Table Of ContentSPRINGERT RACTS
IN MODERN MSICS
Ergebnisse
der exakten Natur-
wissenschaften
52
Volume
Editor: G. Hijhler
Editorial Board: P. Falk-Vairant S. Flijgge J. Hamilton
F. Hund H. Lehmann E. A. Niekisch W. Paul
Springer-Verlag Berlin Heidelberg New York 1970
Malzuscripts for publication should be addressed to:
G. HBHLER, Institut fiir Theoretische Kemphysik der Universitat, 75 Karlsruhe,
KaiserstraBe 12
Proofs and all correspondence concerning papers in the process of publication
should be addressed to:
E. A. NIEKISCH, Kemforschungsanlage Jiilich, Institut fiir Technische Physik,
517 Jiilich, Postfach 365
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
is concerned, specifically those of translation, reprinting, m-use of illustrations, broadcasting, reproduc-
tion by photocopying machine or similar means, and storage in data banks.
Under $54 of the German Copyright Law where copies are madefor other tban private use, a fee ispayable
to the publisher, the amount of the fee to be determined by agreement with the publisher. 0 by Springer-
Verlag, Berlin. Heidelberg 1970, Printed in Germany. Library of Congress Catalog Card Number 25-9130.
The use of general descriptive names. trade names, trade marks, etc. in this publication, even if the former
are not especially identified, is not be taken as a sign that such names, as understood by the Trade Marks
and Merchandise Marks Act, may accordingly be used freely by anyone. Title-No. 4735
Weak
Interactions
Invited Papers presented at the
second international Summer School
for Theoretical Physics
University of Karlsruhe
(July 14 - August 1, 1969)
Contents
A Survey of the Weak Interactions
.S ZClWOROlSAG
Semileptonic Decays
.V F. RELLUM 43
Non Leptonic Decays
B. HCETS 50
Current Algebra and Weak Interactions
.3 RENNER 06
The Decays of the K0--K o System
H. .G HCSOD 97
Questions Raised by CP-Nonconservation
P. K. RIBAK 19
Relations for Semileptonic Weak
Interactions Involving Photons
W. REMMUK 311
Radiative Corrections to
Weak Decays Involving Leptons
NIDDUZAIR 621
Radiative Corrections to Weak Interactions
J. ROTHLEITNER 161
Unconventional Models of
Weak Interactions
.G ~iRGES 171
Weak Interactions at Small Distances
H. NNAMHCSTEIP 193
Physical Symmetries in the
Framework of Quantum Field Theory
j. T. IKS~IAZSUPOL 102
A Survey of the Weak Interactions*
S. GASIOROWlCZ
Contents
Introduction . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . 1
I. The Weak Interactions before 1957 . . . . . . . . . . . . . . . . . . . . 1
II. Parity Nonconservation and the Form of the Beta Interaction . . . . . . . . 5
III. Currents and their Symmetries . . . . . . . . . . . . . . . . . . . . . . 14
IV. Nonleptonic Decays . . . . . . . . . . . . . . . . . . . . . . . . . . 25
V. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 30
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
noitcudortnI
The study of the weak interactions is particularly fascinating, because
these interactions play a dual role in the physics of the fundamental
structure of matter. On one hand they manifest themselves in the interac-
tion of leptons (electrons, muons, neutrinos) which have no strong
interactions; it is thus possible to study them in their "pure" form,
contaminated only by the reasonably well-understood electromagnetic
interactions. On the other hand, the strongly interacting hadrons also
interact weakly, and it is possible to learn something about the strong
interactions with the help of the weak ones as "probes". The degree to
which the weak interactions are understood, and what they have taught
us about the strong interactions, is the theme of this school. The role
of this survey is to show, without going into too much detail, how the
present ideas about the weak interactions have evolved, and how the
various topics treated in the specialized lectures are related to each other.
I. The Weak Interactions before 1957 **
Although the weak interactions, specifically beta decay, were dis-
covered at the end of the nineteenth century, no systematic theoretical
work was possible before the daring hypothesis of the existence of the
* Supported in part by the U.S. Atomic Energy Commission under contract
AT (11-1)-1764.
** The notation that we use is that of S. Gasiorowicz "Elementary Particle Physics"
John Wiley Inc. New York (1966) with one exception: the normalization of the spin 1/2
states is taken to be the same as that of the spin 0 states.
