unification of couplings - frank...
TRANSCRIPT
UNIFICATION OF COUPLINGSRecent high-precision experimental results support thepredictions of the minimal supersymmetric 5U(5) model thatunifies electromagnetism and the weak and strong interactions.
Savas Dimopoulos, Stuart A. Raby and Frank Wilczek
Ambitious attempts to obtain a unified description of allthe interactions of nature have so far been more notablefor their ingenuity, beauty and chutzpah than for any helpthey have afforded toward understanding concrete factsabout the physical world. In this article we wish todescribe one shining exception: how ideas about theunification of the strong, weak and electromagneticinteractions lead to concrete, quantitative predictionsabout the relative strengths of these interactions.
The basic ideas in this subject are not new; they wereall essentially in place ten years ago. For several reasons,however, the time seems right to call them back to mind.Most importantly, the accuracy with which the relevantparameters have been determined experimentally hasimproved markedly in the last few months, making amuch more meaningful comparison between theory andobservation possible. The results of this confrontation, aswe shall see, are quite encouraging and suggestive.
Gauge theoriesIt has been traditional to identify four fundamentalinteractions: strong, weak, electromagnetic and gravita-tional. In the 1960s and 1970s great progress was made inelucidating the principles underlying the first three ofthese interactions (see the articles by Howard Georgi andSheldon Lee Glashow in PHYSICS TODAY, September 1980,page 30, and by David Gross, January 1987, page 39).1 Bycomparison the elucidation of quantum gravity is at acomparatively primitive stage. Except for a few remarkstoward the end, our discussion will be confined to the firstthree interactions—the traditional domain of high-energyphysics.
Savas Dimopoulos is a professor of physics at StanfordUniversity, in Stanford, California. Stuart Raby is a professorof physics at Ohio State University in Columbus, Ohio. FrankWilczek is a professor in natural science at the Institute forAdvanced Study, in Princeton, New Jersey.
To make a very long story short, it was found that acommon mechanism underlies all three of these interac-tions: Each is mediated by the exchange of spin-1particles, gauge bosons. The gauge bosons have differentnames in the three cases. They are called color gluons inthe strong interaction, photons in the electromagneticinteraction, and W and Z bosons in the weak interaction.But despite the difference in names and some othersuperficial differences, all gauge bosons share a commonmathematical description and deeply similar physicalbehaviors. Gauge bosons interact with quarks and leptonsin several ways—mediating forces among them, beingemitted as radiation when the quarks or leptons acceler-ate, and even changing one kind of quark or lepton into an-other.
The original and most familiar gauge theory is alsothe most basic. Quantum electrodynamics is properlyunderstood, in modern terms, to be neither more nor lessthan the theory of a single gauge boson (namely, thephoton) coupled to a single charge, or "color" (namely,electric charge). In mathematical language, it is thetheory of the gauge group U(l).
Chromatic terminology for charges is useful andevocative, but must not be taken too literally. Colorcharges are numerical quantities, which may be positiveor negative integers (or zero). The charges associated withdifferent colors are independent quantities. Thus aparticle may carry blue charge + 1 and yellow charge + 1but green charge 0.
The modern theory of the weak interaction is essen-tially the simplest nontrivial extension of this setup, toinclude two colors. An important new possibility for gaugeboson physics first shows up with two colors: In addition togauge bosons that, like the photon, respond to the colorcharges, there are also gauge bosons that change a unit ofone charge into a unit of the other. In this fundamentalprocess (see figure 1), one kind of particle is changed intoanother carrying different color charge. Color charge is
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Exchange of color-changing gauge bosonscan alter the identities of the quarks andleptons involved. Here, the elementaryprocess underlying ordinary radioactivity isdepicted as a process of weak colortransmutation. Figure 1
conserved overall because the difference in charge be-tween the altered particles is carried by the gauge boson.The W bosons are of this identity-altering type, and theirexchange is the mechanism underlying radioactive trans-mutations of atomic nuclei of one element into those ofanother. The Z boson, acting more like the photon,responds to but does not alter the weak color charges. Inmathematical language, the modern theory of the weakinteraction is the theory of the gauge group SU(2)—the 2here just indicates two colors.
Finally quantum chromodynamics, the modern theo-ry of the strong interaction, is—you guessed it—the theoryof three colors, based on the gauge group SU(3). It involveseight gauge bosons (color gluons), six that alter colors andtwo others that merely respond to them.
