Table Of ContentACTA PHYSICA AUSTRIACA I SUPPLEMENTUM VII
DEVELOPMENTS
-?
IN HIGH ENERGY PHYSICS
PROCEEDINGS OF THE
IX. INTERNATIONALE UNIVERSIT.ATSWOCHEN
FOR KERNPHYSIK 1970 DER KARL.FRANZENS·UNIVERSIT.AT
GRAZ, AT SCHLADMING (STEIERMARK, AUSTRIA)
23rd FEBRUARY-7th MARCH 1970
SPONSORED BY
BUNDESMINISTERIUM FOR UNTERRICHT
THE INTERNATIONAL ATOMIC ENERGY AGENCY
STEIERMARKISCHE LANDESREGIERUNG AND
KAMMER DER GEWERBLICHEN WIRTSCHAFT FVR STEIERMARK
EDITED BY
PAUL URBAN
GRAZ
WITH 134 FIGURES
1970
SPRINGER.VERLAG I WIEN . NEW YORK
Organizing Committee:
Chairman: Prof. Dr. PAUL URBAN
Vorst&nd des Institutee fUr Theoretische Physik.
Universitli.t GrM:
Committee Members: Prof. Dr. P. URBAN
Dr. R. BAIER
Dr. H. KttRNELT
Proceedings: Dr. H. J. FAUSTMANN
Dr. P. hSEC
Secretary: M. PAIL
Acta Physica Austriaca I Supplementum I
Weak Interactions and Higher Symmetries
publisMd in 1964
Acta Physica Austriaca I Supplementum IT
Quantum ElectrodynamIcs
published in 1965
Acta Physica Austriaca I Supplementum In
Elementary Particle Theories
published in 1966
Acta Physica Austriaca I Supplementum IV
Special Problems in HiJlh EnerJly Physics
published in 1967
Acta Physica Austriaca I Supplementum V
Particles, Currents, Symmetries
publisMd in 1968
Acta Physica Austriaca I Suppiementum VI
Particle Physics
published in 1969
AU riilhlll reeerved
No parl. of thl. bool< DUly be translated or reproduced In any form
without written permiMJoD from I:Iprlnger-Verlag
o
11170 by Springer-Verlag I Wlen
SofkGver .... print of the hardcover lit edition 1910
Libra!')' of Co~ Cat.alog Cud Number 17-133409
ISBN 978·3·7091·5337·1 ISBN 978·3_7 091·Ml35· 7 (eSook)
00110.11)071978·3·7091-5335·7
Title-No. 112111
Contents
Urban, P. Introduction................................................. V
Salam, A. Non-Polynomial Lagrangian Theories........................... 1
Atkinson, D. Introduction to the Use of Non-Linear Techniques in S-Matrix
Theory............................................................. 32
Martin, A. Some Exact Results on 7t7t Scattering ......................... 71
Renner, B. Aspects of Chiral Symmetry.................................. 91
Landsholf, P. V. Some Recent Work in Regge Theory: Regge Cuts, The Absorption
Model and Glauber Theory ........................................... 145
Landsholf, P. V. Veneziano-Like Model for a Two-Current Amplitude ......•.. 166
Hite, G. E. Absorption Corrections and Cuts in the Regge Pole Model ........ 180
Phillips, R. J. N. Lectures on Regge Phenomenology ....................... 214
Hlawka, E. Differentiable Manifolds ..................................... 265
Sexl, R. U. General Relativity and Gravitational Collapse ................. , 308
Streit, L. The Construction of Physical States in Quantum Field Theory . . . . .. 355
RiihI, W. The 0(3,1) Analysis of Currents and Current Commutators ........ 392
Kugler, M. Duality..................................................... 443
Flamm, D. Some Remarks on the Quark Model .......................... 548
Kummer, W. Anomalies in Ward Identities? ............................. 567
Pietschmann, H. Structure Effects in Weak Interactions ................... 588
Doebuer, H. D. and T. D. Paley. Realizations of Lie Algebras Through Rational
Functions of Canonical Variables ..................................... 597
Tolar, J. Mass Operator in the SU(3)-Symmetric Strong Coupling Theory with
Pseudoscalar Mesons ................................................ 610
Atkinson, D. Summary-First Week ..................................... 620
Pietschmann, H. Summary-Second Week ................................ 629
Ladies and Gentlemen, dear Colleagues!
With great pleasure I welcome you most cordially to our ninth Internationale
Universitatswochen fUr Kernphysik. The presence of about 200 participants
from 17 countries gives me the hope that this winterschool will maintain its
international reputation among high energy physicists.
So, first of all, thank you for coming. I am also very grateful for the
cooperation with all our lecturers.
It gives me great satisfaction to see among our guests of honour many
distinguished members of our public authorities showing their interest in our
meeting.
