Constraints on parameters of the RF parity model from quark and neutrino mass matrices

Nuclear Physics B 564 Ž2000. 3–18 www.elsevier.nlrlocaternpe

Constraints on parameters of the R F parity model from quark and neutrino mass matrices Jihn E. Kim a

a,b

, Bumseok Kyae a , Jae Sik Lee

b

Department of Physics and Center for Theoretical Physics, Seoul National UniÕersity, Seoul 151-742, South Korea b School of Physics, Korea Institute for AdÕanced Study, Cheongryangri-dong, Dongdaemun-ku, Seoul 130-012, South Korea Received 12 May 1999; received in revised form 17 August 1999; accepted 22 September 1999

Abstract The forms of quark and lepton mass matrices are severely restricted in the R F parity model. We determine the form of the quark mass matrix first and derive the form of the neutrino mass matrix. For this to be consistent with the present experiments, we conclude that the masses of the superpartners of the right-handed down-type squarks and sneutrino vacuum expectation values satisfy, m d˜R ²j n˜m : R 400 GeV 2 or m d˜R ²j n˜t : R 400 GeV 2. We also find that without a sterile neutrino it is difficult to obtain a large mixing angle solution of nm and nt . With a sterile neutrino, we show a possibility for a large mixing angle solution of nm with nsterile . q 2000 Elsevier Science B.V. All rights reserved. PACS: 11.30.Hv; 12.15.Ff; 14.60.Pq Keywords: R parity; Supersymmetric mass matrix; Horizontal symmetry

1. Introduction The strong CP problem has two most attractive possibilities: the spontaneously broken Peccei–Quinn ŽPQ. symmetry w1x and the massless quark solution w2x. For the case of the axion, one can easily construct models for a very light axion w3–6x, since the very light axion is just an addition to the standard model and its effect to the standard model is suppressed by powerŽs. of the axion decay constant Ž Fa ; 10 12 GeV.. For very light axions, there exists a good motivation from the superstring that a model-independent axion should exist w7x. Probably, this is the most attractive theoretical reason for the very light axion w8,9x. On the other hand, it has been very difficult to construct a consistent model for the massless up quark because any new symmetry to make the up quark massless affects the predictions of the standard model sector significantly. It has been conjectured that somehow a symmetry in superstring models would render up 0550-3213r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 Ž 9 9 . 0 0 6 1 3 - 6

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quark massless. But this is also disfavored by the problem of no global symmetry in superstring models except the one corresponding to the model-independent axion w10x. Even if one has a theory for a massless up quark, it will satisfy many constraints related to the fermion mass matrix. Here lies the difficulty of building a massless up quark model. Therefore, it will be very interesting to construct phenomenologically allowable massless up quark models. Indeed, there exist an attempt to build a model toward m u s 0 with a UŽ1. horizontal symmetry w11,12x. Recently, a kind of R parity named ‘‘R F parity’’ was introduced to solve the strong CP problem with almost massless up quark w13x. The R F parity model was constructed such that it satisfies the proton decay constraint and possibility for generating quark and lepton masses w13x. In this paper, we study the R F parity model further to constrain its parameter space from the known quark and lepton mass matrix textures. We find that it is not impossible to satisfy the phenomenological constraints. The R F parity is defined as R F s Ž y1 .

3 BqLq2 S

Ž y1.

2 IF

,

Ž 1.

where F s d f 1 Ž f s 1,2,3., and B, L,S, I and F are the baryon number, lepton number, spin, weak isospin, and the first family number, respectively. In this model the R F parity distinguishes the first family from the second and the third ones. Thus, it is possible to make the first family massless. This is good for up quark but bad for electron and down quark. Therefore, it was necessary to break the R F parity spontaneously by vacuum expectation values ŽVEV. of the sneutrino fields. The sneutrino fields have the same quantum number as the down-type Higgs field H1; thus their VEV’s render electron and down-type quark massive.1 Still up quark remains massless since the quantum numbers of sneutrinos are different from the up-type Higgs field H2 . Another kind of R parity can be defined such that U1c Žthe so-called right-handed up quark singlet superfield. has a different quantum number from the rest of the quark and lepton superfields. This possibility will make up quark massless, but does not explain why the other first family members are light. This possibility will be discussed in Section 5 after exploiting the phenomenological implications of the R F parity model. The R parities responsible for the masslessness of up quark can have a root in superstring models. Presumably these R parities can be a discrete subgroup of UŽ1.R global symmetry w8x. Even though the string theory does not admit a global UŽ1.R symmetry, it can allow discrete groups. Therefore, the study of discrete groups toward massless up quark can have a far reaching extension toward superstring models. In Section 2 we discuss the textures of the quark mass matrix, and obtain bounds the sneutrino VEV’s. In Section 3 we study the lepton mass matrix, and obtain bounds on the masses of zino and down-type squarks. In Section 4 we introduce UŽ1. horizontal gauge symmetry so that the desired mass matrices are obtained from the Froggatt–Nielsen idea w16x. The possibility of the opposite R parity charge for U1c is discussed in Section 5. Finally, we summarize our works in Section 6. In Appendix A we present superpotential terms obeying the UŽ1. X = UŽ1.Z symmetry, and suggest a mechanism of generating the m-term. 1

With a different motivation, similar idea was considered in Refs. w14,15x.

