Leptons and Quarks

By L.B. Okun

This book comprises an introduction to the theory of the weak interaction of elementary particles.The author outlines the current situation in weak interaction theory and discusses the prospects for the coming decade. The reader is familiarized with simple theoretical techniques for the calculation of decay rates, interaction cross-sections and angular and spin correlations.

• Language: English
• Publisher: Elsevier Science
• Release date: May 10, 2013
• ISBN: 9780444596215

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Okun

CHAPTER 1

Introduction

Publisher Summary

This chapter outlines the theory of weak interactions of elementary particles. The standard theory of weak interactions is based on the analogy with the electromagnetic interaction that is produced by the electromagnetic current coupled to the photon. Moreover, in contrast to stronger interactions, namely the strong and electromagnetic, the weak interaction violates a number of conservation laws. It presents an introduction to the quark currents and the color of weak currents. According to quark theory, all known hadrons consist of quarks, which are of five types—u, d, s, c, and b. However, the theoretical arguments point to the existence of a sixth quark—t so that in analogy to the six leptons, the six quarks form three pairs. There are no neutral currents transforming quarks of one type into quarks of a different type. The chapter explains that quarks are characterized by color. It describes various currents and process discussed in the quark theory. There are 12 charged currents coupled to W-bosons and 12 neutral currents coupled to Z-bosons.

The weak interaction is responsible for a large number of physical processes: nuclear β-decay, numerous decays of elementary particles, reactions induced by neutrinos from accelerators and nuclear reactors, and also some subtle effects involving parity violation in γ-decays of nuclei and in atomic optical spectra. All known leptons and hadrons are subject to the weak interaction. It plays an important role in such astrophysical phenomena as the sun’s burning and supernova explosions. Some of the weak processes were already put to use (for example, the angular asymmetry in the muon decay is a promising new tool in chemistry). Mainly, however, our interest in the weak interaction is rooted not in its possible applications but in the hope that its study will ultimately yield a unified theory of elementary particles and of the interactions between them. And although it would be very difficult today to predict any practical consequences of such a unified theory, there can be no doubt of their utmost importance.

In contrast to stronger interactions, namely the strong and electromagnetic, the weak interaction violates a number of conservation laws. Among the quantum numbers that are not conserved are, space parity P, charge conjugation parity C, combined inversion parity CP, strangeness, charm, and some others.

The standard theory of weak interactions is based on the analogy with the electromagnetic interaction which is produced by the electromagnetic current coupled to the photon (see (fig. 1.3). Both the charged and the neutral currents include a leptonic and a hadronic part. At present we know six leptons which are naturally grouped into three pairs:

Fig. 1.1

Fig. 1.2

Fig. 1.3

so that each lepton has its neutrino counterpart. Each lepton enters the charged current j with the appropriate neutrino:

This current emits W+ bosons and absorbs W− bosons. The hermitian conjugate current j +

This follows from the properties of the relevant operators. For example, the operator ē creates an electron and annihilates a positron, while the operator e creates a positron and annihilates an electron. Operators of other particles act in a similar manner.

1.1 Quark currents

Hadrons are represented in weak currents by quarks. According to quark theory, all known hadrons consist of quarks of five types (five flavors): u, d, s, c and b. Theoretical arguments, however, point to the existence of a sixth quark t, so that in analogy to the six leptons, the six quarks form three pairs:

for the π.

, and only very scant indirect information is available on the weak interaction of the b- and t-quarks (see current is responsible for the decay of the Λ-hyperon (fig. 1.5).

Fig. 1.4

Fig. 1.5

If it is assumed that each of the upper quarks can go over to any of the lower quarks, then in the general case the charged hadronic current jh must contain nine terms:

, and so on (see Chapter 2).

1.2 On the color of weak currents

is in fact a sum of three terms:

where suffices 1, 2, 3 stand for yellow, blue, and red colors, respectively. The same is true for other quark currents. Hereafter, unless stated otherwise, summation over color suffices in quark currents is omitted.

1.3 Currents and processes

The theory thus contains twelve charged currents coupled to W-bosons (see .

Fig. 1.6

Fig. 1.7

A symbol du in u d; the same is true for other currents in this figure. In some cases this condensed notation proves convenient.

As each of the twelve currents can interact with each current of the diagram, the total number of possible interactions must equal 78 both for fig. 1.6 and for fig. 1.7. So far only fourteen such current × current interactions are experimentally detected for charged currents, and seven for neutral ones. Let us make a list of these interactions, indicating in square brackets the processes in which they are experimentally observed. These processes are usually classified into three groups: pure leptonic, semi-leptonic (involving both leptons and hadrons), and non-leptonic (involving hadrons only). Four leptonic, seven semi-leptonic, and three non-leptonic interactions were found for charged currents:

(Where the subscripts of neutrinos and the antiparticle bar − were not essential, they were dropped in order to avoid overloading the formulas).

