11
ELSEVIER UCLEAR PHYSICS Nuclear Physics B (Proc. Suppl.) 101 (2001) 410-420 PROCEEDINGS SUPPLEMENTS www elsevier.com/locate/npe Flavor Structure, Flavor Symmetry and Supersymmetry Zurab BerezhianP and Anna Rossi b a Universit£ di L'Aquila, 1-67010 Coppito AQ, and INFN, Laboratori Nazionali del Gran Sasso, 1-67010 Assergi AQ, Italy b Universith di Padova and INFN Sezione di Padova, 1-35131 Padova, Italy We discuss the role played by the horizontal flavour symmetry in supersymmetric theories. In particular, we consider the horizontal symmetry SU(3)H between the three fermion families and show how this concept can help in explaining the fermion mass spectrum and their mixing pattern in the context of SUSY GUTs. 1. Introduction One of the most obscure sides of the parti- cle physics is related to fermion flavor structure. This is a complex problem with different aspects questioning origin of the mass spectrum and mix- ing pattern of quarks and leptons (including neu- trinos) and CP-violation, suppression of the fla- vor changing (FC) neutral currents, strong CP- problem, etc. [1]. Presently, thanks to the new data from the atmospheric and solar neutrino experiments, the flavor problem is getting more intriguing. On the one hand, the experimental data hint to a hierarchical neutrino mass spec- trum, similarly to the case of the the charged lep- tons and quarks. On the other hand, the lepton mixing pattern strongly differs from that of the quarks. In particular, the 2-3 lepton mixing angle is nearly maximal, 0~3 _~ 45 ° in contrast with the analogous mixing angle in quarks, 033 "~ 2 °. The concept of supersymmetry per se does not help in understanding the fermion flavor struc- ture, and in addition it creates another problem, so called supersymmetric flavor problem, related to the sfermion mass and mixing pattern. In the MSSM the fermion sector consists of chiral superfields containing the quark and lep- ton species of three families: qi = (u, d)i, ui, di, li = (v,e)~ and ei (i = 1,2,3). The charged fermion masses emerge from the Yukawa terms in the superpotenial: Wyuk -- ~3 - z3 - - Y ~ u~q3H2 + Y d d~q3H1 + Y~3e*13H1 (1) where H1,2 are the Higgs doublets, with the vac- 0920-5632/01/$ - see front matter © 2001 Elsevmr Science B.V. PII S0920-5632(01)01527-4 uum expectation values (VEVs) vl,2 breaking the electroweak symmetry, (v 2 + v2) 1/2 = v = 174 GeV. The 3 x 3 Yukawa matrices Yu,d,e are not constrained by any symmetry property and thus remain arbitrary. 1 The second aspect of the flavour problem, a specific of SUSY, questions the sfermion mass and mixing pattern which is determinde by the soft SUSY breaking (SSB) terms. These include tri- linear A-terms: A u u, q3H2 + A~3d~jH1 + Ay~i[jH1 (2) (the tilde labels sfermions) and soft mass terms: £m = ~-~'~]~(m~)~J]~, (f=q,-(u),d,l,~) (3) f 2 where Au,d,e and m] are 3 × 3 matrices with dimensional parameters. Theoretical arguments based on the Higgs mass stability imply that the typical mass scale rh of these terms should be or- der 100 GeV, maybe up to TeV. The SSB terms have no apriori relations to the Yukawa constants Yu,d,e. Hence, one can expect that the splitting between the sfermion mass eigenstates is large, order rh, and in addition the sfermion mixing an- gles in couplings to fermions and neutral gaug- inos are large. This situation gives rise to dra- matic contributions in the flavor-changing and CP-violating processes. For example, the decay 1The phenomenologically dangerous R-violating terms can be suppressed by R-parity. In other terms, one can impose the matter parity Z2 under which the fermion superfields change the sign while the Higgs ones are invariant. All rights reserved.

Flavor structure, flavor symmetry and supersymmetry

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Page 1: Flavor structure, flavor symmetry and supersymmetry

ELSEVIER

UCLEAR PHYSICS

Nuclear Physics B (Proc. Suppl.) 101 (2001) 410-420

PROCEEDINGS SUPPLEMENTS www elsevier.com/locate/npe

Flavor Structure, Flavor Symmetry and Supersymmetry Zurab BerezhianP and Anna Rossi b

a Universit£ di L'Aquila, 1-67010 Coppito AQ, and INFN, Laboratori Nazionali del Gran Sasso, 1-67010 Assergi AQ, Italy

b Universith di Padova and INFN Sezione di Padova, 1-35131 Padova, Italy

We discuss the role played by the horizontal flavour symmetry in supersymmetric theories. In particular, we consider the horizontal symmetry SU(3)H between the three fermion families and show how this concept can help in explaining the fermion mass spectrum and their mixing pattern in the context of SUSY GUTs.

1. I n t r o d u c t i o n

One of the most obscure sides of the parti- cle physics is related to fermion flavor structure. This is a complex problem with different aspects questioning origin of the mass spectrum and mix- ing pattern of quarks and leptons (including neu- trinos) and CP-violation, suppression of the fla- vor changing (FC) neutral currents, strong CP- problem, etc. [1]. Presently, thanks to the new data from the atmospheric and solar neutrino experiments, the flavor problem is getting more intriguing. On the one hand, the experimental data hint to a hierarchical neutrino mass spec- trum, similarly to the case of the the charged lep- tons and quarks. On the other hand, the lepton mixing pat tern strongly differs from that of the quarks. In particular, the 2-3 lepton mixing angle is nearly maximal, 0~3 _~ 45 ° in contrast with the analogous mixing angle in quarks, 033 "~ 2 °.

