## Open Questions: Supersymmetry

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Prerequisites: The standard model -- Higgs physics -- Beyond :the standard model

### Introduction

By now, it's a quite familiar fact that symmetry plays a huge role in particle physics. (As it does in other branches of physics.) So if a little symmetry is a good thing, more symmetry should be an even better thing.

Where can we look to find more symmetry? We have already seen in grand unified theories that there should be a symmetry which relates quarks and leptons. How can we go even farther than that?

We recall that the electroweak symmetry relates certain leptons to each other (such as electrons and electron neutrinos), and does the same for quarks (such as up and down quarks). The relevant equations involving these particles are invariant under this symmetry. That is, the equations are still valid if the related particles are interchanged.

In the same way, grand unified theories, which unify the electroweak and strong forces, postulate an analogous symmetry between leptons and quarks. In theories of this sort, the equations are invariant under the symmetry even if leptons and quarks are interchanged. This is a kind of obvious thing to look for, but it has turned out that there are different forms this symmetry can take. There have been a number of difficulties, both experimental and theoretical, in establishing this kind of theory. Nevertheless, it's still an interesting possibility.

Can we go even further? The answer seems to be yes, and in fact the resulting theory is in many ways easier to work with than what has been tried with grand unified theories. It comes from postulating a symmetry which can interchange fermions and bosons. In this theory, there is actually a symmetry between matter and force particles. What other name for such a powerful symmetry could one use -- except supersymmetry?

Postulating a symmetry between particles as dissimilar as those of matter and force might seem like a fairly radical step to take. And it is. The justification, as we shall see, is that a number of theoretical problems are more easily solved with supersymmetry.

Considering just the matter and force particles we already know of, what do they correspond to under supersymmetry? The answer is that none of the known particles are related by supersymmetry. The equations simply don't admit this. Instead, we have to postulate entirely new and unobserved particles to which each of the known ones is related. Thus, every one of the three generations of leptons and quarks, and the various gauge bosons, has a supersymmetric partner to which it is related by the symmetry. If the particle is a fermion, its partner is a boson, and vice versa.

Since these superpartners are entirely new and (as yet) unobserved, we have on our hands a plethora of new particles. This is the first manifestation of how radical a step supersymmetry takes.

Though it's not a theoretical problem, this does presents a practical problem. So many (potential) new particles leads to a need for a lot of new names. To deal with this, the convention is that the name of the partner of a lepton is formed by adding the prefix s- (for "super"). For instance, the supersymmetric partner of an electron is a "selectron". The name of the partner of a boson is formed by adding the suffix -ino. Thus, the partner of a photon is a "photino".

Immediately below is a table of the various known particle types and their supersymmetric partners. For each type of particle, the quantity known as spin is included, because, as we'll see shortly, it's very significant for the theory. Fermions, by definition, have spin values that are half integers, while boson spins are (by definition) always integral. The table also includes one particle and its superpartner that's still conjectural, the graviton -- the quantum of gravity -- and its superpartner, the gravitino. In any quantum theory of gravity, the spin of a graviton must be 2. This is because gravity requires a particle having the mathematical properties of a "tensor", which are different from those of a scalar boson (spin 0) or a vector boson (spin 1).

Supersymmetric partners
Standard model particle Supersymmetric partner
Name Type Spin Name Type Spin
Electron fermion 1/2 Selectron boson 0
Neutrino fermion 1/2 Sneutrino boson 0
Quark fermion 1/2 Squark boson 0
Photon boson 1 Photino fermion 1/2
W boson 1 Wino fermion 1/2
Z boson 1 Zino fermion 1/2
Gluon boson 1 Gluino fermion 1/2
Higgs boson 0 Higgsino fermion 1/2
Graviton boson 2 Gravitino fermion 3/2

Where are all these new particles? Why haven't they been observed? These are very good questions. It is presumed that (assuming the theory is correct) none have yet been observed because they're too heavy to have been produced in current accelerators. Yet, theoretically, they should not be too much heavier than the corresponding standard model partners. The theory doesn't predict their masses (or mass ratios), but in general one does not expect to find many orders of magnitude diffence between the masses of standard model particles and their partners. So the lightest supersymmetric particles are expected to be detectable in the upcoming generation of accelerators.

Anyhow, if supersymmetry is correct, the universe must contain at least twice as many kinds of fundamental particles as those already known. Although this is rather a dramatic and daring prediction, it also means that supersymmetry is readily falsifiable, and hence much more than just an idle metaphysical speculation. Of course, the fact that no evidence has yet appeared for supersymmetry means it is also a theory in some peril of disproof. Since at least some of the superpartners should be light enough to put them in a range accessible with accelerators that will be in operation within the next decade, we shouldn't have to wait very long to get some indications whether supersymmetry is a viable theory.

Supersymmetry is a radical step for a second reason as well. Unlike all other continuous symmetry operations envisioned in elementary particle physics, a supersymmetry transformation is capable of actually changing the spin of a particle. All other symmetries, even in a grand unified theory, effect changes only in various "internal" states of a particle, such as its isospin or electromagnetic charge. A supersymmetry transformation, on the other hand, changes a particle's spin, so it affects nothing less than the way a particle is embedded in spacetime.

One consequence of this is that gravity -- which has until now, alone among the four known fundamental forces, remained aloof from the others -- is incorporated in the theory in a natural way. (Although not in a way, as yet, which seems to be a correct theory of how the world actually works.) This ability to deal with gravity, however incompletely, is one reason that supersymmetry theory has received as much attention as it has for over 25 years.

### Spin

This point about spin is pretty important, since it ultimately leads to a (possible) way to incorporate gravity in the theory. So let's review a little about spin.