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2. The CKM matrix The most general d s 3 superpotential consistent with the R F parity is W0 s f ilj L i Ejc H1 Ž i / 1 . q f iuj Q i Ujc H2 Ž i / 1 . q f idj Q i Djc H1 Ž i / 1 . q l1 jk L1 L j Ekc Ž j / 1 . q lXi1 k L i Q1 D kc Ž i / 1 . q lX1 jk L1 Q j D kc Ž j / 1 . .

Ž 2.

This superpotential gives m e s m d s m u s 0 naturally. To explain the non-zero m e and m d , we introduce a soft R F parity violating term, which is hoped to mimic a general feature in other R F breaking models, W1 s MS S 2 q e 2 S q f S i SL i H2 Ž i / 1 . , Ž 3. where S is a singlet superfield with Y s 0 and R F s y1 and we suppress the d s 4 baryon number violating terms. With this R F parity violation, the electron and the down quark obtain their masses through the vacuum expectation values ŽVEV’s. of the S and sneutrino fields given by

e2 ² S: ;

MS

,

² n˜ i : ;

fS i Õ 2 e 2 Mn˜2i

,

Ž 4.

where Õa ’s are VEV’s of the neutral Higgs fields ² Ha0 : Ž a s 1,2. and Mn˜2i denotes the mass of the ith generation sneutrino field. Let us note that ² n˜ i :’s are not suppressed by 1rMS due to the MS S 2 term in W1. From the above R F parity conserving and violating terms, the mass matrices of upand down-type quarks and charged leptons are given by Õ2 Ž Mu . i j s f jiu '2 Ž j / 1. , Õ1 X ² : Ž Md . i j s f jid Ý lXk ji ² n˜ k :d j1 , '2 Ž j / 1. q l1 ji n˜ 1 Ž j / 1. q ks2,3 Õ1 ² : Ž Ml . i j s f jil Ž 5. Ý l k ji ² n˜ k :d j1 . '2 Ž j / 1. q l1 ji n˜ 1 Ž j / 1. q ks2,3 The vanishing elements of the first column of mass matrix of the up-type quark guarantees the massless up quark at the tree level. And it is obvious that the elements of the first columns of mass matrices of the down-type quarks and the charged leptons are ˇ also zero without the soft R F parity violation, i.e. without sneutrino VEV’s, ² n˜ 2 : and 3. Here, we try to satisfy the phenomenologically known quark mass hierarchy. This hierarchy is expressed in terms of a small expansion parameter, l , 0.22 which is sine of the Cabibbo angle. Using renormalization group equations,2 we have md Ž mt . ms Ž mt . mc Ž mt . mb Ž mt . ; l4 , ; l2 , ; l4 , ; l3 . Ž 6 . mb Ž mt . mb Ž mt . mt Ž mt . mt Ž mt . These ratios of masses indicate that the unitary matrix defining mass eigenstate up-type quarks in terms of the weak eigenstate quarks has the form UL s 1 q O Ž l4 . up to phases.3 Therefore, we consider only the down-type quark mixing from now on, or VCK M s DL q O Ž l4 .. 2 3

Only the strong interaction coupling is dominant at energy scales lower than the top quark mass. We find that the phases and O Ž l4 . term of UL do not change our results.

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In case that only down-type quarks mix, M d and VCKM are constrained by the following relation: † † 2 2 2 VCK M M d M d VCKM s diag Ž m d ,m s ,m b . ,

Ž 7.

which allows us to extract constraints on the parameters of M d from the measured values of the VCK M elements. Another experimental data we take into account in the R F model are the decay modes of Kq w17x, B Ž Kq™ p 0n eq . ( 0.0482,

B Ž Kq™ pq nn . ( 5.2 = 10y9 , from which one can derive a bound on

lXŽ i/1 . 1 j Q 0.012

ž

m d˜R j 100 GeV

/

lXŽ i / 1.1 j

Ž 8.

w17x,

,

Ž 9.

where m d˜R j denotes the right-handed down-type squark mass. Combining this upper bound with constraints from VCK M will lead us to lower bounds on the sneutrino VEV’s. If R F parity is exact, then the mixing matrices of up- and down-type left-handed quarks, which diagonalize Mu† Mu and M d† M d , are given by 4 1 UL s 0

0 uL ,

1 DL s 0

0 dL ,

Ž 10 .

where u L and d L are 2 = 2 matrices. From these, the CKM matrix is given by RF VCKM s

1 0

0 . u†L d L

Ž 11 .

