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Gut model-building with fermionic four-dimensional strings
Volume 205, number 4
PHYSICS LETTERSB
5 May 1988
G U T M O D E L - B U I L D I N G W I T H F E R M I O N I C F O U R - D I M E N S I O N A L STRINGS -AI. A N T O N I A D I S " b c a
a,~, John ELLIS
a,b,
J.S. H A G E L I N c and D.V. N A N O P O U L O S
d
CERN, CH-1211 Geneva 23, Switzerland Stanford Linear Accelerator Center, Stanford University, Stanford. CA 94305, USA Maharishi International University, Fairfield, IA 52556, USA University o f Wisconsin, Madison, W153706, USA
Received 7 January 1988
We report a first attempt at model-building using the fermionic formulation of string theories directly in four dimensions. An example is presented of a supersymmetricflipped SU (5) × U ( I ) model with three generations and an adjustable hidden sector gauge group. The simplest version of the model contains most of the Yukawa couplings required by phenomenology,but not all those needed to give masses to quarks or conjugate neutrinos. These defects may be remedied in a more general version of the model.
It is now well appreciated that the two key ingredients in formulating a consistent string theory are conformal invariance and modular invariance. The former condition fixes the number of degrees of freedom on the worldsheet. It can be satisfied in any number of dimensions d,<<26 (10 for a supersymmetric left- or right-moving sector) if the space-time coordinates X~:/z = 0,1, ..., d - 1 are supplemented by internal degrees of freedom contributing 2 6 - d ( 1 5 - 3 d / 2 ) to the central charge of the Virasoro algebra. Modular invariance then imposes nontrivial constraints on the boundary conditions for these internal degrees of freedom which ensure that counting errors are not made when higher-genus string topologies are summed. It is known that the number of solutions to these modular invariance conditions is restricted, particularly if the number of space-time dimensions d is close to the critical number 26 (or I 0). In particular, in d = 10 there are only two modular invariant heterotic string theories with N = 1 space-time supersymmetry, based on the gauge groups SO (32) and E8 × E~ [ 1 ]. The choice of gauge group for string theories formulated directly in d = 4 dimensions is much more extensive, and is imperfectly understood as yet. An interesting subclass of models is offered by supersymmetric compactifications of the Es × E~ heterotic string. Early attention was focused on Calabi-Yau manifold compactifications [2 ], but has subsequently been extended to other manifolds [ 3 ] and to orbifold compactifications [4 ]. The most systematic approach to this ambiguity in the four-dimensional gauge group would be to use one of the general formulations of string theories directly in d = 4, using either bosonic [ 5 ] or fermionic [ 6-9 ] variables to describe the internal degrees of freedom. The range of choice in four-dimensional theories is embarrassingly generous, and no systematic enumeration of models has yet emerged. The alternative strategy, followed here, is to start from the bottom up, looking for models which contain phenomenologically favored ingredients such as the standard model or a plausible grand unified theory ( G U T ) . Already examples are known of Calabi-Yau ~ and orbifold [ 11 ] compactifications which yield SU (3) c × SU (2) LX U ( 1 )": n/> 2 gauge groups. In this paper we look for models containing the flipped supersymmetric SU (5) × U ( 1 ) G U T whose virtues were recently extolled [ 12,13 ]: natural doublet-triplet mass splitting, a see-saw mechanism for neutrino masses, no cosmologically embarrassing phase tran~r Worksupported by the Department of Energy, contract DE-AC03-76SF00515. On leave of absence from Centre de Physique Throrique, l~colePolytechnique, F-91128 Palaiseau, France. ¢~See ref. [ 10] for a reviewand references. 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
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sitions [14], etc. Of particular importance is the fact that this model does not need any adjoint Higgs representations. No string model with adjoint chiral superfields can be obtained in the fermionic formulation [9 ], and this may well be a general property. Other GUTs require adjoint Higgses to break the initial GUT symmetry and/or realize doublet -triplet mass splitting naturally. Therefore, SU (5) × U ( 1 ) is uniquely favored as a GUT group, as has already been stressed in the more limited context of manifold compactification [ 13 ]. In this paper, our tool for obtaining this uniquely simple GUT is the fermionic formulation of four-dimensional strings, as developed in refs. [ 8,9 ]. We exhibit a choice of boundary conditions for the world-sheet fermions which yields immediately an observable [ SU (5) × U ( 1 ) ] × U ( 1 ) 3 gauge group as well as an d la carte hidden sector gauge group. The model contains three generations of quarks and leptons, additional 10, 10, 5 and representations of SU (5) which could be Higgs fields, and a number of SU (5) X U ( 1 )-invariant fields as advocated in ref. [ 12 ]. Vacuum expectation values of scalar fields break one of the surplus U ( 1 ) 3 gauge generators via the conventional Higgs mechanism, and another is anomalous. The fermionic formulation also tells us what Yukawa couplings are present, and we find all the ones we want except some of those giving masses to quarks and to conjugate neutrinos. These defects may be avoided by a small modification of the simplest consistent choice of fermionic boundary conditions which introduces some additional l0 + 1--0and $ + $ representations. While developing our model, we strive to illustrate some general principles of fermionic model-building which may aid subsequent, more inspired, efforts. We start with a brief review of the main characteristics of four-dimensional heterotic string theories in the fermionic formulation [ 8,9 ]. In the light-cone gauge, in addition to the two transverse bosonic coordinates X ~ and their left-moving superpartners q/U(z), the fermionic content is [ 6 ] 44 right-moving and 18 left-moving fermions q)~ (Z): A = 1, 2 .... ,44 a n d z i ( z ) , y~(z), w~(z): i= 1, 2, ..., 6, respectively. World-sheet supersymmetry is nonlinearly realized among the latter via the supercurrent 6
T~(z)=~,"OzX,,+ E z'y'w'.
(1)
i=l
A four-dimensional string model is defined by specifying a set E of boundary conditions for all the world-sheet fermions, constrained by making the world-sheet supercurrent ( 1 ) periodic (space-time fermions) or antiperiodic (space-time bosons). When all the boundary conditions are diagonalized simultaneously in some general complex basis {f}, the elements of E are vectors a such that every complex fermion f picks up a phase f ~ - e x p [ i n a ( f ) ]f: a ( f ) e ( - 1 ,
1],
(2)
when parallel transported around the string. In this case, -= forms a group under addition (rood 2), and can therefore be generated by some basis B---{b t, b2..... b N}. It has been shown [ 8,9 ] that to every element ot of-= there corresponds a sector ~ in the string Hilbert space Yd, and to every basis element b~of B a fermion number projection:
~=~o,~z,= ~, [exp(inb,.F)=~o,c*(~)]~,
(3)
where F is the vector of all fermion numbers defined: F ( f ) = 1 = - F ( f * ) , the dot product is lorentzian (left minus right), ~, is the space-time fermion parity and the phases c( ~ ) are constrained by multiloop modular invariance. In order for this paper to be self-contained, we now give the explicit form of the constraints on the basis B and on the phases c for generic rational boundary conditions [ 9 ] ,2 ~2 We consider here neither some exceptional cases, nor the nontrivial realizations of world-sheet supersymmetry discussed in ref. [9].
