Grand Unification
The present status of grnnd unified theories (GUTs) are described. Topics include the problems of the standard model that are or arc not solved by grand unification, experimental implications, and features that are likely to survive extensions of the simplest GUTs.
INTRODUCTION
2 The nonobservation of proton decay' · into e + 'TTo has almost cer 3 tainly ruled out the minimal SU5 modeJ .4 as well as many other similar models with only two mass scales. However, such models are only the simplest examples of a large class of theories, many of which have longer proton lifetimes or dominant decays into other modes. Grand unified theories have many very attractive features, in cluding the unification of the basic microscopic interactions, the elegant explanation of the equality of the proton and positron electric charges, the dynamical generation of the baryon-antibar yon asymmetry of the Universe, and the prediction of sin2 0w. On the negative side GUTs have shed little light on the fermion masses, mixing angles, or family structure, have not explained the ex tremely small value of the weak interaction mass scale relative to the Planck mass (the problem is rephrased as the gauge hierarchy 24 problem, which refers to the tiny ratio (Mw1Mx)2 < 10- ), and
Commems Nud Part. Phys (t) 1985 Gordon and Breach, 1985. Vol 15. No 2. pp 41 - 67 Science Publishers. Inc. and OPA Ltd. 01110-2709 /85 i I 51l~-IXl41 /$25 ,011 /ll 1'1 intcd in Great Britain
41 do not incorporate gravity in any fundamental way. On balance it seems unlikely and even naive (and always did) that a model as simple as SU_, could be the ultimate theory of nature below the Planck scale. On the other hand, the advantages of GUTs are sufficiently strong that the basic ideas should not be abandoned lightly. Whatever the ultimate theory may be, it is very likely that it will incorporate many aspects of grand unification.
THE STANDARD MODEL AND ITS SHORTCOMINGS
The standard SU3 x SU2 x U 1 model (QCD plus the Glashow Weinberg-Salam5 electroweak theory) has been spectacularly suc cessful. It is a mathematically consistent renormalizable field the ory that either predicts or is compatible with all known facts in particle physics except possibly for the anomalous jet events re cently observed at the CERN S ppS collider. In particular, the standard model successfully predicted the existence and detailed form of the neutral current interaction, the Wand Z boson masses, and the existence of the charm quark (in order to avoid strange ness-changing neutral currents). The standard model supple mented with classical general relativity is almost certainly an ap proximately correct theory of nature, with a range of validity including ordinary terrestrial and astrophysical conditions and ac celerator energies up to several tens of Ge V. Despite these suc cesses, the standard model cannot be considered the ultimate de scription of nature-it is simply too complicated and arbitrary and leaves too many fundamental questions unanswered. These difficulties can be summarized under five headings. (a) The gauge problem: The standard model gauge group is a complicated direct product of three groups with three dis
tinct coupling constants. Furthermore, because of the U 1 factor the average electric charges for the particles in an SU2 multiplet are essentially arbitrary except for two constraints from the cancellation of anomalies. Hence there is no fun damental explanation for the observed quantization of fer mion and boson charges in multiples of e/3 (or therefore for the equality of the magnitudes of the electron and proton charges).
42 (b) The fermion problem: The fermions are assigned to a com
plicated reducible representation of the SU, x SU~ x U 1 group. No explanation is given for the existence or number of fermion families. Furthermore. neither the fermion masses. which are observed to vary over a range of five orders of magnitude, nor the fermion mixing angles are predicted by the theory: they must be taken from experiment.
(c) The Higgs problem: SU2 x U 1 symmetry is broken by the introduction of one or more fundamental Higgs multiplets
Another characterization of many of these problems is that the standard model with massless neutrinos, three fermion families. and the minimal Higgs structure (one doubler) ha I c fret> param eters. These are three gaug couplings, two P- iula ting O param eters (80 co and ·m analogous but phenomenologically unimportant parameter in l he weak sect r of the theory), nine ferm i n rnassc ·. three Kobayashi-Maskawa-Cabibbo (KMC) mixing angles, one CP-violating KMC phase, and the W and Higgs scalar masses.
43 minus one overall mass scale. If one includes classical gravity one must add the Planck mass and the (observationally tiny) cosmo logical constant to the list. As successful as the standard model may be, it is almost certainly not the ultimate description of nature!
GRAND UNIFIED THEORIES
General Description Grand unified theories are gauge theories with a single gauge cou
pling constant g0 . This can occur only if the gauge group G is a simple group like SU5 or a direct product of identical factors with a discrete symmetry which interchanges the factors. One hopes that the additional synmmetries in G will constrain some of the
arbitrary features of the SU3 x SU2 x U 1 standard model sub group.
The embedding of SU3 , SU2 , and U 1 into G implies that the strong, weak, and electromagnetic interactions are unified: they are simply different parts of a larger underlying gauge interaction. The existence of a single gauge coupling constant leads to a pre diction for the weak angle sin 20w, which is a ratio of coupling constants of the low energy effective theory. Similarly, quarks (q), antiquarks (q), leptons (/), and antileptons (l) are fundamentally similar in most GUTs: the extra symmetries in G typically trans form q, q, l, and l into each other. This will lead to charge quan tization. Furthermore, the gauge bosons associated with the new symmetry generators can usually mediate baryon and lepton num ber violating processes such as proton or bound neutron decay. The simplest and most popular GUT is the Georgi-Glashow SU, model. 3 The fermion representations are still rather compli 5 cated in SU : each family is assigned to a reducible 5* + 10 di mensional representation: 5* 10
(2)
~x,Y~ ~x,v~
44 where the color indices have been suppressed. In addition to the 12 generators and corresponding gauge bosons (W"', Z, "/, 8 gluons) of the standard model, there are 12 new generators associated with transformations between adjacent columns in (2) . These are X;,
i = R, G, B, which carries color and has electric charge ~e; Y1, i = R, G, B, with electric charge k and their antiparticles. The
X and Y bosons are the nearly degenerate members of an SU2 doublet. They can mediate proton and bound neutron decay, as in Fig. 1.
Theoretical Implications GUTs are most successful in dealing with the gauge problem, because there is only one underlying interaction. Charge quantization follows from the fact that the electric charge operator Q is a (traceless) generator of the gauge group. Hence, the sum of the charges of all particles in a multiplet (which generally includes both quarks and leptons) must vanish. For the S* in the
SU5 model this implies Q(ve) + Q(e-) + 3Q(d) = 0, where the 3 is due to the three colors of d. For Q(ve) = 0 this implies the desired result Q(d) = 1Q(e - ) = - k Most GUTs imply some kind of relationship between quark and lepton charges. However, the correct relationship does not follow uniquely in all theories because there may be more than one way to embed Qin the theory6
u------u
FIGURE I Typical proton decay diagrams. Similar diagrams imply bound 1H:utro11 decay or decays into 17 + mesons.