Therefore, mixing of the first family with the other families and a CP phase do not RF appear in VCK M . Thus the R F parity must be broken spontaneously to render phenomenologically acceptable angles. The spontaneous symmetry breaking of the R F parity is achieved by the VEV’s of sneutrino fields, ² n˜ i : Ž i s 2,3.. In the presence of sneutrino VEV’s, let us parameterize the down-type quark mass matrix as

e1 Md s e 2 e3

Ž m2 . 1 Ž m2 . 2 Ž m2 . 3

Ž m3 . 1 Ž m 3 . 2 ' Ž e ,m 2 ,m 3 . . Ž m3 . 3

Ž 12 .

Then M d† M d is given by
e P m 3)

4

e ) P m2 < m2 < 2 m 2 P m 3)

e ) P m3 m 2) P m 3 . < m3 <

Ž 13 .

2

The matrices UL , DL ,u L and d L should not be confused with the particle names with the same notations.

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From Ž7., we obtain the following relations: md < e < s 1 q A2 < z < 2 q O Ž l . f 1 q A2 < z < 2 l m s , l

ž(

(

/

< m 2 < s Ž '1 q A2 q O Ž l . . m s , < m 3 < s Ž 1 q O Ž l3 . . m b ,

e ) P m2 s < e < < m2 <

e ) P m3 s < e < < m2 <

1 q zA2

(

2 Ž 1 q A2 . Ž 1 q A2 z .

zA

ž(

m2 P m3 s < m2 < < m3 <

2

1 q A2 z

ž'

A

0

q O Ž l. ,

/

q O Ž l. ,

/

q O Ž l. ,

1 q A2

Ž 14 .

where z s r y ih and l, A, r , and h are the conventional Wolfenstein parameters w18x. Therefore, for example, one can obtain the following down-type quark mass matrix which has eigenvalues of m d , m s and m b and gives a right form for the CKM mixing matrix:

Md s

O Ž l4 .

O Ž l3 .

l3

l2

z Al )

3

Al

O Ž l4 . O Ž l3 . m b .

2

Ž 15 .

1

From Eqs. Ž5. and Ž15., one can derive the following relations:

lX211² n˜m : q lX311² n˜t : s O Ž l4 . m b , lX212 ² n˜m : q lX312 ² n˜t : s l3 m b , lX213 ² n˜m : q lX313 ² n˜t : s z )A l3 m b .

Ž 16 .

If one of ² n˜m : or ² n˜t : dominates, the relative sizes of lX ’s can be fixed independently of the sizes of the sneutrino VEV’s. The numerical values of the CKM matrix elements are given for l s 0.22 and z )A s 0.34 w19x. Therefore, a typical size of the products lXi1 j ² n˜ i : with i s 2,3 and j s 1,2,3 is Ž0.01 y 0.05. GeV. On the other hand, it is not easy to derive some meaningful information on the other lX1Ž i / 1. j ’s because these couplings contribute to M d together with the conventional Yukawa terms proportional to unknown parameters f idj and Õ 1. From Eqs. Ž9. and Ž16., it is obvious that at least one of ² n˜m : and ² n˜t : should satisfy the following inequality: ² n˜m ,t : R 4 GeV

ž

100 GeV m d˜R j

/

.

Note that the lower bound becomes smaller for a larger squark mass.

Ž 17 .

J.E. Kim et al.r Nuclear Physics B 564 (2000) 3–18

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Fig. 1. The one-loop u quark mass.

Though the up-quark mass is zero at tree level, it can be generated radiatively when S field has a vacuum expectation value as shown in Fig. 1 w13x. The one-loop up-quark mass is given by

d mu ;

f k1u lXi1 k mŽkd . f S i ² S :

Ý i , ks2,3

16p 2

;

MSUSY

Ý is2,3

f S i lXi13 e 2

u m b f 31

16p 2

MSUSY MS

,

Ž 18 .

where mŽ2d . s m s and mŽ3d . s m b and we used ² S : s e 2rMS . The combination fS i lXi13 e 2 is constrained as in Eq. Ž16. through the relation ² n˜ i : ; f S i Õ 2 e 2rMn˜2i . Taking m b s 5 GeV, Õ 2 s 100 GeV and Mn˜2i s 1 TeV gives f S i lXi13 e 2 ; 180 GeV 2 . This one-loop up-quark mass should be small to solve the strong CP problem: d m u - 10y1 3 GeV w8,9x. This leads to u f 31

Ž MSrGeV.

- 2 = 10y11 .

Ž 19 .

This bound is stronger than the rough estimation given in Ref. w13x by one order of magnitude.

