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Relativistic parametrization of the P₃₃ resonance in pion-nucleon scattering

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Relativistic parametrization of the P₃₃ resonance in pion-nucleon scattering
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Tschang, Yin-po, 1944-
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Mathematical constants ( jstor )
Nucleons ( jstor )
Particle resonance ( jstor )
Phase shift ( jstor )
Physics ( jstor )
Pions ( jstor )
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Scattering amplitude ( jstor )
Meson resonance ( lcsh )
Mesons -- Scattering ( lcsh )
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Thesis--University of Florida.
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Bibliography: leaves 60-62.
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Typescript.
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Vita.

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RELATIVISTIC PARAMETRIZATION


OF THE P' RESONANCE IN PION-NUCLEON SCATTERING
33





By

Yin-po Tschang


A Dissertation Presented to the Graduate Council
of the University of Florida
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy




UNIVERSITY OF FLORIDA
1972




RELATIVISTIC PARAMETRIZATION
OF THE P' RESONANCE IN PION-NUCLEON SCATTERING
By
Yin-po Tschang
A Dissertation Presented to the Graduate Council
of the University of Florida
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
UNIVERSITY OF FLORIDA
1972


ACKNOWLEDGMENTS
The author wishes to thank his dissertation advisor,
Professor M.T. Parkinson, for suggesting this interesting
problem and for providing guidance with immense patience and
diligence throughout this work. He would also like to ex
press his deep gratitude for the help and advice of two dear
teachers, Professor A.A. Broyles and Dr. J. McEnnan. They
have helped build a foundation from which the present en
deavor has started.
Thanks are extended to the Department of Physics for
assistantships and an instructorship, to the Graduate School
for a fellowship, and to the College of Arts and Sciences
for a research grant at the University of Florida Computing
Center. The technical help received from Dr. F.E. Riewe and
the U.F.C.C. staff is also to be thanked.
ii


TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ii
LIST OF TABLES v
LIST OF FIGURES vi
ABSTRACT vii
CHAPTERS
I. INTRODUCTION 1
II. PION-NUCLEON SCATTERING 4
1. Introduction 4
2 Kinematics 5
3. The Scattering Amplitude 9
4. The Phase Shift 12
5. The Br eit-Wigner Formula 15
6. The Chew-Low Model 16
7. The Effective Range Approximation l8
III. THE K-MATRIX FORMALISM 20
IV. PHASE SPACE FUNCTION 24
1. The Regular Two-Body Phase Space Function 24
2. Kinematic Singularities of Helicity
Amplitudes 25
3.The Phase Space Function 29
V.PARAMETRIZATION OF THE P* RESONANCE 31
iii


TABLE OF CONTENTS (CONTINUED)
page
1. Introduction 31
2. Basis of Computation j 32
3. Phase Shifts.. 34
4. Left Half-Plane Singularities.. 36
5. Contributing Channels.. 37
6. Computation 39
VI. RESULTS AND CONCLUSIONS 41
1. Comparison with Breit-Wigner Formula 41
2. General Description of Phase Shift Fits.. 42
3. The Resonance Mass 47
4. The Subtraction Constant 47
5. The Coupling Constants 48
6. General Remarks 51
APPENDICES
A. MECHANICS OF CALCULATION ...... 53
B. COMPUTER PROGRAM 54
REFERENCES 60
BIOGRAPHICAL SKETCH 63
iv


LIST OF TABLES
Table page
I. Characteristics of Contributing Channels 33
II. Parameters from Phase Shift Fits 43
III.Phase Shifts Fitted According to Carter et al. 44
IV.Phase Shifts Fitted According to Compilation
Data.. 45
v


LIST OF FIGURES
Figure page
1. Diagrams 17
vi


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
RELATIVISTIC PARAMETRIZATION
OF THE P^3 RESONANCE IN PION-NUCLEON SCATTERING
By
Yin-po Tschang
December, 1972
Chairman: Professor M. T. Parkinson
Major Department: Physics and Astronomy
A relativistic generalization of the Breit-Wigner for
mula is further improved and applied to the study of the P^3
resonance in pion-nucleon scattering. Reasonable fits with
the P^3 phase shift data are obtained. The new parametriza-
tion reduces to a simple Breit-Wigner form in the neighbor
hood of a resonance. Away from the resonance, however, the
new parametrization has been seen to be far superior to the
Breit-Wigner formula. This parametrization opens up a pos
sible way of attacking the problem of hadron dynamics.
In addition, values of the coupling constants C^^
and C^^A are predicted by this parametrization from phase
shift fits.
vii


CHAPTER I
INTRODUCTION '
Direct channel resonances have been assumed to dominate
the scattering amplitudes of strong interactions over a wide
energy range. Empirically, this idea seems to be well sup
ported.1 Theoretically, it is now believed that the duality
between direct channel resonances and cross channel exchanges
allows the resonance-dominance assumption to coexist with the
2
usual exchange mechanisms.
In particular, the dominance of a nearby pole, plus the
requirements of unitarity, have been used to describe the
behavior of the scattering amplitude in the neighborhood of
that pole. The most commonly used parametrization for a pole
is the Breit-Wigner formula. This treatment, while exact at
o
the pole, has the following deficiencies:
(1) It is non-relativistic.
(2) Only open channels are used, nearby singularities can
thus be omitted.
(3) Poles are produced on all sheets of the scattering am
plitude.
(4) The position of the pole is unrelated to partial widths,
thus the parametrization does not say anything concern
ing the dynamical origin of the resonance.
1


2
Previous works by Parkinson and others have succeeded
in building up a relativistic K-matrix formalism which is
very similar to the Breit-Wigner formula in form, but does
O A
not have any of the above objections. **
This work represents an extension of the new parametri-
zation, which has been rather successful in describing the
rho meson, to the case of a high-spin baryon resonance, the
P^ in pion-nucleon scattering. It is intended to be the
first in a series of parametrizations of direct channel poles
to be used as a rigorous check of duality. It is also hoped
that the series of works can shed some light on the signifi
cance of the pomeranchon.
In order to make a systematic study of various reso
nances, the phase space function has been studied in detail.
It is found that if the widths of resonances are ignored,
i.e., if there is no "spread" in the masses of resonances,
it is possible to write down simple general phase space func
tions for all channels involving such particles. In this
work the zero width assumption has been used for certain
channel thresholds. The refinement of non-zero widths will
be attempted later.
Using recent phase shifts of Carter et al.^ and a com
pilation of older results,^ the parametrization has been
used to determine the mass and width of the P^ resonance,
as well as the coupling constants and CttN *A *
General aspects of pion-nucleon scattering and its kin
ematics will be studied in Chapter II. In Chapter III the


3
general parametrization formula will be developed. Chapter
IV deals principally with the phase space function which
plays an important role in the parametrization. Chapter V
studies the problem of parametrization of the P^3* ^ a^-so
includes a discussion of data handling. The last chapter
sums up the findings of this work.
The notations of reference 4 will be followed. The
is sometimes denoted by A for short.