It is a great honour for me to welcome Prof. Dr. A. SALAM, the represent
ative of the International Atomic Energy Agency; Prof. Dr. GUNTER B. FETT
WEIS, the Chancellor of the University of Mining, Metallurgy and Material Science,
Leoben; the Dean Prof. Dr. G. POROD, the representative of the Chancellor of
the University of Graz; Dr. TOPFNER, the representative of the Chamber of
Commerce of Styria; the representative of the Mayor of the city of Schladming
and the members of the local council; Director LAURICH, who had sponsored
all the preceding meetings; and Mr. FRANZ ANGERER, Manager of the Fremden
verkehrsverein in Schladming.
I would like to emphasize especially financial supports from the Ministry
of Education, the provincial government of Styria, as well as the IAEA and
the Chamber of Commerce which made the organisation of this school possible.
Last. not least we have to thank our host, the city of Schladming, for its hospi
tality and help in organising the meeting.
May I continue with a few short remarks about this year's scientific program.
As a consist.ent continuation of the topics of our last school we will try to discuss
the latest developments of the concept of duality and the new attempts in Regge
theory. In addition we will have lectures on general aspects and non-linear
techniques in S-matrix theory. A survey of the status of chiral symmetries and
related topics should round off the present situation in strong interaction physics.
Taking into account the increasing interest in general relativity, we included
lectures on some relevant mathematical techniques and their application in
relativity. Finally, let me mention the lecture on the construction of physical
states in quantum field theory, which will give us some insight into the exciting
recent developments of this branch of physics.
I am sure that this year's meeting also will bring us a small step further in
the understanding of high energy phenomena, and in opening this school on
"Developments in High Energy Physics" I wish you a profitable and successful
two weeks.
PAUL URBAN
Acta Physica Austriaca., Supple vn, 1-31 (1970)
© by Springer-Verlag 1970
NON-POLYNOMIAL LAGRANGIAN THEORIESX
BY
ABDUS SALAM::::
International Centre for Theoretical Physics,
Hiramare - Trieste, Italy
1. INTRODUCTION
Barring lepton electrodynamics, most Lagrangians of
physical interest are "non-renormalizable", the apparent
non-renormalizability arising either from their non-poly
nomial nature or from higher spins. Typical non-polynomial
cases are the chiral 8U(2) x 8U(2) Lagrangian
(d ¢) 2
L = Il (1.1 )
in Weinberg's representation or the gravitational Lagrangian
(L2)
where
The components gllv which enter the expression for g=det gaB
are a ratio of two polynomials in gllV • A typical example
:cpresented at the Coral Gables Conference, Miami, January
23-25,1970,and the IX. Internationale Universitatswochen
fur Kernphysik, 8chladming, February 23 - March 7, 1970.
"". .
.... On leave of absence from Imperial College, London,England.
2
of a higher spin case is the intermediate-boson mediated
weak Lagrangian, e.g., the neutral vector W~ interacting
with quarks Q ,
(1. 3)
So far as non-renormalizability is concerned, this is mani
fested most simply by transforming (1.3) into a non-poly
nomial form. In Stlickelberg variables (Q' = exp{-iy f~} Q
5 K
W = A + 1 d B) an equivalent interaction is given by
~ ~ K ~
- - f
= f Q'Y~(1+Y5)QIA~ + m Q' (exp{iY5~ B}-1)Q'
(1. 4)
It is clear therefore that if Lagrangian theory is to play
any direct role in particle physics beyond that for elec
trodynamics, methods must be developed to extract numbers
from non-polynomial theories. Basically any such methods
must ensure the resolution of the two distinct difficul
ties of non-renormalizable theories, i.e., an infinite
number of distinct infinity types and a high-energy behav
iour which violates Froissart-like bounds.
Problems with conventional treatment of non-renorma
lizable theories
1. An infinite number of infinity types:
Ignoring derivatives for the moment, one may write
Lint in the typical form
Lint = G I v~~) (~)n
where v(n) contain powers of f (typically v(n) m fn).
(We shall call f the minor coupling constant.) A pertur
bation expansion may be written to any given order N in
3
the ~~j2~ coupling constant G and to any desired n order
in the minor coupling f. In this linearized form all con
tributions of fn ~n interactions with n > 4 give rise to
non-renormalizable infinities. To remove these in the con
ventional manner, one would need more and more counter-terms
in each order, reducing very considerably the predictive
power of the theory.
2. Unacceptable high-energy behaviour
The high-energy dependence of individual graphs in
all theories with L « fn~n (n > 4) increases (unacceptably)
as the order increases and is not polynomially bounded.
(One aspect of this is that the counter-terms needed to
cancel infinities must contain arbitrarily high-order deri
vatives of field variables, making the counter-Lagrangians
non-local.)
To my knowledge the first acceptable treatment of
problem 1) was given by S. Okubo [1] as early as 1954 in
a paper which was apparently overlooked by others who sub
sequently worked on different aspects of this problem.