Spin, basically, is a measure of a particle's rotational movement, or more technically, angular momentum. It's like velocity, only in a rotational rather than straight-line sense. Now, angular momentum is intrinsically a vector rather than a scalar quantity. In other words, it can't be described in terms of a single number, because a direction as well as a quantity is involved. That direction is defined by the axis of rotation.
External links on "spin"
• Scientific American: What exactly is the 'spin' of subatomic particles?
• Spin
• Angular momentum, like ordinary linear momentum, is a conserved quantity. This is just Newton's first law of motion: an object in a uniform state of motion tends to remain in that state unless an external force is applied. This is true of both linear and rotational motion. There is an interesting mathematical basis for this conservation law, known as Noether's theorem (after the mathematician Emmy Noether). The laws of physics should not depend on where an observer is. In other words, the laws should by symmetric under the operation of translation. Likewise, the laws should not depend on the direction in which an observer is looking. They should be symmetric under the operation of rotation. Noether's theorem says that corresponding to symmetry operations of this sort, there should exist conserved quantities. For translation, the conserved quantity is linear momentum. For rotation, the conserved quantity is (of course) angular momentum.

The fact that angular momentum is conserved means that it is a stable property of a system (such as a single particle) to which it is associated. It does not just change at random. Any change must be attributable to some specific cause (such as a force).

When you stop to think about it, there's something strange about the fact that an elementary particle, such as an electron, can have an angular momentum due to spin. What's strange is that an elementary particle is supposedly a mathematical point, a zero-dimensional object. How can one imagine that such a thing rotates about some particualr axis? What is even stranger is that force-carrying particles such as photons, which in some sense aren't even matter at all, can also have angular momentum.

The explanation of this situation is that we are dealing with quantum rather than classical objects. The angular momentum -- the spin -- that we are concerned with is a quantum mechanical thing. As with other things in quamtum mechanics, it does little good to ask why it's the way it is. The equations of quantum mechanics describe it exactly, and the main reason it's called "angular momentum" is that it is mathematically analogous to the classical angular momentum of macroscopic spinning objects.

Since it's a quantum mechanical construct, it isn't surprising that spin is quantized in discrete units. Although spin is inherently a vector quantity, there is usually a particular direction that is singled out from all possible directions as important -- for instance the direction in which a particle is moving. In this case, only the component of the spin in that particular direction is of interest, so we are dealing with a simpler scalar quantity.

The units of this quantity, then, are integral or half integral multiples of a number known as Planck's constant. However, it is customary to pick units of mass, length, and time so that Planck's constant has the value 1. Consequently, values of spin are ordinarily quoted as integers or half integers. Bosons are the particles with integer spin, while fermions are the particles with half integer spin.

We have noted that different types of statistics govern the behavior of assemblages of multiple fermions and bosons. Specifically, any number of bosons at a time can occupy the same quantum state, as described by a collection of quantum numbers. But only a single fermion at a time can occupy one quantum state in any coupled system of fermions. That is, if you find multiple fermions together, they must each differ in at least one of their quantum numbers from all the rest. This rule is known as the Pauli exclusion principle. Typically, fermions will differ in their energy levels, as occurs with multiple electrons in an atom. But spin is another quantum value by which separate fermions can be distinguished.

Given this rather fundamental difference between particles having integral and half integral values of spin, we ought to feel a little queasy about trying to relate them by a symmetry operation. Is it realistic to imagine we could have equations which could actually be invariant when particles so different in nature are interchanged?

At first is seemed as though this concern was well-founded, based on considerations from the special theory of relativity. Now, special relativity is founded on the assumption that the laws of physics remain the same not only when viewed from any position and in any direction. They also remain the same for an observer moving with a uniform velocity (i. e., not accelerating) with respect to what is observed. In order to translate between coordinate systems that are moving with respect to each other at a uniform velocity, it is necessary to use what is called a "Lorentz transformation". The collection of all these physics-preserving transformations -- translations, rotations, and Lorentz transformations -- is known as the "Poincaré group" (after the mathematician Henri Poincaré).

There is a mathematical theorem, known as the Coleman-Mandura theorem, which says that any symmetry group which contains the Poincaré group as a subgroup must have a special form that makes it impossible for symmetries to relate particles with different spins. This, then, would seem to kill the idea of supersymmetry immediately.

But there's a way out, obtained by generalizing the notion of symmetry. Ordinarily, in technical mathematical terms, one works with group symmetries by studying certain concrete constructs known as "representations" of the group. These representations, in turn, are studied by looking at analogous representations of a related structure, knwon as the "Lie algebra" of the group.

It turns out that by means of a simple generalization of a Lie algebra, called a Lie superalgebra, it is possible to find representations of the Poincaré group which allow for a relationship between particles with different spin. And that's all it takes to make supersymmetry work -- mathematically. Whether it's correct physics is an entirely different question, of course.

There would be a great payoff if sypersymmetry really works. A combination of two supersymmetry operations is a Poincaré transformation, that is, an actual displacement in spacetime. As we shall see shortly, this leads to a quantum theory of gravity.

### Supersymmetry and Higgs physics

Supersymmetry is strong medicine. Although the concepts seem a little bizarre, soon after the ideas appeared in the mid 1970s, it gradually emerged that supersymmetry might help solve several difficult outstanding problems in particle physics. The incorporation of gravity was one of these. An explanation of the Higgs physics as more than just an ad hoc trick was a second. And the hierarchy problem, which arises out of the nature of Higgs physics, was a third.

Let's briefly review Higgs physics. The idea had its origins outside particle physics -- as a relativistic version of superconducting electrodynamics. It was imported into particle physics on an ad hoc basis to deal with three interrelated problems:

1. What explains the breaking of electroweak symmetry -- the fact that electromagnetism and the weak force appear to be different, with different characteristic ranges and strengths?
2. Where does mass come from, given that Yang-Mills gauge theories are "naturally" theories that work for massless particles?
3. How can the Yang-Mills theory of the electroweak force be made mathematically consistent, i. e., "renormalizable"?
Higgs physics postulates the existence of one (or more) spin-0 quantum fields. Interaction of elementary particles with such a field causes them to have mass. This interaction is called the Higgs mechanism. There are various metaphorical ways to visualize it. One is to think of the Higgs field as a very viscous fluid like molasses. The field impedes the motion of any particle moving through it, giving the particle the appearance of having inertia, and hence mass. Higgs bosons -- the quanta of the field -- constantly interact with other particles, creating a drag effect that slows them down. (All particles would move at the speed of light if they had no mass.)