CHAPTER II
PION-NUCLEON SCATTERING
1. Introduction
The interaction between pions and nucleons is one of the
fundamental problems of subatomic physics.
For nuclear physics, the determination of the nuclear
potential, the long-range component of which is dominated by
the interaction between pions and nucleons, is to be a giant
step forward. It will bring theoretical nuclear physics to
the same level of development as present day atomic and
molecular physics, where the many-body problem is the only
outstanding obstacle.
To particle physics, pion-nucleon scattering is espe
cially important because of the availability of high inten
sity pion beams at a wide energy range. The large amount of
data collected from pion-nucleon scattering not only serves
as a check on most theoretical concepts in hadron physics,
it is also the source of many new ideas.
In particular, phase shift analyses of partial wave am
plitudes for this scattering process have produced evidence
for a series of new particles or resonances, and this dis
covery has been extended to other scattering processes. The
4


5
presence of a large number of resonances, all of which are
considered as essential constituents as the pion and the
nucleon, is a new feature only found in particle physics.
Their presence provides very severe constraints on all
dynamical models of strong interactions.
Various detailed dynamical models have found different
degrees of success in describing experimental results. No
table among low energy models are the Chew-Low theory and
the effective range approximation. Among high energy models
are Regge pole theory and the Veneziano model. The low en
ergy theories will be briefly discussed later in this chap
ter.
2. Kinematics
In the present work only Lorentz-invariant dynamical
variables will be used. The relationship between the invar
iants and ordinary physical variables such as energy and mo
mentum is established here.
In addition, only two-body reactions will be considered
From a practical point of view, this restriction is un
avoidable because three- or more-body kinematics is compli
cated and the kinematic singularities of corresponding heli-
city amplitudes are not known.
A theoretical implication of this restriction will be
taken up later.
A typical reaction is, therefore,


6
1 + 2 3 + 4
where particles ia,l>..)4 have four-momenta p^, rest masses
ek, spins s^, and helicity components X^. If particles 1
and 2 are a pion and a nucleon, corresponding to the initial
state of the scattering, the above reaction will be called
the direct channel or s channel. By contrast, the cross
channels are
1 + 3 -+ 2 + 4 (t channel)
1 + 4 2 + 3* (u channel)
The terminologies will be made reasonable in a moment.
In the direct channel, particles 3 and 4 may be any one
of a number of combinations. They may be the pion and the
nucleon in the case of elastic scattering, or the pion and
the A, the pion and the Pjj resonance, in the case of in
elastic scattering. These different combinations will also
be called "channels", or more correctly, the exit channels.
The basic Lorentz-invariant dynamical variable in the
direct channel is the total center-of-mass energy squared,
2 2
S (Pj + P2) = (p3 + P4)
(1)


7
where the are measured in the direct channel. The momen
tum transfers t and u are defined by
t (Pj P3>2 = (p2 P4)2 (2)
u (pj p4)2 = (p2 P3)2. (3)
The three invariants are not independent,
s + t + u = 2^ m2 (4)
where the summation is over the ir.cident and the exit chan
nels .
If 0 is the scattering angle in the center-of-mass
frame of the direct channel, then
sin 0 = 2[s4>(s,t)]^/S12(s)S34(s)
(S)
cos 0 = [s2 + 2st sZ.m| + (T12U12T34U34^2^/si2^s^S34^s^
(6)
where
- stu B(T,3013T24024)i t(T1212I3434)4 +
- (m2 + m2 m2 m2)(m2m2 m2m2) (7)
Tij = (mi + mj)2
(8)


8
(9)
i
Sij(s) = C(s Tid)(s Uid)]2.
(10)
In the two-dimensional space spanned by s, t and u, not
all points are "physical". In other words, there are regions
in this space that are inaccessible for realistic scattering
processes. This follows from the fact that the four-momenta
are constrained by
2
m, .
(11)
The boundary of the physical region is given by
(12)
( s,t) *= 0 .
All square roots in the above are chosen to be positive '
in the physical region, so that the sine of 0 is positive in
the direct channel.
threshold.
In the center-of-mass frame of the direct channel, the
total energy is
JL
W s2
(13)


9
and the magnitude of the three-momentum of either particle i
or particle j is
kid = Sid(s)/2W. (14)
In the laboratory system, where the nucleon target is
at rest, the total energy of the incident pion is
E = (s m2 ^i2)/2m (15)
where m is the nucleon mass and p the pion mass.
3. The Scattering Amplitude
An experiment in strong interactions consists of ob
serving the initial state of two or more particles, allowing
them to interact, and then observing the final states of the
arbitrary number of particles resulting. The S-matrix ele
ment, is defined such that
Pfi = < f | S | i > (16)
represents the probability of |f> being the final state,
given the initial state |i> .
It is well established that strong interactions have
very short ranges. Since all weak but long-range interac
tions such as electromagnetism and gravitation are ignored


10
in theoretical investigation and subtracted out of experi
mental results, the finite range of strong interactions
guarantees that |f> and |i} are free particle states under
ordinary experimental situations. In the following all par
ticles in the incident and exit channels are regarded as
free, and suitable kinematics is applied.
The helicity components of particles in a given channel
can be used instead of the spin projection along a fixed axis
in space for a complete description of hadron states. When
this is done, a helicity amplitude can be defined in terms
of the S-matrix. In particular, the helicity amplitude h^
is defined by
where
SA|i P4^3\|£|Pi>P2,Al,X2^
The differential cross section is given by
(18)
where k is the magnitude of the center-of-mass three momen
tum in the direct channel, for the incident particles.


11
The helicity amplitude defined here is related to
7
the helicity amplitude of Jacob and Wick by
k f
JW
and to the usual helicity amplitude such as seen in Wang
o
and King7 by
(19)
8
(20)
where q is the magnitude of
in the exit channel.
Partial wave expansion
follows:
the center-of-mass three-momentum
for S
Ap
and %
are defined as
% -k + (2.)
% (2J + l)h^d^(9) (22)
where df (0) is the d function of the rotation matrix ele-
Ap
ment
dAp(0) = (23)
The total cross section, according to equations (l8)
and (22), is therefore given by


12
(24)
4 The Phase Shift
According to equation (l6), and since there must be
unit probability for an initial state to end up in some
final state}
i = zt
= . (25)
Here the completeness of final states |f> has been assumed.
If, in addition, as is customary, the initial state is
normalized, then it follows that
SfS = i (26)
since the initial state \y is arbitrary. This is the
unitarity relation for the scattering matrix.
Combining equations (21) and (26), it is found that the
unitarity relation for the partial wave is
SJV = JU (27)
In the case of elastic pion-nucleon scattering, the


13
partial wave scattering matrix has only four helicity compo
nents:
++ +- -+
where + denotes +5 and denotes -5. By parity invariance,
it is found that the first equals the last, and the second
equals the third. Using these relations, the unitarity re
lation for elastic scattering becomes
+
J* J J* J
S++S+- + S+-b++
0.
(28)
Define
(29)
and the unitarity relation becomes even simpler:
(30)
Conventionally, one insures that this condition is sat
isfied by writing
where the phase shift bj is a real function of W.
(3D


14
In terms of the helicity amplitudes,
4-jrk 2ik
where
1
4ik
[e
2i
J-
2i
+ e
J+
- 2]
- i D + hj]
1 ij(W) ,
h(W) £ e sin J(W).
(32)
(33)
A similar expression can be obtained for h^_.
Above the first inelastic threshold, the phase shift
can no longer be real. The unitarity relation now reads
= l n^. . (34)
The phase shift must therefore have a positive imaginary
part. However, it is conventional to factor out the imagi
nary part, in the form of an absorption parameter t|, and
make the phase shift real even for inelastic processes.
Hence equation (31) becomes
(35)
T|j is equal to unity when the scattering is purely elastic,
and it is smaller than unity when the scattering is partial
ly inelastic.


