Table Of ContentACTA PHYSICA AUSTRIACA I SUPPLE MENTUM VI
PARTICLE PHYSICS
PROCEEDINGS OF THE
VIII. INTERNATIONALE UNIVERSITATSWOCHEN
FUR KERNPHYSIK 1969 DER KARL-FRANZENS-UNIVERSITAT
GRAZ, AT SCHLADMING (STEIERMARK, AUSTRIA)
24th FEBRUARY -8th MARCH 1969
SPONSORED BY
BUNDESMINISTERIUM FOR UNTERRICHT
THE INTERNATIONAL ATOMIC ENERGY AGENCY
STEIERMARKISCHE LANDESREGIERUNG AND
KAMMER DER GEWERBLICHEN WIRTSCHAFT FOR STEIERMARK
EDITED BY
PAUL URBAN
GRAZ
WITH 75 FIGURES
1969
SPRINGER-VERLAG I WI EN . NEW YORK
Organizing Committee:
Chairman: Prof. Dr. PAUL URBAN
Vorstand des Institutes fUr Theoretische Physik,
Universitat Graz
Committee Members: Prof. Dr. P. URBAN
Dr. R. BAIER
Dr. H. KUHNELT
Proceedings: Dr. H. J. FAUSTMANN
Dr. P. PESEO
Secretary: M. P AU.
Acta Physica Austriaca I Supplementum I
Weak Interactions and Higher Symmetries
publi-ahed in 1964
Acta Physica Austriaca ! Supplementum II
Quantum Electrodynamics
published in 1965
Acta. Physica Austria.ca / Supplementum III
Elementary Particle Theories
published in 1966
Acta Physics Austriaca I Supplementum IV
Special Problems in High Energy Physics
published in 1967
Acta Physics. Austriaca I Supplcmcntum V
Particles. Currents. Symmetries
published in 1968
ISBN 978·3-7091-7640-5 ISBN 97s.3-709P638-2 (eBook)
00110.1007/978-3-7091-7638-2
Aile Rechte \'orbellalten
Keln Tell dl_ Buches darf ohne IChrlnll~he Genehmlgung
dCOl Sprl"ller-VerlaHes Ubenetzt oder In Irgendelner Form vervlelfiiltlgt
werden.
C 1969 by Sprl'4ler-Verlalll Wlen
Softcover reprint of the hardcover 1s t edition 1969
Library of Congreaa Catalog Card Number 11-97981
Tltel-Nr. 9261
Contents
Urban, P. Introduction................................................. V
Urban, P. In memoriam Professor GUNNAR KALLEN •••••••••••••••••••••• VII
Barut, A. O. Massless Particles and Analytic Continuation in Mass ......... 1
Gasiorowicz, S. Effective Lagrangians and SU (3) X SU (3) Symmetry Breaking 19
Furlan, G. Equal Time Commutators, Sum Rules and Low Energy Pion Physics 52
Flamm, D. Nonleptonie Hyperon Decays................................. 91
Horn, D. Finite Energy Sum Rules-Use and Interpretation ............... 124
Horn, D. Modified Quark Models ........................................ 157
Klauder, J. R. Hamiltonian Approach to Quantum Field Theory ........... 167
Jacob, M. Duality in Strong Interaction Physics ......................... 215
Dietz, K. Bootstrap of Indefinitely Rising Regge Trajectories .............. 277
Sugawara, M. Off-Mass-Shell Approach to Baryon Resonances .............. 310
Gourdin, M. Miscellaneous Topics Related to the Annihilation of an Electron-
Positron Pair into Meson ............................................ 350
Conforto, G. Experimental Tests of CPT, CP, C and T Invariances ........ 435
Sandhas, W. Composite Particle Collisions in the Non-Relativistic Three-Body
Theory ............................................................. 454
Nachtmann, O. CP Violation and Cosmological Fields ..................... 485
Garczynski, W. Stochastic Approach to Quantum Mechanics and to Quantum
Field Theory ....................................................... 501
I,ukierski, J. Extension of the Conventional Framework of Local Quantum Field
Theory and the Description of Resonances ............................ 518
Gasiorowicz, S. Summary-First Week ................................... 536
Pietsch mann, H. Summary - Second Week ............................... 543
Ladies and Gentlemen!
