Bosonic technicolor in strings

Physics Letters B 284 ( 1992 ) 289-295 North-Holland

PHYSICS LETTERS B

Bosonic technicolor in strings Alex K a g a n ~ a n d Stuart S a m u e l 2 Department of Physics, Cit3 College of City University of New York, New York, NY 10031, USA Received 9 April 1992

It is shown that strings provide a natural theoretical setting for bosonic technicolor. In particular, bosonic technicolor structure arises in a class of strings formulated in four dimensions. An example of a model with three generations and multi-Higgs structure is presented.

1. Introduction

Bosonic technicolor [ 1 ] is a particular combination of technicolor and supersymmetry. It is capable of resolving the hierarchy problem in a more fundamental high-energy theory. It uses technicolor to break S U L ( 2 ) × U v ( 1 ) to UEM(I) and Higgs fields for quark and lepton mass generation. Supersymmetry is invoked to keep the Higgses light. For further details see refs. [ 1-3 ]. The main attraction of bosonic technicolor is its resolution of the hierarchy problem without producing low-energy phenomenological difficulties [4], such as unacceptably large flavor-changing neutral currents or detectable particles which have not been seen. Bosonic technicolor avoids these problems by having a supersymmetry breaking scale, ross, of around 5 TeV. Such a scale for ross suffices to naturally [4] avoid the flavor problem [ 5 - 9 ] and leads to heavy superparticles. Technicolor pseudo-Goldstone bosons, if present, acquire large masses, thereby explaining why they have not been seen. The main drawback of bosonic technicolor is its extensive theoretical structure: it is unaesthetic to have both technicolor and Higgs fields. Furthermore, the probability that Nature possesses both technicolor and supersymmetry is quite small, given that there is hardly any experimental evidence for either of them. ls there a theoretical framework which tenE-mail address: KAGAN @SC1.CCNY.CUNY.EDU. 2 E-mail address: [email protected].

ders more natural the extensive structure ofbosonic technicolor? String theory is one possibility. Consistent string models have supersymmetry. Models are easily constructed which contain SUc ( 3 ) × SUL (2) X U r( 1 ) and have standard model matter including Higgs fields. Often the gauge group contains non-abelian factors beyond SUe(3) and S U e ( 2 ) which may serve as a technicolor group. An important requirement is that the Higgs fields must couple both to technicolor fermions and to the ordinary quarks and leptons. In this letter, we investigate whether strings, formulated in four dimensions, can possess a bosonic technicolor structure. We consider heterotic string models formulated in four space-time dimensions [ 10-15 ] and use the fermion formulation in ref. [ 15 ] based on a non-linear realization of world-sheet supersymmetry. In what follows, we sketchily review the construction method so as to set notation. A string model is defined by supplying 66-dimensional vectors, v~, v2, ..., v~ and spin-structure coefficients, c[',~], both of which must satisfy certain conditions given in ref. [15]. The vectors are used to specify the boundary conditions on the 22 left-moving fermions and the 44 right-moving fermions of the world-sheet. There exist m i n i m u m integers, N,, such that N , v , = 0 mod 2. The physical sectors are associated with linear combinations t,= Y ; ~ m,v, where m, are integers modulo N,. The sector v has a vacuum denoted by [ 0),.. States [ s) ~.are built on I0 ),. by applying creation operators. One only retains states sat-

0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

289

Volume 284, number 3,4

PHYSICS LETTERS B

isfying the generalized GSO conditions. For more details, see ref. [ 15 ].

