- Email: [email protected]

PHYSICS LETTERS B

A new standard-like model in the four dimensional free fermionic string formulation Alon E. Faraggi Center for Theoretical Physics, Texas A&M University, College Station, TX 77843-4242, USA and Astroparticle Physics Group, Houston Advanced Research Center (HARC), The Woodlands, TX 77381, USA

Received 22 October 1991

I present a new three generation superstring standard-like model in the free fermionic formulation with the following properties. The complete massless spectrum is derived and shown to be anomaly free apart from a single anomalous U (l). The DineSeiberg-Witten mechanism is applied to cancel the anomaly, leaving a supersymmetric vacuum. I show that the resulting observable gauge symmetry is SU ( 3 ) × SU (2) × U ( 1)" where n = 1 or 2. All trilinear superpotential couplings have been calculated. I show that of the standard model quarks and leptons, only + 32charged quarks obtain a non vanishing trilevel coupling, which suggests a possible explanation for the heaviness of the top quark relative to the lighter quarks and leptons. The additional, generation independent, U ( 1) symmetry may remain unbroken down to low energies and prevents fast proton decay.

1. Introduction In the last few years great efforts have been m a d e by theoretical physicists to derive the s t a n d a r d m o d e l from the superstring [ 1 ]. Two a p p r o a c h e s can be followed to connect the superstring with the s t a n d a r d model. One is to use a G U T m o d e l with an i n t e r m e d i a t e energy scale. M a n y a t t e m p t s have been m a d e in this direction and most notable are the flipped SU ( 5 ) [ 2 ] a n d the SU (3)3 [ 3 ] models. The second possibility is to derive the s t a n d a r d m o d e l directly from the superstring, without any non-abelian gauge s y m m e t r y at an i n t e r m e d i a t e energy scale [4,5 ]. In ref. [ 5 ] a three generation string m o d e l in the four d i m e n s i o n a l free fermionic formulation [ 6 ] was derived. The observable gauge s y m m e t r y after the application o f the G S O projections is SU ( 3 ) × S U ( 2 ) X U ( I ) B _ L X U ( 1 ) r 3 ~ × U ( 1 ) 6 [5] ,,1 The massless spectrum o f the string m o d e l [ 5 ] contains an a n o m a l o u s U ( 1 ) gauge symmetry. The a n o m a l o u s U ( 1 ) is b r o k e n by the D i n e - S e i b e r g - W i t t e n [ 7 ] m e c h a n i s m in which a potentially large F a y e t - I l l i o p o u l o s Dt e r m [ 8 ] is generated by the VEV o f the dilaton field. Such a D-term would, in general, break s u p e r s y m m e t r y and destabilize the string vacuum, unless there is a direction in the scalar potential ~ = ~.iai~i which is F flat and also D flat with respect to the non a n o m a l o u s gauge symmetries a n d in which y~iQg Io6 12< 0. I f such a direction exists, it will acquire a VEV, canceling the a n o m a l o u s D-term, restoring s u p e r s y m m e t r y and stabilizing the vacu u m [ 9 ]. Since the fields corresponding to such a flat direction typically also carry charges for the non a n o m a lous D-terms, a non trivial set o f constraints on the possible choices o f VEVs is imposed. It is, in general, a non trivial p r o b l e m to find solutions to the set o f constraints. This constraint is found to be especially restrictive in the case o f the standard-like m o d e l s u n d e r consideration. The reason is that the set ofsinglets which do not carry U ( 1 )~_/~ a n d U ( 1 )T3R charges is significantly reduced, thus reducing the set o f fields which can receive a non vanishing VEV. This p r o b l e m is illustrated further in ref. [ 10 ] where the construction o f realistic free fermionic spin structure m o d e l s is discussed in m o r e detail. In this p a p e r I present an explicit example which a d m i t s a s u p e r s y m m e t r i c solution to the F a n d D flatness constraints. This example is an existence p r o o f that supersym~1 In ref. [5 ] and in this paper I take U ( 1)c = ~U( 1)n-L and U ( 1)L----2U ( 1)T3R. Elsevier Science Publishers B.V.

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metric solutions with the desired phenomenological constraints [ 5,11 ] do exist in this class of models.

