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The Physics of Nuclear Reactions, 2nd Edition
GORBACHEV
Nuclear Reactions in Heavy Elements
HODGSON
Growth Points in Nuclear Physics
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Volume 3 — Nuclear Structures, Potentials and Reactions
LANDAU and LIFSHITZ
Course of Theoretical Physics, Volume 4, 2nd Edition
Quantum Electrodynamics
LEITE LOPES
Gauge Field Theories: An Introduction
SINAI
The Theory of Phase Transitions: Rigorous Results
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Pergamon journal available on request from your nearest Pergamon office
Introduction to the Physics of
Electroweak Interactions
by
S. M. BILENKY
Joint Institute for Nuclear Physics, Dubna, USSR
Translated by
G. B. Pontecorvo, Dubna
PERGAMON PRESS
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Copyright © 1982 Pergamon Press Ltd.
All Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system or transmitted
in any form or by any means: electronic, electrostatic,
magnetic tape, mechanical, photocopying, recording or
otherwise, without permission in writing from the publisher.
First edition 1982
Library of Congress Cataloging in Publication Data
Bilenky, S. M. (Samoil Mikhelevich)
Introduction to the physics of electroweak
interactions.
Translated from the Russian.
1. Weak interactions (Nuclear physics)
I. Title
QC794.8.W4 B5413 1982 539.7'54 81-15839
British Library Cataloguing in Publication Data
Bilenky, S.M.
Introduction to the physics of electroweak
interactions.
1. Weak interactions (Nuclear physics)
2. Electromagnetic interactions
I. Title
539.T74 QC794.8.E4
ISBN 0-08-026502-2
This is a translation of
Lektsii po fizike neytrinnakh i lepton — niklonnakh
protsessor
by S. M. Bilenky
Published by Energoizdat, Moscow, 1981
Printed in Hungary by Franklin Printing House
Introduction
RECENT outstanding discoveries (scaling, neutral currents, ^-particles,
charmed particles, etc.) have completely changed the face of high-energy
physics. Enormous progress has been achieved in the understanding of the
structure of hadrons (nucléons, pions, ...). Each year brings in more and
more evidence that hadrons are built up of quarks (particles with fractional
charge) and that the interaction between quarks is due to the exchange of
vector massless particles (gluons).
Great success has been achieved in the understanding of the weak in-
teraction. A few years ago the theory of weak interactions was in a semi-
empirical state. Today we have a consistent renormalizable theory of weak
interactions which is in good agreement with experimental data. The weak
interaction of today appears to be a part of unified weak and electromagnetic
interactions (i.e. an electroweak interaction). A final confirmation of this
theory (there are still to be discovered intermediate bosons, Higgs particles
...) would signify a great stride towards the unification of the various in^
teractions.
An exceptional part in this progress of high-energy physics has been and
is being played continually by the study of lepton-nucleon processes, neu-
trino processes, and e+-e~ processes. This book deals with processes of the
first two types.
In the first two chapters deep-inelastic and elastic lepton scattering on
nucléons (both cases of polarized and nonpolarized initial particles) are
examined. Structure functions and form factors of nucléons are introduced.
Calculations of cross-sections and asymmetries are carried out in detail.
In the third chapter a brief history is presented of the construction of the
phenomenological V-A weak interaction Hamiltonian. The two-component
theory of the neutrino, the lepton number conservation law, various formu-
lations of this law, and other problems are examined.
In the fourth chapter the Glashow-Weinberg-Salam unified theory of
weak and electromagnetic interactions is presented in detail.
The fifth chapter presents the processes ν (ϊ>^+Ν-+■ μ~(μ+)+Χ
μ
vii
viii INTRODUCTION
ν +η ->· μ~+ρ and ν +ρ -*■ μ++η, in which neutrinos take part (charged
μ μ
currents).
Finally, in the sixth chapter processes due to neutral currents, deep-ine-
lastic neutrino-nucleon scattering, elastic neutrino-nucleon scattering, elas-
tic neutrino-electron scattering are examined. In this chapter P-odd effects
in deep-inelastic lepton-nucleon scattering are also discussed.
All calculations are carried out in great detail. I hope this book will be
useful to those who wish to master the techniques for calculating the ex-
perimentally measured quantities (cross-sections, asymmetries). Bearing in
mind the educational aim of the book, I have written extensive appendices
(on the Dirac equation, about "know-how" problems concerning the treat-
ment of the y-matrices, on the comparison of different metrics, on units,
and so on), which I strongly recommend to read before dealing with the
main body of the book.
