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Contemporary , 2000, volume41, number6, pages359± 367

Supersymmetry:what? why? when?

GORDON L. KANE

This article is acolloquium-level review of the idea of and why so many expect it to soon be amajor discovery in physics. Supersymmetry is the hypothesis, for which there is indirect evidence, that the underlying laws of are symmetric between () such as and , and particles () such as and .

1. Introduction (B)In addition, there are anumber of questions we The of [1] is aremarkably hope will be answered: successful description of the basic constituents of matter (i) Can the of nature be uni® ed and (quarks and ), and of the (weak, simpli® ed so wedo not have four indepen- electromagnetic, and strong) that lead to the structure dent ones? and complexity of our world (when combined with ). (ii) Why is the of the Standard It is afull relativistic ®eld . It is now very Model SU(3) ´SU(2) ´U(1)? well tested and established. Many experiments con® rmits (iii) Why are there three families of quarks and predictions and none disagree with them. leptons? Nevertheless, weexpect the Standard Model to be (iv) Why do the quarks and leptons have the extendedÐ not wrong, but extended, much as Maxwell’s they do? equation are extended to be apart of the Standard Model. (v) Can wehave aquantum theory of gravity? There are two sorts of reasons why weexpect the Standard (vi) Why is the much Model to be extended. smaller than simple estimates would suggest?

(A)It does not describe some aspects of nature: Some of these issues are partially or fully explained by (i) The electroweak ( SU(2) ´U(1) symmetry of supersymmetry. Some require the full apparatus of the Standard Model must be broken to allow theory (which probably implies supersymmetry). We will quarks, leptons, and Wand Zbosons to have examine several of these in the following. masses. That breaking cannot be explained, The Hamiltonian, or equivalently the Lagrangian, of the or originate within the Standard Model Standard Model is under some symmetry opera- (though its e Vectscan be parameterized in tions (as mentioned above). In addition to the - the Standard Model), sosome other physics , these include some internal symmetries. For must be present to induce it. these internal symmetries the theory is unchanged if certain (ii) The Standard Model cannot explain the fact particles are interchanged, such as the left-handed that the is made almost completely and its , e m ,or the left-handed up and down L $ e of matter and not , rather than quarks, u d .(In the Standard Model, are L $ L equal amounts of both. classi® ed in SU(2) representations di Verently depending on (iii) The Standard Model cannot describe the cold the handedness, i.e. left-handed (right-handed) for anti- of the universe. parallel (parallel) to . They then interact di Ver- (iv) Studies of have shown that at least ently also. This accounts for violation.) We will see one neutrino has . that supersymmetry implies the interchange of particles too. The perturbative Standard Model theory can be sum- Author’saddress: Department ofPhysics, Randall Laboratory, University marized by asmall setof vertices. Let f = e,l,s ,d,s,b,u,c,t 6 6 6 6 ofMichigan, AnnArbor, Michigan 48109-1120, USA. and ` = e ,l ,s ,U = u,c,t,D = d,s,b and m` = me ,ml,ms .

Contemporary Physics ISSN 0010-7514 print/ISSN 1366-5812 online Ó 2000 Taylor &Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/00107510010001644 360 G. L. Kane

W,Zare the electroweak gauge bosons, c the , and bosons are interchanged in the theory (i.e. in the grepresents the eight gluons that mediate the strong force. Lagrangian). It is remarkable that all of the constraints Then all the phenomena in nature involving fermions that of arelativistic quantum ®eld theory, and consistency with wesee are described by gravity plus the four vertices (for the interactions of the Standard Model+ gravity, can be simplicity wedo not show vertices with Higgs bosons satis® ed when bosons and fermions are interchanged in the (introduced below) since their main e Vectsare to allow a equations. consistent description of mass). Atpresent supersymmetry is still ahypothesis. It was ®rst formulated [2] in 1970 ±71, then demonstrated towork a for four-dimensional relativistic quantum ®eld in 1973 ±74. Its most striking implication is that each of the known particles, each of the quarks and leptons and gauge bosons, must have an associated partner (a `’) that diVersonly by bosons fermions. If the supersymme- $ try wereunbroken, an electron would have apartner that had spin zero but had the mass and electric and weak interactions of the electron. There is indirect evidence (which will be described below) that nature is indeed supersymmetric. None of the has yet [3] been directly observed. Though wecould have been lucky and detected one or more already, the ®rst accelerator that explores amajor part of the region where weexpect to ®nd the superpartner is the upgraded at Fermi National Accelerator Laboratory () westof Chicago. It will begin to take data in 2001. b Aconvenient nomenclature exists to talk about the superpartners. Each superpartner is written with atilde: e $ ~e, c c~,etc. Standard Model fermions get apre-s, and $ Standard Model bosons asu Yx-ino, so selectron, squark, . Assomeone has said, wehave acomplete slanguage. There are also new interactions. Each of the Standard Model vertices above leads to new vertices obtained by replacing Standard Model particles by their superpartners in pairs (changing an even number is obviously required to Asalways with Feynman diagrams, the vertices are conserve ). Thus to the Standard Model combined into all possible Feynman diagrams. For vertex example, the process e+ e Ð ®uuhas acontribution from joining two of these vertices, Each diagram represents apossible physical process. The rules of quantum theory assign the probability for each process to occur. We will seebelow that supersymmetry adds some vertices.

