothersingletI (scaled byMF ). Taking for simplicityyI ≈ yS ≡ y(〈I 〉 ≈ 〈S〉), one readily findsfor the matrix elements ofMU :

(101)(MU)33 ≈m0U, (MU)22 ≈m0

Uyk, (MU)13 ≈m0

Uyn.

For consistency with the phenomenological observation [16] concerning the quark massesof the second and third families,(mc/mt)

2 ≈ (ms/mb)3, comparison of the diagonal

matrix elements ofMU andMD gives, for the case of an even value forr and an oddvalue ofk (see Eq. (97)), a relation of the type

(102)k = 3r + 1.

And, finally, according to the inequality (97), the minimal choice for the powern is

(103)n= 3r + 2.

One can see that the above relations for the powersk, l andn lead to approximately thesame hierarchy in the mass matrices of quarks and leptons for any (even) value of thenumberr. However, we will proceed further withr = 2, thus finally establishing the values

(104)k = 7, l = 4, n= 8 (r = 2)

for the other powers.


According to these values, the quark and lepton mass matrices (78)–(80) and (86) in thescenario B+B∗ acquire the hierarchical forms:

(105)Mf = mf

0 λρf y

6 0

−λρf y6 λ2y4 ρf y4

0 −ρf y4 1

, f =D,N,E

and

(106)MU = mU

( 0 0 σy8

0 αy7 0σy8 0 1

).

Here the parametersρf , α and σ , being ratios of Yukawa coupling constants, are allproposed to be of order unity, while the mass scale factorsmf and mU are as yet“unrotated” masses of the heaviest third family of fermions involved. Also, we haveparameterized the common VEV ratiox̄/�X contained in the matricesMD,N,E as x̄/�X =λy2 (λ≈ 1), according to the behaviour of the small components of the VEVs appearing onthe horizontal tripletsη andζ (see (64)). Note that, in the matricesMf (105), we have notincluded the negligibly small matrix element(Mf )13 = O(y8) stemming generally fromthe “extra” Yukawa coupling (98).10

One can see that, due to the high values of the powers in Eq. (104), even a very smoothstarting hierarchy11 y ≈ (ms/mb)

1/4 ≈ 0.4 leads, as is apparent in the mass matrices (105)and (106), to the observed hierarchy in the masses and mixings of the quarks and leptons.Apart from the values of the weak mixing angles (27) which, as was already shown inSection 2.1, are in a good agreement with experiment, there appear characteristic relationsbetween the quark (and lepton) masses of the type:

mu

mc

≈(mc

mt

)9/7

,md

ms

≈ ms

mb

,mu

mc

≈(md

ms

)9/4

,

(107)mc

mt

≈(ms

mb

)7/4

,

which seem to be successful phenomenologically. Actually, one needs to know only theheaviest mass for each family of fermions—the other masses are then given in terms ofthis mass and the hierarchy parametery. At the same time it is well to bear in mind thatall these mass relations appear at the flavour scaleMF , which could be as high as the GUTscale or even the Planck scale. Doing so, one then needs to project these relations down tolaboratory energies to compare them with experiment [40].

10 This element is, as one can readily see from matrix (105), of order of the lightest massmd for down quarks.Being inessential in any other respect it will, however, contribute to theVub element, thus changing the formula(27) given in Section 2 to a formula of the type|Vub| s13 |√mu/mt + eiγ md/mb|. In the maximal CPviolation case (see Subsection 3.3.6),γ = π/2, this new contribution turns out to be about 10 percent of the mainone, and is neglected below.

11 To avoid confusion we should note that the hierarchy parametery introduced here differs from that used inSection 2.2 (and denoted by the same letter): the oldy appears to be approximately equal to the square of the newone.


We consider finally the form of the polynomialP(I,S,T ) introduced above (seeEq. (88)) and how the VEVs of the basic singlet scalarsI , S andT are determined. Notefirst of all that it is not immediately evident, for theU(1) charges of the scalarsI , S andTalready determined fromWtot, WY1 andWY2 (see Eqs. (91), (93)) and from the hierarchyrequirements (100), (102), (103), that such a non-trivialU(1) invariant polynomial exists.Remarkably, however, just for these charges of the three singlet scalarsI , S and T apolynomial of the type

(108)P(I,S,T )= aABCTAIBSC,

actually exists, where the summation over all the allowed integer powersA, B andC isimposed.12 This polynomial determines the basic mass scaleMF 〈T 〉 in the model, aswell as the characteristic hierarchies related with the two other VEVs〈I 〉 and〈S〉 (all thehigh-order terms inP are scaled by appropriate powers of the massMF contained in thecoupling constantsaABC ).