Suffice it to say, the Higgs mechanism really does work to solve the problems listed above. Unfortunately, Higgs bosons have not yet actually been observed experimentally. Physicists are currently searching with extreme dedication for any evidence of them. But even if the Higgs bosons are found -- especially if they are found -- other questions await answers. In particular, what is the source of the Higgs field itself? Where does it come from? And secondly, why does the Higgs field interact with other particles the way it does (the Higgs mechanism)?

Supersymmetry helps answer these questions. For technical reasons, supersymmetry automatically contains scalar fields like the Higgs field. These fields in fact interact with other fields in the way that the Higgs mechanism requires. The real question is what causes these fields to take on nonzero values, which they must do in order to actually interact with anything in the first place. A Higgs field is only a potentiality -- like any other field -- rather than something present if its field strength is zero.

Supersymmetry deals with this issue. In fact, it deals with it in just the right way. For simplicity, consider just the electroweak force. At a high enough energy, there is complete symmetry between the weak force and electromagnetism. The gauge bosons of the two forces all have zero mass. The Higgs field which would break this symmetry must therefore have a value of zero at this energy scale. Another way to phrase this is to say that -- at the high energy scale -- the state of lowest energy of the Higgs field occurs when the field has a value of zero.

But at some point, as the energy scale drops, this situation must change. In order for the Higgs field to break the symmetry between the weak force and electromagnetism, the Higgs field must be nonzero when it is in its own lowest energy state. This can occur only if the shape of the curve that describes energy as a function of field strength has a minimum at a nonzero field strength.

As it happens, the shape of the energy vs. field strength curve contains a parameter that varies according to the overall energy scale. Let's call the parameter B. If the variable x denotes the Higgs field strength, then the energy in the field is given by an equation like this:

E = x4 + Bx2
where E is the total energy in the field. At a very high energy scale, the parameter B is positive, and so the shape of the curve is roughly like a parabola that has one minimum at zero. But supersymmetry computations show that B decreases as the overall energy scale does, and at some critical point becomes negative. When that happens, the energy curve will have two minima for nonzero values of the field strength x, instead of right at x=0. (Because for x small enough but not zero, the quadratic term in the expression is negative but larger in absolute value than the fourth degree term.)

In this manner, supersymmetry not only provides a way for the Higgs field to become nonzero while in a state of minimum energy, but in fact provides that this will occur starting at some critical point as the energy scale decreases. This is exactly what needs to happen in order that we see the symmetry breaking between electromagnetism and the weak force which in fact occurs in the real world.

In summary, then, supersymmetry produces, automatically, all of the Higgs physics that was simply postulated, at first, to solve the three related problems listed at the beginning of this section. All of the Higgs physics is a necessary consequence of supersymmetry. This means, among other things, that one strong test of supersymmetry is the ability to detect Higgs bosons in accelerator experiments. Supersymmetry even helps put limits on the mass of the lightest Higgs boson, thus sharpening the test.

Higgs physics, however, had other problems apart from lacking a good explanation (before supersymmetry) of where it came from. Chief among these is the hierarchy problem.

### Supersymmetry and the hierarchy problem

If the Higgs mechanism is real physics, then it should affect the strong force as well as the electroweak force. These two forces should be merely separate manifestations of a single force. But the symmetry between the electroweak force and the strong force must be broken at a much higher energy level than the point at which the electroweak symmetry breaks -- more than 1016 GeV vs. a mere 100 GeV.

The huge difference between these two energy scales is the hierarchy problem. This is because any equations which might deal with these forces simultaneously need to be extremely fine-tuned if the quantities involved differ so greatly. In other words, various terms in the equations must cancel out almost exactly if numbers that differ by at least 14 orders of magnitude can come out of the same equations.

The problem shows up, for example, in calculations of the mass of the W and Z bosons. Recall that it is the Higgs fields in the first place that account for these masses. (Without Higgs, W and Z should be massless.) In this case, the mass calculations involve interactions taking place at the 1016 GeV scale. Things would need to cancel out just right to get a reasonable answer.

Without supersymmetry, such cancellation would seem to be simply an incredible accident, like winning the grand lottery 1000 times in a row. With supersymmetry, the cancellations are automatic.

The way that the cancellations come about is simplicity itself. With supersymmetry, terms in the equation corresponding to each standard model particle naturally pair automatically with terms for the superpartner of each partcile -- but with opposite sign. Instant cancellation. Hierarchy problem solved.

Can it really be this simple? Well, almost. Every time one problem is solved, a new one seems to crop up. Ultimately, the only way to tell one is headed in the right direction and not just running around in circles is by verifying theoretical predictions against experiment. The predictions of both the Higgs theory and supersymmetry still need to be verified by observing both Higgs particles and superpartner particles. Such verification -- if it can be done -- is at least on the horizon.

But even if the necessary particles are observed, problems -- new ones -- remain. Most notably, supersymmetry can't be an exact symmetry. It must be broken, simply because particles and their superpartners do not have the same mass. This is a problem for at least two reasons. One may be easy to deal with, the other not.

The potentially easier problem is that if supersymmetry is broken, the necessary cancellations it brings with it are somewhat spoiled. Fortunately, this may be manageable. In the calculation of W and Z masses, for instance, what happens is that the contributions from the very large energy scales occur roughly in proportion to the difference between the masses of particles and their superpartners. In order for the W/Z bosons to have something like their known masses, around 100 GeV, the superpartners need to have masses less than about 1000 GeV. Particles with such masses can be created in the next generation of particle accelerators, so it shouldn't be that long before the superpartners can be detected. The brokenness of supersymmetry, in this regard, isn't then a problem at all, since it assures us the necessary superpartners should be detectable.