2. Bosonic technicolor structure in strings We define a model to have bosonic technicolor structure when the following conditions are satisfied: ( 1 ) The model possesses supersymmetry. (2) There are Higgs scalars. (3) There is an unbroken non-abelian gauge group which is identified with the technicolor group, GTc. (4) There are technicolor fermions (TC-fermions) transforming under GT¢. Some of these also transform under SUe(3) X S U L ( 2 ) × U r ( 1 ) so that bilinears in TC-fermions can have the quantum numbers of Higgs fields. ( 5 ) The model has the appropriate Higgs-Yukawa couplings. These consists of (a) Higgs-TC-fermion-TC-fermion; (b) Higgs-SM-fermion-SM-fermion. A standard model fermion (SM-fermion) is a quark or lepton. Note that a model with bosonic technicolor structure is not necessarily realistic. There are many additional constraints. For example, the technicolor group needs to be asymptotically free and extended supersymmetry is not permitted since chiral representations are required. Our goal, in the current letter, is not to find the perfect realistic model but to uncover bosonic technicolor in strings. The generic nature of bosonic technicolor structure in strings is illustrated below in a simple system, based on three vectors, S, $ and b3. We begin by establishing some notation. As in ref. [ 16 ], the 66 fermions are denoted byz~'Z'yi0)ifl(y'cb'tPk~ ''' in that order, where the indices/z, i, j, k and m, respectively run from 0, ..., 3, 1, ..., 6, 1.... ,3, 1.... ,5, and 1, ..., 8. A bar on a fermion indicates a right-mover. The fermions Z ~', Z', Y', 0)i, ~i and 0)' are real, while 0 j, 5ok and ~7" are complex. A fermion with two superscripts, such as Z ~2, corresponds to two real fermions which have been regrouped into a complex fermion, i.e., Z 12= ( Z ' - i z 2 ) / , , / 2 a n d z ' 2 . = (Z' + i z 2 ) / x f ~. We often denote a vector as a string of fermion variables with coefficients which are the vector entries. To have supersymmetry, it is necessary to have a vector, vl : S : z p Z I 2 z 3 4 Z 5 6 : (11o, 0121044) [12]. The 290

25 June 1992

superpartners of a sector t, are in the sector v + S or v - S . Also required [ 15] is a vector proportional to unity; we take ~ = ~ = (122]144). At this stage, the model consisting of S and ~ has N = 4 supersymmetry and a gauge group of SO (44). Assume there is a sector associated with a third vector, b3, with integer entries, which contains standard model fermions. Without loss of generality, one may take v3=b3= XjtX56(o 16020) 30) 4?]30) i (j) 2(.93G,) 4 ~1 ~t2 ~3 ip4 ~p5. Summarizing,

121 =S=I~/'ZI2~.34X 56 ~-~ ( 110, 012 1044 ) , t'2 = ' ~ = (122 1144), 113 ~ b 3 =XpX560) 10)20)3094~3(~ ) 1(~ 2(2)3(j)4 ~pl ~/2 ~t~3~4 Ip5 = (14, 04, 12, 06, 14, 02104, 12, 06, 14, 02, llo, 016).

(1) The vector b3 reduces N = 4 supersymmetry to N = 2 supersymmetry and breaks SO(44) to S O ( 1 6 ) × SO(28). Besides gravity fields and vector gauge bosons, the null sector, v = 0 , contains scalars associated with an N = 2 scalar super-multiplet, T, transforming as (16,28) under SO(16) × S O ( 2 8 ) . Other scalars, .~ and f/~, in the null sector are part of the gauge multiplet and, consequently, transform respectively as (120,1) and (1,378). In an N = I decomposition of N = 2 supersymmetry, .~f and ,~ are the scalar components of N = 1 chiral fields, H a n d S. The b3 sector contains the fermions of N = 2 scalar multiplets, F a n d F, transforming respectively as ( 128,1 ) and ( 128, 1 ). In this simple model, the massless spectrum is independent of the spin-structure coefficients. For definiteness, we fix the spin-structure coefficients as in table 1. Let us interpret SO ( 16 ) as a G U T gauge group and S O ( 2 8 ) as a technicolor group. Then, T carries both Table 1 Spin-structure coefficients c[,';'] for the model of section 2.

S S "D b3

1 1 1

1]

b3 1 1

-1

-1 -1 -1

Volume 284, number 3,4

PHYSICS LETTERS B

G U T and technicolor quantum numbers and its fermion components serve as technicolor matter. The scalar components of H act as Higgs scalars. In an N = 1 decomposition, there are trilinear couplings in the superpotential of the form

TTH,

FFH,

FFH,

TTS .

