THE EFFES‘ES OF L’NéTARY SYMMEERY UPOPi MESON
PRODUSMQ m PRGWN *ANTiPROTON ANNIHILATION
““m: €232? 'ihe Degree of Ph. D.

Mifjéflafiifi STfiaTE UWERSW
fiahefi G.Ponfim§

1967

 

 

This is to certify that the

thesis entitled

THE EFFECTS OF UNITARY SYMMETRY
UPON MESON PRODUCTION
IN PROTON-ANTIPROTON ANNIHILATION

presented by

Robert G. Ponzini

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Physics

LC an) / Xi”? 04/ I
/ Major professor

Date January 29. 1967

 

0-169

 

THE EFFECTS OF UNITARY SYMMETRY
UPON MESON PRODUCTION

IN PROTON-ANTIPROTON ANNIHILATION

BY

Robert G. Ponzini

AN ABSTRACT OF A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics

1967

ABSTRACT
THE EFFECTS OF UNITARY SYMMETRY
UPON MESON PRODUCTION
IN PROTON-ANTIPROTON ANNIHILATION

by Robert G. Ponzini

A theoretical investigation of meson production
from proton-antiproton annihilation at total center-of-
mass energies in the range 1.88 to 3.00 Bev is carried
out. The Fermi Statistical Model is applied and is gen-
eralized, from an SU2 invariant form, to include invari-
ance under the transformations of the group SU3 .

The calculations were done using the Control Data
3600 Computer. Phase space integrals were obtained with a
relative error of less than 6% using a Monte Carlo tech-
nique. All combinations of mesons from the J=0-, l-, and
2+ nonets, which are allowed by conservation laws, are
included as possible final states. The calculation of the
SU3 statistical weights became increasingly complex with
increasing particle multiplicity. It was therefore
necessary to limit final state multiplicities to six or
less particles under the assumption of SU3 invariance.
This placed an upper limit of approximately 2.5 Bev upon
the energies which could be considered in this case. The
SU3 weights are tabulated. The 802 weights are much

easier to calculate so that, in this case, final states of

Robert G. Ponzini

up to ten particles and energies of up to 3.0 Bev are
considered.

The results indicate that no substantial improve-
ment is made by the generalization from SU2 to SU3
invariance or by the inclusion of the resonances comprising
the vector and tensor nonets. Furthermore, it is still
necessary to introduce unphysically large volume para-
meters into the model in order to obtain results consist-
ent with experiment, a common difficulty in statistical
model calculations. However, the theory adequately
reproduces experimental results for pion distributions,

kaon production rates and charged prong distributions at

a number of energies.

THE EFFECTS or UNITARY SYMMETRY
UPON MESON PRODUCTION

IN PROTON-ANTIPROTON ANNIHILATION

BY
(30“
o( . .
Robert G. Ponzini

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of thsics

1967

ACKNOWLEDGEMENTS

I am most grateful to Dr. J. S. Kovacs for his
suggestion of the topic for this thesis, and for his
support, guidance and encouragement throughout my
graduate studies at Michigan State University.

My special thanks are extended to Dr. Richard
Yoder for his numerous suggestions and help throughout
the course of the calculation. I am also indebted to
Professor Peter Signell who graciously made available
his least squares search routine.

Finally, I wish to thank my wife, Moureen, for
correcting and typing the manuscript and for her many

years of patience.

ii

I.

II.

III.

IV.

V.

TABLE OF CONTENTS

Page

Introduction . . . . . . . . . . . . . . . . . . l
The Statistical Model . . . . . . . . . . . . .

A. Formulation of the model . . . . . . . . .
B. Successes and failures of the model . . . .

\OU'IU'I

The effects of Unitary Symmetry upon
meson production in p-f annihilation . . . . . 11

A. Symmetry groups in physics . . . . . . . . . 11
B. Application of SU symmetry to

strong interactiogs . . . . . . . . . . . . 14
C. The 80 weight factors . . . . . . . . . . . 19
D. The $03 weight factors . . . . . . . . . . . 28
Calculations . . . . . . . . . . . . . . . . . . 47
COHCIuSiOnS O O O O O O O O O O 0 0 O O O 0 O O 76

Appendix A O O O O O O O O O O O O O O O O O O O 7 9
Appendix B . . . . . . . .

References . . . . . . . .

iii

LIST OF TABLES

Table Page

1. PrOperties of the psuedoscalar ( J=0-)
mesonnonet..................16

2. Preperties of the vector ( J=l-)
mesonnonetoeoeeeooeeeoeeeeoe16

3. PrOperties of the tensor ( J=2+)
meson nonet . . . . . . . . . . . . . . . . . . 16

4. 80 weight factors for states of mesons
frgm the 8-dimensional representation . . . . . 40

5. Important decay modes and branching
ratios used in the calculation. . . . . . . . . 49

6. Values of phase space integrals for states
Of 2 to 10 pions o o o o o o o o o o o o o e e 50

7. Results for the statistical model with a
single volume parameter . . . . . . . . . . . . 52

8. Results for the statistical model with a
separate volume parameter for each meson nonet. 53

9. Results for the best fit to the experimental
data for annihilation at rest . . . . . . . . . 57

10. Results for the best fit to the experimental

data for a total center-of-mass energy of
ZOSBevoeooeoeeeeeoeoeeeeoe58

iv

Figure

1.

2.

3.

6.

8.

10.

11.

LIST OF FIGURES

Average number of pions accompanying 0 and
2 kaons under the assumption of SU3 invariance.

Fraction of all annihilations producing states
having one or more K-pairs under the assumption
Of SU3 invariance O O O O O O O O O O O O O O O

The relative probability for the production

of 0,2,4 and 6 charged prongs in the absence
of kaon production, under the assumption of
SU3invarianCe................

The relative probability for the production
of 0,2,4, and 6 charged prongs accompanied by
a K-pair, under the assumption of $03 invari-

ance I O O O O O O O O O O O O O O O O O O O 0

Fraction of annihilations to a K-pair resulting
in the production of l pion under the assump-
tion Of SU3 invariance e e o e e e o o o g g 0

Fraction of annihilations to a K-pair resulting
in the production of 2 pions under the assump-
tion Of $03 invariance e o e e o o e o o o o 0

Fraction of annihilations to a K-pair resulting
in the production of 3 pions under the assump-
tion Of 803 invariance e o e o e e o e o o o 0

Fraction of annihilations to a K-pair resulting
in the production of 4 pions under the assump-
tion of $03 invariance . . . . . . . . . . . .

Average number of pions accompanying 0 and
2 kaons under the assumption of SUZ invariance.

Fraction of all annihilations producing states
having one or more K-pairs under the assumption
Of SUZ invariance e o o e e o o o o e e e e e o

The relative probability for the production
of 0,2,4 and 6 charged prongs in the absence
of kaon production, under the assumption of
SUZ invariance o e e o e e e o o e e e e e e o

Page

.60

.61

.62

.63

.64

.65

.66

.67

.68

.69

.70

Figure

12.

13.

14.

15.

16.

The relative probability for the production
of 0,2,4 and 6 charged prongs accompanied by
a K-pair, under the assumption of SU2

anceoeeeeoee

Fraction of annihilations to a K-pair
in the production of l pion under the

tion of SU2 invariance

Fraction of annihilations to a K-pair
in the production of 2 pions under the assump-

tion of $02 invariance

Fraction of annihilations to a K-pair resulting
in the production of 3 pions under the assump-

tion of $02 invariance

Fraction of annihilations to a K-pair resulting
in the production of 4 pions under the assump-

tion of SU2 invariance

vi

invari-

resulting
assump-

resulting

Page

. 73

LIST OF APPENDICES

AppendixA..................
I . . .
I. Proof that P‘i.‘2i"‘znls independent
Of the order Of £1, 62“" ‘1‘. e e e e o o e o
P 0,: ‘ 1 d t
II. Proof that 7:22."- 1].! is ndepen en
Of the order Of 14140an e o e o e e o e 0

Appendix 8 Outline of a Monte Carlo method of
evaluating phase space integrals .

vii

Page

79

79

81

85

I. Introduction.

The proton-antiproton system and the myriad
of decay modes to which it is strongly coupled, has
been of considerable interest to experimentalists.
With the advent of high-energy accelerators, a variety
of experimental data at various energies has been
obtained.l-1l The theoretical analysis of the system
and attempts to predict experimental results have
been incomplete. There appears to be little hOpe of
deve10ping, in the near future, a quantitative theory
of multiple particle production in strong interactions.
Hence, one is forced to resort to phenomenological
models in order to deal with systems with three or
more particles in the final state.

One such model, which has succeeded in pre-
dicting some of the experimental data, is the Fermi
Statistical Model.12 The model assumes that in a high
energy collision of two baryons in their center-of-
mass system, all the available energy is released into
a small volume surrounding the initial particles. The
energy is then redistributed in a statistical way.
That is, the relative probability for the formation of
a particular final state can be calculated by assuming

that all allowed final states have equal a priori

probability of being formed. The single parameter in
l

the simple model is the volume .0. in which the
interaction takes place. We would naturally expect
this to be of the order of the cube of the range of
the annihilation interaction.

The model, as prOposed by Fermi, allows for
refinements due to conservation laws in Operation and
hence, to consequences of symmetries which the system
possesses. One of the first refinements, due to Fermi
himself, was to build into the model the effects of
isotOpic spin invariance.13 We have extended this by
considering the effects of SU a invariance in apply-
ing the model to meson production in proton-antiproton
annihilation.

The invariance of strong interactions under
the transformations which are generated by the group
803 is currently being investigated. The main suc-
cess of the theory has been in reproducing large
segments of the mass spectrum of the hadrons, the
strongly interacting particles. This spectacular
success is only possible however in a "broken sym-
metry" scheme, that is, it is necessary to include
in the strong interaction Hamiltonian a term which
is not invariant under SUa transformations but
transforms like one of the generators of 803 . In

a simple first-order perturbation calculation, this

term breaks the symmetry by splitting the degenerate
SU3 energy levels to obtain the experimentally
measured masses.

When exact 8U, invariance is applied to the
dynamics of strong interactions, the theory meets
only limited success.l4-17 An obvious explanation
for this failure is that the broken-symmetry term
in the reaction Hamiltonian cannot be ignored in
dynamical calculations. A suggested way of by-passing
this obvious difficulty is to consider interactions
in which the energy available to the reactants is
much greater than the mass differences accounted for
by the broken-symmetry term.18 Proton-antiproton
annihilation into mesons is a reasonable candidate
since, even when the annihilation occurs at rest,
the energy available for meson production is large
compared to the meson mass differences in an $03
multiplet. It is also possible that the effects of
symmetry-breaking upon a many-particle system may
be masked by the higher-order pure 8U; effects.