It is a great pleasure for me to welcome you at our eighth Schladming winter
school. As the organizer of this meeting I would like to thank all of you for
accepting my invitation. About 220 participants from 19 nations, the largest
number in the history of this school, and the presence of high public officials
at this opening ceremony indicate that our school meets with increasing approval
among specialists as well as in the public.
I have the special honor to welcome the representatives of the Ministry
of Education and of the provincial government of Styria, as well as of the
International Atomic Energy Agency and of the Chamber of Commerce. I would
like to ask most heartily all our sponsors to accept my warmest thanks for their
financial support, which made it possible to organize these "Universitatswochen".
It is a great pleasure for me to have among our guests of honour the Rector
of the University of Graz, Prof. Dr. O. BURKARD. Last but not least I welcome
Director LAURICH, Mayor of our host city Schladming, who has helped us in
a dedicated way in organizing this symposium, and, as we have seen already,
has taken great pains to carefully prepare the surrounding slopes to satisfy
also the skiing-fans among us. In this connection I would like to ask the township
the favor of taking care for a constantly fair weather during our stay here.
Finally I welcome all participants of this meeting from this country and
abroad and I hope you will derive great benefit from the lectures and discussions
with regard to your own scientific problems. It is a great privilege for me to
have suceeded in gaining the cooperation of highly outstanding experts and
I want to express my thanks that they have undertaken the trouble of coming
to us, which will be rewarded by the benefit their lectures will render to us.
I~et me now turn to our scientific program. As was the case last year our
program this year, too, deals with problems of elementary particle physics and
in particular with the problems connected with strong interactions. The absence
of a unified theory of strong interactions has led to the establishment of various
theories and models in order to get some new information and to explain the
experimental results available.
One of these attempts is the Regge Pole Theory, which during the last years
has made considerable progress. By combining this theory with dispersion
theoretic statements we arrive at so-called finite energy sum rules, which lead
to the concept of duality in strong interactions. Some other approaches giving
insight into problems of strong interactions are provided by effective Lagrangian
and CuIT ent Algebras. Among other topics we will hear something on experi
mental tests of CPT, CP, T and C and on problems in connection with e+ e-
scattering in storage rings.
Once more I wish to thank all of you for your coming, and in opening this
meeting on "Particle Physics" I wish it every success.
PAUL URBAN
In memoriam Professor Gunnar Kallen
About four months ago, on October 13, 1968, Professor GUNNAR KALLEN
of the University of Lund, Sweden, died in an airplane accident near Hannover.
Undoubtedly Europe lost one of its most prominent theoretical physicists;
moreover, ever since the foundation of our Schladming winterschool, KALLEN
was one of its most regular participants and repeatedly showed his great interest
in this meeting by lectures, critical summaries and by engaging himself in vivid
discussions. His intimate relationship with our school and his continuous friend
ship with the author of this obituary are a special reason for honoring in this
place Professor Kallen and his outstanding scientific successes.
Let me first briefly state the significant dates in his remarkable scientific
career:
Born on February 13, 1926 at Kristianstad, KALLEN studied physics in
Sweden and completed his doctorate at the University of Lund in 1950. Starting
as an assistant professor at Lund from 1950 to 1952, he got to the Theoretical
Study Division of CERN at Copenhagen (1952-1957), worked at NORDITA
(1957 -1958) until he was offered a professorship for theoretical physics at the
VIII
University of Lund. Moreover, KALLEN made several journeys for the purpose
of research and took part in numerous scientific conferences in about 15 countries,
including the USA and the USSR.
Let me now mention some details about his scientific work: Initially he
studied electrical engineering but soon he changed over to physics, especially
to the problems of quantum electrodynamics; in this field he achieved most
important results in the years of 1949 to 1955. His principal aim was the treat
ment of the theory of renormalization using, unlike other authors, consequently
the Heisenberg picture instead of the interaction picture and the relations now
known as Yang-Feldman equations. Considering spectral representations for
two- and three-point functions he succeeded in separating the renormalization
constants of quantum electrodynamics and in expressing them as integrals over
certain weight functions; thus he could precisely formulate and try to solve
the problem of the value of renormalization constants. Indeed, other authors
are in doubt about his famous proof that at least one of the renormalization
constants has to be infinite, but so far no definite answer to this question could
be found.
