Supercosmology revitalized

Volume 197, number 3

PHYSICS LETTERS B

SUPERCOSMOLOGY REVITALIZED B.A. C A M P B E L L i, j . ELLIS CERN, CH-1211 Geneva 23, Switzerland J.S. H A G E L I N Physics Department, Maharishi International University, Fairfield, IA 52556, USA D.V. N A N O P O U L O S Physics Department, Universityof Wisconsin, Madison, 141153 706, USA and K.A. O L I V E School of Physics and Astronomy, Universityof Minnesota, Minneapolis, &IN 55455, USA Received 4 August 1987

Strong coupling effects can avert excess entropy production in supersymmetric models with a large intermediate scale due to symmetry breaking along a flat direction in the effective potential. This is because strong-coupling effects may break supersymmetry giving a large vacuum energy which destabilizes the symmetric field configuration. Alternatively, the reduced number of degrees of freedom in the confined phase destabilize the symmetric field configuration at a high temperature T>> row. This principle is exhibited in a simple supersymmetric flipped SU (5) × U (l) GUT model inspired by the superstring. Compatible scenarios for baryosynthesis are briefly discussed.

G r a n d unified theories ( G U T s ) [ 1 ] p r o v i d e attractive solutions to m a n y p r o b l e m s o f the stand a r d model, b u t bring with t h e m some new ones. P r o m i n e n t among these is the hierarchy problem [ 2 ]. G U T s need at least one high scale ml o f gauge symmetry breaking i n t e r m e d i a t e between rnw and mp, a n d the hierarchies m , / m w and mp/ml >> 1 m u s t be protected from destruction by radiative corrections. A similar p r o b l e m arises in m o d e l s based on the superstring [3], in which the gauge s y m m e t r y at scales just below me tends to be much larger than the S U ( 3 ) c × S U ( 2 ) L × U ( 1 ) y o f the s t a n d a r d model. It is generally believed that the hierarchy p r o b l e m o f i n t e r m e d i a t e scales can only be solved in the context o f s u p e r s y m m e t r y [ 4 ]. R a d i a t i v e corrections which

might destabilize a gauge hierarchy are absent in a s u p e r s y m m e t r i c theory. Moreover, the effective potential in a s u p e r s y m m e t r i c theory has m a n y fiat directions in field space ~ , which offer the hope o f generating d y n a m i c a l l y very different mass scales. Unfortunately, finite-temperature effects usually keep the theory in the s y m m e t r i c phase until T ~ mw, so the d y n a m i c a l relaxation along one o f these fiat directions often takes a long time, a n d is accomp a n i e d by excessive entropy release when it finally occurs [ 5 ]. Excess entropy m a y also be generated by the late decays o f metastable sparticles such as gravitinos [ 6 ]. A n o t h e r difficulty is that the a d d i t i o n a l supersymmetric degrees o f freedom offer new ways for bar-

Permanent address: Physics Department, University of Alberta, Edmonton, Alberta, Canada T6G 2J 1.

~t These also carry the danger of many different phases in different parts of the Universe.

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yons to decay, through dimension-five operators [ 7 ], for instance. Inflation may solve the aforementioned gravitino problem, but only if the reheating temperature TR< 10 ~°-+2 GeV in the most straightforward scenario where m3/2 ~ m w [8 ]. In this case the particles (most probably Higgs triplets H3) whose decays were responsible for cosmological baryosynthesis would need to have masses ~<1012 GeV. These could then lead to excessively rapid baryon decay via dimension-five operators [ 7 ], whilst dimension-six operators would not be a problem if ran3 ~>10 J~GeV [ 9 ]. Nowadays, the conflict between baryosynthesis, entropy generation and baryon stability is less acute, because we have learnt that m3/2 may be >> mw in non-minimal models, such as the no-scale ones [ 10] ~2 rediscovered in the low-energy theories derived from the superstring. If m3/2 >> mw, there is no excessive entropy release when the gravitino or associated Polonyi, dilaton and axion fields decay [ 12 ], so the gravitino constraint TR ~<1012 GeV is removed and the Higgs triplets H 3 may be heavier, thereby alleviating the problem of baryon stability. But the problem remains of excess entropy generation during dynamical relaxation to an intermediate scale of gauge symmetry breaking along a fiat direction in the effective potential [ 5 ]. In this paper, we show how this difficulty is naturally avoided [ 13] in a G U T broken down to S U ( 3 ) c × S U ( 2 ) L x U ( 1 ) r by some VEV (qb) at a high intermediate scale ( q b ) = m . Such a G U T group must be large, and so its coupling in the unbroken phase becomes strong at a high energy scale A~ >> row. Phase transitions in the strong-coupling regime T, ( ~ ) ,% A¢. Therefore a hot universe with ( ~ ) = 0 at T > Ac will make an adiabatic transition to (~b) > A~ when T
29 October 1987