2. What is supersymmetry? In the Standard Model quarks and leptons (both fermions) are the matter particles. The particles that mediate the forces are the `gauge bosons’, c , W, Z, g.Another particle, the Higgs , is introduced to allow aconsistent treatment of the mass of particlesÐ its role is described below. Fermions and bosons are treated very di Verently in quantum theory and in the Standard Model. Supersymmetry is the idea that at the fundamental level the basic laws of nature are invariant if fermions and Supersymmetry:what? why? when? 361

To ®nd the probability of any process involving super- many experiments. Similar remarks could be made about partners one draws all diagrams starting with the relevant other superpartners. initial particles leading to the desired ®nal particle and Broken symmetries are familiar in physics. The electro- calculates the associated probabilities by the normal weak symmetry is spontaneously broken by the Higgs procedures. Each new vertex has the same mechanism in the Standard Model (as mentioned in point strength as its Standard Model counterpart. For example, A(i) above and described further in the next section). the vertex eec is of strength e(the magnitude of the electric Spontaneous breaking does not alter the existence of the charge), and therefore so are the two associated vertices ~e~ec of the particles. The vertices and e~ec~ are also unchangedÐ that is, the Lagrangian of the theory All physics ideas and theories in the past have been is unchanged, but the ground state (the here) does invented to explain data or puzzles or theoretical incon- not respect the symmetry. Afamiliar example of an sistencies. Remarkably, supersymmetry was introduced for analogous system is apermanent magnet. The magnetic purely theoretical reasons. Originally no one imagined ®eld points in some direction, breaking the symmetries the supersymmetry was crucial, or even relevant, to explaining material had before the material cooled into the ground how nature works. Inthe middle 1970s supersymmetry was state. Thus the superpartners can get mass not only from sometimes spoken as `a solution in search of aproblem’.It the electroweak but also from super- was abeautiful mathematical theory, but with no known symmetry breaking. This implies that in looking for the connection to reality. superpartners, and in describing the indirect evidence for Then, as people studied the theory, they realized that it them, wecan assume weknow all their properties except provided explanations for several major physics issues, and their masses. Once superpartners are directly detected, led to new approaches to others. In the `Why’ section below understanding the origin of supersymmetry-breaking will wewill examine some of these. become the central problem of the ®eld, just as under- Supersymmetry can also be viewed as amanifestation standing the origin of the Higgs physics became the central of the quantum, fermionic structure of space-time. The problem of the Standard Model after the 1970s. theory extends to include fermionic coordinates in addition to the usual bosonic coordinates. That is, from the point of view of quantum theory, our 3. Why? familiar space-time coordinates commute and are bosonic Although there is not yet direct evidence for the existence operators. Newcoordinates h a are `superpartners’ of the of superpartners, nearly 10 000 papers have been written usual xl ,leading to aformulation of the theory in on supersymmetry. Supersymmetry is usually viewed as `’.These are not related to the default candidate for physics to strengthen the the extra dimensions of . The latter, while foundations of the Standard Model and extend it. In probably extremely small, can be thought of as tangible this section weexamine the reasons for the widespread extra dimensions. The superspace dimensions are fermio- acceptance. nic and eVectively of zero size. If nature is indeed To understand some aspects of the following results we supersymmetric the study of the superpartners will also need to recall that coupling strengths for any interaction in be astudy of the quantum nature of space-time. If one arelativistic quantum ®eld theory change as the interac- imagines the original formulation of the theory to be in tions are studied at di Verent distance scales or or superspace, then supersymmetry requires the existence of momentum transfers. This e Vectis well understood and fermions. veri® edÐ for example, the ®ne structure `constant’, a, takes Since the supersymmetry transformations a Vect spin, the well-known value a »1/137 at low energies. Aquantum and spin is part of angular momentum, and the generators theory calculation of the value of a at the scale of the Z of angular momentum transformations are part of the boson mass predicts a»1/128, and experiments involving PoincareÂgroup, which is