"It may just take a fews years now until the first Susy particles are discovered. I already said exactly the same 10 years ago... Perhaps it is right to say that it always takes a few years to discover Susy."
(memory quote after H.P.Nilles)
Throughout the 20th century we have gone far from the classical picture of bodies and forces. The first step of this departure has already been well established in physics: we imagine today both bodies and forces as being comprised of elementary particles. Still, to some extent there seem to be the two separate worlds; they are identified with particle spins. In the Standard Model, a constituent particle is one with fractional spin, usually 1/2. These are quarks (6 of them, each in three color versions), electrons, muons, taus and neutrinos, labeled together as fermions. A force is a particle with integer spin. We know of four basic interactions and each of them is imagined as an exchange of a corresponding integer-spinned particle (boson): a photon mediates electromagnetic interactions, 8 kinds of gluons mediate strong and the W and Z bosons - the weak nuclear forces, and finally the hypothetical graviton should perhaps mediate gravity (which is actually not part of the Standard Model). Plus, there is the spin zero Higgs boson, so far only on paper.
The second and final step towards the unification of particles and forces is Supersymmetry. Susy, as called for compactness, is a symmetry that relates particles with different spin: fermions and bosons. Every elementary particle is thought here to exist in two states, a bosonic and a fermionic one. With Supersymmetry being exact, the two states should differ in spin by 1/2, with all other properties being identical. So where are these two states of each particle? Well, that's exactly the question! Obviously, the Standard Model particles cannot be grouped into couples with different spins; not even one such couple can be pointed out! If Susy exists, the number of elementary particles must at least be doubled! One more thing becomes transparent: if Susy exists in nature it must a broken symmetry; the undiscovered supersymmetric partners must be much heavier than all known patricles, if we want to explain why they have not been observed yet. The breakdown of Supersymmetry must be manifest in a split of masses between ordinary particles and their super-partners.
Before we go on, let us notice an interesting Susy feature. Our concept of forces being identified with one or another mediating boson requires a revision. In Susy, plenty of processes can occur without the contribution of any of the known mediating bosons. Either we can talk of plenty new forces, or we need to distinguish between interactions and bosons (a theorist may want to smash me at this point, as this statement doesn't really make much sense).
Under Supersymmetry, every elementary particle is a doublet including a fermion and a boson; the fermionic and bosonic degrees of freedom in the theory must be equal in number. Hence, to every known fermion, two corresponding spin zero partners are predicted. These are denoted by the charming names of squarks, selectrons, smuons, staus and sneutrinos (where the prefix s- stands for scalar, a common name of a spin zero particle). The fact that two rather than one partner are coupled to each fermion relates to two possible spin configurations of a spin 1/2 particle. Similarly, every boson is predicted to have a spin 1/2 partner; they have also nice names: photinos, gluinos, winos, zinos and higgsinos (we can also expect spin 3/2 gravitinos, if gravity is included; gravity is, however, beyond the interest of the minimal supersymmetric extension of the Standard Model).
There are two further complications in this picture. For certain technical reasons, one Higgs boson is not enough to create a consistent supersymmetric theory. In fact, at least five physical Higgs particles are thought to exist, three neutral ones and a charged couple (particle and antiparticle)! In addition, some -inos may not directly reveal their identity. Physical particles with definite mass need not really be the interaction eigenstates, as given above. Gluinos are clearly distinct from the other particles as they carry "colors" (the strong charges); other -inos, however, may freely mix and in general we are left with four neutral and two charged such mixtures. These mixtures are known as neutralinos and charginos. We expect therefore four neutralinos and two charginos, and these are in the end the particles that can be directly observed. We have now multiplied the particle content of the world by more than a factor 2, compared to the Standard Model. It is the minimal spectrum that every supersymmetric model has to include.
I'm sure you've just said: wow! However, it is not really true that Supersymmetry postulates countless new beings. At this level we have only postulated one new being: Supersymmetry itself. All the rest is then bound to appear automatically. After all, nobody questions that the Standard Model cannot be the ultimate theory. The real question is whether evoking Supersymmetry is more or is less plausible than evoking whatever there has to be in the absence of it.
The main reasons that are usually called for to advocate Susy are the following.