Actually, in order for a term in this invariant polynomial to exist, theU(1) symmetrycondition

(109)AQT +BQI +CQS = 3QT ,

for the charges of the scalarsI , S andT should be satisfied. Substitution of the values ofthese charges, expressed (see (Eqs. (91), (93), (100), (102), (103)) in terms of the powerr of the scalarS in the total Higgs superpotentialWtot (90), in this condition gives theequation

(110)A− 3

2(7r + 5)+ 5

3B +C

6r + 5

3r= 0.

There are only a finite number of positive integer solutions of this equation for the powersA, B andC. For the caser = 2 taken in our model, only the following four terms appearin P(I,S,T )

(111)P(I,S,T )= a300T3 + a241T

2I4S + a182T1I8S2 + a0123I

12S3,

while not disturbing the other parts of the total Higgs superpotentialWtot (90). In this caseone can show from the corresponding F-term equations (together with those stemmingfrom Wtot) that in general non-zero VEVs appear for all the singlet scalarsI , S andT .Even a very smooth hierarchy between them could then lead, as was argued above, to theobserved hierarchy in masses and mixings of the quarks and leptons.

3.3.5. Neutrino masses and oscillationsWe have not yet considered the masses of the right-handed neutrinos and the Majorana

mass matrixMNN needed to construct the effective mass matrixMν for the ordinaryneutrinos (33), using the see-saw mechanism [20]. According to the B+B∗ scenario,the right-handed neutrinos should have a mass matrix similar to that of the up quarks,

12 Non-integer or negative values are of course not allowed for the powers in the Higgs superpotential or in theYukawa couplings, due to their generic analyticity.


having symmetric Yukawa couplings with the same structure asWY1 (71), but theantisymmetric couplings are necessarily absent for Majorana neutrinos. As a result, withonly the symmetric Yukawa couplings contained inWY1, one is unavoidably led toone massless right-handed neutrino. Therefore, some additional symmetric coupling isactually needed.13 Fortunately such a coupling, generating right-handed neutrino masses,automatically appears inWY1 once the triplet-sextet coupling term*W2 (70) is includedin the total superpotentialWtot (90). One can then immediately confirm that the horizontalsextetω, with the properU(1) charge value given in (91), is allowed to contribute toWY1together with the triplet pairsζ ζ and ξξ . Thus, the total Yukawa superpotential for theright-handed neutrinos reads as

(112)WY11 =NαRN

βR

[ANN0 ·ω{αβ}SpT +ANN1 · ζαζ β +ANN2 · ξαξβIk

].

The first term inWY11 should, of course, also be included in the Yukawa couplingsWY1(71) for the up quarks. However, this term, while being crucial for neutrino mass formation,turns out to be inessential for the up quarks (for the high value of the powerp of the singletS chosen below).

After substituting the VEVs of the sextetωαβ (66) and the tripletsζ andξ (61), (64) intothe Yukawa couplingsWY11 (112), one immediately comes to the Majorana mass matrixfor the right-handed neutrinos:

(113)MNN =MF

(ANN1z2 0 ANN1zZ

0 ANN2Y2Ik 0ANN1zZ 0 ANN1Z2 +ANN0ωSpT

).

This matrix can be written in a convenient hierarchical form, like the other mass matrices(105), (106), using the already established hierarchy indices (see (104)) and keeping thepowerp as yet undefined:

(114)MNN = mNN

(y4 0 y2

0 γy7 0y2 0 1+ βyp

).

HeremNN , β andγ are appropriate combinations of the masses and coupling constants in(113) and the valueT ≡ T/MF 1 has been used. The physical masses of the Majorananeutrinos are then given by the equations

(115)MNN1 mNNβy4+p, MNN2 mNNγy

7, MNN3 mNN .