In the case of neutral currents, two leptonic, four semi-leptonic, and three non-leptonic interactions were found:

, and the nuclear parity-violating interaction between nucleons. A special analysis is required to separate the contributions of charged and neutral currents to these processes.

Each of the 156 interactions shown in figs. 1.6 and 1.7 generates a number of related processes. We have already mentioned that according to quantum field theory, field operators describe the production and annihilation of both particles and antiparticles. Hence, the same interaction results in several processes. Thus, the interaction (μν)(ud) is responsible for muon capture μ− p → nν, decay of the π-meson π+ → μ+ν, neutrino reactions of the type νμ + nucleon → μ− + hadrons, and so on.

figs. 1.6 and 1.7 obviously invite the following questions:

(i) What are the couplings between different currents?

(ii) What is the space-time structure of the different currents and what are its experimental corollaries?

(iii) What properties are anticipated for W- and Z-bosons?

1.4 About the outline of this book

The first two questions are discussed in Chapters 1 through 17, devoted mostly to a phenomenological analysis of a number of weak processes at low energies (below the production threshold of W- and Z-bosons). The second part of the book treats the high-energy physics of weak interactions (at energies above the production threshold of W- and Z-bosons, i.e. in the range that the experimenters plan to reach in the eighties). The idea of a unified electroweak interaction, as stated in the standard model of electroweak interaction, makes the backbone of this part of the book.

In the low-energy limit, the fact that current interactions are mediated by intermediate bosons, is immaterial. We are thus justified in referring to an effective local interaction between currents. For instance, for muon decay, the non-local graph reduces to a local four-fermion interaction graph (see fig. 1.8). This local (i.e. occurring in a single world-point) four-fermion interaction is characterized by the Fermi constant G ≅ 10−5mp−2. (The system of units employed throughout the book assumes ħ = c = 1.) The same coupling constant characterizes (although with some qualifications) the remaining 155 four-fermion interactions. The next chapter deals with the form of the effective four-fermion lagrangian of the weak interaction in more detail.

Fig. 1.8

The first part of the book is thus devoted to the applications of the theory of four-fermion weak interactions. The second part starts with exposing the weak spots of the four-fermion theory and presents the gauge theory of the unified electroweak interaction. Then we discuss the corollaries of this last theory.

From the purely theoretical point of view it would be more logical, of course, to choose another, deductive approach. Namely, the book should have started with presenting the SU(2) × U(1) symmetry and its spontaneous breaking, writing down the lagrangian of the electroweak interaction and discussing the properties of intermediate bosons, and only then should the results of the theory for low-energy processes have been described. I am of the opinion, however, that the presentation must be based on the phenomena which have been investigated not only theoretically but experimentally as well.

With the plan of the book as chosen here, the reader gradually assimilates important elements of the theory, such as particle states with a definite helicity. He will learn about the experiments from which the main parameters of the theory have been extracted. He will recognize the inherent contradictory nature of the four-fermion theory and the need to introduce the vector and scalar intermediate bosons.

Naturally, the book will unavoidably take on an archaic appearance after the W-, Z-, and especially H-bosons will have been found experimentally. But if it helps to bring us closer to these discoveries, I shall consider its mission fulfilled.

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B mesons containing single b-quarks were discovered in 1980, see footnote on p. 5.

in 1980, with charmed mesons dominating the decay products of B-mesons. The observed decays of B mesons seem to be caused by the interactions (bc) (ev), (bc)(μν), and (bc)(ud). This means that one more current, namely, bc, should be underlined in fig. 1.6.

CHAPTER 2

Structure of weak currents

Publisher Summary

This chapter discusses the quantitative description of weak currents. The low-energy behavior of weak processes is represented by an effective four-fermion lagrangian. The four-fermion weak interaction is sometimes referred to as the universal interaction, which implies that different currents interact with the same coupling constant—G. The chapter also highlights the two peculiarities of neutral currents—first, they are diagonal and transform a particle into itself, and second, they contain both left-handed and right-handed components of spinors. The absence of non-diagonal terms in a neutral current is derived by the fact that experiment does not detect processes which would occur with high decay rates if such terms exist. Therefore, it is an experimental fact that the neutral currents comprise both left-handed and right-handed components.

In the introduction we gave a qualitative bird’s eye view of the weak interaction. Let us discuss now the quantitative description of weak currents. It was mentioned above that the low-energy behavior of weak processes is represented very well by an effective four-fermion lagrangian comprising two