The color charges involved in the strong and weakinteractions are completely distinct. It has becomecustomary, at least in the US, to call the strong colors red,white and blue. The weak interaction gives us anopportunity to soften the chauvinism of this terminologyto some extent, by adding two new colors: Call themyellow and green. It might seem that to complete thestructure we would need a sixth color, for electromagneticcharge. But most remarkably, it appears that havingidentified the five strong and weak colors, we do not needto add a sixth, separate color for electromagnetism.Electric charge is not independent of the other charges. Ifwe make the color assignments indicated in figure 2(whose true significance will emerge only below), then theelectric charge Q of a particle is given in terms of itsvarious color charges (R, W, B, Y and G) according to thesimple formula
Q= -y3 (1)
Unification: Triumphs and challengesThe fact that all three major interactions of particlephysics can be described using the concept of gauge bosonscoupled to color charges hints at some deeper unity amongthem. So too, with more subtlety and power, does equation1. The strong color gluons mediate all possible changesand responses among the red, white and blue colors, while
the weak gauge bosons do the same between the yellowand green colors. What could be more natural than topostulate the existence of gauge bosons corresponding toall possible changes and responses among all five colors?2
Such bosons would include the color gluons, weak bosonsand photon, and also additional gauge bosons that wouldchange (for example) red charge into yellow charge.Altogether 12 new gauge bosons must be added to the 12known ones. The gauge theory for five colors is denotedSU(5). It includes the SU(3)xSU(2)xU(l) gauge theoriesof the strong, weak and electromagnetic interactions—andmore.
This idea, on cursory examination, suggests two lovelyqualitative successes and two quantitative disasters.
First, the successes. If we consider only the gaugebosons, the expansion of the theory appears as anappealing but quite speculative possibility. While itsuggests the existence of new gauge bosons, it does notshed much light concerning the properties of the ones wealready know to exist. However, if we widen ourconsiderations to include quarks and leptons, a wonderfuladvantage of the larger theory comes into view. Asindicated in figure 2, the 15 quarks and leptons within afamily can be grouped into two classes. One class consistsof five particles, each carrying one unit of one of the fivecolor charges. The other class consists of ten particles,each carrying a unit of each of two distinct color charges.Within either of these two classes, transformationscarrying any given particle into any other one can bemediated by appropriate gauge bosons. In other words,the particles within either class are all related to oneanother by the gauge interaction. They are like differentfaces of a single die—inseparable, symmetrical pieces of alarger whole.
In mathematical terms, the particles fall into twoirreducible representations of SU(5): a five-dimensionalvector representation and a ten-dimensional antisymmet-ric tensor representation. By contrast, when we restrictourselves to the transformations of SU(3) X SU(2) X U(l),the particles in a family fall apart into no less than five dif-ferent classes. This striking gain in economy of descrip-tion is one great qualitative success of the simplest SU(5)unification scheme.
The other success concerns equation 1. This marvel-ous equation, in which the electromagnetic, strong andweak charges all come into play, was an encouraging hinttoward unification. Within SU(5), its potential is brilliant-ly fulfilled. Although it is a little too complicated for us toderive here, it is not terribly difficult to show thatequation 1 is an automatic consequence of unification in
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Quarks and leptons collect into two classeswhen assigned strong and weak colors
inspired by SU(5) unification (a). We label thestrong colors red, white and blue and the
weak colors green and yellow. Readingacross each row gives the SU(5) color charge
for each particle. Subscripts L (left) and R(right) indicate the chirality of the particles,
while the superscripts r, w and b indicate thestandard strong-color labels of the u (up) and
d (down) quarks. Overbars indicateantiparticles (b explicitly displays the
antiparticles of the particles in a). Notice thatleft-handed and right-handed versions of the
same particle may be differently colored—thisreflects the violation of parity in the weak
interactions. Figure 2
SU(5). The photon only fits within this symmetry group ifit responds to precisely the combination of color chargesthat occurs in equation 1. Thus unification offers aframework in which the apparently chaotic spectrum ofelectric charges of quarks and leptons can be understoodrationally.
In a more precise treatment we would actually have toworry about the spectrum of weak hypercharges, which iseven worse. One of us (Wilczek) recalls that as a graduatestudent he considered the now standard SU(2)xU(l)model of electroweak interactions to be "obviously wrong"just because it requires such ugly hypercharge assign-ments. That was going too far, but it still seems fair to callthe model "obviously incomplete" for this reason.