15
5. The Breit-Wigner Formula
From the definition of h, equation (33) it can be
found that, if inelasticity can be ignored,
J = I 1
l k cot 6 j i *
(36)
A resonance is said to exist at W if cot 6_. goes
r j *
through zero for this value of W with a negative derivative.
If a power series expansion of cot 6j is made, then in the
neighborhood of
cot 6J (Wr w)/£r r> o (37)
and
r/2
- w ir/2'
(38)
Inserting this result into equation (24) and neglecting the
other partial waves, the resonance cross section is found to
be
O' (W)
res
(j
+ 1)
Ja.
(Wr w)
2 + r2/4
(39)
This is the well-known Breit-Wigner formula.
At-resonance energy, the partial-wave cross section is,


16
according to the Breit-Wigner formula,
= <2J + > <*
which is commonly called the unitarity limit, since this is
the absolute maximum of a partial-wave cross section by
equation (33)
It should be pointed out that in the derivation of
equation (39) the fact that h^ and h*| | (h^_) have different
normalization is used.
6. The Chew-Low Model
The Chew-Low model is the first model that successfully
predicted the gross characteristics of pion-nucleon scatter
ing at pion energies below a few hundred MeV.1 In particu
lar, it predicted the resonance.
A non-relativistic pseudovector interaction between
pions and nucleons is assumed in this model, from which,
with the additional assumption of a nucleon at rest, the
interaction hamiltonian of the following form is obtained:
Ht Z (V a + vV) (41)
j. m m m mm
Vj if( where a^ and a are respectively the creation and annihila-
m m
m
tion operators for single pions, k is the three momentum and


17
G the energy of the pion, and m represents all pion quantum
numbers, and lastly, n(k ) is the Fourier transform of the
nucleon charge density. In order to ensure the convergence
of the necessary integrations n(k ) must be cut off above
some k .
max
By virtue of the pseudovector interaction and the as
sumption that the nucleon remains rigidly at rest, the model
predicts that all scatterings will be in the P-state.
The Low scattering equation is solved with the assump
tion that the only two diagrams of importance are
7T \
IT
TT '
N
N
N
jir
- N
Figure 1. Diagrams
Finally, the following relation is derived for the J*=|-
phase shift
k3
Vcot
-^2 (1 rE)
4*
(43)
where f is the renormalized coupling constant and r a con
stant approximately given by
r 3 f2 Ema* >
E is the pion energy corresponding to k
max max
Relations similar to equation (43) are predicted for
the other P-wave phase shifts. However, for these phase


18
shifts r < 0. This means that there are no resonances in
these partial waves. Experimentally the other phase shifts
are small in the low energy region.
The Chew-Low model was a break-through for field theory.
It was the first calculation involving a strong interaction
which gave any significant agreement with experiment. By its
specific assumption it is limited to low energies. It is
also a particular case of the effective range approximation,
which will be discussed next.
7. The Effective Range Approximation
It is convenient for the discussion of effective range
approximation to use the partial wave scattering amplitudes
h^(w) defined in equation (33)* Here the new variable w,
the square of k, is preferred over W the total energy. From
the definition, it is found that
Im h^(w) = k|h*(w)| (45)
if inelasticity can be ignored. From equations (36) and
(45) it follows that
Im [ ] = -k. (46)
h(w)
It is possible to seek a power series expansion in the
low energy region of w by using analyticity However, the
expansion cannot be made simply for h^(w), for they have a


19
branch point at w=0. The function M(w) is defined as
M^(w) = l/h^(w) + ik.
(47)
M(w) is analytic in the neighborhood of w=0 since Im M = 0
for w real and positive, and therefore by the Schwarz re
flection principle M(w) has no discontinuity across the pos
itive real axis. Furthermore, in the physical region,
(48)
Thus the power series expansion will take the following form,
where the parameters are conventional:
(49)
a is commonly called the scattering length and r the effec
tive range. The effective range approximation is made by
keeping only the first two terms in the expansion.
Because of threshold behavior of helicity amplitudes,
about which a later chapter will be devoted to, equation
(49) is not always the most convenient expansion to make. A
general form commonly seen is
k2J+l co£ c JL -f irw.
U tt
(SO)


CHAPTER III
THE K-MATRIX FORMALISM
In a parameterization of the dynamics of scattering pro
cesses it is most desirable to take care of all kinematic
effects explicitly in the formalism. In a dispersion theory
these effects are generated by kinematic singularities of
the scattering amplitudes.
It will be assumed that these kinematic singularities
factorize, i.e., for the scattering from channel i to chan
nel f, the helicity amplitude can be written as
I 1
hJ(f,i) [p(s)]2 [p^s)]2 (51)
where s) depends only on dynamical variables and charac
teristics of channel x and M^(f,i) contains only dynamical
singularities. The function p^(s) will be called the J-th
partial-wave phase space function for channel x, or simply
the phase space function for channel x if J is understood.
The phase space function is real above the channel threshold
and it vanishes below the threshold.
By their definitions in equations (21) and (22), it is
found that
20


21
sf = 1_ + 2ihf .
(52)
Combining equations (27) and (52), the following relation is
obtained:
Im [ ] -J,. (53)
hJ
If it is defined that
9 <54)
where T is the threshold in channel m and 6(s-T ) is the
m m
step function, then
L1 2 hJ C/]
(55)
()
'dir
J,fi
ApfPi l*W>|2
(56)
IbC i1
(57)
The last equation is the key to the K-matrix formalism. If,
furthermore, we define
(s)
pj(s') ds'
s' s ie +
p.s.
(58)
where p.s. stands for "possible subtraction", then it follows
from equation (57) that a K-matrix can be defined as follows:


22
(59)
and that it is real for positive real s. In
both and V? are analytic in s, so is K**.
can also be written as
JC
1 KJRJ
addition, if
Equation (39)
(60)
Equation (53) has been obtained as a general result of
unitarity. It can be obtained more readily from equation
(46) and the relation between the two helicity amplitudes.
However, the result obtained this way is valid only for the
low energy region.
By definition is a diagonal matrix. The diagonal
elements of this matrix will sometimes be simply denoted by
R(s) .
At this point the following physical considerations can
be made:
(1) For the sake of resonance study, it is only necessary
to use a single incident channel. Mathematically, this means
that both M. and are column matrices. (Note the helicity
indices have been suppressed in favor of channel indices.)
(2) Near a resonance, elements of the K-matrix are assumed
to have the simple form
(61)
r
where s is the position of the resonance, and C is a matrix
r ^


23
of coupling constants that factorizes:
CiCj
(62)
Equation (6l) is the "pole approximation" of the scattering
amplitude and it contains the only dynamical singularity as
sumed for the formalism. The factorization property of the
residue of the pole corresponds to the non-degeneracy of the
resonance.
(3) In the neighborhood of a resonance, the M-matrix ele
ments have a form similar to the expression (6l). That is,
the amplitude should indicate a simple pole at the resonance
energy. To ensure that this is the case, according to equa
tion (59), .R^ should be at least quadratic in (s-s^) in the
neighborhood of s^.
Hence, equation (58) should be changed to be
(s)
[ i (s-sr)2
OO
/*.')ds'
T (s'-sr)2(s,-s-ie)
+ p.s ]
&y (03)
Combining the above considerations, the following pa-
rametrization formula is found:
if'(s)
C.
(sr-s) XjC^R^s) iZjC2p^(s)0(s-Tj)
(64)


CHAPTER IV
PHASE SPACE FUNCTION
In order to have a well-defined parametrization formula,
it remains to find a unique expression for the phase space
function. The phase space function for a two-body channel
is a product of two factors: the regular two-body phase
space function for spinless particles, as required by
kinematics, and the kinematic singularities of the helicity
amplitude, which is caused by the presence of spins.
1. The Regular Two-Body Phase Space Function
The regular phase space function for spinless particles
is well-known.11. It is the Jacobian of the transformation
from the momentum space to the phase space or invariant
space. In a two-body channel, the Jacobian depends only on
a single variable, s.
By definition, in the center-of-mass frame,
dpQ = | d4px d4p2 4(Pj+p2-P) (pj-mj) (p2-m2) (65)
or Po s [(s ~ T12)(s U12)]2 (66)
24