45 7 (for some amusing recent speculations sec Okun er al. ) or because there may be additional fermions in the multiplet. Grand unified theories have only a single underlying gauge cou pling constant. Hence. ratios of coupling strengths such as the electromagnetic to strong ratio ala., = e 2 /g~ or the weak angle sin 2 8w = e2/g 2 are in principle determined. In practice, however. there are two complications.~ The first is that even at sufficiently high energy scales that all symmetry breaking can be ignored. coupling constants will only be equal if their corresponding group generators are normalized the same way. This implies that
2 2 2 sin 8w = Tr(T ) /Tr(Q) 2 (3) 2 2 a/a, = Tr(T,) /Tr(Q) •
where T2 , T3 , and Qare SU2 , SU,, and electric charge generators. respectively. The ratios in (3) depend on the embedding of SU, x SU2 x U 1 in G and can be calculated if one knows the quantum numbers of all of the particles in a complete representation of G. In practice it is usually assumed that the fermions in one standard model family either form a complete multiplet or that any addi tional fermions. which could either be "light" (e.g .. greater than 1 several tens of Ge V) or superheavy (e.g .. 10 -1 Ge V). do not modify the ratios in (3). In that case sin 2 8w and a/a, both approach ~ asymptotically. The other complication is that the various running coupling con stants evolve differently below the unification scale Mx (the typical mass scale of the superheavy gauge bosons in G). Hence predic tions for the low energy values of sin 2 8w and a/a., depend on the detailed pattern of spontaneous symmetry breaking and on the masses and quantum numbers of any new fermions or Higgs bosons between the weak and unification scales. In the special case of models like SU, in which there are only two gauge boson mass scales (Mwand Mx) the normalized standard 2 2 2 2 2 model couplings g3 (Q ) = g,(Q ). g2 (Q ) = g(Q ), and g 1(Q ) = 2 VS13 g'(Q ) all meet at Mx. as shown in Fig. 2. If one further assumes that there are no fermion or Higgs thresholds other than those of the standard model below M x (the great desert hypothesis) then M x can be predicted from the observed low energy value of
46 FIGURE 2 The Q' evolution or the coupling constants. either a/a, or sin 20w. Equivalently, sin20w can be predicted from a/a, as a consistency condition that there are only two mass scales. The major uncertainty in these predictions is the value of the QCD parameter AMS• which determines a/a". A plausible range for AMs is 9
+100 (4) AMs = 160 _ 80 MeV
However, each determination of AMs involves serious uncertainties which are difficult to quantify. The determinations from deep ine lastic scattering suffer from serious higher twist ambiguities, and some analyses yield considerably higher values in the 300 MeV range. The values derived from e+e- ~ jets depend very sensi tively on the Monte Carlo program used to describe the hadron ization. Finally, the values derived from the Y branching ratios are very precise experimentally, but potentially large higher order (in a,.) corrections are a serious problem. Fortunately, the predicted value10
. 2 , [0.16 GeVJ sm 0w(Mw) = 0.214 ± 0.003 + 0.006 In AMs (5)
47 of the renormalized weak angle (in the modified minimal subtrac tion scheme) is insensitive to the precise value of A,ws and is in excellent agreement with the values 0.223 ± 0.007 and 0 .220 ± 0.007 obtained from deep inelastic neutrino scattering" and 1 from the Wand Z masses, " respectively. However, the prediction 111
M ,(GeV) = 2.1 x 10 1-1 x (1.5) "' 1 " x A--cM.i J (6) " l0. 16 GeV depends much more sensitively on AM.I· Grand unified theories have not been very helpful in dealing
with the fermion problem. In the simpler models like SU,, S0 111 , and most versions of Er, each fermion family is assigned to one or
more representations, just as in the SU.i x SU2 x U 1 model, so there is no prediction or explanation for the number of families or for the masses and mixing angles. The only new constraint is that in some models with the minimal Higgs structure (e.g .. SU, with only five- and 24-dimensional Higgs representations) m, and m" are generated by the same Yukawa coupling. The running masses are equal (or related by a known group theoretic factor in some other models) at momenta larger than Mx, and the physical low energy value of m" can be computed. For 3 families. m, = 20 1 GeV, and J\Ms given in (4) one obtains .1 m" = (4.7-5.5) GeV, in excellent agreement with experiment. The agreement would be spoiled in this class of models if there were too large a top mass. One has m, < 180. 145. 20 GeV for J\Ms = 80, 160. 260 GeV. respectively. A fourth family is essentially ruled out for these models unless A Ms < 80 Ge V. Unfortunately. these models also predict m, = 500 MeV, which 1 is larger than most phenomenological determinations -1 (100-200
MeV). Even worse. one has m.1/m" = mµlme = 200, while current algebra determinations yield m,lm" = 20. It has been suggested 15 that small effective interactions associated with the Planck scale could shift the quark masses by a few Me V. which would suffice for m,lm" though not for m.,. A rather inelegant possibility is to consider nonminimal Higgs structures (e.g., a 45 in addition to the
5 and 24 in SU5). In such models m 1,, m", and ms are all free parameters. There have been many attempts to construct larger grand unified theories that incorporate some sort of family symmetry, but despite
48 a great deal of effort no re have reanalyzed a class of 0 18 models in which all known fermion families can be fit into one irreducible 256-dimensional spinor rep resentation. Asymptotically free versions of the model involve four right-handed families, which receive tree level masses of order MW• and four lighter left-handed families, which receive masses at the one loop level. In one very attractive version of the model there are just three ultralight neutrinos, in agreement with limits found in the standard hot big bang cosmological model from the observed abundances of helium and deuterium. 