2. The string model

The string model is generated by the following basis of eight vectors (including the vector 1 ) of boundary conditions for all the world-sheet fermions: S = ( 1 , . . . , 1 , 0 , . . . , 0 1 0 .... , 0 ) ,

(la)

~Uu,ZL--6

b~=(

1.... ,1,

0,...,011 .... , 1 , 0 .... , 0 ) ,

~g~,~ 12,y 3...6,fl3...6

b2=(

1,..., 1,

0,...,0l 1.... ,1, 0, ..., 0) ,

Iff~,)¢34 y I O)5,y 2)~2,(0 6(~ 6,)71(~ 5

b3=(

1,..., 1,

0,...,0[1,..., 1,0 .... , 0 ) ,

7=(

0,..., 0[ 1.... , 1 , 0 , . . . , 0 ) ,

y 3y 6, fD6(2)6 O)3(~ 3,~ 1O)5,(.02(~ 4

~/~I...3 ~1...4

1..... 1,

0 ..... OI 1, ..., 1, O, ..., O ) ,

y I(.t)5,y 5y5,(o 10 1 ~3y6,(2~ 2(~4

~ I...3 ~1 ...,t

1, " " , 1,

(ld)

~1...5 ~3

1,...,1,

a=(

(lc)

~/~I...5,/~2

I//'a,~ 56,O)2(/) 4,0) I(~ 1,0) 3(~ 3,(~12~ 4

fl=(

(lb)

[/~1...5,/~l

0, ..., 0[

O)2£04,y 2j72,y4~4 ,~73y6,y 1¢~ 5

!2~

" " , !2,

1,

i/~1...5 01,2.3,~1 ,~5 ......

(le)

(lf)

"",

1,0 ..... 0)

~3.4

~2~8

•

(lg)

In this notation 1 stands for periodic fermions, 0 for antiperiodic and ½ for those twisted by a phase - i . The vertical line separates real from complex fermions. I have chosen a basis in which all left-movers (~u, zi, yi, o9i: i = 1, ..., 6) are real, among which world-sheet supersymmetry is realized non linearly; 12 right-movers are real (yi, oT) and 16 right-movers (~l...s, 0~, ~ 2 / 7 3 ~1...8) are complex. Pairs of two real fermions that have the same boundary conditions in all the sectors are paired to form complex fermions. To obtain the desired low energy spectrum, I make the following choice of generalized GSO projection coefficients

C~s}=c~bj]=c

-1 S

a

fl

( i , j = l , 2, 3 ) ,

(2a)

o~

Y

c y

with the others specified by modular invariance [ 6 ]. The basic vectors {S, I = 1 + b~ + b2 + b3} define an N = 4 space-time supersymmetry model with an SO (28) × E8 gauge group. The sector S plays the role of the supersymmetry generator since its addition to another sector gives the superpartners. The vectors b~, bz reduce N = 4 to N = 1 supersymmetry. The set {1, S, bl, b2, b3} gives an SO (10) × SO (6) 3× E8 gauge group with 3 × 2 copies of massless chiral fields ( 16 + 4) + ( 16 + 7~), two from each 132