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Basis B. ( a l ) We choose B to be canonical, i.e., any linear combination Y~irnibi=0 iff mi= 0 (mod N~) for some integers N~ (for example, Ni=2 when the fermions are periodic or antiperiodic), and the vector I~B. (a2) For any pair bi, bj of basis elements, one has #3 Nob:bj = 0 (mod 4) where N o is the least common multiple of N~ and Nj, and N~b2= 0 (mod 8) ifN~ is even. (a3) The number of real fermions which are simultaneously periodic under four boundary conditions bl, b2, b3, b4 is even.
Phases c. ( b l ) We choose the c(bb~) for i
c~)=exp(i½rta.,)c*(~),
0
ca
ca
c(fl+y)=c~,~c(#)c(y).
(4)
Physical states from the sector ~ are obtained by acting on the vacuum I0 ) , with bosonic or fermionic oscillators with frequencies ½[ 1 + ca( f ) ] ( ½[ 1 - ca( f ) ] for f* ) and applying the fermion number projections, eq. (3). The mass formula is M2=-½+~ca~ + EVL=--I+~Ca~+EVR, L
(5)
R
where caL (CaR) is the left (right) part of the vector ca and the VL (UR) are frequencies. When some fermions are periodic, the vacuum is a spinor in order to represent the Clifford algebra of the corresponding zero modes. For each periodic complex fermion f there are two degenerate vacua I + >, I - >, annihilated by the zero modes fo and ~, and with fermion numbers F ( f ) = 0, - 1, respectively. Before starting to build a model, we note a simple but very crucial relation between the world-sheet fermion numbers F ( f ) and the U( 1 ) charges Q ( f ) with respect to the unbroken Cartan generators of the four-dimensional gauge group, which are in one-to-one correspondence with the U ( 1 ) currents f*f for each complex fermion f: Q(f) = ½ca(f) + F ( f ) .
(6)
The charges Q ( f ) can be shown to be identical with the momenta of the corresponding compactified scalars in the bosonic formulation [ 15 ]. The representation (6) shows that Q is identical with the world-sheet fermion numbers F for states in a Neveu-Schwarz sector (ca = 0), and is ( F + ½) for states in a Ramond sector (ca = 1 ); note that the charges of the [ + > spinor vacua are + ½. The model we seek in this four-dimensional fermionic string formalism is the supersymmetric flipped SU (5) GUT of ref. [ 12 ]. It contains the following chiral matter superfields: Three generations of matter fields F~= (10, ½), ~, = (~, _ 3 ), and ~ = (1, ~ ); Higgses H = (10, ½), I:I= (i-0, - ½), h = (5, - 1 ), and fi= (5, 1 ); and four singlets ~o.~= (1, 0) o f S U ( 5 ) × U ( 1 ). The superpotential of the model is [ 12] W = 2 1 F F h +22Ft~11 + 2 s ~ C h + 2 4 H H h +,;LsI:II:Ifi +26FI=I¢~ +27hfi¢~ + 2 s t~3 ,
(7)
where the Yukawa couplings 21.2,3.6.7.8 are matrices in generation space. One component each of the H and ft acquire large VEV's breaking S U ( 5 ) X U ( 1 ) ~ S U ( 3 ) c X S U ( 2 ) L X U ( 1 )r, and S U ( 2 ) L × U ( 1 ) r ~ U ( 1 )~m via VEV's of components of the h and ft. Uniquely among all the GUTS known to us, there is no adjoint or larger self-conjugate Higgs representation. The model is also attractive because it solves naturally the Higgs doublet -triplet mass-splitting problem, has a see-saw neutrino mass spectrum, and avoids rapid proton decay [ 12 ]. Our string model is generated by the following basis with eight elements: B = { 1, S, bl .... , bs, ca} where #3 The lorentzian dot-product counts each real fermion with a factor 1/2.
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PHYSICS LETTERSB
S---- ( 1, ...., 1, 0 .... , 0 ) ,
5 May 1988 (Sa)
~gp,ZI .....6
b-(
1.... ,1,
0,...,0;1 .... , 1 , 0 ..... 0 ) ,
~gp Zt.2,y3-~3 .. ,y6 x)6
b2=(
1,..., 1,
0, ..., 0; 1, ..., l ) ,
~g/~,~3,4 ,y 1~2y 1.2 ¢~5.6,6)5,6
b3=(
1..... 1,
0 .... ,0; 1..... 1 , 0 , . . . , 0 ) ,
1,..., 1, 1; .... 1,
0,...,0;-1 ..... 1,0 .... , 0 ) ,
(8e)
~/I ,....5 "ql
0 ..... 0;
~gl~X3,4 CO1-5 ~1.5 y2.6 ~2 6
a= (
(8d)
~ I ... ,5 ,f]3
~gtl,ZI 2,y3 6~3,6 C04,5,~4 5
bs--(
(8c)
~ 1,....5 ,~2
~ ,X5,6 ,m 1,6~1 ,...,~4,6)4 b4~-(
(8b)
~1 ... 5 i]1
l, ..., l,
ZI.....4yl ~ly3~,3y4~,4c02~2
1,...,1,
0,...,0),
(8f)
~1....,5,~2,~1,...,4
0 ..... 0; ½, ..., ½, A ) ,
(8g)
"-------" ~1....,5 ~,,2,3
In the notation used above, all the left-movers ~ga,Zi, yi, o~i,as well as twelve right-movers ~g, oi are real fermions, while the remaining right-movers ~1.....5, faL.2,3and ¢~1,...,8are complex, and the semicolons separate real and complex fermion~:-A few preliminary comments are in order. (a) The presence of the vectors S, bl and b2 guarantees N = 1 space-time supersymmetry [ 8 ]. (b) The restricted set of basis elements { 1, S, bi, b2, b3} alone would yield an S O ( 1 0 ) x S O ( 6 ) 3 observable gauge group, together with 3 × 2 copies of massless chiral fields in (16, 4, 1, 1 ) -i- (16, 7t, 1, 1 ) representations [one for each SO(6) ] and an E8 hidden gauge group. (c) The basis elements b4, b5 and ot break SO(6) 3 to U(1) 3 and SO(10) to S U ( 5 ) × U ( 1 ) . The reduction in the rank is achieved by using the real fermions f , ~, coi and 6~~, and is the maximum