25
where TJ2 and are respectively the threshold and the
pseudothreshold of the channel consisting of particles 1 and
2. The normalization of this function and the normalization
of the coupling constants are closely related. In order to
avoid confusion, has been defined with no numerical coef
ficient here. The coupling constant is going to carry an
overall normalization.
The presence of spins does not change the Jacobian from
the momentum space to the invariant space. But it introduces
kinematic singularities in the helicity amplitudes.
2. Kinematic Singularities of Helicity Amplitudes
The kinematic singularity structure of helicity ampli-
8 o i2 11
tudes has been studied extensively. J It includes
singularities at (a) the boundary of the physical region,
(b) the threshold and the pseudothreshold, and (c) the or
igin of the s-plane.
(a) The Boundary of the Physical Region
The d function of the rotation matrix is related to the
Jacobi polynomials by
<.() E WX+M C- w1^1-
(lx-nl,lx+i*l)
P(J-M) (cOS 0)
(67)


26
where M=max( |X\ y (|| ) and N*=min( |X\, ) From the defini
tion of the partial-wave expansion it can be seen that h^
contains the factors
(68)
By equations (5) and (12), the above expression represents a
series of zeros or singularities at the boundary of the phys
ical region. These are called t zeros or singularities, be
cause for given s, the positions of these zeros or singular
ities are determined by the scattering angle through t, the
momentum transfer in the direct channel*
Since only direct channel kinematic singularities are
of interest in the parametrization, these possible singular
ities at the boundary of the physical region, or cross chan
nel singularities, are ignored in this work.
(b) The Threshold and the Pseudothreshold
The covariant helicity operator approach of King and
o
others is employed here.
The direct channel covariant helicity operators corres
ponding to particles 1 and 2 are defined by the relations
(69)
(70)


27
where /u is a summation index and is the Casimir
operator for the spin of particle i. In the s channel cen-
ter-of-mass frame these operators reduce to the ordinary
helicity operators. Similarly, the t channel covariant
helicity operators for particles 1 and 2 are defined by
F* = 2 p£ / S13(t) (71)
f\= 2 wj2* pj / S24(t) (72)
They also reduce to helicity operators in the t channel cen-
ter-of-mass frame.
The transformations between the eigenstates of the two
sets of operators are given by
^S(pr,X) = Zx, dJxTx(/r)^(pr,X') (73)
a
where X and X' are eigenvalues of the operators Fr and Fr,
respectively, Jn is the intrinsic spin of particle r, and
r=l or 2. y*2 is the angle between -pj and -p^ in the rest
frame of particle 2, while ^ is the angle between -p2 and
p^ in the rest frame of particle 1. The coordinate axes are
always chosen such that In terms of the invariants
s and t, the angles are given by the relations
(s + m2 m2)(t + m2 m2) 2m2M
sl2() Sjji)
cos
(74)


28
cos (s + n2 m^)(t + n2 m^) + 2m2M
where
and
S12(s> S24(t)
M = "4 m2 + ml "3
2m1[<))(s,t) ]2
810 = Si2(s^S13(t)
2m2C4(s,t) ]2
Sn ^ = Sl2(s)S24(t)
(75)
(76)
(77)
From their definitions the operators and F can be
seen to be ill-defined at the threshold and the pseudo-
threshold. The covariant helicity amplitudes, f ^eing
eigenstates of these operators, are singular at these values
of s. Introducing a new set of amplitudes f^(s,t) in which
the states for particles 3 and 4 remain unchanged but the
states for 1 and 2 are now eigenstates of F^ and F^, it is
found that
S\W,t)
dX*X1(;<:i)dK'^2(^2)f*3X4xV
The singularities at the threshold and the pseudothreshold
are now isolated in the d functions, because f ,(s,t)
A3X4X
is regular at these points.
After an analysis of the asymptotic behavior of the d
s o
functions, it is found that the amplitudes f behave as


29
is T34]
'^(J3+J4)
(79)
and
(80)
near the final-state threshold and pseudothreshold respec
tively. Thus the product of the expressions in the above is
the singularity to be included in the phase space function.
However, it should be noted here that the kinematic singu
larity of the helicity amplitude is, by equation (51), the
square root of p^(s), but not p^(s) itself.
(c) The Origin
In the above discussion the origin has not been found
to be a singular point of the helicity amplitude, but it is
often pointed out that the origin is a singular point of the
12
scattering amplitude. Cohen-Tannoudji et al. concluded
that the helicity amplitude does not have singularities at
this point. It is only the linear combinations of helicity
amplitudes commonly used for Reggeization that are singular
here. This is also pointed out by King, since the singular
ity of a linear combination can be different from the singu-
o
larity of a single helicity amplitude.
3. The Phase Space Function
The product of the expressions in equations (66), (79)


30
and (80) is not quite the phase space function yet. The
phase space function has been defined for partial wave ampli-
tudes only. Referring to equation (22), it is seen that the
singularity of d^(0) has to be studied, too. According to
the asymptotic behavior for the d functions,
9,14
() ~ [cos e]J
C(s -T34)(s -U34)]
J
(81)
because of equation (6). Thus, finally,
J>j(s) = i [(s T )(s U )]
jH-j.-j .
3 4
(82)
It must be pointed out that the threshold and pseudo
threshold singularities have been obtained for particles
with well-defined masses, and the parametrization is dealing
with channels that often involve many unstable particles or
resonances. In using equation (82) it is assumed that the
widths of the contributing channels are sufficiently small
to be ignored. This is a limitation of the present formal
ism.


CHAPTER V
PARAMETRIZATION OF THE P^3 RESONANCE
1. Introduction
The parameterization formula gives general representa
tions of all matrix elements of a given partial-wave heli-
city amplitude. The theory does not have the power to pre
dict the exact behavior of individual matrix elements. The
matrix elements can only be determined by "parametrization"
against experimental data.
The P^^ phase shifts, here chosen as the only experi
mental data input, can be expected to determine one particu
lar matrix element. However, because of the way phase shifts
are defined in equation (33)* it turns out that a linear
combination of two matrix elements is determined instead.
4
The P^3 phase shift is related to h_, which is proportional
1 4-
to the linear combination hj j+h* This raises two questions.
First, since singularities of certain linear combina
tions of helicity amplitudes are known, is it not a better
way to start directly with the parametrization of the proper
linear combination? Theoretically, this is an attractive
alternative. In fact, it is not known a priori that a pa
rametrization developed for h^ can produce an adequate ap-
31


32
proximation for because they may have different
kinematic singularities. But practically, direct parametri-
zation of linear combinations is very unattractive because
the kinematic singularities involved are such that a higher
number of subtractions is usually needed, in which case the
calculation may be obscured by the presence of a large num
ber of subtraction constants. Other linear combinations
simply do not have known singularities, so that it is impos
sible to use such an approach even if the subtraction con
stants are not an objection. In pion-nucleon scattering,
both difficulties are present.
J J *
Second, since the linear combination IvT -h^ or h;,
A-H* +
is related to the phase shift, is it not advisable to
make a simultaneous parametrization of the an<* D^3 reso
nances, so that individual matrix elements, i.e., h^ and
h? might be determined? The answer is definitely yes, it
A-H
would seem. But the helicity amplitudes themselves are not
physically important for the strong interaction, because
they are not parity-invariant. The physically important am
plitudes are the parity-invariant linear combinations.
2. Basis of Computation
In Table I are listed the various channels that con
tribute to the resonance, their characteristics, and the
appropriate phase space functions.
The phase space functions for the channels are ob-