Now we must describe two daunting difficulties thatsuch attempts at unification face. The first disaster is thatthe different gauge bosons, although they do similarthings, do not do them with the same vigor. In otherwords, they couple to their respective color charges withdifferent strengths. The strong interaction, as befits itsname, really is much stronger than the weak interaction,which in turn is slightly stronger than the electromagnet-ic. Thus the perfect symmetry among colors required in atruly unified gauge theory doesn't seem to be in the cards.We shall return to this problem below.
The second disaster concerns the processes mediatedby the extra gauge bosons, particularly the ones thatchange strong into weak color charges. It is always atleast slightly worrisome to postulate the existence ofhitherto unobserved particles, but these fellows areespecially objectionable, because their exchange mediates,through the mechanism indicated in figure 3, processescapable of destabilizing protons. However, protons arerather reliably reported not to decay. Even in 1974, whenunified theories of the type we are discussing were firstproposed, the lifetime of the proton was known to beupwards of 1021 years. Since then, systematic experimentshave raised the lower limit to over 1031 years (for mostplausible decay modes).3 Comparing this to the rates forcomparable weak decays, which are measured in micro-seconds, we realize with a start what an enormity is beingperpetrated—these new gauge bosons must be indeed verydifferent from, and in some sense much less potent than,the old (that is, known) ones.
Nevertheless, given the qualitative successes of gaugetheory unification, and its ineluctable beauty, one mustnot give it up without a fight. And indeed both difficultiescan be overcome in triumphal style.
Let us take the second difficulty first. It is actuallynot so difficult to explain this problem away. To do so, we
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must now mention a very important aspect of gaugetheories that for simplicity we have so far neglected:These theories may exist in different phases and exhibitproperties at low energies that differ somewhat from theirsymmetrical high-energy behavior. For our purposes, themost important point is that gauge bosons may becomemassive, through the so-called Higgs mechanism,4 and theheavier the gauge bosons, the rarer are the processesmediated by their exchange. (The Higgs mechanism is in avery direct sense simply a relativistic version of Fritz andHeinz London's superconducting electrodynamics.)
This, by the way, is why the weak interactions aremuch less prominent than electromagnetism, even thoughthe intrinsic strengths of the weak-vector-boson couplings
Proton decay would be caused by theexchange of some of the extra gauge bosons(X) needed for unification. These bosonswould change strong into weak colors andwould lead to processes wherein three quarkschange into an antilepton. Figure 3
PHYSICS TODAY OCTOBER 1991 27
are somewhat greater than those of photons. The weakvector bosons are massive, which not only makes themdifficult to produce and unstable in isolation, but alsomakes the processes they mediate less vigorous. Clearlythen, to exorcise the specter of the dangerous extra gaugebosons we need only suppose that they are very heavy.
What about the first difficulty? Though perhaps lessdramatic, it is more profound. Its resolution involvesanother order of ideas, and is rich in consequences. Tothis, we now turn.
Running coupling constantsThe crucial concept is that of running coupling con-stants—coupling strengths that vary with energy ordistance. This is very similar to the more familiar andintuitive notion of dielectric screening. In dielectricscreening, a positively electrically charged particle withina material tends to pull negative charge toward it, forexample, by distorting (polarizing) neutral molecules.This nearby enhancement of negative charge shields orscreens the effect of the central positive charge, and so theelectric field at large distances due to that charge is lessthan it would otherwise be (see figure 4).
In modern quantum field theory, a similar effecthappens even in empty space. This is because "emptyspace" is not a true void, but rather a dynamical mediumfull of virtual particle-antiparticle pairs that flickerbriefly into existence and then reannihilate before travel-ing very far. (A less poetic but still visually appealing viewof the same thing is afforded by the Feynman diagrams infigure 5.) These denizens of the vacuum can be polarized,no less than molecules in a solid. As a result the chargeand electric field distributions close to a nominal elemen-tary "point particle" are in fact structured: The charge ispartially screened. The vacuum is a dielectric.
Ordinary dielectric screening tends to make theeffective charge smaller at large distances. Conversely, ofcourse, if we work from the outside in, we see the effectivecharge gradually increasing from what we saw from faraway. Virtual quarks and leptons also tend to screen anycolor charge they carry. This effect turns out to be verygeneral: Spin-V2 and spin-0 particles of any hypotheticaltype screen charge.