possible in this approach, whilst the unitary group appears thanks to the complex fermions ~ ' " s in a. The choice of ½for their boundary conditions is the only one whose a-projection leaves in the masgless physical spectrum the full spinor of SO (10), as will become clear below. (d) By choosing appropriately the vector A in eq. (8g) describing twists of ~ ]'-,8 which are consistent with condition (a2) above, it is possible to guarantee that the only massless states transforming nontrivially under the observable gauge group and those from the sectors S, b~, b3, b3, b4 (space-time fermions) and 0, bl +S, b2+S, b3+S, b4+S (space-time bosons). The particular form of A determines at the same time the final form of the hidden gauge group. The conditions (a2) impose the following constraints on the vector A. Defining N~ to be the smallest integer for which N~ot= 0 (mod 2 ), simple inspection of the vector a, eq. (8g), shows that N , = 4 n, where n is an integer depending on A. One must impose (i) n (A 2) = 0 (mod 2 ), corresponding to N~a2=0 ( m o d 8 ) and (ii) nA.B=O (mod 1), corresponding to N,~b~a'bs=O ( m o d 4 ) , where B =- ( 1, 1, 1, 1; 0, 0, 0, 0) for ~,".4 and $5.....8,respectively. One must in addition choose A so as to guarantee that no extra massless states transforming nontrivially under the observable gauge group are introduced. A possible choice of the phases c is c bj = c S = c b~ =c 1 = - 1
( i ~ j ; i , j = l .... , 5 )
(9a)
The first relation ensures the same chirality for all the SO (10) spinors, while the others are a matter of consistent choice. Using property (b2) above, one finds c(~)=+l 462
fori~3,5,
c(~)=-i
for/=3,
(9b)
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and determines other phases not needed here. It is straightforward to derive the massless spectrum of the model (8). In the Neveu-Schwarz sector (0) one has the graviton, dilaton and two-index antisymmetric tensor states: ~¢q/20X~ 10)o, the gauge bosons and some matter fields transforming only under the hidden gauge group [ whose specific form depends on the choice of A in eq. (8g) ], which we will not write explicitly here. One also has: (a) The gauge bosons of the observable gauge group SU (5) X U ( 1 ) X U ( 1 )3:~¢q/2~ / 2 ~ 2 1 0 )o with a, b = 1, .... 5 and ~/2f]~{/2fl~T210)o with ~4 or= l, 2, 3 which we denote by U( 1 )a. (b) Three candidate Higgses h~, ft.: Z1/2~1]1/2111/2 ~ - ~* -~* 10)0 and X1/2~1/211,/2 i -~ -~ ]0)o with i = 1, 2 or 3, 4 with o~= 1 or 2, respectively, and Z~/2~Y2~I3/2 [ 0 )0 and ~ ~,~ ,~3. m / 2 w / 2 . . / 2 [0)o with i = 5, 6. Thus, their quantum numbers are ~5 hl=(5,--l)_l,O,O, h3=(5'-1)°'°'1'
hl=(5,1)Lo,o, h3=(~'
h2=(5,--1)o_Lo,
h2=(5,1)o,l.o,
1)°,°,-! '
(I0)
in a self-explanatory notation. (c) Six SU(5) X U ( ! ) singletsz~/2fl~(/2fl~/2 IO >o,X i~/zrl~/2rl~/z - ~. -~. lO )o ( a #fl) with i= l, 2 or 3, 4 when a , f l = 2 , 3 or 1, 3, respectively, and Z~/z~I/2fl~210)o, m/2 "~ ,~/2 "~* ,~2 .~/z I0 ) o with i = 5, 6. Thus their quantum numbers are (~23 = ( l, 0)0,1,1 ,
~23 -'~ ( 1, 0)0,_ 1,_ 1,
~12 = ( 1 ' 0 ) 1 ' - 1 ' 0 '
~)13= (1, 0)1,0,1,
~13 = (1, 0 ) , 1 , 0 . _ 1 ,
~12 = ( 1 ' 0 ) - 1 ' 1 ' 0 "
(11)
(d) The sectors b. with a = 1, 2, 3 produce out of the corresponding vacua 10),~ three families of quarks and leptons which transform as spinors of O (16). After making the various fermion projections and decomposing with respect to the final gauge group, they are seen to transform as
m~-~M~/2,o.o, M2=Mod/2,0, M3=Mo,o_~/2,
(12)
where each M~---G+I'~ +1~ = (10, ½) + (~, - 3 ) + (1, J ) is a complete Weyl spinor of SO(10). (e) The sector b4 gives rise to the Higgses H~ and IZI.; H~ =(10, 1/2)i/2,o,o,
IZI, = (1--0, -1/2)_1/2.o, o .
(13)
Let us also mention the following massive states in the b~ sector. They are produced by acting on the various spinor components of [ 0 ) b 4 with the oscillators Z]/2, fl~/2 or ~]~Y2 with i=3, 4 or 5, 6 when oz=2 or 3, respectively: H,2=(10,½)_1/2,,.0,
IZI,z=(l-0,-l),/2_t,o,
H13=(10,½)_,/2,0,_,,
IZI~3=(1--lj,-½),/2,0.,.
(14)
These will play an important role in the doublet-triplet mass-splitting. Using the above spectrum and quantum numbers, we now describe some of the properties of the model and compare them with the desired properties of the corresponding field theory model [ 12 ], eq. (7). The superpotential for the model can be derived using the arguments of ref. [ 16 ], and contains the following SU (5) X U ( 1 ) 4 invariant terms: F,F,h,,
F2F2h2,
HiHlhl,
HtHl2h2,
hlfi2~12,
hlfi3~)13 ,
FsF3h3;
~h,,
HiHl3h3; h2fi3(~23;
~21~.~h2, ~'s.~h3;
IZtltZllhl,
filh2¢~12,
I-~IllZIl2fi2, I~lI-~Ii3h3;
filh3(~13,
fi2h3~23;
~)12(~23~13, (~12(~23(~13"
(15)
We note for future reference that no extra terms appear if we relax the requirement of invariance under the ~4 The first U ( 1 ) factor is the combination ¥5'/2 Y~/2t~y210)o. ~5 Note that the four U( 1 ) charges are related to the following world-sheet U( 1 ) currents: Y~= ~t~*alt~ and 1]"°'~" with a = 1, 2, 3.