The concept of supersymmetry per se does not help in understanding the fermion flavor struc- ture, and in addition it creates another problem, so called supersymmetric flavor problem, related to the sfermion mass and mixing pattern.

In the MSSM the fermion sector consists of chiral superfields containing the quark and lep- ton species of three families: qi = (u, d)i, ui, di, li = (v,e)~ and ei (i = 1,2,3). The charged fermion masses emerge from the Yukawa terms in the superpotenial:

W y u k - - ~3 - z3 - - Y~ u~q3H2 + Yd d~q3H1 + Y~3e*13H1 (1)

where H1,2 are the Higgs doublets, with the vac-

0920-5632/01/$ - see front matter © 2001 Elsevmr Science B.V. PII S0920-5632(01)01527-4

uum expectation values (VEVs) vl,2 breaking the electroweak symmetry, (v 2 + v2) 1/2 = v = 174 GeV. The 3 x 3 Yukawa matrices Yu,d,e are not constrained by any symmetry property and thus remain arbitrary. 1

The second aspect of the flavour problem, a specific of SUSY, questions the sfermion mass and mixing pat tern which is determinde by the soft SUSY breaking (SSB) terms. These include tri- linear A-terms:

A u u, q3H2 + A~3d~jH1 + Ay~i[jH1 (2)

(the tilde labels sfermions) and soft mass terms:

£m = ~-~'~]~(m~)~J]~, ( f=q,- (u) ,d , l ,~) (3) f

2 where Au,d,e and m ] are 3 × 3 matrices with

dimensional parameters. Theoretical arguments based on the Higgs mass stability imply that the typical mass scale rh of these terms should be or- der 100 GeV, maybe up to TeV. The SSB terms have no apriori relations to the Yukawa constants Yu,d,e. Hence, one can expect that the splitting between the sfermion mass eigenstates is large, order rh, and in addition the sfermion mixing an- gles in couplings to fermions and neutral gaug- inos are large. This situation gives rise to dra- matic contributions in the flavor-changing and CP-violating processes. For example, the decay

1 The phenomenologically dangerous R-violating te rms can be suppressed by R-parity. In other terms, one can impose the mat te r pari ty Z2 under which the fermion superfields change the sign while the Higgs ones are invariant.

All rights reserved.

Page 2: Flavor structure, flavor symmetry and supersymmetry

Z. Berezhiani, A. Rossi/Nuclear Physics B (Proc. Suppl.) 101 (2001) 410-420 411

rate # --~ e + 3' or the CP-violating parameters

~g and E~ in K ° - ~ 0 system would much ex- ceed the experimental bounds unless r?z is larger than 10-100 TeV. In this case, however, advan- tages of supersymmetry in stabilizing the Higgs mass would be lost. Thus, experimental limits on the FC-processes impose severe constraints on the mass and mixing spectrum of the yet undis- covered squarks and sleptons.

As far as neutrinos are concerned, there is no renormalizable term that can give their masses. However, the Majorana masses of neutrinos can emerge from the lepton-number violating higher order operator cutoff by some large scale M, e.g. the grand unification or Planck scale [2]:

1 v ' J l I ~ 2 y ~ = y T (4) M --v o~3--2 ,

Any known mechanism for the neutrino masses reduces to this effective operator. E.g., in the 'seesaw' scheme [3] it is obtained after integrat- ing out heavy-neutral fermions with the Majo- rana masses ~ M. Hence, modulo the Yukawa couplin constants, the charged fermion masses are

v and the neutrino masses are ~ v2/M which makes it clear why the latter are so small. How- ever, the matrix Y~ remains arbitrary.

The concept of the grand unification provides more constraints on the Yukawa matrices and in this way opens up a possibility for predictive schemes between the quark and lepton masses. In the SU(5) model the all fermion states are unified within 10-plets t~ = (~, q, ~)~ and 5-plets f~ = (d,/), . The minimal structure of the Yukawa terms is the following

Gi3t~f3H + G~t, t3H + M G ~ ] j ~ H 2 (5)

where G~ and G , should be symmetric while the form of G is not restricted. After the SU(5) sym- metry breaking, these terms reduce to the stan- dard couplings (1) with Yd = Yff = G. This implies that the Yukawa eigenvalues are degener- ate between the down quarks and charged leptons of the same generations. Although this prediction for the largest eigenvalues, b - T Yukawa unifica- tion, is a remarkable success of the SU(5) theory, it is completely wrong for the light generations.

The spontaneous breaking of SU(5) to the Standard Model by the adjoint superfield • (24- plet) can be used to remove the unrealistic degen- eracy between down-quark and charged leptons. One can consider the Yukawa couplings matri- ces as operators depending ~, i.e. G = G(~) , ( ~ = G~(<b) etc. which should be understood as expansion series, e.g.

G~J(~)t~fjffI = G~oJtj3H + G~ ~-t~.[-I + ... (6)

where M is some cutoff scale. The tensor prod- uct 24 × 5 contains both 5 and 4-"5 channels and thus can provide different Clebsch factors for the Yukawa entries between the quark and lepton states of light generations. Clearly, such higher order operators can be obtained by integrating out some heavy fermion states with masses order M [4] just like in the seesaw mechanism.

In this way, the concept of GUT provides a more appealing framework for understanding the fermion mass and mixing structures. However, at the same time it makes more difficult the super- symmetric flavor problem. Namely, in the MSSM context natural suppression of the flavor-changing phenomena can be achieved by the SSB terms universality at the Planck scale, which can be motivated in the context of supergravity scenar- ios [5]. However, in the SUSY GUT frames this idea becomes insufficient - the physics above the GUT scale does not decouple and can strongly vi- olate the SSB terms universality at lower scales. In generic SUSY GUTS, the integrating out of the heavy states generally should lead to big non- universal terms [6] which can cause dangerous flavor-changing contributions and thus pose a se- rious challenge to the SUSY GUT concept.