The discovery5 in 1973 that spin-1 gauge bosons havethe opposite effect was a wonderful surprise. It means
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Dielectric screening occurs when a charge ina dielectric medium polarizes the moleculesaround it. This cloud of polarization partiallyhides, or screens, the centralcharge. Figure 4
that a charge that looks large and formidable at largedistances can be traced to a weaker (and calculationallymore manageable) source at short distances. The discov-ery of this dynamical effect—known as asymptotic free-dom—led directly to the identification of SU(3) color gaugetheory, or QCD, as the theory of the strong interaction.6
This came about because the SLAC electroproductionexperiments,7 demonstrating the phenomenon of scaling,indicated that the strong interaction between quarks ismuch weaker at short distances than one would infer fromafar. More precisely, what these experiments indicated isthat rapidly accelerated quarks emit few gluons. In otherwords they behave when they are hit hard as if they areideal structureless point particles: They recoil elastically;they have no "give." This behavior is in contrast to theirappearance when hit softly. Then the more powerful,longer-range aspect of the strong interaction causesquarks to behave not like points but more like thick ballsof virtual gluons, quarks and antiquarks.
The logic of the discovery of hard structurelessparticles—Richard Feynman's partons,8 now identifiedwith quarks and gluons—inside the proton in the SLACexperiments is quite similar to the logic underlying theclassic experiment of Johannes Geiger and Ernest Mars-den. Their observation that alpha particles impinging ongold foil may be violently deflected through large angleswas interpreted by Ernest Rutherford as indicating theexistence of hard, effectively point-like nuclei at the centerof atoms. Replacing alpha particles with electrons, andnuclei with partons, we essentially map the Geiger-Marsden experiment onto the SLAC experiment.
Later experiments, as we shall discuss further below,have confirmed and sharpened the early indications fromSLAC. When quarks are rapidly accelerated they usuallypropagate exactly as ideal structureless point particles,but occasionally radiate one or more color gluons instead.QCD gives a detailed quantitative account of thesematters, and has been very successful in predicting theoutcomes of experiments9 (so much so, that experimenta-lists now rely on it to calculate their backgrounds).
Why do these bosons have the opposite effect fromother particles? The mechanism of screening seems soclear and inevitable that its reverse seems implausible.However, it turns out, roughly speaking, that attractivemagnetic dipole-dipole attractive forces between like-charge gauge gluons outweigh their electric repulsion,leading to an accumulation of the same charge—anti-screening!10
In our present context, it is convenient to considerscreening and asymptotic freedom as functions of energyrather than distance. In a sense that can be made precise,in quantum mechanics high energy or momentum corre-sponds to small distance. Roughly speaking, then, screen-ing corresponds to the coupling's increasing with energy,while asymptotic freedom corresponds to its decreasing.
The coupling of SU(3) is more affected by asymptoticfreedom than are the other couplings, simply becausethere are more strong color gauge bosons. It outweighs theeffect of the quarks. For weak SU(2) the competition ismore equal, while for electromagnetic U(l) there is nogauge boson contribution, and ordinary screening wins.As a result the strong coupling decreases at large energies,while the weak stays nearly constant and the electromag-netic increases. But these are just the directions of changethat can cause the couplings to merge!
This whole circle of ideas is beautifully summarized inthe plot of effective couplings against energy or mass scaledue to Howard Georgi, Helen Quinn and Steven Wein-berg11 (figure 6). The energy scale for the running of thecouplings is logarithmic, so it takes a big change in energy
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to see any change in the couplings. Thus the scale atwhich unification takes place will be very much largerthan what we are accustomed to in accelerator physics.
The logic of prediction from unificationA method for comparing unified theories and reality,using the observed strength of couplings, emerges fromcareful consideration of figure 6. On the left-hand side ofthe plot, we have three measurable parameters: thestrong, weak and electromagnetic couplings. On the right-hand side, we have two unknown parameters: the massscale for restoration of the full unified symmetry, and thestrength of the coupling (there's just one!) when thisoccurs. Since the three measurable parameters aresupposed to derive from two more fundamental (but apriori unknown) ones, they must obey a constraint. Theprimary prediction from the logic of unification is anumerical relationship among the strong, weak andelectromagnetic couplings.