(2)

Note that Hconnects technicolor matter, Twith G U T matter, F, precisely in the form needed for bosonic technicolor. This model demonstrates that four-dimensional strings can have a bosonic technicolor structure; the five conditions given above are satisfied. However, the model is obviously not realistic. There is N = 2 supersymmetry and the gauge group is too big. However, this example is not the final model but the starting point for constructing many four-dimensional strings with bosonic technicolor structure. The deficiencies are eliminated by adding vectors. These vectors, if appropriately chosen, break supersymmetry from N = 2 to N = 1 and break S O ( 1 6 ) × SO (28) to a smaller gauge group. They project out certain parts of the chiral fields T, H and F, so that the particle content of these superflelds is reduced. As long as some pieces of T, H and F remain, their Yukawa couplings are as in eq. (2) because additional vectors leave interactions unaffected. Hence, four-dimensional string models which include S, ~ and b3 among their vectors frequently lead to a bosonic technicolor structure. What is needed to destroy the bosonic technicolor structure in eq. (2)? One must add enough vectors to break the technicolor group down to an abelian group or to reduce the particle content of T, H and F so that they cannot be interpreted as technicolor matter, Higgs fields or standard-model matter. Even if the structure in eq. (2) is eliminated by the addition of vectors, bosonic technicolor may arise in new sectors, as one will see in section 3. Many four-dimensional models in the literature [ 16-18] contain S, and b3. An example is the flipped SU ( 5 ) string [ 16 ], for which the technicolor gauge group under which T transforms has been reduced to U ( 1 ) factors. In other words, except for an unsatisfactory technicolor group, it has "a bosonic technicolor structure".

25 June 1992

3. An example model This section presents a more attractive model with an interesting multi-Higgs structure. At the string scale, we require a net excess of three families. We construct a flipped S U ( 5 ) G U T model [ 19] for which G o u t = SU ( 5 ) × U ( 1 ) and GTC = SU ( 3 ). The advantages of using flipped SU (5) in string model-building have been emphasized in ref. [20]. It is also favored in bosonic technicolor G U T models [21 ] for reasons given in section 4. Flipped SU (5) differs from ordinary SU (5) in that the hypercharge generator is not completely embedded in S U ( 5 ) ; it involves a U ( 1 ) generator outside the group. Furthermore, the roles of " u p " and " d o w n " are interchanged. For more on flipped S U ( 5 ) see refs. [16,20]. We add to v I = S , t'2=~ and t/3=b3, four more vectors:

/'?4=b2 ~.)(ltX34(~ 0 10.)20.)50.)602(~ I(.t)2(~05(~06~'tI ~ ~P3~l~t4~/5

= (14. 02, 12, 08, 12, 02, 12 ] 02, 12, 08, 12, 02, 112,016) , t'5 =b, = Z~,Z34y ly2y 5V602)71)72)75)76~pl ~p2~t3 ~O4~rt5

= (14, 02, 12, 02, 12, 02, 12, 06 ]

02, 12, 02, 12, 02, 12, 06, 110, 016)

,

t'6 = O~ =y60.)6;~1.1 (/~2/~3)ff60j340~6 × 1 ( ~f2 [p2 ~3 Ip4 ~t5~l ~2~3~4~5 ) 58 = (015, 1,05, 1 I

I 12, (~')4, 05,_

1

1,02,12,0, 1, (~)20, 04, 12) ,

/~7=P = y2¢02y2¢02~7465d6~77

= (011, 1,05, 1,041 07, 1,05, 1,020, 18,02) • (3)

The vector bt (or b2) reduces N = 2 supersymmetry down to N = 1 and produces chiral standard model matter. In addition, b, breaks S O ( 1 6 ) to S O ( 1 0 ) 291

Volume 284, n u m b e r 3,4

PHYSICS LETTERS B

× SO (6). The vector o¢ reduces SO ( 10 ) to the G U T group, SU (5) X U ' ( 1 ). The four vectors in eq. (3) break S O ( 2 8 ) to S U ( 3 ) × U ( 1 ) X .... of which the S U ( 3 ) factor is taken to be the technicolor group: Gxc = SU (3). The vector bosons of SU ( 5 ) × U ' ( 1 ) are Z u ~P' ~PJ*(0) I05 o, with 1 ~
b3, b2, bl and S + b3 + b2. The notation is as follows. The symbols T and stand for chiral superfields transforming respectively as 3 and 3 ofGvc. S U ( 5 ) × U ' ( 1 ) × Gxc singlet chiral fields are indicated by S. A superscript D or a subscript U corresponds to a 5 of SU ( 5 ) with a U ' ( 1 ) charge of - 1; a superscript U or a subscript D is a of SU ( 5 ) with a + 1 U ' ( 1 ) charge. Hence, fields with ~t If one wants to make a model with exactly three families, the vector/.,8 =Pt=.vSy60)50)6Ol)~5.156¢05016~6~7~8= (014 , 12, 04, 12

[ 12, 08, 12, 04, 12, 020, 16) can be added. This m a k e s the n u m ber of massless sectors larger and the model a little more complicated. It also eliminates, among things, a 10 and a 10 which might be used to break S U ( 5 ) × U ' ( I ) to SU,.(3)X SUt_(2) × U v ( 1 ).