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of the sectors bl, b2, b3. The vectors a, fl, 7 break the horizontal symmetries to U( 1 ) 3 X U ( 1 )3, which corresponds to the right-moving world-sheet currents ~/20~/2 ( a = 1, 2, 3) and j~3)~6,37~c05, 0)20) 4 respectively (I define (1___(1/x/~) 073+i~6), (2= (1/x/~) 071+ic05), (3= ( 1 / / ~ ) ( O 3 2 + i 0 3 4 ) ) . T h e vectors oL, fl break the SO(10) symmetry to S O ( 6 ) × S O ( 4 ) and the hidden group from E8 to SO(16). The vector 7 breaks SO(6) × S O ( 4 ) - - , S U ( 3 ) c × U ( 1 ) c × S U ( 2 ) L × U ( 1 )L and SO( 1 6 ) - - , S U ( 5 ) H X S U ( 3 ) H × U ( 1 )2. The weak hypercharge is uniquely given by U ( 1 ) r = ~ U ( I ) c + ½ U ( 1 ) L . The orthogonal combination is given by U( 1 )z, = U ( 1 ) c - U ( 1 )L. The U( 1 ) symmetries in the hidden sector, U( 1 )7 and U( 1 )8, correspond to the world-sheet currents ~1 ~1- + ~sq~s* and - 2¢TJ~J*+ q~l~1. _ 4~2q~2" _ ~ 8 ~ 8 " respectively, where summation on j = 5, ..., 7 is implied. After the application of the GSO projections the gauge group is Observable: SU(3 ) c X U ( 1 ) c × SU (2)L× U ( 1 ) L × U ( 1 )6, Hidden ~2: S U ( 5 ) H × S U ( 3 ) H X U ( 1 )2. In addition in the observable sector, the vector S + bl + b2 + o~+ fl gives rise to massless states which transform only under the observable gauge group. The spin structure of eq. ( 1 ) differs from the model of ref. [ 5 ] in the following ways. The basic set {1, S, bl, b2, b3} is common to the two models and to all the realistic constructions in the free fermionic formulation [2,5,12]. As discussed in ref. [ 11 ], this set performs several functions. It splits the hidden sector from the observable sector, gives rise to the chiral generations, and determines their chirality. The vectors a, fl and y are modified. The vectors ot and fl are symmetric with respect to the NAHE set {1, S, b l, b2, b3} [ 2,11 ]. This results in the combination S+bl +b2+a+fl, which gives rise to additional SO(10) singlets that carry only U( 1 ) 1,...,6 charges. These singlets are used to solve the F and D flatness constraints. This type of vector is not present in the model of ref. [ 5 ]. In addition, a vector of the form b4 in ref. [ 5 ], does not appear in the additive group of the current model. The hidden gauge group is extended by the massless states from the sector I. A more extensive discussion of different spin structure models and their phenomenological implications is given in ref. [ 10 ].

3. The massless spectrum

I present the full massless spectrum and emphasize the states of the observable part. To analyze the massless spectrum, I have written a FORTRAN program which takes as input the vector basis, eq. ( 1 ), and the GSO coefficients eqs. (2). The program checks the modular invariance rules, spans the additive group -== Y~jnjbj (j= 1, ..., 8), selects the vectors in E which lead to massless states and performs the GSO projections. It calculates the traces of the U ( 1 ) symmetries and evaluates the trilinear superpotential. The program was tested on the existing examples [ 10 ] in the free fermionic formulation. The following massless states are produced by the sectors b1,2,3, S + bl + b2 + a +fl, O and their superpartners in the observable [ (SU ( 3 ) c, U ( 1 ) c ); (SU ( 2 ) L, U ( l ) L) ] 1.2,3,4,5,6sector, where 1.... ,6 denote the charges under the six extra U( 1 )'s. (a) The b 1,2,3sectors produce three SO ( 10 ) chiral generations. G~ = e ~. + u ~a + N ~ + d~. + Q. + L . ( a = l, .... 3) where e~-[(1,3);(1,1)], N2-[(1,~);(1,-1)],

u~-[(3,-~);(1,-1)],

Q-[(3,½);(2,0)],

(3a,b,c)

d~-[(~, -½); (1,1)] , L-[(1,-I);(2,0)],

(3d,e,f)

of SU ( 3 ) c × U ( 1 ) c × SU (2) LX U ( 1 ) L, with charges under the six horizontal U ( 1 ). From the sector bl we obtain (e~ Ji- L/~) 1/2,o,o,1/2,o,0 .-It- (d~ +NCL)1/2,O,O,_1/2,0,O+

(L)1/2,o,o,1/2,o,o+ (Q)w2.o,o,-w2,o.o,

(4a)

a2 Hidden here means that the states which are identified with the chiral generations do not transform under the hidden gauge group [ll].

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from the sector b2 c c ( e ¢L +UL)o,I/2,0,O,I/2,0 + ( N L¢ "I-dL)o,I/2,0,O,-1/2,0 + (L)0,1/2,0,0,1/2,0 + (Q)o,l/2,o,o,-1/2,0,

(4b)

and from the sector b3 ¢ ¢ (eL¢ + UL)O,OA/2,0,O,I/2 + ( N ¢L +dL)o,oA/2,0,O,-1/2 + (L)o,o,I/2,o,oj/2 + (Q)o,oA/2,o,o,-1/2.