KALLEN's authority at that time in the field of quantum electrodynamics is
well illustrated by the fact that it was he who was requested to write the article
on this topic in the "Handbuch der Physik".
In connection with his work on quantum electrodynamics he began to study
closely the analyticity properties of three- and four-point functions and obtained
a number of important results, partially cooperating with WIGHTMAN and TOLL.
We must not forget a treatment of the Lee model, which KALLEN did together
with PAULI and where they discovered and discussed the possibility of "ghost
states".
KALLEN was not only interested in the development of the general theory,
he also treated many difficult concrete problems, such as in his works on vacuum
polarization of higher order.
During the last years KALLEN performed fundamental work in the field of
radiative corrections to weak decays. Let me please talk in some detail about
the article "Radiative Corrections to {3-decay and Nucleon Form Factors",
published in Nuclear Physics 1967, because it was KALLEN himself who gave
a lecture on the basic thoughts in a preliminary version at our Schladming
meeting 1966 for the very first time:
Taking into account strong interactions, KALLEN tried to find a physically
reasonable solution to the convergence problem for radiative corrections of
order in Fermi decays. As you know, radiative corrections in {3-decay and
(X
also in other semileptonic or leptonic hadron decays depend on a cutoff if one
uses a "hare particle calculation" (i.e. without strong interactions). (Essentially
the lifetime reads:
(1
.!. = ~ + (X f (MF, MGT, GA/GV) In ~ + finite terms),
T To mp
A being the cutoff.)
KALLEN took up the idea proposed by BERMAN and SIRLIN in 1962, namely
to take into account strong interactions by the introduction of suitable form
factors. But there is one crucial difference to BERMAN and SIRLIN:
KALLEN uses on-mass-shell form factors, that means quantities which are
experimentally measurable in principle and can therefore be used as phenom-
IX
enological parameters of the theory. Of course, I can treat the formalism, which
permits the introduction of such form factors in a natural way, only generally.
Essentially one decomposes the fJ-decay matrix element of the current
current Lagrangian into a product of matrix elements of the electron field operator
and the weak hadron current by summing over intermediate states which can
bring about radiative corrections of order ex. As is the case in dispersion theoretical
considerations states of higher mass are neglected. The matrix elements of the
hadrons are calculated first for point particles and are then modified corresponding
to strong interactions by the introduction of only a few form factors; for example,
the matrix element (p, y I jV I n) is considered as
(p, y I jV I n) = ~ (p, y I jV ! n)point particles
with a form factor ~, which can be measured in principle by bremsstrahlung
processes of nucleons in external fields by virtue of eve hypothesis. elearly
we also get the wellknown electromagnetic form factors of the nucleons in the
matrix element (p I jet I p') occurring in the development of the electron operator
in order ex, and also in (p I jV I n) on the account of eve.
By suggesting for the unknown form factor a behavior similar to the usual
ones one obtains higher powers of the photon momentum in the denominator,
infinite integrals do not occur anymore.
A further restriction, which nevertheless ili reasonable and justified is the
following: Since the electromagnetic electron operator renormalization is totally
independent of strong interactions and yields an infinite integral in the usual
Feynman gauge, that means it is gauge dependent, only those gauges are
considered which guarantee a finite result for the renormalization; in summing
up all corrections affecting the lifetime, the gauge dependent contributions
cancel out; KALLEN called this fact "restricted gauge invariance".
There are two important features in this method: first we get finite radiative
corrections (although no exact numerical results can be expected because of
the approximative character of the formalism), and secondly, an estimate of
the cutoff is possible. This estimate shows that the assumption A ~ mp in the
bare particle calculation was a very good approximation.
KALLEN's result-finite radiative corrections by means of strong interaction
is in striking disagreement with works of other authors who included the modern
concept of current algebra in the calculation of radiative corrections in fJ-decays.
