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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Thesis--University of Florida.
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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














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.














TABLE OF CONTENTS


ACKNOWLEDGMENTS.....................................

LIST OF TABLES....................... ...*****.*....

LIST OF FIGURES.....................................e.

ABSTRACT...........*o e.............*****o* ..........O*

CHAPTERS


I. INTRODUCTION..................................

[I. PION-NUCLEON SCATTERING.....................

1. Introduction ............................

2. Kinematics...............................

3. The Scattering Amplitude.................

4. The Phase Shift..........................

5. The Breit-Wigner Formula..............***

6. The Chew-Low Model..........*....... ....

7. The Effective Range Approximation*.......

[I. THE K-MATRIX FORMALISM ........................

[V. PHASE SPACE FUNCTION.......................

1. The Regular Two-Body Phase Space Function

2. Kinematic Singularities of Helicity

Amplitudes............................. .

3. The Phase Space Function.................

V. PARAMETRIZATION OF THE P' RESONANCE.........
33


iii


I


II
I]

3


page

ii

v

vi

vii



1

4

4

5

9

12

15

16

18

20

24

24


25

29

31









TABLE OF CONTENTS (CONTINUED)


page

1. Introduction....**.........,......*...... 31

2. Basis of Computation.............,.i..... 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.... ...e.................................... 60

BIOGRAPHICAL SKETCH...... ............................ 63














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














LIST OF FIGURES


Figure page

1. Diagrams....,.....,.. .... ........,....,,.. 17







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' RESONANCE IN PION-NUCLEON SCATTERING
33


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'
33
resonance in pion-nucleon scattering. Reasonable fits with

the P33 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 CNI,

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

usual exchange mechanisms.2

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

the pole, has the following deficiencies:3

(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.








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

not have any of the above objections.3'4

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

P3 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.5 and a com-

pilation of older results,6 the parametrization has been

used to determine the mass and width of the P'3 resonance,

as well as the coupling constants C NM, C7rA and C N'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*. It also

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 P3

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









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,










1 + 2 --- 3 + 4


where particles i=1,..,4 have four-momenta pi, rest masses

ni, spins si, and helicity components Xi. 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 + --- Z + 4 (t channel)


1 + 4 ---- + 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 PI1 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,


s = (pI + P2)2 (P+ (1)







where the pi are measured in the direct channel. The momen-
tum transfers t and u are defined by


t = (P1 p3)2 2 (P2 P)2 (2)


u = (PI P4)2 (P2 P3)2 (3)


The three invariants are not independent,


11 (4)
a + t + u Z. (4)


where the summation is over the irtcident and the exit chan-
nels.
If 0 is the scattering angle in the center-of-mass
frame of the direct channel, then


sin 0 = 21Cs(s,t)]/S12(s)S34(s) (5)


cos e = C2 + 2st sZim2 + (T12U12T34U34)1 /S12(S)S34 ()

(6)


where
(s,t) stu (T13U13T24U24 T12U1234U3 +


(m2 + 2 m23(m2 m2) (7)


Tij = (i + qm)2 (8)








U (m 2j2 (9)


Sij(s) [(s Tj)(s Uij). (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


S m (11)
Pi mi.


The boundary of the physical region is given by


*(s,t) = 0. (12)


All square roots in the above are chosen to be positive-

in the physical region, so that the sine of e is positive in

the direct channel.

Tij is to be called the threshold of the exit channel

with particles i and j, and Uij is the corresponding pseudo-

threshold.

In the center-of-mass frame of the direct channel, the

total energy is


w s&


(13)








and the magnitude of the three-momentum of either particle i
or particle j is


kij = Sij(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 )/2m (15)


where m is the nucleon mass and u 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, (fISIi> is defined such that


Pfi (16)


represents the probability of If> 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







in theoretical investigation and subtracted out of experi-
mental results, the finite range of strong interactions
guarantees that If) and ji) 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 hx
is defined by


S = (p3 p4',A3,)4pP1 'Pl'2X) + h.A (17)


where 3 = 3 4


= 1 1 2


SAIL"


The differential cross section is given by


(d = h, 2/2 (18)


where k is the magnitude of the center-of-mass three momen-
tum in the direct channel, for the incident particles.