An attractive approach to both flavor prob- lems - fermion and sfermion - is to invoke the idea of horizontal inter-family symmetry. Sev- eral models based on U(1) or U(2) family symme- tries have been considered in the literature [8,9]. However, the chiral U(3)H or its non-abelian part SU(3)H unifying all fermion generations in hori- zontal triplets [10-12] seems to be a most natural candidate for describing the family triplication. In this paper we demonstrate its power to provide a a coherent picture for the fermion and sfermion

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412 Z Berezhiani, A. Rossi/Nuclear Physics B (Proc. Suppl.) 101 (200D 410-420

masses, to explain the origin of the fermion mass spectrum and and mixing structure, and natu- rally solve the supersymmetric flavor problem.

In general, for constructing the realistic situa- tions, we must require that in the horizontal sym- metry limit the fermions should remain massless, so that they can acquire masses only after the horizontal symmetry breaking. In this way, the fermion mass and mixing spectrum could reflect the VEV pattern of the Higgs scalars leading to the spontaneous breaking of the horizontal sym- metry. In other words, the horizontal symmetry should have chiral character. Clearly, the hor- izontal symmetry SU(3)H unifying three gener- ations of quarks and leptons can be the chiral symmetry, in difference from its SU(2)H part.

In particular, in the limit of massless fermions the standard model has a maximal global chi- ral symmetry U(3) 5 = g(3)q x U(3)~ x U(3)d x U(3)t × U(3)e, separately transforming the quark and lepton superfields of three families qi = (u,d),, fi,, d,, l, = (v,e), and ei (i = 1,2,3). The Yukawa terms explicitly break this symme- try. One can suppose, however, that the Yukawa couplings emerge as a result of the spontaneous breaking of this symmetry. Such a situation will be considered in sect. 2. We will show that in this case one could obtain a natural fermion mass hierarchy and mixing pattern. 2

However, the maximal flavor symmetry U(3) 5 is incompatible with the context of the SUSY GUT. Namely, in the SU(5) model the fermions of one generation are unified within two multi- plets t ~ 10 and f ,,~ 5, so that the maxi- mal global chiral symmetry reduces to U(3) 2 = U(3)t x U(3)/ . In the SO(10) model all fermions in one family fit into one multiplet ¢ ,~ 16, there- fore the flavor symmetry reduces to U(3)H. The latter is still chiral, since now all left handed fermions transform as 3 and the right handed ones as 3. Therefore, as far as the GUT framework is concerned, it would be most natural to consider the horizontal symmetry U(3)H or even only its non-abelian factor SU(3)H. In addition it could be the local gauge symmetry emerging from some

2The chiral symmetries of this type can be also very help- ful for suppressing the FC femomena in the TeV scale gravity theories with extra dimensions [14].

more fundamental theory on the same grounds as the GUT symmetry itself.

We consider SU(3)H as horizontal symmetry group in sect. 3. The winning combination SUSY + GUTxSU(3)H acts in the following way: • The spontaneous breaking features of SU(3)H turn the Yukawa constants of the low energy the- ory (MSSM) into dynamical degrees of freedom and fix the inter-family hierarchy in a natural way. Namely, the third generation becomes heavy (Yt "~ 1), while the second and first ones become lighter by successively increasing powers of small parameters. The question how to achieve the de- sired structure of the horizontal symmetry break- ing VEVs will be discussed in sect. 3. • The adjoint Higgs (24-pier) which provides the gauge symmetry breaking SU(5) --+ SU(3) x SU(2) x U(1), should be used in the Yukawa op- erators to remove the unrealistic degeneracy be- tween the down-quark and charged leptons. • Most probably, the horizontal symmetry should exhibite the analogous breaking pat tern SU(3)H ~ SU(2)H x U ( 1 ) H by the adjoint Higgs (octet), with SU(2)H acting between the light (first and second) fermion generations. This could naturally reconcile the observed pat tern of the quark and lepton mixings.

2. M a x i m a l f am i ly s y m m e t r y SU(3) 5

Let us consider the global flavor symmetry SU(3) 3 = SU(3)q × SU(3)a x SU(3)d: of the quark sector. The quark superfields transform as

( u ) ,,~ (3, 1, 1), ~, ~ (1,3, 1), q+ ---- d , dk "-~ (1, 1, 3), (7)

i , j ,k = 1,2,3 are family indices. The quark masses emerge from the effective operators [13]:

j* xk* Xu _ :~_i.~kq, H 1 • M u3 qiH2 + (s)

2 V ~

where X~ ,,+ (:], 3, 1) and Xd "., (3, 1, 3) are the horizontal Higgs superfields in the mixed repre- sentations of SU(3) 3, and M is a cutoff scale (= flavor scale). In the context of renormalizable theory, these effective operators can emerge as a result of integrating out some extra heavy vector- like fermion superfields [4], e.g. the weak isosin- glets U, b" and D , / ) having the same color and

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Z. Berezhtani, A. Rossi/Nuclear Physics B (Proc. Suppl ) 101 (2001) 410-420 413

electric charges as u, ~2 and d, d and transforming in the following representations of SU(3)a:

U,, D, ,~ (3, 1, 1), 0 *,/)* ~ (3, 1, 1) (9)

The latter can get masses from the VEV of some scalar E, which can be a singlet or an octet of SU(3)q, ~ ~ (8, 1, 1). On the other hand, they can mix with the light states via the superfields X~,d and H1,2. The relevant superpotential reads:

fiUX~ + EUU + UqH2 + dDXd + E D D + DqH~ (10)

where order one constants are understood at each coupling. After the fields X~,,d and E gat large VEVs the Yukawa matrices get the form