In order to read this book, a knowledge is sufficient of the fundamentals
of quantum field theory, i.e. quantized fields (scalar, spinor, vector), Wick
theorem, and Feynman diagrams. All these topics are presented in any of
the numerous available books on quantum field theory (see list of references
at the end).
The volume and style of the book have not permitted the presentation of
such recent important achievements of theory as quantum chromodynamics.
Theoretical results are presented here, which were obtained some years ago,
are "settled", and have stood the test of time. Thus, this book is intended for
those who are just beginning to study neutrino and lepton-nucleon physics.
The book is based on lectures given at the Joint Institute for Nuclear Re-
search and at the Dubna branch of Moscow State University. The lecture
style has been partly preserved. There are repetitions in the text, diverse
interpretations of the same problems, etc.
And, finally, two comments. The first has to do with the quoted experi-
mental data. It is well known that such data are subjected to revision (and
sometimes completely altered) at quite a rapid pace. It takes a very long time
to write (and to publish) books. From my point of view data presented in
books are not worth using (this concerns high-energy physics). For this there
exist annually reissued tables (particle properties), preprints, articles, reports
to conferences, etc. For these reasons I do not present here a full review of
data. The presented experimental data have been chosen only for illustration
purposes. The second comment concerns the references. I was not able to
compile the full list of references, on which this book is based (it is enormous).
At the end only a list is presented of the main references, the authors of
INTRODUCTION IX
which have been mentioned within the text of the book. There is also pres-
ented a list of books on particle physics and quantum field theory.
It is a pleasure for me to thank B. Pontecorvo for numerous—exceptionally
important to me—discussions of problems concerning elementary particle
physics and especially neutrino physics.
1.
Deep-inelastic electron scattering on nucléons
1.1« Cross-sections. Structure functions
We begin by examining the process of deep-inelastic electron scattering
on nucléons
e+N — e+hadrons. (1.1)
An exceptionally important part has been and is being played in high-energy
physics by the study of the process (1.1). The first experiments, in which
deep-inelastic electron scattering on nucléons was studied (SLAC, 1968),
actually initiated the present stage of the investigation of the nucléon struc-
ture. The study of this process has led to the discovery of scaling and to the
development of parton ideas.
The experiments, in which process (1.1) was studied, and also analogous
neutrino experiments, have been compared to the famous experiments of
Rutherford on α-particle scattering on atoms. Similarly to the experiments of
Rutherford, which led to the establishment of the existence of nuclei inside
the atoms, experiments on deep-inelastic lepton scattering have made it
possible to establish the existence of quarks inside nuclei.
Here we shall limit ourselves to the derivation of the general expression
for the cross-section of process (1.1). The quark-parton ideas will be pres-
ented in detail in the fifth chapter.
In experiments, designed for the investigation of process (1.1), only final
leptons are detected (energy and angle). Such experiments yield information
on the inclusive cross-section, that is the sum of the cross-sections of all the
possible processes in which the final leptons have a given momentum. Those
events are selected in which the total mass of the final hadrons is greater than
the mass of the most heavy nucléon resonance (this is what is meant by call-
ing such processes deep-inelastic).
1
2 INTRODUCTION TO THE PHYSICS OF ELECTROWEAK INTERACTIONS
We shall start by obtaining the matrix element of the process. Denote the
momenta of the initial and final electrons by k and k\ the momentum of the
initial nucléon by p, and the total momentum of the final hadrons by p'.
We write the initial and final state vectors | /) and | /) as
\i) = c*(k)\p),
\f) = c+{k')\p'). Κ · }
Here c+(k) {c+(k')) is the creation operator of the initial (final) electron,
| p) is the state vector of the final hadrons.* We must find the matrix element
(f\S\i).
The operator S is given by the following general expression
S = r(e-iIa!/Cac)d*), (1.3)
where 7öj is the total interaction Hamiltonian.