33
Table I. Characteristics of Contributing Channels
Meson
Baryon
T2 [MeV]
1
U2 [MeV]
P
pion
nucleon
1077.9
843.7
1
S
C(.-T)(s-0)]3/2
pion
P1
33
1373.6*
1094.4*
i [(s-T)(s-U)*
pion
P1
*11
1609.6
1340.4
1
a
C(s-X)(s-U)]3/2
varied in actual computation as the P'. mass is changed.
tained from equation (82). While for the pion-P^ channel
only two subtractions are required to make the dispersion
integral convergent, in the other channels three are needed.
Thus the dispersion integral for the second channel has the
form of equation (63), without the p.s. term, the same inte
grals for the other channels are of the following form
p^(s) ds*
R^(s) £ (s s )3 -
3 (s' s )3(s' s i£)
Ai r
(83)
The real part of the denominator in equation (64) is
denoted by D(s), and the imaginary part by G( s) They are:
D(s) = sr s + z(s sr)2 C2Rj( s) (84)
G(s) = Zj C2p(j)0(s T.)
(85)
where z is an overall subtraction constant that is made nec
essary by the extra subtraction, and 6(s) is a step function


34
The phase shift is given by
tan 6j(s) = (86)
Because of the fact that most coupling constants in
volved in the parametrization of the are unknown at the
moment, one of the objectives of this work is to make cer
tain predictions on their values. The procedure is to use
phase shift data and the parametrization formula to find a
best fit, with the coupling constants and the subtraction
constant, as well as the mass of the resonance, as free pa
rameters. The computer routine PARFIT at the Department of
Physics and Astronomy of the University of Florida is used
for this purpose.
3. Phase Shifts
Two sets of phase shifts have been used. Set A is tak
en from Carter et al.^ Set B is compiled from older re
sults.
The Cavendish-Rutherford collaboration of Carter et al.
recently reported an extensive series of cross section meas
urements at the CERN synchrocyclotron in the energy range
around the Pj^ resonance. The reported error bars are much
smaller than all previously available results. An interest
ing feature of the new data is an apparent shift of the peak
of the cross section to a lower mass. The phase shifts re-


35
ported by this group are extracted directly from the total
cross sections.
Set A is used to yield more accurate information on the
resonance: its mass and width, and possibly also its
coupling constants.
In Set B, values of phase shift reported by Moorhouse,
Kirsopp, Johnson, Donnachie, Bareyre et al,.1'* and a set from
Berkeley called Path 1^ are taken on equal footings and
their average determined. Since there is no available basis
to prefer any one set of these over any other, no weight
factors have been assigned in the average.
Since different listings cover different energy ranges,
the number of entries at any particular energy can vary con
siderably. In the lower range where entries typically num
ber two or three, the standard deviation is determined by
the following observation:
In the Saclay isovector pion-pion phase shifts,the
quoted uncertainty roughly forms a band of constant width
about the mean when plotted graphically.
In the absence of further information, this observation
has been applied to assign standard deviations for set B. In
the higher energy range where data entries are more numerous,
the standard deviation is taken to be one half of the maxi
mum difference among the sets, generally. And in order to
achieve some kind of uniformity throughout the energy range,
the width of the band in the lower range has been determined
at the upper energy range.


36
The results are listed in column 2 of Table IV.
4. Left Half-Plane Singularities
In the complex s-plane, the partial-wave helicity am
plitude has dynamical singularities at poles and branch cuts
corresponding to all "communicating1' channels or possible
exit channels as required by unitarity.1^ These are known
as the right-hand singularities for they occur in the right
half-plane. There are also left-hand singularities, occurr
ing in the left half-plane, because of cross channel right-
hand singularities.
The N/D method of Ball and Parkinson, which is equiva
lent to a K-matrix formalism using the R-matrix of equation
(63), which in turn approximates the contribution of right-
hand singularities, uses one form of the effective range ap-
3
proximation for the left-hand singularities. With an in
creasing number of subtractions in the R-matrix, it has been
found that more left-hand poles are generated. In particu
lar, the R-matrix of equation (83) generates a pair of com
plex conjugate left-hand poles. The symmetric poles about
the left-hand branch cuts along the negative real axis is a
better approximation than a single pole. A wider range of
validity can be expected than the effective range approxima
tion .
This line of reasoning has been used in works on the
approximation of a Regge cut by a pair of complex conjugate


37
17
Regge poles in the J-plane.
In this respect, it should be noted that if the parame-
trization is to extend to higher and higher energies, reso
nances of higher and higher J values will have to be in
cluded. Then, by equation (82), the phase space function
will be so modified that more and more subtractions are nec
essary in the R-matrix. And this indeed is going to extend
the region of validity of the parametrization to higher and
higher energies.
Returning to the present problem, it is seen that the
energy range is relatively small. Thus a two-pole approxi
mation should be sufficient.
5. Contributing Channels
In equation (64) the summations are over all contribut
ing channels. These are also called communicating channels
or possible exit channels. In principle they cannot be ex
hausted, unless experimentalists find an upper limit for re
sonance production. Practically, the choice is made in the
following manner.
From the set of well-established particles, i.e., the
ones listed by the Particle Data Group with complete quantum
l8
number and mass-width specifications, all meson-baryon
pairs that have the right quantum number combinations are
selected, and their thresholds and pseudothresholds, accord
ing to equations (8) and (9), calculated. Most thresholds


38
lie beyond the top of the energy range of phase shift data.
Referring to equation (64) it is found that channels
whose thresholds are higher than the top of the energy range
under consideration contribute only to the function D(s),
through the R-matrix. Elements of the R-matrix are rela
tively small for those channels with high thresholds, as can
be seen from equation (83)* Here a high threshold means a
large s'-s in the denominator of the integrand, and conse
quently a small matrix element. Thus all channels with very
high thresholds are not considered important, and a total of
eight channels are finally selected as possible contributing
channels. These are listed in Appendix B as a comment in
the computer program actually used in the calculation.
The maximum energy for the phase shift data is at 172
MeV. Arranged in order of increasing thresholds, the fourth
channel threshold is at (1660 MeV)^ and the eighth at (1783
MeV)^. In fitting the phase shift data it is found that all
channels higher than the third can be ignored without affect
ing the chi-square of the fits. Thus, for the energy range
under consideration, there are only three contributing chan
nels. These are listed in Table I.
The practical aspect of two-body kinematics has been
studied in Chapter II. From a theoretical point of view,
the exclusive use of two-body channels is an application of
the idea of direct channel resonance dominance. That is,
three- or more-body channels, though experimentally observed
to be dominating at times, are themselves dominated by two-


39
body channels.
As an example, in pion-nucleon scattering the pion pro
duction process of two pions and a nucleon in the exit chan
nel are often observed. Rather than trying to solve a three-
body problem, the view is taken that the "extra" pion has
come mostly from either a rho meson or any one of the baryon
resonances. (The word "come" must be understood in a very
loose sense.) Thus, the inclusion of pion-baryon resonance
and rho meson-nucleon channels should describe the three-
body channel adequately.
Direct channel dominance, used in this manner, is quite
similar to Feynman diagrams in field theory, though there is
a subtle difference. In Feynman diagrams, the scattering is
assumed to have gone through a virtual intermediate state.
In the S-matrix theory there is merely a statement of domi
nance in the amplitude, and a subsequent substitution.
6. Computation
The general procedure of actual computation is as fol
lows:
A set of values for the resonance mass and the cou
pling constants in various channels, plus the subtraction
constant, is supplied to the program PARFIT, which uses equa
tion (86) to determine the chi-square value on the basis of
experimental phase shifts and standard deviations. PARFIT
also automatically minimizes the chi-square by adjusting the


40
free parameters, that is, the coupling constants and the sub
traction constant. At the end of computation PARFIT produces
the best fit, with all relevant data.
The mass of the is then shifted and a new best fit
found. The best overall fit with the mass as a parameter is
taken to be the final result.
In addition, the number of contributing channels can be
varied in each chi-square calculation, by varying the value
of PAR.
Thus, the K-matrix formalism provides the most general
form for the parametrization. The number of contributing
channels is determined by the sensitivity to chi-square fit,
and the actual parameters determined by best fit against ex
perimental data. And a complete parametrization for the P^
is obtained.
The mechanics of actual calculation are contained in the
Appendices. The routine PARFIT is not included.