The harder problem is theoretical. It is that there are no good ideas on how to explain supersymmetry breaking itself. It would actually be easier to explain this symmetry breaking if the mass difference of superpartners were very large. But that can't be, as just noted. Despite many attempts, it doesn't seem like something analogous to the Higgs mechanism of symmetry breaking will work here either. The net result is that if supersymmetry is correct, the mechanism of its breaking is still a mystery.

### Supergravity

We've seen that supersymmetry is not only strong medicine. It's also good medicine for several tricky problems in particle physics -- Higgs physics and the hierarchy problem. How about gravity?

Quantum theory and the theory of general relativity (which is all about gravity) were the two greatest accomplishments of 20th century physics. But they are completely different theories which have very little in common. Not only that, they do not work well together in those cases where both are needed -- at the time of the big bang or in the interiors of black holes, for instance. Physicists have been trying to unify them for over 75 years. Einstein spent more than half his adult life futilely trying to construct a "unified field theory".

It isn't that hard to make a start. If gravity is to be a quantum field theory, there needs to be a field quantum. This is called, naturally enough, a "graviton". It needs to be a boson, with integral spin. And it needs to be massless, since gravity is a long-range force. Einstein's equation of general relativity is mathematically an equation involving objects called "tensors", so a quantum gravitational field must be a tensor field, meaning simply a field which at each point of spacetime has a value that is a tensor. As a consequence, the field quantum -- the graviton -- needs to have a spin of 2. (It can also be shown that any force which is mediated by a boson whose spin is an odd number must be repulsive between like particles, but gravity is an attractive force.)

All this is largely ordained by the combined requirements of quantum mechanics and general relativity. There's no other way to do it. Having this, it becomes possible to try to make calculations of what happens when gravitons interact with other particles, much as one does with photons in the successful theory of quantum electrodynamics -- the quantum theory of the electromagnetic field.

The result is a disaster. The calculations are made using a technique called perturbation theory. One starts with an approximate result and tries to improve it by adding extra terms that correspond to successively more complex forms of particle interaction. When this is done, the results do not "converge". Successive new terms do not become small. If all the terms were added up the result would be infinite -- a physical impossibility.

This problem arose originally in quantum electrodynamics also. It was overcome by finding a suitable redefinition of the quantities involved, such as the charge and mass of an electron. This process is called "renormalization". No way was found to do the same thing with quantum gravity. The theory was considered to be non-renormalizable.

When the idea of supersymmetry got started, the connection to gravity was of interest, because, as noted above, the combination of two supersymmetry operations is an operation belonging to the Poincaré group -- a Lorentz transformation, a symmetry operation which obeys the rules of special relativity. Now, the collection of all supersymmetry operations form a group, and the fact just indicated says that the Poincaré group is a subgroup.

By that time (early 1970s) physicists had learned that interesting things happen when one considers the results of applying different symmetry operations of the same type locally at every point of spacetime, instead of applying the same operation globally. The result is that a new field, called a gauge field appears. And along with that field comes a gauge particle which is the quantum of the field.

It was already known that when the Poincaré group is used as a local symmetry group, the field that results is precisely the gravitational field, and the field quantum is the graviton. This differs significantly from other quantum field theories in several ways. As already mentioned, the field quantum has a spin of 2. Also, the symmetries involved are of spacetime itself, rather than being "internal" states of a particle, such as some sort of charge. Mathematically, the means that the group is not "compact".

Given the larger group of supersymmetry operations, which contains the Poincaré group, it's only natural to ask what happens when that is regarded as a local symmetry group. The answer is that you get a really strange field whose quantum is a particle with a spin of 3/2. Since the Poincaré group is a subgroup, you get the gravitational field and the graviton too. And guess what -- the graviton and that spin-3/2 particle are supersymmetric partners, so the latter has to be called a gravitino. The theory of the graviton and the gravitino that results is called supergravity.

Graivtinos are odd. They are the only particles that can be said to come up "naturally" with a spin of 3/2. That makes them fermions, so they are also the only quanta of a symmetry-induced field that aren't bosons. In spite of that, is there nevertheless a force that they mediate? The answer is no. Since they are fermions, they obey the Pauli exclusion principle. They don't combine to produce a macroscopic force of any kind.

So what good are gravitinos then? Well, in supergravity gravitinos are legitimate particles just like any others. Virtual gravitinos can be exchanged between other types of particles just like gravitons. So they will affect calculations of the gravitational forces arising from the exchange of gravitons.

Because gravitons and gravitinos are sypersymmetric partners, they tend to appear in calculations with the same magnitude but opposite sign. In short, they mostly cancel each other out -- very much the same sort of cancellation that successfully solves the hierarchy problem.

Does this mean that supergravity successfully removed all the infinities that had plagued quantum gravity? The initial indications were encouraging. But that didn't last long. Some infinities remained. Yet the theory was too promising to be abandoned so soon. Was there anything else that could be done? Yes, there were some interesting further alternatives that could be explored. But dealing with the infinities was not the only area in which supergravity theory needed more work, so let's look at another issue first.

### Extending supergravity

Neither supersymmetry nor supergravity is by any means a unique theory. Both can be varied and extended in a number of ways. Supersymmetry, for instance, can be added to other particle theory models, such as the standard model and grand unified theories. That approach is relatively straightforward, since it assumes that all the elementary particles and forces corresponding to known physics are already included. The particles are already arranged in symmetry multiplets, and supersymmetry has little effect on these, since it doesn't alter anything about a particle except its spin.

With supergravity, the situation is slightly different. All the theory has to begin with are the gravitons, gravitinos, and their associated quantum fields. In a class of early supergravity theories particles could be added "by hand", almost at will, in superpartner doublets of spin-1 and spin-1/2 particles or spin-i/2 and spin-0 particles. Such theories had immediate problems with infinities, because they were not fully unified. That is, not all particles were related to every other particle of the theory by a symmetry operation. The addition of the gravitons and gravitinos to the theory made it necessary that they be related to standard model particles by a symmetry operation, and this wasn't an easy thing to do.