25 June 1992

U and D labels carry the S U ( 5 ) × U ' ( 1 ) quantum numbers of Higgs fields and are denoted by H if they are GTc-singlets. The chiral fields 1, 5, 10 indicate the S U ( 5 ) content obtained from a 16 of SO( 10); they are associated with standard-model fermions. An anti-family corresponds to 1, 5, 1~. Superscripts _+ denote a global U( 1 ) charge associated with o~34; it is used to help distinguish some fields. A subscript in parentheses indicates in which sector a field appears. The model contains several bosonic technicolor systems. For example, the remnant of the superpotential ofeq. (2) contains

TsT'uH~,

U-+ H35(3) 10?3).

TsTDH~,

We can identity the fermions of the b3 sector with the third family. The top quark obtains its mass from the last term in eq. (4) when ( H 3 U ) # 0 due to ( ~7~tj 5 4: 0. The bottom quark and tau lepton do not have trilinear couplings and their masses might arise from higher-dimensional operators, explaining why they are suppressed relative to the top quark mass. Such isospin breaking, in which only an up or down quark gets a mass, happens often in 4D string models [181. There are two interesting mechanisms for giving masses to the lighter quarks and leptons. One is by non-renormalizable string-induced interactions. In the usual flipped SU ( 5 ) string [ 16 ], five- and higherdimensional interactions play an important role in generating the fermion mass hierarchy [22]. Another possibility is a multi-Higgs system. In fact, our model has such interactions. There are Higgs fields, HU~23) ± and H~2~) D+ in the bz+b3 sector, which connect the family in the b3 sector to the family in the b2 sector via

Table 2 Spin-structure coefficients c[',',j] for the model of section 3. t's

t', S

S

-

292

b2

1

-1

--I

l

-1

1

-I -1

1

-l

b3 b2 b, P

b3

~

-i

(4)

b~ -1

1

-1

at -1 1

-1 1

-1

1

1

-1 -1

i

i -1

-1 1

-1

1

-1 -1

1

--1

1

-i -1

-1

1

-1

P

-i

Volume 284. number 3,4

PHYSICS LETTERS B

25 June 1992

Table 3 Some interesting states in the model of section 3. Superscripts ""b" run from 1..... 3. Superscripts "i" and "i~" run from 1..... 5. When lx~o or more "'t~'"appear, the indices must be distinct. Left-handed chiral superfields

SU ( 5 ) × SU ( 3 )u¢~) quantum numbers

TD

HD tit' tt D 2 T3 $23 '~23 T2

7"2 1(3) +

z 12 9i* 6b(O)tO)o z '2 SO'6h*(0)I0)o

3,1)1

Z 56 S°i 0 3 * ( 0 ) I 0 > o

5, 1)_1 3,1)~ 5, 1)_~ 1, 3)0 1,3)o 1,1)o 1,1)o 1, 3)0 1,3)o

X56 SOi*r/3(O) I0)o Z34 Sv#02"(0) 10)0 X34 SO~*r/2(O) I0)0 X34¢>* 03(0)I0)0 X 34 6 t' 0 3 * ( 0 ) I O ) o

;¢2 r/2 ~13.(0) i 0)0 Z '2 02* t13(0) 10)o X566 ~ 02(0)10)o ~,566b ~12.(0) I0)o I0)~, W"* ~v,~*7"* ¢P"*(O)I0)~

5, 1 )-3/2

I0~) I(2)

10, 1 )1/2

~Jll* [/Jt2* (034* (.O34.(0) I O ) b 3

1, 1 )5/2

5(2)

5, 1 )-3/2

~12"(0) 10>,= SO;'* SO':*SO'* SO'~*q2*(O)10>~

10(2)