connected to gravity, super- the Zagree with this value to good accuracy. (Appendix I symmetry has physical connections to gravity. The precise has abrief explanation of this.) way this connection enters the theory is not yet clear, but one reason supersymmetry is an attractive property of a theory is that its presence suggests there will be ways to 3.1. Uni® cation of forces unify the Standard Model forces with gravity. Since Coulomb showed that the electrical force had the Supersymmetry cannot be an unbroken symmetry, or the same form as the gravitational force, many physicists have superpartners would already have been detected. That’s thought about how to unify the basic forces, perhaps into obviousÐ if there werea spin-zero particle with the mass one that manifests itself in several ways as the forces we and of the electron it would have formed apparently see. In the Standard Model all the forces arise bosonic and been easily produced and detected in by the same gauge invariance mechanism, so the idea they 362 G. L. Kane could be related is reinforced. Afterthe Standard Model 3.2. Higgs physics was formulated, theorists studied how the forces behaved Probably the most important result explained by super- when their strengths werecalculated at shorter distances, or symmetry is the `Higgs Physics’ of the Standard Model. higher energies, and found that the Standard Model forces The theory of the Standard Model appears to be correct approached one another in strength, though they did not only if all quarks and leptons and gauge bosons have zero actually meet. In the early 1980s it was realized that if mass. This result follows from the symmetries of the superpartners wereincluded in the calculation the forces Standard Model. The SU(2) symmetry says the theory is came together rather precisely, at asomewhat higher invariant under interchanges of whatever particles are energy, about 2 ´1016 GeV. Atthat time the measured placed in SU(2) representations, for example the electron values of the coupling strengths werenot very accurately and the , or the up and down quarks. Such known, so data did not distinguish between the Standard an invariance can only hold if the interchangeable particles Model and the supersymmetric Standard Model versions. have the same mass. It turns out that to hold in all cases, Data from LEP in the early 1990s showed that the the only mass they can have is zero mass. supersymmetric version led to asigni® cantly better But particles do have mass. Itis the interaction with the uni® cation. Higgs ®eld that allows all particles to have mass, and each a This uni® cation occurs at ascale somewhat below the diVerent mass. The Higgs ®eld is assumed to have anon- scale, where gravity is expected to become strong. zero value in the ground state of the universe (called its Atthat scale, the strength of the gravitational force is , vev), and particles that interact similar to the strength of the other forces. [From the with the Higgs ®eld obtain mass proportional to that vev. universal quantities h, c, and G (Newton’sconstant) one Technically this works perfectly well. The vev of His can construct quantities with units of length, time, and argued to be non-zero as follows. Suppose the potential mass or energy. These are then expected to be the natural energy of H is V = l2H2+ kH4. k must be positive for the units for the most basic theory. They are ( hG/c3)1/2» potential not to be unbounded from below. Take l2 to be 10 Ð 35 m, (hG/c5)1/2»10 Ð 44 s, and (hc/G)1/2»1019 GeV. negative. Then the potential looks like this, ®rst pointed these units out, soon after he de® ned h.The gravitational force between two particles, e.g. having the Planck energy, is about equal to the V electrical force between them, encouraging us to expect that at these energy or distance scales the forces are related.] Thus it is very encouraging that in asupersymmetric world it may indeed be possible to unify all the forces. This uni® cation works if the superpartners are but m2 > 0 not if they are too heavy. Unfortunately, it is not very H accurate, so it implies they have to be lighter than about ten 0 m2 < 0 times the masses of W,Z,and the top quark, but doesn’t distinguish between superpartners with masses about the same as the Zand those several heavier. This uni® cation says that physics is simpler at ascale 16 >~ 10 GeV, even though nothing in the Standard Model requires that to happen. It has two major implications. First, unless it is simply an irrelevant accident, it strongly suggests that nature is indeed supersymmetric, with super- partners having masses below aTeV. Second, because the and the ground state will indeed occur for avalue of H ®eld theory calculations of the ways the force strengths vary diVerent from zero. are perturbative calculations (through two loops), it Conceptually this approach