Examination of process (1.1) requires that the following be taken into
account: (1) the electromagnetic interaction of electrons; (2) the electro-
magnetic interaction of hadrons; (3) strong interaction (obviously, weak
interaction does not give any significant contribution to (1.1)). For the in-
teraction Hamiltonian 7uj(x) we have correspondinglyî
SBKJC) = 7Ö<>y(x)+7Öh'>y(x)+76h(x). (1.4)
Here
W-*(x) = ieë(x)y e(x)Mx) (1.5)
a
is the interaction Hamiltonian of the electron e(x) and the electromagnetic
fields (e is the charge of the proton),
9Ôh'.r(x)=-ieUx)A (x) (1.6)
a
is the interaction Hamiltonian of the electromagnetic and hadron fields
(j (x) is the electromagnetic hadron current), and 76h is the strong interac-
a
tion Hamiltonian.
t In (1.2) many indices have been omitted (although understood to be present)—the
electron helicities, the helicity of the nucléon, and the momenta and helicities of the final
hadrons.
i We use the Pauli metric: x = (x, ix), x2 = * .* = x2-xl; γ+ = y , yWß+yp* =
0 α α e
2δ .
αβ
1. DEEP-INELASTIC ELECTRON SCATTERING ON NUCLEONS 3
Let us obtain the matrix element of process (1.1) in the lowest order in e.
Obviously in the expression for the 5-matrix one must take into account
only the term linear in 76e>γ and 76h'v. We have
S = T\(l+(-i)i'Me-y(x )dx + ...)
1 1
X(l _ j^;^ ...)e-'i^')^j
+( 0 2)dx2+
^ 1 +(-02rJ {7ûe>y(xï) 7ùh>y(xfc~^ ***>**) d*i àx . (1.7)
2
We now proceed to calculate the matrix element of the operator (1.7).
Applying the conventional rules of quantum field theory, we obtain1
<p' | c{kf) T(ê( ) γΛχύ) SU+(*) | p)
Xl
= </>' | c(fc') #-Kxd 7«e(+)(*i) €+&)%, I p)
= AWV*e-i(*-*>* ü(k') γ φ) (ρ' 121« | p). (1.8)
α
(The electromagnetic and hadron field operators enter into 9ί.) Furthermore,
α
in the Γ-product of the two electromagnetic field operators only the con-
traction must be taken into account (there are no photons in the initial and
final states):
(1.9)
Taking into account these comments, we obtain
(1.10)
t Indeed, we have é+\x) c+(k) I p) = {[e{+\x\ c+(k)] -c+(k) e{+)M) I p). The oper-
+
ator e{+\x) is represented by the expansion e{+\x) = I N'U(kf) cifyé*'* dk' ÎN> —
k k
■==]. The creation c+(k) and the annihilation c(k') operators obey the
(2TT)3/2 ^/2k'J
anticommutation relation [c(k'\ c+(k)] = <5(k-k'). From this relation we obtain
+
[é?(+)(jt), c+(k)] = Nu(k)eikx. Further, taking into account that e{+)(x) | p) = 0
+ k
(in the state described by the vector | p) there are no electrons) we find that
e{+)(x) c+(k) 1 p) = NkUik)^** \ p). Hence, applying Hermitian conjugation and multi-
plying on the right by y we further obtain (// | c(k')e{~\x) = {ρ' \ 7VVi/(£')e~tt'*.
4
4 INTRODUCTION TO THE PHYSICS OF ELECTROWEAK INTERACTIONS
(
N = 7= is a standard normalizing factor. 1 Integration over
k
the variables JCI and κ does not present any difficulties. We have
1
</1 (S-1) I i> = ieW,,N vihf) u(k) ψ
k Va
X Je^</>' | T{Mx)c-iSgcH*dx,)\p)dx, (1.11)
where
q = k-k'
is the four-momentum transfer. The Feynman diagram for the process (1.1)
is represented in Fig. 1.1. The hatched part of the diagram corresponds to
the hadron part of the matrix element.
p'
FIG. 1.1. Diagram of the process e+N -*· e+X.
The hadron part of the matrix element (1.11) may be written in a consid-
erably more compact form in the Heisenberg representation. Namely, one
can show that
</>' IS T{Mx)e-iSg6hMdx') \p) = </,' | /.(*) \p) (1.12)
out in
(all quantities in the right-hand side of (1.12) are in the Heisenberg represen-
tation).
Henceforth, while examining processes involving hadrons, we shall always
write the hadron part of the matrix elements in the Heisenberg representation
(as is generally done). We shall, therefore, present a very detailed derivation
of the relationship (1.12).
The state vector | φ(ή) describing any hadron system satisfies, in the
interaction representation, the equation
i ^> = #W(0>, 0-13)
