Introduction to High Energy Physics: Particle Physics for the Beginner

551 Pages·2022·25.018 MB·English

by  Lee G. Pondrom

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Library of Congress Cataloging-in-Publication Data
Names: Pondrom, Lee G., 1933– author.
Title: Introduction to high energy physics : particle physics for the beginner /
Lee G. Pondrom, University of Wisconsin--Madison, US.
Description: New Jersey : World Scientific, [2022] | Includes bibliographical references and index.
Identifiers: LCCN 2021033762 | ISBN 9789811222092 (hardcover) |
ISBN 9789811223013 (paperback)
Subjects: LCSH: Particles (Nuclear physics)
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Preface
ElementaryParticlePhysicshasdevelopedoverthelast75years,and
is now a mature field. A theoretical framework called the “Standard
Model” has been with us about 40 years, and has been confirmed
by many experimental tests. There are to be sure some soft spots.
Strong binding of light quarks, and the concept of quark mass, is a
difficulty that gets some exposure in this book. The neutrino sector
is afield of active experimental work, that is notgiven fulltreatment
here. Confirmation of the expected properties of the newly discov-
ered Higgs boson, discussed in Chapter 8, is ongoing at the CERN
Large Hadron Collider, and there may be future storage rings dedi-
cated to this subject.
Particle physics is a very broad subject, and any text is bound
to reflect the taste of the author in the choice of material to be
covered. This text is devoted to what might be called “First-Order
Electroweak Theory”. The consequences of unification of weak and
electromagnetic forces have been established by many beautiful
experiments since its theoretical development in the 1970s. Most of
the detailed predictions of the theory are accessible to the begin-
ning student, and it is the goal of this book to inspire the reader
to understand and appreciate the subtleties and powers of the first-
order theory. Parity violation in the weak interactions leads to many
interesting polarization phenomena involving spin 1/2 and spin one
particles. Polarization calculations play a major role in this book.
v
March23,2022 10:35 IntroductiontoHighEnergyPhysics-9inx6in b4353-fm pagevi
vi Introduction to High Energy Physics: Particle Physics for the Beginner
The definitions of the Dirac spinors for electrons and positrons, and
the treatment of the spin one W± bosons, and how all of this works
in a parity violating theory are impressive. In some problems the
cover is removed fromthemachinery, soyou canseethewheelsgoing
around, and admire the structure that has been created for us.
QCD, the theory of the strong interactions, is not ignored here,
but plays only a peripheral role. There are at least two reasons for
this. Perturbative QCD is mathematically more difficult than per-
turbative electroweak theory, and because of the strong coupling,
corrections tothefirst-ordercalculations arenotsmall, makingQCD
lessaccessible. TheseproblemshavebeenlargelyovercomebyMonte
Carlo programs like PYTHIA, that accurately model strong interac-
tion phenomena, and are very useful in the interpretation of exper-
imental data from hadron machines. But a several hundred page
program in FORTRAN or C++is noteasy to handlein an introduc-
tory class. Computer calculations appear occasionally in this text in
the problem sets, but you are free to choose what program language
youwanttouse. InthisregardIshouldmentiontheprogramROOT,
a free download from CERN, that combines a C++ interpreter with
a graphics package, so you can calculate and plot curves in the same
environment. ROOT was used to make many of the illustrations in
this book.
One of the first choices in a relativistic theory is called the “met-
ric”, that is the convention for handling the fact that in four dimen-
sional space–time the length that is Lorentz invariant is ds2 =
c2dt2 − d(cid:2)x2, usually handled by introducing a metric tensor, and
upper and lower indices for the four vectors, or the opposite sign
ds2 = d(cid:2)x2 −c2dt2, that is often accommodated by imaginary time
idt. Various authors have defended their choice by accusing others
of being feeble minded. Emotions can run high when discussing the
metric. Sakuraiinhiswell-knowntext“AdvancedQuantumMechan-
ics” uses imaginary time, and calls anyone who would use a metric
tensor in Special Relativity, where you do not really need one, defi-
cient. Martinus Veltman has adopted the same convention as Saku-
rai, and has dismissed the use of a metric tensor as misguided, if not
worse. Thisbookusesthemetrictensor,andupperandlowerindices
March23,2022 10:35 IntroductiontoHighEnergyPhysics-9inx6in b4353-fm pagevii
Preface vii
on the four vectors. We do have company. This metric was popular-
ized by Bjorken and Drell in Relativistic Quantum Mechanics, and is
the choice of two texts that are extensively referred to in this book:
Quarks and Leptons by Halzen and Martin, and Collider Physics by
Barger and Phillips. Upper and lower indices can be a nuisance, but
so can imaginary numbers. There is no easy escape from the minus
sign. OurLorentztransformations arereal, andthelength of aphys-
ical particle four vector is also real: p2 = E2 −p(cid:2)2 = m2. Physical
observables are of course metric independent, but intermediate for-
mulas can look very different. This means that in order to check the
steps in a particular calculation by referring to another source, be
sure that all of the conventions are the same.
Our units are called “natural units”, where (cid:2) = c = 1, c being
the speed of light, are both dimensionless, so that mass, momen-
tum, and energy all have the same dimensions. Since c is dimen-
sionless, length and time have the same dimensions also, namely
x ∼ 1/E. The uncertainty relations ΔxΔpx ∼ (cid:2) and ΔEΔt ∼ (cid:2)
are useful in determining which dimensions go where. Conversions
