Contemporary Physics, 2000, volume41, number6, pages359± 367 Supersymmetry:what? why? when? GORDON L. KANE This article is acolloquium-level review of the idea of supersymmetry and why so many physicists expect it to soon be amajor discovery in particle physics. Supersymmetry is the hypothesis, for which there is indirect evidence, that the underlying laws of nature are symmetric between matter particles (fermions) such as electrons and quarks, and force particles (bosons) such as photons and gluons. 1. Introduction (B) In addition, there are anumber of questions we The Standard Model of particle physics [1] is aremarkably hope will be answered: successful description of the basic constituents of matter (i) Can the forces of nature be uni® ed and (quarks and leptons), and of the interactions (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 gravity). (ii) Why is the symmetry group of the Standard It is afull relativistic quantum ®eld theory. 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 masses 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 cosmological constant 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 string 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 invariant under some symmetry opera- (though its e Vectscan be parameterized in tions (as mentioned above). In addition to the space-time the Standard Model), sosome other physics symmetries, 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 electron that the universe is made almost completely and its neutrino, e m ,or the left-handed up and down L $ e of matter and not antimatter, rather than quarks, u d .(In the Standard Model, fermion 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 spin anti- dark matter of the universe. parallel (parallel) to momentum. They then interact di Ver- (iv) Studies of neutrinos have shown that at least ently also. This accounts for parity violation.) We will see one neutrino has mass. 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 photon, 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 theories 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 `superpartner’) 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 charge and weak interactions of the electron. There is indirect evidence (which will be described below) that nature is indeed supersymmetric. None of the superpartners 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 energy region where weexpect to ®nd the superpartner is the upgraded collider at Fermi National Accelerator Laboratory (Fermilab) 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, photino. 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 angular momentum). 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 boson, 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 coupling 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 quantum number of the particles. The interaction 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 vacuum 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 symmetry breaking 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 special relativity 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.