The purpose of this calculation is then two-
fold. First we wish to investigate the effect of

SU3 invariance upon a dynamical system of the type

discussed above. The secondary aim is to carry out a

fairly complete calculation applying the Fermi
Statistical Model to mesons produced in p-‘F
annihilation emphasizing the effects of including
recent experimental data on meson resonances. In
this vein, the following calculation differs from
those done previously in that all allowable combina-
tions of 27 mesons, including the JP= 0", 1"

and 1+ nonets, have been included as possible
final states. A Monte Carlo routine has been used
to evaluate the phase space integrals. Final states
of up to 6 particles have been included under the
assumption of SUa,invariance and states of up to

10 particles under the assumption of SU1 invariance.
All calculations have been done at total energies of
1.9, 2.3 and 2.5 BeV in the center-of-mass system.
assuming invariance under both SU2 and $03 ,and at

3.0 Bev for SUzvinvariance only.

II. The Statistical Model
A. Formulation of the Model.

The Statistical Model was prOposed by Fermi
in l950 as an order-of-magnitude method of dealing
with multiple particle production resulting from
collisions between nucleons. The model attempted to
give estimates of relative cross-sections for pro-
duction of mesons and baryons from an initial two-
nucleon state.12 The model was quite crude in its
original form, taking into account only energy con-
servation. Ordinary momentum was conserved only ap-
proximately by considering all particles to be either
extremely relativistic (mesons) or non-relativistic
(baryons) and then allowing the non-relativistic
particles to carry off momentum. Subsequent to its
introduction, the model was modified to include momentum
conservation,19 isotOpic spin conservation13 and final-
state interactions.20 In 1960, Hagedorn reformulated the
model in a rigorous framework starting from S-matrix
theory.21 An important result of this formulation is
Hagedorn's interpretation of the volume parameters. Under

the assumption that the S-matrix dependence upon the

isotOpic spins, momenta and masses of the individual

particles may be neglected, the product of the
volume parameters for a specified final state can
be interpreted as the mean value of the square of
the S-matrix element. This is a reasonable assump-
tion since, in the statistical theory, the main
dependence of the relative cross-section for the
production of any state upon the individual masses
and momenta is exhibited by the phase space integrals,
and the dependence upon individual iSOtOpiC spins
is exhibited by the weight factor. Hence, Hagedorn
concludes, the product of the volume parameters
should be dependent, at most, upon the total energy
and the number of particles in the final state
irrespective of their isotopic spins, masses or
momenta. We shall refer to this conclusion later.
According to the modified statistical model,
the unnormalized probability for the formation of a
specific state consisting of n mesons from p- F
annihilation at energy E, in the center-of—mass

system, is:

‘na 4 fl
Swamp-«Mn: {IT 313(qu ‘3
ll
mm ) 5'
l __.._

1r Vt,“ (1W)“'3

where:
Vvtg, is the mass of the 5*! particle.
‘01. is the number of (identical) particles
“‘I

k
of the h‘i' kind so that t2 71L= Y?- .
II

~51; is the spin of' the it“ kind of particle.
at is the volume parameter associated with
the at” kind of particle.
Ggunis a weight factor arising from the in-
variance of the interaction responsible for
the production under the Unitary group SUn .
is; is the familiar phase space integral.
In this calculation we shall consider invariance under
the groups 80; and SD, . In the above formula, one
must consider all the particles which fit into a
single representation of SUn_ as identical. We have
assumed throughout that all particles which fit into
a single irreducible representation of SUz’, i.e.,
all members of the same isotOpic Spin multiplet, are
identical. In the case of SU3 invariance we con-

sider the above assignment to be a consequence of

symmetry breaking.*

 

*If Sua invariance were strictly true, all
particles in an irreducible representation of Su‘a
would have to be considered identical. However, the
members of a given ”SU3 multiplet' are actually split
into its isotOpic spin ($01,) multiplets by the

symmetry-breaking term.

B. Successes and failures of the model.

The refined statistical model has had moderate
success in reproducing experimental data on distributions
for the production of pions and charged prongs from
annihilation of antiprotons. Its most drastic failure
has been that it predicts too large a rate of kaon pro-
duction. This discrepancy, of course, can be minimized
by introducing a volume parameter for kaons that is much
smaller than those for other mesons. Also, in order to
fit the experimental data, it has been necessary to intro-
uce volume parameters of the order of the pion volume,*
whereas general field theoretic arguments show the range
of the annihilation potential to be of the order of the
nucleon Compton wave-length.23 Assuming that the volume
parameter must be of the order of magnitude of the inter-
action volume, we would expect it to be of the order of
7350 S11? . As pointed out by Hagedorn, the requirement
of unphysically large volume parameters may be due to the
omission of possible fépal states, in particular, to those

containing resonances. Therefore, we have included as

possible final states all combinations of the 27 mesons

 

*The pion volume is the volume of a sphere whose

3
radius is the pion Compton wave-length SI": §W(%g) ,

10

comprising the "so-called” psuedoscalar, vector and tensor
Inonets which are allowed by conservation of energy, charge,
isotOpic spin, baryon number and hypercharge. (The hyper-
charge Y is a quantity equivalent to the strangeness S

and related to it and the baryon number B by Y = B + S.

The isotopic Spin projection quantum number is also con-
served in strong interactions. Its conservation is
guaranteed by the conservation of charge and hypercharge,
the three quantities being related by Q a T + 1/2 Y.)

A major source of difficulty with the model has
been the accurate evaluation of the phase Space integrals.
Except for special cases, the integrals become so
complex for n greater than three that approximations are
unavoidable and the use of a high speed computor is
necessitated.24 In this calculation, the phase space
integrals were evaluated to within approximately 6% rela-
tive error bylg Monte Carlo technique develOped by Cerulus

and Hagedorn. This technique, together with an error

analysis, is briefly outlined in Appendix B.

III. The effects of Unitary Symmetry upon meson
production in p-P annihilation.

A. Symmetry groups in physics.

It has always been the hOpe of physicists to be
able to describe matter as a composite of a few funda-
mental constituents. Thus, discovery of new particles
frequently leads to attempts to explain their existence
as a part of a general scheme or Spectroscopy. With the
development of high energy accelerators has come the
discovery of a large number of new "particles". Among
the many attempts to classify these, the most successful
and fruitful has been the model known as the "8-fold way".
The basic assumption of this model is the invariance of
a thus-far undiscovered super-strong interaction under
the symmetry group SU3 .

In the 1950's, experimental evidence for the
charge independence of the two nucleon force led to the
postulation of the conservation of isotOpic Spin in strong
interactions.25 In group theory language, this can be
expressed as the invariance of the interaction under the
special unitary group in two dimensions, SU;Q . SUZJ
is the group of all unitary, unimodular (determinant = 1)
2 by 2 matrices. This is, in a sense, a generalization
of the rotation group, having representations of all even

dimensions (corresponding to half-integral values of

11

12

angular momentum) in addition to the odd dimensional
representations (corresponding to integral values of
angular momentum), which it has in common with the rota-
tion group.

In applying the theory of group representations
to physical systems,* symmetry with respect to a particu-
lar group is meant to imply that the Hamiltonian of the

system remains invariant under all transformations of the

form:
M
00 M . n [2‘ 9 J;
R(".°a."' 9.021. -‘- ‘eaJA) = C M “h
”.0 YL'. In
where:

:1; are Operators which are called generators of
the group. ThSSe may be chosen Hermitian in
which case they are related in the usual way to
physical observables. Of course, only those
generators which mutually commute may be
simultaneously diagonalized, the number of which
is called the rank of the group.

€31,are continuous parameters, the values of

which Specify the particular transformation.

 

*In all that follows we limit ourselves to a con-
sideration of irreducible representations of simple Lie

groups.

13

An n-fold degenerate state of the system is
represented by the basis states of an n-dimensional irre-
ducible representation of the group, one basis state
corresponding to one physical state.* There exists, for
a group of rank .2 l ,2 irreducible fundamental representa-
tions. These are fundamental in the sense that all irre-
ducible representations of the group appear in the Kronecker
products of the fundamental representations with themselves.
An important aim of most models is to fit the basic
constituents into the fundamental representations and then
to form the structured physical systems from these accord-
ing to the rules of group theory. For example, consider
the isotOpic Spin group, SU2 , as applied to nuclear physics.
The single fundamental representation of the rank 1 group
SUZ is a doublet (I ==VL9, the two basis states of which
represent the neutron and the proton. Nuclei, of all
possible 2 and N, fit into the higher dimensional irreducible
representations which are formed by taking Kronecker pro-

ducts of the fundamental representation with itself.

 

*We note that, in considering the theory of symmetry
groups in physics, there is first the group invariance
prOperties and then the correSpondence of the mathematical
quantities with the physical system which we shall call the

model.

14

B. Application of $03, symmetry to strong
interactions.

In 1953, Gell-Mann and Nishijima, on the basis of
the empirical evidence then available, postulated a new
additive quantum number called the strangeness which is
conserved in all strong interactions.26 This additional
quantum.number could not be accounted for by SUg,
invariance alone. SU1,, being a rank 1 group, allows for
only a single additive quantum number, this being identi-
fied with the 6-Spin projection I; . Hence, there could
either be a product symmetry at work or the strong inter-
action could be invariant under a higher rank group which
contains 8U; as a subgroup. A rank 2 group, having two
mutually commuting generators, could account for both the
additive quantum numbers S and I; . An obvious candidate
(among others) for this role is SO; , the group of 3 by 3
unitary, unimodular matrices.

There were a number of different models suggested,
each corresponding to a particular way of fitting the
known particles into the representations of the group. The
most famous of these was the (IV- A.) triplet model of
Sakata.27 In l961, Gell-Mann and Ne'eman independently
prOposed the model which has come to be known as the
"eightfold-way".28'29 This model differed from that of

Sakata in that no attempt was made to fit observed

15

particles into the fundamental representations of the
group. Instead, it was recognized that a large number of
the experimentally observed particles could be fit into
certain higher dimensional irreducible representations of
803 . The real strength of the model was that a simple
modification of the 803 symmetry could yield known mass
differences to a high degree of accuracy. It was later
postulated that, consistent with the eightfold-way and
the observed sprectrum of particles, three hypothetical
particles called ”quarks" and their antiparticles could
be associated with the fundamental representations of
SU3,.* The lower mass baryons are then formed by triplets
of quarks according to the prescription 6®3®38100q03h01
and mesons from a quark-antiquark pair according to
3&3“: 80.1. . This adequately describes the known
baryon decuplet, octet and singlet and the three meson

nonets ( see Tables 1. 2. 3 ). (However, a second baryon

octet has not yet been detected). In addition, the search

 

*SUa , being a rank 2 group, has two fundamental
representations. These are each of dimension 3 and are

conjugate to each other.

16
Table l. Preperties of the psuedoscalar (J=0-) meson nonet.

 

Mesons Average mass IsotOpic Hypercharge Dimension
of the I-spin spin of $03 rep-
multiplet (Mev) resentation

TI 13? 1 Q

[(3 496 1/2 1 8
O

fif- 496 1/2 -1 8

V1. 549 o 0 a

Vt‘ 959 0 o 1

 

Table 2. Properties of the vector (Js1-) meson nonet.

 

 

Meson Average mass IsotOpic Hypercharge Dimension
of the I-Spin spin of 8U, rep-
multi let (Mev) resentation

r“ 1

K‘: 891 1/2 1 e
*2

K 391 1/2 -1 8

4? 1020 0 o 3*

u)’ 783 o 0 1+

 

Table 3. Properties of the tensor (J=2+) meson nonet.

 

 

 

Meson Average mass IsotOpic Hypercharge Dimension
of the I-Spin spin of $0 rep-
multi let (Mev) resen ation

A «g I a 1 T
‘-

K 3 1405 1/2 1 8
“:

K 1405 1/2 -1 8
’ :-

1253 0 0 8

I
‘ r

f. 1500 0 0 7 l
*

In order to fit the masses of the mesons in the
vector and tensor nonets within the framework of the eight-
fold way, it is necessary to introduce mixing between the
two isotOpic Spin singlets. Therefore, neither can be as-
signed tc a definite irrcmucible representation. For sim-
plicity, we assigned the particles as shown.

17

for quarks, which in this model must be of non-integral
charge, has been fruitless.