The form of this relationship is conveniently ex-pressed in terms of the weak angle #w defined by theexpression tan#w = g1 /g2, where gj and g2 are the couplingstrengths of the U(l) and SU(2) gauge bosons. Thecoupling g2 is directly the coupling strength of W bosons,analogous to the electromagnetic coupling e for photons.The physical photon is a mixture of the fundamental U(l)and SU(2) gauge bosons, and one finds e = g2 sin(9w =S\gil (st + S% )1 /2 • Given the experimental values of thestrong coupling g3 and the electromagnetic coupling e, thelogic of unification allows one to predict the value ofsin20w, which is the quantity experimentalists generallyreport. The precise value predicted depends on thespectrum of virtual particles that enters into the calcula-tion of the running of the couplings. We shall elaborate onthe numerical aspect of these predictions and theircomparison with experiment in a moment.
The predicted constraint on the observed couplings,however, does not exhaust the interest of this calculation.If the observed couplings do obey the constraint, we willalso obtain definite predictions for the mass scale ofrestoration of symmetry and for the value of the couplingat this scale. Together these allow us to predict the massof the dangerous gauge bosons whose exchange destabi-lizes protons and to obtain a rate for proton decay throughthis mechanism, which in principle (and perhaps inpractice) can be compared with experiment. In thecontext of cosmology, these parameters determine thetemperature at which a phase transition from unbroken tobroken unified gauge symmetry occurred during the BigBang.
Slightly more subtle but perhaps in the long run evenmore important for the future of physics is another aspectof the unified coupling and scale. A classic problem ofphysics for the past several decades has been the meaningof the numerical value of the fine-structure constanta = e2/4irfic. This pure number largely controls thestructure of the world (that is, all of chemistry and most ofphysics, as Dirac described the domain of quantumelectrodynamics). Many attempts, ranging from crackpotnumerology to serious efforts by leading physicists, havebeen made to calculate its value from deeper principles.None has succeeded. Unified theories radically alter theterms of this problem but do not remove its substance. Itbecomes, if anything, grander. The fine-structure con-stant no longer appears as a simple primary ingredient offundamental theory. Rather it, together with the strongand weak couplings, derives from the primary unifiedcoupling at short distances by processes of renormaliza-tion and symmetry breaking. The right problem, it seems,is not to try to calculate a but rather to explain why the
Empty space is a dielectric medium inquantum field theory and can screen charge.These Feynman diagrams represent theinteraction between a bare charge and aphoton (a) and the effect of virtual particlescoming between the photon and thecharge (b). Figure 5
unification coupling and scale are what they are.Finally, the value of the scale of unification has
important implications for the eventual reconciliation ofgravity with the other, now unified interactions. We shallreturn to this point below.
Comporison with experimentBut first let us return from these rarefied heights backdown to earth, to discuss experimental measurements andtheir confrontation with models in a more concrete way.
The value of the strong interaction coupling (at somedefinite scale) can be measured in many ways. Perhapsthe most intuitively appealing is to use electron-positronannihilation into hadrons. The fundamental processunderlying the annihilation is production of a virtualphoton, which converts to a quark-antiquark pair (figure7). Of course one does not see actual quarks in thelaboratory, but only the hadron showers or jets theyinduce. At high energies the dressing process, whereby abare quark is converted into physical hadrons, is soft. Thismeans that the hadrons in the jet are all moving in verynearly the same direction as the underlying quark, andthat the total energy and momenta of the particles in thejet add up very nearly to the underlying quark's energyand momentum (figure 7a).
However, there is also a small probability—propor-tional to the strong coupling—for the quarks to radiate ahard gluon, that is, one with substantial energy andmomentum of its own. In that case one should observeevents with three jets, as shown in figure 7b. Such eventsare indeed observed. That their angular distribution is
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Electromagnetic:9
ENERGY OR MASS SCALE
observed to agree with the predictions of QCD providessplendid evidence for the existence of spin-1 gauge bosonscoupled to color charge with the same space-time struc-ture as the coupling of the photon to ordinary charge.Most importantly for our purposes, the ratio of three-jet totwo-jet events gives a direct quantitative measure of thestrength of the strong coupling. Four-jet events are alsoobserved at the expected (low) rate.
Conceptually, the most straightforward way to mea-sure the weak coupling is simply to measure the mass ofthe W boson. Indeed, the rate of all weak processes at lowenergy, including, for example, the easily-measured rateof muon decay, is governed by the ratio of the weakcoupling to the mass of the mediating W boson. (Thisfollows from graphs like that shown in figure 1, using theelementary Feynman rules.) In practice other, morecomplicated measurements, involving the mixing of the Zboson with the photon, are more easily made accurate.
The electromagnetic coupling, of course, has beenknown with extreme accuracy for a long time.
