The standard model of particle physics, developed in the 1960s and 1970s, has stood for 30 years as “the” theory of particle physics, passing numerous stringent tests. In fact, while many people believe that the standard model is not a complete description of particle physics, it is expected to be, at worst, incomplete rather than wrong; that is, the standard model is at worst a subset of the true theory of particle physics.

For this reason, a good working knowledge of the standard model and its phenomenology is essential for the modern particle physicist. The goal of this book is to provide all the tools for a working, quantitative knowledge of the standard model, with the minimum of formal developments. It presents everything needed to understand the particle spectrum of the standard model, and how to compute decay rates and cross sections at leading order in the weak coupling expansion (tree level). We assume a solid quantum-mechanics background, up to and including canonical quantization and the Dirac equation, but we do not assume familiarity with formal quantum field theory (renormalization, path integrals, generating functionals).

As we see it, this book fills two gaps in the existing literature. The first of these concerns the balance between theoretical sophistication and phenomenological utility. Most treatments of the standard model appear at the end of quantum field theory books.

The previous chapters have revealed the standard model as a remarkable theory which successfully describes the vast majority of all of modern particle physics. Better yet, its implications are very robust since it is the most general theory which is consistent with general principles (like Lorentz invariance and unitarity) plus the two assumptions: (i) renormalizability and (ii) the given particle content – all of which are seen in experiments (modulo confinement for the case of quarks and gluons) save the as-yet-undiscovered Higgs scalar.

In this chapter we encounter the first (and, as of this writing, only) known case where there is good evidence that the standard model does not provide a good description: the phenomenon of neutrino oscillations. After describing the conceptual issues and the evidence for the standard model's failure, we briefly describe what can be said at present about which of the two core assumptions must be relinquished.

The standard-model prediction which has gone sour – encountered in Subsection 2.5.2 – asserts the separate conservations (up to negligible corrections involving the electroweak anomaly) of the three lepton numbers, Le, Lµ, and Lτ. Even considering the electroweak anomaly, the quantities Le − Lµ and Lµ − Lτ are anomaly free and so should be exactly conserved. As a consequence the theory predicts exactly massless and stable neutrinos, νe, νµ, ντ, with νℓ only participating in charged-current weak interactions together with its corresponding charged lepton, ℓ−.

A great deal of particle phenomenology, including both the properties of the observed particles and their reactions in many accelerators, deals with energy scales that are very small in comparison with the mass of the weak vector bosons, MW or MZ. A technique that has been used to good effect at various points in the previous chapters is the expansion of low-energy scattering amplitudes in inverse powers of the W- or Z-boson masses. This expansion greatly simplified the corresponding calculations and was justified in each case by the fact that the typical energies involved in the amplitudes under consideration were much smaller than MW and MZ.

Concrete examples where this type of expansion is justified are given by the weak decays of a light meson such as the muon, as was computed in Chapter 5, since the energy scales involved are much smaller than the mass of the virtual W boson that mediates these decays. A similar simplification is justified in Chapter 6 in the amplitudes for electron–positron annihilation at energies that are low compared to the Z-boson mass.

All of these examples furnish special cases of the general technique of low-energy expansions. This technique appears ubiquitously throughout physics because many physical systems have the property that they involve two (or more) degrees of freedom that each have very different masses.

The preceding chapters have described bosonic strings as well as type I and type II superstrings. In the case of the bosonic string, one was led to 26-dimensional Minkowski space-time by the requirement of cancellation of the conformal anomaly of the world-sheet theory. Similar reasoning led to the conclusion that the type I and type II superstring theories should have D = 10.

In all of these theories the world-sheet degrees of freedom can be divided into left-movers and right-movers, though in the case of open strings these are required to combine so as to give standing waves. In the case of the type II superstring theories, the left-moving and right-moving modes introduce independent conserved supersymmetry charges, each of which is a Majorana–Weyl spinor with 16 real components. Thus, the type II superstring theories have two such conserved charges, or N = 2 supersymmetry, which means that they have 32 conserved supercharges. The type IIA and type IIB theories are distinguished by whether the two Majorana–Weyl spinors have the same (IIB) or opposite (IIA) chirality. In the case of the type I theory, as well as related theories whose construction involves an orientifold projection, the only conserved supercharge that survives the projection is the sum of the left-moving and right-moving supercharges of the type IIB theory. Thus these theories have N = 1 supersymmetry in ten dimensions.