The mass matrixMν for the physical neutrinos can now readily be constructedaccording to the see-saw formula (33), with the neutrino Dirac mass matrixMN (105)taken in its “unrotated” form (73)

(116)MN = mN

(λ2y4 0 λy2

0 λ2y4 ρNy4

λy2 −ρNy4 1

)

13 One might imagine introducing a symmetric mixing term containing the non-diagonal horizontal tripletcombinationηζ in the Yukawa couplingWY1 (71). However, as we have seen in the above, such a term is strictlyprohibited by theU(1) symmetry.


and the Majorana mass matrixMNN given above (114). The inverse of the Majorana massmatrixMNN is readily formed

(117)M−1NN = 1

βy4+pmNN

(1+ βyp 0 −y2

0 βγ−1y−3+p 0−y2 0 y4

),

and gives rise to an effective light neutrino mass matrixMν . To simplify its form we take, asin the general phenomenological case (see Section 2.2), for all the order-one parametersλ,ρN andβ contained in the matricesMN andMNN , an extra condition of the type (38):

(118)|∆− 1| � y4 (∆≡ λ,ρN ,β),

according to which they are supposed to be equal to unity with a few percent accuracy(see footnote 11). After this simplification and the choicep = 8, the matrixMν takes atransparent symmetrical form (ignoring some inessential higher-order terms)

(119)Mν = mν

(y4 −y2 −y2

−y2 1+ γy−1 1+ γy−1δ

−y2 1+ γ δy−1 1+ γy−1(1+ δ2

)),

where the matrixMν has been first constructed from matricesMN (116) andM−1NN (117)

according to the see-saw formula (33) and then brought to the appropriate physical basis,by making a rotation in the (1-3) block (equal to that already applied to the matrix of thecharged leptonsME so as to bring it to the form (105)). Here the mass parameter

(120)mν = − m2N

mNN

γ−1y

essentially corresponds to the mass of the heaviest neutrino, while the parameterδ ≡ (λ− 1)/y4 satisfies|δ| � 1 according to the condition (118) above.

One can see that the matrixMν is crucially dependent on the parameterγ relatedto the triplet–triplet termξξIk in the Yukawa superpotentialWY11 (112). While all thedimensionless coupling constants inWY11 are proposed (just as for the other Yukawa andHiggs coupling constants in the model) to be of the natural orderO(1), the parameterγshould be considerably smaller to have an experimentally acceptable mass interval betweenthe masses of the second and third family neutrinos. In particular, if we takeγ ≈ y4 (whichis just of the orderO(ms/mb), see the matrices (105)), we come to an attractive picture forneutrino masses and oscillations. Actually, after diagonalization of the matrixMν , one canreadily see that the ratios of the neutrino masses are14

(121)mν1 :mν2 :mν3 = 1

2y7 : 1

4y3 : 1,

and the small-mixing angle MSW solution (SMA) for neutrino oscillations naturallyemerges, with the following predictions for the basic parameters:

(122)sin2 2θatm 1, sin2 2θsun� 2y4, Ue3 y2

√2,

*m2sun

*m2atm

y6

16.

14 For simplicity, we takeδ = 1 here. For the other limiting case,δ = 0, the mass ratios are a little changed,mν1 :mν2 :mν3 = 1

4y7 : 1

2y3 : 1.


Here, in deriving the limit on the contribution to sin2 2θsun from the neutral lepton sector,we have used the inequality (118) for the(Mν)12 element obtained after successive (2-3)and (1-3) block diagonalizations.

For the phenomenological value of the hierarchy parameter in our model,y ≈ (ms/mb)

1/4 ≈ 0.4, the predictions (122) turn out to be inside of the experimentallyallowed intervals for the SMA solution (49). Note that for the case when the above pa-rametersλ, ρN andβ are strictly equal to unity the (1-2) mixing completely disappears,while the (2-3) mixing determining the value of sin2 2θatm is left maximal. Note also that inthe SMA case one must expect, as we mentioned above (see Section 2.2), a sizable contri-bution to sin2 2θsun stemming from the charged lepton sector (see Eq. (31)). Remarkably,this contribution taken alone

(123)sin2 2θsun 4 sin2 θeµ = 4me

mµ

gives a good approximation to the SMA solution.So, one can conclude that theSU(3)F theory of flavour successfully describes the

neutrino masses and mixings as well. Remarkably, just the neutrino oscillation phenomenaalready observed could give the first real hint as to where the scaleMF of the SU(3)Fflavour symmetry might be located. Actually, if the right-handed neutrinos are not sterileunderSU(3)F (as they are under all other gauge symmetries) then their masses shouldbe directly related to the flavour scale, just like the masses of the quarks and leptons arerelated to the electroweak scale. We treat the right-handed neutrinos, like the other quarksand leptons, as tripletsNα

