OiTS-6 17

.

OSURN-318

UTEXAS-HEP-96-18

DOE)3R~075?587

Texture of fermion mass matrices in partially d e d theories

'Present address, on eabatical leave at the University of Texas at Auotin.

1

Since the existence of a grand unifying scale is always well motivated, several attempts

have been made to UIllfY the SU(3), x s U ( 2 ) ~ x U ( l ) y in bigger groups e.g SU(5), SO(lO),

& dc. But it is not dtcisive yet which one among them is "the grand unfyjng group". A h it is not clear whether there is no other d e , with some kind of symmetry being broken,

in between the weak scale and the grand unifying s d e . Then can $ways be 8 partial

unificathon happenning in between the two scales, since the existme of a partially uni&ng

sca& around lom - loi3 GeW in the SUSY theories i8 also w d l motivated for difbent reasons. For example, orie can naturdly get a neutrino masa Sa tthe int-ng region of

3-10 eV, which wuld serve aa hot dark matter candidate. Su& a aeUt.rin0 115888 may also be

needed to explain tht large a d e structure of the Upivase [11. Tht aindov, 10" - 10u GeV, is ale0 the right zdzt for a hypothetical PQ symmetry to be brdocn m ae to dvt the crtrong

CP problem without creeting any phcmmenological or c o e m o ~ c a l problem (21. This d e

also gives rise to lepton fiavor violatingprcesses like, p * ey (31 snd dectric dipole moment ofdectrcm andneutron (4 which may &it poeaibto detect thia tmle in theneerhtum TbeS? kptun flaw vidatiom 6nd the& get gascrstedin thae llpodcb without heving

to make eny assumption regarding the location ofthe soat SUSY breakhg terms. They COLI

be anywhrere between the GUT scale and the P h d c or striag d. "he GUT 4 (with MSsM being the symnretry below the GUTscale) ir lower tbm theatriqgscde by a factar

of 25. Hoaiever using the intermdate scalc around lou GeV, it is possible to push up the

GUT& to thestring scde [SI. Also recently, a n i n U t e d e at - 10" G e V h beep dwxated [6] to produce monopoIcs that d d explain the high energy cosmic ray

spec- The partially unified models like SU(2)L x SU(2)R x SU(4)c gvea rise to such Inmo@ee naturally.

It is dwap a big challenge to find B justitication for the the mm distribution of the

fermions. So fiu stkmpts have ban made fronz the GUT d e and the weak scale 17-91.

In this letter we try to perate the mast predictive textures with minimum nuxnber of

patarnetera at a s d e 1012 GtV where the theory is pertially uniki- We do not use any

effect of any particUier grand unifpg group, bowever for the partial unification p u p we

2

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, make any warranty, express or implied, or assumes any legal liabili- ty or resporsibility for the accuracy, completeness, or usefulness of any information, appa- ratus, product, or pnnmss disdosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily com-tute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessar- ily state or reflect those of the United States Government or any agency thereof.

Portions of this document m y be illegible in electronic image products. Images are produced from the best available original document

,

use s U ( 2 ) ~ x s U ( 2 ) ~ x SCI(4)c symmetry. We write down the eupcrpotentid for these mass

matrices and also wc write m e new relations among the m8sses and the mixjng angles.

Wc use the bottom up approach as developed in the reference [9] in order to derive the

mass matrices for the quarks and leptons at the scale - 1Ol3 GeV. In thia method one starts from the known and/or presumed h w n quark and lepton maeses and the mixing angles at

the iow energy d e and then evolve these parameters, using the approprfste RGEe for the MSSM [lo}, to the desired scalc which in thh c89e is situated at 10"GeV. After evolving

these parameters, the complex symmetric matrices I@ and MD for the up and down are constructed with texture zeros Ill]. Far example

sane hermitian xnakriz U&g Syivcrrtet's theorem the - d o n for aH is giwm m:

where V is the CKM xnattix, and t$s are its eigenvalues wbich are kept mdegenerate.

The parameter x can be varied from 0 to 1 to obtain Merent fsxtures, The dements of

those taure are then compared to nmoye the lowest ones in order to get the textunswith

maximum& afwcls. After constntcting these textures, themasm anddxjnganglee

arc derived and then are e d d to the d scale to be CQmpPrcd with the txpaimental

values.

Ram c n ~ analysis described above, we have found only taro types of textures (at the

partial tmfication scale), fix the quark mass matrices, that are consistent with the low

energy data. Both type b at most four tatue zeros. Qpe 1 in the symbolic form is:

w = ( ; ; &); (4) 3

and

Here all the parameters are 1-4. The w e 2 lookn Illre:

MU-($ 4) W O

For this type we also find the additional relation:

(7)

G~mequedy we have one less parameter in this type and obtain one extra prediction. So

we will discus this type 2 m the rest of* paper.