( 0 X~ ) ( 0 Xd ) (11) //2 M u ' H1 MD

where Xu.d "" (Xu,d) and M~y,D "~ (E). The the effective operators (8) emerge after integrat- ing out the heavy states in the so called seesaw limit X~,d _< M. Namely, diagonalizing the ma- trices (11), we see that the states ~, 0 and d , / ) are mixed so that the actual light states which couple q via H1 a n d / / 2 become:

e '_~ ~ + X ~ M u 1 0 , d' ~_ d + X ~ M D ~ D (12)

Therefore, in this situation the Yukawa constants of the Standard Model are nothing but

Y ~ = X ~ M u 1, Y d = X d M D 1 (13)

The heavy fermion mass matrices are SU(3) in- variant, MU, D "~ M if E is a singlet, or MU, D " MAs if E is an octet with a VEV towards the ~s generator of SU(3)q. In either case they have rather democratic structure and cannot give rise to the fermion mass hierarchy. Thus, the Yukawa matrices just reflect the form of the horizontal symmetry breaking pat tern by the VEVs of X~,a.

Let us consider now the VEV pattern for the horizontal Higgs superfields. In order to be able to write superpotential terms for Xv~,d, one has to introduce also the superfields in the conjugated representations )(u "" (3, 3, 1) and -X'd ' ~ (3, 1, 3). The latter do not couple the fermion sector and just play the role of spectators in the fermion mass generation. The most general renormaliz- able superpotential does not contain, by SU(3) 3

symmetry reasons, any mixed term between X~ and Xd, and so it has a separable form W = W ( X u ) + W ( X d ) , where:

- 3 w ( x , , ) = ~,x,,2~, + x~ + x~,, W(Xd) = #dXd)(d + X 3 + 2 3, (14)

The detailed consideration of the superpotentials of these type is given in Appendix. By means of the bi-unitary transformations the VEV of X~ can be always chosen in the diagonal form, X~ = X D ---- diag(x~, x~, x~), while Xd ----- x D v t, Xd ----

d d d (xl ,x2, x3) , where the unitary matrix V defines the relative orientation of two matrices X~ and Xd in the SU(3)q space and it is nothing but the CKM mixing matrix of quarks, V = VCK M.

It is shown in Appendix that in the supersym- metric limit the VEV eigenvalues of X,, (Xd) are not fixed, but only the product of all three eigen-

u u u d d d values x l x 2 x 3 (XlX2X3). The vacuum degener- acy should be removed by the SSB terms. Then for a certain range of the coupling constants the

X u largest eigenvalues of these fields ( 3, xd) can be order cutoff scale M while others are smaller.

The above can be interpreted in the following manner. In the view of operators like (8), the MSSM Yukawa constants as well as the CKM mixing angles become dynamical degrees of free- dom. In particular, the first operator in (8) im- plies

Yu = diag(Y,,,Yc, Yt) ,,~ Mdiag(S1,S2,,.q3) (15)

In the exact supersymmetric limit the values of the constants Yu,c,t are not fixed - they have fiat directions where only their product is fixed Y~,YcYt -- (#u /M) 3. However, the SSB terms could naturally fix the Yukawa constants so that Yt ~ 1. (For related discussion, see also ref. [15].) The same is true for the constants Yd,~,b. In this way, one can naturally obtain the following hi- erarchy for the upper and down quark Yukawa eigenvalues:

2 Yt : Yc : Y~ ~ 1 : ~ :eu , E~ = ~,,/M Yb : Y~ : Yd "~ l : ed : ~ , ed = # d / M (16)

which for e,, ,,~ 1/200 and Sd "-~ 1/20 well de- scribes the observed spectrum of quark masses.

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414 Z Berezhiani, A. Rossr/Nuclear Physics B (Proc. Suppl.) I01 (2001) 410-420

The question remains what are the CKM mix- ing angles. In the SUSY limit the latter are flat directions as far as the superpotentiai is factor- ized and so the VEV orientation of X~ and Xd remain arbitrary. However, also this degeneracy should be removed by the SSB terms. In partic- ular, one has to consider the following effective operators in the form of D-terms [13]:

M 2

A1Tr (X~tX~tz )

where z - rh02, 2 = rh0 ~ are supersymmetry breaking spurions, rh being typical SUSY break- ing mass. Clearly, such effective operators always emerge in the loop corrections after the supersym- metry breaking. If ~ is a singlet, then the VEVs of Xu and Xd are alligned in the SU(3)q space and thus no CKM mixing can emerge. However, if ~ is an octet of SU(3)q, then, for positive A's, there is a parameter range for which the VEVs Xu and Xd are not alligned anymore and so non- zero CKM mixing can emerge [13].

It is needed to remark that in this model the pattern of the CKM mixing angles is not related to the hierarchy of quark mass eigenvalues, and in general they should be large. The 1-2 mixing angle is indeed order 1, s12 ~- 0.22, while the 2-3 mixing is small, s23 ~- 0.04, for which some fine tuning of the parameters is required. 3

Let us consider now the squark mass and mix- ing pat tern in this model. By the SU(3)H sym- metry reasons, the soft mass terms of the states q, ~, d as well as of the heavy states U, 0-, D, D should be degenerate between families, while the trilinear A-terms should have the structures pro- portional to the Yukawa couplings (10). There- fore, after integrating out the heavy states, the pat tern of the soft mass terms (3) should be the following, the states q = (u, d) do not mix the heavy fermions, so the soft masses of the left- handed squarks maintain the SU(3)H degeneracy (at the decoupling scale M). As for the states and d, they mix the heavy states according to

3Interestingly, then the third mixing angle is predicted as s13 "~ (ms/mb)2(s12/s23), in a good agreement with the experimental data [13].