is unsatisfactory. We don’t 2 suggests that as the energy scale is increased no strong know the physical origin of the Higgs ®eld, or why l < 0. non-perturbative e Vectsenter that would cause the Supersymmetric theories can explain this `Higgs Physics’. couplings to run di Verently. It does not imply, contrary The Standard Model has no scalar ®elds apart from the to what is sometimes stated, that there is adesert as energy assumed Higgs ®eld. Supersymmetric theories, on the other increases, but only that whatever is in the desert (e.g. right- hand, automatically have scalars, and uni® ed supersym- handed heavy neutrinos, additional quarks and leptons metric theories naturally have scalars with the properties of perhaps with di Verent SU(2) ´U(1) properties, etc.) does the Higgs ®eld. More importantly, in the supersymmetric 2 not interact strongly with our three families and the l is not assumed to be negativeÐ rather, that is bosons. derived! Supersymmetry:what? why? when? 363

That derivation works as follows. The basic input is that wherever it is. Thus the Standard Model is really nature is simpler near the uni® cation scale, as described in incomplete as abasic theory. the previous section. The various masses of superpartners, In the late 1970s it was noticed that supersymmetry could etc., do not have any special values. But just as coupling provide asolution to this problem. In the supersymmetry strengths change when examined at di Verent distance or case, the particles and their superpartners both enter the energy scales, so do masses. In the supersymmetric case the loops. The standard sign of fermion loops in ®eld theory is entire is calculable from the basic opposite to that of boson loops, so if the particles and Lagrangian. One identi® esthe (mass) 2 quantity that superpartners have the same masses and couplings (as they corresponds to the l2 parameter of V = l2H2+ kH4, and do in the supersymmetric limit), their contributions cancel calculates how that quantity changes as it is examined at exactly. When the supersymmetry is broken, the divergent 2 2 2 2Ð 2 2 lower energies. The appropriate quantity indeed goes K /Mh is replaced approximately by g (m~ m )/Mh , negative, and the shape of the e Vective potential looks like where m~ is asuperpartner mass, m is atypical Standard the one sketched above. This automatically happens at Model mass, and g the electroweak coupling strength. If m~ energy scales in the tens or hundreds of TeVregion as it is not very large, such corrections eliminate the hierarchy must for the results to numerically, so long as the problem. This is yet another clue that the superpartner spectrum contains afermion signi® cantly heavier than the masses are not too large, but again it is too qualitative to W,which it does (the top ). Further, in the tell us precise predictions for the masses. It suggests the supersymmetric case the quantity corresponding to k in V masses should be below aTeV, perhaps much below. is determined and the shape of Vcan be calculated approximately. From that shape one can predict the mass of the lightest to be about 90 6 40 GeV in a 3.4. Cold Dark Matter large class of models. In the most general cases that mass In the late 1970s it was realized that the vertices of the can get somewhat larger. supersymmetric Standard Model, as written above, allowed When this derivation of the was ®rst all superpartners todecay into lighter ones, except that the recognized in the early 1980s, the mass was not lightest superpartner (LSP) was expected to be stable by known. It was thought to be afewtimes the bmass, while it energy conservation. Soon thereafter people calculated the turned out to be about 35 times the bmass. This successful relic densities of various possible choices for the LSP, and prediction that mtop would be large reinforces our saw that in typical models the LSP could provide the Cold con® dence that this is the correct explanation of the Higgs Dark Matter of the universe. At that time astronomers mechanism. knew that dark matter was needed to understand the It is very important to understand that the electroweak amount of matter in galaxies and galaxy clusters, but did symmetry-breaking of the Standard Model cannot be not know if dark baryonic matter such as Jupiters could explained or generated within the Standard Model. Some provide what was needed. That the LSP could provide the new physics is required. The success of supersymmetry in Cold Dark Matter was aprediction of supersymmetry. providing the explanation for the breaking of the electro- The relevant calculations are qualitatively straightfor- weak symmetry via the Higgs mechanism is the most ward. Just after the all kinds of particles are in important reason why weexpect supersymmetry to be part equilibrium. All heavier ones decay as the universe cools, of our description