Though no true ambiguities arise if one uses thetheory to calculate physically meaningful quantities,quantitative comparison with experiment requires greatcare. For instance it is not at all trivial to define thecouplings properly and consistently, because they all runand in any physical process there are a variety of ways onemight choose the nominal "scale." Once the experimentalmeasurements are properly translated into values of thecouplings, it becomes possible to confront them with thepredictions of different unified models.
This involves at least two criteria. First, we mustdemand that the constraint on the observed couplings issatisfied. Second, we must demand that the predicted rateof proton decay through gauge boson exchange is not toolarge. Different models of unification will contain differ-ent numbers and kinds of virtual particles, which willcause the couplings to run differently. Therefore, ingeneral, different models will lead to different constraintson the observed couplings and to different values for theunification scale and coupling.
It is good scientific strategy to check the simplestpossibility first. The simplest unified model is the onebased on SU(5) unification as described above. Theminimal version of this model does not require any
The running of the strong, weak andelectromagnetic couplings, extrapolated tovery high mass scales, can result in theirmeeting at a point. This occurs if these threeinteractions, as observed at low energy, allresult from the spontaneous breakdown of aunified theory at a large mass scale. Thegravitational coupling, shown schematicallyon this plot, starts out very much smaller thanthe other couplings but would join them ifextrapolated to about 1019 CeV. Figure 6
unobserved particles with mass significantly less than theunification scale beyond those already needed inSU(3)xSU(2)xU(l) without unification (that is, the topquark and the Higgs boson). This minimal model doesamazingly well by the first criterion. Until quite recentlythe measured values of the couplings did satisfy theconstraint imposed by this minimal unified model, withinexperimental uncertainties. This is a truly remarkableresult and greatly encourages us to think that there ismuch truth captured within this circle of ideas. Theminimal SU(5) unified model has difficulties, however,meeting the second criterion. It predicts too large a rate ofproton decay or, equivalently, too small a unification scale.The predicted scale is roughly 1015 GeV, and the predictedlifetime is roughly 1029 years. This is not quite acceptable,as we mentioned before. On the other hand the predictedproton lifetime is not absurdly short—certainly it is a vastimprovement on a microsecond!—and in fact special,heroic experimental efforts were required to rule it out.
Given these results, it seems wise merely to tinkerwith the basic ideas rather than simply junk them. Arethere compelling alternatives to the minimal unifiedmodel? Do they manage to retain its successes whileremedying its shortcomings?
SupersymmetryOnce we wander from the straight and narrow path ofminimalism, infinitely many silly ways to go wrong lieopen before us. In the absence of some additional idea, justadding unobserved particles at random to change therunning of the couplings is almost sure to follow one ofthese. However, there are a few ideas that do motivatedefinite extensions of the minimal model and are suffi-ciently interesting that even their failure would be worthknowing about.
Surely supersymmetry12 is in this class. In a well-defined sense, supersymmetry is the only possible way tounify the description of particles with different spins.Indeed, it is a symmetry whose basic operation is totransform particles or fields with one spin into otherparticles or fields whose spins differ by the minimal unitfi/2. In the process it transforms bosons into fermions, andvice versa. As yet there is no direct sign of supersymmetryin nature (the developments reported below are probablythe nearest thing so far), and if supersymmetry is relevantto the description of nature it must be broken. However, abroken symmetry can still be rich in consequences if itsbreakdown occurs in a mild and orderly way.
Perhaps the most appealing idea in this direction isthat the breakdown of supersymmetry is spontaneous.13
This means that it remains a valid symmetry of theunderlying laws of physics but is broken in the course ofthe evolution of the state of the universe. This process issimilar to the way the alignment of spins in a ferromagnetspontaneously breaks rotational symmetry as the magnetis cooled through its Curie point: Rotational symmetry isstill valid in a fundamental sense, even in a magnet, but
3 0 PHYSICS TODAY OCTOBER 1991
the stable configurations of spins within the magnet do notrespect it. The fundamental laws have more symmetrythan any of their stable solutions.
Supersymmetry is a necessary ingredient in severalother theoretical ideas. There are many hints that it mayhelp to elucidate the gauge hierarchy problem—that is,the vast difference between the unification scale and thescale of electroweak symmetry breaking.1415 When onepromotes supersymmetry to the status of a local gaugesymmetry, one finds that Einstein's general relativity is anecessary consequence, thus finally bringing that theorywithin the circle of ideas used to describe the otherinteractions of particles.16 And, of course, supersymmetryis a necessary ingredient in superstring theory, the mostpromising concrete approach to unifying the three interac-tions of particle physics with gravity currently known.