(4c)

The vectors bl, b2, b3 are the only vectors in the additive group E which gives rise to spinorial 16 of S O ( 1 0 ) . This is in contrast to the case in which the SO (10) symmetry is broken, by the spin structure, to SU (5) × U ( 1 ) [ 2 ] or SO (6) × SO (4) [ 12 ]. Then, the spectrum includes additional 16 and 16 multiplets. The fact that there are exactly three generations in the standard-like models and no mirror generations is closely related to the choice of SU ( 3 ) c × U ( 1 ) c × SU (2) L × U ( 1 ) L as the observable gauge symmetry at the level of the spin structure. (b) The S + bl + b2 + a + fl sector gives

h45 -

[ ( 1, 0 ); (2, 1 ) ] _ 1/2,- 1/2,0,0,0,0, D45 -= [ (3, - 1 ); ( 1, 0) ] _ 1/2,- 1/2,0,0,0,0,

~)45 ~-" [ ( 1 ,

0);

(1, 0) ]--1/2,--1/2,--

~

1,0,0,0 ,

• f -= [(1, 0); (1, 0) ]-1/2,1/2,0,0,±1,0,

~

-- [ ( I , 0); (1, 0) ]_1/2,1/2,0,± 1,0,0 , -= [(1, 0); (1, 0)]-1/2,1/2,0,0,0.±1

(5a,b) (5c,d) (5e,f)

(and their conjugates h-45, etc. ). The states are obtained by acting on the vacuum with the fermionic oscillators ~4,5, ¢ 1,...,3, r~3, ~3 + i376, )71 + i03 5, 032 + i034, respectively (and their complex conjugates for h-45, etc. ). (c) The Neveu-Schwarz O sector gives, in addition to the graviton, dilation, antisymmetric tensor and spin 1 gauge bosons, the following scalar representations: Electroweak doublets: hi - [ ( 1, 0); (2, - 1 ) ]o,l,o,o.o,o,

12 -4,5* - I X l / 2 ~ / / 1 / 2 ~] 112 1 0 > 0 '

(6a)

h2 ~ [ ( 1, 0); (2, - 1 ) ]o,o,l,o,o,o,

~ 3 4 ,;~4.5".~2 1/2 ~o, 1/2 i / 1 / 2 1 0 ) 0 ,

(6b)

h3 = [(1, 0); (2, - 1) ]o,o,l.o,o,o,

~ 5 6 )r~4,5" ~ 3 1/2 u~°'1/2 '11/2 [ 0 > 0 ,

(6C)

Singlets: ~ 2 3 ~'~"[(1,

0); (1, 0) ]o,1,-1,o,o,o,

X1/2rh/2,tl/210)o I 2- 2 43*

,

(7a)

@13 --- [(1, 0); (1, 0) ] l.o,-t,o,o.o,

)~34 - 1

~3"

(7b)

• 12 = [(1, 0); (1, O) ] 1,-1,o,o,o,o,

~56

~2"

(7C)

1/21]1/2,11/2 1 0 > 0 , -1

1/2~]1/2,11/2 1 0 > 0

(and their conjugates h-l, etc.). Finally, the Neveu-Schwarz sector gives rise to three singlet states that are n e u t r a l u n d e r a l l t h e U ( 1 ) symmetries. • (~,2,3.. X~/2wl/2091/210)o, 12 - 3 -6 4.34 . 5 1/2 t~ I:~11/2 IO)o,ZI/2Yl/2Yl/210)o. 56 -2 -4 .~ 1/2.Y The sectors b~+2y+ (I) ( i = 1.... , 3 ) give vector representations which are S U ( 3 ) c × S U ( 2 ) L × U ( 1 ) L X U ( 1 )c singlets (see table 1 ). The vectors with some combination o f (bl, b2, b3, or, fl) plus 7+ (I) (see table 2) give representations which transform under S U ( 3 ) c x S U ( 2 ) L × U ( 1 ) L X U ( 1 )C, most of them singlets, but carry either U ( 1 ) r or U ( 1 )z' charges. Some of these states carry fractional charges + ½ or + ]. There are no representations that transform nontrivially both under the observable and hidden sectors. The only mixing which occurs is of states that transform nontrivially under the observable or hidden sectors and carry U ( 1 ) charges under the hidden or observable sectors respectively. Some immediate observations of the above spectrum are: (a) There are only three generations and all the fields needed to break the extra U ( 1 )'s down to the standard model are available. (b) There are overall ten U ( 1 ) symmetries, eight in the observable part and two in the hidden part. Out of those, four are anomaly free and six are anomalous: TrUt=24, 134

TrU2=24,

TrU3=24,

TrU4=-12,

TrUs=-12,

TrU6=-12.