(12), so that the soft mass terms of the right- handed squarks should get the forms:

-2 t -2 t 2 2 = rh2u + m 2 u Y u y ~ + m 3 ~ ( y u y u ) mfi

- 2 t - 2 t 2 m~ : rh~d + m2dYdY d + m3d(YaYa) (18)

where the factors are order rh 2. Hence, the latter are not degenerate, but they are fully ailigned to the Yukawa matrices Y= and Yd respectively. In the similar way, one can easily see that the trilin- ear terms Au,a (2) are also fully ailigned to the matrices Vu,d. Thus, in this theory no FC con- tributions emerge at the flavor scale M. Clearly, the initial conditions for the SSB terms are dif- ferent from the universal ones usually adopted in the MSSM. For computing the SSB terms at the electroweak scale, the above expressions should be evolved down by the renormaiization group equations. However, all FC effects will remain un- der controll and the observable FC rates should be of the same order as in the "universal" MSSM.

The similar considerations can be applied in the lepton sector, for which the maximal chiral flavor symmetry is SU(3) 2 = SU(3)t x SU(3)e:

l ~ = ( u ~ ~ ( 3 , 1 ) , ~ , - ( 1 , 3 ) (19) \ ] e

Therefore, the effective operators for the charged lepton and neutrino masses are

Xe eal, H1 + --M-~l,lyH~, (20) M where Xe ~ (2, 3) and X~ -~ (6, 1) are horizon- tal Higgs superfields and M is the flavor scale. These operators can emerge from the integrating out of the heavy charged leptons E, E and neu- tral leptons N , /~ (right-handed neutrinos) in the following representations of SU(3)2:

E,, Ni "~ (3, 1), /~,,/~i ..~ (:], 1) (21)

The relevant superpotential terms read:

eEXe + E,1H~ + EEE,+ N2X~ + N1H2 + £NN-(22)

where ~ is some scalar which can be a singlet or octet of SU(3)I. Therefore, for the lepton Yukawa matrices we obtain:

Y~ = X e M ~ 1, Y , / M = M N I X , M N t (23)

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Z. Berezhiani, A. Rossi/Nuclear Physics B (Proc. Suppl.) 101 (2001) 410--420 415

where X~,~, = (Xe,~,), and ME,N '~ (~) are the mass matrices of the heavy states. Once again, the VEV of X~ can be always chosen in the di- agonal form, X . = X D while Xe -- x D v I , where the unitary matr ix V~ defines the relative orien- tat ion of two matrices X~ and X~ in the SU(3)l space and it is related to the neutrino mixing ma- trix as Vl = VMNS- As in the case of quarks, the hierarchy between the mass eigenstates can find a natural origin in the solution of the Higgs super- potential for X~ and X~, while the mixing pat tern will be fixed by the SSB terms analogous to (17). As we remarked above, the large neutrino mix- ing angles can be obtained in this situation in a rather generic case.

3. SU(5) × SU(3)H m o d e l

Let us consider now the grand unification case. In SU(5) model the fermions of one generation are unified within two multiplets, 10-plets t -- (~2, q, ~) and 5-plets f - - (d, l). Let us assume the horizontal symmet ry SU(3)H which unifies three fermion families as:

j~ ~ (5, 3), t, .,, (10, 3), (24)

(i = 1,2,3 is SU(3)H index), while the Higgs superfields are singlets of SU(3)H, H ,.~ (5, 1) and

~ ( 5 , 1 ) . Since the fermion bilinears t ransform as 3 x

3 -- :] + 6, their "standard" Yukawa couplings to the Higgses are forbidden by the horizontal symmetry. Hence, the fermion masses can be induced only by higher order operators involv- ing a set of "horizontal" Higgs superfields X i3 in two-index representations of SU(3)H: sym- metric X] 3 ----- S {~3} ~ (1,8) and ant isymmetric X~3 - A[~3] = e~3kAk ,,~ (1,3): 4

S~3 + A~3 S~3 - - 2 t~t,H + fit3[-1 + -M--~f, f3H (25)

M M

where M is some large scale (flavor scale). In this way, the fermion mass hierarchy can be naturally

4The theory may also contain conjugated Higgses "Y~3 in representations S ~ (1, 6) and A ... (1, 3). These usually are needed for writing non-trivial superpotential terms in order to generate the horizontal VEVs (see next Section). These fields, however, do not couple the fermions (24) and thus do not contribute to their masses.

linked to the hierarchy of the horizontal symme- t ry breaking scales [11,12]. Needless to say, by SU(5) symmet ry reasons, the ant isymmetr ic Hig- gses A can part icipate only in second term.

In particular, let us assume tha t the horizon- tal Higgses include a sextet S and one or more triplets A. Without lose of generality, the VEV of S can be taken diagonal:

( $ 1 0 0 ) ( S i J ) = 0 82 0 , Sa>>S2>>,-ql, (26)

0 0 83

while the triplet A i.i - e'JkAk in general can have the VEVs towards all three components: (0 (A'3) = - .As 0 A1 , A I> A2>A3 (27)

- A 2 - A 1 0

Therefore, in the low-energy limit the operators (25) reduce to the Yukawa couplings which in terms of the dimensionless VEVs S = ( S ) / M and A = ( A ) / M read as:

Y~ = S, Vd, y T = pS + A, Y~ = ~/S (28)

where p and 77 are the proportionali ty coeffi- cients related to different coupling constants in (25). This predictive texture, so called Stech ansatz [11,17], is completely excluded on the phe- nomenological grounds. However, its realistic modifications are possible as discussed below.