R underSU(3)F (see (51)) which acquire their masses from theSU(3)F invariant couplingsWY11 (112), when the horizontal scalars develop VEVs andflavour symmetry breaks. Due to the chiral nature ofSU(3)F , there appear no mass termsfor the Majorana neutrinosNα

R when the symmetry is unbroken. Thus, it is apparent thatthe mass of the heaviest Majorana neutrino, appearing from the Yukawa couplingWY11,should be of the same order as the flavour scaleMF , since the basic Yukawa couplingconstants are all proposed to be of order unity. Using now Eq. (120), with the heaviestDirac neutrino massmN taken as large as the top quark massmt , the heaviest physicalneutrino massmν determined from the experimentally observed value of*m2

atm [19],

mν √*m2

atm 0.07 eV and the hierarchy parametery ≈ (ms/mb)1/4, one comes to

an approximate value for the flavour scale

(124)MF ≈ mNN = −m2N

mν

γ−1y m2t

mν

(mb

ms

)3/4

1016 GeV,

from which the masses and mixings of the quarks and leptons are basically generated. Dueto the enhancement factorγ−1y appearing in theSU(3)F model considered, this scaleturns out to be as large as the well-known SUSY GUT scale [41]. The high scaleMF

of the flavour symmetry found above (see (124)) would mean that theSU(3)F flavoursymmetry could be part of some extended SUSY GUT, either in a direct product form likeSU(5)⊗ SU(3)F [8], SO(10)⊗ SU(3)F [34] andE(6)⊗ SU(3)F [34] or even in the formof a family unifying simple group likeSU(8) GUT [35].


3.3.6. Spontaneous CP violationIt seems that the whole picture of flavour mixing of quarks and leptons, including the

hierarchical pattern of the masses and mixing angles, can arise as a consequence of thespontaneous breakdown of the family symmetrySU(3)F considered. We show, in thissection, that even CP violation can be spontaneously induced in theSU(3)F theory offlavour. Furthermore, when it occurs, one is unavoidably led to maximal CP violation.

The essential point is that, in general, the horizontal scalars (55) develop complex VEVs(61), (66) leading to CP violation, even if all the coupling constants in the superpotentialWtot (90) and the Yukawa couplings inWY1 (71) andWY2 (72) are taken to be real. Letus consider this in more detail. For simplicity, one can always choose real VEVs forthe singlet scalarsI , S and T , among the possible VEV solutions stemming from theinvariant polynomialP(I,S,T ) (108). Then, as follows from the triplet VEV equations(59), (64), the phases of the large and small components (61) of their VEVs should satisfythe conditions

δX + δ�X = 0, δY + δ�Y = 0, δZ + δ�Z = 0, δX + δY + δZ = π,

(125)δx̄ = π − δX + δY + 2δZ, δz = π − δX + 2δZ, δz̄ = π − δX.

There is one more non-trivial condition, stemming from the imaginary part of the tripletF-term Eqs. (58) containing also the singlet scalarS. When one uses the previousconditions (125), this extra condition reads as follows:

(126)−δX + δY + 3δZ = π · k, k = 0,1.

Note now that, due to the exactU(1) symmetry of the total superpotentialWtot (90) and thepolynomialP(I,S,T ) (108), one can always choose the phase of theU(1) transformationin such a way as to make one of the phasesδX, δY or δZ in the Eqs. (125), (126) vanish.Thus, takingδY = 0, one finally comes to two non-trivial solutions for the phases of thelarge and small components of the triplet VEVs:

δX = π/4, δZ = 3π/4,

(127)δx̄ = π/4, δz = π/4, δz̄ = 3π/4

and

δX = π/2, δZ = π/2,

(128)δx̄ = 3π/2, δz = 3π/2, δz̄ = π/2,

depending on the two possible values ofk, k = 0 andk = 1, respectively. We show belowby applying these triplet VEV phase values to the quark and lepton mass matrices that,while the second solution (128) is CP-conserving, the first one (127) leads to maximalCP-violation. There are no other solutions for the phases of the triplet VEVs in theSU(3)Fmodel considered.15

15 As to the VEVs of the horizontal sextetsχ andω they always can be chosen real. Actually, by a properchoice of the phase of theSU(3)F transformations (particularly itsλ8 subgroup transformation) one can take thephase of〈χ〉 to be zero. Then, as follows from the sextet F-term Eq. (65), the phase of〈ω〉 turns out to vanish aswell.


