Volume 207, number 4 PHYSICS LETTERS B 30 June 1988

COMMENT ON THE QUARK MASSES IN THE SU(5) X U( 1) MODEL DERIVED FROM THE 4D FERMIONIC SUPERSTRING

G K LEONTARIS CERN CH-1211 Geneva 23 Swtzerland

Received 14 March 1988

In this letter, the problems of the up-quark and neutrmo mass matrix, as well as the lepton non-conservation are dlscussed wthm the N= 1 supersymmetric SU (5 ) x U ( 1) model derived from the four-dlmenslonal fermlomc superstrmg

Nowadays, it 1s widely recognized that superstrmg theories appear to be the only candidates for a um- fied theory of all known interactions [ 1 ] Thus, dur- mg the last three years much effort has been devoted to the construction of realistic superstrmg models m four dimensions using either fermlomc or bosomc formulations [ 2,3 ]

Although string theories are very promlsmg and seem to incorporate all the known particles, the road from the Planck scale to low-energy physics is not an easy one Thus it 1s important to answer the question whether superstrmg theories can lead to a realistic low-energy gauge group that 1s consistent with the well-known world of elementary particles and their

interactions In this paper, we will concentrate our attention on

an N= 1 supersymmetric model based on the group SU ( 5 ) x U ( 1) which can be derived from the four- dImensiona formulation of superstrmgs

This model has been reported [ 4,5 ] as an attempt at model building usmg the fermlomc formulation of strmg theories m four dimensions Although this model does not seem to be a unified one, like the standard SU ( 5 ) , it has some very nice features which make it quite attractive Indeed, first of all, the fer- mlon fields as well as the Hlggs fields needed to break the group down to the standard SU (3 )c x SU (2 ) L xU( 1) y belong only to the 5, 5, 1, 10 and 10 repre- sentations of SU ( 5 ) Secondly, the doublet-triplet mass splitting 1s solved easily m the above model by a missing partner mechanism Indeed, one can use

the ten-dimensional Hlggs representation of SU (5 )

to give a large mass to the triplet components of 5

and 5 Hlggs fields, while the doublet components re-

mam massless at the first stage of symmetry break-

mg The latter are those which will realize the second stage of symmetry breaking from SU (3),x

SU(2),xU( l)ydown to SU(3),xU( l)EM

Finally, we notice here the absence of the adJoint

representation as well as any higher ones However,

as we will soon realize, one encounters some defects

of the above “GUT” model [ 41 when trying to ob-

tam the necessary Yukawa couplings from the 4D

fermlomc superstrmg [ 51 To be more precise, the

fermlomc formulation informs us which Yukawa

couplings can be present m the superpotential In our

case, one can have all the necessary Yukawa cou-

plings to give masses to the fermlons, but not those

which can give the necessary Kobayashl-Maskawa

mixing between the first and the other two genera-

tions of quarks [ 5 ] In principle, one could generate

the necessary Yukawa terms by modlfymg the

boundary condltlons which generate the strmg model,

but one can always do it at the price of the appear-

ance of some new extra Hlggses

Here, we will make an attempt to generate the nec-

essary Kobayashl-Maskawa mixing m the above

string model We will see that this can be dome as-

suming one-loop corrections to the up-quark matrix

Further, one can also generate a neutrmo mass term

between the left- and right-handed neutrinos at the two-loop level

03%2693/88/$03 50 0 Elsevler Science Publishers B V (North-Holland Physics Publishing Dlvlslon )

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Volume 207, number 4 PHYSICS LETTERS B 30 June 1988

To start with, we remind the reader of the salient

features of the model [ 4,5 ] The possible Yukawa terms allowed by the SU (5 ) X U ( 1) symmetry in the superpotential are the following

+A,FJ!I$+A,hii@+&@3 (1)

The SU ( 5 ) x U ( 1) quantum numbers of the various fields appearing m eq ( 1) are the following

F,=(lO,;), F,=(s, -;), Q;=(l,$),

H=(IO, i), H=(lO, -$),

h=(5, -l), h=(s, I), Qo,,,=(l,O) (2)