1 Springer Tracts Modern Physics 52
2 S. Gasiorowicz:
neutrino, made by Pauli* in 1930 and the discovery of the neutron by
Chadwick in 1932. The identification of the basic reaction as
n--,p+e- +v
made possible the quantum field theory formulation of beta decay by
Fermi .)4391( Fermi proposed to describe the interaction by the Hamil-
tonian
~H = C l d3x{ (~)n(x) ~Ipp(X)) (/~v(x) 7"//)e(X))
)i-i(
+ (~(%) 7~.(%)) )X(eT~( f~(x))} (cid:12)9
Here the quantities ~,(x), tpp(X), ... are field operators. The operator
,)X(pp~ for example, annihilates a proton or creates an antiproton, and
similarly for the other operators. This dual role of the operators is
necessary for the relativistic invariance of the theory, and it implies that
the same interaction Hamiltonian also describes the reaction
e- +p~n+v
which is observed in the capture of electrons from atomic orbits about
nuclei. The Hamiltonian also describes positron decay
p~n+e + +v
which is energetically possible in nuclei, and the reactions
v+p--*e + +n,
v+n~e- +p,
which can be studied in the laboratory < with high energy neutrino beams.
Fig. .1 Graphical representation of the Fermi interaction for the reaction n~p + e- + .v
Related reactions are described by a reorientation of the lines with the interpretation that a
change from outgoing to ingoing line requires a change from particle to antiparticle
As was already noted by Fermi, the Hamiltonian in (I-1) is not
unique. If one assumes the same local form (which has the diagrammatic
representation given in Fig. ,)1 does not include derivatives of field
* A very interesting historical review may be found in C. S. ,uW "The Neutrino",
in Theoretical Physics ni the Twentieth Century, M. Fierz and .V F. Weisskopf (Ed.), Inter-
science Publishers, New York (1960).
A Survey of the Weak Interactions
operators* and if one assumes tha~ the interaction is invariant under
space and time inversions (something which was totally taken for
granted before 1957), then the most general form that one can write is
H t = ~ dax {Cs(~.(x ) ~pp(x))(~(x) ))X(e)/~l "~ h.c.
+ Cv(~,(x ) 7~p(x)) (O,(x) 7~pe(X)) + h.c.
+ (cid:1)89 CT(~,(X) a~plpv(X)) )X(~pT( ))X(ep~P~a + h.c. (I-Z)
-~- CA(~)n(X) )Y51Op(X))('~v(X) ))X(e~15)~tc~ -~- h.c.
~- Cp(~)n(X) 5V ll)p(X)) )X(v~/,t( 5)~ II)e(X)) -~- h.c.}.
The inadequacy of (I-1) in view of the experimental situation was pointed
out by Gamow and Teller (1936). Abandoning space inversion (parity)
invariance allows for the addition of terms of the form
~d3x {C~(~n(X ) lPv(x))(~)v(x) 75 Ipe(X)) + h.c.
+ C'v(Fp,(x) y,~p(x)) (~)~(x) ))x(~p~5y~7 + h.c. (I-3)
+...}.
Time reversal invariance requires all the C's to be real. One does not
need to include terms like
Jd3x{Cs(~n(x) lpe(X))(~)v(X) ))X(p)ll -I- h.c. +-..} (1-4)
because of a general theorem of Fierz (1937) based on the completeness
of the sixteen 4 x 4 Dirac matrices, according to which
Y c,(olr,~2) (rv3r'~,) = Z D~(~r~4)(03rqp2) (I-5)
1
where ~F - ,1 ?,, ~ ,a~r~ 7,7s, 5~ and
V z
t i( 4 6 4 lsc(ti
D v -2 0 2 - C v
Dr -- ~ 0 -2 0 Cr (cid:12)9 (I-6)
D A 2 0 - 2 - C a
DI, -4 6 -4 Cp
Given the interaction (I-2), it is a fairly straightforward matter to derive
the experimental consequences for the allowed transitions in beta decay
of nuclei, i.e. for the transitions for which the matrix elements (flvTp Fi ~, Ii),
~F( = ~o, ~k75) do not vanish (the other matrix elements are very small
* Early misinterpretations of the electron spectrum measurements appeared to require
a theory involving derivatives. Such a theory was discussed by iksniponoK and kcebnelhU
.)1491(
4 .S :zciworoisaG
when the nucleon momenta are nonrelativistic). A simple calculation
shows the following
d2F
{Cs2 ( 1 ~P)(I>' 2
oc
dpedpv
(
+lEvi 2 1 + EeEv }i 21>1<1
+ ICrl 2 1 + 3 E~ 21>~<1 (I-7)
+ICA 2 1 3 EeE-~ I(a>12
2me m e }
+ ~ CsCvl<l>12- e~E 21>~<lrCAC "
In particular the terms proportional to CsCv and AC C r show a marked
energy dependence; such an energy dependence is not observed from
which one deduces that
CsCv ~ ,O
(I-8)
AC Cr ~ .O
The choice between S and V for the transitions involving the nuclear
matrix element (1> = (fw~% ,>i and between T and A for the transi-
tions involving (a> = (f +p~ ,o~-o i) can be made by means of measure-
ments of electron-neutrino correlations. Such experiments are very
difficult* and by 1957 the choice implied by (I-8) had not been made.