Reason 1. Why not? And yes, that's quite a good reason! Supersymmetry is a mathematically elegant and philosophically challenging theory, so it must be true. Paraphrazing Einstein, God would have been a fool, had he not made use of such a beutiful idea. No serious physicist, however, would dare to postulate all this enormous new phenomenology for philosophical reasons only. The real origin of the concept of Supersymmetry is in fact quite technical.
Reason 2. In the Standard Model, we expect an elementary Higgs boson of a mass not exceeding a few hundred GeV. Theoretical calculations based on Quantum Electrodynamics leave us with a serious problem, though: the mass of a scalar particle will go to infinity, due to interaction with fermions. It is true unless there is a not-too-heavy boson to conterpart each fermion. This is a place which explicitly begs for Supersymmetry!
Reason 3. The strengths (coupling constants) of the three Standard Model interactions are well known to be energy dependent functions. Moreover, it has been noticed that they tend to approach each other as energy rises, leaving way for a Grand Unification. Ever from the time it was first noticed, it has been considered a very physically attractive concept that electromagnetic, weak and strong forces become one force at some very high energy scale. Many Grand Unification Theories have been proposed. Precise measurements at CERN and Fermilab revealed, however, that the three interaction strengths will not meet at one point unless something modifies their behavior in between the Standard Model energy scale (~100 GeV) and the Grand Unification scale (~10^16 GeV). Supersymmetry is a plausible candidate and, in fact, most (but not all) Grand Unification Theories are based on Susy.
Reason 4. The largeness of the Grand Unification energy scale strongly contrasts with the smallness of the Standard Model energy scale. This is called the "hierarchy problem" and technically it is manifest by the need of an enormously fine tuning in the parameters of a Great Unification Theory in order to give back correctly all the Standard Model as a low energy approximation. It is hard to imagine two numbers being tuned to each other up to the fourteenth digit after comma for pure chance. Unless we want to evoke miracles, a plausible explanation is the existence of an extra symmetry which in principle ensures the two number are tuned to each other completely, while it is by the breakdown of that symmetry in nature that the tuning is actually not infinite. The existence of Susy solves the hierarchy problem.
Reason 5. An outstanding problem of particle physics is gravity. In the Standard Model, its very existence is simply ignored. It is clear, however, that a good theory must some day include gravity, too. Even though the minimal supersymmetric extension of the Standard Model tells nothing about the subject, it is an amazing and physically compelling Susy feature that it leaves room for a natural inclusion of gravity in the picture. Such concepts are known under the name of local Supersymmetry, or Supergravity. Supergravity is far from commonly accepeted, but even in the alternative String Theories one can hardly live without Supersymmetry. One way or another, Susy is likely to be there!
Reason 6. Astronomical observations have proved beyond any doubt that the known forms of matter can account for merely a fraction of the total matter that forms the Universe. This is obvious both from measurements of the galaxy rotation and the more from cosmological considerations based on the very popular these days inflationary theory of the Big Bang. The Universe seems filled up with some mysterious dark matter of unknown origin. And here goes Susy again, with its new particles, this time to provide good candidates for such dark matter.
Up to now, you may have got the impression of me being a convinced Supersymmetry partisan. Wrong! So let me be critical now. Susy weaknesses are likely to be the following.
There is no definite theoretical argument that we really need it as an extension of the Standard Model. It is possible to rebut one by one, all the arguments cited above in favor of Supersymmetry. There are other than Susy possible explanations of dark matter in the Universe. There are other possible solutions of the hierarchy problem and of Grand Unification. Supergravity may be wrong, while String Theories require Supersymmetry being relevant at roughly the same energy scale at which String Theories are relevant themselves. Not even the "why not?" argument is valid to prove that Susy has anything to do at the energy scale of the Standard Model. As for "reason 2", one may well say there is absolutely no problem as long as there is no elementary Higgs boson found. Actually, the argument may be even reversed against the existence of an elementary Higgs boson!
The Standard Model is thought unnatural for having more than 20 free parameters, i.e., fundamental quantities whose values it does not predict. In this sense, the supersymmetric extension is nothing but a large step backwards: it has easily a hundred free parameters.