It is still possible to graft supergravity onto the standard model or one of its grand unified extensions, but the symmetry problems remain messy. There is also no plausible way to pick which unified model to add onto, as there are many models and no experimental reason to choose one over another. It would be much nicer to have some means to include particles corresponding to those already known in a more principled way. Fortunately, there is another class of supergravity theories that has a much more constrained form. In this class of theories, there are only eight possibilities. They correspond exactly to the number of gravitinos in the theory, which can vary from 1 to 8, but no more. There is always just a single spin-2 graviton.

The theory is referred to by this number, denoted by N. In the "N=1" theory there is just one gravitino and one graviton, and no other particles at all. This, of course, is not a realistic theory. Known particles would need to be added in an ad hoc way. Doing that would defeat the purpose of finding a better explanation for exactly which particles should exist. It would also destroy potential unification and therefore not help in dealing with infinities.

The N=8 theory provides for 8 spin-3/2 gravitinos, 28 spin-1 (gauge) bosons, 56 spin-1/2 fermions, and 70 spin-0 (scalar) bosons. Even this might not have been enough to account for all the known spin-1/2 and spin-1 particles (when particles were distinguished by the various possible kinds of charge they could carry). But, as usual, there were tricks that could be applied to make things fit.

For a time around 1980, many physicists thought N=8 supergravity might actually be the long-sought "theory of everything". This hope was based on the guess that an intriguing old trick might be redeployed in order to explain naturally what kind of particles and forces should exist, and to conquer the problem of infinities in one fell swoop.

### Higher dimensions of spacetime

The trick is called Kaluza-Klein theory, an idea that had been investigated in the theory of relativity more than 50 years earlier.

In 1919 -- just three years after Einstein published his general theory of relativity -- an obscure young German physicist named Theodor Kaluza discovered something very curious and interesting. Specifically, if the theory of general relativity were formulated in four space dimensions instead of three (in addition to the one dimension of time), then Maxwell's equations of electromagnetism would fall out of the theory, as if by magic, right alongside Einstein's equation of general relativity. This was not only very elegant, but hinted strongly that the two forces, the only fundamental forces known at the time, could in some sense be unified by regarding both types of force as an effect of spacetime geometry.

Kaluza sent his results to Einstein, who was quite impressed, but who also had some reservations. Two and a half years later Einstein decided the idea was OK after all, so Kaluza's paper was published in 1921.

A little later, in 1926, Oskar Klein in Sweden investigated whether Kaluza's theory was compatible with the then-new quantum mechanics, so he formulated Schrödinger's equation in five variables instead of four and came up with reasonable results. The combination of the two physicists' work became known as Kaluza-Klein theory. Unfortunately, in spite of its elegance, the theory was unable to make useful new predictions, so it languished for more than 50 years.

In the late 1970s when physicists were struggling with the new ideas of supersymmetry and supergravity, they also had two additional forces to deal with -- the weak and strong nuclear forces. Various physicists realized that adding even more extra dimensions could accommodate those new forces, in addition to electromagnetism and gravity.

More specifically, if one postulated the existence of not one but seven additional spatial dimensions, and if one also postulated that the additional spatial dimensions were "curled up" into a very small size, then several good things happen:

1. When general relativity is formulated in the resulting 11 spacetime dimensions, the equations governing all four of the fundamental forces fall out of the theory.
2. There is "room" for more symmetry, and hence there can be more cancellation of infinities.
3. Macroscopically, spacetime still appears to have only four dimensions, because the other seven dimensions are all smaller than what is possible to observe experimentally.
And not only that, but it turned out that while 11 is the smallest number of dimensions that suffices to produce the four known forces in a Kaluza-Klein theory, it is also the largest number of dimensions in which a theory of supergravity can be formulated. Note that supergravity does not intrinsically require -- or predict -- the existence of spacetime dimensions beyond the four that are obvious. It's just that the theory is most elegant and seems to have the best chance of success when there are higher dimensions.

The combination of N=8 supergravity and Kaluza-Klein theory in 11 spacetime dimensions therefore seemed natural and irresistable. It accounted for both gravity and the three other traditional quantum forces. The extra dimensions even seemed to be of some help in dealing with the bothersome infinities.

Exactly how is it that compact extra dimensions seem to give rise to gauge forces? It depends on the geometry of those extra dimensions. Spacetime may be thought of as consisting of the four normal dimensions, and at each point there is another space consisting of all the compact dimensions. (Mathematically such a composite space is called a "fiber bundle".) In the original Kaluza-Klein case, there was just one extra dimension, which was topologically a circle. The symmetry group of the circle is just the rotation group, U(1), which is the gauge group of electromagnetism. If the extra dimensions have the form of a higher dimensional compact manifold, the symmetries of that object yield a non-Abelian gauge theory, such as the theories of the weak and strong forces. In this way, particle physics arises from spacetime geometry right along with gravity.

This all seemed too good to be true. And it was. There were several fatal problems that just wouldn't go away. So, at best, the theory could be regarded as a very elegant failure which could give a good approximation of reality in some cases. The problems included:

1. There is a version of the cosmological constant problem that causes even the four macroscopic dimensions to have such extreme curvature that the entire observable universe would be less than a meter in size.
2. Efforts to show that infinities cancelled each other eventually failed as soon as Feynman diagrams with seven or more closed loops were considered. (Cases with fewer closed loops could be handled as a result of the seven extra compact dimensions.)
3. Worst of all, perhaps: the theory predicted that W bosons (and the electroweak force) would necessarily couple with right-handed as well as left-handed particles -- which contradicts experimental facts.

So supergravity was ultimately an unsuccessful theory. But it is worth remembering for a couple of reasons. First, the two key ideas of supersymmetry and higher dimensions of spacetime live on as very viable theoretical possibilities. They are, in fact, foundational ideas in the theory of superstrings. (We will discuss both superstrings and higher dimensions elsewhere in much more detail.) Second, 11-dimensional supergravity re-emerges, with better underpinnings, as one specific type of superstring theory.