of nature. Every e Vort to consider new leaving stable particles ( c s, ms, uand dquarks, electrons, physics beyond the Standard Model should be evaluated in and LSPs). The c s and msremain, but comprise only atiny termsof its ability to explain the breaking of the part of the energy density of the universe. The quarks bind electroweak symmetry. into protons and , which in turn bind into nuclei, which combine with electrons to make atoms. Matter composed of and electrons provides at most afew 3.3. The `’ per cent of the energy density of the universe. All There is another conceptual problem for the Standard superpartners originally present end up as an LSP. Some Model. If one calculates loop corrections to the Higgs of the LSPs annihilate each other as the universe evolves. If boson mass in the Standard Model, or the Standard Model the masses of the superpartners (including the LSP mass) extended to include most kinds of new physics, the resulting wereknown, the number that annihilate could be calculated integrals are in® nite and the theory is not de® ned. If one since the Lagrangian of the supersymmetric theory is says that some new physics enters so one can de® ne the known. Using models for the masses leads to estimates for theory by parameterizing the new physics, one ®nds the number of relic LSPs. Multiplying that number by their 2 2 corrections of order K /Mh , where K is the scale of the mass gives an estimate of the LSP energy density. One ®nds new physics. Such quadratic corrections essentially move anumber consistent with the LSP providing all the Cold the Standard Model scale up to the new physics scale, Dark Matter of the universe, though the range resulting 364 G. L. Kane from the calculation is large. Thus supersymmetry provides non-supersymmetric versions led to predicted decay acandidate for the Cold Dark Matter of the universe. The rates that weretoo rapid compared to data, but the Standard Model cannot account for the Cold Dark Matter. supersymmetric ones are not in contradiction with experi- To con® rmthat the LSP is indeed the Cold Dark Matter ment. the LSPwill ®rst have to be observed directly. That will Another is the goal of understanding why the universe is require acombination of experiments. If superpartners are made of baryons rather than an equal mixture of baryons discovered at , the LSP will have been discovered in and anti-baryons. This is afundamental property of the laboratories on Earth. But there are two subtleties. LSPs universe that the Standard Model cannot explain. There is could appear to be stable at colliders, but still not live as not aunique, agreed on explanation; rather, there are long as the universe before decaying (it only takes afew several approaches. All of them require supersymmetry. billionths of asecond to traverse adetector). Several kinds Athird is in¯ation, avery rapid and large expansion of of experiments are searching explicitly for the relic LSPs, the universe before the big bang. Before in¯ation is fully and could ®nd asignal in the next fewyears. There are accepted as part of the explanation for the observed three kinds of searches. Perhaps the most promising uses properties of the universe, it will be necessary to identify the the fact that LSPs would be spread uniformly throughout physical particles whose interactions led to the expansion. the galaxy. Aswe move in the galaxy anucleus in adetector Supersymmetry provides scalar ®elds that can play this would occasionally recoil from an LSP and deposit tens to role, and supersymmetry-breaking is expected to give hundreds of keV in the detector. Clever detectors have been masses to the scalars of string theory, to determine the developed to observe such signals. The other two potential that ®xes the dynamics of in¯ation. This is a approaches use the decay products from LSP . newer area of activity, and apromising one. Detectors in balloons, and one (acronym AMS) that will ¯y Afourth is CPviolation, asmall ( ~10 Ð 3)but extremely on the space station, look for an excess of or important eVectsuch that particles and their CPconjugates that could arise from LSP annihilation. The have slightly di Verent interactions. The Standard Model other uses the fact that LSPs, being heavy and only can describe CPviolation in the system, but it cannot interacting weakly, will concentrate at the centre of the sun explain the , which requires CPviola- and the earth. Then LSP sometimes lead to tion. We have recently shown that supersymmetry can energetic neutrinos, which then interact in the earth provide auni® ed explanation of all CPviolation. The to produce which in form can be detected in large Standard Model CPviolation is not required. Whether the underground detectors. supersymmetric version of CPviolation is correct can be In addition, all of the supersymmetry parameters that tested at