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combination U ( 1 )1 + U ( 1 ) 2 - U ( 1 )3 of extra hypercharges. We now make the following comments on the basis ofeq. (15). (a) The HI, tZll scalars, eq. ( 13 ), are candidates for the Higgses which break SU (5). Their VEV's would also break the symmetric combination of U ( 1 ) 1 and the generation-independent U ( 1 ), leaving a low-energy gauge group of the form SU (3) ¢× SU (2) L× U ( 1 ) r × U ( 1 ) 3. The Hill,h1 and H IH 1hl superpotential couplings in eq. (15) then realize the elegant doublet-triplet mass-splitting mechanism of ref. [ 12 ] by giving a large mass to the triplet components of hi, ill. The massive states HI2, I7112, H13, I2113, eq, (14), have HlHI2h2+HIHI2h2 and H t H l a h 3 + H i H13h3 superpotential couplings which would likewise provide through mixing large masses of order M~UT 2 I M p ,.., > 1013 GeV for the triplet components of h 2 3 and h2,3 #6, while leaving the doublet components of h2,3 and fi2.3 light. (b) The h~, fi~ scalars, eq. (10), are candidates to be the Higgses of the standard model. In the presence of soft supersymmetry breaking they could acquire nonzero VEV's which would break SU ( 3 ) c × SU (2) L× U ( 1 ) r to SU (3) c× U ( 1 )em at the weak scale. The F~Fihi and i';1~hi couplings in eq. ( 15 ) could then provide masses for the charge-] quarks and leptons. However, a big defect of the model is that the couplings which could give masses to the charge+ ] quarks and conjugate neutrinos are forbidden by the extra U ( 1 ) charges. (c) The 0120231~13 and the ~12~23013 terms in the superpotential leave fiat directions in the effective scalar potential in the directions 10J21= 17~121~ 0 or 10J31 = 10131# 0 or 10231= 1523 ] # 0 with the other field VEV's vanishing in each case. If any one of these pairs develops a VEV, it reduces the gauge symmetry to S U ( 3 ) c × S U ( 2 ) L X U ( 1 ) r × U ( 1 )2. Such a pair of VEV's also give large Dirac masses to two hifii doublets. However, one (hk, fik ) pair would still have light doublet components available to break SU (2)L and give masses to charge - ~ quarks and leptons (e.g., to those in FI, FI and ~ if 1023[ = [~)23 I 5~0). (d) One combination of the generation-dependent U ( 1 )'s is anomalous [ 18 ], namely U ( 1 ) ~+ U ( 1 ) 2 - U ( 1 ) 3. It has been shown [ 19 ] that consistency of the string theory implies that string loop corrections generate Dbreaking of such an anomalous U ( 1 ) . This mechanism would reduce the gauge symmetry to SU (3) ~× SU (2) L× U ( 1 ) r × U ( 1 ). However, as noted above, no extra superpotential terms are allowed by the breaking of the U ( 1 ) l + U ( 1 ) 2 - U ( 1 ) 3 gauge symmetry. (e) The remaining surplus U( 1 ) may be broken at low energies by some small V E V o f a 0ij field along a nonfiat direction. (f) While it is unsatisfactory that the model described here contains no masses for charge + ] quarks, it is not necessarily a disaster that there are no masses for the conjugate neutrinos. This is because there are also no Dirac neutrino masses, so the conventional and conjugate neutrinos are all massless. The conjugate neutrinos are sufficiently weakly coupled that they neither disturb primordial nucleosynthesis nor are produced easily at accelerators. (g) One can modify slightly the above model, eqs. ( 8 ), to become more realistic, at the price of the appearance of extra Higgses. For example, if one modified the vector bs, eq. (8e), by making ~1,...4 antiperiodic, one would obtain an extra (10, ½) + (1-0, - ½) pair as well as an extra 5 and 5. Their renormalizable superpotential couplings yield 2 charge + ~ quark mass, as well as some Cabibbo-Kobayashi-Maskawa mixing. If all 10, ~Higgses and 023 acquire large VEV's all extra U ( 1 ) ' s are broken so that the low-energy gauge group is just SU (3)c × SU (2)L× U ( 1 )r. We have reported in this paper on a first attempt at model-building with the fermionic formulation of fourdimensional strings [ 8,9 ]. We arrived at a supersymmetric flipped SU (5) GUT [ 12 ] which is distinguished by its natural doublet-triplet mass-splitting mechanism, and the absence of any adjoint (or higher) Higgs representations. We have found a fermionic string theory which almost reproduces the desired model, modulo questions about the charge + ] quark masses and the conjugate neutrino masses. One possible line of future research would be to tune up the model presented here. Another would be to look for other phenomenologically appealing ~6As in ref. [ 17], these compensatefor the effectsof light Higgs doublets in the renormalizationgroup equations, so that acceptable values of sin20w,etc., are obtained. 464
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m o d e l s in t h e f e r m i o n i c f o r m u l a t i o n o f f o u r - d i m e n s i o n a l strings. H o w e v e r , e v e n i f a v e r y a t t r a c t i v e string t h e o r y w e r e f o u n d , t h e r e w o u l d still be t h e p r o b l e m o f d e f i n i n g h o w a n d w h y N a t u r e c h o s e that p a r t i c u l a r string theory. T h e w o r k o f D . V . N . was s u p p o r t e d in p a r t by D e p a r t m e n t o f Energy g r a n t D E - A C 0 2 - 7 6 E R 0 0 8 1 a n d in p a r t by the U n i v e r s i t y o f W i s c o n s i n R e s e a r c h C o m m i t t e e w i t h f u n d s g r a n t e d by the W i s c o n s i n A l u m n i R e s e a r c h Foundation.