In order to account for the mass splitting of the
particles fitting into an irreducible representation of
$03 , (with pure SU 3 invariance, all particles fitting
into a given irreducible representation have equal masses)
it is necessary to introduce a symmetry-breaking term
into the Hamiltonian. The term is chosen to transform like
one Of the generators of $03 . This choice is in analogy
to the breaking of SU,~ symmetry by the electromagnetic
force where a term transforming like the isotOpic spin
projection Operator 1:. splits a degenerate set of particles
Of a given isotopic Spin into their observed charge multi-
plets. With this choice, a perturbation calculation then
yields the mass Splittings of the mesons and baryons to
reasonable accuracy. Up to the present, this has been the
most impressive success of the "eightfold-way".

An obvious extension of this theory is to consider
the effects of SU3 invariance upon dynamical systems
undergoing strong interactions. Using the Simple rules
for the expanding of a Kronecker product in terms of
irreducible representations Of the group, with a table of
Clebsch-Gordan coefficients for 803 , one can predict rela-

tionships between various cross-sections. In.this crude

18

analysis, the symmetry-breaking term is always neglected.
AS pointed out by Levinson, Lipkin and Meshkov, this is
reasonable only if the energy available to the system is

much greater than the mass splittings of the particles
18
involved in the interaction. This analysis has shown,
14-16
for the most part, poor agreement with experiment.

19

C. The SU;_weight factors.

It is our objective to extend the statistical
theory to include the effects of invariance under the
unitary groups 80,. and SUa . As pointed out in the pre-
vous chapter, invariance under these groups give rise to
a multiplicative factor C;3oain the expression for the

relative production probability for a given final state:

he“ "
P(.q'...n")s {Tr fl‘h(2.5‘+l)“‘ (’50.. f“ ("n.wg°"“ln:E-)
Kat <1")3’l-5 11. “*1

where, in the :1 particle final state, there are ‘1).

“‘1

identical particles of the it" kind with n = *2. "a .

We now proceed to derive an expression for G‘ulusing
an approach very close to that of Cerulus.

Starting from S-matrix theory, with a definite
initial state qlé,the transition probability to a set

F of possible final states \p; is given by:

P: 2 |<¢;Iswa>n"

58F
._._ Z Z J-"j Jr.Jp.-~drnl<¢:|5'¢a)lz

U! “#5533 unsunsus

20

We assume that the wave functions are eigenfunctions of
the momenta, masses, spins and isotOpic spins of the
individual particles (two nucleons in the initial state,
Vt mesons in the final state). We represent all quantum
numbers, except those associated with the isotOpic spin,
by a single parameter,aL . In what follows, we shall
represent the isotOpic Spin of a single particle by C
and its projection by it .

For simplicity, we shall assume that the initial
state is a “pure” isotOpic Spin state 4’“ l1; 19.; 3 41'). -
If it is an admixture of isotOpic spin states, one merely

has to average over its components as follows:
(Li: % a1. lIA Iz* 3‘L)

an... % lax)“ :L.

The final state is to be made up of YL single particles
each of which has a definite value of isotOpic spin

“1' LL.) “'1. Ll..>“" I i... £2“), We transform from the
set of product wave functions to those which are eigen-
functions of the total I-Spin by a unitary transformation.

We choose a scheme where the intermediate isotOpic spins:

. -9 L
f: = (I: “' La.)
A,“ A A A

( (II'LIX+ (3),.

F4
v»
u

-h" '9 '9 t‘ l
In... a (l: + LLI '” + Ln-l)

21

are also diagonal. The transformation is:

'3:in 111‘», I 117" 2 C: ‘1 91.1 013,93 I3

‘1: £41" ' l.‘ 1 II" ‘15123

Wyn-war hamlet.» Hum)

Ila-1 ‘t a Ii

where the C's are the familiar SU2_ Clebsch-Gordan

coefficients. The above expression is generally written

as:

'11,,13' ”In-4,111): Z (11;... 10")IIZlL' A in)

L L7. “'41
Ll. up. L2,.-. 1‘ " "
X (It.‘z.>lt;¢zx> °°°° "10“?)

where the term in brackets is called the recoupling
coefficient. We will make use of the fact that the re-
coupling coefficient is real since it is a product of

Clebsch-Gordan coefficients. We also note that

. , ' 2.
Z (ILVIS: "I”... .) 1:11 t. it" 8"“) = '1 '

‘1,Lz,_ L2.
L1! ‘22..” ‘13:.

22

We want to calculate the transition probability to
a final state of 7» particles IL. L1.)I (1,1..11)‘ in (in)
the admixture of which in the preperly constructed
final state [116:3 MI,“ 31 I1) is given by the

square of the recoupling coefficient

' 2.
Ln
<I&,Ia’... In-‘3 I IL'L' i1. L:L LL51 .

Assuming the initial state is a pure isotOpic Spin
state qq‘ = II: ILC 3 0&2) J our n particle state

yields a contribution to the S-matrix element of:

(44‘ 5‘ ¢é> = (Iain ”1,.-. ) Iqllflu. .. §u>

I L11: L191,

<I.Ia-~In-.,IsI-uHASIIchc 3*: ).

Conservation of isotOpic spin demands that the

S-matrix element be zero unless I; = I; and
In; -’- Izg .

Therefore:

<45 ‘6' WI.) = <ILIJH'I-Yl-l.)Il:IZC|'L' EL ”ML'L)

‘1' ‘11. ‘1».

X (ILIQ'”I1D-I,I£Ili 34$ ‘5']:5 I102 5‘“)-

23

Of course, the final state consisting of particles
‘ ("(1') | Ltiz‘) I fit“) contributing to a
total isotOpic spin (Ig’Izg ) may be realized
through a number of sets of intermediate isotopic

spins (Ih’I‘,...In_.) . We must therefore sum

over these:

<¢9IS‘WL>=Z <Iz'I3"°I“-|°’-£~, Ilc f;'mt¢>

(In
It. 113 Ifl—l

x (Law-Imus; a-k; \S‘Ich; r“).

In order to obtain the transition probability we must
take the absolute square of the term above. We now
make a statistical assumption and neglect all inter-
ference terms by taking the absolute square within
the summation.

‘<¢;|S|wc)l’“ z. z (ampulmfilclzdg. >2.

L1. £1.“
Itt)°”IV\-I '

a,
X‘<IL13"'In-',Icizc 3 4; lSlIiI-zi 3*c>l

The basis for this approximation is that if enough
states are considered, there will tend to be an

equal amount of positive and negative contributions

24

to the square of the S-matrix from the cross terms.
This is essentially a random phase approximation.
In order to completely isolate all isotopic
Spin dependence into a single multiplicative factor,
we shall assume that any isotOpic spin dependence
left in the S-matrix element is negligible
|<:r. 13.1..-.)1-115 34.15lIché; 4;)I~\<49|5H.; )‘

so that we have:

Kq’fifl‘szz {é (I 1,-- 1“,;1 ‘ 11 ‘fl. i:._”°€iz‘n)%}

MI“.

x Kdt|5‘fl(i>‘
PI? . |<4;|5\«La>lt

t1. czzm LG

where, following the notation of Cerulus:

IF i i L 1'
81.21;,“ Lln E 15.53:. I3 .“In-..’ Iirzé ‘2'. £13.." ‘1';
Finally, we must take into account the fact that the
expression above assumes a definite order of the
individual particles (as Specified by their isotOpic

spin variables c: C; ) in the final state.

I

25

We want to find the transition probability to a final
state consisting of 10., particles of the first kind, ,4;
of the second kind, etc. regardless of their order. We
must therefore sum up the contributions to the S-matrix
element due to final states which are distinct permuta-
tions of the state above. In doing this we make use of

I
the fact that the value of P-

Lz.,Lz.,.-- LG 19

independent of the order of the {_iz) . (See Appendix A).
Hence all the contributions are the same and we simply
multiply the term above by the number of distinct permuta-
tions of the set Ii, iz,>| L;LZ;) l {.3 (za)--- , in tzn),
This factor is:

in...
11‘

. m- '
L" all-H1, n2? "k = n;
*m; I 'l (I!

11:: n*'

where:

 

YZL is the number of particles with a given
value of isotopic spin.
)1), is the number of identical particles of the
3'93 kind. Particles are considered identical if
they have the same quantum numbers ( 5, (z; ).
We now have the following expression for the
square of the S-matrix element from an initial state of

isotopic spin (Ii, 11;) to a final state consisting

26

of .R m, sets of YL‘g identical particles of isotopic

8p1n(£k,izh):

IT
Z. |<wswc>\"= 'L—-- P19 . KAHSHO“:

‘ L L O. L
6PM
UR I\ROL£$

Substituting this into the expression for the transition

probability gives:

 

rp= Tg’m'. ”PW-1.“. g j" dem’P‘Z'“ 4,0: I<“flS’4£>I.
T‘ ‘1 ‘14. t .
,h “kl. 1 ngR'iflBLEb

We recall that the symbol 4k represents mass, momentum
and spin coordinates. It can be shown rigorously (see

21
the article by Hagedorn) that the terms

2..
Z injapz 4,3... 4,3,, /<4;l$I-(£>/
33.3mm

give rise in the statistical model to the factors

1 wvvn “urn .E
{f‘kh‘ A (asp i)" *} flingi..- .13....) .
'I

A comparison of the expression for the transition
probability given above to that given previously, allows

us to identify the 80‘, weight factor for transition to

27

a final state consisting of individual particles

IL, £1,>/L" ‘.l.1>”'/‘.‘I (1") from an initial state

of isotopic spin ( Ih',1&[ ) as:

 

I' , . c L La

P ‘ . - 2' (12.13" Inn ; It 1'1; (7:,12L'” (1.. >
Lll LIL". tin. I; I’UIq...

For the case where the initial state is not a pure

isotOpic spin state but an admixture of the form:

¢:= 2. “J'IJIZJ>
J
where: 1‘
Z laJ-l - 1
a
the SUI, weight factor becomes

1.
FM! 14 d
L IQJ’ Pi-Z‘izz."'£2'! .

C754»: 1T u! 3
R

In the case of p- F annihilation, the initial state

 

(consisting of a proton and antiproton) is an equal

admixture of isotOpic Spins l and 0 so that:

T." We! ' I-1 I'°
I.

so» I 1. Lz.,11..¢zn 1. :1, 12;...Lz‘

 

28

D. The $03 *weight factors.

We now proceed to generalize the preceeding
discussion to obtain the weight factor for the case of
invariance under the group SUJ . SU3 differs from SUJ’
in that it is not a simply reducible group. This gives
rise to two basic changes in the Racah algebra for the
group:

1. In general, an irreducible representation is not
equal to its complex conjugate representation.

2. In a Kronecker product, a given irreducible
representation may appear more than once.

We will represent the pair of eigenvalues of the
Casimir Operators of 803 by the single symbol* U and the
basis states of the irreducible representation by 49:02, y
The symbol Y is necessary only if ¢ is a state belong-
ing to a representation which appears more than once in
a Kronecker product; I is the eigenvalue of the Casimir
Operator of the subgroup SUI, (it is the isotOpic spin);

11 and Y are the two additive quantum numbers for SU3

 

*SU; is a rank two group and therefore has two
Casimir Operators. The eigenvalues of these, in a
particular order, uniquely identify the irreducible

representations.

29

and are respectively the eigenvalues of the Operators

representing the isotOpic spin projection and the hyper-

 

charge.
We couple states in SU3 as follows:
U‘ - ( M; . A; U, ) + A! + U,‘
IIlY 21“. (2..., M £2. j. ‘2. Luja. III Y ‘I 1:3! utzzfla.