The helicity amplitude hxy defined here is related to
the helicity amplitude of Jacob and Wick7 by


h = k f
\y. Xy4


(19)


and to the usual helicity amplitude such as seen in Wang
and King9 by


h = -
hxf 27


where q is the magnitude of
in the exit channel.
Partial wave expansion
follows:


[ 9q] (20)


the center-of-mass three-momentum


for S and h are defined as


S 4 Zj(2J + 1)S d (0) e-i('-)


h, ZJ (23 + 1)h d J () e^i(-
\ 7~A t,


(21)


(22)


where d. (9) is the d function of the rotation matrix ele-
ment
sent


dJ(e) = (Jhle-iRJj>) .


(23)


The total cross section, according to equations (18)
and (22), is therefore given by









7 2Lj(2J+ 1) Ih (24)



4. The Phase Shift


According to equation (16), and since there must be
unit probability for an initial state to end up in some
final state,

1 = Zf (il! If >f lSli>


= iIS.tli). (25)


Here the completeness of final states If) has been assumed.
If, in addition, as is customary, the initial state is
normalized, then it follows that


S = 1J (26)


since the initial state ji) 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


SS = .* (27)


In the case of elastic pion-nucleon scattering, the








partial wave scattering matrix has only four helicity compo-
nents:
S s3 s sJ
++ S-_ --

where + denotes +j and denotes -~. 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


SI+I + SI=- 2 1
(28)

S+s_ + S~ + 0.

Define SJ= S + S3
++ + + --
(29)
S s= s
+ ++ 4+-


and the unitarity relation becomes even simpler:

| 2 = 2
s = = 1. (30)


Conventionally, one insures that this condition is sat-
isfied by writing

2i 6
S e (31)


where the phase shift 6 is a real function of W.







In terms of the helicity amplitudes,




4wrk 2ik

2i6 2i6j+
o "' +e 2
41k


(h + h1] (32)

i JS(w)
where h (W) = e sin bj*(W). (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


)ilsli) = 1 ni . (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 r1, and
make the phase shift real even for inelastic processes.
Hence equation (31) becomes

SJ =- j. e2i (35)


j1 is equal to unity when the scattering is purely elastic,
and it is smaller than unity when the scattering is partial-
ly inelastic.
I.









5. The Breit-Wigner Formula


From the definition of hJ, equation (33), it can be
found that, if inelasticity can be ignored,


S=1 (36)
h- k cot 6j, i (36)


A resonance is said to exist at Wr if cot 6j goes
through zero for this value of W with a negative derivative.
If a power series expansion of cot 6 is made, then in the
neighborhood of W


cot 6j* = (Wr w)/ r r>0 (37)


and
h = i r/2 (38)
I kW r W ir/2'


Inserting this result into equation (24) and neglecting the
other partial waves, the resonance cross section is found to
be


e. (W) A4 (2J + 1) r2/ (39)
res k2 (W2 W)2 + r2/4



This is the well-known Breit-Wigner formula.
At-resonance energy, the partial-wave cross section is,








according to the Breit-Wigner formula,


s(Wr) = ( 2J + 1) (40)
res r k2


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 h7 and hj (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.O1 In particu-

lar, it predicted the P33 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:


HI = (Vmam + V*m m) (41)


VI if(*Ek/C2E] )T n(k2) (42)
I m


where am and am are respectively the creation and annihila-

tion operators for single pions, k is the three momentum and








E the energy of the pion, and m represents all pion quantum

numbers, and lastly, n(k2) is the Fourier transform of the

nucleon charge density. In order to ensure the convergence

of the necessary integration n(k2) must be cut off above

some kmax

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



r I **. --.%.I

N N N N

Figure 1. Diagrams


Finally, the following relation is derived for the I- Ju=

phase shift
Scot (1 rE) (43)
SP33 4f2


where f is the renormalized coupling constant and r a con-

stant approximately given by

r f2 Ema 0. (44)
max


Emax is the pion energy corresponding to kmax*

Relations similar to equation (43) are predicted for

the other P-wave phase shifts. However, for these phase







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
hJ(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) = klh(w)l (45)


if inelasticity can be ignored. From equations (36) and

(45) it follows that
Im [ J ] -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







branch point at w=0. The function M(w) is defined as


MJ(w) = 1/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,


M(w) k cot 6&j. (48)