(8)

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Table 1 Massless states and their quantum numbers. Vindicates that these states form vector representations of the hidden group. F

SEC

SU(3)G×SU(2)L

Qc

QL

Qi

Q2

Q3

Q4

Vt I/2

bl+2fl+(l )

(1,1) (1,1) (1,1) (1,1)

0 0 0 0

0 0 0 0

0 0 0 0

½ ½ ½ ½

½ ½ ½ ~

(1,1) (1,1) (1,1) (I,1)

0 0 0 0

0 0 0 0

½ ½ ½ ½

0 0 0 0

(1,1) (1,1) (1,1) (1,1)

0 0 0 0

0 0 0 0

½ ½ ½ ½

½ ½ ½ ½

V3 V,

I/5 V6

b2+2fl+(1)

I/7 Vs

I/9

b3+2fl+(1)

V,o V,, V,2

Q5

Q6

SU(5) xSU(3)

Q7

Q8

½ ½ -½ -½

0 0 0 0

0 0 0 0

(1,3) (1,3) (5,1) (5,1)

--½ ½ -½ ½

--3 -~

½ ½ ½ ½

0 0 0 0

½ ½ -½ -½

0 0 0 0

(1,3) (1,3) (5,1) (5,1)

-½ ½ -½ ½

-~ -~

0 0 0 0

0 0 0 0

0 0 0 0

½ ½ -½ --½

(1,3) (1,3) (5,1) (5,1)

-½ ½ -½ ½

-3 -~

The two trace U ( 1 )'s, U ( 1 )L and U ( 1 )c, are anomaly free. Consequently, the weak hypercharge and the orthogonal combination, U ( 1 )z,, are anomaly free. Likewise, the two U ( 1 )'s in the hidden sector are anomaly free. O f the six anomalous U ( 1 )'s, only five can be rotated by an orthogonal transformation and one combination remains anomalous and is uniquely given by: UA= k~j [ Tr U ( 1 )j ] U ( 1 )~, where j runs over all the anomalous U ( 1 )'s. For convenience, I take k = ~ , and therefore the anomalous combination is given by UA =2U~ +2U2 -I-2 U3 - U4 - U5 - U6,

Tr QA = 180.

(9a)

The five orthogonal combinations are not unique. Different choices are related by orthogonal transformations. One choice is given by U'~ = U~ - U2,

U~ = U~ + U2 - 2 U3,

(9b,c)

Ut3=U4-U5,

Ut4=U4-{-U5-2U6,

(9d,e)

u~ = Ul + u2 + u3 +2u4 +2u5 + 2 u 6 .

(9f)

Together with the other four anomaly free U ( 1 )'s, they are free from gauge and gravitational anomalies. The cancelation o f all mixed anomalies a m o n g the five U ( I )'s is a non trivial consistency check o f the massless spectrum o f the model.

4. The superpotential The non vanishing trilevel terms in the superpotential o f the model are

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Table 2 Massless states and their quantum numbers. F

SEC

SU(3)cXSU(2)L

Qc

H~ H2 n3 Ha

bl+bE+a+fl+?+(1)

(1,1) (1,1) (1,1) (1,1)

½ -I -½ l ~ ½ -~ -~ -½ l

Hs //6 H7 Hs

b,+b~+a+fl+_y+(1)

(1,1) (1,1) (1,1) (1,1)

QL

al

~

-I

Q3

Q,

Q5

-¼ l -l l

1 -½ -½ l -½ - l -½

-½ -½ -½ -½

-l

~-

½ -l 1 _~ _½ l _1 l 1 ½ -l l -l __~ _½ 1 -I l 3

H9 bz+b~+a+fl+y+(1) H~o HII H,2

(1,1) (1,1) (1,1) (1,1)

-~ ½ l -¼ -½ - l ¼ ½ l

-I

-½

H,3 Hi,, H,5

(1,1) (I,I) (1,2)

-I

½ -l --½ l -½ - l

b,+b3+a+?+(l)

02

-l

-l 1 l l

-~1

-½ -½ (1,3) -½ -½ (1,3) --½ --½ (1,1) -½ -½ (1,1)

l -~ -l _] 15 1 - ~15

0

(1,1)

l

0 (1,1) 0 (1,1)