H: H).

where: Ix .- L1H“ LIL

Y’- fl‘+ag.

where the term in parentheses is the SU 3 Clebsch-Gordan
coefficient. We again start with the expression for the

transition probability:

P: 2 Z jujagdfiu-JP} l<'~\*;|9\‘k>lz

503 Bfifl
ungmgggs ORR! ABLES

and, as before, we shall attempt to eXplicitly to perform the

summation (approximately) over the SUJ variables.

We assume that the wave functions are eigen-
functions of the momenta, masses, spins and of the SU3
variables of the individual particles filling the states.
The $03 variables consist of the eigenfunctions Of the

803 Casimir operator A , of the SU; Casimir Operator L.

30

(the isotOpic spin), of its projection L1, and of the
hypercharge g . We again represent all other eigenvalues
by the single parameter A .

We assume that the initial state is a pure SU;

state of the form:

.-. Ui
*1 ¢IL,I1;)YC°

We shall later generalize to the general case of an
initial state which is an admixture of various SU;5
states. The final state is to contain 71 single particle

states each Of which is an SUa eigenstate:

‘ (rt. L1 3:7’ Cr"; £13.31) 0‘ +°n°2n3n>

We recall that the final state is to consist of mesons
which fit into either the 8 or the 1 dimensional
irreducible representation of SUJ each with a given
value of isotOpic spin L , projection Cz and hypercharge

.3 . (See Tables 1, 2, 3)

The final state is to be formed as an eigenstate

f: .
o U-SY: M.OH&®H30-°-$Mn

- +

if; = (13.9.. .34.”... an)

31

which will be some linear combination of our product
states above. We transform from the system of product
states to those which are eigenfunctions of (J4. and 1;}
by a unitary transformation. We choose a scheme in
which:

LLLY== 11,!) AhL

11*: (C + 13)“

(J37‘3 .X‘ldb JLLGD’LQ

. .... 1.8

. 1" L
Ia = (L|+L1~+L3)

UM, = who 41.0 4, ® 44"-.

—A L if . :8 . L
I‘-' = (L'+ Li, + La + "‘ + Ln-|)

are also diagonal. The transformation is:
‘ U" 037 ...UVMY ~U¥y 7 = E (.f' (1.!"
L
It ’ Ia ) In-'Y ’I*I2;Y; I 2,3: «(£1131
3 \ Lb;
X LILY «1, L1; 213 I", Izfl,

O n.
. . Un-I ; 111 ‘V \¢£u' i1d> ”'\ +311“ in» .

In... Ila-IYHJ (‘ (In a” I; Iqu

‘1‘! ‘)
Iilksfl.

 

We again identify the product of Clebsch-Gordan

32

coefficients as the (real) recoupling coefficient and

write this as

< U11 U3, ... Ufl-l Y U4, 1“ A, ”L 11.1,
) ) . ooo .
I; I: I“... ’1‘?” ‘I I (JV-l. La ‘41.

where we have introduced the standard notation

7/ E (£1 I .7)
also
U47 U37 Un-‘r . Ufr 11' ’da' ..
2 | ) 2'" ‘V (V, t. V» (.an =1
was/rum. I" I3 1"" I; 4' ‘1

We now identify the recoupling coefficient above as the

admixture of our product state

H? >.l+::’:a «“3139

in the correctly constructed final state

Ulf U37)... Unqr _ U9),
I,_’ 1:, I.-. J 1.14 .

 

 

Assuming the initial state to be of the form

WC :: 43::ch E. ‘I\:M> , our n meson state

has an S-matrix element equal to:

Uix Uar Un-u, 0%} ‘31. ...An'>

<q’4‘s’¢t7= <11. Ll:3 ' In-u 31:;14 ("fl/n

U U U U; . .
X <1: 1' :12"r "1&3A’F'S‘IM A”).

33

Invariance of the interaction under SU;5 demands that the
transition probability be zero unless:

U-Fr= UC

I+ 3 1;

Vi-

]bf' (This implies the two equalities

Iz+-Iz¢ and Y4: Yc )
( Y' may take on more than one value since L);
may be contained 1' times in the Kronecker
product A,$Jl,_® u-oxxn)

Therefore:

<U l} (1" U- Ah ll
<wswe> = <5: I? 11.: :41 all?

x U1): UJI'” Un-I' Ué‘. 45‘8‘ UL . o4.>
I; ’1, I.-. ’Iflf’ I‘d/4’ " .

Again, we must sum over intermediate states to obtain the
transition probability to our TL meson final state. How-
ever, in addition to summing over the intermediate 50:3
representations ‘Jy' , we must sum over all intermediate
isotOpic spin states I: which are contained in a given
irreducible representation [Jr . We now have:

<~Ms I w > z’< ‘42:: a 153% 151'. 51>

Uzi UZJIY _ Uri-Ir
IL I; 111-:

U U Ufl-l
x 1:): I?" I.-.“ 151/. "MS‘IU: 1’. ’4 >

34

where the prime on the summation is meant to indicate that
there is to be a sum over the I associated with U" if X
takes on more than one value.

We now take the absolute square of the S-matrix element
and make the statistical assumption by neglecting the
interference terms.

U U-. ; 14.1.1 ...“ L
K4d5|¢c>i~ 21’ <::*» grit. f; .9.)

”gr. .U“" 3'
1&1; 7In- -I

IJ In‘. )IL-V‘.

Xl<0:r Uar" Una, UL, .4;15|I£V3.(->’L.

In order to isolate all SU}3 dependence in a single term,
we assume any further SU~3 dependence in the S-matrix

element is negligible:

I<”:: g/ ~<.|S\11/,9I?|'<°‘¥|5'°“>lf
So that we have

I<+.Is|I».-)’“z if,” 3” its. - ”f waif-3.7m?
x1<otelsHé>r

131:“. KAISHOV

 

 

   

35

where we have defined a P-factor for SUa in analogy to
that for 9-50,. .

Finally we must take into account the fact that a definite
order has been assumed for the individual particles in

the final state. Since we want the transition probability
to a final state of n Specified individual particles
regardless of their order, we must sum the contributions
due to states which are distinct permutations of the state
above. It is shown in Appendix A that the sum over inter-
mediate states of the recoupling coefficient is independent
of the order of the constituents in the set. { :3i,¢ 5 ,
Therefore all the contributions are the same and we must

multiply the exPression above by the number of distinct

rmutations of th set ‘j>m /A 3?
pa e le, 14> >/(J,_1/ inn/n

This factor is:

where:
'7b is the number of particles in a given
irreducible representation of SUJ . For the
weights we have calculated,ޢ is the total number
of particles in the final state.
71* is the number of identical particles of the
At” kind.

36

In this case, particles are considered identical if they
have the same quantum numbers (.a, 6, £1, .1):

We therefore have the following expression for
the square of the S-matrix element (summed over SU_3
variables) from an initial state Specified by the quantum
numbers ( U6, Ii, Iz£,.Y¢,' ) to a final state consist-
ing of km sets of 7L; identical particles with quantum

numbers (11)., £4" £11, 3); ):
. L ‘nl U£,I}
é.» |<¢4\S|\h7‘ = W “*1 13.4v,_:{,...¢~

Vau‘nues "
x KO‘AFIS' “OIL,

 

Substituting this into the expression for the transition

probability gives:

If) e ———YL! {21’ 0"” 0""! ”5r .‘u' ’d’” >1}

1T “.3! U4: ”.Un-.r IL .Ifl-l ’ 1‘7" ("7, H. (”f/n
‘R I). Ila-I
.5 .s a 1.
x Z j~~fWrwm Nassau»: .
‘OPIN

Osman.“

As in the previous section a comparison with the general
expression for the transition probability allows us to

identify the SUa weight factor 6-503 . The following

37

weight factor is for a transition to a final state con-

sisting of individual particles '4»? V )I? V7 'W V)
Ln n

U .
from an initial state ¢IfVa
O.

._ n1 UgI;
GSU.’ .- I. ‘PZV‘H'VAJ

It only remains to generalize this expression for the

case where the initial state is a linear combination of
'pure' 503 states. We construct our initial state by
coupling the states representing the proton and the anti-
proton using SU3 Clebsch-Gordan coefficients. We recall
that the proton and the antiproton, respectively, fit into
the (I' 'la. , 1'1: V1,, Y3 4- ) and the (I:'/a.,Il=-'I&’ Yr-i)
slots of 8-dimensional irreducible representations of 50;.

They therefore couple to I r O or 1. with Iz-Y= 0 . Our

initial state is:

=% a‘kCI-o) +3; + J-‘(r-i) 4510
where 0* an.d ,5; are SU; Clebsch-Gordon coefficients
satisfying the relation %' lafilx 4' 'l’h'x =
The index 1 runs over the g irreducible representations

in the Kronecker product 80 8 I- 17 0 IO 0 “”98. 9 8‘61 .

Substituting the Clebsch-Gordan coefficients into the

 

*The 8-dimensional representation is equal to its

complex conjugate, Le. 8' 8‘

38

expression above yields:

=‘U2‘ ($8 'ltl : " "a. ‘1

U). ) +01:
00° 00°
U1...
8

+ "flak-1 '/i."/a.'1 loo) +Ioo.

For simplicity we may cOmbine the two terms and write:

a s \
q’t - a: Z ('lt'hl ‘lg-Vz'i ‘IBOO/ ¢lilac)

 

 

UR In

.. U U

- 2’ 2" at: I‘oor
tin 1;; “ .

While U; runs over the set {-77, ’0; ’0'. 5:, 82., 1} ; It
takes on each of the values 0 and 1. Using this initial

state, the weight factor becomes:

= __.. u 2,
65°: 1“ €40“ i lazy
1.. "
‘ 1‘- In-") it

2..
<14, 03'“ U“.., . U) y‘ .11. ,:u~ ’... .A‘ 7

X I Ia I“... ’Ifiv’h ‘Iv' (‘7‘ (“1"
where the prime has been dropped. Also we must eXplicitly

sum over both of the 8 dimensional representations in

the set {UL} . We express the final result in terms

of the SU; P-factor:
Y1.

U; l. 0)., I;
Cysu, "' 11' 11124 2.. “11* 137, 1"...)40
A ”A 017 0,.-.
- I). I" ' '1‘:

39

where we recall:

'PV Uh It a I <04? U31.”Un-|y . d1, ”1 “,Afl. 2“
my... v... 2 I“ I» I»: Ina. im’ am

We also define the product of the P-factor and the permuta-

tion factor as: '
*U,I - mo U)I

VI VI'“ in, = E ”*1 V, y" My .
The SU 3 weight factors and terms P:u V." IV") used in the
calculation are listed in table 4 for given values of
(1’, $34, Va) ). A check on these is obtained by noting

'6
that the sum 2‘ 33%;)" If , “m is equal to the number
-myk

of times that the irreducible representationéfi appears
in the Kronecker product 41,0 #;0- . - O A...» . The
sums agree with these values to within 1%. The discrep-
ancy is annoying. It may be due to a misprint in the

table of SUa, Clebsch-Gordan coefficients* or, less likely,

to computer round-off error.

 

* The table of SU3 Clebsch-Gordan coefficients
that we have used is:
Kuriyan, Lurie' and Macfarlane. Jour. of Math.

Phys. 9 722 (1955).

 

40

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41

 

 

 

 

 

 

 

 

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IV. Calculations.