17 Also, the proton decay rate is suppressed relative to the minimal SU5 model (M xis larger 32 1 because of the extra fermions) to the interesting level 10 "' yr. and sin 2 ew(Mw) = 0.215. In grand unified theories the Higgs problem is renamed the gauge hierarchy problem. This refers to the incredibly tiny ratio M{-,)M;, which is< 10- 24 in most theories. The existence of two such different mass scales is not a natural feature of GUTs. It is necessary to adjust the parameters in the Higgs potential, including 24 the loop corrections, to one part in 10 • Possible solutions to the Higgs/hierarchy problem include: (i) Populating the desert between Mw and Mx with many new thresholds. Such an approach is somewhat contrary to the spirit of grand unification. Also, one generally loses predictive power (e.g., for sin20w), and many such models have difficulty in ac counting for the observed cosmological baryon asymmetry. (ii) A very attractive possibility for the Higgs problem is some form of dynamical symmetry breaking. In technicolor models, for example, the analogues of the Higgs scalars are composite with a compositeness scale of a TeV or so. Unfortunately, the simple technicolor models do not generate fermion masses, and extended technicolor theories (which have extra gauge bosons that couple ordinary and technicolor particles) have serious problems with flavor changing neutral currents and are very difficult to embed in grand unified theories. (iii) The Higgs/hierarchy problem is one of the strongest moti vations for supersymmetry. Although most supersymmetric models 49 do not explain the ongm of the gauge hierarchy, they at least prevent its renormalization because of cancellations between fer mion and boson loops. (Supersymmetry breaking effects do lead to a renormalization of the MwlMx ratio, but the effects are finite and sufficiently small if the scale of supersymmetry breaking is less than a few TeV). GUTs therefore shed no light on why weak scale is so much smaller than M x or the Planck scale . The analogous question for the strong interaction scale (e.g .. A Ms) is considerably less prob lematic. Because of the slow (logarithmic) variation of ex, with Q2 , any small value of cx,(M~) (e.g., - 0.024 in SU, and similar models) will lead to an exponentially small ratio AM.\IMx = exp(-c/cx,(M7>;)) <<< l, (7) where c is a group theoretic constant of order unity. CP violation 18 is even trickier in grand unified theories than in the standard model because the generation of the cosmological baryon asymmetry requires an adequate amount of CP breaking in the early Universe when the temperature was comparable to Mx. Weak CP breaking and the baryon asymmetry can be induced by complex Yukawa couplings. As in the standard model. however. the strong CP parameter Boen is arbitrary. The limit Boen < 10 - 9 from the neutron electric dipole moment requires that the bare value Bb"'" must be fine-tuned to cancel an apparently independent correction of order 10- 1 from the phases in the quark mass matrix. 19 The Peccei-Quinn mechanism. in which a global U 1 symmetry is imposed on the theory. which ensures that the minimum of the Higgs potential is at a location for which the necessary cancellation occurs, can be incorporated into GUTs. However. cosmological and astrophysical constraints strongly suggest that the scale of Pec cei-Quinn spontaneous symmetry breaking must be in the range 108 -10 12 GeV, which is not a natural scale in most GUTs. Supersymmetric theories require extra Higgs multiplets and are therefore good candidates for the Peccei-Quinn mechanism. In globally supersymmetric models. R invariance can be the necessary U 1• In supergravity, however. it is difficult to have both an R symmetry and a vanishing cosmological constant. at least in the simpler models. It is possible to introduce a Peccei-Quinn sym- so metry unrelated to R invariance, but only at the price of intro ducing extra fields. 20 Another approach to the strong CP problem is to break CP spontaneously, so that 00 co is finite and calculable. A number of rather complicated models have been proposed in which 0oco is sufficiently small. However, it is difficult to incorporate sponta neous CP breaking into G UTs because if the breaking occurs at the weak scale it will most likely be restored in the early Universe so that no baryon asymmetry could be produced. This problem is avoided in an interesting class of models recently suggested by Nelson 21 and generalized by Barr.22 These are GUTs in which CP is spontaneously broken at M x rather than M w, so that it is mar ginally possible to generate a large enough baryon asymmetry. 23 CP violating phases are generated in the effective low energy the ory by fermion mixing effects. Such models may still have diffi culties with cosmological domain walls, however, Finally, ordinary (i.e., nonsupersymmetric) grand unified the ories still do not incorporate gravity. Fortunately, the unification scale Mx usually turns out to be far enough below the Planck scale that a partial unification without gravity is at least internally con sistent. Simple grand unified theories therefore solve the gauge problem but contribute little to the other theoretical shortcomings of the standard model. The simplest SU, model (which does not generate an adequate baryon asymmetry unless extra Higgs multiplets are introduced) actually has more free parameters than the standard SU, X SU~ X U I mode] (24 compared to 2J, including classical gravity). The extra parameters mainly involve superheavy Higgs particles, however. Experimental Implications The most dramatic implication of grand unified theories is baryon and lepton number nonconservation. In particular, almost all GUTs predict proton and bound neutron decay. Unfortunately, the life time and, to a somewhat lesser extent, the branching ratio pre dictions depend rather sensitively on the pattern of spontaneous symmetry breaking and on the gauge group G. 51 One has the most predictive power for models such as minimal SU5 which: (i) have only two mass scales. Mw and Mx. with an SU, x SU 2 x U 1 invariant desert in between; and (ii) have no light (O(Mw)) bosons or fermions other than those of the standard model (i.e .. for the models which naturally lead to the successful prediction in (5) for sin28w). Even for minimal SU,. however. there are serious uncertainties from our inability to calculate strong in teraction effects quantitatively. The unification mass Mx depends approximately linearly on the poorly known QCD scale AMs· as in (6). Furthermore. the matrix elements of the four fermion op erators generated by the diagrams in Fig. l are not directly related to known quantities and are difficult to compute reliably. There have been many theoretical calculations.24 based on relativistic and nonrelativistic quark models. the MIT bag. PCAC. vector meson dominance, chiral Lagrangians, QCD sum rules, etc. Even for the same value of Mx these calculations vary in their predictions for the proton lifetime by as much as two orders of magnitude. A 0 reasonable range for the partial lifetime (Tl B) into e + TI is T(p ___,, e+7Tll) = 4.5 x 1Q29±ll.7 x [-2-.1---~-~j"'"~--l -G_c_·V- r yr (8) 0 7 1 where the theoretical uncertainty 10± · ~ (5) ± is a guess based loosely on the spread of theoretical estimates. Combining this with (6) one obtains A- T(p ___,, e+7TO) = 4.5 x 1Q29±0 9 x MS ]-i yr (9) [ 0.16 GeV which is less than 2.3 x 1031 yr for AMs in ( 4) and less than 1.4 x 1032 yr for the more conservative assumption A Ms < 400 Me V. The present experimental lower limit1 T(p ___,, e+7r0) > 1.2 x 10'2 yr therefore essentially rules out minimal SU5 . However, it is hard to make an absolute statement to that effect because the uncer tainties in AMs and the matrix elements are large and difficult to quantify. Other