_1 l

-~

0

~

0

-~ ~

0 0

~

0 0 0

o o o o o o 0 0

o o o o o o 0 0

(5,1) (LI) (1,1) (1,1) (1,1) (1,1) (l,l) (1,1)

-½ _~ 1 - I -½ ½ ½ ½ ~ 1 1 ½ -½ -½ - I -½ -~ - I - I ½ -½ ½ 1 ½ l l 1 -½ ½ -½

(1,3) (1,3) (1,1) (1,1)

,] ½ --¼ --~ --l ~- -½ l ~ 1

(1,1) (1,1) (3,1) (.],1) (1,1) (1,1) (1,1) (1,1)

½ 1 -½ l ½ -l -½ --I ½ k -½ --I ½ l -½

(1,1) (1,1) (1,1) (1,1)

-I

-~

-I

--~ -~ -~ -~

-~ -~

-~ -~ ---

0

~

- ~5

!~ -~ ---~

0

(1,1) (1,1)

0 (1,3) 0 (I,.]) 0 (1,1)

l -l -I

0 0

--l

0 0 0

¼ _l~

0 0

-I

H,v H,s

b,+b2+b~+a+fl+y+(1)

(1,3) (1,3) (1,1) (1,1)

-~

0

l

4~

Hz7 Hzs H2, H3o

0-½ 0 -½ 0 -½ 0 --½

l -l -~

l

l

-l

(1,2)

b2Wb3"Fot+-~+(I)

(1,3) (1,3) (1,1) (1,1)

Qs

0 0 0

½

-l

0 0 0 0

0 0 0 0

07

1

] -7~

l

l l l

H~ 6

Ht9 H2o H:~ H22 Hz3 H~. H~s Hz6

_½ -½ -½ -½

Q6 SU(5) XSU(3)

1 ¼ --I --~ -I -~ ,~ 9 2 15 15 15

1__5 4

W = [ (u~,~ Q~ h-l +NCL,L1 [[~ + U~L2Q2f[2+NCc2L2~/2 + UCL3Q3[[3+ N~,3L3 h-3)

-l- hlg2 (i~,2 +h,fi3cbt3+hzfi3~23-'l-fil h2 ~12 -4.-fil h3 ~D,3 -I--fi2 h3 ti1523 + ¢ h 3 ~ 3 4 h 2 + q523¢'13 45,2 + 45,2(q5 + 45 i- + ~ - ~i- + ¢'+ ¢ ) ; ) +¢'lA¢'i- ¢ '+ +q:'f q~+ + O ; q ~ ) + ½~3(q:hs q54s +h4sh-4s +D45/)45 + O + q5 ~- + O F (/5 F + O ~ - q5 + +O~- qSi- + O + qS~- + ¢ ) ; ~ - )

+ h~fi4~ 04~ + fi3hA~ 4i4sl + {~ [~1 (HI9H20 +H21H22 + H23H24 + H25H26) + ~2 (HI3HI4 +HLsHI6 + H 1 7 H I s ) ] + q523Hz4H25 + O23Hz3H26 +h2H16Hl7 +[[2H15H, s + eCr,HloH27 +eCL2HsHz9

+ ( VLH9 + V2Hli )//27 + V6HsH29 + q54sH17H24 + D45HtsHzl + h45H16Hz5 } , 136

(lo)

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where a common normalization constant x / ~ is assumed. Next, I discuss the pattern of symmetry breaking. The "anomalous" U( 1 )A is broken by the Dine-SeibergWitten mechanism [7 ]. The set of constraints is summarized in the following set of equations: Da = ~ Qg IZk 12=

--g2 exp(~D)

D~=~Q'~lzklz=O,

1927~2

j=l,...,5,

k

Tr(QA) ,

(1 la)

Dj=~]CTklXkl2=0,

j=C,L, 7,8,

(llb,c)

k

W= 0 W = 0 ,

(lid)

where Xk are the fields that get a VEV and Q'k is their charge under the U ( 1 )j symmetry. The set {q~}is the set of fields with vanishing VEV. In the case of the model of eqs. ( 1 ), (2), it is observed that there is a large set of states with vanishing hypercharge and a smaller set of fields with both U( 1 )c and U( 1 )e vanishing. Choosing a set of fields from the first set breaks the observable gauge symmetry to SU (3)c × SU (2)L × U ( 1 )r- Choosing a set from the second set leaves the observable symmetry with SU (3)c × SU (2) L× U ( 1 )L × U ( 1 )c, and U ( 1 ) z' remains unbroken. An example for the second case is given by the set {~13, g545, qSi-, ~ , ~P~-}- The set ofeqs. ( 1 1 ) has the solution I cP4z12=31 g31312=31 g5 7 12=31 q~2- 12=31 cP~- 12= 161r g22 .