Let us consider the CP-conserving case (128) first. As one can quickly confirm, themass matricesMU (86) andMD (78) of the up and down quarks, in this case containingthe phases of the horizontal triplet VEVs given by Eq. (128), are strictly Hermitian withreal diagonal elements and pure imaginary off-diagonal ones. Such matricesMU andMD ,with five and three texture zero structure respectively, cannot lead to CP violation—thephase of the corresponding Jarlskog determinant [42] vanishes identically.

By contrast, the CP-violating case (127) results in non-Hermitian mass matricesMU

andMD . One can find their explicit form, in terms of the masses of the up and downquarks and the triplet VEV phases (127), by forming their Hermitian productsMUM

+U and

MDM+D and then using the conservation of their traces, determinants and sums of principal

minors. Doing so, we find for the matricesMU andMD :

(129)MU = 0 0 −√

mumt eiπ/2

0 mceiπ/2 0√

mumt eiπ/2 0 mt −mu

and

(130)MD = 0 −√

mdms e−i3π/4 0√

mdms e−i3π/4 ms

√mdmb e

i3π/4

0 −√mdmb e

i3π/4 mb −md

,

where we have removed the inessential common phase factors.Notably, while the diagonalization of the non-Hermitian matricesMU and MD

requires different unitary transformations for the left-handed and right-handed quarks, theHermitian constructionsMUM

+U andMDM

+D are diagonalized by unitary transformations

acting on just the left-handed quarks. These are precisely the transformations whichcontribute to the weak interaction mixings of quarks collected in the CKM matrix. Thus,one must diagonalize theMUM

+U andMDM

+D first and then construct the CKM matrix.

Proceeding in such a manner, we immediately find that the diagonalization ofMUM+U and

MDM+D is carried out by unitary matrices of the type

(131)VU =RU13Φ

U, VD =RD12R

D23Φ

D.

They in fact contain the same plane rotationsRU13 (s13 √

mu/mt ),RD12 (s12 √

md/ms )andRD

23 (s23 √md/mb ) as those used for the diagonalization of the phenomenological

Hermitian mass matricesMU (22) andMD (2) considered in Section 2.2. However, thephase matricesΦU andΦD now become

(132)ΦU =(eiπ/2 0 0

0 1 00 0 1

), ΦD =

(1 0 00 e−i3π/4 00 0 1

).

Finally, substituting all the above rotations and phases into the CKM matrixVCKM =VUV

†D (14) and properly rephasing thec quark field (c→ ei3π/4c), one comes to

(133)VCKM =(

c12c13 − is12s13s23 −s12c13 − is13s23c12 −is13c23s12c23 c12c23 −s23

−s13c12 − is12s23c13 s13s12 − is23c12c13 −ic13c23

).


While this is in substance a new parameterization for the CKM matrix, it is quite close tothe standard parameterization [16], as one can show by comparing the moduli of the matrixelements in them both. In such a way, one readily concludes that the CP violation phase inVCKM (133) is in fact maximal,δ = π/2, while all the mixing angles are rather small beingbasically determined by the masses of the lightest quarksu andd (see Eq. (27)), as wasexpected.

4. Conclusions

The present observational status of quark flavour mixing, as described by the CKMmatrix elements [16], shows that the third familyt andb quarks are largely decoupledfrom the lighter families. At first sight, it looks quite surprising that not only the (1-3) “farneighbour” mixing (giving theVub element inVCKM) but also the (2-3) “nearest neighbour”mixing (Vcb) happen to be small compared with the “ordinary” (1-2) Cabibbo mixing (Vus)which is determined, according to common belief, by the lightestu andd quarks. This ledus to the idea that all the other mixings, and primarily the (2-3) mixing, could also becontrolled by the massesmu andmd and that the above-mentioned decoupling of the thirdfamily t andb quarks is determined by the square roots of the corresponding mass ratios√mu/mt and

√md/mb respectively. So, in the chiral symmetry limitmu = md = 0, not

only does CP violation vanish, as argued in [1], but all the flavour mixings disappear aswell.