gauge groups G such as 5010 , E 6 , etc., give identical 2 predictions as SU5 for sin 0w provided they break directly to SU3 x SU2 x U 1 at Mx and have no additional light particles. They 52 typically predict slightly faster decays into c 1 '1T 11 and enhanced branching ratios into neutrino modes because of the additional superheavy gauge bosons. Although the simplest SU, model is apparently ruled out. the basic ideas of grand unification are still alive and well. Many rel atively simple modifications of minimal SU, lead to a longer proton lifetime, although unfortunately they also tend to have less pre dictive power for sin 28w. The models with longer proton lifetimes fall into three principal classes. The first possibility is to add struc ture in the desert between M wand Mx so that the renormalization group equations which determine Mx and sin 28w are modified. Approximately degenerate new G multiplets of fermions or spin () bosons do not affect the Mx predictions to lowest order. (How ever, if the new multiplets are too large asymptotic freedom will be lost for QCD so that the perturbative calculation will not be a good approximation.) On the other hand, split multiplets which have some light (e.g., O(Mw)) and some heavy (e.g., O(Mx)) 2 particles can affect T" and sin 8w significantly in either direction. For example, a desired pattern can be achieved25 by the addition to minimal SU5 of a HJ + 10 of fermions, which contains ( U, D). C, and£+, where Uhas the same charge and color as the u quark. etc. T" can be increased by~ 10-' while changing sin 28w by< 0.01 if E + is superheavy and the others light. Similarly, one can assume the existence of light or intermediate mass scalars. 26 such as color octets. It should be emphasized that such multiplet splitting models are usually quite ad hoc, and most splittings lead to unacceptable values of Mx and sin 28w. Also, it is hard to understand how the mass splittings come about, although one splitting (of a five di mensional Higgs representation in minimal SU5 ) is needed in any case to generate the SU2 x U 1 scale without having very fast pro ton decays mediated by the partners of the ordinary Higgs doublet. Theories with a low energy supersymmetry are considerably less arbitrary. In such models all of the ordinary particles are auto matically accompanied by new superpartners with masses less than a TeV or so, with all of the necessary splittings implied by the gauge hierarchy (M wl M x) and supersymmetry. These new particles increase Mx significantly and decrease the proton decay rate due to heavy gauge boson exchange to an acceptable level without changing sin28w too drastically. It is also possible to have more than two symmetry breaking 53 (i.e .. gauge boson mass) scales. For example. S0 111 might first 14 break to SU3 x SU21. x SU211 x UI. at a mass Mx >> 10 GeV. 27 and then to SU3 x SU2 x U 1 at a scale M 11 • Then one has M.\ 112 14 - M5 (M •.,IMR) >> M5 • where M5 = 2 x 10 GeV is the SU 5 scale. Sin20w depends on the mass ratios in such schemes. so one has no real prediction. If one requires ~sin 2 flw < ll.01 then Ml<> 12 15 T , 10 GeV and M.\ < 2 x 10 GeV. so that 1 can be increased by about four orders of magnitude. Another possibility is that the proton decay rate is suppressed by mixing angle effects. 4 It is possible. though not very natural. that in models with complicated Higgs structures the light quarks and heavy leptons. for example. are associated together in mul tiplets. In this case the dominant amplitudes could be into ener 0 getically forbidden channels such asp--'> T' 'TT . In most cases the lifetime is expected to increase by no more than sin· 28, - 20. however. One exception to this limitation is models in which the proton is made absolutely stable by imposing an ad hoc new quan tum number on the theory. Such models generally involve a dou bling of the fermions and can lead to the dramatic signal of baryon number violation at accelerators. Finally, effective nonrenormalizable operators28 such as Tr(Fµ,,Fµ" 54 Most models leading to !::i.B = 2 interactions involve heavy Higgs particle exchange. In many cases the Higgs particles are introduced in a rather ad hoc manner. Futhermore, T.," is proportional to M". where M is the typical mass of the particles mediating the inter action. Only for M in the rather narrow range I0'-106 GeV are neutron oscillations experimentally relevant. This range has not emerged in a natural way in any theory, and in particular a number 2 of authors-' have shown that it does not occur for any viable S0 111 symmetry breaking pattern if one makes the reasonable require ment that M be associated with a gauge boson mass scale. The theoretical estimates are therefore not very optimistic, but never theless it seems well worth pushing the experiments as far as pos sible. All grand unified theories predict the existence of superheavy (mass = Mxla<; = 10 16 GeV) magnetic monopoles.-'-' In addition to their electromagnetic interactions GUT monopoles may catalyze baryon number violating processes such as Mp ~ Me+Tro with strong interaction strength (much larger that the naive geometric 06 estimate rr ~ Mx - 2 ~ 10- cm 2). Searches for monopole catalysis 13 in proton decay detectors yield flux limits in the range FM < (1o- - 10- 16) cm- 2 sr- 1 s- 1 for U' ~ (1-100) mb and the monopole velocity in the expected 10--'c region. Much more stringent limits 22 2 1 1 (typically FM< 10- cm- sr- s- for U' = I mb) come from x ray emission from neutron stars, although there are many com plications and uncertainties concerning the astrophysics. The strongest limit not relying on catalysis is the Parker bound FM < Io- 16 cm - 2 sr- 1 s - 1, based on the requirement that the GUT monopoles not deplete the observed galactic magnetic field faster than it can be regenerated. The possiblity of magnetic monopoles is not unique to grand unified theories (the existence of non-GUT monopoles would be an alternate explanation for charge quantization). Nevetheless, the direct detection of a superheavy monopole would very strongly support the ideas of grand unification. Direct terrestrial monopole search experiments may eventually be sensitive to fluxes compa rable to the Parker bound, but such experiments have little hope if the neutron star limits are correct. A very speculative alternative is to look for the spectacular annihilation products that could result from monopolonium-'4 (MM bound states), which may have been produced in the early Universe. 