Thus the power series expansion will take the following form,

where the parameters are conventional:


k cot 6 = + Irw + ... (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+1 cot 6j + irw. (50)
J a













CHAPTER III

THE K-MATRIX FORMALISM


In a parametrization 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


h(fi) = [pJ(s)]2 Mf,i) [p2(s)] (51)


where p (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









S = 1+ 2ihj. (52)


Combining equations (27) and (52), the following relation is
obtained:
Im [ ] -1. (53)
h

If it is defined that


C P Cp(s)] mn, e(s T) (54)


where Tm is the threshold in channel m and 9(s-Tm) is the
step function, then


p= I h : ] (55)


(d I i IJ (f,i) 2 (56)
J,fi k

In C ] = J. (57)


The last equation is the key to the K-matrix formalism. If,
furthermore, we define

1, P?(s') dsa
R. (a) [ i + p.s. 6j (58)
ii

where p.s. stands for "possible subtraction", then it follows
from equation (57) that a K-matrix can be defined as follows:










+ R (59)
K M


and that it is real for positive real s. In addition, if
both J and are analytic in s, so is Kj. Equation (39)

can also be written as

N = K (60)
1- KJRJ


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 RJ is a diagonal matrix. The diagonal

elements of this matrix will sometimes be simply denoted by

R1(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 K 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
K= (61)
Sr


where ar is the position of the resonance, and C is a matrix







of coupling constants that factorizes:


cj. = cic. (62)


Equation (61) 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 (61). 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), Rj should be at least quadratic in (s-s ) in the
neighborhood of sr.
Hence, equation (58) should be changed to be


1 2 (*s) ds'
R (s) = (s-s)2 J )(s) i + p.e. ] 6 (63)
Ti (s'-)2(s'-s-i)


Combining the above considerations, the following pa-
rametrization formula is found:


W(s) = s 2 i.2.. (64)
(sa-s) C2R(s) i C (s)0(s-T
r- i J J A(~e~-r













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, po, 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,


dPo d4P1 d42 64(p +2-P) S(p1) 2 (p2 2) (65)


or o C(s T)( U I (66
or po !lP(s T12)(s U12)] (66)








where T12 and U12 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, po 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-
tudes has been studied extensively.8,9,12,13 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


d (e) = J I J-M)1 CIcos e3 C[sin i3] )


P (-M) (cos e) (67)







where MEmax( IX,IJ I) and N=min( Xl,l V). From the defini-

tion of the partial-wave expansion it can be seen that hxV
contains the factors


[ cos .e ]' sin 1e ] (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

others is employed here.9
The direct channel covariant helicity operators corres-
ponding to particles 1 and 2 are defined by the relations


F 2 (1) / S2(s) (69)
S= 2 W p / 12(s) (70)

F2 (2) 14 /AS12(s) (70)
2 e2W~ p11








where p is a summation index and W)W()F 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 W(A) 13(t) (71)


F 2 W(2) pJ/ S2(t). (72)
2 4 24


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


(PX) d )rX(r)/$t(P,') ) (73)


where X and X' are eigenvalues of the operators Fr and F ,
respectively, J is the intrinsic spin of particle r, and
r=l or 2. z2 is the angle between -p and -p4 in the rest
frame of particle 2, while C1 is the angle between -2 and

P3 in the rest frame of particle 1. The coordinate axes are
always chosen such that O0&Cir. In terms of the invariants
s and t, the angles (r are given by the relations

( + m2 m2)(t + m2 m2) 21m2M
cos1 1 2 1 (74)
12(s) S13(s)








( + 2 m2 + 2 + 2m2M
cos +2 .2..(t +M2- (75)
S12(s) S24(t)

where M m2 m2 + m2 3
and I
z2mi[l(s,t)]2
sin 1 S12(s)S13) (76)