In such a way the Lightest Family Mass Generation (LFMG) mechanism for flavourmixing of quarks was formulated in Section 2.1, with two possible scenarios A and B. Wefound that the LFMG mechanism reproduces well the values of the already well measuredCKM matrix elements and gives distinctive predictions for the yet poorly known ones, inboth scenarios A and B. One could say that, for the first time, there are compact workingformulae (especially compact in scenario B) for all the CKM angles in terms of quarkmass ratios. The only unknown parameter is the CP violating phaseδ. Taking it to bemaximal (δ = π/2), we obtain a full determination of the CKM matrix that is numericallysummarized in Section 2.1. The LFMG mechanism was extended to the lepton sector inSection 2.2, again with two possible scenarios A∗ and B∗. These were shown to be quitesuccessful phenomenologically as well, naturally leading to a consistent description of thepresently observed situation in solar and atmospheric neutrino oscillations. Remarkably,the same mechanism results simultaneously in small quark and large lepton mixing.

From the theoretical point of view, the basic LFMG mechanism arises from the genericproportionality condition (1) between diagonal and off-diagonal elements of the massmatrices. One might think that this condition suggests some underlying flavour symmetry,probably a non-abelianSU(N) symmetry, treating theN families in a special way. Indeed,for N = 3 families, we have found that the localSU(3)F chiral family (or horizontal)symmetry [8], considered in detail in Section 3, seems to be a good candidate. TheSU(3)Ffamily symmetry is accompanied by an additional globalU(1) symmetry, which seemsto be very important for the recovery of the final picture of flavour mixing of quarksand leptons. There is a whole plethora ofSU(3)F invariant Yukawa couplings available.However, due to the extra protecting symmetryU(1), one is able to choose them in a


special form which leads to the minimal flavour mixing of quarks and leptons. There arefour pairs of experimentally allowed scenarios (see Section 2) for the combined quark andlepton flavour mixings: A+A∗, A+B∗, B+A∗ and/or B+B∗. We found that theSU(3)Ftheory only admits the scenario B+B∗ which, fortunately, seems to provide the best fitto the experimental data on quark mixing. It also leads to the SMA solution for the solarneutrino problem. Besides its crucial role in selecting the pattern of flavour mixing amongthe four scenarios involved, theU(1) symmetry turns out, as we have shown, to determinethe form of the hierarchy in the quark and lepton mass matrices.16 Actually, one only needsto know the heaviest mass for each family of fermions—the other masses are then given interms of this mass and the input hierarchy parametery.

The high scaleMF of the flavour symmetry found here from neutrino oscillationsmeans that the local flavour symmetrySU(3)F might be interpreted as part of someextended SUSY GUT; this could appear in a direct product form likeSU(5) ⊗ SU(3)F ,SO(10) ⊗ SU(3)F andE(6) ⊗ SU(3)F or even in the form of a simple group like theSU(8) GUT unifying all families.

And the last point, which is especially noteworthy, is that the symmetry-breakinghorizontal scalar fields, triplets and sextets ofSU(3), in general develop complex VEVsand, in cases linked to the LFMG mechanism, transmit a maximal CP violating phaseδ = π/2 to the effective Yukawa couplings involved. Apart from the direct predictabilityof δ, the possibility that CP symmetry is broken spontaneously like other fundamentalsymmetries of the Standard Model seems very attractive—both aesthetically and becauseit gives some clue to the origin of maximal CP violation in the CKM matrix.

So, anSU(3) family symmetry seems to be a good candidate for the basic theoryunderlying our proposed LFMG mechanism, although we do not exclude the possibilityof other interpretations as well. Certainly, even without a theoretical derivation of Eq. (1),the LFMG mechanism can be considered as a successful predictive ansatz in its own right.Its further testing could shed light on the underlying flavour dynamics and the way towardsthe final theory of flavour.

Acknowledgements

We should like to thank H. Fritzsch, H. Leutwyler, D. Sutherland and Z.-Z. Xing forinteresting discussions. J.L.C. is grateful to PPARC for financial support (PPARC grantNo. PPA/V/S/2000/) during his visit to Glasgow University in June–July of 2000, whenpart of this work was done. Financial support to J.L.C. from INTAS grant No. 96-155 andthe Joint Project grant from the Royal Society are also gratefully acknowledged. C.D.F. andH.B.N. would like to thank the EU commission for financial support from grants No. SCI-0430-C (PSTS) and No. CHRX-CT-94-0621. Also J.L.C., C.D.F. and H.B.N. would like toacknowledge financial support from INTAS grant No. 95-567.

16 Note, however, that we had to take the parameterγ in the Majorana mass matrix (114) to be small (γ ≈ y4),while all the parameters in the quark and lepton mass matrices are supposed to of the orderO(1).