b-factories. If this is avalid approach, it has enter the calculation of the relic density of the LSP will interesting implications for how string theory relates to the have to be accurately measured so that the relic density can real world. be determined to indeed be essentially equal to the actual Cold Dark Matter. When LSPs are initially discovered it should be possible to estimate their relic density toafactor 3.6. Successful predictions from supersymmetry of perhaps afew. When the parameters are well-measured it Since superpartners have not yet been directly observed, should eventually be possible to calculate the amount of sometimes people say supersymmetry has not yet made LSP Cold Dark Matter to better than twenty per cent successful predictions. But in fact at least three experi- accuracy from laboratory data. mental results werepredicted by supersymmetry. One is the heavy top quark. In the early 1980s people weresearching for top quarks afewtimes heavier than the 3.5. Newapproaches from supersymmetry b-quarks, 15 GeVor so. Amajor justi® cation for the The three examples just described are the clearest where Tristan collider was to ®nd the top quarkÐ it could only supersymmetry explains phenomena that the Standard search up to 30 GeVtop mass. Asmentioned above in the Model cannot explain, or provides stimulating new explanation of the Higgs mechanism, supersymmetry approaches to basic questions. There are several more provides this explanation if the top quark is heavy, where areas where supersymmetry may explain basic phenomena, heavy means compared to MW .Uncertainties in other but results are not yet strong enough to be sure. Aswith its quantities meant the prediction could not be preciseÐ could earlier successes, supersymmetry was not invented to not distinguish between (say) 1.2 MW and 2 MW , but Mtop explain any of these phenomena. had to be large compared to the typical estimates of the One is the rate at which protons decay. In some versions time, as it indeed was. of uni® ed theories quarks can make transitions into leptons Asecond prediction is the value of the quantity called 2 and vice versa. Then the quarks in the proton can turn into sin h W (essentially determined by MW /MZ )in the Stan- lighter leptons, and the proton can decay. The simplest dard Model. From the Standard Model perspective, Supersymmetry:what? why? when? 365

2 4. When will Higgs bosons and superpartners be detected? sin h W is an unexplained parameter. In auni® ed theory 2 sin h W can be predicted. The uni® ed Standard Model Since supersymmetry is abroken symmetry, wedo not 2 theory predicted sin h W =0.215, while the uni® ed super- know the masses of the superpartners. We do know how 2 symmetry theory predicted in 1982 that sin h W = 0.231, they interact, so for agiven choice of masses wecan consistent within errors with the value later measured at the calculate the production cross-sections and decay branch- CERNLEP collider, in Geneva, , and the ing ratios. Consequently, wecan systematically look for SLAC SLC collider in Stanford, CA. signals of superpartners at any collider, and setlower limits Athird prediction was made for the LEPstudy of 20 on masses if no signal is found. million Zdecays and precision measurements. All super- The current limits are not very strong. and symmetry eVectscould only enter through loop corrections. squarks feel the strong force, so their production cross- The loop corrections could only be large enough to lead to sections are largest. But model-independent limits for them deviations from the Standard Model if the superpartners are only about the top quark mass. Since superpartners werelight. Therefore either the superpartners would be obtain mass from both the supersymmetry-breaking and light enough to directly detect at LEP, or no precision the electroweak symmetry-breaking, it is not surprising if measurements should deviate signi® cantly from their the squarks and gluinos are heavier than the top quark. Standard Model values. Other approaches to extending Limits for partners of Ws and sleptons are only alittle the Standard Model generally suggested that large devia- larger than MW ,again not surprising. There are no general tions should be seen. The superpartners werenot directly limits for the electrically neutral partners. In speci® cmodels produced, and no signi® cant deviations from the Standard the limits can be stronger. Model wereobserved, consistent with the supersymmetric What superpartner masses should weexpect? There is prediction. only one aspect of supersymmetry that can quantitatively normalize the masses for us, and even this requires some interpretation. That aspect is the derivation of the Higgs 3.7. Window on the Planck scale mechanism from supersymmetry. Since the Wand Zobtain If nature is indeed supersymmetric, the most important mass from the