The Standard Model of particle physics is tremendously successful because it can accurately predict the outcomes of experiments from the atomic scale (about 10^-10 m) all the way down to the shortest distances that can be probed in the laboratory (about 10^-18 m). However, particle theorists are deeply dissatisfied with the model and are racking their brains to find a theory that can go beyond it. Why are they doing this and what is wrong with the Standard Model?

First let us review the Standard Model. In many ways it is analogous to the periodic table of the elements in that the known fundamental particles of matter – the six quarks and the six leptons (i.e. the electron, muon, tau particle and the associated neutrinos) – can be arranged in a table according to their various quantum numbers such as electric charge, colour, flavour and spin.

However, the Standard Model goes much further than the periodic table because it can describe exactly how the quarks and leptons interact with each other through the exchange of “gauge bosons” (i.e. gluons for the strong force, W and Z bosons for the weak force and photons for the electromagnetic interaction). A feature of the model is that the matter particles – the quarks and leptons – all have “spins” of h-bar/2, where h-bar is Planck’s constant divided by 2φ, whereas the particles that carry forces have spins of h-bar. This means that all the fundamental matter particles are fermions, while the particles that carry forces are all bosons.

Since the top quark was discovered in 1995, three complete “families” of quarks and leptons and all the different gauge bosons have been seen in experiments. Moreover, their interactions have been measured very precisely and everything is in perfect agreement with the theory.

Let there be mass
The final ingredient of the Standard Model – the Higgs mechanism – describes how the fundamental particles obtain their masses. This mechanism, which is named after Peter Higgs of Edinburgh University, was discovered independently in 1964 by Francois Englert and Robert Brout, and by Gerald Guralnik, Dick Hagen and Tom Kibble. Higgs and the others – building on earlier work by Julian Schwinger and Phil Anderson – showed how electroweak symmetry could be broken, thus allowing particles to have mass. The photon, however, is massless because the symmetry of the electromagnetic force is not broken in nature. (Note that most of the mass of particles that are not fundamental, such as neutrons and protons, comes from the binding energy of the strong force that holds the quarks together, and not from the masses of the quarks themselves.)

According to the Standard Model, the vacuum in which all particle interactions take place is not actually empty, but is instead filled with a condensate of Higgs particles. The quarks, leptons, and W and Z bosons continuously collide with these Higgs particles as they travel through the “vacuum”. The Higgs condensate acts like molasses and slows down anything that interacts with it. The stronger the interactions between the particles and the Higgs condensate are, the heavier the particles become.

The Higgs mechanism is an essential part of the Standard Model. Without it the quarks and leptons – and also the W and Z bosons – would all be massless and the world as we know it could not exist. However, the physics behind the Higgs mechanism is the least tested aspect of the Standard Model. Although we have much circumstantial evidence for the Higgs particle, given that fundamental particles have masses that are consistent with the Higgs mechanism and from indirect measurements at CERN and Stanford (so-called precision electroweak data), Higgs particles have never been directly produced and observed in collider experiments.

Nevertheless, this is not the reason why theorists are dissatisfied with the Standard Model. In fact, most theorists are actually convinced that the Higgs will be discovered in this decade either at the Tevatron at Fermilab or at the Large Hadron Collider (LHC) at CERN.

This prediction – which implies that the Higgs is light enough to be produced in collisions at the Tevatron or the LHC – can be understood as follows. Like quarks and leptons, the Higgs particle also derives its mass from coupling to the Higgs condensate. Furthermore, precision electroweak measurements indicate that the strength of the Higgs’ coupling to the condensate is not much larger than the corresponding coupling of the top quark. Therefore the Higgs particle cannot be much heavier than the top quark (which has a mass of 174.3 ± 5.1 GeV). More precisely, we expect it to have a mass above 114 GeV – as masses below this level have been ruled out by experiments at CERN – and below a few hundred GeV. By comparison, the mass of the proton is approximately 1 GeV.

The hierarchy problem
So what is the problem? The issue concerns the internal consistency of the theory – the Higgs sector of the Standard Model contains an instability that arises from quantum-mechanical interactions. In addition to the mass it acquires from interactions with the condensate, the Higgs particle can also gain mass as a result of interactions with virtual particles. The uncertainty principle of quantum mechanics allows pairs of short-lived virtual particles to “appear” from the vacuum and then disappear again. Although they have extremely short lifetimes, these virtual particles can have a significant impact on the properties of real particles.

Unfortunately for the Standard Model, these contributions grow with the energy of the virtual particles, and since virtual particles with arbitrarily large energies are allowed in quantum mechanics, it seems that quantum corrections make the mass of the Higgs particle arbitrarily large as well. This is clearly in contradiction with the requirement that the Higgs be lighter than a few hundred GeV. This is often called the “hierarchy problem”. (This issue only afflicts the Higgs because it has zero “spin”: it is not a problem for particles with non-zero spin, such as the gauge bosons.)

How can this problem be solved? The hierarchy problem tells us that we must modify the Standard Model into a new theory that takes over at energies of about 1000 GeV or 1 TeV. At first this might seem like bad news, but it is not because the Standard Model has only been experimentally tested at energies below about 1 TeV, so we have no real reason to believe that it provides a valid description of particles with higher masses and energies.

Any new theory must contain new particles beyond those in the Standard Model, and also new interactions that will somehow conspire to cancel the exceedingly large quantum corrections to the Higgs mass that are caused by the particles in the Standard Model. This conclusion by itself is very important and exciting: the hierarchy problem predicts that there is new physics beyond the Standard Model that is likely to be within the energy reach of the LHC. Thus not only can we expect to see Higgs particles produced at the LHC, we are also likely to discover the new particles that we need to solve the hierarchy problem.

What are these new particles? Well, nobody knows, and that is why we need to do experiments. However, it is interesting to speculate. Using the hierarchy problem as a guide, we can try to infer properties of these new particles. At energies above 1 TeV, calculations of the quantum corrections to the Higgs mass will include contributions from virtual particles from the Standard Model and also from particles in the new theory. We know that the contribution from the particles in the Standard Model grows with energy and quickly becomes too large. Since we do not know what the new particles are, we cannot compute their contributions. However, we know that in order to solve the hierarchy problem, the particles must precisely cancel the corrections due to the Standard Model particles.

For a long time the only known example of a theory in which this cancellation takes place was supersymmetry. Supersymmetry relates each Standard Model particle to a “superpartner” with opposite spin statistics – the superpartner for a fermion is a boson and vice versa. When computing quantum corrections to the Higgs mass in supersymmetry one discovers an amazing result: each Standard Model particle and its superpartner give equally large contributions, but they are opposite in sign and so cancel each other out exactly! Unfortunately, no superpartners have been observed in experiments, which implies that supersymmetric particles – if they do exist – must be heavier than the current experimental bound, which is of the order of 100 GeV.

Given this absence of experimental evidence for supersymmetry, some particle theorists have returned to the drawing board in a search for alternatives. Last year, in a major theoretical breakthrough, Nima Arkani-Hamed of the University of California at Berkeley, Andrew Cohen of Boston University and Howard Georgi of Harvard University discovered a new class of theories with the desired cancellation of quantum corrections (www.arXiv.org/abs/hep-ph/0105239). A significant number of theorists are now working on this new approach.

Enter the little Higgs

Similar to supersymmetry, these “little Higgs” theories – so-called because they generate a Higgs particle with a relatively small mass – also predict new particles with masses near 1 TeV. However, little-Higgs theories are based on a different symmetry principle and predict new particles with quantum numbers that are different to those predicted by supersymmetry.

Contrary to what happens in supersymmetry, in little-Higgs theories the cancellation of the quantum corrections occurs between fields of the same spin: fermions cancel fermions and bosons cancel bosons. Consequently, the new particles in the theory include fermionic partners for quarks and leptons, and also bosonic partners for gauge bosons.

One might wonder why it was so difficult to construct such models. The answer lies in the fact that it is not sufficient to simply postulate partners for each Standard Model particle in order to be able to obtain a precise cancellation. Moreover, there needs to be some reason for the quantum corrections from Standard Model particles and their partners to be of the same magnitude with the opposite sign: in other words a symmetry is required.

In little-Higgs theories this symmetry, the analogue of symmetry between bosons and fermions in supersymmetric theories, is a so-called nonlinearly realized symmetry. The importance of similar nonlinear symmetries for the cancellation of mass contributions was discovered in 1961 by Jeffrey Goldstone of the Massachusetts Institute of Technology. However, Arkani-Hamed, Cohen and Georgi are the first theorists who have been able to successfully incorporate these symmetries into an extension of the Standard Model and solve the hierarchy problem. In this theory a nonlinear symmetry unifies the Standard Model particles with heavy new particles. This unification relates the couplings of virtual particles to the Higgs in such a way so as to ensure that the quantum corrections cancel.

Little-Higgs models have only recently been discovered and the hunt for the simplest and most elegant little-Higgs model is still going on. In addition, particle theorists are beginning to investigate detailed experimental signatures. While the precise masses and other properties of the new particles in the little-Higgs theories are model dependent, some robust predictions can be made. First, little-Higgs theories predict one or several Higgs particles with masses at or below a few hundred GeV. Second, they predict at least one new heavy fermion particle with a mass of less than about 2 TeV – this particle is needed to cancel the very large quantum correction to the Higgs mass caused by top quarks. Third, they also predict new gauge bosons with TeV-scale masses that cancel the Higgs-mass corrections from weak and electromagnetic interactions.

It is impossible to tell from theory alone which new model – little Higgs, supersymmetry or something completely different altogether – will prove successful in the long run. Experiment has to decide. Fortunately, these experiments will be carried out. With luck, hints of these new particles may soon be discovered at Fermilab, but we will probably need to wait for the LHC to see the particles themselves. Construction of the LHC is scheduled to be completed by 2007 – a date that is eagerly awaited by theorists and experimentalists alike.