References

[1] H. Fritzsch, Z.-Z. Xing, Prog. Part. Nucl. Phys. 45 (2000) 1.[2] C.D. Froggatt, H.B. Nielsen, Nucl. Phys. B 147 (1979) 277.[3] M. Leurer, Y. Nir, N. Seiberg, Nucl. Phys. B 398 (1993) 319;

M. Leurer, Y. Nir, N. Seiberg, Nucl. Phys. B 420 (1994) 468.[4] L.E. Ibanez, G.G. Ross, Phys. Lett. B 332 (1994) 100;

J.K. Elwood, N. Irges, P. Ramond, Phys. Lett. B 413 (1997) 322;N. Irges, S. Lavignac, P. Ramond, Phys. Rev. D 58 (1998) 035003.

[5] D.S. Shaw, R.R. Volkas, Phys. Rev. D 47 (1993) 241.[6] M. Dine, R. Leigh, A. Kagan, Phys. Rev. D 48 (1993) 4269;

L.J. Hall, H. Murayama, Phys. Rev. Lett. 75 (1995) 3985.[7] R. Barbieri, L.J. Hall, Nuovo Cimento A 110 (1997) 1;

R. Barbieri, L.J. Hall, S. Raby, A. Romanino, Nucl. Phys. B 493 (1997) 3.[8] J.L. Chkareuli, JETP Lett. 32 (1980) 671;

Z.G. Berezhiani, J.L. Chkareuli, Yad. Fiz. 37 (1983) 1043;F. Wilczek, Preprint NSF-ITP-83-08 (1983);Z.G. Berezhiani, Phys. Lett. B 129 (1983) 99;Z.G. Berezhiani, Phys. Lett. B 150 (1985) 177.

[9] C. D Froggatt, G. Lowe, H.B. Nielsen, Nucl. Phys. B 414 (1994) 579;C.D. Froggatt, G. Lowe, H.B. Nielsen, Nucl. Phys. B 420 (1994) 3;C.D. Froggatt, H.B. Nielsen, D.J. Smith, Phys. Lett. B 385 (1996) 150.

[10] P. Ramond, R.G. Roberts, G.G. Ross, Nucl. Phys. B 406 (1993) 19.[11] H.P. Nilles, Phys. Rep. 110 (1984) 1.[12] J.L. Chkareuli, C.D. Froggatt, Phys. Lett. B 450 (1999) 158.[13] K. Matsuda, T. Fukuyama, H. Nishiura, Phys. Rev. D 60 (1999) 013006;

K. Matsuda, T. Fukuyama, H. Nishiura, Phys. Rev. D 61 (2000) 053001;D. Falcone, Mod. Phys. Lett. A 14 (1999) 1989;T.K. Kuo, S.W. Mansour, G.-H. Wu, Phys. Rev. D 60 (1999) 093004;T.K. Kuo, S.W. Mansour, G.-H. Wu, Phys. Lett. B 467 (1999) 116;S.-H. Chiu, T.K. Kuo, G.-H. Wu, Phys. Rev. D 62 (2000) 053014;D. Falcone, F. Tramontano, Phys. Rev. D 63 (2001) 073007;B.R. Desai, A.R. Vaucher, Phys. Rev. D 63 (2001) 113001.

[14] H. Fritzsch, Phys. Lett. B 70 (1977) 436;H. Fritzsch, Phys. Lett. B 73 (1978) 317.

[15] H. Fusaoka, Y. Koide, Phys. Rev. D 57 (1998) 3986.[16] Particle Data Group, Eur. Phys. J. C 15 (2000) 1.[17] B.H. Behrens et al., Phys. Rev. D 61 (2000) 052001.[18] LEP Working Group on|Vub|, http://battagl.home.cern.ch/battagl/vub/vub.html.[19] The Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 81 (1998) 1562;

The Super-Kamiokande Collaboration, Y. Fukuda et al., Phys. Rev. Lett. 85 (2000) 3999.[20] M. Gell-Mann, P. Ramond, R. Slansky, in: P. von Nievenhuizen, D.Z. Friedman (Eds.), Supergravity, North-

Holland, 1979;T. Yanagida, in: O. Sawada, A. Sugamoto (Eds.), Proceedings of the Workshop on Unified Theories andBaryon number in the Universe, KEK, Japan, 1979;R.N. Mohapatra, G. Senjanovic, Phys. Rev. Lett. 44 (1980) 912.