metric flipped SU(5) G U T [14] ~3 inspired by the superstring. We also sketch how baryosynthesis may be achieved [ 15 ] in this model by the decays of particles with mass < 1015 GeV without causing rapid baryon decay via dimension-five operators. First we sketch our general scenario and discuss entropy generation. Models with large intermediatescale symmetry breaking acquire it along a direction in field space which is D-flat and F-flat, with the only field-dependence given by a supersymmetrybreaking term ~ r h 2 ~ 2 where rh2=O(m~v) and is negative for small qb, and possibly non-renormalizable terms ~eP4+"/m~,. The VEV ( t ~ ) ~ m I is of order of the scale ~o at which rh 2 changes sign to become positive for large ~ , or O(rh2m~) ~/2÷" if there are non-renormalizable terms and this estimate is smaller than ~o. In a weakly-coupled theory, finite-temperature effects ~ ~ 2 T 2 overcome the rh2~ 2 term and keep the origin stable until T ~ I rh [ ~ m w [ 5 ]. However, in a grand unified theory, the asymptotically-free gauge coupling becomes strong [ 13 ] for temperatures T and values of qb less than some condensation scale Ac. At and below Ac the theory is expected to form bilinear condensates, breaking gauge and global symmetries dynamically. We expect that this symmetry breaking will proceed according to a straightforward extension of the most attractive channel (MAC) hypothesis [ 16], which we call stumbling [ 17 ] #4 via SMAC. In some cases, the resulting bilinear condensates break supersymmetry, giving rise to a large vacuum energy ~A~ in the strong-coupling phase of the theory when T,
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fewer light degrees of freedom than in the spontaneously broken Higgs phase where q~ > A~ [ 13 ]. In general, there is a negative contribution to the free energy at finite temperature which is proportional to the number of light degrees of freedom with rn << T AF=-~zt2Ne~T

a,

N¢rr=NB +~NF .

(l)

The smaller number of degrees of freedom in the strong-coupling phase will cause it to be disfavoured energetically in comparison with the Higgs phase at large qb [ 13 ]. Hence both scenarios can prevent the universe from being trapped in the symmetric phase until T < O(rh) ~ mw as in theories without strong coupling. In our case, the rh2q~2 term in the effective potential is insignificant for q~ ~A¢, and the theory makes a transition to the Higgs phase when T ~ A ~ . Very possibly, as we show in some toy examples below, there may be no barrier between the dynamically broken phase and the Higgs phase, in which case • just rolls or fluctuates into the new phase. However, we cannot exclude the possibility that the strong dynamics may provide a barrier of height A V~ A 4 whet, T ~ A~. Since thermal and strong-coupling fluctuations are both ~A~ at this epoch, and since the difference in heights on either side of the putative barrier is ~ T 4 ~ A 4, we expect an O(1) probability for completion of the transition to the Higgs phase when T ~ A~ without any excess entropy generation. Clearly an exact estimate of the transition probability depends on unknown details of the potential in the non-perturbative regime. Given the occurrence of such a strong-coupling phase transition, we can then estimate how much entropy will be generated as • subsequently evolves from a value ~A~ when T ~ T¢ to its low-temperature value ~ rn~. The avoidance of excessive entropy generation will constrain allowable values of A¢ and m~. The evolution of q) at lower T is governed by the equation

4; + 3Hob + rh2q~= V'(q0, w ( ,~) - a v / a , ~

.

(2)

As in the case of the axion [ 18 ], the oscillations which start at T~~ Tc with a stored energy density p~ _~ th 2m 2 evolve adiabatically with an energy density PO ~- f f l 2 m l ( R i / R ) 3 ,

(3)

29 October 1987

for R > Ri where Ri is the value of the cosmological scale factor when qb oscillations begin. These oscillations eventually decay when H ~ p ~ Z / m p ~ F ~ erh3/m~, at which time the energy density is dominated by p~. Decays occur at R =RD where

D'i/OD'~ - -

~Ht~" ! / "Jt~" "

t2/3 ~ 4 / 3 ~t t2~ p: 3 1 ~/ t i 2l l •. lt~

.