### String theory

String theory is an approach to the study of fundamental forces and particles based on principles rather different from any considered here so far. In particular, it supposes that fundamental particles are not point-like objects, having a mathematical dimension of 0, but are string-like, 1-dimensional objects. There are good theoretical reasons for this choice. It is not just an arbitrary difference made for the purpose of being different. But we'll have to go into the explanation elsewhere.

The string theory approach goes back quite a way in time. It originated in 1968 in work of Gabriele Veneziano who was trying to describe the strong force. He noticed that a function known as Euler's beta function could describe important properties of strongly interacting particles. A little later, in 1970, several other physicists found that if particles were modeled as vibrating 1-dimensional strings then the description of their interactions in terms of Euler's beta function came about naturally. In 1974 John Schwarz and Joel Scherk discovered that one of the particles predicted by this early string theory could be identified with the graviton.

However, in this same time frame, a more robust theory of the strong force came together in the form of the QCD gauge theory, involving traditional point particles. So string theory received rather little attention.

Research on string theory proceeded anyhow. The original theory dealt only with particles having integral spin -- bosons -- so it was sometimes called bosonic string theory. In 1971 Pierre Ramond figured out how to get fermionic vibrations out of strings. It soon appeared that bosonic and fermionic vibrational modes came in related pairs. The idea of supersymmetry actually arose from this insight. In other words, string theory was in some sense the progenitor of supersymmetry. In 1973 Julian Wess, Bruno Zumino, and others realized that supersymmetry was also applicable to point particle theories, so supersymmetry took on a life of its own.

Meanwhile, string theory itself turned out to have problems of its own. For one thing, some bosonic string vibration modes turned out to have a mass whose square was negative and which (therefore) necessarily must have velocities always greater than that of light. Such particles, called tachyons, did not fit well with accepted physics. Another difficulty was that string theories led to technical quantum mechanical problems known as "anomalies".

In 1976 it was realized that if supersymmetry became a required feature of string theory -- which would therefore be called superstring theory -- then tachyons could be ruled out because (as it happens) tachyons can't have superpartners. So supersymmetry fairly quickly solved one of the major problems in string theory.

Finally, in 1984, John Schwarz and Michael Green showed that the 10-dimensional superstring theory is also free of quantum mechanical anomalies, provided the theory incorporates a gauge group which is either SO(32) or E8xE8. This boon from supersymmetry finally attracted substantial attention to string theory, so much so that it became known as the (first) "superstring revolution".

Today, superstring theory sees far more activity than supersymmetry, but that's a long story to be told elsewhere. We'll let it suffice for now to say that the benefits of supersymmetry in string theory certainly did not end with the 1984 results of Schwarz and Green. Just to cite one example, not long ago (1996) physicists Andrew Strominger and Cumrun Vafa enumerated the number of quantum states for certain types of black holes. This work relied on supersymmetry and recent developments in string theory such as "D-branes". The computation was especially interesting since it agreed perfectly with a classical computation of black hole entropy due to Stephen Hawking and Jacob Beckenstein. The results may be relevant to a major problem in the theory of black holes known as the "black hole information problem".

### Testability of supersymmetry

Supersymmetry is a very testable theory, since it makes some quite specific predictions. It is true that many of the tests require accelerator capabilities slightly beyond what is currently available. But definitive results should be forthcoming within five to ten years.

Supersymmetry (just like any other physical theory) is tested by figuring out what effects and results it implies. These effects are the predictions of the theory. They are necessary conditions for the theory to be true. So one must test for these predicted effects. If any of the tests fail, the theory must be false. (Although perhaps it can be modified.)

On the other hand, none of these predicted effects is a sufficient condition for the theory to be true. Other theories, or a modification of the one under consideration, may account for the same effects. So a theory can't be proven correct in an absolute sense simply by testing all its predictions. The predictions of general relativity, for instance, are still being tested.

However, if tests verify all predictions -- and especially predictions for which there is no other known explanation -- one has pretty good evidence for the theory.

#### Detecting superpartners

Existence of all the superpartners is the most obvious prediction of the theory. The theory gives good guidance about the kinds of interactions the superpartners would be involved in, but it is much more vague about how massive they might be. The mass is a key factor, because before any particle can be "detected" it must be created in the first place. Since particles can be created only together with their antiparticles (which have the same mass), at least twice as much energy as that contained in a particle's mass must be available. Furthermore, not all the energy that an accelerator is capable of generating can be put to use creating particles. Considerations like this make the game difficult.

As noted above, although supersymmetry is broken, it can't be broken too badly, or else it wouldn't work to deal with the hierarchy problem. This puts a limit of something like 1000 GeV on the mass of the lightest supersymmetric partner. So far, no superpartners have been detected, and that 1000 GeV limit is still well above the mass of particles which can be created in current accelerators. But it should be within reach of the Large Hadron Collider (LHC), which is expected to begin work around 2007.

#### Higgs physics

In our discussion of supersymmetry and the Higgs field, we noted that the equation
E = x4 + Bx2
describes the energy in the Higgs field for a certain parameter B. B is not a fixed constant, but instead varies slowly with the energy scale. Under certain conditions, it is possible to calculate using supersymmetry that B at some point becomes negative as the energy scale decreases (or equivalently, as the length scale increases). If B does indeed become negative at some point, then the Higgs mechanism works to give mass to the W and Z bosons of the electroweak theory -- and a Higgs boson (possibly more than one) must exist.

A sufficient condition for B to become negative in this calculation is that the heaviest quark be at least as heavy as the W/Z bosons, since mass measures the strength of interaction with the Higgs field. At the time this was realized, the top quark had not been detected, and its mass was unknown. In fact, it might have had a rather small mass, since the next heaviest quark, the botton quark, weighs in at only around 5 GeV -- compared to about 80 and 90 GeV for the W and Z. The existence of the top quark was confirmed only in 1995 -- and it had a mass of 175 GeV. This is far more than sufficient.