CHAPTER VI
RESULTS AND CONCLUSIONS
1. Comparison with Breit-Wigner Formula
The K-matrix can be made relativistic simply by using
phase space functions and dynamical variables that are all
Lorentz invariant. This has been done in Chapters II, III,
and IV. Thus the parametrization is relativistic. And this
removes the first objection.
It should be remarked here that it is not the aesthetic
aspect of non-relatiyistic nature of the Breit-Wigner formu
la that prompted the objection. Rather, it is the fact that
non-relativistic mechanics restricts all formalisms to low
energies, or narrow energy ranges, and this restriction is
quite undesirable.
The summation in the parametrization formula, equation
(64), in the real part of the denominator specifically, in
cludes both open and closed channels for any particular en
ergy. Hence nearby singularities can influence the dynamics
of the scattering.
The zeros of the denominator in equation (64) do not
appear on all sheets of the scattering amplitude, in gener-
41


42
The zero of the denominator in equation (64) is deter
mined, in part, by the expression
(87)
The relative contribution of each channel toward producing
3
the resonance is then given by''
CjRj(sr). (88)
This gives a rough indication of the relative importance of
each channel in the dynamical origin of the resonance.
2. General Description of Phase Shift Fits
Set A
Reasonable fits to phase shift data have been obtained.
The best chi-square value for 14 points is 15*3 correspond
ing to a resonance mass of 1230.4 MeV. Results of the pa-
rametrization fit are listed in Table II. Table III contains
the phase shifts and the values of chi-square at all data
points, together with the experimental data used.
It is of interest to note that the lowest data point
carries the worst chi-square. If this data point is deleted,
the best chi-square value would be 10.7 for 13 points. Pre
viously, the data point at the other end of the spectrum has
l8
been considered a bad fluctuation.


43
Table II. Parameters from Phase Shift Fits
Set A
Set B
resonance mass
1230.4 MeV
1235.0 MeV
£-
resonance width
120 MeV
122 MeV
SrNA
0.1551*0.0002
0.1543*0.0010
CttAA
0.0830*0.0073
0.1029*0.0073
CrrN'A
0.8048*0.0202
1.018 *0.031
subtraction constant
-0.0488*0.0003
-0.0506*0.0016
chi-square
15.31
63.68
mean chi-square
1.09
1.63
* full width at half
4
maximum.
With a total of five de facto parameters (three cou
pling constants, one subtraction constant, and the resonance
mass), the fit to set A has not been very good. But it
should be pointed out that different channels contribute
differently to the resonance, and not all coupling constants
affect the quality of the fit equally significantly. In
fact, the introduction of some more channels and their cou
pling constants does not change the chi-square value at all.
The parametrization used here is not an optimum mathematical
approximation.
Set B
The best chi-square value for the 39-point set B is


44
Table III.
Phase Shifts Fitted According to Carter et al
5
W [MeV]
Exp. [deg.]
Theory
1139
11.87*0.15
11.54
4.75
1160
21.41iO.06
21.42
0.01
1177
33.170.10
33.36
3.50
1178
34.36db0.20
34.26
0.25
1190
45.64iO.ll
45.66
0.04
1193
47.83iO.ll
47.81
0.04
1206
62.82iO.l8
62.61
1.31
1210
67.730.21
67.48
1.37
1215
73.51dbO.30
73.63
0.15
1227
84.940.98
85.97
1.10
1244
102.05i0.51
102.3
0.21
1261
114.41iO.23
114.6
0.45
1280
124.03i0.17
124.2
0.85
1301
13l.960.15
131.8
1.27
63.7* This result is as good as can be expected, because
systematic errors among different sources are expected to be
important. For this very reason results of this fit should
not be taken too seriously, especially those pertaining to
the resonance. In Table II are listed main findings of
this fit. In Table IV are the phase shifts.
Similar to the parametrization of the rho meson,^ in
this relatively wide energy range the theoretical values de
viate from experimental phase shifts significantly near the


45
Table IV. Phase Shifts Fitted According to Compilation Data
W [MeV]
Esq). [ deg. ]
Theory
1094
1.4 0.9
0.88
0.33
1104
2.6 0.9
2.01
O.42
1109
3.6 0.9
2.76
0.88
1113
4.2 0.9
3.44
0.71
1127
7.6 0.9
6.66
1.32
1160
21.0 1.0
19.79
1.47
1177
31.8 1.1
31.12
0.39
1185
37.9 1.1
37.77
0.00
1195
45.9 1.3
47.28
1.12
1197
48.2 1.4
49.32
O.64
1202
54.2 1.4
54.59
0.08
1213
65.8 1.4
66.71
0.52
1216
69.4 1.4
70.05
M

O
1231
85.1 1.4
86.09
0.61
1235
90.1 1.4
90.04
0.00
1247
100.6 1.4
100.7
0.04
1252
105.2 1.3
104.7
0.15
1254
107.5 1.2
106.2
1.20
1258
109.4 1.0
109.0
0.07
1268
117.7 l.P
115.4
5.16
1275
119.8 1.0
119.3
0.23
1291
126.9 0.6
126.8
0.02
1320
136.7 1.3
136.8
0.01
1362
144.9 0.6
146.5
7.33
1390
149.2 1.0
151.2
3.86
1416
153.3 1.1
154.6
1.81
1442
156.3 1.1
157.5
1.28
1470
160.2 1.4
160.2
0.00
1481
l6l.l 1.7
161.1
0.00
1500
164.5 2.5
162.6
0.59
1512
166.0 0.9
163.4
*fr
H

GO
1524
166.1 1.0
164.2
3.45


46
Table IV. continued
W [MeV]
Exp. [deg.]
Theory
9C2
1543
168.3 + 2.2
165.4
1.69
1572
170.3 + 2.8
167.1
1.31
1601
172.2 + 3.0
168.6
1.41
1617
171.2 + 1.7
169.3
1.30
1629
173.2 + 4.0
169.0
1.11
1658
174.1 + 3.6
167.6
3.24
1672
175.5 + 2.6
166.7
11.56
top of the range. The chi-square for a 38-point fit, for
example, would be 52.1, a significant improvement. The
introduction of higher contributing channels does not change
this top-of-the-range deviation.
Since set B covers a wide energy range, in the calcula
tion of the chi-square values the number of contributing
channels has been varied. The result is that only the low
est three channels contribute, the same channels as present
for phase shift data set A. Altogether eight channels, with
O
thresholds up to (1783 MeV) have been tried. It may be
conjectured that a certain channel with still higher thresh
old contributes significantly by virtue of a huge coupling
constant. At least this cannot be ruled out yet. And this
very high threshold may solve the difficulty at the higher
end of the range.