3. Neutrino mixing and sterile neutrino Due to the VEV’s of the sneutrino and S fields in our case, three neutrinos mix with ˜ ˜ 3 , H˜10 , H˜20 ,S . four neutralinos and one S field in the 8 = 8 mass matrix. In the Ž n i ; B,W basis, the mass matrix is given by M0 s

ž

0

mD

mTD

M

/

,

Ž 20 .

J.E. Kim et al.r Nuclear Physics B 564 (2000) 3–18

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where we neglected the contributions coming from the one-loop diagrams. Here, 0 is the 3 = 3 matrix with 0 entries and m D is a 3 = 5 matrix

°y 1 g ²n˜ : e

2 1

¶

1

X

2 1

X m D s y g ² n˜m : 2 1 y g X² n˜t : 2

2 1

¢

2

g ² n˜e :

0

ym 1

g ² n˜m :

0

ym 2

fS 2

g ² n˜t :

0

ym 3

f S3

0 Õ2

'2

,

Ž 21 .

Õ2

'2 ß

where m 2,3 s f SŽ2,3.² S :. M in Eq. Ž20. is a 5 = 5 mass matrix of the neutralinos and S field

°

Ms

cM2

0

0

M2

1 y g X Õ1 2 1 X g Õ2 2 0

1

2 1 y gÕ 2 2 0

¢

gX Õ2

0

1 y gÕ 2 2

0

0

ym

0

ym

0

Ž fS 2 q fS3 . ² S :

0

Ž fS 2 q fS3 . ² S :

MS

gÕ1

1

¶

1 y g X Õ1 2 1 gÕ 2 1

2

,

ß Ž 22 .

where c ' M1rM2 s Ž5r3.tan2u W , 0.5, assuming the unification relation. Since a typical scale for M is much larger than that of m D , it is enough to use the see-saw formula to find the following reduced neutrino mass matrix w20x:

mneff s ym D My1 mTD s

y

Õ 22 2 MS



cg 2 q g X 2

0 0

0 f S22

0

f S 2 f S3

D



L2e

L e Lm

L e Lt

L e Lm

Lm2

Lm Lt

L e Lt

Lm Lt

Lt2

0 1 f S 2 f S3 q O , MS2 2 f S3

0

ž /

0 Ž 23 .

where L i and D are given by

L i s m ² n˜ i : y Õ1 m i , D s 2 m y2 cM2 m q Õ 1Õ 2 Ž cg 2 q g X 2 . .

Ž 24 .

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Neglecting the O Ž1rMS2 . contributions, the neutrino mass matrix, Eq. Ž23., has one zero eigenvalue and two non-zero eigenvalues mn 1 s 0, mn 2 s y

mn 3 s

Ž fS 2 Lt y fS3 Lm .

2

2 q Ž f S22 q f S3 . L2e Õ 22

2 L 2 MS

Ž cg 2 q g X 2 . L2

y

Ž fS 2 Lm q fS3 Lt .

2

Õ 22

2 L 2 MS

D

,

,

Ž 25 .

where L2 s L2e q Lm2 q Lt2 . Naively, one would expect that Lm,t are smaller than L e because L e comes from the R F-parity conserving parts. But, this expectation is not true because ² n˜m ,t : given in Eq. Ž4. is not suppressed by MS . Namely, the sneutrino n˜e , preserving the R F parity, and sneutrinos n˜m,t , violating the R F parity, can have comparable VEV’s, or even ² n˜e : can be smaller than ² n˜m,t : depending on the sizes of the corresponding soft SUSY breaking terms. This is because below the R F breaking scale their VEV’s are determined by the respective potentials. Then, the mixing matrix Vn which diagonalizes mneff satisfies VnT mneff Vn s diagŽ mn 1,mn 2 ,mn 3 ., and has the following form:

LtrLX Vn s



y Ž L e Lm . r Ž LLX .

L erL

X

L rL

0 X

yL erL

0

LmrL q O X

y Ž Lm Lt . r Ž LL .

LtrL

1

ž / MS

,

Ž 26 .

where LX 2 ' L 2e q Lt2 . On the other hand, the mixing matrix of the left-handed charged leptons 5 Ž L L . which satisfies L†L Ml†Ml L L s diagŽ m e ,mm ,mt . does not mix the first family with the other family members without R F-parity violation as the left-handed quark mixing matrices in Eq. Ž10. do not mix the first family members. Therefore, in the presence of R F parity violation we obtain

ne s Ž L†LVn . 1 k n k f Ž LtrLX . n 1 y Ž L e Lm . r Ž LLX . n 2 q Ž L erL . n 3 ,

Ž 27 .

where n i Ž i s 1,2,3. are the mass eigenstates. From the lower bounds on ² n˜m,t : given in Eq. Ž17., one can obtain the following lower bound on mn 3 : mn 3 0

cg 2 q g X 2 D

m n˜m ,t :2 ; 80 KeV 2²

ž

300 GeV M2

1 TeV

/ž / m d˜R j

2

,

Ž 28 .

where we take c s 0.5 as given by the unification condition. This means that Lm andror Lt are significantly greater than L e to avoid the present mass bound on the electron neutrino: mn - O Ž1. eV. e

5

The matrix L L should not be confused with the left-handed lepton superfield.