In view of the renormalizable theory, the opera- tors (25) can be obtained as a result of integrating out some heavy fermion states with mass order 111. It is natural to assume tha t this scale itself emerges from the VEVs of some fields which can be in singlet or octet representations of SU(3)H. In this case, all terms in the superpotential can become trilinear terms, and one can impose a dis- crete T¢ symmet ry under which all superfields as well as the superpotential changes the sign. In particular, operators (25) can be obtained by in- tegrating out the following heavy states:

T ~ ~ (10, 3), T, ~ (]-0, 3)

F-~ ~ (5, :]), F, ~ (5, 3)

N ' ~ (1, :]), N , ,-, (1, 3) (29)

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416 Z Berezhiani, A. Rossi/Nuclear Physics B (Proc. Suppl) I01 (2001) 410-420

from the superpotential terms of the form:

WT = tTH + f T + K-IY:TT + TtS,

WE = f F A + E F-F + -ff t [-I

W y = ] N H + EN--N + S-N 2 (30)

If E is a singlet, then one immediately obtains the ansatz (28). However, one can assume that E contains also the SU(3)H octet with the VEV towards the As component, and in addition the antisymmetric scalars A contain also an adjoint of SU(5), A ~ (24, 3). The form of superpotential (30) can be motivated by some additional symme- try reasons, which differently transform S and A.

Let us consider now the superpotential of the horizontal Higgses S and A. Due to the different quantum numbers with respect to additional dis- crete symmetries, one can easily have a situation when the superpotential of these fields has a fac- torized form, W -- W(S) + W(A). In addition, the discrete 7Z symmetry requires the superpo- tentials of the form:

W ( S ) = Z ( h 2 - S S ) -]- Z 3 ~- S 3 -]- ,~3 (31)

and

W(A) = Z'(A 2 - Aft.) (32)

where Z and Z' are some singlet superfields. Clearly, if Z has a non-zero VEV, there is a solu- tion when the sextet S has a diagonal VEV with non-zero eigenvalues (S) = diag(Sl ,S2,Sa) . In this case, the term Z S S plays the role of the mass term/*SS. From the second term, the field A gets a non-zero VEV which orientation with respect to S will be determined by the SSB terms pattern.

We see that Yt ~ 1 implies 83 '~ M, close to cutoff scale, which can naturally arise from the Higgs sector. Similarly one can expect that also Yb,~ ~ 1 which would require large tan/3 regime. However, in realistic schemes also moderate tan/3 can be naturally accommodated [20]. The flavor scale in the theory are comparable and can be consistently thought to be close to the GUT scale Ma "~ 1016 GeV.

This theoretical background allows us to moti- vate the following Yukawa matrices [20]:

Y~ = S, Yd = pS + b - l A d ,

y v = r /b- iS , y T = pS + b - l A e , (33)

where S is a symmetric matr ix which can be taken diagonal, p and ~ are some proportionality coeffi- cients, Ae,d are antisymmetric matrices with 24- plet dependent entries inducing different Clebsch factors for the down quarks and charged leptons, and b = Diag(1, 1, b), where b is an asymme- try parameter induced by the SU(3)H symmetry breaking due to the interplay of the singlet and octet VEVs. Clearly the above pat tern represents an extension of the Stech-like texture considered in [20]. A careful analysis proves that the above Yukawa pattern provides a successful description of fermion masses and mixing angles (for details see [20]). Alternatively, in another set up, one could obtain also a different predictive pattern:

Yu = S, Y d = b - I ( p S + Ad) ,

Yv = ~ b - l S , y T = b - l ( p S + A~), (34)

Both these patterns have a remarkable property. Namely, they offer the key relation to understand the complementary mixing pat tern of quarks and leptons. The origin of this relation in fact can be traced to the coincidence of the Yukawa matrices Yd = y T in the minimal SU(5) theory. In the textures (33) and (34) this relation is not exact, but is fulfilled with the accuracy of the different Clebsch factors in Ad and A~. Explicilty this means that the 2-3 mixing angles in the quark and lepton sectors respectively should be:

tan 0~a _~ b_l/2 rn/-~, tan013 _~ b l / 2 ~ / - ~ (35) V mb V m r

where b is a dimensionless parameter which in the range ~ 10 correctly fixes the mixing angles. From here one immediately obtains the product rule [19]:

tan 0~3 tan 0t23 ( rnt, rns ~ 1/2 ~_ - - ( 3 6 ) \ mrmb /

This product rule indeed works remarkably well. It demonstrates a 'see-saw' correspondence be- tween the lepton and quark mixing angles and tells us that whenever the neutrino mixing is large, tan 0~3 ~ 1, the quark mixing angle comes out small and in the correct range, tan 9~3 .,~ 0.04.

Let us remark, however, the patterns consid- ered above relay on the fact that the sextet S

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has a VEV with non-zero eigenvalues (S) = diag(S1,82, 83). Such a solution of the Higgs su- perpotential, with a hierarchy $3 >> ,-¢2 >> 81, is indeed possible if the horizontal symmet ry •U(3)H is a global symmetry. However, this so- lution dissapears if SU(3)H is a local gauge sym- metry, since it is not compatible with vanishing gauge D- terms of SU(3)H.

In this case, however, one can make use of another solution (S) = (S) = diag(0 ,0 ,S) , wich is compatible with the SU(3)H D-term flatness (see Appendix). As for triplet fields, the most general VEV pat tern is /A) -- (A) -- diag(B, 0, .4) (see Appendix).