The 16 fermlons are found m F,, F, and Q; (where I= 1, 2, 3 generation index) while the necessary

Hlggses H, 8, h, h belong to the 10, 10, 5 and 5 rep- resentations of SU ( 5 ) respectively The Q0 ,n are four SU( 5) singlets They are necessary to provide the hli mixing and the see-saw mechanism for the neutn- nos through the term A,FI%$, If this were the case, however, all fermlons could acquire masses from the VEV of the h and ii fields However, if one desires to produce all the above terms from the superstrmg one must be careful To be precise, the string model un- der conslderatlon [ 5 ] IS generated by a basis of eight elements, five of them yielding an SO ( 10 ) x SO ( 6 ) 3 observable gauge group together with 3 x 2 copies of massless choral fields m ( 16, 4, 1, 1) + ( 16, 4, 1, 1) representations and an Es hidden gauge group The remaining basis elements break down SO( 10) to SU(5)andS0(6)3toU(1)3 TheextraU(1)3sym- metry will modify the previous picture of the model, since one cannot generate all the Yukawa couplmgs of the superpotential ( 1) In fact, one can generate most of the terms of (1) but not those which give masses to the up-quarks and the neutrinos Indeed, m the string model under conslderatlon the three families of quarks and leptons transform under U ( 1) 3 as

M,=(;,O,O), M,=(O,;,O),

M?=(O, 0, -4)) (3)

where each M, decomposes according to the 16 spl-

448

normal representation of SO( 10) under SU (5)

xU(l),ie

M,-tF, +f, +!Z; ,

where F,, fl and !J: are given m (2) Furthermore, there are six singlet fields @,,, i$, m the model [ 5 ] and the following massless Hlggs fields

h,=(5, -1)Lloo, hz=(5, -l)o-lo,

h3=(5, -l)oo, >

H,=(lO, f),,zoo, n=(n, -t>oo-I (4)

Further, four massive states are mentioned m this model m order to provide large masses for the triplet components of the h fields

From the above, it 1s now clear that the term

A,FE which gives massess to the up-quarks IS no longer allowed Nevertheless, one can still modify the above model at the price of some extra Hlggses [ 61 In that case, however, one can generate non-renor- mahzable Yukawa terms which would be sufficient to give a 2 x 2 mlxmg mass matrix for the up-quarks

but leave the other quark unmixed This 1s so because m the low-energy theory there still remains a discrete symmetry which does not allow the appearance of these terms One could nevertheless generate the missing terms m the up-quark mass matrix radla- tlvely, usmg only the allowed terms ;1 ,FFh and I13hFQc of the superpotential Thus, if one assumes non-zero VEV for the scalar partners of the neutrinos [ 71, then one can generate the one-loop diagram of fig 1 In- deed let us suppose that the remaining discrete sym- metry leaves the quark, which belongs to the M3 generation [ 5 1, unmixed with the other two Then, from the diagram of fig 1, one can generate through the VEVs of the sneutrmos, contrlbutlons to all the diagonal elements of the up-quark mass matrix Fur- ther, through the VEV of the field e13 and using the

h-,, -X- . . ;’ / \

--+‘-+x+-‘4- U dC i,, uc

Fg I Radlatlve correctlons to the up-quark matrix

Volume 207, number 4 PHYSICS LETTERS B 30 June 1988

non-renormahzable terms of the superpotentlal [ 6 1, one can have mixing (off-diagonal terms) of the form u ,u;, uzuS However, one should calculate the contn- butlon of the above graphs To this end, an estimate is needed of the dCDh and e”h mixing Thus, if one

considers the allowed terms of the superpotential ( 1)) then one can generate the matrices

ec ii

0

A($)

(6)

The above matrices can be dlagonahzed by separat- mg L and R unitary transformations S,, S,,

S~mmtSL=d2=S~mtmSR (7)