55 Unfortunately, grand unified theories do not make any firm predictions for the monopole flux. Monopoles may have been pro duced prolifically during phase transitions in the very early Uni verse. Early estimates suggested that the present mass density from relic monopoles is too large by many orders of magnitude. How ever, a number of cosmological scenarios have been suggested-'' in which the monopole flux is greatly reduced. For example, Sal omonson et al. 35 have recently argued that for a large parameter range in the SU5 model the SU_, symmetry would not have been restored even for T > > MA, therefore avoiding the monopole producing phase transition. The relic monopole problem was one of the primary original motivations for the inflationary Universe models,-'6 in which the density of primordial monopoles could have been diluted to essentially zero during a very rapid period of ex ponential expansion of the size of the Universe. Inflationary models have the potential for solving many other cosmological problems as well. Most grand unified theories other than minimal SU, predict nonzero neutrino masses. Many models naturally lead to a very small mass (l0- 9 -10- 4 eV) for v .. , but with some difficulty larger masses (e.g., 10 + 2 e V) can be accommodated. The neutrino masses are usually Majorana (i.e., lepton number violating) but recently it has been shown37 that some more complicated models can lead to light Dirac neutrinos as well. The experimental situation is com 38 plicated. Lubimov et al. have reported a nonzero v e mass in the 20-45 eY range. This has not yet been confirmed, but a number of sensitive experiments are underway. There is no positive evi dence for neutrinoless double beta decay (which occurs only for Majorana neutrinos), and neutrino oscillations (which depend on mixing angles as well as masses) have not been established. There is a possible indication of an effect in the Bugey reactor experi ment. This is marginally consistent with the Gi:isgen reactor two distance limits but in conflict with the stronger but less reliable Gi:isgen limits using reactor electron spectra. Stable neutrinos with masses in the tens of e V range could dom inate the energy density of the Universe. One has an upper limit of 100 e V on the sum of the light neutrino masses from limits on the large scale deceleration, suggesting the need for an inverted hierarchy (mv, > mvµ > mvJ if the Lubimov et al. result is correct. 56 Massive neutrinos arc also candidates for the dark matter in gal axies and clusters, although that view runs into difficulties in de tail.·19 Even tiny neutrino masses (e.g., > 10 - 5 eV) could, for large enough mixing angles, lead to oscillations which would reduce the predicted flux of solar neutrinos. One of the most attractive features of grand unification is that the baryon number violating interactions can dynamically generate the observed baryon asymmetry (10) where n 13 , fl7J, and n'I are the present average number densities of baryons, antibaryons, and microwave photons, respectively, in the first 10- 35 s after the big bang40 when the temperature was com parable to Mx. Jn addition to B violation, the necessary ingredients are nonequilibrium (otherwise nn = n7i since B and B are degen erate by CPT) and C and CP violation (to distinguish baryons from anti baryons). In most models the actual mechanism is the decay of superheavy Higgs particles H and their antiparticles H. CP violation allows the relative rates for H ~ qq and ql to differ from those for H ~ qq and ql. In the minimal SU5 model with three fermion families the asymmetry requires a three loop diagram and is too small 20 (-10- ). However, an adequate asymmetry can be generated for 2: 4 families or if additional superheavy Higgs multiplets are added to the theory. Structure in the Desert Most theories that have a longer proton lifetime than the minimal SU5 model and most suggested solutions to the Higgs/hierarchy problem involve new Higgs, fermion, or gauge thresholds in the region between Mw and Mx. The possibility of new particles is very exciting, especially if they occur in the experimentally relevant range between M w and a few Te V. However, the new thresholds can also cause several serious problems. The first difficulty is that new particles generally mess up the prediction of sin20w, which was after all one of the major attrac tions of simple SU5-like models. Many models have been con- 57 structed in which the changes in the sin 28w prediction are kept small either by limiting the possible values of new spontaneous symmetry breaking scales or by cleverly choosing representations for which the effects cancel. Such models may be phenomenolog ically viable, but sin28w must be regarded as more an input than a prediction. Models with a low energy supersymmetry are an exception because the representations of the new particles are related to those of the ordinary particles; sin 28w increases by a barely tolerable 0.02 in the simplest case. The cosmological baryon asymmetry is a very serious problem for any model in which CP, C, or a related symmetry is sponta neously broken at a scale v < < M x· Except for a few very com plicated cases, the symmetry is restored at temperatures T >> v, so a baryon asymmetry cannot be generated during the GUT ep och. The baryon asymmetry also introduces a crucial constraint on inflationary models: any baryon number density is diluted to neg ligible values during the period of exponential growth. The Uni verse must therefore reheat to a high enough temperature follow ing the inflationary period to generate the presently observed asymmetry. In view of these difficulties it is important to investigate the minimum temperature at which the asymmetry could have been produced. The conventional scenario4u involves the baryon number violating decays of a slightly out of equilibrium particle. In order to produce enough of the decaying particles thermally the initial temperature (e.g., the reheating temperature after inflation or the scale v at which C and CP are broken) must be comparable to the particle's mass. However, in order to be out of equilibrium the particle's interaction and decay rates must be comparable to or slower than the Hubble expansion rate of the Universe. The latter is very small at low temperatures (it varies as T 11m,,), so in most theories the relevant temperature and mass scale must be greater than 10 12 -1013 GeV. There are several possibilities for evading or weakening this limit. Recently, Kosower et al. 41 have shown that it is indeed pos sible to generate an adequate baryon asymmetry with decaying particles as light as 1 Te V; however, the decaying particle must be a weakly coupled gauge singlet, and a complicated supporting array of exotic fermions and scalars is required. Another possibility 58 is that there is a nonthermal production mechanism for the heavy particles. In particular, at the end of an inflationary period the Universe is far from equilibrium, with the energy density in the form of coherent scalar field oscillations. Ovrut and Steinhardt42 have recently found a class of supergravity inflation models (in which the curvature of the Higgs potential at the global minimum varies by many orders of magnitude as a function of direction) in which superheavy particles with masses much greater than the 10 12 reheating temperature (10 -10 ) can be prolifically