What was known from the decay rates was that the magnitude of the
largest of the C's (which have dimension M-2) was
ICI = a0 -5 mp 2 (I-9)
During the late 1940's information about other weak interactions
began to accumulate. The newly discovered muon was found not to
interact strongly; its electromagnetic interactions were consistent with
it being a heavy electron (this now seems to be true to a very high degree
of accuracy) and it had interactions which led to reactions like
p- +p~n+v
and
e,----+~I -+ + v + v.
Allen
* For a review see (1959).
A Survey of the Weak Interactions 5
A very important observation made by a number of people (Klein, 1948;
Puppi, 1948; 7iomno and Wheeler, 1949;Lee, Rosenbluth, and ,gnaY )9491
was that if the Hamiltonian (I-2) was also used to describe the above
reactions (with the obvious substitutions) then the C's appeared to be
of the same order of magnitude as those for beta decay. This observation
was generalized to a principle of universality of the weak interactions,
and its formulation at the time was that in the form
E Ci(~)lrilP2)(u~3 ri )4)i1 (I-10)
,S=i P...
the various weak interactions could be described by taking for the pairs
,1( )2 and ,3( )4 pairs of fields ,n( ,)p ,v( )e and ,v( .)# This notion proved
very fruitful. It provided a mechanism from the comparison of the reac-
tion rates for ~ ~.u + v and cr ~ e + v via the mechanism
+ # { , v +
n+ ,gnorts p( +~) kRow
e++v
Although the simplest Feynman graph (Fig. )2 led to a divergent expres-
sion, the universality of the (#v) and (ev) couplings implied that informa-
tion about the relative magnitudes of aC and pC could be obtained from the
n
P
Fig. .2 Simplest Feynman graph for the decay of a charged pion
ratio F(r~ev)/F(rc--*ltv) (Ruderman and Finkelstein, 1949). Here too
the information was not conclusive and the whole field seemed to be in
a state of confusion (Michel, 1956). The confusion was soon removed
by a whole reevaluation of the known experimental information and
by a great deal of new experimental activity stimulated by a paper of
Lee and Yang )6591( entitled "Question of Parity Conservation in Weak
Interactions".
II. Parity Nonconservation and the Form of the Beta
Interaction
In the early 1950's a rather cozy, if intractable, picture of a hadronic
world consisting of protons, neutrons and pions had to be scrapped
because of the discovery of the "strange" particles. Although their
6 S. Gasiorowicz:
peculiar interactions were soon systematized by the introduction of the
"strangeness" quantum number by Gell-Mann (1953) and Nakano and
Nishijima (1953), very little was understood about them. They could, in
principle be fitted into the "universal" picture of (I-10) by adding the
pair (A, p) to the set of pairs, but no attempt at any theoretical work was
possible before the spins and parities of the particles were determined.
In trying to establish these properties for what seemed to be a single
particle (same mass and lifetime within the considerable error limits) an
apparent contradiction was found. In the decays
K + ~ +er ~o
the lanif states could only have spin and parity 0 ,+ 1 ,- 2 + , ... whereas the
evidence from the decay
K+ ~+ +~z+ +7c -
(Dalitz, 1953; Fabri, 1954; Orear et al., 1956) strongly suggested the spin
and parity assignment 0-. As the masses and lifetimes of the "two"
particles came closer and as the artificiality of ad-hoc explanations of the
~-~K spectrum became more evident, Lee and Yan9 raised the question
of parity nonconservation in the weak interactions. They noted that
except for this puzzle there was no evidence on this point, since all rates
and correlations that had been measured until that time were scalar
observables*. To see that this is insufficient, note that parity non-
conservation implies that a matrix element must contain both even and
odd parity terms
M=M+ +M_
where _+MP P- 1 = + M+. Thus in a reaction rate which measures
IM 2 = M+ 2 + M_ 2 +2 ReM+M*_
the presence of both M+ and M_ can only be definitely established by a
measurement of ReM+M*_, which is odd under parity and whose
nonvanishing definitely proves parity nonconservation. A measurement
of a decay rate, for example, involves an averaging over spins (or integra-
tion over momenta) which never detects the presence of the pseudoscalar
term. Lee and Yan9 then suggested the measurement of some pseudoscalar
observables in a weak decay. Most of their tests reduced to a measurement
of quantities like (S. p>, where S is an axial vector, e.g. the polarization
of a nucleus, and p is a polar vector, e.g. the momentum of the electron
* For an interesting exception see the report of L. Grodzins in the Proceedings of the
National Academy of Sciences, Washington, 45, 399 (1959).