References [ 1] M.B. Green and J.H. Schwarz, Phys. Lett. B 149 (1984) 117; D.J. Gross, J.A. Harvey, E. Martinec and R. Rohm, Nucl. Phys. B 256 (1985) 468; B 267 (1986) 75. [2] P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B 258 (1985) 46. [3] E. Witten, Nucl. Phys. B 268 (1986) 79. [4] L. Dixon, J.A. Harvey, C. Vafa and E. Witten, Nucl. Phys. B 261 (1985) 678; B 274 (1986) 285. [5] K. Narain, Phys. Lett. B 169 (1986) 41; W. Lerche, D. Liist and A.N. Schellekens, Nucl. Phys. B 287 (.t987) 477. [6] I. Antoniadis, C. Bachas, C. Kounnas and P. Windey, Phys. Eett. B 171 (1986) 51. [7] H. Kawai, D. Lewellen and S.-H.H. Tye, Phys. Rev. Lett. 57 (1986) 1832; Nucl. Phys. B 288 (1987) 1. [8] I. Antoniadis, C. Bachas and C. Kounnas, Nucl. Phys. B 289 (1987) 87. [9] I. Antoniadis and C. Bachas, Nucl. Phys. B 298 (1988) 586. [ 10] J. Ellis, CERN preprint TH.4840/87 (1987). [ 11 ] 1. Ib~ifiez,H.-P. Nilles and F. Quevedo, Phys. Lett. B 187 (1987) 25; B 192 (1987) 332; L. Ibfi~ez, J.E. Kim, H.-P. Nilles and F. Quevedo, Phys. Lett. B 191 (1987) 282. [ 12 ] I. Antoniadis, J. Ellis, J.S. Hagelin and D.V. Nanopoulos, Phys. Lett. B 194 (1987) 231. [ 13] B.A. Campbell, J. Ellis, J.S. Hagelin, D.V. Nanopoulos and R. Ticciati, Phys. Lett. B 198 (1987) 200. [ 14 ] B.A. Campbell, J. Ellis, and J.S. Hagelin, D.V. Nanopoulos and K.A. Olive, Phys. Lett. B 197 (1987 ) 355. [ 15] I. Antoniadis, C. Bachas and C. Kounnas, CERN preprint TH.4863/87 (1987). [ 16] I. Antoniadis, J. Ellis, E. Floratos, D.V. Nanopoulos and T. Tomaras, Phys. Lett. B 191 (1987) 96; S. Ferrara, L. Girardello, C. Kounnas and M. Porrati, Phys. Lett. B 192 (1987) 368; B 194 (1987) 358. [ 17] C. Kounnas, A.B. Lahanas, D.V. Nanopoulos and M. Quir6s, Nucl. Phys. B 236 (1984) 438. [ 18] R. Barbieri, S. Ferrara and D.V. Nanopoulos, Z. Phys. C 13 (1982) 267; Phys. Lett. B 116 (1982) 16. [ 19 ] M. Dine, N. Seiberg and E. Witten, Nucl. Phys. B 289 ( 1987 ) 589.
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G U T M O D E L - B U I L D I N G W I T H F E R M I O N I C F O U R - D I M E N S I O N A L STRINGS -AI. A N T O N I A D I S " b c a
a,~, John ELLIS
a,b,
J.S. H A G E L I N c and D.V. N A N O P O U L O S
d
CERN, CH-1211 Geneva 23, Switzerland Stanford Linear Accelerator Center, Stanford University, Stanford. CA 94305, USA Maharishi International University, Fairfield, IA 52556, USA University o f Wisconsin, Madison, W153706, USA
Received 7 January 1988
We report a first attempt at model-building using the fermionic formulation of string theories directly in four dimensions. An example is presented of a supersymmetricflipped SU (5) × U ( I ) model with three generations and an adjustable hidden sector gauge group. The simplest version of the model contains most of the Yukawa couplings required by phenomenology,but not all those needed to give masses to quarks or conjugate neutrinos. These defects may be remedied in a more general version of the model.
It is now well appreciated that the two key ingredients in formulating a consistent string theory are conformal invariance and modular invariance. The former condition fixes the number of degrees of freedom on the worldsheet. It can be satisfied in any number of dimensions d,<<26 (10 for a supersymmetric left- or right-moving sector) if the space-time coordinates X~:/z = 0,1, ..., d - 1 are supplemented by internal degrees of freedom contributing 2 6 - d ( 1 5 - 3 d / 2 ) to the central charge of the Virasoro algebra. Modular invariance then imposes nontrivial constraints on the boundary conditions for these internal degrees of freedom which ensure that counting errors are not made when higher-genus string topologies are summed. It is known that the number of solutions to these modular invariance conditions is restricted, particularly if the number of space-time dimensions d is close to the critical number 26 (or I 0). In particular, in d = 10 there are only two modular invariant heterotic string theories with N = 1 space-time supersymmetry, based on the gauge groups SO (32) and E8 × E~ [ 1 ]. The choice of gauge group for string theories formulated directly in d = 4 dimensions is much more extensive, and is imperfectly understood as yet. An interesting subclass of models is offered by supersymmetric compactifications of the Es × E~ heterotic string. Early attention was focused on Calabi-Yau manifold compactifications [2 ], but has subsequently been extended to other manifolds [ 3 ] and to orbifold compactifications [4 ]. The most systematic approach to this ambiguity in the four-dimensional gauge group would be to use one of the general formulations of string theories directly in d = 4, using either bosonic [ 5 ] or fermionic [ 6-9 ] variables to describe the internal degrees of freedom. The range of choice in four-dimensional theories is embarrassingly generous, and no systematic enumeration of models has yet emerged. The alternative strategy, followed here, is to start from the bottom up, looking for models which contain phenomenologically favored ingredients such as the standard model or a plausible grand unified theory ( G U T ) . Already examples are known of Calabi-Yau ~ and orbifold [ 11 ] compactifications which yield SU (3) c × SU (2) LX U ( 1 )": n/> 2 gauge groups. In this paper we look for models containing the flipped supersymmetric SU (5) × U ( 1 ) G U T whose virtues were recently extolled [ 12,13 ]: natural doublet-triplet mass splitting, a see-saw mechanism for neutrino masses, no cosmologically embarrassing phase tran~r Worksupported by the Department of Energy, contract DE-AC03-76SF00515. On leave of absence from Centre de Physique Throrique, l~colePolytechnique, F-91128 Palaiseau, France. ¢~See ref. [ 10] for a reviewand references. 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
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sitions [14], etc. Of particular importance is the fact that this model does not need any adjoint Higgs representations. No string model with adjoint chiral superfields can be obtained in the fermionic formulation [9 ], and this may well be a general property. Other GUTs require adjoint Higgses to break the initial GUT symmetry and/or realize doublet -triplet mass splitting naturally. Therefore, SU (5) × U ( 1 ) is uniquely favored as a GUT group, as has already been stressed in the more limited context of manifold compactification [ 13 ]. In this paper, our tool for obtaining this uniquely simple GUT is the fermionic formulation of four-dimensional strings, as developed in refs. [ 8,9 ]. We exhibit a choice of boundary conditions for the world-sheet fermions which yields immediately an observable [ SU (5) × U ( 1 ) ] × U ( 1 ) 3 gauge group as well as an d la carte hidden sector gauge group. The model contains three generations of quarks and leptons, additional 10, 10, 5 and representations of SU (5) which could be Higgs fields, and a number of SU (5) X U ( 1 )-invariant fields as advocated in ref. [ 12 ]. Vacuum expectation values of scalar fields break one of the surplus U ( 1 ) 3 gauge generators via the conventional Higgs mechanism, and another is anomalous. The fermionic formulation also tells us what Yukawa couplings are present, and we find all the ones we want except some of those giving masses to quarks and to conjugate neutrinos. These defects may be avoided by a small modification of the simplest consistent choice of fermionic boundary conditions which introduces some additional l0 + 1--0and $ + $ representations. While developing our model, we strive to illustrate some general principles of fermionic model-building which may aid subsequent, more inspired, efforts. We start with a brief review of the main characteristics of four-dimensional heterotic string theories in the fermionic formulation [ 8,9 ]. In the light-cone gauge, in addition to the two transverse bosonic coordinates X ~ and their left-moving superpartners q/U(z), the fermionic content is [ 6 ] 44 right-moving and 18 left-moving fermions q)~ (Z): A = 1, 2 .... ,44 a n d z i ( z ) , y~(z), w~(z): i= 1, 2, ..., 6, respectively. World-sheet supersymmetry is nonlinearly realized among the latter via the supercurrent 6
T~(z)=~,"OzX,,+ E z'y'w'.
