As this is being written, particle physics stands on the threshold of a new era, with the commissioning of the Large Hadron Collider (LHC) not even two years away. In writing this book, I hope to help prepare graduate students and postdoctoral researchers for what will hopefully be a period rich in new data and surprising phenomena.

The Standard Model has reigned triumphant for three decades. For just as long, theorists and experimentalists have speculated about what might lie beyond. Many of these speculations point to a particular energy scale, the teraelectronvolt (TeV) scale which will be probed for the first time at the LHC. The stimulus for these studies arises from the most mysterious – and still missing – piece of the Standard Model: the Higgs boson. Precision electroweak measurements strongly suggest that this particle is elementary (in that any structure is likely far smaller than its Compton wavelength), and that it should be in a mass range where it will be discovered at the LHC. But the existence of fundamental scalars is puzzling in quantum field theory, and strongly suggests new physics at the TeV scale. Among the most prominent proposals for this physics is a hypothetical newsymmetry of nature, supersymmetry, which is the focus of much of this text. Others, such as technicolor, and large or warped extra dimensions, are also treated here.

Even as they await evidence for such new phenomena, physicists have become more ambitious, attacking fundamental problems of quantum gravity, and speculating on possible final formulations of the laws of nature.

In previous chapters, we have seen that string theory at the classical level shows promise of describing the Standard Model, and can realize at least one scenario for physics beyond: low-energy supersymmetry. But there are many puzzles, most importantly the existence of moduli and the related question of the cosmological constant. At tree level, in the Calabi–Yau solutions, the cosmological constant vanishes. But whether this holds in perturbation theory and beyond requires understanding of the quantum theory.

In studying string theory, we have certain tools:

1. (1) weak coupling expansions,

2. (2) long-wavelength (low-momentum, α′) expansions.

We have exploited both of these techniques up to this point. In analyzing string spectra, we have worked in a weak coupling limit. There are corrections to the masses and couplings, for example, in string perturbation theory, and most of the states that we have studied have finite lifetimes. At weak coupling, these effects are small, but at strong coupling, the theories presumably look dramatically different.

In asserting that Calabi–Yau vacua are solutions of the string equations, we used both types of expansions. We wrote the string equations both in lowest order in the string coupling, and also with the fewest number of derivatives (two). Even at weak coupling and in the derivative expansion, we can ask whether Calabi–Yau spaces are actually solutions of the string equations, both classically and quantum mechanically. For example, we have seen that, at lowest order in both expansions, there are typically many massless particles. We might expect tadpoles to appear for these fields, both in the α′ and in loops.

One of the original reasons for interest in supersymmetry was the possibility of dynamical supersymmetry breaking. So far, however, we have exhibited models in which supersymmetry is unbroken, as in the case of QCD with only massive quarks, or models with moduli spaces or approximate moduli spaces. In this section, we describe a number of models in which non-trivial dynamics breaks supersymmetry. We will see that the dynamical supersymmetry breaking occurs under special, but readily understood, conditions. In some cases, we will be able to exhibit this breaking explicitly, through systematic calculations. In others, we will have to invoke more general arguments.

Models of dynamical supersymmetry breaking

We might ask why, so far, we have not found supersymmetry to be dynamically broken. In supersymmetric QCD with massive quarks, we might give the index as an explanation. We might also note that there is not a promising candidate for a goldstino. With massless quarks, we have flat directions, and as the fields get larger, the theory becomes more weakly coupled, so any potential tends to zero. This suggests two criteria for finding models with dynamical supersymmetry breaking.

1. (1) The theory should have no flat directions at the classical level.

2. (2) The theory should have a spontaneously broken global symmetry.

The second criterion implies the existence of a Goldstone boson. If supersymmetry is unbroken, any would-be Goldstone boson must lie in a multiplet with another scalar, as well as aWeyl fermion. This other scalar, like the Goldstone particle, has no potential, so the theory has a flat direction. But by assumption, the theory classically (and therefore almost certainly quantum mechanically) has no flat direction.

In the previous chapter, we were forced to face the fact that string theory, if it describes nature, is not weakly coupled. On the other hand, the very formulation we have put forward of the theory is perturbative. We have described the quantum mechanics of single strings, and given a prescription for calculating their interactions order by order in perturbation theory in a parameter gs. There is a parallel here to Feynman's early work on relativistic quantum theory: Feynman guessed a set of rules for computing perturbative amplitudes of electrons. In this case, however, one already had a candidate for an underlying description: quantum electrodynamics. It was Dyson who clarified the connection. For Abelian theories, the non-perturbative theory probably does not really exist, but in the case of non-Abelian gauge theories it does. The field theoretic formulation provides an understanding of the underlying symmetry principles, and access to a trove of theoretical information.

A string field theory would be a complicated object. The string fields themselves would be functionals of the classical two-dimensional fields which describe the string. The quantization of such fields is sometimes called “third quantization.” Much effort has been devoted to writing down such a field theory. For open strings, one can write relatively manageable expressions which reproduce string perturbation theory. For closed strings, infinite sets of contact interactions are required. But apart from their cumbersome structure, there are reasons to suspect that this is not a useful formulation. There would seem to be, for example, vastly too many degrees of freedom.

Even as the evidence for the Standard Model became stronger and stronger in the 1970s and beyond, so the evidence for general relativity grew in the latter half of the twentieth century. Any discussion of the Standard Model and physics beyond must confront Einstein's theory at two levels. First, general relativity and the Standard Model are very successful at describing the history of the universe and its present behavior on large scales. General relativity gives rise to the big bang theory of cosmology, which, coupled with our understanding of atomic and nuclear physics, explains – indeed predicted – features of the observed universe. But there are features of the observed universe which cannot be accounted for within the Standard Model and general relativity. These include the dark matter and the dark energy, the origin of the asymmetry between matter and antimatter, the origin of the seeds of cosmic structure (inflation), and more. Apart from these observational difficulties, there are also serious questions of principle. We cannot simply add Einstein's theory onto the Standard Model. The resulting structure is not renormalizable, and cannot represent in any sense a complete theory. In this book we will encounter both of these aspects of Einstein's theory. Within extensions of the Standard Model, in the next few chapters, we will attempt to explain some of the features of the observed universe. The second, more theoretical, level, is addressed in the third part of this book. String theory, our most promising framework for a comprehensive theory of all interactions, encompasses general relativity in an essential way; some would even argue that what we mean by string theory is the quantum theory of general relativity.