The relative production rates for all allowed states
consisting of mesons (from those listed in tables 1, 2 and
3) produced in 9-5 annihilation at a total center-of-mass

energy B, were computed from:

S ("nut-"Mg, E) =

has: 1 "A 6800. (”V m", E)
11- n 2.6 +1)- -
i *3! c h } (.117) ‘-' x” "3.!

 

The total rate (the sum of the individual relative rates)
was normalized to unity.

The weight factors 950.; were computed according
to the technique set up in chapter III. The $03 weights
are listed in table 5. The SUZ"weights have not been
listed; a table listing some«of these is included in the
paper by Cerulus.31 Because of the complexity of the
numerical calculation, the sub weights were obtained
only for states containing up to a maximum of six particles;
the 802' weights, which are more tractable, were obtained
for states of up to ten particles. It was therefore not
possible to carry out calculations, under the assumption
of 803, invariance, at total energies exceeding approxi-
mately 2.5 Bev (in the center-of-mass). At center-of-mass

energies above 2.5 Bev, the rate of production for final

states of seven or more particles begins to become

47

48

appreciable. Reference to table 6 shows that the phase
space integrals for states consisting of all pions are
indeed negligible for 131 at 2.5 Bev but are not negligible
at 3 Bev. From table 6 one also sees that it is reasonable
to assume that rates of states consisting of more than 10
particles are negligible at 3 Bev. We have therefore
limited ourselves to total center-of-mass energies less than
or equal to 2.5 Bev involving final states of up to 6
particles (with resonances included) assuming $03 invari-
ance and energies less than or equal to 3 Bev for final
states of up to 10 particles assuming 80‘ invariance. The
calculations presented in figures 1 to 16 have been carried
out at total center-of-mass energies of 1.88, 2.29 and

2.50 Bev for $03 invariance and at energies of 1.88, 2.29,
2.50 and 3.00 Bev for $01, invariance. The latter corres-
pond to laboratory antiproton kinetic energies of 0, 0.92,
1.45 and 2.92 Bev respectively.

The phase space factors 39,. were calculated with a
relative error of less than or equal to 6% using a Monte
Carlo technique outlined in Appendix B, (the method of
estimating the relative error is also described there).
Resonant particles had to be specially treated as they are
not observed directly but decay rapidly into pions and
kaons. These were treated by first calculating the
production rate for the state containing the resonant

particles and then "allowing it to decay". The rate for

49

Table 5. Important decay modes and branching ratios used
in the calculation.

 

Particle Mass (Mev) Important decay modes

Partial modes* Fraction %

7777 38.7

W 549 SW‘ or Wu?) 30.8

17*11" 11° 25.0

Tr"'11"'(¥) 5.5

W’ 959 VI 11". 76.0

“*Tr (Y) 24.0

f “765 2,11’ 100

K“ 891 00" 100

4: 1020 K’R° 38.0

K’K' 30.0

11'? or JV 32.0

1» 783 W‘tf'fr" 100

A2. 1324 11"? 100

K' 1405 KT? 50.0

#1" 50.0

f 1253 217' 100

3" 1500 ' K‘E’ 30.0

K° K’ 30.0

K K‘ (891) 40.0

 

*Branching ratios into charge states, if not
explicitly shown, were obtained from isotopic spin
invariance.

50

Table 6. Values of phase space integrals for states of 2
to 10 pions.

The integrals have been calculated from:

. N .3 N ,‘ * I"
In (3579*; f‘“/ a'lJFI-mdfir {(24, PC)I(E'%.(P£+"£ ) )

Those obtained from the Monte Carlo routine have
errors attached.

 

 

ToEaI energy Kinetic energy N Phase Space integral f

 

in CM. (Bev) of F in lab (in pion mass units
(Bev) with‘b - c = 1)
1.88 0
(annihilation at rest) 1.16
1.981 .08
.6811: .027

.0721: .0029
.00266: .00016
(3.36: .20)x1o’5
(1.87: .11)x10"§0
(2.54: .15)x1o"
(7.94 :.48)x10"14

...-l

01000.10”):wa oxoooqo‘mbuw O‘DQQO‘UI-bww

2.50 1.45 2.06

8.85: .35

7.84: .31

2.43: .15

.314: .019
.0172: .0010
(5.77: .35)x10'4
(6.97: .42)x10'5
(4.42: .27)x10'8

3....

3.00 2.92 2.98

22.7: .9

34.9: 1.4

19.0: 0.8

5.16: .31

.6651 .040
.050:.003
.00148: .00009
(3.00: .18)x10"5

H

 

51

this was then divided among the various final states accord-
ing to the individual branching ratios. The branching
ratios for the necessary decay modes are listed in table 5.
Two ”models" were used in the calculations, each
corresponding to a particular way of specifying the volume
parameters. It was found that if the volume parameters
are chosen to be the same for all particles and equal to
the pion volume, the model predicts that the percentage
of annihilations at rest in which a.K-pair is produced
(kaons from annihilation are always produced in pairs due
to conservation of strangeness) is approximately 40%.
Experiment shows it to be of the order of 4%. Furthermore,
the theoretical value could not be substantially improved
by varying the volume parameter (see table 7). Another
obvious approach, consistent with SUa invariance, is to
specify that all mesons in a given nonet have the same
volume parameter. This leads to even more abundant kaon
production (as shown in table 8). It was therefore deemed
necessary to introduce something into the volume parameters
which would suppress kaon production. One way to do this
is to allow the volume parameter for a particular particle
to vary as the reciprocal of the mass of the particle
(in pion units) to some power. Hence, in this model,

32
which was originally introduced by Kalbfleisch, the

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P3 m: .... Pikes P5 P5... P3 P3 P3 P5 P3 P3
P3 so x 558?? 5.. .35 .P5 8.... 2L 5.... :5 3.3
P5 Ma 3.».me .95 9.... ..m... PS ..5 ..5 ...E ..3
P5 $3.. 8.. P355 P3 P3 P5 P5 P5 P5 P5 P5
P5 .N... 5.5.U 5.5 2L .3... 5.5 m5... 8P5 3.5 3.5

 

565m mxcmHMSmban <mHsmm menmm on ~.wo wm< mHm monamwww won m nonmw nooanIOHI
59mm momno< macaw no ~.aw mm<.

mxpmnwsmsnmw <mwswm mnoau m wmbms on mp. pawn. wm<. ppm. Hqu .Hm¢ov

v n. w. 5365. we? om zoo. 633. M... 3m .5:

emUHm m. wmmcwnm non firm mnmnwmflwnmw Scamp Swwr m mmvmnmnm <owcam wmnmamnmn mom mmn:
ammo: bound.

Him 40:56 vmnmamnmnm mum wnovonnwosmw no pralau . zsmnm June Hm mmwwoxwamdmww
mncmw «0 «sm w<mnmmm Smmm om firm vmnnwnwmm w: «Sm scum". >HH nmwncwwnwonm 3m<m
6mm: moam 659mm dam mmmcawnwo: om mcu ws<wnwmanm.

by?" by PIN.» pail.0h.

.u .3»

 

53

Hanna. 0: ocgnwnw mxvmnwsmsnmw»b n.3,.“ .9...» .98\ up...» .00.»... up... u .80.»...
msmfiow <mwcm .u n u
nwmi apv+u.ou< .ph+...oowd upb+u.oooud bk... .8OOUQ
P3 >3. panama on 325 who: Pom“... .Xm P3 P3 P3 P2.
95mm twnbocn wmoamnmav
H.mm w<m. asaUmH Om vwoam GHOI II H.¢o .mam .moo .mmw
anoma— Swg M meSmAflpa. 9.5
H.mm wmnnmsn. om wbeSMHmnwosm ».oh.w.om mm.m wq.m ~m.~ mm.H
wngcownfi ”mo—um A, .3
PP. fl: mkubu ..mm PK P5 P5
P8 4A.... Px P. .+. Lu P3 P3 P5 P2
P3 flax SLuPHu 2h mPu ”3% PP:
P3 #4 Fab? P3 PS f3 PS
P2 42.... .... Pm“... .Nu P3 P3 P3 P5
P8 fix Sway mob firm 3L wPo

 

term mxvmnwamanww <mwcmm menmm mu ~.mo mm< mHm monfimwww Mon m nonmw omaanIOHI
5mmm mbmnow macaw no n.5u wm<.

mxvmnwambflmw <muamm mnoau m>oam£ mn mp. vswm. ww<. ppm. quw Awmmo.

cm. x. 3:6? we: 9.. zoo. 2%? M... 8m 3::

54

volume parameter is given by:

--S\.L‘= SDL..($%&)"
where Q. is measured in units of the pion volume. The
other model which we have used is one in which all the
volume parameters are the same except the one which is
associated with kaons and kaon resonances. The production
of kaons may then be reduced by decreasing the kaon volume
parameter.

The results of the calculations are shown in
tables 9 and 10 and in figures 1 to 16. In all cases,
the parameters were chosen to give the ”best fit" to the
experimental data.*

The results at rest and at a total center-of-mass
energy of 2.5 Bev are shown in tables 9 and 10. The
agreement appears to be quite good, with the values of the
volume parameters under the assumptions of SU’a invariance
and of $01 invariance being roughly the same. A minor
difference is that the value of the parameter B in the

Kalbfleisch model is about 1 for invariance under SUz'and

 

*In doing this, use was made of a search program
written by Professor Peter Signell which varies the
parameters to minimize the value of )L1 for the experi-

mental data to be fit.

55

about 2 for invariance under SU3 , resulting in smaller
volume parameters for all particles (except the pions)
under the assumption of an SU45 invariant matrix element.
The only general conclusion we draw from these results
is that the volume parameters decrease with energy and
that there appears to be no marked difference between the
two assumptions (SD 1, or SUg. invariance) as far as the
present overall experimental data are concerned.

‘ Also shown in tables 9 and 10 is the fraction of
all annihilations resulting in the production of states
containing certain resonances. These are, in most cases,
consistent with experimental values.

Figures 1 to 14 show various quantities plotted as
a function of energy. Since the results shown in tables
9 and lO seem to indicate that the volume parameters
decrease with energy, it was decided to modify the models
above by including a multiplicative factor of (zuP/E¢.u, )
in each volume parameter. This is the Lorentz contraction
term introduced in the original theory by Fermi. It was
found that this modification slightly improved the fit.

To sum up, one can vary the volume parameters to
get very good agreement with experiment at a single energy
and fairly good agreement at a number of energies. In

general one has equal success assuming invariance under

56

80; or 803 . However, the values of the volume para-
meters necessary to fit the data are, in all cases, at
least an order of magnitude greater than the ”physical“
value of approximately .001.51“.consistent with the size

of the range of the annihilation potential.

57

Table 9. Results for the best fit to the experimental data
for annihilation at rest.