10, 1 )~/2 1, 1 )-5/2

~i,* SOl:* ~ 5 056 q2*(O ) i O ) b 2

5, I )3/2

Soi* (t) l ( 0 5 ( 0 ) [ 0 ) b 2

10, 1 ) _ t/2 1, 1 )5/2

SO"* 9 ~:* SO'* 0)' 0)6(0)t0)b2 r/2*(O) ]O)b, ~,t* SO,a* SO,3*SOi4*02*(0)iO)b ' SO"* SO"*95)'6 02*(0)[0)t,, I'll,5 r/2*(O) ]O)b, t/t;,* SO,:* SO,s*SO,4*O,p5 tl2*(O)IO)b, 7s,,. 7j,:..~).6 ~12.(0) IO)t, SO¢r/3*(O)IO)s+tj+n: SOt 0) 34* (2) 34* 035 0)6 q3.(O) iO)s+~3+b2 SO,.0)5 ~b~q3.(O ) IO)s+~,+b~ SO,* 0) 34* (034* q3.(O)i 0 )S+b3+b2 de. 0)5 0)6 q2.(O ) I0 )s+,,+~., ()b* 0) 34* 0) 34* q2.(O) i O)s+o~n2 6~ q2*(O)IO)s+o,+~2 d~ 0)34.0)34. 0)5 {o6 q-'*(O) I0 )s+,~+o~

](2) 5(2) 10(2)

1(~) 5(1)

(3, 1 )-3/2

I0(~) I(,) 37,)

(10, 1)1/2

10(t)

(10, 1)t/.,

( 1, 1)~/2 (3, 1 ) _ 3 / 2

u+ H(23) uH(23) D+ 11(23) D H(23) + T(23)

(L1), (5, I)) (5, 1) 1 (5, 1 ) _ l

(1,3)o (I, 3)o (1,3)o (1,3)o

T [-2~) 7"~23, T(23)

l) H(23) 3(2) 1 (+3 ) , LI+

5,3)_, 3,3),

1, 1 )s/2

-+ 5(3)

State

H(23) 3(2) 10~3),

H~25) I(2 ) 3~-3), H D+ (23) I0(~) - i0(-3) -+

HU25) 10(2) 5(3).

(5)

Such i n t e r a c t i o n s p e r m i t a see-saw m e c h a n i s m for s e c o n d - g e n e r a t i o n f e r m i o n masses, a p o t e n t i a l explan a t i o n of why the s e c o n d g e n e r a t i o n is lighter t h a n the third. T h e Higgses, H(23) u+ a n d H(2g), D+ might obtain suppressed e x p e c t a t i o n values from n o n - r e n o r m a l i z a b l e t e r m s which mix t h e m with H ~ a n d H ~ . Eq. (5) c o m b i n e s with

,

± J~l u (23)

T ~(23) Tu,

HU± (23) H~2~ ) $23,

D± -H(23) T~23) TD,

T~23)T(23)$23

(6)

to p r o d u c e an interesting a l t e r n a t i v e b o s o n i c technicolor system. In this case the top q u a r k c o m p o n e n t s , tR a n d tL, a p p e a r in either 3(2) a n d 10(-3) or in 3~-3) a n d 10(2) #2. Note that the origin o f this b o s o n i c tech#2 In determining which vacuum is selected and which fermions correspond to the top quark, the Dine-Seiberg-Witten mechanism for anomaly cancellation is important [23,24]. 293

Volume 284, number 3,4

PHYSICS LETTERS B

nicolor structure is different from that arising in eq. (2). Eqs. (5) and (6) illustrate how the addition of vectors can lead to further sources of bosonic technicolor. There are Higgs fields, H~ J and H p , which couple to the leptons and d-quarks in the sectors b~ and b2. These Higgses also couple to the technicolor chiral superfields, T2, 7~u, 7~2and TD, thereby forming a third bosonic technicolor system. However, it is more natural to assume that technicolor condensation does not occur for this system, (7~uT2) = 0 and (7~2TD) = 0 , and that ( H U ) and ( H P ) are small or zero so that the first generation is light. If ( H U ) = ( H D ) = 0 , first generation masses might arise from radiative corrections of higher-dimensional non-renormalizable terms. The model contains additional fields and interactions which an interested reader may request by writing to the authors. The structure in eqs. ( 4 ) - ( 6 ) is attractive and suggestive. It is similar to the multiHiggs system proposed in ref. [25] to resolve the family-mass hierarchy problem in a bosonic technicolor model.