J.E. Kim et al.r Nuclear Physics B 564 (2000) 3–18

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Let us assume Lt 4 L e and the off-diagonal elements of charged lepton mass matrix Ml are zero, then we obtain ne ; n 1 ,

nm ; Ž Lt n 2 q Lm n 3 . rL , nt ; Ž yLm n 2 q Lt n 3 . rL .

Ž 29 .

If one of Lm and Lt dominates, then nm or nt will be the mass eigenstate n 3 whose mass is greater than ; 100 KeV. This implies that it is hard to explain the observation 2 of the large mixing and Datm , 5 = 10y2 eV given by the Super-Kamiokande in terms of nm y nt oscillation w21x. To explain the Super-Kamiokande observation, let us introduce the sterile neutrino N which is a singlet superfield with Y s 0 and R F s y1. The relevant additional superpotential is given by

(

2

W2 s MN N q MNS NS q f N i NL i H2 Ž i / 1 . q

l˜XXN i jk MP

NUi c Djc D kc ,

Ž 30 .

where M P is the Planck mass. The VEV of N field ² N : induced by ² S : is ² N :;

e 2 MNS MN2˜

q

e 2 MN MNS MS MN2˜

q

Ž fS q f N . MN ² n˜ i : Õ 2 i

i

MN2˜

.

Ž 31 .

If MN and MNS are negligible, then ² N : , 0 and hence we can avoid the fast proton decay, arising from the last term of Eq. Ž28.. The explicit mixing between S and N gives two mass eigenstates SX and N X with MS X f MS

2 MN X f MN y MNS rMS .

and

Ž 32 .

With appropriate MN and MNS values, one could obtain almost vanishing sterile neutrino mass MN X . 2 Assuming < mn2m y mn2 N < , Datm and Lt 4 Lm , from Eqs. Ž4. and Ž25., we obtain mnm ; mn 2 ;

f S22 Õ 22

;

2 MS

Mn˜42² n˜m :2 2 e 4 MS

.

Ž 33 .

(

2 Taking Datm , mnm assuming almost vanishing sterile neutrino mass, we obtain

MS ; 10 10 GeV

Mn˜ 2

4

ž /ž e

² n˜m :2 1 GeV 2

/

.

Ž 34 .

For Mn˜ 2 , 10 TeV, e , 100 GeV, and ² n˜m : , 10y3 GeV, the mass parameter MS should be around 10 12 GeV which is the intermediate scale needed in supergravity and invisible axion w22,23x. Also, the large mixing between N and nmŽ; n 2 . could be obtained taking f N 2 Õ 2 f mn 2 with f N 2 4 f N3 . Note that the assumed hierarchy between the sneutrino VEV’s ² n˜t : 4 ² n˜m :,² n˜e : is consistent with the neutrino data since we introduced a sterile neutrino. In this case, nt can decay to a lighter neutrino plus photon, which may affect the evolution of the universe. The neutrino interaction with electromagnetic fields due to non-zero transition magnetic moment is described by f iX m B Ž n i smn nt . F mn q H.c. ,

Ž 35 .

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where m B denotes the Bohr magneton. Then the partial lifetime of the tau neutrino is w24x

tnt ™ n i ,

2p

y3

mn3t f iX 2m2B

f 4.5 = 10

ž

100 KeV mnt

3

10y7

/ž / f iX

2

s.

Ž 36 .

The decay, nt ™ n i q n j q n k , is negligible compared to the above photon mode. Note also that our assumed hierarchy on the sneutrino masses that the third generation VEV is much smaller than those of the first and second ones, Mn˜ 3 < Mn˜ 1,2 , is possible in the so-called effective SUSY framework w25–29x.6

4. Horizontal symmetry The mass hierarchy of quarks and charged leptons can be explained by introducing an abelian horizontal symmetry UŽ1. X = UŽ1.Z in our framework. This kind of horizontal symmetry at high energy scale, presumably at the Planck scale, might be necessary to introduce an expansion parameter in the mass matrix w16x. In string models, there appear many gauge UŽ1. symmetries which act as horizontal symmetries in models distinguishing all fixed points w30x. In this case, the expansion parameterŽs. is the ratioŽs. of VEVŽs. of SM singlet fieldŽs. and the string scale. In this spirit, we try to search for a possibility of introducing the mass matrices discussed in the previous sections. Because it is very difficult to study the general mass matrix, we are guided first by the phenomenologically plausible relation Ml f M d Žbut with a modification a la Georgi and Jarlskog w31x., and a phenomenological hierarchy m2c Ž m t .rm2t Ž m t . ; l8. As explained before, then UL s 1 q O Ž l4 .; and we can take the following forms for the mass matrices using l as an expansion parameter: 0 Mu ; 0 0

l6 l5 l4

l2 l mt , 1

l7 Ml ; l6 l6

l6 l5 l5

l7 l6 m t . l3

l7 M d ; l6 l6

l6 l5 l5

l7 l6 m t , l3

Ž 37 .