2m2 z (s,t) ]
sin 2 12()S24t) (77)


From their definitions the operators FO and Fs can be
seen to be ill-defined at the threshold and the pseudo-
threshold. The covariant helicity amplitudes, f:, being
eigenstates of these operators, are singular at these values
of s. Introducing a new set of amplitudes fat(s,t) in which
the states for particles 3 and 4 remain unchanged but the
states for 1 and 2 are now eigenstates of F1 and P2, it is
found that


f (st) = 3s (st)

31 3 2 fat
7X )AI dXX(I)d A 2QC2) f 3 l,(st). (78)


The singularities at the threshold and the pseudothreshold
are now isolated in the d functions, because f ,(s,t)
X3 x4 A
is regular at these points.
After an analysis of the asymptotic behavior of the d
functions, it is found that the amplitudes fs behave as
'P









[s T] 3+J4 (79)

and
a[s U34] (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 pJ(s), but not pj(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
scattering amplitude. Cohen-Tannoudji et al.12 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-
larity of a single helicity amplitude.9


3. The Phase Space Function


The product of the expressions in equations (66), (79)








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 (O) has to be studied, too. According to
the asymptotic behavior for the d functions,9'14


d (e) ~ [cos e]J

-3J
S[( T34)(s U34)] (81)


because of equation (6). Thus, finally,


T i J+-J -JW 3
p (s) = [(s T34 )(s U34)]. (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 parametrization 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 P33 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.

The P33 phase shift is related to h_, which is proportional

to the linear combination h1+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 hJ" can produce an adequate ap-








J J
proximation for h +h ,.. 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.
SJ J
Second, since the linear combination h -h or h

is related to the D33 phase shift, is it not advisable to

make a simultaneous parametrization of the P' and D3 reso-
33 33
nances, so that individual matrix elements, i.e., h and

h -, might be determined? The answer is definitely yes, it

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 P3' resonance, their characteristics, and the

appropriate phase space functions.

The phase space functions for the J=0 channels are ob-
2








Table I. Characteristics of Contributing Channels


Meson Baryon T2 [MeV] U2 [MeV] p
pion nucleon 1077.9 843.7 1 E(s-T)(s-U)]3/2
pion P3 1373.6 1094.4* 1 (s-T)(s-U)J]
pion Pt 1609.6 1340.4 1 [(s-T)(s-U)]3/2

varied in actual computation as the P3 mass is changed.



trained from equation (82). While for the pion-P' channel
33
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'(at) da.
R(s) = 1 (s s)3 (83)
S(s' Sr)3(s' s i)

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) r + z(s a)2 Zj CR (s) (84)


G(s) p( j)O(s Tj) (85)


where z is an overall subtraction constant that is made nec-
essary by the extra subtraction, and e(s) is a step function.








The phase shift is given by



tan 6 (s) (86)



Because of the fact that most coupling constants in-

volved in the parametrization of the P' are unknown at the
33
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.5 Set B is compiled from older re-

sults.6,15

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 P' resonance. The reported error bars are much
33
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-








ported by this group are extracted directly from the total

cross sections.

Set A is used to yield more accurate information on the

P' 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.5 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:
16
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.








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 "communicating" channels or possible

exit channels as required by unitarity.14 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-

proximation for the left-hand singularities.3 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








Regge poles in the J-plane.17

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

number and mass-width specifications,18 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








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 1672

MeV. Arranged in order of increasing thresholds, the fourth

channel threshold is at (1660 MeV)2 and the eighth at (1783

MeV)2. 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-








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 P3 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








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 P' is then shifted and a new best fit
33
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'
33
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-relativistic 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-

al.19








The zero of the denominator in equation (64) is deter-

mined, in part, by the expression


Zj cR (s). (87)