47
3. The Resonance Mass
Set A
The mass of 1230.4 MeV for the resonance is lower
18
than most values reported. The lone exception is the "nu
clear" result reported by Carter et al.^ themselves. The
shift to lower values is in agreement with most recent works.
However, it is worth noting that various forms of the Breit-
Wigner formula, all based on the same experimental data,
18
consistently give higher masses.
The full width at half maximum is 120 MeV.
Set B
The resonance mass is high compared with the one for
set A. This is typical of older results. It is believed
that, due to higher systematic and statistical errors, in
formation concerning the extracted from this fit is of
very limited value.
4. The Subtraction Constant
The subtraction constant is very small, both for set A
and for set B. It is believed that the magnitude of the
subtraction constant is a measure of the goodness of fit,
too. For example, in the successful rho meson parametriza-
tion, it has been found that not only is the chi-square low,
the subtraction constant is practically zero. A physical


48
interpretation is the following: the subtraction contributes
to offset any deficiency in the knowledge of kinematic and
dynamic effects. Had the last two effects been properly
handled* there would be no need for a non-zero subtraction
constant. This is reinforced by a mathematical interpreta
tion of the subtraction: it is merely a free term whose
function is to reduce the chi-square.
5. The Coupling Constants
A. The Pion-Nucleon-P^^ Coupling
Each term in the expansion for -G(s) can be identified
with a certain partial decay width for the resonance; if it
is evaluated at the resonance energy. In particular; using
the value of obtained in the parametrization; we have
r cK2 H + v2)l2 H v2}^
m*
0.81
(89)
in pion mass unit. This can be compared with the full width
at half maximum of 0.87. They represent two different ways
in which the width can be defined.
21
According to Mathews; this partial decay width is
also given by


49
*2
SttNA 2
47T 3 p
3 (mA + ->'
- a.
(90)
mA
Comparing the last two expressions, it can be seen that
2 / >2 2
,2 JTNA (mA + "n> ~ "V
'ttNA
4tt
48 m3
(91)
g:
'irNA
Or that a of 0.1551 corresponds to a ^
of 0.37, in
dimensionless unit.
20
Ebel et al. report a
g
fl-NA
47T
of 0.33, which they main
tain is inferior to the value of 0.34 obtained from non-
relativistic spinless Born approximation. By inference, it
would seem that the value of 0.37 is quite acceptable.
B. The Pion-Pl^-P^ Coupling
22
According to Rushbrooke, the partial width correspond
ing to the decay of a resonance into a pion and a
33
resonance is given by
g
'TTAA
47r 3m
C3(
m2 + m2 m2
a A ir
2m.
^ + 3^
2p2 m2 + m2 m2
/A a 7r
2mA
e om
ttAA _1 rr
4-rr m 2m.
2mA) ]
(92)
since the center-of-mass momentum of the decay product is
very small. According to the K-matrix formalism, the same
quantity is given by


Combining the two we have
2 mi
4ir 4mA *
(94)
Our value of C,^ =0.083 therefore corresponds to the value
ry
0.243 for SirAA/47T in dimensionless unit.
2 3 2
Hori and Kanki reported that is approximately
o
one ninth of g^^. The relativistic SU(6) model of Sakita
2 a 2
and Wali, ^ on the other hand, predicted that g is about
2 2 5 26
nine times of g Sutherland and Michael predicted
1TNA
2
that g-rrv\a is even larger by a factor of about two. Our re-
2
suit is not in close agreement with any of the above.
2
is shown here to be two-thirds of g^^.
C. The Pion-Pj^-P^ Coupling
The K-matrix formalism developed here cannot be applied
to the decay of the PJj into a pion and a resonance,
though the appropriate branching ratio and total width have
been measured experimentally. Thus no comparison can be
made at this point.
It is worth noting that a similar problem occurs in
reference 22. The decay of a spin| particle into a spin--|
particle and a spin-0 particle warrants special equivalence
relations for satisfactory results.


51
6. General Remarks
In spite of its many known and suspected deficiencies,
the formalism developed here not only points to a way by
which some experimental results can be derived through a
theoretical model, it also opens up an approach by which the
whole problem of hadron dynamics may be attacked.
Traditionally, hadron physics has two distinct compo
nents, the low energy and the high energy theories. Duality
is a way to reconcile the two components. The present ap
proach, on the other hand, is to extend the low energy theo
ry into higher and higher energies. If the K-matrix parame-
trization scheme is successful there, a unified theory of
hadron physics is in sight. And it may eventually be able
to check the validity of duality, and of many other inter
esting concepts popular in high energy physics.
Of more practical concern, and related to the quality
of these phase shift fits, is the fact that phase shifts
must be deduced from "direct" experimental data through some
kind of parametrization first. Existing parametrizations
commonly used are all based on the Breit-Wigner formula. In
this light the chi-square values mentioned in this work can
not be too small. The present work offers an alternative to
the variations of the Breit-Wigner formula. Because of the
many objections of the latter as studied in Chapter I, and
their removal in the K-matrix formalism, it is believed that
many difficulties associated with present-day description of


52
the phase shift results can be removed.


APPENDIX A
MECHANICS OF CALCULATION
The basic principal integral is defined by^
OO
f(s,T,U) P j [(s' T) (s' U) ] 2 (Al)
T
so that
_i _! 1
for s>T, f(s,T,U)=-2[(s-T)(s-U)] 2Coth C (^ 3 (A2)
2 1 || '2
for T)s>U, f(s,T,U)= 2[(T-s)(s-U)] tan C(fE) 3 (A3)
1 1T 2
for U>s, f(s,T,U)= 2[(T-s)(U-s)] tanh C(^Ef) 3 (A4)
and the derivative of f(s,T,U) with respect to s is
-1
f'(s,T,U)=[(s-T)(s-U)] [-l+|(T+U-2s)f(s,T,U)3 (AS)
which may be used iteratively to produce higher derivatives
with respect to the same variable.
53


APPENDIX B
COMPUTER PROGRAM
c
channel # meson
baryon
psf
T
U
c
1
pion 140
ncln
940
3
1080
800
c
2
pion 140
delta
1234
1
1374
1094
c
3
pion 140
N1470
H70
3
1610
1330
c
4
pion 140
N1520
1520
1
1660
1380
c
5
pion 40
N1535
1535
3
1675
1395
c
6
kaon 496
sigma
1190
3
1686
694
c
7
rho 765
ncln
940
1
1705
175
c
8
eta 549
delta
1234
1
1783
685
c
1. energy unit in
pion mass for
all
calculations
C 2 input energy unit is the MeV
C 3. normalization of energy by variable "unit
C 4* A(par) is the subtraction constant
C 5* numerical differentiation is used
C 6. exp data input: UCRL20030 compilation
implicit real*8 (a-h,k-z)
real x
integer psf
common/data/unit,m,T(8),U(8),psf(8)/intg/w(39,8),energy(
+39)/stpdf/fx,x(400)/mass/pion,kaon,rho,eta,ncIn,delta,
+N1470,N1520,N1535,sigma
read (1,1) eta,kaon,pion,rho,delta,N1470,N1520,N1535>
4-ncln, sigma
1 format (I0d8.l)
read (1,2) psf
2 format (8i7)
do 4 i"l>39
54


