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Grand Unification

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Grand Unification 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 <P. which should have mass2 parameters µJ, that are not too much larger in magnitude than M~. However, µ,T, receives quadratically divergent corrections from gauge, Higgs, and fermion loop diagrams of order (I) where g, '/\,and h represent gauge. quartic Higgs. and Yu­ kawa couplings. respectively. and A is the next highest mass scale in the theory abov~ the weak scale (at which the other­ wise divergent integrals are presumably cut off). In the standard model the only higher mass scale is the Planck mass m,, = GN - 112 = 10 19 GeV. Hence, an incredibly accurate cancellation (fine-tuning) between the bare value of µ,T, and the correction is needed. (d) The strong CP problem: CP and T violation associated with nonperturbative instanton effects in the strong (QCD) sector of the theory will lead to an unacceptably large value for the neutron electric dipole moment unless a parameter 8oco is fine-tuned to a value < 10 - 9 ( e) The graviton problem: The standard model does not incor­ porate quantum gravity. 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.
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