(4)

The entropy produced in the q~ decays is then given by SD ~--- [ p ( R D )

] 3/4 ~ •3'2 m9/2

m3/Z/m31 ,

(5)

to be compared with the entropy in radiation, which is given by si(RD) = T 3 ( R i / R ) 3 ,

(5')

where Ti is the temperature of the radiation when oscillations begin. Comparing eqs. (5) and (5'), we see that the entropy produced by the q~ oscillations is /I_~SDItSi( R D ) ~ ~ 1 / 2 ~ 31,,,

,mel/2 'T'3ai •

(6)

A conservative upper bound on A subsequent to baryosynthesis is A < 10 +6, though this depends somewhat on the specific mechanism for generating the baryon asymmetry [ 15 ]. This corresponds in eq. (6) to ( T i / m l ) > lO-2( fft/ernp)1/6 ~ 10-4.7E-1/6

(7)

In the absence of inflation, or if after inflation TR>A¢, the temperature will then decrease until T ~ A c , where strong-coupling effects destabilize the

symmetric field configuration as we have previously discussed. Then, once T drops to a value T~2Ac, where 2 is the superpotential coupling which gives a mass oc (q) ) to the lightest superheavy particle, the • field starts oscillating. Hence, one should associate T~ in eqs. ( 5 ) - (7) with L//c. The same holds even if AA¢ < TR < A¢ provided either (a) TR > 2q~R (where q~R is the q~ field expectation value immediately after inflation), in which case qb will find itself temporarily at a minimum between A~ and T/2, or (b) TR > ~ X / ~ , in which case qb will fluctuate out of its local minimum at q~ = rnx and into the global minim u m between Ac and T/2, where it will remain temporarily until T
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oscillating immediately after inflation and Ti should be identified with TR. In any case, TR ( >I T~) is always subject to the above constraint on Ti. An additional constraint on this type o f model is that the temperature of the universe after ~b decay must be greater than O(1 ) MeV so that nucleosynthesis occurs during a radiation-dominated epoch, rather than during the @ oscillations, The temperature o f the radiation at R=RD is just

T, ~-Sg 3 ~" El/2 t~13/2m l / 2 / m l

•

0 /

F'~'t¢'¢%

"6 ~

0

" ~ F,~',l¢,¢'m

/

r,T.~. _

g

~FT®~

Fig. 1. Diagram contributing to a D-term generating a decay rate o~23/m~ for the • field in the flipped SU(5)×U(1) GUT model.

(8)

W=).~F, Fjh + 2~F,~h + )~3fi~h + 24HHh + 25fiflfi

Thus we obtain the additional constraint

mt <,%C/2 th3/2m~,/2/O(lO) MeV "~ ~i/2.1017 G e V ,

(9) on any intermediate scale model. We illustrate the above general analysis with a specific supersymmetric flipped S U ( 5 ) × U ( 1 ) G U T [ 14] inspired by the superstring. It contains three generations o f chiral matter fields with the SU (5) X U(1 ) transformation properties F,--(10,1),

t'~(5,-3),

U)u ~_=(1,5),

(10)

and the particle assignments

{: _

F=[

e

0

de

de

-d c

0

--U

--U

d

d

u

,

--V c

Ue

}'= u e

,

etc.,

(11)

for the quarks and leptons o f each generation. The minimal version o f the model also contains two conjugate pairs o f Higgs supermultiplets H=- (10,1), h_ (5,-2),

IZI- ( 1 0 , - 1 ) , fi_-_-( 5 , 2 ) ,

(12)

as well as four singlets 0 , ~ - (1,0) ~5. The superpotential for the model is ~5 Inflation or baryosynthesis might motivate the inclusion of more fields. 358

The G U T symmetry is broken by a VEV along the unique F- and D-flat direction ( H ) = ( I : I ) ~ 0 #6. These VEV can be rotated into the v ~ ( g h ) directions, indicating that the G U T breaking is

su(5)xU(1) , SU(3)e × SU(2)L xU(I

)r.