Symmetry group 5U 3 5U; 5 U3 5 0,,
Model Kalbfluoah emu K paranohr
Values of parameters 51.. 5,45 n.IJ,KG 11. c 7.53 J), c 5, 57
3- I.“ s e .96 42...: I.“ $1181.15

Quantity Exp. Theoretical values
value

Average pion
multiplicity 4.88:"..24“l 4.67 4.70 4.93 4.83

Percent of annih. a
producing K-pair 4.01:1.0 4.6 4.6 4.6 4.9

Fraction of annih. producing
prongs (no kaons):

o prongs .032: .oosb .029 .024 .025 .019
2 prongs .4261- .011b .432 .437 .439 .441
4 prongs .4581” .010b .466 .467 .462 .468
6 prongs .0381'.020b .028 .028 .040 .030

Production of resonances:

Fraction resulting

inf production 2 .250b .083 .238 .422 .505

Fraction resulting b

inc» production 2.045 .015 .037 .098 .096

Fraction resulting b

in Vt production 2,.014 .086 .070 .300 .148

 

Experimental values from:
aAgnew, Phys. Rev. 118, 1371 (1960)

b
c. Baltay, Phys. Rev. 145, 1103 (1966)

58

Table 10. Results for the best fit to the experimental data
at a total center-of-mass energy of 2.5 Pev.

defluseh 5m.“ K parameter
Jt: 1.59 31,-1.76 .0.- .50q 53.: .419
‘B s L90 8 81.01 .9." . lo; .51.: .MB

Symmetry group
Model
Values of parameters

 

Quantity Exp. Theoretical values
value*

Ave. pion multplcty

without kaons 5.4 1’- .35a 4.65 5.10 5.33 5.19

Ave. pion multplcty

with K-pair 2.6t .20":1 2.59 2.59 2.73 2.63

Percent of annih.

producing K-pair 131 3a 13.8 13.7 19.3 14.5

Fraction of annih. producing

pions with K-pair: b
0 pions .01 t .01 .0016 .0061 .0153 .0160
1 pion .051:.03 .068 .081 .075 .093
2 pions .41 t .17 .387 .365 .262 .313
3 pions .453: .17 .429 .421 .491 .426
4 pions .091 .09 .107 .116 .128 .131
5 pions .01 r .01 .006 .010 .023 .016

Fraction of annih. producing (pion)

prongs with K-pair: b
0 prongs .061: .02 .059 .076 .060 .061
2 prongs .62 t .23 .533 .516 .388 .429
4 prongs .321: .21 .395 .395 .536 .499
6 prongs .0051: .005 .001 .013 .015 .012

Production of resonances:

Fraction of annih. resulting

in 9 production .182 .460 .619 .739

Fraction of annih. resulting

into production .032 .081 .333 .225

Fraction of annih. resulting

in \_ production .110 .105 .281 .200

 

'ru-‘I w" rum“- ‘5 1

*All experimental values are for a total CM energy
of 2.43 Bev. The calculation was done at 2.50 Bev.

Experimental values from:

as. R. Lynch, Rev. of Mod. Phys. 33, 395 (1961)

' bG. R.

Kalbfleisch, UCRL-9597 (1961)

59

Reference to figures 1 to 16.

The experimental values appearing on the graphs are
from the following sources:

All experimental values for pion multiplicities and
charged prongs accompanied by a kaon pair are from:

G. R. Kalbfleisch, UCRL-9597 (1961)

Other values used, listed by energy (kinetic energy
of the antiproton in the laboratory), are:

0 Bev

0.47 Bev

0.92 Bev
1.26 Bev

2.86 Bev

2.99 Bev

C. Baltay, P. Franzini, G. Lutjens, J.C.
Severiens, D. Tycko, and D. Zanello,
Phys. Rev. 145, 1103 (1966)

S. Goldhaber, G. Goldhaber, W. Powell,
and R. Silberberg, Phys. Rev. 121, 1525
(1961)

G. R. Lynch, Rev. of Mod. Phys. 33, 395
(1961)

T. Ferbel, A. Firestone, J. Sandweiss,

H. Taft, M. Gailloud, T. Morris, W. Willis,
A. Bachman, P. Baumel, and R. Lea,

Phys. Rev. 113, 1096 (1966)

C. Baltay, J. Lach, J. Sandweiss, H. Taft,
N. Yeh, D. Stonehill, and R. Stump,
Phys. Rev. 142, 932 (1966)

60

-&

03

Average number of pions
p»

k

 

 

mwocnm H. ><mnmom saavmn Om chosm
moooawmbwwco c mam N mebm canon "so
mmmcaanOU Om mabwc<muwmbom.

 

 

0 63019

 

 

N X30970
waomwmwmor Sonny awn: be" fico an >3:
IIIIII mamHH wwoc Bonny 8:: ab.» ...Nh bin .r.:
..6. >6 >m.

xwamnwo mcmnm< om munwwnonoa H5 firm Hmconwnon< Awm<v

61

{FD

a“

nnihilations

Fraction of a

\
O

 

 

wwmanm N. mnmnnMOU 0m mwp macwrwwmnwocm cnomcowbm mnmnmm sm<w=o
own on Bonn xnvmwnm canon «so wmmcavwwoc Om mau.w5<mnwmcom.

\
\
\
\
\I
\-
\
\
\
\
\
\

wmwomwmwmns aommw SHE; ban Q. 00 Nu ..V».

 

I l llll mam—Hp .80: 5003. Sun: 10.» ....uv. .D-Rn .0da

 

 

.64 >6 5%.

chmnwn mbmno< on «so mbflwvnonoc M: nan Hmoonmnon< Awm<v

62

Relative production probabilit

Y .
3*

5° 2»

05—0—4

mwmcnm w. Hum HmHmnw<m pnovwcwwwn< Mon Hum whomconwoc 0m o.~.o. mam m ormuomo
chosen M5 «to mommcnm om wwoc whom—nonwoc. gown. gm mmmcavnwoc 0m mcfi
H=<mnwm=om. F

W lllllllllll It» cacao 0

       

 

“
II.“
‘
“
\
\

 

 

wwwcmwmwmn: aoamw town: ..Pou 3.5 an rd».

IIIII want. when .893 in: b." ...uu. bin. v.3

 

 

 

 

||||||||||| (WI 'l'llllllflo UHObo D

I’

ooh. ‘00 \c“!
xwamnwo muonm< On «to abnwwnonoc Mu firm Hmoonmnonw Amm4.

63

Relative production probability

A

t

b

‘

in

 

mwmcno a. eao nowmnw<o vnovmowwwn< mon n o wnoncnntva 0m
canon «so mmmcav woa Om

. non cnoamm woooavoawon on m nummwn.

mcb . anwwano .

l
“
‘II‘
“

o.-n. man m

“I. ‘.

4. A UHODQ o

 

xmwcmwowmoa aonon. Erna bio" SH» ware».

Hrau.ehh

mam“: wwoa Sonon. sauna up" 3??"

 

 

 

00K! 0
xwaonwo oaonnn cm «so manwvnonoa Ma nao Hmaonmnonw “vo<.

\O

 

/

w pnoam D

I.

 

 

\\um anoam I
anoam D

\.un

64

Fraction of annihilations

.01 wwmcno m. wnmonwoa 0m waawawwmnwoam no m Nlomwn nomcwnwao wa nao
anoncnnwoa Om H cwoa canon nao wmmcafinwoa 0m mcu Ha<mnwwaoo.

wwwcmwowmna Bonow Sauna bye» 3 cu m n x. Q».

.u mamww wmoa aonon. in: ID: u r... uh 19.x u . 6.3

 

 

 

at“: \to \tl‘

Nwaonwn oaonn< Om «no can. Unonoa Ha nao Hmconmnonc Eon:

65

Fraction of annihilations

2;:

i»

i»

o
\

 

cnoncnnwoa cm.P cwoaucanon nao ommcacnwoa cm mcuwa<mnnmanw.

/////Mwocno o. mnmonwoa om maawaiwmnwoam no a anamwn nomcwnwam Ma nao

 

 

 

wwwcmwowmna aonon. Erna .910" V. on an ..QF

mam“: xmoa Sonow in: HF" ....vh. bx u . hJ¢

 

 

. o §WI 11 o \c o \o “I
xwaonwo oaona< 0» «co manwcnonoa Ha nao Hmconwnon< Awo<v

66

Fraction of annihilations

cs

 

 

nwmcno q. nnaonwoa cm maawawwmnwoam no a xlcmwn nomcwnwan w nao
unonconwoa 0m w vwoao canon nao concafinwoa on man nacmnwwao .

 

 

 

 

meUmHonoa Bonon. sauna «Pan 900 Darn».
llllll mam—5. xmoa Bonon. twna up: ...: «a. hHXu.h4J
o.“ \00 \oq

Nwaonwo oaonmw Om nao manwvnonoa Ma nao Hmaonmnon< “wocv

67

Fraction of annihilations

.0

.P

L

on 5 vwoam canon nao mmchvnMoa Om mcu Ma<anwoaoo.

 

 

 

 

 

 

nwmcno m. nnmnnwoa cm maawcwwmnwoam no u xlcmwn nomcwnwam wa nao cnoncnnwoa

166333: sonny in: (P... Now a... ...a».

llllll mam—Hp xmoa BonoH 3.9 13:.» c. 9%. bun .644

 

9&. so-
wwaonwo oaonn< Om nao manwnnonoa wa nao Hmconmnonv

Awocv

>h.

68

\0

“pions

0?!

number of

3%

Avera e
P 9

 

wwocno o. aconooo acacon 0m wwoam moooavmawwam 0 man w rmoam
canon nao mmmcawnwoa om man. wa<mnwmaoo.

o nooam

 

 

 

wmwcmwowmoa SonoH awn: be a h»... D u .460
lllllll mamz. wmoa Sonon. and: .b. u ..A. bu u . v.09
>6 .. . wro 196

 

finaonwo oaonow 0m nao manwwnonoc wa «so Hmoonmnonw Amocv

69

Fraction of annihilations

.Pb

 

O

nwocno Ho. unwonwoa Om mHH maawawwmnwoam cnoncowam mnmnom amcwao
oao on Bono xlnmwnn canon nao mmmgwnwoa an we». wacmnwmaoo.

hmHUmHonoa Bonon. sauna be u «....NQ a u . can

 

I III! mam—HP rmoa BonoH town: by... ...: 1?.» u . “00

 

 

 

\§° « ”0° “0°
xwaonwo oaonm& 0m nao wanwwnonoa wa nao Hmconmnonw Awo<v

70

)W

*o-b-H-o-c

Relative production probability
P 2.. '

C
‘

‘

A

 

 

mwmcno Hp. aao nonmnw<o Unocmvwwwnw non nao wnoncnnwoa o o.~.h. man m
namnoon vnoaom Ma nao mcmoaoo om Kmoa anoncnnwoa. canon n o mmmcavnwoa
on mcu.wa<mnwoaoo. 1.1. llllll

I

 

 

 

I I I G v". Q
I a,

IV,

1— vnoao 0
I
l
. /
/
wwwcmwowmna aonow twna bin ho..— .. .43/ afimooao o
.I \
I \
mawww rmoa aonow awn: .kru fa. .krzpzhd. ..IV2W
w /
\ ’JI
\
\
\
\
\\

\ N wnoao D
(II. - (nllllIIIIIIII ownoaofl

\Qo “so “to

Nwaonwn oaonmw 0m nao manwvnonoa wa «no Hmvonmnonw “wocv

71

Relative production probability

 

 

 

waUmHonoa Lwnow 5.25 b.- " (“ha NI . ‘b D

 

mam—E. wmoa So on. «39 b.» ...: but. 0‘00

 

 

 

 

(

I E 4
.0
xwaonwn oaona on R% manwmnonoa Ma nao HmUOWononw Amoi

 

 

72

'4:

ilations

Fraction of annih

 

 

 

mwmcno Hm. unwonwoa om maawawwmnwoam no m xnvwwn nomcwnwan Ma nao
vnonconwoa on n. awoa canon nao ommcamcnwoa om ma». wacmnwmaoo.