Unitary gauge, as described in Chapter 5, has several disadvantages that make it inappropriate for most calculations that go beyond tree level in the perturbative expansion. One of these difficulties is that the spin-one propagator does not fall to zero for large momenta, p → ∞, thereby making the ultraviolet behavior of the theory appear to be worse than it really is. It is therefore usually more convenient to use in these computations a gauge in which the proper ultraviolet behavior is more manifest. A one-parameter family of such gauges is given by the Lorentz-covariant ξ-gauges.

Although no loop graphs are attempted in this book, the modification of the Feynman rules appropriate for ξ-gauges are included here for the sake of completeness. There are three new types of propagator that arise in ξ-gauge. The first of these is a modified spin-one boson propagator that was given in Chapter 5.

The particular cases ξ = 1 and ξ = 0 are respectively known as Feynman–'t Hooft Gauge and Landau Gauge. There are also two other types of unphysical particles that arise in the ξ-gauge graphs. These are the unphysical scalars and the Fadeev–Popov–DeWitt ghosts. Their role is to cancel the contributions within loops of the various unphysical components of the vector boson propagator. Neither the unphysical scalars nor the ghosts ever appear in external lines in scattering amplitudes.

String theory is one of the most exciting and challenging areas of modern theoretical physics. It was developed in the late 1960s for the purpose of describing the strong nuclear force. Problems were encountered that prevented this program from attaining complete success. In particular, it was realized that the spectrum of a fundamental string contains an undesired massless spin-two particle. Quantum chromodynamics eventually proved to be the correct theory for describing the strong force and the properties of hadrons. New doors opened for string theory when in 1974 it was proposed to identify the massless spin-two particle in the string's spectrum with the graviton, the quantum of gravitation. String theory became then the most promising candidate for a quantum theory of gravity unified with the other forces and has developed into one of the most fascinating theories of high-energy physics.

The understanding of string theory has evolved enormously over the years thanks to the efforts of many very clever people. In some periods progress was much more rapid than in others. In particular, the theory has experienced two major revolutions. The one in the mid-1980s led to the subject achieving widespread acceptance. In the mid-1990s a second superstring revolution took place that featured the discovery of nonperturbative dualities that provided convincing evidence of the uniqueness of the underlying theory. It also led to the recognition of an eleven-dimensional manifestation, called M-theory.

During the “Second Superstring Revolution,” which took place in the mid- 1990s, it became evident that the five different ten-dimensional superstring theories are related through an intricate web of dualities. In addition to the T-dualities that were discussed in Chapter 6, there are also S-dualities that relate various string theories at strong coupling to a corresponding dual description at weak coupling. Moreover, two of the superstring theories (the type IIA superstring and the E8 × E8 heterotic string) exhibit an eleventh dimension at strong coupling and thus approach a common 11-dimensional limit, a theory called M-theory. In the decompactification limit, this 11-dimensional theory does not contain any strings, so it is not a string theory.

Low-energy effective actions

This chapter presents several aspects of M-theory, including its low-energy limit, which is 11-dimensional supergravity, as well as various nonperturbative string dualities. Some of these dualities can be illustrated using low-energy effective actions. These are supergravity theories that describe interactions of the massless fields in the string-theory spectrum. It is not obvious, a priori, that this should be a useful approach for analyzing nonperturbative features of string theory, since extrapolations from weak coupling to strong coupling are ordinarily beyond control. However, if one restricts such extrapolations to quantities that are protected by supersymmetry, one can learn a surprising amount in this way.

Since critical superstring theories are ten-dimensional and M-theory is 11-dimensional, something needs to be done to make contact with the four-dimensional space-time geometry of everyday experience. Two main approaches are being pursued.

Kaluza–Klein compactification

The approach with a much longer history is Kaluza–Klein compactification. In this approach the extra dimensions form a compact manifold of size lc. Such dimensions are essentially invisible for observations carried out at energy E ≪ 1/lc. Nonetheless, the details of their topology have a profound influence on the spectrum and symmetries that are present at low energies in the effective four-dimensional theory. This chapter explores promising geometries for these extra dimensions. The main emphasis is on Calabi–Yau manifolds, but there is also some discussion of other manifolds of special holonomy. While compact Calabi–Yau manifolds are the most straightforward possibility, modern developments in nonperturbative string theory have shown that noncompact Calabi–Yau manifolds are also important. An example of a noncompact Calabi–Yau manifold, specifically the conifold, is discussed in this chapter as well as in Chapter 10.

Brane-world scenario

A second way to deal with the extra dimensions is the brane-world scenario. In this approach the four dimensions of everyday experience are identified with a defect embedded in a higher-dimensional space-time. This defect is typically given by a collection of coincident or intersecting branes.