The relative contribution of each channel toward producing

the resonance is then given by3


CR (a) (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

been considered a bad fluctuation.18








Table II. Parameters from Phase Shift Fits


resonance mass

resonance width

CKNA
C.AA


CrN 'A
subtraction constant

chi-square

mean chi-square


Set A

1230.4 MeV

120 MeV

0.1551*0.0002

0.0830*0.0073

0.8048o0.0202

-0.0488t0.0003

15.31

1.09


Set B

1235.0 MeV

122 MeV

0.1543*0.0010

0.1029*0.0073

1.018 *0.031

-0.0506*0.0016

63.68

1.63


Full width at half maximum.4


With a total of five de fact parameters (three cou-

pling constants, one subtraction constant, and the resonance

mass), the fit to set A has not been very good.18 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








Table III. Phase Shifts Fitted According to Carter et al.5


W [MeV] Exp. [deg.] Theory 72

1139 11.87*0.15 11.54 4.75
1160 21.41*0.06 21.42 0.01

1177 33.17*0.10 33.36 3.50
1178 34.36*0.20 34.26 0.25
1190 45.64*0.11 45.66 0.04

1193 47.83*0.11 47.81 0.04
1206 62.82*0.18 62.61 1.31
1210 67.73*0.21 67.48 1.37
1215 73.51*0.30 73.63 0.15
1227 84.94*0.98 85.97 1.10
1244 102.05*0.51 102.3 0.21
1261 114.41*0.23 114.6 0.45
1280 124.03*0.17 124.2 0.85

1301 131.96*0.15 131.8 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 P' resonance. In Table II are listed main findings of
33
this fit. In Table IV are the phase shifts.
Similar to the parametrization of the rho meson,4 in
this relatively wide energy range the theoretical values de-
viate from experimental phase shifts significantly near the








Table IV. Phase Shifts Fitted According to Compilation Data


w CMeV]

1094
1104
1109
1113
1127
1160
1177
1185
1195
1197
1202
1213
1216
1231
1235
1247
1252
1254
1258
1268
1275
1291
1320
1362
1390
1416
1442
1470
1481
1500
1512
1524


Exp. [deg.

1.4 0 0.9
2.6 0.9
3.6 + 0.9
4.2 0.9
7.6 o 0.9
21.0 1.0
31.8 1.1
37.9 1.1
45.9 1.3
48.2 4 1.4
54.2 4 1.4
65.8 1.4
69.4 4 1.4
85.1 1.4
90.1 1.4
100.6 1.4
105.2 1.3
107.5 : 1.2
109.4 1.0
117.7 1.0
119.8 1.0
126.9 0.6
136.7 1.3
144.9 0.6
149.2 1.0
153.3 1.1
156.3 1.1
160.2 1.4
161.1 1.7
164.5 2.5
166.0 0.9
166.1 1.0o


Theory

0.88
2.01
2.76
3.44
6.66
19.79
31.12
37.77
47.28
49.32
54.59
66.71
70.05
86.09
90.04
100.7
104.7
106.2
109.0
115.4
119.3
126.8
136.8
146.5
151.2
154.6
157.5
160.2
161.1
162.6
163.4
164.2


2

0.33
0.42
0.88
0.71
1.32
1.47
0.39
0.00
1.12
0.64
0.08
0.52
0.21
0.61
0.00
0.04
0.15
1.20
0.07
5.16
0.23
0.02
0.01
7.33
3.86
1.81
1.28
0.00
0.00
0.59
8.14
3.45








Table IV. continued


W [MeV] Exp. Edeg.] Theory c2

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

thresholds up to (1783 MeV)2, 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.








3. The Resonance Mass



Set A

The mass of 1230.4 MeV for the P' resonance is lower
33
than most values reported.1 The lone exception is the "nu-

clear" result reported by Carter et al.5 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,

consistently give higher masses.18

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 P' extracted from this fit is of
33
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








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
33
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 C NA obtained in the parametrization, we have


r = C NA-1 c { ("N + )2 m2 ( mN a )
MA

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.