(12)

This solution leaves one horizontal U ( 1 ) symmetry unbroken. The superstring vacuum is SU (3)c × SU (2)e × U ( 1 )L × U ( 1 )c × U ( 1 )h × hidden. Choosing other sets will break the remaining horizontal symmetry. The last stage of symmetry breaking can be achieved by using one of the singlets with vanishing hypercharge. Using the set {H23, H18, ~13, ~45, q~z3, q~-, ~ - }, eqs. ( 1 1 ) have the solution

IH2312=lHls12=~lrPas12=31~1312=31~2312=½1cP~12=l~;12-16n2. g2

(13)

This set breaks the observable symmetry to SU (3)¢ × SU (2)L × U (1)r. In the case of standard-like models, it is in general non trivial to find solutions to the set of eqs. ( 11 ), especially solutions of the type of eq. (12), which leave U ( 1 ) r and U ( 1 )z, unbroken. The reason is that, in general, the set of fields which do not transform under SO (10) is reduced relative to other examples [ 2,12 ]. Thus fewer fields are left which can be used to cancel the D equations. This problem will be further discussed in ref. [ 10 ]. There several examples will be presented which do not allow a solution of the F and D fiat constraints. For example, in the model of ref. [ 5 ] such solutions were not found. A possible scenario is that only standard-like models in which one type of Yukawa coupling is obtained as trilevel admit a solution of the set of F and D constraints eqs. ( 11 ). Therefore, in these standard-like models a connection may exist between the heaviness of the top quark, which is observed at low energies, and the requirement o f f and D flatness, via the Dine-Seiberg-Witten mechanism, at the string level.

5. Discussion and conclusions

The choice of VEVs in eq. (13) leaves//1, h45 to give masses to the quarks and leptons. The trilinear term u~, Q~//1 gives mass to the top quark. From eq. (10) we observe that of the standard model quarks and leptons, only + -~ charged quarks obtain a trilinear mass term. This suggests an explanation for the heaviness of the top quark relative to the lighter quarks and leptons. At the string level only the top quark obtains a trilevel mass term and the remaining mass terms are obtained from nonrenormalizable terms. Nonrenormalizable terms are 137

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expected to be suppressed by at least an order of magnitude relative to trilevel terms [ 13 ]. In ref. [ 10 ] I will show how the assignment o f boundary conditions in the vector 7 selects either + 2 charged quark trilevel Yukawa coupling or - ] charged quark trilevel Yukawa coupling, for the states from a sector b~, b2 or b3. For example, in the model of ref. [ 5 ] trilinear mass terms are obtained for - ] charged quarks and charged leptons as well as for + ] charged quarks. A possible scenario is that models which admit a supersymmetric solution, like eqs. (12), (13), allow trilevel Yukawa couplings only for + ] charged quarks. Nonrenormalizable terms are expected to play a decisive role in low energy phenomenology, such as, giving rise to the entries in the seesaw mass matrix [ 14 ]. Eq. (10) indicates that some of the fractionally charged states obtain a Planck mass by giving a VEV to ~ , ~2, without destroying F or D flatness. Nonrenormalizable terms are expected to give rise to the rest of the fractionally charged states. Preliminary studies o f quartic level terms indicate that such non vanishing terms do not exist. Another way to avoid fractionally charged states is by enlarging the hidden gauge group. Examination o f table 2 reveals that all the problematic states fit into 4 and 7~ representations o f SU (4). By modifying the vector 7, the hidden SU ( 3 ) symmetry may be enlarged to SU (4), and the fractional charges are confined. Nonrenormalizable terms are also expected to give rise to effective quartic operators, r/l (u~rd~d~N~)~+q2(d~QLNCL)~. A combination o f fields, qb, fixes the string selection rules [ 13 ] and gets a VEV o f O (mp~). If we assume that the B - L symmetry is broken by a VEV of the righthand neutrino, N ~ , these terms would produce effective dimension-four operators and together mediate very rapid proton decay [ 15 ]. In superstring G U T models, like the flipped SU (5), N~ is the component in the 10 o f SU (5) which is used to break the G U T symmetry. It is seen that the scale of the B - L symmetry breaking controls the rate of proton decay. Because o f the nonperturbative nature o f the string, one must ascertain that such terms are not induced to all orders of such nonrenormalizable terms. Indeed, in the flipped SU (5) [ 2 ], such terms have been shown to exist [ 16 ]. This consideration motivates keeping the additional U ( 1 )z, gauged down to low energies [ 15,17 ]. In this paper I have presented a three generation superstring standard-like model. The massless spectrum is anomaly free apart from a single anomalous U ( 1 ). Application o f the Dine-Seiberg-Witten mechanism cancels the anomaly and restores supersymmetry. The resulting observable gauge group may extend the standard model gauge symmetry by an additional U ( 1 ). If the extra U ( 1 ) is gauged down to low energies, it protects the proton from decaying through effective dimension-four operators which, in general, may arise in superstring models from nonrenormalizable terms. I will expand upon the phenomenology derived from this model in future publications.