Some elements arise as higher order corrections. For example, the first and the second rows of Mu and Ž1,i . and Ž2,3. elements of M d arise as higher order corrections. Note that the lower right 2 = 2 submatrix of Mu gives a determinant of order l5 m2t . However, this does not imply m c ; l5 m t since Mu must be diagonalized by a biunitary transformation, implying this determinant is not invariant. To get eigenvalues, we diagonalize Mu Mu† in which case we obtain m c ; l4 m t . 6

This effective SUSY was proposed to suppress the SUSY contributions to low-energy flavor changing neutral current ŽFCNC. or CP-violating processes.

J.E. Kim et al.r Nuclear Physics B 564 (2000) 3–18

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Table 1 UŽ1. X =UŽ1.Z charges QX QZ QX QZ

Q1

Q2

Q3

D1c

D 2c

D 3c

U1c

U2c

U3c

9r2 6

0 y1

y5 1

y2 y9

y2 y6

4 y3

1r3 y1

1r3 2

1r3 5

L1

L2

L3

E1c

E2c

E3c

H1

H2

S

4 2

y1r2 y5

y11r2 y3

y3r2 y5

y3r2 y2

9r2 1

y1 y5

14r3 y6

x z

Let us introduce three SM singlet fields, u 1 , u 2 and u 3 , to explain the above mass hierarchies, which seems to be our minimal choice. We further assume a universal VEV to simplify the analysis, ² u 1 :3 f ² u 2 :3 f ² u 3 :3 f l M 3 , Ž 38 . where M is the energy scale where non-renormalizable interactions are introduced but still the UŽ1. X = UŽ1.Z symmetry is preserved below M. In string models, one can identify M s Mstring . We find that the UŽ1. X , Z charge assignments shown in 1 are enough to explain the hierarchies of the mass matrices given in Eq. Ž37., where Q X and QZ denote the UŽ1. X and UŽ1.Z charges, respectively. The above charge assignment is valid for tan b ; O Ž1.. For large tan b , a somewhat different charge assignment will be obtained. Here, we have not fixed the charges of the S field. Its charges can be fixed once we know the size of MSrM. One has to cancel anomalies. Since we are considering a low energy effective theory, we will ignore the UŽ1. X , Z related anomalies except the UŽ1. X , Z –SUŽ3.c –SUŽ3.c anomaly. This is because an introduction of colored scalars at intermediate scale is dangerous for proton stability. If the symmetry breaking scale of UŽ1. X , Z is near the grand unification scale, then the proton stability problem reduces to that of supersymmetric models w32,33x. Anyway, we assume that there are no colored scalars below the scale M. The other anomalies can be canceled by introducing Q X and QZ carrying color singlet superfields which we do not try to specify due to our ignorance of the super high energy physics. Note that as in any hierarchical model of this type, the ratio of charges of horizontal symmetry UŽ1. X , Z can include a large number. The Yukawa couplings rendering quark masses are of the form Q i Ž D c or U c . Ha Ž a s 1 or 2.. Among these, only Q 3U3c H2 is allowed by the SUŽ3.c = SUŽ2.L = UŽ1. Y = UŽ1. X = UŽ1.Z symmetry. To introduce other Yukawa couplings through non-renormalizable terms, we need at least three singlet superfields u 1 , u 2 and u 3 which are assigned with the charges given in Table 2. Then the needed SM invariants become neutral through Žnon.renormalizable interaction with the u i ’s. The above charge assignments do not allow the interactions which can give the non-zero elements in the first column of Mu . All the other elements of Mu and Table 2 UŽ1. X =UŽ1.Z charges of superheavy singlets QX QZ

u1

u2

u3

0 1

y1 y1

1 0

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M d,l anticipated in Eq. Ž37. can be obtained through interactions which are neutral under the UŽ1. X = UŽ1.Z horizontal symmetry. These interactions are shown in the Appendix A. Note that the above charge assignment suggests the relation ² u 1 : s ² u 2 : s ² u 3 : since only the combination of u 1 u 2 u 3 is neutral. The superpotential of u fields is given by 2

Wu s l1 Ž u 1 u 2 u 3 . q l2 Ž u 1 u 2 u 3 . q . . .

Ž 39 .