Starting with the matrix (5 ), it IS enough for our pur- poses to consider the mixing only between the first

two states, thus one gets the elgenvalues

PI x mM,/Jm, ,u2 z Jm, (8)

1 e , one light elgenstate identified with the down quark and a heavy one, namely the ,u2 = 0 (MGUT) We notice here that M, 1s of the order of MGUT, while AJ2 _ 0 ( 1 013 GeV) The mixing matrices are

sLR= CL R SLR

--LR CLR

with

tan8,=2M,Mz/(M:--Mf),

tan 0,=2mM,/(M:+M$) (10)

In a similar way, one can write down the elgenmasses and mixing angles of the matnx m, Thus we have (assummgthat m’, -cm;-m;)

.uU; =m’,m;l(M:-M:),

pL;=J_,

and

tan8~=2m$m;l(m;Z-m~2),

(11)

tanB~=2m’,m;/(m~2-tm;2) (12)

In an analogous manner, one can also construct the corresponding boson mass matrices [ 81 Then from fig 1, one can estimate the contnbutlon to the up- quark matrix which 1s found to be

m, z (A113/32n2) m sin 2&ln (M:/M2) (13)

In the above formula, M, IS a supersymmetrlc mass m the range of 100 GeV to 1 TeV The mass M char- acterizes the mass scale dm which 1s of the order of MGUT Finally, an 2/& can be as large as 1 0w2 as one can see from the relation ( 12 ) Thus, from the

above formula, one concludes that we can have a contrlbutlon m$ m the range of few MeV to 0 1 GeV This contnbutlon could be enough to generate the desired mixing between the third and the other two generations of the up-quarks

In a similar manner, one can also assume radiative contrlbutlons to the neutrmo mass matrix One can use the couplmg generated above to construct the diagram of fig 2 This contnbutlon is, however, small thus there is no conflict with the experimental mfor- matlon In any case it will be added to the off-dla- gram terms of the general neutrmo mass matrix [ 4- 6] to realize the see-saw mechanism necessary to op- erate m this model

Finally, we briefly discuss the lmphcatlons of the model of the lepton number, and m particular for the electron number The same couplings used m fig 1 can also generate the Feynman diagrams of fig 3 They can lead to the neutrmoless double-beta decay process which violates the lepton number by two units

Fig 2 Dlrac neutrmo masses

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Volume 207, number 4 PHYSICS LETTERS B 30 June 1988

al bl

Fig 3 Contrlbutlon to double-beta decay

Diagram 3a gives a contrlbutlon which is charac-

terized by the parameter 12 [ 9 ] which for the photmo mediated case (which turns out to be the most lm- portant ) 1s given by the formula [ 10 ]

ny= (A, an 28R/2GFm:)2za( 1 GeV/m?) (14)

Using now experimental information [ 111, as well as theoretical estimations [ 121 of the relevant nuclear matrix elements, one can obtain the limit

,i,sm20&51 3~10-~(rn,/lOOGeV)~, (15)

where we have used mp = 5 GeV Diagram (3b) gives the contrlbutlon [ 12 ]

ng= fna,(l, sm 2BR/2G,m&)2( 1 GeV/mg) (16)

However, we already know from ( 11) that the mix- mg 1s tmy and thus this contrlbutlon becomes irrelevant

In conclusion, we have seen that one can obtain the necessary mixing and the light up-quark masses m the SU ( 5 ) x U ( 1) model through radiative corrections We notice, however, that one can generate the corre- sponding off-diagonal terms also m the down-quark mass matnx through the same kmd of diagrams, while m the case of charged leptons, this 1s possible m the two-loop level In addition, one can get a Dlrac-neu- trmo mass matrix at the two-loop level Fmally, we

have seen that the vlolatlon of the lepton number lies safely within the experimental bounds m this model, for natural values of the parameters involved in the relevant amplitudes

I would like to thank I Antomadls for fruitful dls- cusslons and useful suggestions I have also benefited from dlscusslons with G Lazandes, D V Nanopoulos, C E Vayonakls and Q Shafi

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