Femtophysics. A Short Course on Particle Physics

206 Pages·1990·8.731 MB·English

by  M. G. Bowler (Auth.)

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F E M T O P H Y S I C S
A Short Course on Particle Physics
by
M. G. BOWLE R
Department of Nuclear Physics
Oxford University
PERGAMO N PRES S
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Copyrigh t © 1990 M. G. Bowler
All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system or transmitted
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or otherwise, without permission in writing from the
publisher.
First editio n 1990
Reprinte d with correction s 1991
Library of Congres s Catalogin g in Publicatio n Data
Bowler , M. G.
Femtophysics : a shor t cours e on particl e physics : by
M. G. Bowler ,
p. cm.
Include s bibliographica l references .
1. Particle s (Nuclea r physics ) I. Title
QC793.2.B6 8 1990 539.7'2--dc2 0 89-2883 3
British Librar y Cataloguin g in Publicatio n Data
Bowler , M. G. (Michae l George )
Femtophysics .
1. Elementar y particle s
I. Title
539.7'2 1
ISBN 0-08-036943- X Hardcove r
ISBN 0-08-036942- 1 Flexicove r
Front cover illustration
A u quar k in a proto n radiate s a hard gluon and
subsequentl y annihilate s with an antiquar k in an anti-
proto n to form a massiv e boso n Z°. The Z° decay s into
an e + e- pair. The intermediat e boson s of the weak
interactio n were first observe d at CERN throug h suc h
processes .
Printed in Great Britain by BPCC Wheatons Ltd, Exeter
PREFACE
Atoms are structures of size ~ 10~"8cm, consisting of electrons bound to a
positively charged nucleus. The physics of these composite structures became
well understood only half a century ago, with the development of quantum me-
chanics. The crude technology of atoms is prehistoric in origin — fire, metallurgy
and more recently chemistry. A precision atomic technology is emergent, with
the development of techniques for manipulating and imaging single atoms.
The nucleus of the atom is again a structure, of size ~ 10~13cm (lfm)
—10~12cm. The components are nucleons, protons and neutrons. Nuclear tech-
nology exists, but there is no control on a scale of 10~12cm and present day
nuclear technology may be compared with atomic technology at the level of
mastery of fire. The physics of nuclei was largely understood over thirty years
ago. At a fundamental level, study of nuclear structure led to the discovery of
fields of short range which involve the exchange of internal quantum numbers
between the sources.
Just as nuclear physics emerged from the periodic table of the elements and
the probing of the nucleus with beams of particles, the rich structure exhibited
by the strongly interacting particles, hadrons, among which are to be found
the constituents of the nucleus, suggested many years ago that these particles
are also composite structures. The probing of hadrons with beams of electrons,
muons and neutrinos has revealed the structure and identified the constituents.
Hadrons are structures of size & 10~13cm composed of either quark-antiquark
pairs (mesons) or three quarks (baryons). The quarks have spin J , electric
charge + | and — J (in units of the electron charge) and are bound into hadrons
through the operation of eight vector fields, quantum chromodynamics. There
is as yet no technology on the scale of particle physics, < 10~13cm.
The particle physics of today is the physics of spin | particles, leptons and
quarks, interacting through vector and axial vector fields, the free field quanta
of which have spin 1. These fermions and bosons are at present treated as el-
ementary and should they have structure the scale must be < 10~16cm. The
theoretical structure known as the standard model embraces quantitatively all
known aspects of particle physics, and over the last ten years has been tested
with increasing precision and has emerged with distressing success. The funda-
mental physics that has emerged from the study of particles is the physics of
self-interacting vector fields, covered by the principle of gauge invariance.
Very many books have been written on the subject of gauge field theory.
The majority have been written by theoretical physicists at a level which makes
them inaccessible to most undergraduates. There are relatively few books which
cover the general field of particle physics at (advanced) undergraduate level.
In the four years 1986-1989 I lectured on particle physics to Oxford un-
dergraduates in their final year who had chosen to take an advanced optional
course in Nuclear and Particle Physics. The subject is my own and I took the
opportunity to make a fresh study of the subject in the light of the advances
of the last ten to fifteen years. I developed a new course and here it is. I have
restricted myself to those aspects of the subjects which I regard as established
and likely to stand, regardless of developments in the future. I am as confident
í
vi PREFACE
of the existence of quarks and gluons as I am of the existence of electrons and
photons, less confident that the vacuum is inhabited by a Higgs field and I have
no confidence that any of the tentative developments to which I allude briefly
under Final Remarks will survive. With the exception of those remarks, I
have confined myself to topics which I believe I understand. I do not under-
stand grand unified theories in any detail, nor supergravity and superstrings at
all. However, a publisher's referee remarked that superstrings should at least
appear in the index ... hence much of Final Remarks.