55
read (1,3) energy(i)
3 format (2x,d8.l)
4 continue
do 5 1=1,400
5 x( i) =0.1
call callit
stop
end
double precision function f(A,L)
implicit real*8 (a-h,k-z)
real rgm,A(l),duml,dum2
integer psf,L,par
cnvr=l80./3.14159265358979dO
common/data/unit,m,T(8),U(8),psf(8)/intg/w(39,8),energy(
+39)/stpdf/fx/srchit/rgm(50),duml(2554),par,dum2(9),inx
+/mass/pion,kaon,rho,eta,ncln,delta,N1470,N1520,N1535,
+sigma
if (inx.ne.l) go to 15
inx=2
read (1,11) delta
11 format (dl2.4)
write (3,20) delta
unit=pion
T(l)=((ncln+pion)/unit)**2
U( l)=((ncln-pion)/unit)**2
T(2)=((delta+pion)/unit)**2
U( 2) *=( ( delta-pion)/unit)**2
T(3)=((N1470+pion)/unit)**2
U(3)=((N1470-pion)/unit)**2
T(4)((Nl520+pion)/unit)**2
U(4)-((N1520-pion)/unit)**2
T(5)=((N153 5+pion)/unit)**2
U(5)=((N1535~pion)/unit)**2
T(6)=((sigma+kaon)/unit)**2
U( 6) =( ( sigma-kaon) /unit) *#2


56
T(7)=((ncln+rho)/unit)**2
U(7)((ncln-rho)/unit)**2
T( 8)=( ( delta+eta) /unit)#*2
U(8)((delta-eta)/unit)**2
m(delta/unit)**2
isum!=,par-l
do 14 il,39
s=(energy(i)/unit)#*2
do 12 jl,isum
12 w(i,j)=v( j,s)
write (3,13) (w(i,j), jl,isum)
13 format (8gl6.3)
14 continue
15 i=rgm(l)
sm(rgm(2)/unit)**2
g=0 .dO
isum^par-l
do 16 j^ljisura
if (s.lt.T(j)) go to 16
g^g-Aij)**2*fcn(psf(j) ,s,T( j) ,U(J))
16 continue
d=m-s+A(par)*(m-s)#*2
do 17 3=1,isum
17 d=d+A(j)**2*w(i,j)
h=dabs( d)
if (h.lt.l.d-20) go to 19
god*=g/ d
if (god.lt.0) go to 18
fdatan(god)*cnvr
f x**f
return
18 fl80.0+datan( god)*cnvr
fxf
return
f=90.dO
fx~f
19


57
return
20 format ('l'jgSO.)
end
double precision function fen (dmml,dmm2,dmm3,dmm4)
implicit real*8 (a-h,o-z)
integer dmml
go to (21,22,23), dmml
21 fcn"dsqrt((dmm2-dmm3)*(dmm2-dmm4))/dmm2
return
22 fcn^O.dO
return
23 fcn(dsqrt((dmm2dmm3)*(cimm2-clmm4)))**3/din2
return
end
double precision function v(i,r)
implicit real*8 (a-h,m-z)
integer psf
common/data/unit,m,T(8),U(8),psf(8)
d=psf(i)
P*T(i)
z=U(i)
am-p
bm-z
e*m-r
x*=r-p
y=r-z
Pi3.14159265358979dO
cutoff99d25
go to (41*42,43), j
41 g=a*b/m/e
gpg*(1.dO/a+1.dO/b-1.dO/m-1.dO/e)
v*>( -p*z#e**2*q(0.dO,p,z,cutoff) /r/nr**2+x*y*q( r ,p,z,cutof f
+)/r+e##2*gp*q(m,p,z,cutoff)+e**2*g*qp(m,p,z,cutoff))/pi
return


58
42 v=0. dO
return
43 ha**2*b**2/m/e
hp=h*(2.dO/a+2.dO/b-1.dO/m-1.dO/e)
hpphp*(2.dO/a+2.dO/b-1.dO/m-1.dO/e)+h*(1,d0/m**2+l.dO/
+e**2-2.dO/a**2-2.dO/b**2)
v-(p**2*z**2*e**3*q(0.dO,p,z,cutoff)/r/m**3-x**2*y**2*q(
+r,p,z,cutoff)/r+e**3*hpp*q(m,p,z,cutoff)/2.d0+e**3*hp*qp
+(m,p,z,cutoff)+e**3*h*qpp(m,p,z,cutoff)/2.dO)/pi
return
end
double precision function q(s,t,u,lmda)
implicit real*8 (a-z)
f actor(lmda-t)/(lmda-u)
if (s.lt.u) go to 32
if (s.eq.u) go to 33
if (s.lt.t) go to 31
q-l./dsqrt((s-t)*(s-u))*dlog((1.+dsqrt((s-t)/(s-u)/
+f actor))/(1.-dsqrt((s-t)/(s-u)/f actor)))
return
31 q2./dsqrt((t-s)*(s-u))*datan(dsqrt((s-u)/(t-s)*factor) )
return
32 q=>l ./dsqrt( ( s-u)*( s-t) )*dlog( ( 1 .+dsqrt( ( s-u) /( s-t)*
+f actor))/(1.-dsqrt((s-u)/(s-t)*f actor)))
return
33 q^ ./(t-u)*dsqrt( factor)
return
end
double precision function qp(s,t,uflmda)
implicit real*8 (a-z)
if (s.eq.u) go to 34
factor*dsqrt( ( lmda-u) *( lmda-t) ) /( s-lmda)
qp(factor+((t+u)/2.-s)*q(s,t,u,lmda))/((s-u)*(s-t))
return


59
34 f actor=dsqrt((lmda-t)/(lrada-u))
qp=2,/3.*factor/(t-u)*(2./(t-u)+l./(lmda-u))
return
end
double precision function qpp(s,t,u,Irada)
implicit real*8 (a-z)
if (s.eq.u) go to 35
qpP(((t+u)/2.-s)*qp(s,t,u,lmda)-q(s,t,u,Irada)*(1,+((t+u
+)/2.-s)*(1./(s-t)+l./(s-u)))-dsqrt((lmda-t)*(lmda-u))/(s
+-lmda)*(1./(s-t)+l./(s-u)+l./(s-lmda)))/((s-t)*(s-u))
return
35 factor=dsqrt((lmda-t)/(lmda-u))
qpp=4,/l5.*factor/(t-u)**2*(8./(t-u)+4/(lmda-u)-3.*(t-u
+)/(lmda-u)**2)
return
end


REFERENCES
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BIOGRAPHICAL SKETCH
Ji- ,
Yin-po Tschang, in Chinese was born in Chungking,
China on June 20, 1944 He received his early education in
Taipei, Singapore, and Hong Kong, before entering Chung Chi
College of the Chinese University of Hong Kong on a four-year
Hong Kong Government Scholarship. In 1966 he graduated with
the degree of B.S. cum laude and since then has been in the
Graduate School of the University of Florida. In December,
1972, he received the degree of Doctor of Philosophy with a
major in physics.
He is married to the former Miss Kwok-lan Chan,
of Hong Kong. They have a seven-month-old boy, Yuan Tschang,
63


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Michael T. Parkinson, Chairman
Assistant Professor of Physics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Mkrt.
Arthur A. Broyles ^
Professor of Physics and Physical Sciences
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Zoran R. Pop-Stojattovic
Professor of Mathematics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Samuel B.
Assistant
Tricke;
Professor
Physics


I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Jjahn R. Sabin
Associate Professor of Physics & Chemistry


This dissertation was submitted to the Department
of Physics and Astronomy in the College of Arts and
Sciences and to the Graduate Council, and was accepted
as partial fulfillment of the requirements for the
degree of Doctor of Philosophy.
December| 1972
Dean, Graduate School


Full Text
UNIVERSITY OF FLORIDA
3 1262 08554 5357



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