produced by their coupling to the oscillating field. Affleck and Dine43 have made an even more radical proposal, viz., that in some supersymmetric GUTs the scalar partners of the quarks and leptons may have large vacuum expectation values after a period of inflation. The sub sequent evolution and decay of such fields can yield enormous 3 asymmetries as large as n8 /n "Y ~ 10 at temperatures orders of magnitude below the GUT scale. One could then tolerate sub stantial entropy generation in later cosmological events. Despite these loopholes, the baryon asymmetry remains a se rious problem for many models with low mass scales. Yet another difficulty with many low mass scale theories is the possibility of topologically stable domain walls, which can exist for any theory with a spontaneously broken discrete symmetry. Such domain walls could be produced in the early Universe during the phase transition in which the discrete symmetry is spontaneously broken, and would in general contribute far too much energy den sity and anisotropy to the present Universe. 44 Similar problems are likely to occur in theories in which the discrete symmetry is the remnant of a continuous symmetry broken at a much higher scale. 45 In that case the walls are quantum mechanically unstable but their lifetime is very long compared to the age of the Universe. They may be infinite or bounded by strings, and there is a possi bility that their density could be reduced by interactions with strings if there are enough of the latter left over from earlier phase tran sitions. Domain walls are a serious problem for many axion models, 19 all models with spontaneous CP violation, and all models with a low energy left-right symmetry. One possible solution to un wanted domain walls is to invoke a period of inflation after the walls are produced. As already discussed, however, baryosynthesis must occur after the inflation, and in most models an adequate 59 asymmetry can only be generated if the reheating temperature is greater than Hl 12 GeV. This in turn implies that it is hard to elim inate domain walls in this way unless they are associated with a spontaneous symmetry breaking scale > I 0 12 Ge V. This is marginal for axion models because other astrophysical and cosmological constraints favor the range 108 -1() 12 GeV. Much attention has been devoted to SU21, x su2R x u I models with a discrete left-right symmetry to interchange the SU21, and SU2R factors. Low energy particle physics constraints allow a mass scale MR for the Wt? as low as a few TeV.46 However, when con sidered in the context of GUTs and cosmology. left-right sym metric models suffer from all of the problems described above. For example, when SU21. x SU21? x U 1 is embedded in SO Ill or a similar model, sin20w comes out much too large (typically ~ 0.28) unless MR is very large27 (> l0 12 GeV) or additional split fermion or Higgs multiplets are invoked.47 Similarly. the left-right symmetry generally leads to unacceptable domain walls and pre vents the generation of a baryon asymmetry48 unless MF? > [() 12 GeV or so. All of these difficulties may be avoided in a class of models49 in which the left-right symmetry breaking scale Mu? is decoupled from the su2L x SU2R x u l breaking scale MR· This may come about if additional Higgs multiplets break the left-right symmetry 12 18 at MLR ~ (10 -10 ) GeV, allowing MR as small as a few TeV. Such models may be viable, but at the price of abandoning most of the elegance of and motivations for introducing SU2L x SUm x U 1 in the first place. Supersymmetry In addition to their theoretical elegance, supersymmetric models50 partially solve the hierarchy problem by cancelling the worst di vergences in the renormalization of the Higgs mass scale. Fur thermore, local supersymmetric models (supergravity) involve a nontrivial unification of the other interactions with gravity. Su pergravity does not by itself render quantum gravity finite or re normalizable, but recent developments with superstring theories in ten space-time dimensions are promising. The simplest (N = 1) supersymmetry theories have a single fermionic supersymmetry generator. Such models require a dou- 60 bling of the particle spectrum, with a new fermionic (bosonic) partner for each ordinary boson (fermion). In order for the re maining (finite) radiative corrections to the Higgs mass to be suf ficiently small the typical splitting between ordinary particles and their superpartners should not be much more than a few Te V, which is encouraging for accelerator physics. In fact, it is quite possible that anomalous S ppS mono jet events are due to the pro duction and decay of relatively light scalar quarks, gluinos, etc. In addition to the spectrum doubling it is necessary to have two light Higgs doublets (and their fermionic partners) in order to generate masses for both the charge - *and + ~ quarks. Most viable supergravity models assume that the supersymmetry is spontaneously broken in a hidden sector, coupled to ordinary matter only through supergravity, at a scale F - 10 11 >-10 11 GeV. The effective theory relevant to the ordinary particles and their superpartners is then identical to a globally supersymmetric theory with specified (up to several parameters) soft supersymmetry 2 breaking mass and cubic terms. These are of order m v 2 - F /m" 4 - 10-10 GeV, where m 312 is the mass of the gravitino, the spin- 3/2 partner of the graviton. SU 2 x U 1 breaking can be induced at a scale within a few orders of magnitude of mv2 by radiative cor rections associated with the Yukawa coupling that generates the top quark mass, provided m, is greater than about 20 GeV. In most such schemes F, which eventually determines m 312 and M w is an input scale. Supersymmetric GUTs have a unification scale MsysY as well as m", F, and Mw. The spectrum includes the superheavy gauge and Higgs bosons, their fermionic partners, the ordinary low mass particles, and their superpartners. The latter modify the renor malization group equations that determine M5xusY and sin28w. For models with two light Higgs doublets and supersymmetry breaking in a hidden sector, M5xusY is a factor 30 or so larger than the 1 ordinary nonsupersymmetric unification scale (i.e., - 10 (' Ge V). The proton lifetime due to ordinary X and Y gauge bosons is therefore increased by - 106 to (probably) unobservable levels of order 1035 yr. Sin28w is increased by about 0.02 compared to Eq. (5), which is somewhat large, but possibly acceptable, and the mb prediction is essentially unchanged. In an alternative set of models51 supersymmetry and the grand unified gauge symmetry are broken in the same sector. These models always predict the existence of 61 new chiral supermultiplets (fermions and scalars) which have the same SU3 x SU2 x U 1 quantum numbers as the gluons and elec troweak bosons (these are in addition to the gluinos, winos, zino, and photino); their masses should be in the TeV range or