References [ 1] M. Green, J. Schwarz and E. Witten, Superstring theory, Vols. 1, 2 (Cambridge U.P, Cambridge, 1987). [2] I. Antoniadis et al., Phys. Lett. B 231 (1989) 65. [3] B. Greene et al., Phys. Len. B 180 (1986) 69; Nucl. Phys. B 278 (1986) 667; B 292 (1987) 606; R. Arnowitt and P. Nath, Phys. Rev. D 39 (1989) 2006; D 42 (1990) 2498; Phys. Rev. Lett. 62 (1989) 222. [4] L.E. Ibafiez et al., Phys. Lett. B 191 (1987) 282; A. Font et al., Phys. Lett. B 210 (1988) 101; Nucl. Phys. B 331 (1990) 421; D. Bailin, A. Love and S. Thomas, Phys. Lett. B 194 (1987) 385; Nucl. Phys. B 298 (1988) 75; J.A. Casas, E.K. Katehou and C. Mufioz, Nucl. Phys. B 317 (1989) 171. [ 5 ] A.E. Faraggi, D.V. Nanopoulos and K. Yuan, Nucl. Phys. B 335 (1990) 437. [6] I. Antoniadis, C. Bachas and C. Kounnas, Nucl. Phys. B 289 (1987) 87; I. Antoniadis and C. Bachas, Nuel. Phys. B 298 (1988) 586; H. Kawai, D.C. Lewellen and S.H.-H. Tye, Phys. Rev. Lett. 57 (1986) 1832; Phys. Rev. D 34 (1986) 3794; Nucl. Phys. B 288 (1987) 1; R. Bluhm, L. Dolan and P. Goddard, Nucl. Phys. B 309 (1988) 330. [7] M. Dine, N. Seiberg and E. Witten, Nucl. Phys. B 289 (1987) 585. [8] P. Fayet and J. lliopoulos, Phys. Lett. B 51 (1974) 461. 138

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[9] S. Ceeotti, S. Ferrara and M. Villasante, Intern. J. Mod. Phys. A2 (1987) 1839. [ 10 ] A.E. Faraggi, in preparation. [ 11 ] A.E. Faraggi and D.V. Nanopoulos, Texas A&M University preprint CTP-TAMU-78, ACT-15. [ 12 ] I. Antoniadis, G.K. Leontaris and J. Rizos, Phys. Lett. B 245 (1990) 161. [ 13 ] S. Kalara, J. Lopez and D.V. Nanopoulos, Phys. Lett. B 245 ( 1991 ) 421; Nucl. Phys. B 353 ( 1991 ) 650. [ 14] A.E. Faraggi, Phys. Lett. B 245 ( 1991 ) 435. [15] R.N. Mohapatra, Phys. Rev. Lett. 56 (1987) 561; A. Font, L.E. Ibafiez and F. Quevedo, Phys. Lett. B 228 (1989) 79. [ 16] J. Ellis, J. Lopez and D.V. Nanopoulos, Phys. Lett. B 252 (1990) 53; G. Leontaris and T. Tamvakis, Phys. Lett. B 260 ( 1991 ) 333. [ 17 ] A.E. Faraggi and D.V. Nanopoulos, Mod. Phys. Lett. A 6 ( 1991 ) 61.

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