ml = ( H ) = ( l q ) =mGUT

The 24 and 25 superpotential couplings and this VEV pair the d~(a~) with the Dh(IS)n) in massive Dirac states o f mass ~,4(25)mGUT. Likewise, the 26 superpotential term and this VEV pair the v7 and three o f the ~m to form Dirac states o f mass 26mGtrT- All the other states have small masses ~ ( h , f i ) ~ ffz~ 1 TeV. Thus the Higgs phase o f this theory has 45 light quarks and leptons, four light Higgs fields, one singlet from among the 0m, one field • which is a linear combination o f the vh and 9~, and twelve gauge multiplets, for a total o f 63 light superrnultiplet degrees o f freedom. Note that the decay o f the field in our model occurs via the diagrams o f fig. 1, which generate effective D-terms [ v ~ F t F ] o , etc. The decay rate is given by F ,,, ( 92~,2,3,7/ 2048nS)(ma, m~.r,LO~,,,,/m2),which implies that some o f 2t,2,3.7 and/or m~,~.Lc~ must be quite large in order ~6 We expect this scale to be of order the supersymmetry-breaking, compactification or string excitation scale, whichever is the smaller. It is not, strictly speaking, an intermediate scale of the type discussed in ref. [ 19]. The arguments in the third paper of ref. [ 19] suggest that it is only possible to generate such a large VEV, if supersymmetry-breaking scalar masses are present initially as well as gaugino mass terms.

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to be consistent with the nucleosynthesis constraint (9). We now analyse the strong-coupling spectrum [20] using the SMAC hypothesis that the condensation of two supermultiplets can be predicted from the quantum number dependence of the single gauge boson exchange potential

Vocg2AC,

AC=Cc-C,-C2,

(14)

where, g is the running gauge coupling and Cc, C~ and C2 are the quadratic Casimirs of the composite channel and the two elementary supermultiplets, respectively. It has been proposed [ 16 ] that a condensate will occur in the composite channel at an energy scale A~ where

g2(Ac)AC=O(1 ) .

(15)

This proposal has been supported by lattice gauge theory simulations both in the quenched approximation and with dynamical fermion loops [ 21 ]. In particular, an analysis of SU(2) lattice gauge theory in the quenched approximation with vector-like ferm i o n s i n t h e I = ~, ~ 1,x3 and 2 representations suggests Casimir scaling of the type (I 5) with condensation occurring at a scale Ao where

g2(Ac)AC.-,4 .

(16)

Whilst the critical value ofg2AC at which dynamical symmetry breaking occurs could depend on the gauge group, and there could easily be violations of Casimir scaling (15) for greatly differing values of C, we will take (16) as our criterion for stumbling. According to our SMAC hypothesis, the first condensate to form would be the SU(5) gaugino condensate 24X24-,1, for which AC=10. Such a gaugino condensate would signal supersymmetry breaking, leading to a positive vacuum energy ~A 4. Since this vacuum energy does not fall with temperature a s T 4 like Veff, any such supersymmetry breaking would rapidly destabilize the symmetric field configuration causing a transition to the Higgs phase, as we have previously emphasized. However, it is possible that this naive application of SMAC is incorrect, and that supersymmetry may forbid such gaugino condensation, as is known to occur in certain theories [22]. Therefore we now present an alternative stumbling scenario assuming

29 October 1987

unbroken supersymmetry ~7. When we supplement our SMAC hypothesis with this additional assumption, we find that the first condensate to form is 10i X 1--6n~1~ ( i = 1,2,3,H), which has A C = ~ . This condensate breaks the full flavour chiral symmetry ~8 U(4) v,H x U ( 1 )n × U(3)~ XU(1 )~ x U ( 1 )h x U ( 1 )R/U(1 ) XU(1 )A,

(1 7)

to U(3)~XU(1)X... and also breaks the R symmetry, giving rise to eight Goldstone bosons which sit in the four bound-state chiral supermultiplets L. The next SMAC is 5hXS~-~l, which has AC=-~ and breaks U(1)hXU(1)~ in (17) down to U(1). This results in one massless Goldstone boson, which sits in the bound-state chiral supermultiplet 1 together with its real supersymmetric partner, which is not a Goldstone boson of any broken compact symmetry and therefore corresponds to a non-compact degeneracy of the vacuum [23]. All of the above Goldstone bosons from the li and the 1 acquire masses when @ acquires a VEV, since they each contain a massive D or d c constituent. This will be important when we consider the detailed shape of Verf(~, T) in the strong coupling phase of the theory. Since the 1~ and 1 condensates give masses to all components of 10, 1--0, 5 and 5 representations, the chiral symmetry associated with the remaining light gauge non-singlet fields is just U(3)v × U (3)r/U (1) X U (1)A. According to the naive SMAC hypothesis, the next condensate to form should be 10~Xl0j~5o, which also has A C = ~ . Such a condensate would necessarily break sUpersymmetry by a D-term. If this were present, the vacuum energy would be ~A~ and the symmetric vacuum immediately destabilized as discussed earlier. However, if we continue to assume that supersymmetry overrules SMAC, we must assume that such 50 conden#7 We still anticipate the formationofcolour-singletbound states