My concern throughout has been with what I conceive to be the underlying
physics of the subject and I have dispensed with formal mathematics wherever
possible. There are no mathematics more advanced than volume integration and
matrix multiplication. I have explicitly constructed the Clebsch-Gordan coeffi-
cients for SU(2) and SU(3) and made no appeal to formal group theory. I believe
the book will be accessible to the advanced and enthusiastic undergraduate and
that it will also be useful to graduate students. The first six chapters introduce
the subject at an elementary level — and could be used as the basis for a short
course — the remaining chapters are much more technical. The reader who finds
the first chapters trivial should press on.
The first chapter introduces the fundamental particles and their interac-
tions. The experimental significance of large centre of mass energy is discussed.
Massive scalar fields and the (field theoretic) reinterpretation of negative energy
solutions are introduced. The strength of the strong interaction is considered
and this is further illustrated in Ch.2, which opens with a calculation of scat-
tering through a Yukawa potential acting once. We thus obtain the propagator
for virtual meson exchange and the connection between real and virtual parti-
cles is discussed in terms of Fourier components and annihilation and creation
operators. Charge exchange interactions are introduced and nuclear charge in-
dependence is used to calculate the isospin coupling coefficients for 1 ® §.
The muon and the strange quark were the first identified members of the
second generation of quarks and leptons. Both decay weakly and introduce
Ch.3. Dimensional considerations which are essential in discussion of strange
particle decays lead naturally to the violation of conservation of probability by
the pointlike Fermi interaction and hence to the intermediate bosons W* 1 and
the correct way to construct a dimensionless coupling for the weak interactions.
Ch.4 is a gallop through the properties of the forest of hadrons, including
some discussion of how (over many years) these properties have been established.
The singlet, octet and decuplet structures uncovered gave rise to the elementary
quark model, which is introduced and criticised. In Ch.5 it is shown how the
concept of confined colour augments the quark model and gives rise to verifiable
(and verified) predictions in addition to solving the problems raised at the end
of Ch.4.
Ch.6 is concerned with the evidence for the existence of tiny grains of
momentum and energy, carrying electro-weak charges, within hadrons. An ele-
mentary discussion of form factors, deep inelastic scattering and the Drell-Yan
process is given. (The evidence that these partons are light quarks is deferred
until Chs.9 and 10.)
In Ch.7 the Fermi Golden Rule is derived within time-dependent pertur-
PREFACE vii
bation theory and its validity and use without this framework discussed. The
time evolution of an explicitly decaying state is used to find the width. The
Breit-Wigner formula is obtained by a simple extension. The relation between
resonance formulae and boson propagators in the time-like region is examined.
The chapter closes with the introduction of invariant matrix elements and phase
space.
The essential evidence for the existence of light quarks of spin \ cannot
be understood without the Dirac equation. In Ch.8 I have constructed an ex-
plicit representation — that of Weyl — in the simplest way that I know, by
starting with the Dirac equation for zero mass. The eigenstates can be rep-
resented by two-component spinors and are easily constructed; the rotational
properties of spin | emerge through this construction. Vector (or axial vector)
interactions couple left-handed particle to left-handed particle It is easy to
stitch together two Weyl equations and then reintroduce mass. The result is
the Weyl representation of the (4 x 4) Dirac matrices, admirably adapted for
calculation when mass is small or may be neglected. (I found to my delight
that many relatively complicated problems in nuclear â decay are easily solved
in this representation by a little matrix multiplication, even though the mass of
the electron may not be neglected.) I have tried to distinguish carefully between
the properties helicity and handedness: a left helicity particle contains a
right handed piece, which is lost only in the limit í —• c. I think that lack of
a clear distinction between these properties is responsible for much confusion
when attempting to understand the V — A structure of the weak interactions.
The material of Ch.8 is put to work in Ch.9 in order to calculate explicitly
the matrix elements for fermion-fermion interactions through vector (or axial
vector) fields. The coupling recipe, together with the rotational properties of
spin |, yields the essential features of e+e" annihilation to a fermion-antifermion
pair and of fermion-fermion scattering, without even 2x2 matrix multiplication.