Susy needs some hand work or risks being immediately abolished by everyday observation. In general, it leads to a huge number of new interactions and processes which do not exist, e.g. proton decay. To avoid unwanted processes, R-parity has been introduced. In short, R is thought a new quantum number, equal to 1 for all ordinary particles and to -1 for their super-partners. The additional requirement of R conservation induces an interesting phenomenological feature that supersymmetric particles can only be produced in pairs and vanish in pairs. It is not known to me if theorists can attribute a deeper significance to R-parity. The fact is, it has been introduced ad hoc to render the theory plausible. Moreover, Susy has in the general case a sort of CP problem. There are many new possible sources of CP violation in the theory, enough to produce a CP disaster. Some stringent relations between Susy parameters are therefore required (like the one of squark masses being nearly identical within the first two generations, just to focus on that one).
It is experiment which, in the end, establishes or abolishes a theory. Searches for Susy have been carried out for the last ten years, if not more, in laboratories all over the world. So far, they have been a continuing source of disappointments. Instead, new and new limits are given. Some experiments even boast of what limits they can achieve in the future...
Despite the lack of experimental success, Susy still survives, which in the end means nothing but a very weak predictiveness and poor practical chances of falsification. Many new particles are predicted, but the masses of most of them can be whatever between zero and over 1000 GeV. There is enough freedom in the parameter space that we cannot be certain, in the general case, what to expect, or how to interprete the results, in case we see a positive signal. Only non-standard versions of supersymmetric models, like the light gluino hypothesis, are easily testable and have been ruled out.
To summarize, whether or not one believes in Supersymmetry seems to depend mainly on his aesthetic intuition.
The first part of the question would require writing a book to answer. The general outstanding idea for the last 10 years has been: increase the energy! At some point we should detect directly the super-partners of ordinary particles. From all the predicted Susy signatures in big accelerators, one particularly interesting feature outstands. Due to the requirement of R-parity conservation, Susy particles can only decay to a lighter Susy particle plus an ordinary patricle. There is then some lightest supersymmetric particle (it is in fact the subject of so much discussion, that a dedicated acronym, LSP, has been created!) which has nothing more to decay to. LSP is a stable particle, it moreover can interact with ordinary matter only via the exchange of heavy particles, which means, extremely weakly (LSP is usually thought to be a neutralino, although other scenarios are still possible). As there is no chance to actually see the LSP, a common experimental signature of a lot of supersymmetric processes is missing energy and transverse momentum. Further details are impossible to summarize briefly and perhaps not even very elucidating.
There are many possible indirect methods of observing Susy, although correct interpretation may be very difficult. Most of them consist of testing the Standard Model in every possible detail; not to prove it right, actually, but to prove it wrong. LEP experiments at CERN provide high precision data on electromagnetic and weak processes, to which theoretical fits, with any free parameters the model requires, are done. Unfortunately, fits within the Standard Model have been extremely successful so far and there is no known way to improve anything by adding Supersymmetry. Predictions for particular processes can be also verified. There was much rumor around 1995, for instance, based on new results from LEP, which suggested that particles containing the b quark seem produced too often in electron-positon collisions, compared to Standard Model expectations. Explanation in means of additional Susy contributions was considered; the whole effect was then proved just a rumor...
Another good path is connected to the physics CP violation. In the Standard Model, all CP violating effects must be related to one sole parameter. A possible inconsistency of various experimental results, if observed, may reveal the need of a supersymmetric extension. In principle, it should be even possible to find Susy effects in proton structure function studies, like the ones carried out in DESY. Gluinos, if they exist, may influence the obtained results, only it is far from clear how.
Finally, the LSP, being the natural candidate for dark matter, can be detected by astronomical observations. In fact, a positive effect has been recently reported by the DAMA group at Gran Sasso. However, nobody knows for sure what they really see.
To directly prove or rule out the existence of Supersymmetric particles, we would, in most cases, have to scan all the energy range up a few TeV (1 TeV = 1000 GeV). With one notable exception, though, and this one has a large chance to become the experimental result of the decade. Certain relations between boson masses require at least one Higgs boson being really light, as it should be found below 150 GeV. Masses below 100 GeV are already excluded from LEP data, which means there is actually not much room left! It also means that, disregarding pathological scenarios which require some miraculous coincidences, the next planned large accelarator at CERN, the LHC, is bound to provide either a positive or a negative answer. No light Higgs boson means no Susy.
The positive conclusion of it all is that the present state cannot last forever. There is good chance that we will definitely find out who was right and who was wrong still within the present decade. Let it be!