Pioneering attempts17 to incorporate supersymmetryinto realistic models of particle physics ran into variousdifficulties. Finally a consistent phenomenological modelwas found, using the idea of soft symmetry breaking.18
Ten years ago, we pointed out that extension of theminimal model to incorporate supersymmetry had impor-tant implications in the context of the ideas discussedabove.15 The most important change suggested by super-symmetry is that one should include many additionalparticles with masses of 104 GeV or less, whose propertiesare predicted with sufficient definiteness to allow ameaningful analysis of their effect on the running ofcouplings. Roughly speaking, one should expect a dou-bling of the spectrum of elementary particles at theseenergies. This doubling occurs because supersymmetrytransforms particles into their superpartners, differing inspin by fill but with closely related couplings, and none ofthe known quarks, leptons or gauge bosons can beidentified with the superpartner of any other. The massestimate for the superpartners is not quite firm, but ifsupersymmetry is to help address the hierarchy problem itseems necessary that its breaking (of which the massdifference between superpartners is a measure) not be toolarge.
What are the effects of adding these superpartners?The main effect is to raise the scale of unification withoutmuch disturbing the successful SU(5) relation amongcouplings. Indeed, the main reason superpartners tend toraise the scale of unification is that the gluinos, the spin-V2partners of the color gluons, partially cancel the asympto-tic-freedom effect of the gluons themselves. Thus it takesa longer run in energy for the biggest difference betweencouplings—the anomalously large strength of the stronginteraction—to get wiped away. On the other hand thegroup theoretic structure of the calculation, which con-trols the ratio of couplings, is not much affected by the newsuperpartners. This is because the new superpartnersoccur in the same symmetrical pattern as their knowncounterparts. (Indeed supersymmetry relates particleswith different spins but the same gauge—that is, color—charges.) Thus, roughly speaking, the running of each
coupling is slowed down by the same factor.Of course, by raising the scale of unification, the
supersymmetric unified models made it seem less certainthat the proton would decay on schedule. As time went onand no decays were observed, it became clear that thismight be just what the doctor ordered.
Until recently it was appropriate to emphasize thatincorporating supersymmetry into simple unified modelsdoes not drastically change the relation that they predictamong coupling constants, since this prediction wasconsistent with the available data. However, smalldeviations from the nonsupersymmetric predictions doexist, because it is not quite true that particles (and theirsuperpartners) occur in a completely symmetrical pat-tern.161819 The "bad actors" are the scalar fields intro-duced to implement electroweak symmetry breaking—theHiggs fields. In constructing the standard electroweakSU(2)X U(l) theory one must introduce a complex doubletof Higgs fields carrying the weak color charges. However,on phenomenological grounds one must not introducetheir counterparts carrying strong color charge. Indeedexchange of the strongly colored Higgs particles candestabilize protons, and it leads to catastrophic rates forproton decay unless the mass of these particles isextremely large. There is no compelling understanding ofwhy the strong-color Higgs particles are so heavy com-pared to their weak-color counterparts; this is one aspectof the gauge hierarchy problem. (Actually what ispuzzling is not so much the heaviness of the strong-colorHiggs particles but the lightness of the ordinary ones.) At
Antiquark Quark \ / Antiquark
Electron-positron collisions at high energycan produce an elementary quark-antiquarkpair, which materializes as two jets ofparticles moving in opposite directions (a).More rarely, the rapidly accelerated quarksradiate a color gluon, which produces a thirdjet (b). Figure 7
PHYSICS TODAY OCTOBER 1991 31
present it seems wise to be pragmatic and simply acceptnature's unequivocal indication that this is so. How doesthis mass difference affect the unification of couplings?
The normal, weak-color Higgs fields influence (to firstorder) only the running of the weak and, to a lesser extent,the electromagnetic couplings. They tend to make thesecouplings increase with energy. The inclusion of fer-mionic superpartners accentuates these effects. Further-more, for technical reasons it turns out that in theminimal supersymmetric model one must introduce notjust one but two weak Higgs doublets. The contribution ofall the Higgs fields and their superpartners to the runningof couplings is quantitatively small compared with thecontribution of quarks, leptons and gauge bosons, butrecent accurate measurements (especially the beautifulresults from LEP20) can resolve the small corrections theHiggs fields are predicted to make for the constraint on thecouplings. Ugo Amaldi, Wim de Boer and HermannFiirstenau21 conclude that the minimal supersymmetricmodel gives an excellent fit to the data, whereas theminimal nonsupersymmetric model is definitely excludedby many standard deviations. Figures 8 and 9 show plotsof effective coupling versus energy in the minimalnonsupersymmetric and supersymmetric SU(5) models,extrapolated from the latest data.