4. Discussion

In this letter, we constructed four-dimensional strings with bosonic technicolor structure. We argued that such structure arises in a wide class of models. Hence, four-dimensional strings offer an explanation of why a theory might have both supersymmetry and technicolor. Technicolor models require a custodial S U ( 2 ) symmetry to maintain t h e p parameter near I [26]. Flipped S U ( 5 ) offers a novel way of achieving this without using families of technicolor matter. If T a n d Tu are technicolor matter transforming respectively as (l, N T c ) o a n d (5, ATvc) i with ( T u T ) ¢ 0 , i t i s necessary to have a partner, T', for which (-)(i) is a custodial S U ( 2 ) doublet and for which ('FsT') = ( 7~u T ) . In flipped SU (5), a candidate for T' is a field transforming as the charged lepton singlet of a family with quantum numbers ( 1, NTC)+5/2. Hence, a singlet and a 5 of SU (5), together with their mirrors, suffice. Such an economical technicolor system assists in satisfying S-parameter constraints [ 27-29 ] 294

25 June 1992

and in achieving asymptotic freedom for SU,. (3) and Gwo

In 4D string models, T' is usually accompanied by

a (5, Nvc) 3/2 state. Unless vectors are added so that the generalized GSO projection eliminates it, there will be at least two SU (5) 5's and 5's of technicolor matter at low energy. This would spoil asymptotic freedom for Q C D or technicolor. Another possibility, which is perhaps more natural from the point of view of strings, is as follows. Frequently in models we have analyzed, in the sectors of the form v, where v . . . . _+ ½ ( 17[/2(7~J2~E/3tT~4(TU5 ) .... there appears technicolor matter, with SU (5) u.( ~} quantum numbers of 1+5/4, 51/4, and 3_~/4. A system of technicolor matter fields, T, T' and 7~u, transforming as (1, Nwc) 5/4, (1, ATC)S/4, and (5, NT(),/4, together with their mirrors, can couple to Higgs fields and satisfy the p-parameter constraint. The color-singlet components of T, T' and Tu have electromagnetic charges of + ½. Fractionally charged states always appear in potentially realistic d = 4 string models [ 30 ]. Often, models are built in which such states are confined by a hidden-sector gauge group [ 16 ]. Technicolor can play a role in confining these states. Curiously, the color-singlet components of T, T' and Tu coincide with the well-known minimal technicolor system consisting of an SUL(2) doublet and two SUL(2) singlets. In short, flipped-SU(5) bosonictechnicolor string models potentially offer an elegant solution of the p-parameter constraint. If a technicolor 10~/2 (respectively, 10_3/4) is present, doublet-triplet splitting is possible for the technicolor 5 ~ (respectively, 51/4). This works the same way as the Higgs doublet-triplet splitting mechanism of ref. [20] via a trilinear Yukawa coupling with a Higgs 10~/2. This can assist in achieving asymptotic freedom for technicolor and QCD. The mechanism is not possible for technicolor matter transforming as (B, Nvc) 3/2. Although the model in section 3 is unlikely to survive many of the phenomenological constraints of an ideal or realistic theory, it is just one possibility among a large class of bosonic technicolor string models. More importantly, it illustrates that four-dimensional string theories can have various types of bosonic technicolor structures as well as attractive multiHiggs systems. Such structures can potentially resolve the G U T hierarchy problem and lead to realis-

Volume 284, number 3,4

PHYSICS LETTERS B

tic q u a r k - l e p t o n m a s s e s a n d m i x i n g angles, w i t h o u t i n t r o d u c i n g u n a c c e p t a b l y large f l a v o r - c h a n g i n g n e u tral c u r r e n t s o r u n s e e n d e t e c t a b l e particles. T h e p h e nomenologically attactive framework of bosonic t e c h n i c o l o r s h o u l d b e o f i n t e r e s t for f u t u r e f o u r - d i mensional string model-building.

Acknowledgement We thank Ignatios Antoniadis for valuable discussions. T h i s r e s e a r c h was s u p p o r t e d in p a r t b y t h e U S Department of Energy under the grant DE-FG02-92ER40698, by the National Science Foundation under the grant PHY90-20495, and by the PSC Board of H i g h e r E d u c a t i o n at C U N Y u n d e r t h e g r a n t 6-62361.