These include Arnowitt and Oeser, Fradkin, Efimov, Fein
berg and Pais, Glittinger, Volkov, Fried, Lee and Zumino,
Fivel and Mitter in addition to Delbourgo, Strathdee,
Boyce and Sultoon, and Koller, Hunt and Shafi [2]. I shall
review the earlier results and also state some new ones
particularly relating to renormalization constants. These
are jOint work of Trieste and London groups.
The basic idea in dealing with problem 1) is that
for a fixed order in the major coupling constant GN one can
Borel-sum the ~g~i~~_E~~~g~Q~~iQn_§~~i~§_~Q_~11_Q~g~~§_ig
.
i~ Formally this is an asymptotic series with each term
given by an infinite expression. These Borel sums have the
remarkable property that the summation automatically
4
g~~~~h~~_~~~~_~!_~h~_!~!!~!~!~~. (This is perhaps not too
unexpected a result when one considers that the Lagran
gians of the type
1
1 + f 4>2
visibly appear to possess a built-in damping factor for
higher frequencies.) For some Lagrangians this quenching
is so strong that all matrix elements are rendered finite,
offering thus the possibility of computing even self
masses and self-charges. For others some few infinities
still survive and these need renormalizing.
There are a number of different formulations of the
summation procedure - several variants - ~.,hich fall basic
ally into tvlO classes: !:!:!~_~=~J?~9~_!!)~!:b5:&~L~!}~L!:!:!§U2=2I?~9~
!!)~!:!:!Qg~. The results obtained using either method are
equivalent. The chief problem is to ensure that the Borel
sums i) possess the requisite analyticity properties in
p-space, ii) satisfy unitarity and iii) are unique. Since
good reviews [3] of these methods exist, I shall not attempt
to make this report comprehensive; I shall confine myself
to a statement of results. In respect of problem 1), these
are: i) the requirements of analyticity and unitarity are
most likely met by these asymptotic sums, though unique
ness seems to need additional criteria; ii) for a large
class of non-polynomial Lagrangians, a consistent renormal
ization programme can be devised where all infinities can
be incorporated into acceptable counter-Lagrangians.
Regarding problem 2), which concerns the high-energy
behaviour of Borel sums in the minor coupling constant, we
obtain a perfectly acceptable behaviour for space-like mo
menta. For time-like momenta the cross-sections computed
to order G2 in the major coupling constant increase un
acceptably fast with energy. It appears, however, that a
further summation, !:!:!!§_!:!~~_!~_!:!:!§_!!)~jQ!_9Q~J?1!!}g_9Q!}2!:~!}!:
9, of sets of chain-graphs alters this, just as is the case
5
in conventional theory ",here, for example, a summation of
ladder type perturbation diagrams produces Regge asymp
totic behaviour.
2. A RAPID EXPOSE OF THE HETHODS
The basic ideas of the summation methods can per
haps be rapidly illustrated by considering [4]
a) The formal series expansion for amplitudes
Formally an expectation value like
equals the asymptotic series:
co
I n., f2n lIFn ( x1-X2) • (2.1)
n=o
Each term is infinite. Indeed as n increases, the singu
larity of lin (x) (1/x2)n gets worse and worse. We shall
co
use the Borel method to sum the series. Ultimately we
are interested in the Fourier transform of this sum:
(2.2)
The criterion for an acceptable summation technique is
that F(p2) should exhibit conventional p-space analyticity.
b) The euclidicity postulate
To guarantee this consider the Symanzik region in
p-space (p2 < 0). (~Jhen more than one external momentum Pi
is involved, the Symanzik region is the region for which
P1f ~ 0 , p.p. ~ 0 . Certain other restrictions on momenta
1 J
6
are also placed but the heart of the matter is that all
momenta can be simultaneously chosen such that PiO=O.)
For p2 < 0, choose the frame where Po=O. Clearly we may
a Wick rotation without altering the value of
~ake xo~ ix~
F • Thus for the Symanzik region of p-space one needs to
consider ~n(x) for euclidean vectors x2 only. (For a zero
mass field ~(x) = -1/4w2X2, where x2 = -x2 - -x2 and is
~
real and positive.) For p-space regions outside the
Symanzik region we analytically continue (2.2). (It can
not be emphasized strongly enough that for divergent ser
ies of the type (2.1) one is not starting by "proving"
the validity of the Wick rotation. Rather, euclidicity
is a basic postulate - part of the process of defining
the theory. One it for the Symanzik region in
~99~E~2
p-space: outside this region one makes an analytic conti
nuation in the momenta.)
c) Borel summation
To give meaning to the divergent sum use Borel
F(~),
transforms and write:
m m 2
F(~) = ~ f e-~(f ~~)n (2.3)
n=O 0
using the identity:
m .
n
nl = f e-~ d~
~
0
d) The x-space method
The x-space method consists of inverting integration
and summation in (2.3) and writing it as
m
f
F(A) = d~ e-~(1-~f2A)-1 • (2.4)
o
The expression (2.4) defines the amplitude F(~). For zero
mass particles (m=O) this equals:
