As this book is being completed, the Large Hadron Collider (LHC) at CERN, and its two large detectors, ATLAS and CMS, are nearing completion. The center of mass energy at this machine will be large, about 14 TeV. The center of mass energies of the partons – the quarks and gluons – within the colliding protons will be larger than 1 TeV. The luminosity will also be very large. As a result, if almost any of the ideas we have described for understanding the hierarchy problem in Part 1 of this book are correct, evidence should appear within a few years. For example, if the hypothesis of low-energy supersymmetry is correct, we should see events with large amounts of missing energy, and signatures such as multiple leptons. Large extra dimensions should be associated with rapid growth of cross sections for various processes, again with missing energy; the warped spaces suggested by Randall and Sundrum should be associated with the appearance of massive resonances. Technicolor, similarly, should lead to broad resonances. Assuming some underlying technicolor model can satisfy constraints from flavor physics and precision electroweak measurements, one might expect to find some number of light (compared with 1 TeV), pseudo-Goldstone bosons, many with gauge quantum numbers. If any of these phenomena occur, distinguishing among them in the complicated environment of a hadron machine will be challenging. It is conceivable that there will be competing explanations, and that choosing between them will require a very high-energy electron–positron colliding beam machine. Such a machine is under consideration by a consortium of nations, and is referred to as the International Linear Collider, or ILC.

While motivated in part by the hopes of building phenomenologically successful models of particle physics, we have uncovered in our study of supersymmetric theories a rich trove of field theory phenomena. Supersymmetry provides powerful constraints on dynamics. In this chapter, we will discover more remarkable features of supersymmetric field theories. We will first study classes of (super)conformally invariant field theories. Then we will turn to the dynamics of supersymmetric QCD with Nf ≥ Nc, where we will encounter new, and rather unfamiliar, types of behavior.

Conformally invariant field theories

In quantum field theory, theories which are classically scale-invariant typically are not scale invariant at the quantum level. QCD is a familiar example. In the absence of quark masses, we believe the theory confines and has a mass gap. The CPN models are an example where we were able to show systematically how a mass gap can arise in a scale-invariant theory. The breaking of scale invariance in all of these cases is associated with the need to impose a cutoff on the high-energy behavior of the theory. In a more Wilsonian language, one needs to specify a scale where the theory is defined, and this requirement breaks the scale invariance.

There is, however, a subset of field theories which are scale invariant. We have seen this in the case of N = 4 supersymmetric field theories in four dimensions. In this section, we will see that this phenomenon can occur in N = 1 theories, and explore some of its consequences. In the next section we will discuss a set of dualities among N = 1 supersymmetric field theories, in which conformal invariance plays a crucial role.

Very quickly after Einstein published his general theory, a number of researchers attempted to apply Einstein's equations to the universe as a whole. This was a natural, if quite radical, move. In Einstein's theory, the distribution of energy and momentum in the universe determines the structure of space-time, and this applies as much to the universe as a whole as to the region of space, say, around a star. To get started these early researchers made an assumption which, while logical, may seem a bit bizarre. They took the principles enunciated by Copernicus to their logical extreme, and assumed that space-time was homogeneous and isotropic, i.e. that there is no special place or direction in the universe. They had virtually no evidence for this hypothesis at the time – definitive observations of galaxies outside of the Milky Way were only made a few years later. It was only decades later that evidence in support of this cosmological principle emerged. As we will discuss, we now know that the universe is extremely homogeneous, when viewed on sufficiently large scales.

To implement the principle, just as, for the Schwarzschild solution, we begin by writing the most general metric consistent with an assumed set of symmetries. In this case, the symmetries are homogeneity and isotropy in space. A metric of this form is called Friedmann–Robertson–Walker (FRW). We can derive this metric by imagining our three-dimensional space, at any instant, as a surface in a four-dimensional space. There should be no preferred direction on this surface; in this way, we will impose both homogeneity and isotropy. The surface will then be one of constant curvature.

The last chapter developed the general principles for writing down a relativistic quantum field theory. It showed what types of fields are possible, and explained that spin-one fields can only appear in an interacting, renormalizable theory if they are coupled via the gauge principle.