. QUO

 

\\

wwwvmwowmoa Sonow awn: .pon bung S

IIIII mam“: rmoa Sonon. an: .9: u >3 ban . «.60

I l
I ' l
I
l l...
J"

. . a o . . N30 0.0
xpaonwo oaono< om nao manwwnonoa Ha «Jo Hmconmnonc .no4v

73

Fraction of annihilations

  

 

 

 

     

 

.a nwmcno Ha. nnmnnwoa on maawawwwnwoam no a xlvmwn nomcwnwao Ma nao

cnonconwoa on ~ whoam canonanao mmmcawnwoa cm may. wacmnwwano.
.k

T;
.0
1
LP
I
I
L xmwcmwowmoa aonon. an: ..Pon human an . 4am I I I
I... ....c

lllll mam—E. wmoa aonon. sauna up» ...: ban .moa /

6 >6 v.6 Mo

xwaonwn oaonmw on nao manwvnonoa ma nao Hmconmnon< «nocv

74

1::

Fraction of annihilations
F’ (9

.\

L

   

 

nwmcno Hm. nnmonwoa Om maancwwonwoam no m nlwmwn nomcwnwbm Ma nro
wnonconwoa on w wwoam canon nao mmchvnHoa om mc».wa<mnwmaoo.

 

 

 

nowcmwowmoa aonon. £55 19...» «who 6.... . Jaw

mam—Hp wooa Honour and: .9." ...: 13.x» .mom

 

 

>M. N6 0.6

xwaonwo oaonow Oh «no manwvnonoa Ma nco Hmuonmnon< Amocv

75

Fraction of annihilations

 

 

nwmcno Hm. nnmonwoa on maawawwmnwoam no a xlvmwn nomcwnwam Ma nro
mnonconwoa on 6 vwoam canon nno ammcavnwoa Om mc Macmnwmaoo.

353656: aommc in: b... h»... on how

 

IIIIII want .36: comma in: pr. ...: b..." .mom.

 

N6) hto 6.6
xwaonwo oaonow Om nwo manwwnonoa Ha nao vaonmnonw Awo<v

 

V. Conclusions.

We have attempted to reproduce experimental
results for the production of kaons, pions and charged
prongs from p-F annihilation using a simple statistical
model with the modifications mentioned below. Previous
statistical calculations were plagued with over copious
production of kaons and required unphysically large volume
parameters in order to obtain results consistent with
experiment. It was hoped that the inclusion of a large
number of resonances (which have been observed to appear
quite frequently among the annihilation products) would
alleviate these difficulties. Including the resonances
seemed reasonably promising since,to get the same final
answer, the volume parameters required can always be
reduced by including more final states which are involved
in the statistical process. The second modification
consisted of building into the model the assumption of
invariance under SU3 . However, the results presented
show that the improvement, based mainly upon whether the
experimental values can be fit with ”small" volume para-
meters, is slight. The fit to the experimental data is
fairly good: however, this is achieved only after the
introduction of modifications to suppress kaon production

and still involves rather large volume parameters.

76

77

It is difficult to isolate the effects of SU3
invariance upon the model. As might be expected, the
differences between the results obtained under the
assumption of SUa, invariance and those obtained under
$01, invariance show up predominately at low (2 and 3)
particle multiplicities and are averaged out in a
statistical calculation where multiplicities of up to
six or more particles are considered. One can see from
the results that no substantial difference is incurred in
this calculation by the generalization from 80; to $0 3
invariance.

Finally, it is worthwhile to mention a suggestion
which may help to alleviate the difficulties mentioned
above. If one uses a single parameter statistical model
(with the volume parameter of the order of 0.1.Slw- ) to
calculate the fraction of annihilations at rest producing
a kaon pair, one obtains a result which is approximately
an order of magnitude larger than the experimental value.
Furthermore, when the volume parameter is reduced to a
value of the order of the "physical” range (.Olg.‘ 5.0010...)
the kaon production is not changed significantly while
the average pion multiplicity falls well below the experi-
mental value. Maglic' has suggested a model in which kaon

pairs are still produced in abundance but, at low relative

78

33
energy for example, immediately annihilate into pions.

This effect would not only reduce the kaon production, but
would also substantially raise the average pion multiplic-
ity so that reasonable results might be expected with
small volume parameters. This line of thought can be
investigated by including in the model a K- E final

state interaction. The preliminary calculations made

by Maglic' indicate that a reasonable modification along
these lines give results which go in the direction of

further improving the model.

APPENDIX A:

1. Proof that Pain“ is independent of the
order of £1“ 6'“. in. ‘

We wish to interchange the it” and ((+1)"-"
particles going from a scheme in which:

15 ..x

A
Ii-’ + L" a: I"
It + (in : I¢'+I

to one in which:
....L A ...h

..x .9- ..s
I 4' Li. = Iifil .

The basis states transform via the éJ symbol:

'IL+I,M1+I>I.= Z (11‘.+,)’/’- (114.1)V1

I.
I- é- I'
X L4 .L t ‘14:“, Mt“):
IL-H Lei-H I

where II, M>I' implies the final state (I) M) is formed
via the intermediate value I'. We then introduce this

change of basis into the recoupling coefficient in the

I .
P‘faCtor PLZ. 11).... (.2 IL 6

In what follows we write the intermediate state in
, . + 9* -"
parentheses; L, a; (I...) implies t, + ‘1, = I". .

79

80

I4 .. 4- I I ...I I I " L" 1
pint“..- it“. Qfiml‘nf 1, 3 n-n f 1" ltzfi’. L1,” Czn>‘

-2. [u 2..
I

1,. 1,..1

I '- «1- I- . . .
X [1‘ ' (if I‘} <5: ‘z.(Ii), I; ‘5 (1:5)” Ii-I ‘c'fl (1’))1‘} “(H)“
in H:

. .. Ifl'l L" (I?) "2. £164., I :l‘ ’ ‘1nn> I 1.]
We may write the above square as a sum over I and I' and

isolate the sum over Ii:

= 2" 2. (aI+1)"’~ (1141)";

12.13" ’IC-I , Ic'fl, "In-a I) I,
X (Loni-3.0;.) "' Iii-I ££+|(I):I it (IQ!) ml”, 9" mfg")<"1 (1‘)”
I ‘0'! € 000 L
. I“ -_I LL41 (I ) I Lzo (1.1”) ”In I ‘0: (If),(.z, . (.Zul’ :2" (1")

n
I -g l: I; IL-I £6 1'
IIL +1 L L . C
X g: ( ) {I Ii-H Law-T I}{ICH LCM II}

From the unitarity condition on the 6j symbols we have:

I 5' 1M 5' .I' ...L. J
2.1-4-1 “" ‘ ‘ ‘ = 1'1".
2 ~ 4 ina- M

It“ ("WI ‘61-: 1’

After performing the sum over I' by using the 6° , we have:

PI? . 3 25 §<9 5(1).) 1211"”: (1’),"

t1. LZL... LG Iz'°’ Ii-"IC+| .00 In-'

' 2.
”I" (rim-H) 144541;), ‘c-H H ,... .0»: >

L1: ‘1‘“ +1 ‘1‘“ ‘1»: '

81

If we relabel I as I") it is clear that the right hand
side is equal to P1? with ifl’ and (oi-Ufl' states inter-

changed. Therefore:
Is - p1;

L‘l ”‘1'” ‘1‘: ‘16” Lin ‘1: ‘.l:."' ‘Z¢'+/I‘Z¢,' "° 41’! .

11. Proof that Pfl‘fijd‘ is independent of the
order of 3’, Va. 1’", .

We wish to prove that the sum over intermediate
states of the recoupling coefficient for SU 3 is independ-
ent of the order of the individual states.

We wish to go from a basis in which:

Uri-1 a At: Ur

L

...L f5
to one in which:
A .5

Ufcfl 0 Ac}: 3 U1 It—l + ((+1 3

LHt

Ur <9 A; = Ur“, I + i; = In...

In this case, the transformation will have both 503 and
Sun, 6j symbols. The notation used for the SU 3 transforma-
tion is according to Sharpe and Derome.34 We again write

the intermediate states in parentheses. The SH 3 and 80 L

parts transform respectively as:

82

‘0‘” ‘4‘ ‘0"): vi “6+: (Wad) .-.-. %‘ CUJ'I" [Utl'h

x {32" j :1} =- ’ |U”'u“'w’a).U/‘«'(U‘*'63>
' 6+: («H 1

in Y.‘

Va. '1.
‘ 1'1"“ (It): I‘ Lin (1:41)) a % [Ii] LI] I

It“ HZ-H

X {Ii-u id I} ‘1.“ (6+: (I), I (‘- (Ian) >

where summation is implied over repeated 3’ indices and [.1]
represents the dimension of the irreducible representation

labeled by I.
The P-factor for $03 invariance is:

L
U431? - I 0:: ”,UI-lr 0‘? l’u' ....An
E’c’im”: E: $033}, I“ ’3"? "‘4 ‘~"“ -

I; I"...
We suppress the indices on thex's wherever possible: the
T's above are, in general,all different. The ’ indicates
a summation over the T associated with U; . We introduce

the 6j symbols into the equation above:

83

=2 [l 2 [M] [u 1"" 1111"" [I 1""

0" On“,
It. .. In-c ‘ a". 4". 06* x '3 It}: ‘1' If
U (+0 duh-a U' I"... Y‘ If“ (in I

x (“a “‘(u‘r)“ . 0‘. .Ir 4",, (”Y’)’ I)” I": (”Oi/r )
l'I '4 (1;) If. I ‘c-H (I) I ‘O'

¢ (Inn) ,4! a,
097-! J U}. ) 1‘: J“. . . "
1m: 6:: ((1.?) “VI, (‘16, ° in“, >'
and write the square as a double sum:

I

=- u 2‘ Z EUJ "‘- E 0’] ""Lu.-J £13 ”" [IT/4 11;]
‘1' ”Jr... 0m, u u!
II. I", Ifl-n I II

fi
u... A. an N. a. 4‘. an} I... .,
0“" ”(fin U.‘ yd. g? {0:3, “(flu {1}" K It?» ‘H11 1'”, ‘m-I}
' :‘H

x <MI out-7d (w (01).”, .. 3:,» WW .-., Al“, ((1:1). “('1'

It-l ((+0 (I) I". -, ‘L+I (I, "ll“!

(where there is no sum over the Y... 's and the X‘- ’3 ).
First consider the SD), part. The sum over I; is

done first, yielding a term 7sz J13." from the unitar-

ity of the 6j symbols. This gives us, isolating the sum

over U; :

z z: 2. mm Ira-(... (”0) .3.)
03V ... UP')‘ Uttlru 04- a, U U, n
1" I"! 1”; Int-l

..(u' ) M 0:} m
..HIC ’m‘u (3'3") >°I[Uq, UV." "“3” u‘ r“. Y‘.

u... da c). ‘} 6’ ”a 1
x 1
via}, 44:?” U N (in n. J

From the unitarity of.the 803 6j symbols, we can show

84

I

J’ 3
(see below)* that the sum over U; yields: {5- ‘UN’ ‘1 J? .

Hence, we have:

[>VU5]:M :3 ‘ézzl

day .. Jt-I r U¢+Iru ”'1‘! r
13., IL-I In” Ifl-l

2.
06-1, 1’54, (Ur) I)” ll, (1):”) . Hun-o, .11.. (U; ”[1: ... ”6+! ,5" “#5)
It-I ‘1'1H (I) I t (Ii-H) In-: in (15' " cu""”“v ‘0’)

Relabeling U-v Ug I +1, (3 —r n ,

J, J(:4) ,
2“ <0” 6:41;)”

HCM

we see that the right hand side is the 803 P-factor with

the ("M and the (in) fl states interchanged.