I have attacked the content of deep inelastic scattering by supposing that the
nucleon consists of light quarks obeying the Dirac equation with minimal electro-
weak coupling and obtaining the differential cross sections which are known
(experimentally) to apply. In Ch.10 it is shown how comparison of deep inelastic
electron (and muon) scattering with deep inelastic neutrino scattering establishes
that the partons which couple to the electro-weak interactions are quarks.
Chapter 11 reverts to hadron structure and isospin. It is concerned first
with distinction between those aspects of the formalism which are essential and
those which are only conventional. The decay of hadrons into other hadrons must
take place by creation of quark antiquark pairs from energy stored in the colour
field, and the physical origin of isospin invariance is to be found in the amplitudes
for creation of uu being (almost) identical with the amplitudes for creation of
dd. A systematic implementation of this rule leads to explicit construction of
the isospin coupling coefficients (SU(2) Clebsch-Gordan coefficients) for 0 ® 0,
1 ® 0, 1 ® 1. The famous rule that (1,0) ® (1,0) does not couple to (1,0) emerges
as an exact cancellation of amplitudes. It is easy to extend such construction
to take account of charge conjugation and hence extract the almost forgotten
quantity G parity. Perhaps this is doing things the hard way, but after I had
devised this treatment (for a graduate course which I gave some years ago) I felt
viii PREFACE
that I actually understood isospin (and G parity) for the first time.
Chapter 12 establishes the content of an SU(3) of colour. First, the effects
of interaction between quark-quark, quark-antiquaxk with two colours, colour
exchange forces and colour independence are studied. This is little more than a
relabelling of isospin. Three colours are then introduced, with colour exchange
forces and colour independence. The relations among the couplings required for
colour independence are solved and for vector colour fields both quark-antiquark
and qqq ground states are colour singlets. With colour confined within a flux
tube a mechanism exists for driving all coloured states to infinite mass. A
three quark colour singlet is antisymmetric under permutation of colour: the
spin-space configurations of the quarks are dictated. Hyperfine splitting by
chromomagnetic interactions is worked out for the low lying mesons. In Ch.13
this is extended to the three quark systems and the baryon magnetic moments
are calculated in terms of the quark structure. The chapter ends with a discus-
sion of the mixing of quark antiquark pairs through annihilation into colourless
configurations of the colour fields.
Chapter 14 harks back to Ch.3: it is concerned with the properties of the
massive r lepton and the quarks 6 and á The Bohr atoms of QCD, cc and 66
systems, are discussed in elementary but quantitative terms. The lowest lying
states consisting of a heavy quark and light antiquark are considered briefly.
Chapter 15 is concerned with the weak interactions of quarks. The curious
pattern of weak interactions connecting quarks in different generations is studied,
leading to the famous GIM mechanism for the suppression of weak processes
which change hadron flavour but do not change hadron charge.
Chapter 16 contains a discussion of weak isospin and the entanglement of
the electromagnetic and weak interactions. It is shown how the assumption that
a singlet gauge boson (a proto-photon) and the neutral member of a triplet W± c
are mixed leads to the identification of the couplings of the quarks and leptons
to the neutral intermediate boson Z°. The predictions are tested and they work.
The mechanism whereby weak isospin is broken, the proto-photon and Z° mixed
and W*, Z° acquire mass is introduced in general terms. This is an introduction
to the Higgs mechanism but the apparatus of gauge field theory is left alone.
The structure of the Weinberg-Salam model is discussed, again in elementary
terms. I have devised a diagrammatic representation of the interaction of gauge
bosons with a robust vacuum screening current which enabled me to understand
(for the first time) the content of more formal development of the theory, and
which is easily adapted to obtain the predictions of the theoretical structure for
more complicated Higgs sectors than that of the standard model.
These sixteen chapters were originally in the form of sixteen lectures, each
lasting one hour. I had to maintain a cracking pace to cover only the original
material, and in preparing this book I have added more. Even so, some topics
are omitted entirely. I have devised Problems, which appear at the end of every
chapter, and many of these are invitations to explore omitted topics and can
be solved with the techniques developed in preceding chapters. Those problems
which are particularly difficult or time consuming have been indicated with a
star.