It follows, therefore, that all of the Higgs physics is actually a consequence of supersymmetry. In particular, the existence of Higgs bosons is a prediction of supersymmetry. So detection of Higgs bosons is a necessary condition for the validity of supersymmetry.

Supersymmetry itself helps place limits on how heavy the lightest Higgs boson can be. That limit is around 180 GeV and could be as little as 135 GeV. Consequently, a Higgs boson (if such exists) is likely to be detected before any superpartners. Conversely, failure to detect a Higgs boson with a smallish mass would be a rather negative indication for supersymmetry.

#### Supersymmetric effects on other processes

There is a loophole in the requirement for how much energy is required to create particles. The loophole is provided by quantum mechanics: virtual particles (in pairs) can come into existence anytime, in spite of the law of conservation of energy, provided they don't last long before annihilating each other. Consequently, the "vacuum" is actually swarming with virtual particles of all possible kinds. How long a virtual particle can exist depends on how massive it is. The more massive it is, the shorter its lifespan must be, by the Heisenberg inequality.

But such virtual particles are real for some period of time, however short. So they can potentially interact with other particles and affect the rate at which all kinds of particle interactions take place. Not only that, but they affect other particle properties like mass and charge. Indeed, the "bare" mass and charge of any particle is quite different from the mass and charge which can be measured, all as a result of the constant flux of virtual particles in the vacuum.

Virtual superpartners will very much play a part in this process -- if they exist. In fact, the assumption that they do exist, together with computations that can be made on the basis of supersymmetry, contribute decisively to predictions that the theory of supersymmetry makes. For instance, given the assumption that the coupling constants of the electromagnetic, weak, and strong force all become the same somewhere around 1016 GeV, one can calculate what the coupling constants at accessible energy levels (100 GeV) would need to be -- and the answer is very close to what is measured.

Of course, we can't verify that the coupling constants really do become the same at very high energy. That was just an assumption. But in other theories -- such as grand unified theories -- that assumption is a prediction. Consequently, any such theory together with supersymmetry does correctly predict the observable coupling constants. This is evidence. But it's somewhat weak evidence, since the amount of influence by very massive particles which is required to get the coupling constants right isn't too large.

However, there are many other computations along the same lines that can be done. For instance, there are interactions known as "flavor-changing neutral currents" which cause a bottom quark to decay into a strange quark (plus other particles). Such a process is thought to be suppressed in the standard model, but allowable (with small probability) if heavier particles (like the superpartners) exist. Consequently, searching for interactions of this sort may provide evidence for supersymmetry -- if sufficiently precise measurements can be made to attribute the effect to supersymmetry.

#### Dark matter

Cosmology has fairly well established that most of the matter in the universe is dark matter -- not part of visible stars, gas clouds, galaxies, or the like. And most of the dark matter -- upwards of 80 to 90% -- is not baryonic (i. e., composed of quarks). The great bulk of matter, therefore, must consist of "weakly interacting massive particles". This component of dark matter could well consist of a variety of different sorts of particles. But superpartners are among the leading candidates.

In particular, it is fairly probable that the lightest superpartner (LSP) is stable. This is because of the conjectured conservation of a quantity known as R-parity. R is simply a number associated with any particle. It is 1 for ordinary particles and -1 for superpartners. Since the LSP must be more massive than any decay products, they would all have to be ordinary particles. But that would violate R-parity conservation. If supersymmetry is correct, a large portion of dark matter should consist of LSPs, created in the big bang.

The problem is how to distinguish supersymmetric dark matter from other possible types -- assuming we can detect any of them somehow. If we are able to detect dark matter directly, we will also need to be able to make observations of its properties (mass, types of interaction, etc.) If we are lucky, such observations may help us tell whether or not we have an LSP.

### Successes of supersymmetry

Given that there is as yet no solid, direct observational evidence for supersymmetry, has the theory actually accomplished anything? Yes, quite a bit.

To date, supersymmetry is "only" a theory, but it is a very good one. Unlike many other theories and models that have been investigated in high-energy physics, supersymmetry solves a number of theoretical problems without introducing any significant ones of its own. (For example, supergravity -- the extension of supersymmetry -- does have serious problems, as discussed above.) That a theory solves problems without introducing new ones is a welcome thing, and a very good sign. We can summarize the successes in several areas.

#### Relating physics at different energy scales

Because many terms automatically cancel out in supersymmetry calculations, it is relatively easy to calculate quantities at one energy scale give known values at another scale. The calculation of the electroweak mixing angle is a good example. This number basically represents the relative strength of the electromagnetic and weak forces. Since (by the unified theory of the electroweak force) the forces must have equal strength at very high energy, one can compute the relative strengths at low energies where the results can actually be measured. The outcome is a very good fit between prediction and theory.

If one then considers the strong force and takes its known strength at low energies relative to electromagnetism and the weak force, its strength at high energy can be computed. The outcome is that the strong force becomes equal to the other two at exactly the same energy level where the others become equal. This is a pretty striking result.

#### Higgs physics

The Higgs mechanism for explaining the masses of the weak force bosons and of the standard model quarks and leptons works very well. As part of the deal it also explains the breaking of the electroweak symmetry and makes the theory mathematically consistent as well. But it also introduces new problems of its own.

The first problem is that the Higgs fields with appropriate properties are ad hoc entities. There is no obvious explanation of where the Higgs fields themselves come from. Supersymmetry provides at least a partial explanation, as indicated above.

#### The hierarchy problem

The second problem with the Higgs physics is the hierarchy problem, the difficulty of explaining how nature is so precisely fine-tuned that the masses of the electroweak bosons are so small -- something like 14 orders of magnitude smaller than might be expected. Although the problem seems abstract, it can be compared to picking two cars at random off the street and, upon weighing them, finding that they weigh the same to within a hundred millionth of a gram (10-8 gm) -- about the weight of 1000 bacteria.