Now we will show analytically how the clements of this texture gim by eqn.(S) and

eqn. (7), can be related to the m899c8 and the CKM angku and whether there is any

further relation among tbe masses and themixing angles. Thc clcments oh the texture has

a hierarchy, c >> b - V >> a. Also the matrix M D csn be rewritten by remwing the phaees: MP = P ’ M D Q ~ , where P and Q are the phs# matrices. ’Ib fhd the quark

at the scale loJ2 GeV, the matrices Mu and MD need to be dfsgpnalizcd. We use the orthogonal tmmhmtmn . RMR” = M* to get the eigenvduea q,q,-~llc(,),m,(b). So the elenrmts of the textures can be written in terms of the znaaes:

where r is the pattial unification scale, lou GeV. F b n Eqn. (8) and Eqn. (9), wc obtain,

4

The matrix R ioob like :

For thc up quark matrix, we obtain

and

and

For the down Bettor RV have

Using Ekln.3 (lo), (13) and (15), we obtain

All the above equations are valid at the partial mfication scale, t. So far, 0th than &, and Vd and the phase p, dl the 0th- elements ham been detenm'scd in terms of the masse^.

Since b"s are involved in the expteasions for 92's we use the CKM matrix elements. For the

determination of phasee we &o use the CKM. At the d e r V a w is given by:

5

i

where 0 = -20 and I = -8. Then, in terms the transforming angles and the phases, the CKM element %(r) can be written as:

IVd(T)/ = I S f P - s;esq

The eqn. (19) can be approximated as:

W h e r e

mcmb k=-- mt m a

Using the magnitude of VJj.), we can soh far 4 , since k is dmadyknown in tem3s the mass ratios at the intermediate scaie. For the deterrmns ' tionofthepbeweuaetEbeCKM

elemmt V&):

Thus we have determhed all the parametem ofthe model in termr dthe maims and some

of the CKM elements. The mmining CKM angles are the pradidione ofthe model. Fhlr

example, the model predicts:

For the parametrfzstion-invariant CP vialatian quantity J fl2], we obtain,

whue J is detesmiDed at the weak scale. Now, we uae the mpcrhmtal d u e of the quark

masses and the CKM elements \Vdl and \V,l at the weak d e to determine the parameters

6

of the model. We use m, = 180GeV, and the light quark ma88 ratios [13] 2 = 0.55, = 18.8 along with m,,(lGeV) = 5.1MeV,m(lGeV) = 9 . 3 M e V , ~ = 1.27GeV,mm,(lGeV) = 0.175CeV, mb(m) = 4.32GeV, m(m,) = 1.78GeV. The ratio of the value of the Yuhwa

eou& at the mt scale to the 1 GeV or to the corresponding p i e m 8 b ~ sade is given by q.

The values ofthe r) we use are q, = 2.4,rjd = 2.4,q, = 2.4,% = 2 . 1 , ~ = 1.0158. The strong coupbg a, is taken to be 0.118 dong with &aa& to be 2321 md Q =1/127.9 at the Mz scale. Ale0 we we IVdl= 0.040 and IV,l = 0.22 at the mr de. Nar we need to d v e these mama and mixing dements to determine tbcir d u e s at the d e r . If wewe a-

tan @(tan B = 48) scena;rio where d the third Blen(eraton hkawa ampling ie top, bottom and 7 are same, weget themiises as folkam

~ ( r ) = 158.4GeV; %(r) = 0.33GcV; m&) = 0.001GeV; (W m(r) = 0.0019GeV; m,(r) = 0.043GeV; m(t) = 3.30GeV. (W

At the sale r, IVdl becoants 0.03 and IV,l is 022. W e u a e h tosolvehr Y Sndthephrrar!

8. Weobtain

i

under the 2 L 2 R k symmetry in order to produce a reasonable lepton ma88 matrix and s's

are the singlet Higgs auperfieldso The quantum numbera for the field under the discrete

symmetry we invoke can be as follows:

4 and & transform in the same way under the discrctesymmctry. There cau be other & of choices of the quantum numbers. However for this particular choice of quantum numbers

the 11 duneat of thc up and down quark matrices get generated for tbe 10th powar of al,

which can easily be negkxttd.

In conclusion, we have derived twlo types of textures at the d e - lo1' GeV which can give rise to the correct values for the masees of the quarks and the leptons and the CKM

angles and the CP vio&tim parameter J . These textures axe dif famt hnthoseobtained

at the GUT scale. Out of these tmm types we found one haa k numbad parametera and some new reiations anmng the mixing angieS and rnatims. This peticulsr texture also loola

l&C the %tzec& W h h the "22" -t beiRgwaro. wk& W&C SSlIPerpO- tentid for this texture supported by a mockl with string unification and an intamediate

scale around 1Ol2 GeV.

One of u8 (SN) wish= to thank Duane Dicun afthe Universiv adTexas at A w t h for a very warm hospitality and support during this sabat id leave. This work was eupported in

part by the US Department of Eaugy Grants No. DEFG-0294ER 40852 and DEFGOG-

854ER 40224.

8

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