by Jim Baggott
As Noether discovered, continuous transformations of spacetime symmetries — of time, space and rotation — are associated with the conservation of energy and linear and angular momentum. An important theorem developed in the late 1960s suggested that this was it — there could be no further spacetime symmetries. However, it soon became apparent that the theorem was not watertight; it contained a big loophole. It assumed that in any symmetry transformation, fermions would continue to be fermions and bosons would continue to be bosons.
This brings us to an important assumption.
The supersymmetry assumption. In essence, SUSY is based on the assumption that there exists a fundamental spacetime symmetry between matter particles (fermions) and the force particles (bosons) that transmit forces between them, such that these particles can transform into each other.
It is essential to our understanding of what follows that we grasp the significance of this last sentence.
The standard model of particle physics and the great variety of experimental data that has been gathered to validate it offer no real clues as to how its rather obvious failings can be addressed and corrected. Physicists have no real choice therefore but to go with their instincts.
And their instincts tell them that at the heart of the solution must lie some kind of fundamental symmetry. As Gordon Kane, a leading spokesperson for supersymmetry, explained:
Supersymmetry is the idea, or hypothesis, that the equations of [a unified theory] will remain unchanged even if fermions are replaced by bosons, and vice versa, in those equations in an appropriate way. This should be so in spite of the apparent differences between how bosons and fermions are treated in the standard model and in quantum theory … It should be emphasized that supersymmetry is the idea that the laws of nature are unchanged if fermions [are interchanged with] bosons.6
Assumption, idea, hypothesis. Call it what you will, SUSY is basically an enormous bet. With no real clues as to how physics beyond the standard model should be developed, physicists have decided that they must take a gamble.
Okay, but if we’re going to bet the farm on Lucky Boy, running in the 3.15 at Chepstow, we will typically need a damn good reason. Why pin all our hopes on a fundamental spacetime symmetry between fermions and bosons? For the simple reason that a symmetry of this kind offers exactly the kind of cancellation required to fine—tune the Higgs mass.
Stephen Martin again:
Comparing [these equations] strongly suggests that the new symmetry ought to relate fermions and bosons, because of the relative minus sign between fermion loop and boson loop contributions to [the Higgs mass] … Fortunately, the cancellation of all such contributions to scalar masses is not only possible, but is actually unavoidable, once we merely assume that there exists a symmetry relating fermions and bosons, called a supersymmetry.7
Superpartners and supersymmetry-breaking
In fact, supersymmetry is not so much a theory as a property of a certain class of theories. There are therefore many different kinds of supersymmetric theories. To keep things reasonably simple, I propose to explore some of the consequences of supersymmetry by reference to something called the Minimal Supersymmetric Standard Model, usually abbreviated as the MSSM. This was first developed in 1981 by Howard Georgi and Greek physicist Savas Dimopoulos,* and is the simplest supersymmetric extension to the current standard model of particle physics.
So, what are the consequences of assuming a symmetry relation between fermions and bosons in the MSSM? The answer is, perhaps, relatively unsurprising. In the MSSM, the various particle states form supermultiplets, each of which contains both fermions and bosons which are ‘mirror images’ of each other.
Every particle in the standard model has a superpartner in the MSSM. For every fermion (half-integral spin) in the standard model, there is a corresponding so-called scalar fermion, or sfermion, which is actually a boson (zero spin). For every boson in the standard model, there is a corresponding bosino, which is actually a fermion with spin ½.
To generate the names of fermion superpartners we prepend an ‘s’ (for scalar). So, the superpartner of the electron is the scalar electron, or selectron. The muon is superpartnered by the smuon. The superpartner of the top quark is the stop squark, and so on.
To generate the names of the boson superpartners we append ‘-ino’. The superpartner of the photon is the photino. The W and Z particles are partnered by winos* and zinos. Gluons are partnered by gluinos. The Higgs boson is partnered by the Higgsino (actually, several Higgsinos). Once you’ve grasped the terminology, identifying the names and the properties of the superpartners becomes relatively straightforward. The superpartners of the standard model particles are summarized in Figure 6.
This all seems rather mad. And yet we’ve come across something very similar once already in the history of quantum and particle physics. There are precedents for this kind of logic. Recall that Paul Dirac discovered — from purely theoretical considerations — that for every particle there must exist an anti-particle. A negatively charged electron must be ‘partnered’ by a positive electron. The positron was discovered shortly after Dirac’s discovery, and anti—particles are now familiar components of the standard model.
The symmetry between particle and anti-particle, between positive and negative and negative and positive, is an exact symmetry. This means that a positron has precisely the same mass as an electron and is to all intents and purposes identical to an electron but for its charge. Similarly for all other particle—anti-particle pairs.
Figure 6 Supersymmetry predicts that every particle in the standard model must possess a corresponding superpartner. Matter particles — leptons and quarks (which are fermions) — are partnered by sleptons and squarks (bosons). Force particles (bosons) are partnered by bosinos (fermions), such as the photino, wino, zino and gluinos. The Higgs boson is partnered by the Higgsino.