containing gauge bosons and/or gauginos. Althoughthe natural mass-scalefor such bound states is A¢,the presence of an R symmetry would imply that chiral supermultiplets made out of gauginoswere massless. However,since R symmetryis broken by 10i X 10n ~ li condensates,we do not expectthese states to be massless in the confinedspectrum. ~8This chiral symmetryexcludes SU(5)-singlet fields, whichdo not form condensates. The U(1) factors in the numerator of ( 17) includeone anomalousU(1 ) factorand one gaugedU ( 1) factor, which are therefore divided out. 359

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sates do not form until the next SMAC condensate 10iX 5j= 5i~ with AC= ~ also forms, yielding a combined D-fiat direction. If they are all aligned, as is energetically preferred, these 5~j, 5gj condensates break the gauge symmetry to SU(4) ×U(1 ), and also break all the remaining chiral symmetries. Since the required Goldstone bosons are among the 5~j and 5ij bound state superfields, it is reasonable and conservative to assume that all the 5ij and 5tj bound states form. Taking account of the Higgs mechanism these bound states include five massless 71representations of SU(4) from among the 5ij [which transform as (6,1) under SU (3) F X SU (3)~], eight 4's from among the 5,j [which transform as (3,3) under SU(3)F × SU(3)r], and 14 SU(4)-singlets 14, 9 of which contain a vc and hence receive masses oc < • >. All the components in the 10i and 5j participate in these condensates and thus get masses, except for the 3 SU(4) singlets lj within the 5j, which must be counted among the massless states in the strong-coupling phase. Finally, the bound-state 71 and 4 representations are expected to form immediately El,.× 4.--, 1,.. condensates, since A C = -~ which is larger than ~ . All of the possible 40 l,.. bound states are expected to have masses ~A¢, since there are no gauge or chiral symmetries to forbid such mass terms from appearing in the superpotential. The total number of light states in the strong-coupling phase is therefore expected to be at most 4(1,) + 1 (1) + 14(14) + 3(1j) + 4 ~ , , + 3 ( ~ ) =29, of which 17 get mass oc ~ and 12 do not. As expected, this is fewer than in the Higgs phase, where there are 107 total supergield degrees of freedom, of which 44 acquire masses oc < qb > and 63 do not. To estimate when the bulk of this dynamical symmetry breaking occurs, we use the middle-of-the-road estimate AC= ~ which, when substituted into (16) corresponds to a =-g2/4n = 0.065. A renormalization group analysis of the low-energy effective theory yields the estimate a ~ u T = 0.043. Using the leading order formula aGUT

(18)

oL(#)-~ 1 - ( 5/2n)aGUT ln(mGVT//~) '

29 October 1987

0

I TIA¢= 0.1

a)

-10 /

T / A t = 0.5 -2O >~ -30

-40

T/At= 2 T/At= 5

-50 0.1

0

,

,

,

,

02

. . . .

0.5

,

,

1 ¢/A c

2

,

,

. . . .

5

10

I b)

-10

-20

-30

-40

-50

,

01

0.2

,

,

....

05

,

,

1 • /A c

2

,

, ,

....

5

10

Fig. 2. Sample movies of the evolution with temperature of the effective potential Verr(~,T) in strongly-coupledSU(5) XU(1 ) when T/Ac=O(1), with (a) p= 1, (b) p=2. which is consistent with the lower limit (7) based on the entropy generation constraint. Clearly our understanding of strong-coupling dynamics is incomplete, and the estimate (19) of Ac could be off by orders of magnitude in either direction ~9, but at least hope is still alive. We have made a computer simulation of the evolution of the effective potential when T ~ Ac which is shown in fig. 2. To model the qb dependence of the

we calculate Ac = 2 × 360

10-4mGvx ,

(19)

,9 Indeed, it has been argued [24] that supersymmetricGUTs may effectivelybe in a strong-couplingregimeat mcux.