(1)
i=l
A four-dimensional string model is defined by specifying a set E of boundary conditions for all the world-sheet fermions, constrained by making the world-sheet supercurrent ( 1 ) periodic (space-time fermions) or antiperiodic (space-time bosons). When all the boundary conditions are diagonalized simultaneously in some general complex basis {f}, the elements of E are vectors a such that every complex fermion f picks up a phase f ~ - e x p [ i n a ( f ) ]f: a ( f ) e ( - 1 ,
1],
(2)
when parallel transported around the string. In this case, -= forms a group under addition (rood 2), and can therefore be generated by some basis B---{b t, b2..... b N}. It has been shown [ 8,9 ] that to every element ot of-= there corresponds a sector ~ in the string Hilbert space Yd, and to every basis element b~of B a fermion number projection:
~=~o,~z,= ~, [exp(inb,.F)=~o,c*(~)]~,
(3)
where F is the vector of all fermion numbers defined: F ( f ) = 1 = - F ( f * ) , the dot product is lorentzian (left minus right), ~, is the space-time fermion parity and the phases c( ~ ) are constrained by multiloop modular invariance. In order for this paper to be self-contained, we now give the explicit form of the constraints on the basis B and on the phases c for generic rational boundary conditions [ 9 ] ,2 ~2 We consider here neither some exceptional cases, nor the nontrivial realizations of world-sheet supersymmetry discussed in ref. [9].
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Basis B. ( a l ) We choose B to be canonical, i.e., any linear combination Y~irnibi=0 iff mi= 0 (mod N~) for some integers N~ (for example, Ni=2 when the fermions are periodic or antiperiodic), and the vector I~B. (a2) For any pair bi, bj of basis elements, one has #3 Nob:bj = 0 (mod 4) where N o is the least common multiple of N~ and Nj, and N~b2= 0 (mod 8) ifN~ is even. (a3) The number of real fermions which are simultaneously periodic under four boundary conditions bl, b2, b3, b4 is even.
Phases c. ( b l ) We choose the c(bb~) for i
c~)=exp(i½rta.,)c*(~),
0
ca
ca
c(fl+y)=c~,~c(#)c(y).
(4)
Physical states from the sector ~ are obtained by acting on the vacuum I0 ) , with bosonic or fermionic oscillators with frequencies ½[ 1 + ca( f ) ] ( ½[ 1 - ca( f ) ] for f* ) and applying the fermion number projections, eq. (3). The mass formula is M2=-½+~ca~ + EVL=--I+~Ca~+EVR, L
(5)
R
where caL (CaR) is the left (right) part of the vector ca and the VL (UR) are frequencies. When some fermions are periodic, the vacuum is a spinor in order to represent the Clifford algebra of the corresponding zero modes. For each periodic complex fermion f there are two degenerate vacua I + >, I - >, annihilated by the zero modes fo and ~, and with fermion numbers F ( f ) = 0, - 1, respectively. Before starting to build a model, we note a simple but very crucial relation between the world-sheet fermion numbers F ( f ) and the U( 1 ) charges Q ( f ) with respect to the unbroken Cartan generators of the four-dimensional gauge group, which are in one-to-one correspondence with the U ( 1 ) currents f*f for each complex fermion f: Q(f) = ½ca(f) + F ( f ) .
(6)
The charges Q ( f ) can be shown to be identical with the momenta of the corresponding compactified scalars in the bosonic formulation [ 15 ]. The representation (6) shows that Q is identical with the world-sheet fermion numbers F for states in a Neveu-Schwarz sector (ca = 0), and is ( F + ½) for states in a Ramond sector (ca = 1 ); note that the charges of the [ + > spinor vacua are + ½. The model we seek in this four-dimensional fermionic string formalism is the supersymmetric flipped SU (5) GUT of ref. [ 12 ]. It contains the following chiral matter superfields: Three generations of matter fields F~= (10, ½), ~, = (~, _ 3 ), and ~ = (1, ~ ); Higgses H = (10, ½), I:I= (i-0, - ½), h = (5, - 1 ), and fi= (5, 1 ); and four singlets ~o.~= (1, 0) o f S U ( 5 ) × U ( 1 ). The superpotential of the model is [ 12] W = 2 1 F F h +22Ft~11 + 2 s ~ C h + 2 4 H H h +,;LsI:II:Ifi +26FI=I¢~ +27hfi¢~ + 2 s t~3 ,
(7)
where the Yukawa couplings 21.2,3.6.7.8 are matrices in generation space. One component each of the H and ft acquire large VEV's breaking S U ( 5 ) X U ( 1 ) ~ S U ( 3 ) c X S U ( 2 ) L X U ( 1 )r, and S U ( 2 ) L × U ( 1 ) r ~ U ( 1 )~m via VEV's of components of the h and ft. Uniquely among all the GUTS known to us, there is no adjoint or larger self-conjugate Higgs representation. The model is also attractive because it solves naturally the Higgs doublet -triplet mass-splitting problem, has a see-saw neutrino mass spectrum, and avoids rapid proton decay [ 12 ]. Our string model is generated by the following basis with eight elements: B = { 1, S, bl .... , bs, ca} where #3 The lorentzian dot-product counts each real fermion with a factor 1/2.