But supersymmetry cannot be an exact symmetry. If the symmetry between fermions and bosons were exact, then we would expect the superpartners to have precisely the same masses as their standard model counterparts. In such a situation, we wouldn’t be talking about assumptions, ideas or hypotheses, because our world would be filled with selectrons and massless photinos.* These super—particles (or ‘sparticles’) would already be part of our experience. And, consequently, they would already form part of our theoretical models.
You can guess where this is going. The fact that the world is not already filled with superpartners that impress themselves on our experience and shape our empirical reality suggests that the exact symmetry between partners and superpartners no longer prevails. Something must have happened to force a distinction between the standard model particles and their superpartners. In other words, at some time, presumably shortly after the big bang, the supersymmetry must have been broken.
We assume that the superpartners gained mass in this process, such that they are all (rather conveniently) much heavier than the more familiar particles of the standard model, and have so far stayed out of reach of terrestrial particle colliders.
Cynicism is cheap, and not altogether constructive. It’s certainly true that we seem to be building assumptions on top of assumptions. If superpartners exist then they must be massive. But, to a certain extent, suggesting that there must exist — from purely theoretical considerations — massive superpartners for every particle in the standard model is no different in principle to Dirac’s discovery of antimatter.
Before we go on to consider what the MSSM predicts and how it might be tested, it’s useful to review the potential of the theory to resolve many of the current problems with the standard model.
SUSY and the hierarchy problem
It is already apparent that one of the most compelling arguments in favour of SUSY is that it provides a perfectly logical, natural resolution of the hierarchy problem, at least in terms of stabilizing the electro-weak Higgs mass.
Recall from the last chapter that the hierarchy problem has two principal mani
festations. There is the inexplicable gulf between the energy scale of the weak force and electromagnetism and the Planck scale. And then there is the problem that quantum corrections to the Higgs mass should in principle cause it to mushroom in size all the way to the Planck mass, completely at odds with the electro-weak energy scale and recent experiments at the LHC which suggest a Higgs boson with a mass of 125 GeV.
At a stroke, SUSY eliminates the problems with the radiative corrections. The Higgs mass becomes inflated through interactions particularly with heavy virtual particles, such as virtual top quarks. The top quark is a fermion, and in the MSSM we must now include radiative corrections arising from interactions with its corresponding sfermion, the stop squark.
Now, I have very limited experience and virtually no ability as a mathematician. But what experience I do have allows me the following insight. If, when grappling with a complex set of mathematical equations, you are able to show that all the terms cancel beautifully and the answer is zero, the result is pure, unalloyed joy.
And this is what happens in SUSY. The positive contributions from radiative corrections arising from interactions with virtual particles are cancelled by negative contributions from interactions with virtual sparticles.*
The cancellation is not exact because, as I mentioned above, the symmetry between particles and sparticles cannot be exact. This is okay; the theory can tolerate some inexactness. However, force-fitting a light Higgs mass and broken supersymmetry does place some important constraints on the theory. Most importantly, the masses of many of the superpartners cannot be excessively large. If they exist, then they must possess masses of the order of a few hundred billion electron volts up to a trillion electron volts. Much heavier, and they couldn’t serve the purpose of stabilizing the Higgs at the observed mass, and hence establishing the scale of the weak force and electromagnetism.
This has important implications for the testability of the theory, which I will go on to examine below.
SUSY and the convergence of subatomic forces
In 1974, Stephen Weinberg, Howard Georgi and Helen Quinn showed that the interaction strengths of the electromagnetic, weak and strong nuclear forces become near equal at energies around 200,000 billion GeV. The operative term here is ‘near equal’. As Figure 7(a) shows, extrapolating the standard model interaction strengths up to this scale shows that they become similar, but do not converge.
We tend to assume that the subatomic forces that we experience today are the result of symmetry-breaking applied to a unified electro-nuclear force shortly after the big bang. If this is really the case, then it seems logical to expect that the interaction strengths of these forces should converge on the energy that prevailed at the end of the grand unified epoch. Although this is a sizeable extrapolation — we’re trying to predict behaviour at energies about 14 orders of magnitude beyond our experience — the fact that the standard model interaction strengths do not converge is perhaps a sign that something is missing from the model.
The MSSM resolves this problem, too. The effects of interactions with virtual sparticles change the extrapolation such that the strengths of the forces smoothly converge on a near—single point, as shown in Figure 7(b). This is much more like it. Now the extrapolation is strongly suggestive of a time and an energy regime where a single electro-nuclear force dominated.
If we are prepared to draw conclusions from such calculations, then it does seem that something significant is missing from the standard model, and that something could be SUSY.