[21] H. Fritzsch, Z.-Z. Xing, in: B.A. Kniehl, G.G. Raffelt, N. Schmitz (Eds.), Proceedings of the RingbergEuroconference on New Trends in Neutrino Physics, World Scientific, 1999, pp. 137–146, hep-ph/9807234;C.D. Froggatt, M. Gibson, H.B. Nielsen, Phys. Lett. B 446 (1999) 256;R. Barbieri, L.J. Hall, A. Strumia, Nucl. Phys. B 544 (1999) 89;G. Altarelli, hep-ph/0106085.

[22] K. Matsuda, T. Fukuyama, H. Nishiura, see Ref. [13].[23] J.N. Bahcall, P.I. Krastev, A.Yu. Smirnov, Phys. Rev. D 58 (1998) 096016;

J.N. Bahcall, M.C. Gonzales-Garcia, C. Pena-Garay, hep-ph/0106258.

http://battagl.home.cern.ch/battagl/vub/vub.html


[24] E.J. Chun, S.K. Kang, C.W. Kim, U.W. Lee, Nucl. Phys. B 544 (1999) 89.[25] J.L. Chkareuli, I.G. Gogoladze, A.B. Kobakhidze, M.G. Green, D.E. Hutchcroft, Phys. Rev. D 62 (2000)

015014;J.L. Chkareuli, C.D. Froggatt, Phys. Lett. B 484 (2000) 87.

[26] F. Quevedo, Nucl. Phys. (Proc. Suppl.) 62 (1998) 134.[27] Z.G. Berezhiani, J.L. Chkareuli, JETP Lett. 37 (1983) 338;

Z.G. Berezhiani, J.L. Chkareuli, Sov. Phys. Usp. 28 (1985) 104.[28] J.L. Chkareuli, JETP Lett. 50 (1989) 255;

J.L. Chkareuli, Phys. Lett. B 246 (1990) 498.[29] Z.G. Berezhiani, J.L. Chkareuli, in: A.N. Tavkhelidze, V.A. Matveev, A.A. Pivovarov (Eds.), Proceedings

of the International Seminar QUARKS 84, Moscow, 1985, pp. 110–121.[30] R.D. Peccei, H.R. Quinn, Phys. Rev. D 16 (1977) 1891.[31] Z.G. Berezhiani, G.R. Dvali, M.R. Jibuti, J.L. Chkareuli, in: A.N. Tavkhelidze, V.A. Matveev, A.A. Pivova-

rov (Eds.), Proceedings of the International Seminar QUARKS 86, Moscow, 1987, pp. 209–223.[32] Z. Berezhiani, Phys. Lett. B 417 (1998) 287.[33] T.M. Bibilashvili, G.R. Dvali, Phys. Lett. B 248 (1990) 259;

J.L. Chkareuli, JETP Lett. 54 (1991) 189;J.L. Chkareuli, JETP Lett. 54 (1991) 295;J.L. Chkareuli, Phys. Lett. B 272 (1991) 207;D. Spergel, U.-L. Pen, Astrophys. J. 491 (1997) L67;S.M. Carroll, M. Trodden, Phys. Rev. D 57 (1998) 5189.

[34] J.C. Wu, Phys. Rev. D 36 (1987) 1514.[35] J.L. Chkareuli, Phys. Lett. B 300 (1993) 361.[36] J.L. Chkareuli, JETP Lett. 41 (1985) 577.[37] M. Green, J. Schwarz, Phys. Lett. B 149 (1998) 117.[38] M. Dine, N. Seiberg, E. Witten, Nucl. Phys. B 289 (1987) 587.[39] H.P. Nilles, hep-ph/0003102.[40] C.H. Albright, M. Lindner, Phys. Lett. B 213 (1988) 347;

K.S. Babu, Q. Shafi, Phys. Rev. D 47 (1993) 5004.[41] S. Dimopoulos, H. Georgi, Nucl. Phys. B 193 (1981) 150;

N. Sakai, Z. Phys. C 11 (1981) 153;S. Weinberg, Phys. Rev. D 26 (1982) 287;N. Sakai, T. Yanagida, Nucl. Phys. B 197 (1982) 533.

[42] C. Jarlskog, Phys. Rev. Lett. 55 (1985) 1039.

Minimal mixing of quarks and leptons in the SU(3) theory of flavour

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