Together with the previous indications from protondecay, these new results provide highly suggestive, ifcircumstantial, evidence for virtual supersymmetry.They also greatly reinforce the case for color unification.A minimal supersymmetric model is certainly not the onlyway to reconcile the existing data with color unification.More complex unification schemes, typically involvingnew particles with exotic quantum numbers and morecomplicated symmetry-breaking patterns, are also con-tenders.22 At the moment these other contenders seemless compelling than the minimal supersymmetric model.
ProspectsIf we take these indications of unification of couplings andvirtual supersymmetry at face value, they both brilliantlyconfirm old ideas in particle theory and augur a brightfuture for particle experimentation. Within the next yearor so the electron-proton collider HERA should begathering data that will both test QCD and refine thedetermination of the strong coupling, constant, whoseuncertainties are currently the most important factorlimiting comparison of theory and experiment. If virtualsupersymmetry is operative well below the unificationscale, Nature would be perverse not to use it inaddressing the gauge hierarchy problem. If Nature is notperverse in this way, the masses of the superpartnerscannot be too large, and real supersymmetry should notelude the next generation of acclerators (CERN's Large
102 104 10 10s 1010 10':
MASS SCALE (GeV)101 1016
The most recent measurements of the low-energy couplings (a, =gf /4-rrfic) clearly fail tomeet at a point when they are extrapolated tohigh energies by computations incorporatingthe particle content of the minimalnonsupersymmetric SU(5) [or simplyS U ( 3 ) x S U ( 2 ) x U ( D ] model . This indicatesthat the observed low-energy couplings arenot consistent wi th a unified model having justthis particle content. The thickness of thelines indicates the experimental uncertainties.(Adapted from a figure provided by UgoAmaldi , CERN.) Figure 8
Hadron Collider or the SSC).There is also good news regarding the search for
proton decay. Taken at face value, the best fits of minimalsuperunified models to the couplings predict a unificationscale of about 1016 GeV and a proton lifetime of about 1033
years through gauge boson exchange.21 This lifetime isslightly outside the reach of existing experiments, but nothopelessly so. In supersymmetric models there areadditional mechanisms for proton decay—involving decayinto virtual scalar quarks—that do not occur in nonsuper-symmetric models.23 The rate of decay through thesemodes depends on details of aspects of the models that arepoorly understood, and so it cannot be predicted withprecision. However, in a wide class of models these modesdominate the decay and lead to extremely unusual finalstates.24 Thus if the proton is ever observed to decay, thenature of its decay modes may give strong clues as to thenature of the unified theory underlying its demise.
Finally, we would like to make some simple observa-tions about how gravity might fit into this picture. Thecoupling of the graviton may also be considered to run, andmuch faster than the other couplings (see figure 6).Because it is characterized by a dimensional coupling—Newton's constant—rather than the dimensionless cou-plings that characterize the other interactions, it increases(to a first approximation) linearly with energy, ratherthan logarithmically. However, it starts out so small thatits extrapolation only meets the other couplings atapproximately • 1019 GeV, the Planck mass. This iscomparable to, but definitely greater than, the unificationscale. An important implication of this is that gravitation-al corrections do not drastically affect the running ofcouplings at or below the unification scale, on which thepreceding discussion was based. The ratio of thesescales—the Planck mass and the grand unification scale—is another fundamental dimensionless number, whosecalculation presents an inspiring challenge to theoreticalphysics.
3 2 PHYSICS TODAY OCTOBER 1991
104 106 10s 1010 1012
MASS SCALE (GeV)1014 1016 101
Minimal supersymmetric SU(5) model doescause the couplings to meet in a point. Whilethere are other ways to accommodate thedata, this straightforward, unforced fit isencouraging for the idea of supersymmetricgrand unification. (Adapted from a figureprovided by Amaldi.) Figure 9
We wish to thank Ugo Amaldi, Wim de Boer, John Ellis, HermannFiirstenau and Luis Ibanez for valuable discussions.
References1. Other good semipopular introductions to many of the main
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