Since this superpotential is symmetric under the exchanges of Ž1,2,3., it may hint at the same VEV’s of u 1 , u 2 , and u 3 . But at this stage only the hyperplane u 1 u 2 u 3 is guaranteed. The D-flat conditions are 3

3

Ý Q Xa < ua < 2 s 0, as1

Ý QZa < ua < 2 s 0,

Ž 40 .

as1

where Q Xa and QZa are the UŽ1. X and UŽ1.Z charges of ua , respectively. Therefore, u 1 s u 2 s u 3 is the D-flat direction. In addition, the introduction of a universal soft term in the potential m 23r2 Ž u 12 q u 22 q u 32 . would not destroy the relation ² u 1 : s ² u 2 : s ² u 3 :. From this consideration we conclude that u fields have the same magnitude of the vacuum expectation values ² u 1 : s ² u 2 : s ² u 3 : ' l1r3 M ,

Ž 41 .

where l is the expansion parameter. Let us consider the Ž2,2. element of Mu of Eq. Ž37. as an illustration. The relevant term which is neutral under the horizontal symmetry is

Q 110Q 25 Q2 U2c H2 ,

Ž 42 .

where the dimensionless parameters Q i are defined as Q i ' u irM and we assume an O Ž1. coupling. The VEV of u i of order of l1r3 and ² H2 : s Õsin b give

l5 Õsin b U2 U2c f l5 m t U2 U2c ,

Ž 43 .

5.

which is the O Ž l given in Eq. Ž37.. Similarly, other elements of Mu can be obtained through interactions which are neutral under the UŽ1. X = UŽ1.Z horizontal symmetry. Now, let us consider the down-type quark mass matrix M d . For example, the terms which give the Ž1,1. element of M d are

Q 16Q 33 L3 Q1 D 1c and Q 110Q 22 L2 Q1 D1c ,

Ž 44 .

which are responsible for a non-vanishing down quark mass. After the Q i ’s develop VEV’s of order l1r3 and assuming ² n˜ 2 : f 0 and 3ˇ f l4 m t , we obtain

l3 3ˇ D 1 D 1c f l7 m t D 1 D 1c .

Ž 45 .

Similarly, other elements of Mu could be obtained through interactions which are neutral under the UŽ1. X = UŽ1.Z horizontal symmetry. Of course, the above dimensionless coupling constants are given at very high energy scale, e.g. at M. Note that our UŽ1. X = UŽ1.Z symmetry includes the result of the R F parity model w13x, namely the UŽ1. X among UŽ1. X = UŽ1.Z does not allow R F parity violating terms. But the UŽ1. X = UŽ1.Z is more restrictive than the R F parity: it dictates the fermion mass hierarchy through the mass matrix equation Ž37..

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5. The opposite R parity charge for U1c Another choice of making the up quark massless is to distinguish U1c from the others. Thus, an R parity called R U can be defined such that only U1c, H1 and H2 have different quantum numbers from the rest of the quark and lepton superfields. For this purpose, let us assign the R U quantum numbers as R U Ž U1c . s R U Ž H1 . s R U Ž H2 . s q1 and R U Ž other fields . s y1. Ž 46 . Then, the R U -parity conserving superpotential is given by WR U s f ilj L i Ejc H1 q f iuj Q i Ujc H2 Ž j / 1 . q f idj Q i Djc H1 q lXXi j U1c Djc D kc q m H1 H2 ,

Ž 47 . lXXi j s ylXXi j .

where This superpotential gives the following Qem s 2r3 and Qem s y1r3 quark mass matrices: 0 0 Ž2r3. M s H2 H20



0 H20 H20

0 H20 , H20

0

H10

H10

H10

M Žy1r3. s H10

H10

H10 ,

H10

H10

H10



0

Ž 48 .

where we have suppressed the Yukawa couplings. The rows count the singlet anti-quarks, and the columns count the doublet quarks. The charged lepton matrices have the same form as the Qem s y1r3 quark mass matrices. It is obvious that Det M Ž2r3. s 0, implying a massless u-quark. But, it does not explain why the other first family members are light. Note that the R U parity prevents the proton from decaying into ordinary particles such as to eqq p 0 . However, the proton can decay if gravitino or axino is lighter than proton w34–36x. In this case, lXXi j should be severely constrained for some reasons. 6. Conclusion We have explored the phenomenological constraints of the R F parity model which gives m u , 0 consistent with the strong CP solution. In addition, we put an ansatz for the hierarchical fermion mass matrices. Within this framework, we first required that the R F parity model describes the charged lepton and quark masses and mixing angles properly. In particular, from the observed down-type quark mass matrix we derived a plausible form for the sneutrino VEV’s 2 m d˜R ²j n˜m ,t : R 400 Ž GeV . .

Ž 49 .