Higgs mechanism the result of that consequence will be to provide awindow to seethe Planck derivation is an equation of the form: scale. We expect afundamental theory that describes the 2 ~2 most basic law(s) of nature to be formulated at the Planck MZ Cimi ˆ i scale. But wedo experiments at the `collider’ scale, many orders of magnitude lower in energy, or larger in distance. Here m~i are various superpartner masses, and Ci are How can wetest the formulations of the ultimate theory? coeYcients that are calculated; some Ci are larger than one, We have learned that our description of the physical world and some are negative. can be organized in termsof `e Vective theories’,each for If all the m~i wereindependent, in order to avoid taking part of the picture, e.g. . Aswe probe smaller diVerences of larger numbers (`® ne tuning’)wewould distances, wemove to the theory of the nucleus, of protons expect several m~i to be of order mZ itself, so that signals and neutrons, of quarks and leptons (i.e. the Standard should be in the reach of present colliders, and surely Model). Then there is alarge gap to the ultimate theory accessible to the upgraded Fermilab that takes near the Planck scale. In this e Vective theory sense Ithink a data in 2001. However, when there is atheory of super- good name for the ultimate theory is the `primary theory’, symmetry-breaking it is expected that there are relations since all the e Vective theories coexist together yet this one among the m~i,socancellations may occur. We have studied will be the most basic. this by constructing models where the m~i are all related, and Most eVective theories break down at higher energies or studying how long the m~i can be made without invoking too shorter distances. Supersymmetry has special properties so much ®ne tuning. While the results are somewhat that it can remain valid down to the primary theory. Ifthe subjective, we® nd that after gathering luminosity for a world is supersymmetric, it will be possible to write fewyears the upgraded TeVatron should de® nitely have theories near the Planck scale and make predictions for observed asignal if supersymmetry indeed explains the colliders and rare decays, or conversely write the Higgs mechanism. Ofcourse wecould be lucky and one Lagrangian based on data at the collider scale and could be found in 2000 at LEP, or early at Fermilab. The extrapolate to near the Planck scale. With supersymmetry Large Collider, LHC, under construction at wecan expect to have extensive guidance from experiment CERN, will be asuperpartner factory. toward formulating the primary theory. That does not, of There is an analogous situation for Higgs bosons. Since course, mean that nature is supersymmetric, but it does light Higgs bosons are anecessary consequence of super- help understand why many physicists are excited by the symmetry, they must be detected. Studies show that they possibility. can be found at Fermilab too, though there is asignal/noise 366 G. L. Kane problem soit may take several years of running to observe uni® ed special relativity and quantum theory to predict the them. LHCwill be aHiggs factory, though observation will existence of anti-particles, adoubling of the number of be challenging. Again, with luck aHiggs boson may be particles. Luckily the was light, and it was found at LEP in 2000. observed afewyears later. Suppose the positron had not Unfortunately, the very nature of supersymmetry implies been light. Toobserve antiprotons anew facility had to be that the nature of the signal for superpartners will not be an built. They wereobserved 27 years after the prediction of unambiguous, clear e Vect. That is because the superpartners antiprotons. must be produced in pairs (seethe vertices described earlier), Supersymmetry is also abeautiful theory. It uni® esbosons so after decays two LSPs occur in every . The LSPs do and fermions, forces and matter, and also predicts a not have colour or electric charge, so they only interact doubling of the number of particles. It was written for weakly, essentially as neutrinos. Therefore, they escape the relativistic quantum ®eld theories about 1974. If weadd 27 detectors. That means that the missing energy for super- years, wewould be right on schedule if superpartners were partner events will be larger than for Standard Model detected in 20001 ±2003 at Fermilab (or even before at LEP). events, which is one of the main signals for supersymmetry. It also means that because the energy is carried away by two particles rather than one it is very di Ycult toreconstruct any Acknowledgements that show dramatic e Vects. Establishing the Iwould like to thank M.Sanders for encouraging the existence of superpartners, and measuring their properties, writing of this review and for signi® cant assistance in will require careful, detailed study. improving it.