According to Mathews,21 this partial decay width is

also given by








%2 2
r -p T. S (90)
A

Comparing the last two expressions, it can be seen that

2 (m2 _) m2
2 NA (m + mN2 (91)
1NA 4r 48 ma
2
Or that a CA of 0.1551 corresponds to a -- of 0.37, in
dimensionless unit. 2
Ebel et al.20 report a grN of 0.33, which they main-
4-
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-P1 -P' Coupling
33 33-`-
According to Rushbrooke,22 the partial width correspond-
ing to the decay of a P' resonance into a pion and a P'
resonance is given by

2 p 2 +2 m2 2p 2 + 2 2
S.m + m+ mA+mA -
r r- [3m 2 mA) + 2m2r 2mA)



4"w mA 2mA


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









2 2p
r= c (93)


Combining the two we have

2 2
2 wrA mr
c = (94)
AA 4- 4m (94)


Our value of C1j =0.083 therefore corresponds to the value
22
0.243 for g2 ./4r in dimensionless unit.
Hori and Kanki23 reported that g2 is approximately
2
one ninth of g NA. The relativistic SU(6) model of Sakita
24 2
and Wali,24 on the other hand, predicted that g is about
2 2
nine times of g NA. Sutherland25 and Michael26 predicted

that g2,, is even larger by a factor of about two. Our re-

sult is not in close agreement with any of the above. g2AA
is shown here to be two-thirds of g2N 2


C. The Pion-P' -P' Coupling
11--33
The K-matrix formalism developed here cannot be applied

to the decay of the Pi into a pion and a P3 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-4 particle into a spin-j
particle and a spin-O particle warrants special equivalence

relations for satisfactory results.








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 by3


f(s,T,U)


=P ((s' -
T


T)(s' U)


so that


for s>T,


for T>s>U,


for U>s,


c1- -1 A
f(s,T,U)=-2C(a-T)(s-U)] coth ( ) ]


f(s,T,U)- 2C(T-s)(s-U)


-tan1 )-U
tan


(A2)


(A3)


(A4)


f( U) 2(T-s)(U-)] tanh
f(s,T.U)= Z2(T-s)(U-s)]- t -anh UE-s 1
(-s" J


and the derivative of f(s,T,U) with respect to a is

-1
f'(s,T,U)-[(s-T)(s-U)] E-1+2(T+U-2s)f(s,T,U)]


(AS)


which may be used iteratively to produce higher derivatives
with respect to the same variable.


ds'I
Sa a


(Al)













APPENDIX B


COMPUTER PROGRAM


channel # meson
1 pion 140
2 pion 140
3 pion 140
4 pion 140
5 pion 140
6 kaon 496
7 rho 765
8 eta 549
1. energy unit in


baryon
ncln 940
delta 1234
N1470 1470
N1520 1520
N1535 1535
sigma 1190
ncln 940
delta 1234
pion mass for


psf T U
3 1080 800
1 1374 1094
3 1610 1330
1 1660 1380
3 1675 1395
3 1686 694
1 1705 175
1 1783 685
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,rhoeta,ncln,delta,
+N1470,N1520,N1535,sigma
read (1,1) eta,kaon,pionrho,delta,N1470,N1520,N1535,
+ncln,sigma
1 format (10d8.1)
read (1,2) psf
2 format (8i7)
do 4 i-1,39







read (1,3) energy(i)
3 format (2x,d8.1)
4 continue
do 5 i=1,400
5 x(i)-0.1
call callit
stop
end

double precision function f(A,L)
implicit real*8 (a-hk-z)
real rgm,A(l),duml,dum2
integer psf,L,par
cnvr=180./3.14159265358979d0
common/data/unitmT(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,rhoeta,nclndelta,N1470,N1520,N1535,
+sigma
if (inx.ne.1) go to 15
inx=2
read (1,11) delta
11 format (d12.4)
write (3,20) delta
unit=pion
T(1)( (ncln+pion)/unit)**2
U(1) =((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) ( (N1520+pion)/unit)**2
U(4)=((N1520-pion)/unit)**2
T(5)=((N1535+pion)/unit)**2
U(5)=((N1535-pion)/unit)**2
T(6)( (sigma+kaon)/unit)**2
U(6) ((sigma-kaon)/unit)**2