So, we can s tar t from the effective operators

A*J~ ~ Ai3~ ~ f, tjH(37) (-~-+ M2 ]t,t3H, (-~-+ M 2 ]

which structures can be obtained after integrat- ing out the heavy states from the appropriate su- perpotential . In this way, one can obtain the the following Yukawa textures [16]:

Y / = - A I 0 B) , f = u , d , e , u (38) 0 -Bf C/

which resembles the familiar Fritzsch ansatz [18]. The lat ter in fact corresponds to the particular

r = B/ , which can be obtained if the there case B f is no SU(3)H breaking by the octet representa- tion, i.e. the scalar ~ is a singlet [12]. 5 This sit- uation by now is completely excluded by the ex- perimental data. However, for the case B~ ¢ B/ , one can achieve a very good description of the quark and neutrino mass and mixing patterns.

One can take more general approach and con- sider the effective operators which also incorpo- rate the SU(5) adjoint Higgs (I):

(X*_~ +--M -5-+X~'@ X~"~2~M3 ] tJ, H,

( X*~_~ + ---M -~ X~J~2 ~ ] (39)

5remark, that in this case the neutrino mass matrix can have only 33 non-zero element C~.

where Z0,1,2 a re the horizontal scalars, which can be symmetr ic or antisymmetric. This pat- tern can be mot ivated by some additional sym- metry reasons, which differently transform the horizontal Higgses X0,1,2. E.g. one can con- sider a Z3 symmet ry acting on the superfields as k9 --. ko exp ( i~Q~ , ) , and take the corresponding charges as Q(Xo) = O, Q(X1) = 1, Q(X2) = 2 and Q((I)) = - 1 .

The relevant superpotential can have a form:

tTH + fTH + T.T:F + Xo:Ft XITT1 + E1T1T1 + '~Tlt X2TIT2 + ~2T2T2 + , ~ 2 t (40)

where Z, T~l, T~2 are some superfields in singlet or octet representations of SU(3)H (in the fol- lowing, singlets will be denoted as I and octet as T.), with the VEVs order M. These VEVs can emerge from the Higgs superpotential includ- ing linear terms A2Ik for singlets, with Ak "~ M, and all possible trilinear terms consistent with the symmet ry (among those can be also the ones like I(I) 2 and (1)3). In this way, all singlet and ad- joint fields can get order M VEVs. s There is also a possibility to generate some of these VEVs by means of the anomalous U(1)A symmetry, which also leads to the scale M slightly bigger than the grand unification scale M c ~ 1016 GeV.

The observed pa t te rn of the fermion masses clearly requires tha t the "leading" horizontal scalar should be a SU(3)H sextet, X0 -- So. As for other fields X1,2, these can be sextets or triplets. In the lat ter case one should ob- tain the Frittzsch-like textures ([18]) already con- sidred above. It is interesting to consider also the case when these fields are sextets, $1,2.

Let us consider now the superpotential of the horizontal Higgses. Due to the different quantum numbers of S0,1,2 with respect to additional dis- crete symmetries, one can easily have a situation when the superpotent ial of these fields has a fac- torized form, W = ~ W(Sk), where

w ( & ) = Zk(h 2 - Sk&) + S~ + ~3 (41)

6The linear terms could effectively emerge from the trilin- ear couplings IQQ, with the fermions Q,t~ of some extra gauge sector in the strong coupling regime.

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418 Z Berezhiam, A. Rossi/Nuclear Physics B (Proc. Suppl.) 101 (2001) 410-420

In this case, each of the VEVs (Ski can have only one non-zero eigenvalue, however these VEVs can have non-trivial orientations with respect to each other in the SU(3)H space, with generically large angles determined by the SSB terms. In this case one obtains the Yukawa matrices of the following structures:

Y / = P3 + G'P2 + E~P1, f = u, d, e (42)

where P1,2,3 are rank-1 matrices with O(1) ele- ments which can be chosen as

P3 = (0, 0, 1) T • (0, 0, 1),

P2 = (0, c, s) T • (0, c, s),

P1 = (x, y, z) T • (X, y, Z), (43)

and z f ,~ (~P}/M are the Clebsch factors for the different types of fermions projected out the cou- plings of ~. Yukawa matrices of such structures have been considered in ref. [21], and earlier, in the context of radiative mass generation mecha- nism, in ref. [22].

All these considerations not only lead to a gen- eral understanding of the fermion mass and mix- ing pattern, but also can lead to the predictive schemes. In addition, the schemes constructed by the heavy fermion exchanges like (40), exhibite the remarkable allignment of the sfermion mass matrices to the Yukawa terms, and thus are nat- ural as far as the supersymmetric flavor problem is concerned [16].

The following remark is in order. The horizon- tal symmetry could guarantee the automatic R- parity. The SU(3)H symmetry does not work for this, though in certain context it could suppress some R-violating terms [23]. However, the auto- matic R-parity can be achieved in the context of the horizontal symmetry SU(4)H [24].

4. Appendix: Hor izonta l V E V structures

Consider the following superpotential terms in- cluding the superfields S = Si3 and S - ~3:

W S = - ~ S S + S 3 Jc ~3 , (44)

where # is some mass parameter, S 3 = l_~jk_~bco ~, o (similarly for ~3), and order 1 coupling constants are absorbed. Observe also

that this superpotential is manifestly invariant under Z3 symmetry: S --* e x p ( i ~ ) S and S --* e x p ( - i ~ ) S .