 

*The sum over LN is almost identical to the
unitarity condition given by Sharpe and Derome. One can
easily rewrite the summation above in the standard form
by first lowering the K indices using the A matrices and
then switching the order of the indices using the X's .

This gives:

* .. ' ’ r! I; I

2 [J J {31..."(6u063 Y '3 { Up” 4:" dc, 'j . .5
c , I .
m", dig-H ’ ((+1 X6 U13” 41:” u n.“ ’7‘.

2: [U] { 06: ”5“, u}* {M a": '4‘.” U,
c
u‘. ”(H 4”! U‘ r Yin Y; t, 0“" (j‘ ("31 Ii ’3’

.L J S ’J r.’
= Lu] my r r.

by unitarity of the 6j symbols.

APPENDIX B: Outline of a Monte Carlo method of
evaluating phase space integrals.*

All calculations which follow are
done in dimensionless units.
The phase space integral for n particles with total

energy E and total momentum P is:

g. (5.9) = fix?“ 1'43» JAN Jpn orb-é. Pr) J (Eng: e‘-)

where f; , e; are respectively the individual particle
momenta and energies. In what follows, we dr0p the yu-rfi)""
and write the integral as f: ( E, P) .

The directional part is isolated and related to the

well-known 'random.walk' problem.
a?” P‘I'Jf‘ Jég 6'5' = 3’” 9" J9! 4*i

5’: (a m-j-- .:]n r. fls- :5. mspnush

('I

x}...f{(p 2, F‘é‘. ) dance :16.

C“

We separate the last integral as follows:

wn(P;P.p,,-v Mai-fl} far 2. as? He mcfen,

 

*What follows is an outline of the paper:
P. Cerulus and R. Hagedorn: Nuovo Cimento Supplement 3,
646 (1951). This article is referred to throughout the

discussion as I.

86

The function U44 is the probability that, for fixed
magnitudes of the momenta p,,P¢---Fn but random directions
A A a A
€. ,€,-'- C." the total momentum (g; P; 6.; lies in the
'I
neighborhood JP of P . For given arguments ( P; I’: P»),
no“, may be calculated using the theory of random walk.
We note that uh. is a function only of the magnitude of
the momenta. We have now reduced the phase space integral
t0: .0 .o n.
as 1' t... 1' ..
race,» = (”r-Lin a n. H5 747, PM»)
X a)“ C P; F. P‘oooPn) aP. aPa"'aPn .

In what follows, we restrict ourselves to the

C.M. system so that 5' O . Let us change the variables

I

of integration from momenta to energies by:
Pia. 1' Ci," “1“

Our intejral beecMQS:

.9 .o
P: (E)°) = (*W)“’J ”.Jm 8' 53‘”. “.1. c‘. JQL " m: as.
M. n
n
.cnm: %(P=°; e...) {(E-&e¢-)&e.---&en

where:
‘U'nLPs'O; e. era-eus wn,(P=O; P.‘Je.’-+m."--Pn).

 

Let: J1£(e): 3531:-”1“ 82%;
fl

* ‘
f“ (5,0) = (1417)“"J‘ ... A. (3.) #1‘ (64)”‘fln(en\
I lay;

x 11:,l go; e....e.‘)J (5- 28;) ac, &¢L kn .

87

A transformation to kinetic energies is made:
*6: e£-mL.o

Let: I (t3 ta.” ta.) " 11(t1" ml)a1(t2.+m'a.) "' Jul tn+"3n)

... 9:2,. (E) O) 3 (4fl)nj.?°13(t,mtn)J(T-'2ftd) "ti ... gt”-

We now order the kinetic energies:

T-I 3 ti
T2,: tl+t1

‘Tm.’ ‘t:*"t1'*""p'tna

The inverse transformation is:
t,* 17
£41.: Til-'7:

{4:3 'Th-‘Th-u

. *- W. p at- .0
.. fit— (Mr) a MIA team/Taft!

x I (T. , Ta-T. WT" - Tm.) J ( T- m)

where, it is clear that

057-157; 5 "' .4. Tno/ 5T"...
We carry out the integration over :2 Tm making use of the
6' function. The above relation then applies that the

upper limits on all the integrals may be changed to 1’ .

T
T 7'
f3: (wr‘j ant/T av; jam, 30573-77, ---T- TM)
9 o

Tn-L.

88

and we have: 0 6 T, 4. T; e 5 Tm: s T.
We are now ready to apply simple Monte Carlo

theory to the integral above. For integrals of the form:

‘I ‘3, bn.
I: a, c1771“; 47; ...L ’ &'r;‘ {(7:71...7;_)f(‘n73...Tu)

where fflt‘rruTn) is an )1) dimensional density function
defined, in the usual way, by:

1) POTWT") 20 over the allowed range of("T,‘71---Tn)
19 ba. 5.

2 ' ... ‘1’ 0.0 -

)d.d1:j%a7; jhamfmj "m- 1

and f (T. , Tana-73': ) i335 a weight function.

It can be shown that,

N . .
1'” ..L, (c) u)”. (n
1:101,» Né‘FCT‘ )T") T" )
where the set(1}--VTk)is chosen randomly but weighted
according to the distribution function fir Ta. mTw) .

We may rewrite our integral as:

f;(z,o) = (4w3”13".[j:atj;ar,_ IT NM

 

(Vt-I). Tm.
I
(31,-!) .
x Bun-11, “'T’Tn-I —-—-—-—T M]
we make the correspondence:
(n-n)!
fvr‘ 1" ...Tnd) = Constant = .1— n—I

This represents a uniform distribution in ( u-t) dimen-

sions where the intervals for each variable are chosen as:

[0,7'3 for T.
LT..TJ tor Ti

LT'a'le $0? Tn-l

89

It is clear that the variables satisfy the relation:
Oé-néTabé"' :T‘-,§ TO
We must show that f> ,integrated over the limits of our

integral, yields unity.

v.
f. ./f(T,- -7’.,-1)"7 170.3, jg‘fi/ZT A": 197ml 72-! I

If we allow all the lower limits to be replaced by C), we
introduce a factor'z;£3! into the integral above. (It is
clear that a set chosen randomly has a probability {7&331
of being in increasing order). Therefore, the integral

over the density function is equal to:

‘T T, 7 .JL. -
If!” ’TN'JTn—I =jo d’TI/a ’7;- ”A find 711-! “ ‘1 '

Hence, simple Monte Carlo calculations give

N (i) TtlT (a)
5’:(E.O)= (waft. #LIWJ ‘ T. 7'73- 1

A

(.8!

where the set CIT-NT“) is generated by choosing n-I random.
numbers uniformly from the interval LO,T] and relabeling
them so that:
'17“; Tf’e .4. Taf.”
In order to actually calculate the phase space

integrals, it is convenient to express all quantities in

terms of the individual energies. We have:
I(T|)T1’T|""T'Tn-I)= e. 56.1-m "an Kent—m: 1r". (0; 6‘" en.)

We recall that the function“U1u is evaluated using the

90

theory of the random walk. Refering to I, the result is:
1

fl.

_ - —L— n+a '
- (mourns refit-me W

nrb
. 6“”) FW)
:45) 216;; ”‘6. [33(14‘.“ 1(U'
0"!

where: all 52 (except €T:-+a.) take on the values 4.asJ-n1.

51“); +1 ‘f'or X>o
. 1 +9? X60-

Hence, we get: ) .
c: m“ ”'cm'" ”9;: £2, 5b,.(et‘e cg”)

 

*
Yn(E,O)= \n-')l (n- 3)! L'I
where
. _ . 5'0- .4712”
wnugfnen) - .— 9.9,, £{éanfin-‘fi T5j<f~ due:— ’1, )

x [2.0: may 3
The method of calculation is:
1) Generate a set of kinetic energy variables {T}
by obtaining m4 random numbers in the interval
0 to T.
2) From these calculate the individual energies
8. g 7;- Tm +m, 6-1, m (7;: o, 'T..=7').
3) Knowing the energies and masses, the function
(figs, e,"- e...) is calculated.
4) Steps l-3 are donehl times, and the phase Space

integral is obtained by multiplying the average

91

(E_“)n-l (11211-1
(Vt-n)! (rt-.5)!

value of these by the factor

We now proceed with a brief outline of the error
analysis.

It is clear that, as one increases the value of hJ ,
the value calculated for the integral approaches the true
value. The error analysis has been done in reference I and
is quite lengthy. The method used for estimating the
error is briefly outlined below.

Before starting the calculation, one chooses an
integer ”Jo which is much greater than one but small
enough so that No contributions are not expected to give
a good approximation to the integral. We then do the .
calculation, checking the convergence after successive sets
of N. terms.

Let us consider a calculation done in >\ steps of No
contributions each, so that N = >‘Ng . We define the
partial sums as follows:

5 = 2““ +(Ln(0"'a:“'” emf“)

4 c-(uq)N.+1

where A = I, 2.) A.

We then have:

X .
?>= .5. 2" 3A fem-tar a iii"; 5,).

x 14:!

where: A: £E_M)n-I (am-W“
m—a)‘. m-WTNo'

 

92

We want to approximate the integral by some value
,2; calculated with >s>>1 so that N: XN. contributions
yields a good approximation to the true value f .

At this point a problem is encountered that is a
consequence of the Monte Carlo method. We want to
approximate the exact phase space integral by the average
of A partial sums P" and hope to obtain some information
about the error in fix . The straight-forward approach to
this is to calculate ‘5‘ many times independently and
from these values calculate an RMS error. Yet we calculate
F) but one time and do not wish to do so again; indeed,
if we did so we would use the value .1) as a new A, and
get a more accurate approximation ’3}! . Hence, we work
toward obtaining an estimate of the (probable) error in
without having to calculate it more than once for a given
value of X .

We assume that for fixed, large enough )\ , the
approximate values ,3“ are normally distributed about the
exact value j: . The error analysis yields the result

that the standard deviation (5?) xdivided by the mean value
,3 is:
1.

- X
S) ( - ~ 'I " a swig)
(ean 9 ~ *(F‘J ( K

 

 

93

where — (5) _|_ 4* S
S = ‘k (17“. A 0

We note that the expression above is easily calculated,
along with fix , for successive values of A . It is also
expected that, as X increases, the standard deviation
(89))\ and therefore (i); will decrease.

We now make use if the following prOperty of the
Gaussian distribution. If fx is normally distributed
about F with a standard deviation (593mm, for an

arbitrary value f)‘ , the probability that:
‘f'fi | é (J?)A Is a: .68

‘f-f)‘$ 1({r))~ I6 '3. ‘75

‘F- F“ 5 59?), Is 2:. .999!

We can now arrive at a solution to the problem
above. The particular value we calculate for Pl is assumed
to be one of a set of values {9):} which are normally
distributed about the true value f . Using the formula
above, we calculate the approximate (JF))_/f and use this
to obtain limits on the relative error in our calculated

value Pk . Let us define:

2» Him

r

94

then:

. Exr- El 5 Zx wH-L, a: 681; Car+aun+j

I FX -f ‘ é Z’ZX usl‘i'h z 96" certamt-y
F

|‘)‘-i_‘ 562% wc+k=~¢ 99.95%
V ¢€rt11n£j

In our calculation, we chose No3 lo and varied >~
in increments of .1 starting with NI! until the value of z)
became 5:. 06 . Hence, with N = )~ No terms, we obtained
phase space integrals with a relative error less than

6% with 68% certainty or less than 12% with 95% certainty.

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HWIIlI

31293 03177 891

((fo11)?“