smaller. These and possibly other light supermultiplets lead to M~usv close to the super Planck scale m"lvs:IT = 2.4 x 1018 GeV, with ac ceptable values of sin20w possible. It is possible to construct variant models in which extra split multiplets lead to predictions similar to SU5 for proton decay and sin20w. Nevertheless, in most supersymmetric GUTs the unifica tion mass is so large that gauge boson mediated proton decays into 0 the standard SU5 modes like e + 1T are unobservably slow. There is another source of proton decay in most such models, however: new diagrams involving the superpartners of the superheavy gauge and Higgs bosons, such as Fig. 3. The diagram of Fig. 3(a) involves the exchange of a light scalar quark. It is not suppressed by any power of M~usv and, if present, would lead to a disastrously short proton lifetime. Fortunately, it can he forbidden by imposing discrete symmetries on the Lagran gian. The colored Higgsino (H) exchange in the left half of the box diagram in Fig. 3(b) leads to the dimension 5 operator L - h2 qqql! M (11) ( 0) q 1 c 3---H ""cl~ -ICw"" q c q ------q ( b) FIGURE 3 New proton decay diagrams that can occur in supc rsymmctric GUTs. (a) A dimension 4 diagram that must h.: forbidden to <1void a disastrously short lifetime. (b) A dimension 5 diagram. 62 where h is a Yukawa coupling and M - Mx is the Higgsino mass. After exchanging a light wino50 or gluino52 these operators lead to proton decay with a lifetime of order M 2 (rather than the usual Mi). This appears disastrous, but in fact there are a number of suppressions to the decay rate (M~usv > Mx, 1/lfrrr2 from the loop, two small Yukawas, smaller higher order enhancement fac tors). The expected lifetime is typically in the range 1026-1031 yr (with large uncertainties from the superpartner masses) for models with hidden sector breaking, and - 10 14 yr in the models with supersymmetry and G UTs broken in the same sector. The favored modes for proton decay from these dimension 5 operators are usually vK, but models can be constructed-'2 in which v"TT dominates because of cancellations. It is possible to find models in which the dimension 5 operators are absent, but for the most supersymmetric GUTs proton decay should be observable. The cosmological gravitino problem-'3 is a serious problem for supergravity models. Cosmological constraints based on nucleo synthesis and the present energy density conflict with the expected density of primordial gravitinos from the very early Universe if their mass is in the range 1 Ke V- 104 Ge V, which includes the value m 312 - M w favored by most models of radiative SU2 x U 1 breaking. These gravitinos can be diluted to a negligible density by a later period of inflation, but new gravitinos will be produced subsequently if the reheating temperature Tu is too high. It has recently been argued54 that too much D and 3He will be produced by the radiative breakup of 4He (by the photons produced in the gravitino decay process g312 ---'> 'Y + ~unless TR(GeV) < 2.5 x JOH (LOO GeV/m 312). (12) This is much too low to produce the observed baryon asymmetry by the thermal production of heavy Higgs bosons, but as has al ready been described there are some nonstandard scenarios that might get by. The limit in (12) assumes tha the photino is lighter than the gravitino, as occurs in most models. If this is not true then the upper limit on T11 is increased by four orders of magnitude, eliminating the difficulty. Another possible solution of the gravitino problem are the no scale models. 55 These are particular versions of supergravity in which the tree level potential is flat. The supersymmetry breaking 63 and weak scales are therefore determined by radiative corrections = , so that M w m 1 exp[ - 0( I )/ex]. Since only the ratios of other mass scales tom" are observable there are no input scales in the theory and the hierarchy problem is solved! Furthermore, the grav itino mass can be quite a bit different from Mw in the no-scale models. Versions with very light (<< I KeV) and very heavy (> 104 GeV) gravitinos have been constructed;"<' both of which satisfy the cosmological constraints. Also, the strong CP problem is solved in the light gravitino case because there are dynamical degrees of freedom which cause 80 c D to relax to zero. The no scale models therefore have many exciting features. The major problem is that they are based on a very specific and apparently ad hoc assumption concerning the form of the K~ihler potential. which determines the dynamics. Fortunately there is a good chance that they may emerge naturally in superstring theories. 07 SUMMARY The standard SU, x SU2 x U 1 model combined with classical gravity has been remarkably successful in describing essentially all known physics. Nevertheless. it is too complicated and has far too much arbitrariness to be taken seriously as the ultimate theory of nature. Grand unified theories are a very promising step towards the goal of embedding the standard model in a simpler underlying theory. Perhaps their best feature is the fact that the three known interactions relevant to microscopic physics are elegantly unified, and there is no fundamental distinction between quarks, leptons, antiquarks, and antileptons. These features lead to the explanation for charge quantization, proton decay, the possibility of explaining the cosmological baryon asymmetry dynamically, and the success ful prediction of sin26w. With the possible exception of the sin26w prediction, which could turn out to have been a lucky (or unlucky!) accident, it is very likely that these aspects of GUTs will survive. On the other hand. GUTs have not solved the fermion, Higgs/ hierarchy, CP, and graviton problems, and of course proton decay has not been observed at the level predicted by the simplest models. Grand unified theories, or at least some aspects of GUTs, are almost certainly embedded in some still more fundamental struc ture. 64 One possibility is that the fermions and/or Higgs bosons and/or gauge bosons are composites. This is actually a far more drastic proposal than it first appears because, unlike all of the previous levels of compositeness, the binding energies must be comparable to the rest masses of the constituents. Composite fermions could possibly explain the complicated family structure and spectrum of the fermions, and composite Higgs particles could resolve the Higgs problem. It is not clear what the advantages of composite gauge particles are (perhaps they would help with the Higgs problem), and in fact one would have to abandon the many attractive features of gauge theories. In any case, no one has succeeded in constructing a very compelling composite model. The other popular possibility is supersymmetry. Supergravity is the most promising approach for unifying gravity with the other interactions, and there is reason to hope that it can lead to a successful inflationary cosmology. Also, it helps considerably with the Higgs/hierarchy problem. The main theoretical drawback is that the viable models are very complicated. In particular, most versions of supersymmetry breaking require a rather ad hoc hidden sector that is very weakly coupled to ordinary particles. Perhaps everything will be resolved in a ten-dimensional superstring theory, in which all interaction and parameters are in principle determined! PAUL LANGACKER Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19104 References I. F 65 S. L. Glashow, Nucl. Phys. 22, 579 (1961); S. L. Glashow, J. Iliopoulos and L. Maiani, Phys. Rev. D 2, 1285 (1970). 