SUSY, the LSP and the problem of dark matter
The problem of dark matter demands a solution that lies beyond the current standard model of particle physics. By definition, dark matter must consist of a particle or particles that are not to be found among the known families of quarks, leptons and force carriers. Dark matter candidates can only be found among new particles predicted by theories that extend the standard model in some way.
It probably won’t come as much of a surprise to learn that SUSY predicts the existence of particles with precisely the right kinds of properties.
Figure 7 (a) Extrapolating the strengths of the forces in the standard model of particle physics implies an energy (and a time after the big bang) at which the forces have the same strength and are unified. However, the forces do not quite converge on a single point. (b) In the Minimum Supersymmetric Standard Model (MSSM) the additional quantum fields change this extrapolation, and the forces more nearly converge.
Remember, there is a category of candidate for cold dark matter particles known as WIMPs. These are neutrino-like (neutral, not subject to the strong force or electromagnetism) but much, much heavier. By design, the MSSM predicts superpartners that must be heavy (as they haven’t been observed yet) and several of these are neutral. To take an example, the photino — the heavy superpartner of the photon — readily fits the bill.
But many potential candidate superpartners are believed to be relatively unstable. They will decay rapidly into other sparticles. We can conclude, with some sense of justification, that the cold dark matter responsible for governing the shapes, structures and rotation speeds of whole galaxies is likely to be much more enduring than this. The attentions of theorists have therefore turned to the sparticle most likely to lie at the end of the chain of decays. This is referred to as the Lightest Superpartner, or LSP. Simply put, once the sparticles have decayed into the LSP, there’s nowhere else for it to go. The LSP is stable.
Now, the neutralino is not, as might first be thought, the superpartner of the neutrino.* In the MSSM, neutralinos are formed through the quantum superposition of the zino, the photino and a couple of electrically neutral Higgsinos. This mixing occurs because these particles have similar properties (such as charge and spin), and there is an unwritten law in quantum theory that if particles with mass cannot be distinguished by their other quantum properties, then they are likely to suffer an identity crisis and combine in a superposition. The mixing produces four neutralino states, the lightest of which is a candidate for the LSP and therefore a candidate WIMP.
Calculations suggest that if the MSSM is right and neutralinos do exist, then their ‘relic density’ — the density of neutralinos left over after the big bang — is consistent with the observed density of cold dark matter in the ACDM model.
Supergravity
It does seem as though we’re getting quite a lot in return for our willingness to suspend disbelief and embrace the idea of heavy superpartners. We solve the hierarchy problem. We enable convergence between the fundamental forces that operate at atomic and subatomic levels. We gain some candidates for cold dark matter particles.
There is yet more.
In SUSY, a spacetime symmetry transformation acting on a fermion or boson changes the spin of the particle by ½. Fermions become sfermions, with spin zero. Bosons become bosinos, with spin ½. This kind of change affects the spacetime properties of the particle that is transformed, such that it is slightly displaced.* Conventional standard model forces such as electromagnetism cannot displace particles in this way. They can change the direction of motion, momentum and energy of the particle but they cannot displace it in spacetime. In fact, this displacement is equivalent to a transformation characteristic of the gravitational force.
It is possible to conceive a supersymmetry theory that is also therefore a theory of gravity. This is actually quite remarkable. The standard model itself does not accommodate gravity at all, and attempts to create a quantum field theory of gravity have led to little more than a hundred years of frustration. By extending the standard model to include supersymmetry, it seems that we open the door to gravity.
Such theories are collectively called supergravity. They introduce the gravitino, the superpartner of the graviton, the notional force carrier of quantum gravitation. It seems that supersymmetry not only offers the promise of illuminating the path to a grand unified theory, but may also be a key ingredient in any theory purporting to be a theory of everything.
One of the first theories of supergravity was developed in 1976 by American Daniel Freedman, Dutch physicist Peter van Nieuwenhuizen and Italian Sergio Ferrara. They discovered that some of the problems associated with renormalizing a quantum field theory of gravity were somewhat relieved if supersymmetry was assumed. It seemed that the contributions from terms in the equations that had mushroomed to infinity could be partly offset by terms derived from the gravitino. There was indeed some cancellation, but the problem didn’t go away completely.
For a relatively short time, excitement built up around a version of supergravity based on eight different kinds of supersymmetry. Such theories include not only the particles of the standard model and their superpartners, but many others as well. But these theories could not be renormalized, and by 1984, interest in them was waning.
The reality check
Obviously, SUSY has a lot going for it, and many contemporary theoretical physicists are convinced that nature must be supersymmetric. Of course, the theory will stand or fall on whether or not sparticles are observed in high-energy particle collisions, or supersymmetric WIMPs are detected.
The big problem with SUSY is that the supersymmetry must be broken, and it is not at all obvious how this is supposed to happen. Using a Higgs-like mechanism which ties symmetry-breaking to a real scalar field and a real scalar boson gives incorrect particle masses. Mechanisms for ‘soft’ supersymmetry-breaking have been devised, but these tend to introduce yet more fields and yet more particles.