In this book, the reader has had the opportunity to follow the development of the electroweak theory and the discovery of several new phenomena predicted by the theory. We have analyzed several of them in various chapters. The richness of the field and its high level of accuracy have been achieved with the help of several very large, very accurate, and refined experiments.

The potential for discovery has not yet been exhausted because the theory must be completed with the discovery of the Higgs particle(s) or some other symmetrybreaking scheme. Experiments at the LHC will either discover the Higgs boson or find new interactions indicating another mechanism for the breaking of symmetry. There is another observation in addition to symmetry-breaking that demands extension of the theory: the mixing and mass differences of neutrinos. Most of our colleagues speculate about a larger theory. Grand unified theories will unify the electroweak with the strong interactions, at some high energy bringing the three coupling constants together. The main issue here is to find predictions unique to the grand unified theory that will be verified by experiment. The symmetry of the new theory and the particle classification within the group remain open issues.

An alternative theory is supersymmetry, with many more particles. Supersymmetry is a symmetry relating fermions to bosons and must be broken. There are many different symmetry-breaking schemes. Predictions of the minimal supersymmetric theory will be tested at the LHC.

A parallel development has been the association of weak and electromagnetic interactions with astronomical and cosmological phenomena.

Introduction

When quarks were introduced into physics, they were considered to be light, like the up, the down, and the strange quarks. Their bound states were light mesons, such as the π and K mesons. In fact, light states were interpreted as the Goldstone bosons of the SU(3) symmetry with several of the aspects of Goldstone particles discussed in Chapter 5.

The next quark is the charm quark, which was formulated in order to suppress flavor-changing-neutral couplings in the K mesons (Glashow et al., 1970) (GIM henceforth). Since the charm quark is much heavier than the proton, its existence gave rise to the possibility that additional heavy quarks may exist. The expectation was confirmed with the discoveries of the bottom and top quarks when accelerators of higher and higher energies began operating.

The precise definition of quark masses is a delicate topic and for this reason we shall discuss some of the issues involved. Masses of fermions appear in the electroweak Lagrangian after the breaking of the symmetry, i.e. when the Higgs field acquires a vacuum expectation value. Masses for particles are measured through their interactions with an external field; for example, the bending of an electron beam in a magnetic field determines the ratio e/m. The interaction contains higher-order corrections, which must be included. For leptons the masses are defined as poles of the propagators. For quarks the situation is more complicated because they never appear as free particles, but are always confined within hadrons. The masses of quarks must include radiative corrections from the forces which confine them. On the energy scale of the heavy quarks the strong coupling constant is small enough to allow perturbative calculations.

Introduction

The quark model arose from the analysis of symmetry patterns observed when particles were grouped together according to their spin and parity. When the eight mesons with Jp = 0- are displayed in a strangeness (S) versus isospin (I3) plane, they form the octet of Fig. 3.1. An identical pattern emerges for the eight vector mesons with Jp = 1- also shown in Fig. 3.1. The vector mesons are excited states of the particles in the Jp = 0- octet. The symmetry pattern was interpreted as a generalization of the isospin group SU(2) to the group SU(3) which incorporates both isospin and strangeness. Gell-Mann and Neeman (1964) proposed that the eight baryons with Jp = ½+ also belong to an octet of SU(3), thus establishing a parallelism between meson and baryon states. Finally, many static properties of the particles exhibit the SU(3) symmetry.

Since the fundamental representation of the group SU(3) is a triplet, it is natural to try to interpret the hadronic states in the octets as bound states of triplets or of triplets with antitriplets. If the fundamental fields also carry baryon number, the product of triplet ⊗ antitriplet would be mesons with zero baryon number. The product of three triplets carries baryon number and contain octets and a decuplet as was required by the observed states of baryons. This is the quark model of Gell-Mann (1964) and Zweig (1964).

Feynman rules

We gave in Chapter 6 the Lagrange function for the fermions and the gauge bosons. In the previous chapter we defined the physical bosons with definite masses. It is now a straightforward exercise to rewrite the Lagrange density in terms of the physical bosons and read off the Feynman rules. For the rules, it is necessary to introduce quantized fields in order to keep track of the combinatorics and other factors, especially for diagrams with closed loops. The canonical quantization method in terms of Wick's theorem does not work for non-Abelian gauge theories because there are ambiguities that arise from gauge transformations. The appropriate discussion at this point is the quantization in the path-integral formalism. This will be a long digression and will delay us from arriving at physical results. We adopt a compromise. We consider the fermionic part of the Lagrange function in terms of the physical fields and read off the relevant vertices. The interested reader can compare this method with the procedure used in textbooks of quantum electrodynamics. In this way we obtain an extensive set of Feynman rules for vertices and propagators, in terms of which we discuss many physical processes.

Later on, we repeat this procedure for other parts of the Lagrangian, which include Higgses and gauge bosons. The rules that we obtain suffice when we calculate tree diagrams to any order. Difficulties occur when loop diagrams are computed, beginning with one-loop diagrams. The difficulties are solved by introducing additional diagrams with scalar particles: the Faddeev–Popov ghosts.

We saw in the previous chapter that the neutral gauge fields mix among themselves.

The aim of the book is to introduce the electroweak theory and the methods that have been developed for calculating physical processes. To this end it was decided to divide the book into three major parts.

The road to unification

This part gives a general view of early developments when the theory was based on numerous empirical rules. These topics are extensively discussed in older books on weak interactions, and I selected a few topics among them, such as form factors, CVC, and PCAC, in order to give a general impression of how the field developed. It should serve as an introduction to a few topics from the early period of weak interactions and as a guide to articles and texts. Appreciation of the first part requires familiarity with the methods developed at that time. The readers who find this part too brief or difficult may proceed to the second part, where gauge theories are introduced.

Field theories with global or local symmetries

This part presents field theories based on continuous symmetries and guides the student to the electroweak theory based on the group SU(2) X U(1). Special effort has been made to present it in a simple and pedagogical way. For this reason the chapters are short and accompanied by references that the reader or lecturer can consult.

Experimental consequences and comparisons

The third part of the book covers some of the exciting discoveries that took place in the process of verifying the electroweak theory.