6. M. Gell-Mann. P. Ramond and R. Slansky, Rev. Mod. Phys. 50, 721 (1978). 7. L.B. Okun, M. B. Yoloshin and V. I. Zakharov, Phys. Lett. 1388. 115 (1984). 8. H. Georgi. H. R. Quinn and S. Weinberg. Phys. Rev. Lett. 33. 451 (1974). 9. A. J. Buras, 1981 In!. Symp. on Lepton and Photon lllteractions. p. 636; B. L Ioffc. XXllth Int. Conf. on High Energy Physics, Vol. ll. p. 176; I. Savin. ibid .. p. 251. 10. W. J. Marciano. Fourth Workshop on Grand Unification. eds. H. A. Weldon et al. (Birkhiiuscr. Boston. 1983). p. 13 and Aspen Winter Physics Conference. Jan. 1985. BNL preprint. 11. C. Gcweniger. Neutrino '84 . Dortmund. 1984. 12. J. Rohlf. XX/Ith Int. Conf. 011 High Energy Physics. Vol. IL p. 12; P. Bagnaia and J.P. Repellin. ibid .. p. 21; W. J. Marciano. Ref. HJ. 13. D. V. Nanopoulos and D. A. Ross. Phys. Lett. 1088. 351 (1982); J. Oliensis and M. Fischler. Phys. Rev. D 28. 194 (1983). 14. S. Narison and E. de Rafael. Phys. Lett. 1038. 47 (1981); J. Gasser and H. Leutwyler. Phys. Rep. 87. 77 (1982); L. J. Reinders and H. R. Rubinstein. Phys. Lett. 1458. 108 (1984); B. L. Ioffc. Ref 9. 15. J. Ellis and M. K. Gaillard. Phys. Lett. 888. 315 (1979). 16. J. Bagger et al .. Nucl. Phys. B 244. 247 (1984); Phys. Lett. 1458. 211 (1984); Phys. Rev. Lett. 54. 2199 ( 1985). SLAC-PUB-3441. and references therein. 17. For reviews. see A. M. Bocsgaard and G. Steigman. Bartol BA-85-6; G. Steigman , Inner Space/Outer Space. 18. For a review. see P. Langackcr. Aspen Wi11ter Physics ConferC'nce. UPR-0276T. 19. For a review. sec P. Sikivie. Fourth Workshop 011 Cra11d U11i(icatio11. p. 214. 20. P. Moxhay and K. Yamamoto. Phys. Lett. 1518. 363 (1985) and relerences therein. 21. A. Nelson. Phys. Lett. 1348, 422 (1984); 1368. 387 (1984); 1438. 165 (1984) . 22. S. Barr. Phys. Rev. Lett. 53. 329 (1984); Phys. Rev. D 30. 1805 (1984). 23. G . Segre. Phys. Lett. 1458. 231 (1984). 24. For reviews sec Rd. 4 and M. Goldhaber and W. J. Marciano. Brookhaven preprint BNL 35787; W. Lucha. Vicnn;1 preprint UWThPh-l 984-26; K. Enqvist and 0. V. Nanopoulos. CERN-TH.4066/84. 25. A. Yu. Smirnov, JETP Lett. 31. 737 (1980): P. Frampton and S. L. Glashow. Phvs. Lett. 1318. 340 (1983); and 1358. 515(E) (1984); S. Nandi. Phvs. Lett. 14iB. 375 (1984). . 26. W. J. Marciano. Ref. 10; K. Hagiwara et al .. Phys. Lett. 1418. '1>72 (1984); K. S. Babu and E. Ma . Phys. Lett. 1448, 381 (I 984): J.C. Wu. Pittsburgh prcp1int. 27. See. for example. Y. Tosaetul.. Phys. Rev. D 28. 1731 (1983) and rcl'erenccs therein. 28. C. T. Hill. Phys. Lett. 1358. 47 (1984); Q. Shafi and C. Wctterich. Phys. Rev. Lett. 52. 875 (1984) . 29. For a review. see G. Fidecaro. Fourth Workshop on Grnnd Unification. p. 196. 30. C. B. Dover. A. G;1I and J.M. Richard, BNL-35749. 31. T. W. Jones et al .. Phys. Rev. Lett. 52. 720 (1984). 32. For example. sec P. Majumdar et al .. Phys. Lett. 1378. 181 (1984) and ref erences therein. 33 . For recent reviews see V. A. Rubakov. XX/Ith Int. Con/ on High Energy Physics. Vol. I. p. 79: J . Stone, Inner Space/Outer Space, Michigan preprint UM HE 85-02; J. Preskill, CALT-68-1224 and ARNPS 34. 461 (1984); E.W. Kolb and M. S. Turner. Appl. J. 286 . 702 (1984). 34. C. T. Hill , Nucl. Phys. B 224 . 469 ( 1983). 66 J5. P. Salomonson, D.·S. Slqgcrslam and/\. Stern. Phys. Lcll. iS IB, 243 ( 1985). 6. For n;vicws. ~ce P. J. Slci11 lrnrrl 1, Comments Nud. 1'arl. Phys. XU A, 273 ( 11 84 : i\. D. Linde. XXJ/tli Int. Co11f Hi 1'1 Enngy f'liysit:s. Vol. U, p. 125. 37 . M. R 111::\tlc ll i and D. Wyl ·r. Phys. I.cu. 1338. 325 ( 198~) ; P. Roy ct al. , Phys . Rev. L It. 52, 7 13 ( 1984); Phys. Rev. D 30. 1949 ( 1.984) ; D 31, 2385 (E) ( 1984); and TrFR/TI l-fM-6. -20; O. Shanker. Nud. Phys. B 250, 351 ( 1985) ; .I . liensis and C. H. Albright, ANL-HEP-PR-85-21. 38. For a review, see V. A. Lubimov, XX/Ith Int. Conf. on High Energy Physics, Vol II, p. 108. 39. G. Steigman, Ref. 17. 40. For a review, see E. W. Kolf and M. S. Turner, Ann. Rev. Nucl. Part. Sci. 33, 645 (1983). 41. D. A. Kosower et al., Phys. Lett. 150B, 436 (1985); M. Claudson et al., Nucl. Phys. B 241, 309 (1984). 42. B. A. Ovrut and P. J. Steinhardt, Phys. Rev. D 30, 2061 (1984); Phys. Lett. 147B, 263 (1984). 43. I. Affleck and M. Dine, Nucl. Phys. B 249, 361 (1985). 44. Ya. B. Zel'dovich, L. B. Okun and I. Yu. Kobzarev, Sov. Phys. JETP 40, 1 (1974). 45. T. W. B. Kibble, G. Lazarides and Q. Shafi, Phys. Rev. D 26, 435 (1982); A. E. Everett and A. Vilenkin, Nucl. Phys. B 207, 43 (1982); Phys. Rev. Lett. 48, 1867 (1982). 46. For a review, see P. Langacker, XXl/th Int. Conf. on High Energy Physics, Vol. II, p. 215. 47. D. Cocolicchio and G . L. Fogli, Phys. Rev. D 30, 2391 (1984); J. Maalampi and J. Pulido, Phys. Lett. ISOB. 165 (1985). 48. V. Kuzmin and M. Shaposhnikov. Phys. Lett. 92B, 115 (1980). 49. D. Chang et al., Phys. Rev. Lett. 52, 1072 (1984); Phys. Rev. D 30, 1052 (1984); Phys. Lett. 142B, 55 (1984); Phys. Rev. D 31, 1718 (1985); J. M. Gipson and R. E. Marshak. Phys. Rev. D 31, 1705 (1985); M. K. Parida and J. C. Pati. Phys. Lett. 145B, 221 (1984); T. G . Rizzo, Phys. Lett. 1508, 173 (1985). 50. For reviews. see D. V. Nanopoulos, XX//th Int. Conf. on High Energy Physics, Vol. II, p. 36 and Ref. 4. 51. B. OvrutandS. Raby, Phys. Lett. 138B, 72(1984);8. Ovrut, NATA Advanced Study Institute. Bonn. August 1984. 52. Gluino exchange is important if there are off-flavor-diagonal gluino vertices and a splitting between the u and d scalar quarks, both of which are expected in most models. See J. Milutinovic, P. Pal and G. Senjanovic, Phys. Lett. 140B. 324 ( 1984); S. Chadha et al .. Phys. Lett. 1498, 477 (1984); R. Arnowitt, A. H. Chamseddine and P. Nath, HUTP-84/A094. 53. H. Pagels and J. R. Primack, Phys. Rev. Lett. 48, 223 (1982); S. Weinberg, Phys. Rev. Lett. 48. 1302 (1982). For recent references, see J.E. Kim et al., Phys. Lett. 1458, 187 (1984) and Ref. 50. 54. J. Ellis. D. V. Nanopoulos and S. Sarkar. CERN-TH.4057/84. See also J. Ellis, J.E. Kim and D. Y. Nanopoulos. Phys. Lett. 1458, 181 (1984) and D. Lindley, Fermilab-Pub-84/l 15A. 55. E. Cremmer et al .. Phys. Lett. 133B. 61 (1983); N. P. Chang, S. Ouvry and X. Wu, Phys. Rev. Lett. 51. 327 (1983). 56. J. Ellis. K. Enqvist and D. V. Nanopoulos. Phys. Lett. 1518, 357 (1985) and references therein. 57. E. Witten . Princeton preprint: W. Lang. J. Louis and B. Ovrut. Pennsylvania preprint UPR-0280T. 67