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effective potential when ~ , T~Ac we use a simple monotonic interpolating function exp{ - [ ( 0 2 + TZ)/A~lP},

(20)

to match the finite-temperature effective potential oo

-2g

0

) A~) phases o f the theory. The coefficient p in eq. (21 ) is an unknown and model-dependent power which is presumably O(1 ). In evaluating (20) we have taken ,~4,5,6=O(1 ), as seems appropriate in the strong-coupling domain. Fig. 2a shows the effective potential for T,O ~A~ if we take p = 1, whilst fig. 2b is for p = 2. We see that there is no barrier to trap q~ close to the origin. Although it might be possible by varying parameters to form an apparent barrier of height O(A 4) when T = O(Ac), we expect, as was explained above, that even in this case strong-coupling and thermal effects would easily make an end run around any such barrier when T~A~. To make this scenario complete we must still add cosmological inflation and baryosynthesis. We expect that inflation is produced by additional S U ( 5 ) × U ( I ) singlet fields ~u. If we dissociate this model from superstring theories, then we are free to use existing acceptable models of inflation [25] based on S U ( N , I ) supergravity. These models, however, contain either linear or quadratic superpotential couplings and as such cannot be used in the context of the superstring, where all couplings must be at least trilinear. The production of a net baryon asymmetry is much less problematic in this flipped SU( 5 ) × U (1) model [14]. The magnitude o f the asymmetry will, however, depend on the amount o f reheating during inflation. A detailed treatment of baryosynthesis will be given elsewhere [ 15 ] and here we simply outline the basic mechanism. In the absence o f strong reheating: TR <,~c, where 2 is the Yukawa coupling which gives mass oc ~ to the lightest particle capable o f generating a baryon asymmetry, we must rely on the decay of the inflaton q/ to 0 whose subsequent

29 October 1987

decay will lead to a baryon asymmetry by its out-ofequilibrium decay. The resulting asymmetry is approximately given by

nB/s=(TR/mH or*,

(22)

where e' is the net baryon number per decay. On the other hand, if TR > Ac, thermal equilibrium is restored and we must depend on a small decay rate o f either H or Q,n. For example, if we consider the decay of H to the combination FFFi', we can estimate the baryon asymmetry to be

na/s~- ~' ( m~2 m~,/2/ m 3) ,

(23)

and clearly an asymmetry o f order 10- ~t is possible. We have seen in this paper that the entropy problems o f intermediate-scale models can be avoided by strong-coupling effects, in an action replay o f the previous "supercosmology" scenario [ 13] in conventional supersymmetric S U ( 5 ) GUTs. This scenario cannot rescue intermediate-scale models with small low-energy group factors ___SU( 3 ),but is applicable to a recently proposed [ 14] flipped supersymmetric S U ( 5 ) × U ( 1 ) GUT. We would like to thank L. Corners, A. Deans, S. Kelley, R. Parker, R. Ticciati and G. Veneziano for valuable conversations. B.A.C. thanks the CERN Theory Division for its kind hospitality. His research was supported by N S E R C Canada under grant A8863. The work o f D.V.N. was supported in part by D O E grant DE-AC02-76ER0081 and in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation. The work o f K.A.O. was supported in part by D O E grant DE-AC02-83ER-40105.

References [1] P. Langacker, Phys. Rep. 72 (1981) 185. [2] E. Gildener and S. Weinberg, Phys. Rev. D 13 (1976) 3333; E. Gildener, Phys. Rev. D 14 (1976) 1667; A.J. Buras, J. Ellis, M.K. Gaillard and D.V. Nanopoulos, Nucl. Phys. B 135 (1978) 66. [3] M.B. Green, J.H. Schwarz and E. Witten, Superstring theory (Cambridge U.P., Cambridge, 1987). [4] E. Witten, Nucl. Phys. B 188 (1981) 513. [5] K. Yamamoto, Phys. Lett. B 168 (1986) 341; K. Enqvist, D.V. Nanopoulos and M. Quir6s, Phys. Lett. B 169 (1986) 343. [6] S. Weinberg, Phys. Rev. Lett. 48 (1982) 1303. 361