In this chapter, we write down specifically what the field content of the standard model is. The interactions will then follow as the most general set of renormalizable interactions, compatible with that field content. We then explore what the vacuum and the particle content are, and write down the complete interaction Hamiltonian in the particle basis.

We will not attempt to motivate theoretically why the particle content of the standard model is what it is. We have no deep understanding of why the gauge group is SUc(3) × SUL(2) × UY (1), for instance. We just take the field content as observed fact, and present it. The exception is the Higgs boson, which has not been observed. This is the weakest part of our understanding of the standard model. Note however that the field content of the standard model is not completely arbitrary; once the gauge group is known, the fermionic field content is somewhat constrained by the requirement of anomaly cancellation, which we discuss at the end of the chapter.

Up until this point quarks and gluons have been treated in basically the same manner as have leptons, with little acknowledgement of the plain fact that quarks and gluons are never directly seen in experiments in the same way as are electrons, protons, and pions. This voluntary blindness has been possible to the extent that we have always considered processes that have the following two important properties:

1. (i) they never involve any strongly interacting particles in the initial state, and

2. (ii) they never involve any specific combinations of strongly-interacting particles in the final state – so-called exclusive processes.

The only reactions involving hadrons that have been contemplated are inclusive ones, i.e. those for which we have summed over all possible combinations of hadrons that could be produced, sometimes subject to some general flavor-conservation rules.

This is obviously a fairly serious handicap, since the vast bulk of the reactions that are seen in experiments involve strongly-interacting particles of one sort or another. In order to be considered a success, the standard model must provide at least a qualitative, but preferably also a quantitative, picture of these processes. The model does indeed provide such a framework. This chapter and Chapter 9 are devoted to outlining to what extent predictions and post-dictions can be made and to what extent they are successful. The goal is to focus here on those calculations that can be made with the minimum of modeling of the unknown dynamics of strongly-coupled physics.

The standard model – augmented (say) with dimension-5 operators to account for neutrino oscillations – explains all particle physics experiments performed to date (2006). Yet there are a number of reasons to believe that it is incomplete, and should be regarded at best as being the effective theory describing particle physics at the energy scales which have been probed experimentally (roughly several hundred GeV).

This chapter aims to summarize these reasons, with an eye to identifying the main themes which govern the searches for the standard model's replacement. These themes typically revolve about “puzzles,” which either center around attempts to explain the values of some of the standard model's couplings, or around speculations about what kinds of new particles might exist at very large masses, and what their implications might be for experiments at accessible energies. Our goal in this summary is not to be exhaustive, but rather to provide a conceptual framework for further reading of the many research directions within the literature on the broad topic of physics “beyond the standard model.”

The organizing theme for our discussion is the assumption that any particles which have not yet been discovered must be heavy compared with the energies to which we presently have experimental access. This assumption has three motivations, not least of which is the outstanding success of the standard model itself.

The bosonic string theory that was discussed in the previous chapters is unsatisfactory in two respects. First, the closed-string spectrum contains a tachyon. If one chooses to include open strings, then additional open-string tachyons appear. Tachyons are unphysical because they imply an instability of the vacuum. The elimination of open-string tachyons from the physical spectrum has been understood in terms of the decay of D-branes into closed-string radiation. However, the fate of the closed-string tachyon has not been determined yet.

The second unsatisfactory feature of the bosonic string theory is that the spectrum (of both open and closed strings) does not contain fermions. Fermions play a crucial role in nature, of course. They include the quarks and leptons in the standard model. As a result, if we would like to use string theory to describe nature, fermions have to be incorporated. In string theory the inclusion of fermions turns out to require supersymmetry, a symmetry that relates bosons and fermions, and the resulting string theories are called superstring theories. In order to incorporate supersymmetry into string theory two basic approaches have been developed

• The Ramond–Neveu–Schwarz (RNS) formalism is supersymmetric on the string world sheet.

• The Green–Schwarz (GS) formalism is supersymmetric in ten-dimensional Minkowski space-time. It can be generalized to other background space-time geometries.

These two approaches are actually equivalent, at least for ten-dimensional Minkowski space-time.