5. Concluding remarks Appendix I The Standard Model of particle physics provides awell- In aquantum theory, all observables change with the tested and theoretically robust description of the physical distance scale probed. Basically this is simply the e Vect of world. It is here to stay. For the reasons outlined earlier, we the familiar perturbative expansion of any element, know it is not the ®nal answer. It is an e Vective theory, as aBorn termplus asum over intermediate states. The valid at the energy scale of order 100 GeV. termsinvolving intermediate states depend explicitly on the It seemsvery likely that some form of string theory (or energies of the external particles, so at higher energies more generally M-Theory) will be the primary theory. But (corresponding to larger momentum transfers or shorter that is formulated in 10 (or 11) space-time dimensions at distances), more intermediate states can come into play. the Planck scale and has unbroken supersymmetry. It is In particular, for couplings one can have an expansion so unlikely that theorists can guess how to break super- the true coupling is apoint-like piece, the Born term, plus symmetry and how to compactify to four dimensions loop diagrams with intermediate states, without guidance from experiment. Even if they do guess (or already have), without data tocon® rmit they will not be con® dent they got it right. If nature is indeed supersymmetric, supersymmetry can mediate between these extremes. It provides astronger foundation for the Standard Model. It can allow us to relate data to string ideas, to testapproaches to compacti- ®cation and supersymmetry-breaking against data, or even Ata given energy, all intermediate states with masses to have data point the way to solve these problems. below or near that energy can contribute signi® cantly to the Supersymmetry is the penultimate e Vective theory. loop term. This is the origin of most of the Lamb-shift, and We have seen in this review that supersymmetry provides this same eVectallowed the top quark to be `detected’ as a explanations for anumber of the basic questions raised in loop contribution before any collider had enough energy to the Introduction (A(i),(ii), B(i)), and string theory can produce it directly. The result is usually presented as an address some of these and others (B(i) ±B(vi)). It is not yet equation relating any coupling, such as the ®ne-structure known if string theory plus supersymmetry can explain `constant’ a,at one energy scale to its value at others. neutrino masses, but this is an active research area. The important question is then what particles go in the Together supersymmetry and string theory can explain loop. If the Standard Model particles werethe only most of our deepest questions about understanding our particles in the world, they would be the only ones in the universe and why it is the way it is. loop. If superpartners exist, they go in the loop too, and Ahistorical analogy provides auseful perspective on the change the way the coupling varies as the energy scale time-scale. Dirac in 1929 was led by abeautiful theory that changes. The results are described in the text. Supersymmetry:what? why? when? 367

References Gordon Kane receivedhis PhDfrom the Uni- [1]A treatment ofthe Standard Model at thelevel ofthis review is versityof Illinois in1963, under J. D.Jackson. available from Kane, G. L.,1993, Modern Hehas been at the University of Michigan since Physics,updated edition,Addison-Wesley. A review ofrecent tests 1965,except for sabbaticalsand leaves at ofthe Standard Model isRiles,K., 1998, Testing the Standard Model, RutherfordLab and Oxford, Stanford Linear ,Volume 39, pp.1 ±11. AcceleratorCenter, and CERN. For overtwo [2]The historyof early supersymmetry can be traced from Shifman,M. decadeshe has studied particle physics beyond (ed.),1999, Many Faces of the Superworld ,Singapore, World theStandard Model, focusing on howexperiment Scienti® c. andtheory can inform each other to extend the [3]A detailedreview ofthe experimental limitson superpartner, and other StandardModel and strengthen its foundations. aspects ofsupersymmetry and , can be foundin Kane, His work hasemphasized supersymmetry and G.L.(ed.),1998, Perspectives on Supersymmetry ,WorldScienti® c, Singapore, 1998. For the experimental situationsee the chapter by Higgsphysics. In additionto numerous research Carena et al.Itis possible to write more complicated versions of papers,he has published and edited several supersymmetryÐ forexample, inthis review weare assuming there isa books,including ModernElementary Particle conserved quantum number distinguishingSM particles and super- Physics thatgives a modernexplanation of the partners called -parity;the chapter by H.Dreiner describes the StandardModel of particle physics at thesenior theory ifR-parity isnotconserved. For astudy ofthepotential of the level,and two booksfor thegeneral public, The upgraded Fermilab colliderto detect superpartners and Higgs bosons, ParticleGarden and,very recently, Supersymme- see the report ofthe RunII Working Group,ed. J. Lykken, inpress. try.