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[7] S. Weinberg, Phys. Rev. D 26 (1982) 187; N. Sakai and T. Yanagida, Nucl. Phys. B 197 (1982) 533. [ 8 ] J. Ellis, A.D. Linde and D.V. Nanopoulos, Phys. Lett. B 118 (1982) 59; J. Ellis, J.E. Kim and D.V. Nanopoulos, Phys. Lett. B 145 (1984) 181; J. Ellis, D.V. Nanopoulos and S. Sarkar, Nucl. Phys. B 259 (1985) 175; M. Kawasaki and K. Sato, Phys. Lett. B 189 (1987) 23. [9] B.A. Campbell, J. Ellis and D.V. Nanopoulos, Phys. Lett. B 141 (1984) 229. [ 10] E. Cremmer, S. Ferrara, C. Kounnas and D.V. Nanopoulos, Phys. Lett. B 133 (1983) 61; J. Ellis, A.B. Lahanas, D.V. Nanopoulos and K. Tamvakis, Phys. Lett. B 134 (1984) 429; J. Ellis, C. Kounnas and D.V. Nanopoulos, Nucl. Phys. B 241 (1984) 406; B 247 (1984) 373; Phys. Lett. B 143 (1984) 410. [ 11 ] A.B. Lahanas and D.V. Nanopoulos, Phys. Rep. 145 (1987) 1. [ 12] J. Ellis, D.V. Nanopoulos and M. Quir6s, Phys. Lett. B 174 (1986) 16; J. Ellis, D.V. Nanopoulos and K.A. Olive, Phys. Lett. B 184 (1987) 37. [13]D.V. Nanopoulos and K. Tamvakis, Phys. Lett. B 110 (1982) 449; M. Srednicki, Nucl. Phys. B 202 (1982) 327; B 206 (1982) 132; D.V. Nanopoulos, K.A. Olive and K. Tamvakis, Phys. Lett. B 115 (1982) 15; D.V. Nanopoulos, K.A. Olive, M. Srednicki and K. Tamvakis, Phys. Lett. B 124 (1983) 171. [ 14] I. Antoniadis, J. Ellis, J.S. Hagelin and D.V. Nanopoulos, Phys. Lett. B 194 (1987) 231; see also S.M. Barr, Phys. Lett. B 112 (1982) 219;

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J.-P. Derendinger, J.E. Kim and D.V. Nanopoulos, Phys. Lett. B 139 (1984) 170. [ 15] B.A. Campbell, J. Ellis, J.S. Hagelin, D.V. Nanopoulos and K.A. Olive, in preparation (1987). [ 16 ] S. Raby, S. Dimopoulos and L. Susskind, Nucl. Phys. B 169 (1980) 73. [ 17] H. Georgi, L. Hall and M.B. Wise, Phys. Lett. B 102 (1981) 315;B 104 (1981) 499(E). [ 18 ] L.F. Abbott and P. Sikivie, Phys. Lett. B 120 (1983) 133; J. Preskill, M. Wise and F. Wilczek, Phys. Lett. B 120 (1983) 127; M. Dine and W. Fischler, Phys. Lett. B 120 (1983) 137. [ 19] M. Dine, V. Kaplunovsky, M. Mangano, C. Nappi and N. Seiberg, Nucl. Phys. B 259 (1985) 519; B.R. Greene, K.H. Kirklin, J.P. Miron and G.G. Ross, Phys. Lett. B 180 (1986) 69; Nucl. Phys. B 274 (1986) 574; and Oxford preprint (1987); J. Ellis, K. Enqvist, D.V. Nanopoulos and K.A. Olive, CERN preprint TH. 4713 (1987). [20] A. Bouquet, Paris preprint LPTHE 85-2 (1982). [21 ] J. Kogut, M. Stone, H. Wyld, S. Shenker, J. Shigemitsu and D. Sinclair, Nucl. Phys. B 225 (1983) 326; J. Kogut, J. Polonyi, H. Wyld and D. Sinclair, Phys. Rev. Lett. 54 (1985) 1980. [22] D. Spector, Harvard University preprint HUTP-86/A070 (1986). [23] I. Attleck, M. Dine and N. Seiberg, Phys. Lett. B 137 (1984) 187. [24] H. Goldberg, Phys. Lett. B 139 (1984) 45; J. Kapusta, D.B. Reiss and S. Rudaz, Nucl. Phys. B 263 (1986) 207. [25] J. Ellis, K. Enqvist, D.V. Nanopoulos, K.A. Olive and M. Srednicki, Phys. Lett. B 152 (1985) 175.