Then we applied this constraint to the recent neutrino oscillation data. We find that without a sterile neutrino a large mixing angle solution of nm and nt is not possible in the R F parity model. However, introducing a sterile neutrino, it is possible to generate a large mixing angle solution, if the parameter MS is of order intermediate scale and ² nm : is of order GeV. Since the hierarchical fermion mass matrix in the R F parity model is given in Eq. Ž37., we introduced a horizontal UŽ1. X = UŽ1.Z symmetry such that the fermion mass matrix arises naturally. This kind of horizontal symmetry can appear in string models frequently. We also pointed out another kind of R parity, R U parity, rendering the up-quark massless, but its phenomenological consequences are not explored.

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Acknowledgements J.E.K. and B.K. are supported in part by KOSEF, MOE through BSRI 98-2468, and J.E.K. is also supported by Korea Research Foundation. Appendix A. (Appendix A) In this appendix, we present the interactions which are neutral under UŽ1. X = UŽ1.Z horizontal symmetry. These are given by Ø QD c H1 type

Q 17Q 32 Q3 D 3c H1 , Q 112Q 33 Q2 D 2c H1 ,

Q 112Q 23 Q2 D 3c H1 ,

Q 110Q 38 Q3 D 2c H1 ,

Q 113Q 38 Q3 D 1c H1 ,

Q 115Q 33 Q2 D 1c H1 .

Ž A.1 .

c

Ø LE H1 type

Q 17Q 32 L3 E3c H1 , Q 112Q 33 L2 E2c H1 ,

Q 112Q 23 L2 E3c H1 ,

Q 110Q 38 L3 E2c H1 ,

Q 113Q 38 L3 E1c H1 ,

Q 115Q 33 L2 E1c H1 .

Ž A.2 .

c

Ø QU H2 type Q3U3c H2 ,

Q 110Q 25 Q2 U2c H2 ,

Q 17Q 25 Q2 U3c H2 , Q 16 Q3U1c H2 ,

Q 13 Q3U2c H2 , Q 113Q 25 Q2 U1c H2 .

Ž A.3 .

c

Ø LQD type

Q 16Q 33 L 3 Q1 D 1c ,

Q 13Q 33 L 3 Q1 D 2c ,

Q 13Q 23 L3 Q1 D 3c ,

Q 110Q 22 L2 Q1 D 1c ,

Q 17Q 22 L 2 Q1 D 2c ,

Q 110Q 28 L2 Q1 D 3c ,

Q 110Q 22 L1 Q2 D 1c ,

Q 17Q 22 L1 Q2 D 2c ,

Q 110Q 28 L1 Q2 D 3c ,

Q 16Q 33 L1 Q3 D1c ,

Q 13Q 33 L1 Q3 D 2c ,

Q 13Q 23 L1 Q3 D 3c .

Ž A.4 .

c

Ø LLE type

Q 16Q 33 L1 L3 E1c , Q 110Q 22 L1 L2 E1c ,

Q 13Q 33 L1 L3 E2c , Q 17Q 22 L1 L2 E2c ,

Q 13Q 23 L1 L3 E3c , Q 110Q 28 L1 L2 E3c .

Ž A.5 .

Note that the U c D c D c-type terms are not allowed, which ensures the proton’s stability. Note that the so-called m-term, m H1 H2 , is not allowed in the model presented above. However, we need an electroweak scale m-term toward a successful SUŽ2. = UŽ1. symmetry breaking w37x. One example for the m-term is through F 1 H1 H2 , where F 1 represents the color singlet superfields introduced to cancel anomalies in Section 4. For example, one can introduce the appropriate m-term by taking Q X ŽF 1 . s y11r3, QZ ŽF 1 . s 11 and ²F 1 : s m. This kind of charge assignment does not generate the dangerous term U c D c D c, but can generate the L1 H2 term m Q 27Q 32F 1 L1 H2 s l3m L1 H2 f L H . Ž A.6 . 100 1 2 Note that the relative size of the induced m 1 to m is independent of the charges of F 1 since the factor l3 is given by the differences between charges of H1 and L1. This size

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of the induced m 1 could break the relation Lt 4 L e , which we needed to avoid the present mass bound on mn e. Therefore, to suppress L e , we require some cancellation between ² n˜e : and l3 Õ1 or m 1 and the induced m 1. To give a VEV to the F 1 field, one can introduce a field F 2 with opposite UŽ1. X = UŽ1.Z charge, Q X ŽF 2 . s q11r3, QZ ŽF 2 . s y11 and then the relevant superpotential and scalar potential are given by W s fFF 1 H1 H2 q MF F 1F 2 , V s M12
Ž A.7 .

where M12 s MF2 1 q MF2 , M22 s MF2 2 q MF2 , and MF2 1, MF2 2 , AF and BF are the soft SUSY breaking parameters. Running M12 can be driven negative by the large Yukawa coupling fF . This radiative breaking, which is the same mechanism as the radiative electroweak symmetry breaking, generates the VEV’s for F fields as ²F 1 : s ²F 2 : ; MF w38–41x.

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