The essential point in this alternative method is the contraction of the photon
in the matrix element (:71;0, y I jV I :70), hereby obtaining a retarded commutator
of two currents. This commutator can be calculated in case of equal times by
the relations of current algebra. The result is a logarithmically divergent factor
in the vector coupling constant in spite of strong interactions, as long as the
vector part of the weak interaction is considered. Because of the model-dependence
of the axial vector part the authors of the current algebra method get finite
radiative corrections by constructing a composite model of hadrons in such
a way that the resulting axial vector divergence compensates the logarithmic
vector divergence.
Though KALLEN did not solve the problem of the influence of strong inter
actions on the convergence of radiative correction" in weak decays (by means
of his form factor method) it turned out to be an important controversial
question in this way, still lacking a final satisfactory solution.
KALLEN was one of the first who used reduction formalism, dispersion cal
culations and spectral representations in all his works, methods which became
x
standard tools in modern physics. Surely lULLEN'S works have contributed much
to the fact that field theory is applied in elementary particle physics more
than ever.
Furthermore KALLEN has earned considerable merits in the field of elementary
particle physics as the author of an excellent book in which many problems
of strong and weak interactions are treated. Here, just as in his conference lectures,
KALLEN proved his outstanding peda.gogical talent. In his book he has shown
excellently how much about mathematical methods and detailed calculations
should be presented, enough to clear up the connection between theory and
experiment, but not so extensively that the presentation could be spoiled.
Ladies and gentlemen, I hope that to a certain extent I have been successful
in doing justice to the personality of GUNNAR KALLEN and his position in science.
You will certaiuly agree if I emphasize again that his early death undeniably
has left a gap among the most outstanding theoretical physicists of Europe.
PAUL URBAN
MASSLESS PARTICLES AND ANALYTIC CONTINUATION IN MASSt
By
A.O. BARUTtt
International Centre for Theoretical Physics
Trieste
I. INTRODUCTION
The fundamental importance of the massless particles
in physics is seen from the fact that almost all known
interactions are dominated by a massless particle (with
an extrapolation to soft mesons (m+o) in the case of
strong interactions) (Table 1).
Interaction "Massless" Particle
Gravitational graviton j = 2
Electromagnetic photon j = 1
--------------------------------------
t
Weak (leptonic) neutrino j =
--------------------------------------
Strong pions (kaons) j = 0
Table 1 - Massless Particles and Funda
mental Interactions.
An exception seems to be the non-leptonic weak interac
tions of strange baryons: whether any massless virtual
t Lecture given at the VIII. Internationale
Universit~ts
wochen fUr Kernphysik, Schladming, February 24-March 8,1969.
ttOn leave from the University of Colbrado,Boulder,Colorado.
Acta Physica Austriaca, SUl'pl. VI
2
states enter into these processes is not clear.
The strength of the interaction decreases with increas
ing spin for bosons : j = 0 (strong),j = 1 (em) , j = 2 (gra
-!
vitatti on), j = 3 (no) and for fermions j = (\'leak),
=
j (no).
II. GAUGE CONDITIONS
It is wellknown that the limit from a massive particle
to a massless particle is a discontinuous one: there are
(2j+l) states of polarization for m>o, but only one state
of polarization for m = 0, no matter what the spin j is.
This is a direct consequence ~f the properties of the uni
tary irreducible representations, labelled by [m,j], of
the restricted Poincare group. One could treat therefore
the properties of the scattering amplitudes for massive
and massless particles quite 'independently taking into
account from the beginning the condition m=o. There is
however a device which allows one precisely to perform
the discontinuous limit m+o • It is important to use
this limiting process, because we can then study the ana
lyticity of the amplitudes or coupling constants as a
function of the masses, in particular in the neighborhood
of the points m=o. If the various elementary particles are
intimately related to each other one would expect that the
amplitudes for different processes should be related to
each other by analytic continuation in the quantum numbers,
in particular in mass and spin. There arise even the fur
ther, more difficult process of analytic continuation of
an amplitude with N particles to another one with N-l (or
more generally to one with M) particles, which involves
the vanishing of all quantum numbers and of momentum
p~
of a particle. This is indeed quite a stretch of imagina
tion as to what the analytic continuation should be able