References [ 1 ] S. Samuel, Nucl. Phys. B 347 (1990) 625. [2] A. Kagan and S. Samuel, in: Proc. Conf. Quarks, symmetries and strings: a symposium in honor of Bunji Sakita's 60lh birthday (World Scientific, Singapore, 1991); Bosonic technicolor and the flavor and family-mass-hierarchy problems, in: Proc. PASCOS-91: S. Samuel, in: Proc. Conf. Beyond the standard model II (World Scientific, Singapore, 1991 ). [3] A. Kagan and S. Samuel, Intern. J. Mod. Phys. A 7 (1992)

1123. [4 ] M. Dine, A. Kagan and S. Samuel, Phys. Left. B 243 (1990) 250. [ 5 ] J.F. Donoghue, H.P. Nilles and D. Wyler, Phys. Len. B 128 (1983) 55. [61M.J. Duncan, Nucl. Phys. B 221 (1983) 285. [7] A. Bouquet, J. Kaplan and C.A. Savoy, Phys. Lett. B 148 ( 1984 ) 69. [8] L. Hall, A. Kosteleck~, and S. Raby, Nucl. Phys. B 267 (1986)415.

25 June 1992

[9] H. Georgi, Phys. Lett. B 169 (1986) 231. [10] K.S. Narain, Phys. Lett. B 169 (1987) 41. [ 11 ] H. Kawai, D.C. Lewellen and S.H.-H. Tye, Phys. Rev. Lett. 57 (1986) 1832; Phys. Rev. D 34 (1986) 3794; Nucl. Phys. B288 (1987) 1. [ 12 ] I. Antoniadis, C. Bachas and C. Kounnas, Nucl. Phys. B 289 (1987) 87. [ 13 ] W. Lerche, D. Liist and A.N. Schellekens, Nucl. Phys. B 287 (1987) 477. [14] R. Bluhm, L. Dolan and P. Goddard, Nucl. Phys. B 309 (1988) 330. [ 15 ] I. Antoniadis and C. Bachas, Nucl. Phys. B 298 ( 1988 ) 586. [ 16] 1. Antoniadis, J. Ellis, J.S. Hagelin and D.V. Nanopoulos, Phys. Lett. B231 (1989) 65. [ 17 ] A.E. Faraggi, D.V. Nanopoulos and K. Yuan, Nucl. Phys. B 335 (1990) 437; I. Antoniadis, G.K. Leontaris and J. Rizos, Phys. Lett. B 245 (1990) 161: A.E. Faraggi, preprint CTP-TAMU-49/91 (June 1991 ). [18]A.E. Faraggi, preprint CTP-TAMU-65/91 (September 1991 ). [19] S. Barr, Phys. Lett. B 112 (1982) 219. [20] I. Antoniadis, J. Ellis, J.S. Hagelin and D.V. Nanopoulos, Phys. gen. B 194 (1987) 231; J. Ellis, J.S. Hagelin, S. Kelly and D.V. Nanopoulos, Nucl. Phys. B 311 (1988) 1. [ 21 ] A. Kagan and S. Samuel, work in progress. [22] I. Antoniadis, J. Ellis, J.S. Hagelin and D.V. Nanopoulos, Phys. Lett. B 208 (1988) 209; J.L. Lopez and D.V. Nanopoulos, Phys. Len. B 251 (1990) 73. [23] M. Dine, N. Seiberg and E. Winen, Nucl. Phys. B 289 (1987) 1839. [ 24 ] S. Ceconi, S. Ferrara and M. Villasante, Intern. J. Mod. Phys. A 2 ( 1 9 8 7 ) 527. [25] A. Kagan and S. Samuel, Phys. Lett. B 252 (1990) 605. [26] L. Susskind, Phys. Rev. D 10 (1979) 2619; S. Weinberg, Phys. Rev. D 19 (1979) 1277. [27] M.E. Peskin and T. Takeucbi, Phys. Rev. Len. 65 (1990) 964. [ 28 ] W.J. Marciano and J.L. Rosner, Phys. Rev. Len. 65 (1990) 2963. [29] M. Golden and L. Randall, Nucl. Phys. B 361 ( 1991 ) 3. [30] A.N. Schellekens, Phys. Lett. B 237 (1990) 363.

295