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S---- ( 1, ...., 1, 0 .... , 0 ) ,
5 May 1988 (Sa)
~gp,ZI .....6
b-(
1.... ,1,
0,...,0;1 .... , 1 , 0 ..... 0 ) ,
~gp Zt.2,y3-~3 .. ,y6 x)6
b2=(
1,..., 1,
0, ..., 0; 1, ..., l ) ,
~g/~,~3,4 ,y 1~2y 1.2 ¢~5.6,6)5,6
b3=(
1..... 1,
0 .... ,0; 1..... 1 , 0 , . . . , 0 ) ,
1,..., 1, 1; .... 1,
0,...,0;-1 ..... 1,0 .... , 0 ) ,
(8e)
~/I ,....5 "ql
0 ..... 0;
~gl~X3,4 CO1-5 ~1.5 y2.6 ~2 6
a= (
(8d)
~ I ... ,5 ,f]3
~gtl,ZI 2,y3 6~3,6 C04,5,~4 5
bs--(
(8c)
~ 1,....5 ,~2
~ ,X5,6 ,m 1,6~1 ,...,~4,6)4 b4~-(
(8b)
~1 ... 5 i]1
l, ..., l,
ZI.....4yl ~ly3~,3y4~,4c02~2
1,...,1,
0,...,0),
(8f)
~1....,5,~2,~1,...,4
0 ..... 0; ½, ..., ½, A ) ,
(8g)
"-------" ~1....,5 ~,,2,3
In the notation used above, all the left-movers ~ga,Zi, yi, o~i,as well as twelve right-movers ~g, oi are real fermions, while the remaining right-movers ~1.....5, faL.2,3and ¢~1,...,8are complex, and the semicolons separate real and complex fermion~:-A few preliminary comments are in order. (a) The presence of the vectors S, bl and b2 guarantees N = 1 space-time supersymmetry [ 8 ]. (b) The restricted set of basis elements { 1, S, bi, b2, b3} alone would yield an S O ( 1 0 ) x S O ( 6 ) 3 observable gauge group, together with 3 × 2 copies of massless chiral fields in (16, 4, 1, 1 ) -i- (16, 7t, 1, 1 ) representations [one for each SO(6) ] and an E8 hidden gauge group. (c) The basis elements b4, b5 and ot break SO(6) 3 to U(1) 3 and SO(10) to S U ( 5 ) × U ( 1 ) . The reduction in the rank is achieved by using the real fermions f , ~, coi and 6~~, and is the maximum possible in this approach, whilst the unitary group appears thanks to the complex fermions ~ ' " s in a. The choice of ½for their boundary conditions is the only one whose a-projection leaves in the masgless physical spectrum the full spinor of SO (10), as will become clear below. (d) By choosing appropriately the vector A in eq. (8g) describing twists of ~ ]'-,8 which are consistent with condition (a2) above, it is possible to guarantee that the only massless states transforming nontrivially under the observable gauge group and those from the sectors S, b~, b3, b3, b4 (space-time fermions) and 0, bl +S, b2+S, b3+S, b4+S (space-time bosons). The particular form of A determines at the same time the final form of the hidden gauge group. The conditions (a2) impose the following constraints on the vector A. Defining N~ to be the smallest integer for which N~ot= 0 (mod 2 ), simple inspection of the vector a, eq. (8g), shows that N , = 4 n, where n is an integer depending on A. One must impose (i) n (A 2) = 0 (mod 2 ), corresponding to N~a2=0 ( m o d 8 ) and (ii) nA.B=O (mod 1), corresponding to N,~b~a'bs=O ( m o d 4 ) , where B =- ( 1, 1, 1, 1; 0, 0, 0, 0) for ~,".4 and $5.....8,respectively. One must in addition choose A so as to guarantee that no extra massless states transforming nontrivially under the observable gauge group are introduced. A possible choice of the phases c is c bj = c S = c b~ =c 1 = - 1
( i ~ j ; i , j = l .... , 5 )
(9a)
The first relation ensures the same chirality for all the SO (10) spinors, while the others are a matter of consistent choice. Using property (b2) above, one finds c(~)=+l 462
fori~3,5,
c(~)=-i
for/=3,
(9b)
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and determines other phases not needed here. It is straightforward to derive the massless spectrum of the model (8). In the Neveu-Schwarz sector (0) one has the graviton, dilaton and two-index antisymmetric tensor states: ~¢q/20X~ 10)o, the gauge bosons and some matter fields transforming only under the hidden gauge group [ whose specific form depends on the choice of A in eq. (8g) ], which we will not write explicitly here. One also has: (a) The gauge bosons of the observable gauge group SU (5) X U ( 1 ) X U ( 1 )3:~¢q/2~ / 2 ~ 2 1 0 )o with a, b = 1, .... 5 and ~/2f]~{/2fl~T210)o with ~4 or= l, 2, 3 which we denote by U( 1 )a. (b) Three candidate Higgses h~, ft.: Z1/2~1]1/2111/2 ~ - ~* -~* 10)0 and X1/2~1/211,/2 i -~ -~ ]0)o with i = 1, 2 or 3, 4 with o~= 1 or 2, respectively, and Z~/2~Y2~I3/2 [ 0 )0 and ~ ~,~ ,~3. m / 2 w / 2 . . / 2 [0)o with i = 5, 6. Thus, their quantum numbers are ~5 hl=(5,--l)_l,O,O, h3=(5'-1)°'°'1'
hl=(5,1)Lo,o, h3=(~'
h2=(5,--1)o_Lo,
h2=(5,1)o,l.o,
1)°,°,-! '
(I0)
in a self-explanatory notation. (c) Six SU(5) X U ( ! ) singletsz~/2fl~(/2fl~/2 IO >o,X i~/zrl~/2rl~/z - ~. -~. lO )o ( a #fl) with i= l, 2 or 3, 4 when a , f l = 2 , 3 or 1, 3, respectively, and Z~/z~I/2fl~210)o, m/2 "~ ,~/2 "~* ,~2 .~/z I0 ) o with i = 5, 6. Thus their quantum numbers are (~23 = ( l, 0)0,1,1 ,
~23 -'~ ( 1, 0)0,_ 1,_ 1,
~12 = ( 1 ' 0 ) 1 ' - 1 ' 0 '
~)13= (1, 0)1,0,1,
~13 = (1, 0 ) , 1 , 0 . _ 1 ,
~12 = ( 1 ' 0 ) - 1 ' 1 ' 0 "
(11)
(d) The sectors b. with a = 1, 2, 3 produce out of the corresponding vacua 10),~ three families of quarks and leptons which transform as spinors of O (16). After making the various fermion projections and decomposing with respect to the final gauge group, they are seen to transform as
m~-~M~/2,o.o, M2=Mod/2,0, M3=Mo,o_~/2,
(12)
where each M~---G+I'~ +1~ = (10, ½) + (~, - 3 ) + (1, J ) is a complete Weyl spinor of SO(10). (e) The sector b4 gives rise to the Higgses H~ and IZI.; H~ =(10, 1/2)i/2,o,o,
IZI, = (1--0, -1/2)_1/2.o, o .
(13)
Let us also mention the following massive states in the b~ sector. They are produced by acting on the various spinor components of [ 0 ) b 4 with the oscillators Z]/2, fl~/2 or ~]~Y2 with i=3, 4 or 5, 6 when oz=2 or 3, respectively: H,2=(10,½)_1/2,,.0,
IZI,z=(l-0,-l),/2_t,o,
H13=(10,½)_,/2,0,_,,
IZI~3=(1--lj,-½),/2,0.,.