Supersymmetry, again due to the magical cancellations that it introduces into the computations, explains this seemingly incredible coincidence in a "natural" way.

#### Quantum gravity

Supersymmetry does not, unfortunately, yield a satisfactory quantum theory of gravity. But it comes significantly closer than previous attempts. As we discussed above, when the Poincar&eactute; group is used as a local symmetry group, the gauge boson that results is the graviton. And when the same is done with the supersymmetry group, which contains the Poincaré group as a subroup, the gravitino -- supersymmetric partner of the graviton -- emerges.

When gravitons and gravitinos as both accounted for in the field theory equations, once again wonderful cancellations result. Although they are not sufficient to remove all the infinities due to higher order Feymnan diagrams, they do show that another remarkable circumstance -- the astonishing weakness of the gravitational force -- can be accounted for, at least in part. This weakness, specifically, is the fact that the gravitational force between two protons in an atomic nucleus is 1036 times as weak as the electromagnetic force (of repulsion).

#### Dark matter

Because supersymmetry predicts the existence of superpartners of all the known fermions and bosons, and because all of these superpartners must be quite massive, the theory provides an abundance of candidates for the cold dark matter which is rather definitely known to be necessary to make up the bulk of the mass of the universe. Most of the superparters, to be sure, are likely to be unstable, but the lightest among them, at least, must be stable.

While there may be other massive particles which interact at most very weakly with "normal" matter and could make up the dark matter, all are at least as exotic as superpartners, and all have so far evaded direct detection as well.

#### Superstring theory

Last, but hardly least, supersymmetry is an essential part of the theory of superstrings. The various theoretical virtues of supersymmetry have fit together harmoniously and effectively with string theory.

It appears that there is even a place for that ugly duckling -- 11-dimensional supergravity. The most recent extensions of superstring theory -- known as M-theory -- allow for supergravity as one type of approximation of the more complete (and still shadowy) edifice.

### Recommended references: Web sites

#### Site indexes

The Net Advance of Physics: Supersymmetry and String Theory
An index of tutorial and research articles located at the physics preprint archive. Topics include supersymmetry, duality theory, mirror symmetry.

#### Sites with general resources

Gordon L. Kane
Home page. Contains links to some lectures, talks, and publications related to sypersymmetry.
CU High Energy Physics Supersymmetry Studies page
Research group at the University of Colorado at Boulder.
Supersymmetry Studies: University of Colorado-Boulder
Contains a little information about the university's program and a few external links.

#### Surveys, overviews, tutorials

Supersymmetry
Supersymmetry Phenomenology
A collection of educational overviews and tutorials on supersymmetry by Hitoshi Murayama. Includes a good single-page overview: Introduction to Supersymmetry, based on a longer pedagogical review. The author has other, more technical papers on Supersymmetric Field Theories.
In SUSY we trust: What the LHC is really looking for
November 2009 New Scientist article. "If the theorists are right, before it ever finds the Higgs, the LHC will see the first outline of something far bigger: the grand, overarching theory known as supersymmetry. SUSY, as it is endearingly called, is a daring theory that doubles the number of particles needed to explain the world. And it could be just what particle physicists need to set them on the path to fresh enlightenment."
Introduction to Supersymmetry at the NLC
Discussion of the standard model and supersymmetry, and how they will be investigated at the Next Linear Collider facility. By David Wagner (University of Colorado at Boulder).
A survey on supersymmetry and supergravity
Short article by Paulo Vargas-Moniz.
Composite Models and Supersymmetry
Lecture notes by Steve Lloyd, from a course on Elementary Particle Physics.
Supersymmetry
Several very brief pages by John Schwarz as part of his superstrings site.
Evidence for supersymmetry found
August 1999 news article concerning manifestations of supersymmetry in atomic nuclei.
An Introduction to Supersymmetry
Technical introductory paper on supersymmetric field theories in general and the minimal supersymmetric Standard Model (MSSM) in particular, by Manuel Drees.
Superspace, or One thousand and one lessons in supersymmetry
Long 1983 book on superspace and supersymmetry, by S. J. Gates, M. T. Grisaru, M. Roček, and W. Siegel. (Very large PDF file.)
Supersymmetry stands the test
October 1999 article from Physics World, by Piet Van Isacker, about strong evidence for the existence of nucler supersymmetry.
Evidence for supersymmetry found
August 1999 news article from Physics World about experimental evidence that supersymmetry exists in atomic nuclei.

### Recommended references: Magazine/journal articles

Uncovering Supersymmetry
Jan Jolie
Scientific American, July 2002
Desperately Seeking SUSY
Hans Christian von Baeyer
The Sciences, September/October 1998, pp. 10-13
The theory of supersymmetry, which has been developing slowly for many years, may be near experimental verification. Among other things, it can account for the force of gravity, as an especially weak force, due to the cancellation of much larger opposing supersymmetric quantities.
Is Nature Supersymmetric?
Howard E. Haber; Gordon L. Kane
Scientific American, June 1986, pp. 52-60
Supersymmetry postulates a new symmetry relationship between fermions and bosons. The theory should be testable in accelerators that should become availble in the foreseeable future.
Supergravity and the Unification of the Laws of Physics
Daniel Z. Freedman; Peter van Nieuwenhuizen
Scientific American, February 1978, pp. 126-143
Supergravity is a theory in which gravity arises from a symmetry between matter and energy particles (fermions and bosons). If the theory is successful, it could provide a unified theory of the four known forces of nature.

### Recommended references: Books

Gordon Kane -- Supersymmetry: Unveiling the Ultimate Laws of Nature
Perseus Publishing, 2000
In this relatively short volume the author presents a number of themes from particle physics and the Standard Model by focusing specifically on the notion of supersymmetry. He explains what the theory of supersymmetry is and how it can be tested. From there he considers additional topics, such as the physics of Higgs particles and the relationship of supersymmetry to string theory.
[Book review]