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-1
do 14 i=1,39
s( energy(i)/unit)**2
do 12 j-l,isum
12 w(ij)uv(j,s)
write (3,13) (w(i,j), j-l,isum)
13 format (8gl6.3)
14 continue
15 i=rgm(1)
a=( rg( 2) /unit) **2
g-O.dO
isum-par-1
do 16 j=lisum
if (s.lt.T(j)) go to 16
gg=g-A( )**2*fcn(psf(j) ,s,T(j) ,U( ))
16 continue
d=m-s+A( par) ( m-s) **2
do 17 j=ilisum
17 d=d+A(j)**2*w(ij)
h=dabs(d)
if (h.lt.l.d-20) go to 19
godd-g/d
if (god.1t.0) go to 18
f=datan(god)*cnvr
fx-f
return
18 f-180.0+datan(god)*cnvr
fx-f
return
19 f-90.dO
fx-f








return
20 format ('1',g50.6)
end


double precision function fcn (dmmidmm2,dmm3,ddmm4)
implicit real*8 (a-h,o-z)
integer dmal
go to (21,22,23) drml
21 fcn-dsqrt((dam2-dmm3)*(dam2-dmim4))/dma2
return
22 fcni=.dO
return
23 fcn-(dsqrt( (dmm -dmm3)*(dmn2-dma4)))**3/dmm2
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)
j-psf(i)
p-T(i)
z4U(i)
a=m-p
bem-z

x-r-p
y=r-z
pi-3.14159265358979d0
cutoff-9.9d25
go to (41,42,43), j
41 g=a*b/a/e
gp-g( 1 .dO/a+1.dO/b-1 .dO/m-l .dO/e)
v( -p*z* e**2*q( 0. dO ,p, z,cutoff)/r/m~*2+x*y*q( r p, z ,cutoff
+)/r +e*2*gp*q(m,p ,z cutoff)+e**2*g*qp( m,p,z,cutoff) )/pi
return








42 v=0.dO
return
43 haa**2*b**2/m/e
hp=h*(2.dO/a+2.d0/b-l.d0/m-l.dO/e)
hppmhp*(2.dO/a+2.dO/b-1 .dO/m-l .dO/e)+h( 1. d0/m**2+1 .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,ppzcutoff)/r+e**3*hpp*q(m,p,z,cutoff)/2.dO+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,ulmda)
implicit real*8 (a-z)
factor-(lmda-t)/(Imda-u)
if (s.lt.u) go to 32
if (s.eq.u) go to 33
if (s.lt.t) go to 31
q=-1./dsqrt((s-t)*(s-u))*dlog(( 1 .+dsqrt((s-t)/(s-u)/
+factor))/(1.-dsqrt((s-t)/(s-u)/factor)))
return
31 q-2./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)
+factor))/( 1.-dsqrt( (s-u)/(s-t) factor)))
return
33 q=2./(t-u)*dsqrt(factor)
return
end

double precision function qp(st,ulmda)
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(a,tu,lda))/(( -u)*(s-t) )
return








34 factor-dsqrt((lmda-t)/(lmda-u))
qp2 ./3.*factor/(t-u)*(2./(t-u)+1./(lmda-u))
return
end

double precision function qpp(s,tu,lmda)
implicit real*8 (a-z)
if (s.eq.u) go to 35
qpp=(((t+u)/2.-s)*qp( s,t,u, mda)-q(s,t ,ulmda)*( 1.+((t+u
+)/2 .-s)*( 1./(s-t)+1./( s-u)))-dsqrt(( Imda-t)*( Iida-u))/(s
+-lmda)*(1./(s-t)+1./(s-u)+1./(s-lmda)))/((a-t)*(s-u))
return
35 factor=dsqrt((lmda-t)/(lIda-u))
qpp-4./15.*factor/(t-u)**2*( 8./(t-u)+4./(mda-u)-3.*(t-u
+)/(lmda-u)**2)
return
end













REFERENCES


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BIOGRAPHICAL SKETCH


Yin-po Tschang, in Chinese Pat, 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, an s,

of Hong Kong. They have a seven-month-old boy, Yuan Tschang,









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.




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-Stoja ovic
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. Tricke
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.




Jn 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








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