(14)
These will play an important role in the doublet-triplet mass-splitting. Using the above spectrum and quantum numbers, we now describe some of the properties of the model and compare them with the desired properties of the corresponding field theory model [ 12 ], eq. (7). The superpotential for the model can be derived using the arguments of ref. [ 16 ], and contains the following SU (5) X U ( 1 ) 4 invariant terms: F,F,h,,
F2F2h2,
HiHlhl,
HtHl2h2,
hlfi2~12,
hlfi3~)13 ,
FsF3h3;
~h,,
HiHl3h3; h2fi3(~23;
~21~.~h2, ~'s.~h3;
IZtltZllhl,
filh2¢~12,
I-~IllZIl2fi2, I~lI-~Ii3h3;
filh3(~13,
fi2h3~23;
~)12(~23~13, (~12(~23(~13"
(15)
We note for future reference that no extra terms appear if we relax the requirement of invariance under the ~4 The first U ( 1 ) factor is the combination ¥5'/2 Y~/2t~y210)o. ~5 Note that the four U( 1 ) charges are related to the following world-sheet U( 1 ) currents: Y~= ~t~*alt~ and 1]"°'~" with a = 1, 2, 3.
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combination U ( 1 )1 + U ( 1 ) 2 - U ( 1 )3 of extra hypercharges. We now make the following comments on the basis ofeq. (15). (a) The HI, tZll scalars, eq. ( 13 ), are candidates for the Higgses which break SU (5). Their VEV's would also break the symmetric combination of U ( 1 ) 1 and the generation-independent U ( 1 ), leaving a low-energy gauge group of the form SU (3) ¢× SU (2) L× U ( 1 ) r × U ( 1 ) 3. The Hill,h1 and H IH 1hl superpotential couplings in eq. (15) then realize the elegant doublet-triplet mass-splitting mechanism of ref. [ 12 ] by giving a large mass to the triplet components of hi, ill. The massive states HI2, I7112, H13, I2113, eq, (14), have HlHI2h2+HIHI2h2 and H t H l a h 3 + H i H13h3 superpotential couplings which would likewise provide through mixing large masses of order M~UT 2 I M p ,.., > 1013 GeV for the triplet components of h 2 3 and h2,3 #6, while leaving the doublet components of h2,3 and fi2.3 light. (b) The h~, fi~ scalars, eq. (10), are candidates to be the Higgses of the standard model. In the presence of soft supersymmetry breaking they could acquire nonzero VEV's which would break SU ( 3 ) c × SU (2) L× U ( 1 ) r to SU (3) c× U ( 1 )em at the weak scale. The F~Fihi and i';1~hi couplings in eq. ( 15 ) could then provide masses for the charge-] quarks and leptons. However, a big defect of the model is that the couplings which could give masses to the charge+ ] quarks and conjugate neutrinos are forbidden by the extra U ( 1 ) charges. (c) The 0120231~13 and the ~12~23013 terms in the superpotential leave fiat directions in the effective scalar potential in the directions 10J21= 17~121~ 0 or 10J31 = 10131# 0 or 10231= 1523 ] # 0 with the other field VEV's vanishing in each case. If any one of these pairs develops a VEV, it reduces the gauge symmetry to S U ( 3 ) c × S U ( 2 ) L X U ( 1 ) r × U ( 1 )2. Such a pair of VEV's also give large Dirac masses to two hifii doublets. However, one (hk, fik ) pair would still have light doublet components available to break SU (2)L and give masses to charge - ~ quarks and leptons (e.g., to those in FI, FI and ~ if 1023[ = [~)23 I 5~0). (d) One combination of the generation-dependent U ( 1 )'s is anomalous [ 18 ], namely U ( 1 ) ~+ U ( 1 ) 2 - U ( 1 ) 3. It has been shown [ 19 ] that consistency of the string theory implies that string loop corrections generate Dbreaking of such an anomalous U ( 1 ) . This mechanism would reduce the gauge symmetry to SU (3) ~× SU (2) L× U ( 1 ) r × U ( 1 ). However, as noted above, no extra superpotential terms are allowed by the breaking of the U ( 1 ) l + U ( 1 ) 2 - U ( 1 ) 3 gauge symmetry. (e) The remaining surplus U( 1 ) may be broken at low energies by some small V E V o f a 0ij field along a nonfiat direction. (f) While it is unsatisfactory that the model described here contains no masses for charge + ] quarks, it is not necessarily a disaster that there are no masses for the conjugate neutrinos. This is because there are also no Dirac neutrino masses, so the conventional and conjugate neutrinos are all massless. The conjugate neutrinos are sufficiently weakly coupled that they neither disturb primordial nucleosynthesis nor are produced easily at accelerators. (g) One can modify slightly the above model, eqs. ( 8 ), to become more realistic, at the price of the appearance of extra Higgses. For example, if one modified the vector bs, eq. (8e), by making ~1,...4 antiperiodic, one would obtain an extra (10, ½) + (1-0, - ½) pair as well as an extra 5 and 5. Their renormalizable superpotential couplings yield 2 charge + ~ quark mass, as well as some Cabibbo-Kobayashi-Maskawa mixing. If all 10, ~Higgses and 023 acquire large VEV's all extra U ( 1 ) ' s are broken so that the low-energy gauge group is just SU (3)c × SU (2)L× U ( 1 )r. We have reported in this paper on a first attempt at model-building with the fermionic formulation of fourdimensional strings [ 8,9 ]. We arrived at a supersymmetric flipped SU (5) GUT [ 12 ] which is distinguished by its natural doublet-triplet mass-splitting mechanism, and the absence of any adjoint (or higher) Higgs representations. We have found a fermionic string theory which almost reproduces the desired model, modulo questions about the charge + ] quark masses and the conjugate neutrino masses. One possible line of future research would be to tune up the model presented here. Another would be to look for other phenomenologically appealing ~6As in ref. [ 17], these compensatefor the effectsof light Higgs doublets in the renormalizationgroup equations, so that acceptable values of sin20w,etc., are obtained. 464
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m o d e l s in t h e f e r m i o n i c f o r m u l a t i o n o f f o u r - d i m e n s i o n a l strings. H o w e v e r , e v e n i f a v e r y a t t r a c t i v e string t h e o r y w e r e f o u n d , t h e r e w o u l d still be t h e p r o b l e m o f d e f i n i n g h o w a n d w h y N a t u r e c h o s e that p a r t i c u l a r string theory. T h e w o r k o f D . V . N . was s u p p o r t e d in p a r t by D e p a r t m e n t o f Energy g r a n t D E - A C 0 2 - 7 6 E R 0 0 8 1 a n d in p a r t by the U n i v e r s i t y o f W i s c o n s i n R e s e a r c h C o m m i t t e e w i t h f u n d s g r a n t e d by the W i s c o n s i n A l u m n i R e s e a r c h Foundation.
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