to usual quantum mechanical cgs units are supplied throughout the
book and in the Appendix. (cid:2)c = 197.3eV-nm is especially useful, as
itgivesthecorrespondencebetweenenergysquaredandcrosssection:
1 GeV−2 = 0.389 millibarns.
Choice of variables is important. Can you imagine attempting to
solve the wave equation for hydrogen in Cartesian coordinates? The
Mandelstamvariabless= (p1+p2)2,t = (p1−p3)2,andu = (p1−p4)2
fora2to2scatteringwithfourmomentap1+p2 = p3+p4 areuniver-
sally used in this text, and lead to formulas that are generally much
simpler than they would be in terms of other choices of kinematic
variables. StanleyMandelstamintroducedthevariablesthatbearhis
nameinapaperaboutpion–nucleondispersionrelations,thatalmost
nobodyremembers. Butthevariableshaveproventobeagreatboon
in simplicity for a wide variety of kinematic formulas, and texts that
do not use them exclusively are more difficult to read. The basic for-
mulas for e−e− (Møller) scattering, e−e+ (Bhabha) scattering, and
e−e+ annihilation are related by permutation of the Mandelstam
variables, so if you solve one of them, you have solved them all.
March23,2022 10:35 IntroductiontoHighEnergyPhysics-9inx6in b4353-fm pageviii
viii Introduction to High Energy Physics: Particle Physics for the Beginner
This text would not exist without the concept of using a diagram
todisplayascatteringordecayprocess,andattachingtothediagram
specific rules for writing down the quantum mechanical amplitude.
We owe this development to Richard Feynman. Julian Schwinger is
credited with the remark that Feynman has given field theory to the
masses. We masses are grateful for the contribution. The formula
for Compton scattering of X-rays by electrons was first derived by
Klein and Nishina soon after Dirac’s relativistic electron theory was
introduced. It took the authors about six months to work it out.
It is derived in Chapter 4. The Klein-Nishina formula is one of the
more difficult derivations in the chapter, but even so it only takes a
few pages, and maybe a day or two. That is one of many examples
of the labor saving techniques that have been developed to facilitate
calculations in quantum electrodynamics. If the number of steps in
a calculation can be reduced, the chance for making a mistake is
diminished. The traditional form for the Klein–Nishina formula has
been retained, so it is one of the few cross sections in the text not
written in the Mandelstam variables.
Chapter 3 covers some field theory formalism. In quantum
mechanics thespacecoordinates(cid:2)xareobservablesthatobeycommu-
tation relations with the time derivatives, and time is a parameter
supplied by a clock. In the relativistic theory space and time are
mixed, and the distinction is removed by assigning all coordinates
parameter status. Thefield itself replaces thespace coordinates, and
the usual canonical commutation rule becomes [Φ((cid:2)x,t),Φ˙((cid:2)y,t)] =
iδ((cid:2)x − (cid:2)y). We introduce the concept of a Lagrangian density, and
functional derivatives for the wave equation. We then use the for-
malism in Chapter 4 to derive a simple example both with the field
theory and the Feynman rules to demonstrate that the two proce-
dures are the same, although Feynman’s approach requires fewer
manipulations. Having done that, the formalism is not used again
until Chapter 8, where we need it to explain the non-Abelian gauge
theory and the Higgs mechanism that are necessary for predictions
regardingthe propertiesof theHiggs boson. Non-Abelian gauge the-
ory is not easy, but we only take it far enough to obtain important
parts of the electroweak theory.
March23,2022 10:35 IntroductiontoHighEnergyPhysics-9inx6in b4353-fm pageix
Preface ix
The quark model appears throughout the text. Chapter 1 has
an introduction to the SU(3) group, and the original three quarks
(u,d,s). The heavy charmed and bottom quarks are also introduced
in Chapter 1. The production and decay of the interesting 175 GeV
mass top quark are discussed in Chapter 7. The light quarks gave
birth to the quark model, but it is the heavy quarks that confirm its
reality.
Quark binding and CP violation are covered in Chapter 9. We
harken back to the non-relativistic Schr¨odinger equation to discuss
positronium, and use QED to calculate the rate of 1S0 positron-
ium decaying into two γ rays. We introduce the relation between
the non-relativistic limit of the relativistic matrix element calcu-
lated by the Feynman rules and the corresponding potential in the
Schro¨dinger equation. Then we apply similar ideas to charmonium
and bottomonium, where the effective potential is not known. Sev-
eralnon-relativistic potentials arediscussedusingtheWKBapproxi-
mation. Applicationsofordinaryquantummechanicstopositronium
appear in the problems to Chapter 9. CP violation in the K0K¯0 sys-
tem is described in terms of the mass mixing matrix, and the BB¯
heavy meson system is analyzed in the same way. The heavy quark
analysis is based on a complex phase in the Cabibbo–Kobayashi–
Maskawa 3 × 3 unitary quark mixing matrix. The predictions are
compared to recent experimental results from asymmetric e−e+
colliders.
The last half of this book was written duringthe COVID-19 pan-
demic, that was totally disruptive to ordinary life. It is still rag-
ing as I write this in December 2020. The pandemic has shut down
everything, and this has had effects on the book. A positive effect
has been the forced isolation with minimum distractions to work on
the manuscript. The negative effects have been the lack of contact
with colleagues, and the access only to publications available online.
Twocolleagues fromtheUniversity ofWisconsin, VernonBarger and
Richard Prepost, have helped clarify several points in the course of
the book. I owe thanks to Cyrena, my wife, who has had to put up
with many weeks of absence in a long career in experimental particle
physics.