Without loss of generality, the VEV of S can be chosen in the diagonal form, (S) -- Diag(S1,82,83). Then the condition of vanish- ing F-terms Fs, F~ = 0 implies that (S) is also diagonal, (S) = Diag(31, $2 ,33) , and

,.qvS 3 = ttei3k3 k, S~S 3 = ~E~JkSk, (45)

So, in the exact supersymmetric limit the VEV pattern of S and S is not fixed unambiguously and there remain flat directions which represent a two-parameter vacuum valley. In other words, the six equations (45) reduce to four conditions:

8 1 3 1 : 8 2 3 2 = 8 3 3 3 -- ~2, 818283 = tt 3 (46)

while the others are trivially fulfilled (e.g. equa- tion 818283 =/~3 follows from eqs. (46)). Thus, in principle the eigenvalues 81,2,3 can be different from each other, say 83 > 82 > 81. Then eqs. (45) imply that 31,2,3 should have an inverse hi- erarchy, 33 < 32 < ,~1. More precisely, we have 31 : 3 2 : 3 3 = 8 ~ - 1 : 8 ~ 1 :S~ -1.

The fiat directions of the VEVs are lifted by the soft SUSY breaking D-terms: T

£ = - / d 4 O z ~ , [ o ~ T r S f S + ~ - ~ ( T r S f S ) 2 +

7 Tr(StSStS) + ...] (47) M 2

having a similar form also for S. Here z = rh82, = ~ 2 are supersymmetry breaking spurions,

with rh ,-~ 1 TeV. The cutoff scale M is taken as the flavor scale, i.e. the same as the one in super- potential terms (25), and we assume that M > #. The stability of the scalar potential associated with (47) implies that /3 > 0 and 7 > - f l , whereas a can be positive or negative. In the former case the minimization of the potential, under the con- ditions (46), would imply that 81 = 82 = 83 = #, i.e. no hierarchy between the fermion families. In the latter case, however, the largest eigenvalue of

7The F-terms ,.~ f d2OzW are not relevant for the VEV orientation as far as they just repeat the holomorphic in- variants like SS and det S whose values ave already fixed by the conditions (46).

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Z Berezhiani, A. Rossi/Nuclear Physics B (Proc. Suppl.) 101 (2001) 410-420 419

S and 5', respectively $3 and $1, grow up above the typical VEV size # and reach values of the order of the cutoff scale M:

( o 83, $1 = 2(/~ q- ~/) M ~ M (48)

Then it follows from (46) that

$ 2 , 8 2 = / z ,~ e 8 3 , 81 , ~-~3 : ~2 - - ~'~ e 2 8 3 , (49) $3

where e ~ ]~/M. Let us consider now modification (31) of the

superpotential (44) when we assume a discrete symmetry T~ under which all superfields in the theory as well as the superpotential changes the sign. In this case we have two solutions:

(i) (Z) # 0: clearly, in this case the F - t e rm conditions are the same as in (46) apar t of the fact tha t the mass scale tt should be substi tuted by the VEV (Z) = #. The latter is then fixed by the condition Fz = 0 which implies (Z) = A. Thus we still have flat directions which will be stabilized by the soft SUSY breaking terms, so that we can obtain a hierarchy of the eigenval- u e s S 3 : S 2 : S 1 ' ~ 1 : e : e 2, where e = A/M. This solution exists if SU(3)H is a global sym- metry, however it is no more valid if SU(3)H is local. The reason is simple: the inverse hierarchy of VEVs S I , 2 , 3 and $1,2,3 is not compatible com- patible with the D-flatness condition of the gauge terms D~ = ~--],~ X t T ~ X , where T ~ are SU(3)H generators, a = 1,..8, unless 33 = 82 = 31. This solution, however, directly contradicts to the fermion mass hierarchy, s

(ii) (Z) = 0: in this case the condition Fz = 0 tells tha t ~--:~SiS ~ -- A 2, while the conditions Fs, ~ = 0 yield S 1 S 2 S 3 : 0 and ~ 1 ~ 2 ~ 3 : 0.

Therefore, the VEVs of S, S can have only one non-zero eigenvalue, which can be chosen e.g. as 83 and ~3, so that $3S 3 = A 2. Clearly, this so- lution is compatible with the D-flatness condi-

Sin principle, one could imagine the theory containing addi t ional " spec ta to r" superfields in different represen- ta t ions of SU(3)H and their VEVs are or iented so tha t to cancel the contr ibut ions of (S) in gauge D- terms of SU(3)H. In this way, not very appeal ing though, one could save the hierarchial V E V solution also in the case of local SU(3)H.

tion. The requirement D~ = 0 simply fixes that 83 ---- ,~3 = A.

Thus, in the case of local SU(3)H we ob- tain the non-degenerate solution (S) = (S) = A • diag(0, 0, 1). In this case the operators like (25) involving S can induce only the third gener- ation masses and so for obtaining the order one Yukawa couplings we need to take A ~ M, which is a pret ty natural assumption.

Now consider superpotent ial (32) for anti- symmetr ic Higgs superfield A. In the exact super- symmetr ic limit the ground state has a continu- ous degeneracy (fiat direction) related to unitary transformations A ~ UA with U C SU(3)H. In other terms, the superpotent ial W = Ws + WA has an accidental global symmet ry SU(3)s x SU(3)A, with two SU(3) factors independently transforming the horizontal superfields S and A.

Similarly to the case of the Higgs fields S, S, now the conditions FA, F A = 0 can only fix the values of holomorphic invariant A.4 = A 2, while D- te rm flatness require that (A) = (.A). By uni- tary t ransformation A ---+ UA (U C SU(3)A), one can choose a basis where the VEV of A points towards the first and third components. The rel- ative VEV orientation between S and A should be fixed from the soft D-terms:

1 / M2 d4Ozz [aTr(StSAtA)+

/3Tr(StSEtE) + A2Tr(AtAEtE)] (50)

with z = rh02 and 5 = rh02 being supersymmetry breaking spurions. 9 Namely, for a, B, 7 all posi- tive, one can have (A) = (B, 0, .A) with B and Jt both non-zero.

R E F E R E N C E S

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