The reader will need to be acquainted with electromagnetic theory, relativ-
PREFACE ix
ity and (non-relativistic) quantum mechanics. Some prior knowledge of nuclear
and particle physics would help.
The style of these extensively edited lectures is essentially oral: I have
tried to eliminate the inevitable infelicities without ruining the pace and impact
of what is intended to be a compact introduction to what is known today of
particle physics — physics at a scale of less than 10~13cm. It is remarkable that
ancient notions of the nature of space and time and the relativistic quantum
theory of fields have — so far — survived unscathed to a level of ^ 10~~16cm,
10-3fm.
In the second printing I have corrected a few typographical errors, inserted
some newly determined numerical values and provided a brief note on some
recent results.
1. AN INTRODUCTION .
1.1 What is a particle?
The subject of this book is called variously elementary particle physics,
particle physics and high energy physics. It is the physics pertaining to dis-
tances less than or of the order of 10~13cm and the denizens of this world are
by definition particles. We distinguish particles and elementary particles: our
working definition of a particle is anything smaller than 10~13cm. This unit is
known as the fermi and is also one femtometer; 10~15m = 10~13cm = lfm.
The vast majority of particles—the strongly interacting particles known
as hadrons—are not elementary but are bound states of quarks. The particles
still regarded as elementary, or fundamental, comprise leptons, quarks and the
quanta of the gauge fields which mediate interactions among them.
The leptons are spin | fermions which interact via electromagnetism and
the weak interactions. There are three with electric charge q = — 1 (in units of
the electron charge) and three neutrinos:
q = -1 e 0.511 MeV ì 106 MeV r 1.784 GeV
q= 0 u <18eV í <0.25 MeV v <35 MeV
e ì T
The masses are given in units of rest mass energy.
The quarks are spin \ fermions which interact via electromagnetism, the
weak interactions, and the colour fields. Each may carry one of three colours,
usually denoted r, 6, and these colours act as sources and sinks for the colour
field:
+l
q = ti ~ 1 MeV c - 1.5 GeV t > 89 GeV
-f d~6MeV s ~ 200 MeV 6 ~ 5 GeV
There is little doubt that the quarks exist only as constituents of hadrons.
The ordinary world is made up from the first of the three generations
alone—the others are transients (with the possible exception of the neutrinos).
1.2 Fundamental interactions.
The photon (7 ) couples to the electromagnetic current and is the quan-
tum of the electromagnetic field. There are three gauge bosons mediating the
weak interactions—W±,Z°. The charged bosons W± mediate nuclear â decay
and change leptonic charge but not leptonic generation. The interactions are
represented by the diagrams:
Fig.1.1
The coupling to leptons is universal and does not depend on the generation.
Between the quarks, the charged current interaction predominantly operates
within a given generation, for example
1
2 Femtophysic s
Fig.1.2
but the processes
(a) (b)
Fig. 1.3
mix generations. The cross generation links attenuate as generation number
increases: the rates for the processes illustrated in fig.l.3(a),(b) are suppressed
relative to fig.1.2 by factors of 0.05, 0.005 respectively.
The photon has mass m < 10~15 eV and does not distinguish between
7
left and right handed fermionst. The W± (Mjy± ~ 80 GeV) couple only to left
handed fermions, thus violating parity to the maximal extent. The Z° couplings
(Mz* ~ 90 GeV) are complicated, change neither charge nor flavour, involve
only left handed neutrinos but both left and right handed massive fermions.
The colour couplings among the quarks seem to be universal, independent
of flavour, and there are eight gauge bosons, for example
[3 <g> 3 = 8 È 1. Experimentally there is no 1]
1.3 Confinemen t
Colour seems to be absolutely conserved and confined within a region of
space of dimensions ~ lfm.
The hadrons are colourless composites with (minimal) composition qq and
qqq. These composites are the so called strongly interacting particles. The (qq)
set axe bosons (integral spin) and strongly interacting bosons are called mesons;
the (qqq) set are fermions (half integral spin) and strongly interacting fermions
are called baryons. Note that with the interactions already discussed and no
others, there must be at least one stable baryon; baryon number is conserved.
The composites have excited states and these are the particles which ap-
pear as resonances in hadron-hadron interactions. You (apparently) never get
I In the high energy limit a left handed fermion has spin oriented opposite to
its direction of motion, a right handed fermion has spin parallel to its direction
of motion.